a probabilistic-based design approach with game theoretical ...

A PROBABILISTIC-BASED DESIGN APPROACH WITH GAME
THEORETICAL REPRESENTATIONS OF THE ENTERPRISE
DESIGN PROCESS
A Thesis
Presented to
the Academic Faculty
By
Gabriel Hernandez
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science in Mechanical Engineering
Georgia Institute of Technology
June 1998

A PROBABILISTIC-BASED DESIGN APPROACH
WITH GAME THEORETICAL REPRESENTATIONS
OF THE ENTERPRISE DESIGN PROCESS
Approved:
Farrokh Mistree, Chairman
Professor
Mechanical Engineering
Janet K. Allen
Senior Research Scientist
Mechanical Engineering
Soumen Ghosh
Associate Professor
School of Management
David W. Rosen
Assistant Professor
Mechanical Engineering
Mark L. Spearman
Associate Professor
Industrial and Systems Engineering
Date Approved
ii

DEDICATION
iii
A mi abuelo, Jose M. Perez Bravo

ACKNOWLEDGMENTS
Only passions, great passions, can elevate the soul to great things
iv
Denis Diderot
Since I was child I have been taught that the labor we accomplish shapes our souls. It
must be taken with pride and passion because it is an essential feature of who we are. This
thesis is a result of that belief, but I know it is only the first step, and probably the easiest one.
I have also been taught that no accomplishment is possible in loneliness, but only with the
help, inspiration and support of other people, such as the ones that I have been blessed to meet
during this time. Specifically, I sincerely thank my advisors Dr. Farrokh Mistree and Dr. Janet
K. Allen for their extraordinary guidance and patience. I also appreciate the insightful
recommendations of my other members of the committee, Dr. David Rosen, Dr. Mark L.
Spearman, and Dr. Soumen Ghosh.
My beholden thanks to my family for the unconditional love and teaching that have driven
me through life.

I am extremely grateful to Mr. Antonio Moratinos for his long-standing and sincere
friendship to me and my family. Without his support I would not be writing these lines today.
I give heartfelt thanks to the friends who have accompanied me during this time in Mexico
and Europe, and whose constant communication has helped me to face bad times I have
encountered: Javier Romo and his family, Celia Sánchez, Margarita Aviña, and Luis A. Calvillo
and his family. I also thank those who built me a second home in Atlanta: Iván Zeron, Nicola
Giberti, Guillermo and Raquel Rodríguez, and Eileen Lai. I apologize to all the good friends that
the brevity of space does not allow me to name. My deepest gratitude to all of you.
I thank the members of the Systems Realization Laboratory for their welcome and
friendship. I give special thanks to Jesse Peplinski and Tim Simpson, whose vision and feats
have greatly inspired me. They are the kind of people that make a difference in the world.
I gratefully acknowledge the help of Dr. Eduardo Bascarán, Director of The Engineering
Center of Carrier Mexico, for providing assistance for the realization of the present research. I
hope this work is just the beginning of a long-term friendship.
I thank Dr. George A. Hazelrigg, Director of Design and Integration Engineering of the
National Science Foundation, for his valuable commentary on this thesis.
I recognize and thank the Consejo Nacional de Ciencia y Tecnología of Mexico,
CONACYT, for the economical support that made possible this endeavor. I shall work hard to
pay back my country this assistance. Financial support from NSF grant DM1-96-12327 is
v

gratefully acknowledged. The Systems Realization Laboratory of the Georgia Institute of
Technology has underwritten the cost of computer time.
vi

TABLE OF CONTENTS
DEDICATION iii
ACKNOWLEDGMENTS iv
TABLE OF CONTENTS vii
LIST OF TABLES xiii
LIST OF FIGURES xvi
NOMENCLATURE xx
SUMMARY xxiii
CHAPTER 1
FOUNDATIONS FOR DESIGNING WITHIN AN ENTERPRISE DESIGN
PARADIGM 1
1.1 MOTIVATION 2
1.2 INTELLECTUAL FOUNDATIONS 4
1.2.1 Systems Thinking in Design and Manufacturing 4
1.2.2 Design as a Science of the Artificial 6
1.3 FRAME OF REFERENCE: DECISION-BASED DESIGN 12
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1.4 A DECISION-BASED ENTERPRISE DESIGN 12
1.4.1 The Concept of Enterprise Design 12
1.4.2 A Decision-Based Method for Enterprise Design 14
1.5 RESEARCH QUESTIONS AND HYPOTHESES 20
1.6 ORGANIZATION OF THE THESIS 25
CHAPTER 2
ENTERPRISE DESIGN AS AN EXTENSIVE GAME 29
2.1 OVERVIEW OF ENTERPRISE MODELING: ARCHITECTURES,
TOOLS AND METHODS 30
2.1.1 Modeling Architectures 30
2.1.2 Modeling Tools 32
2.1.3 Modeling Methods 33
2.1.4 Enterprise Models and Enterprise Design 35
2.2 A GAME THEORETICAL REPRESENTATION OF THE ENTERPRISE
DESIGN PROCESS 37
2.2.1 Decisions in a Manufacturing Enterprise 37
2.2.2 Decisions and Game Theory 39
2.2.3 Fundamentals of Game Theory in Design 41
2.3 ENTERPRISE DESIGN AS AN EXTENSIVE GAME 47
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2.3.1 Extensive Form Games 47
2.3.2 Enterprise Design as an Extensive Game 52
2.4 RELEVANCE OF MODELING AN ENTERPRISE DESIGN PROCESS
CHAPTER 3
AS A GAME 59
FORMULATION AND SOLUTION OF ENTERPRISE DESIGN PROBLEMS 60
3.1 ENTERPRISE INTEGRATION AS COORDINATION OF DECISIONS:
IMPLICIT AND EXPLICIT COORDINATION 61
3.2 GAME THEORY AS A FOUNDATION FOR COORDINATION OF
DESIGN DECISIONS 68
3.3 IMPLICIT COORDINATION THROUGH A COMMON PARADIGM
FOR DECISION SUPPORT 71
3.4 EXPLICIT COORDINATIO THROUGH A UNIFIED DECISION
MODEL: THE COMPROMISE DSP 73
3.5 FORMULATION OF DESIGN PROBLEMS IN ENTERPRISE DESIGN 81
3.5.1 Approximation in Engineering Design 81
3.5.2 The Cooperative Formulation using Compromise DSPs 84
3.5.3 The Non-cooperative Formulation using Compromise DSPs 88
3.6 A PROBABILISTIC-BASED DESIGN APPROACH FOR ENHANCING
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COORDINATION WITH FLEXIBLE SOLUTIONS 94
3.6.1 Why a Probabilistic-Based Design Approach? 94
3.6.2 Preference Functions for Design Requirements 96
3.6.3 Ranged Design Solutions 98
3.6.4 Design Preference Index 100
3.6.5 Constraint Feasibility Evaluation for Ranged Solutions 101
3.6.6 A Probabilistic-Based Multiobjective Design Model 105
3.7 TYING IT ALL TOGETHER: A MATHEMATICAL FRAMEWORK
CHAPTER 4
FOR ENHANCING COORDINATION IN ENTERPRISE DESIGN 109
CASE STUDY: INTEGRATED PRODUCT AND PROCESS DESIGN OF
AN ABSORBER/EVAPORATOR MODULE FOR A FAMILY OF
ABSORPTION CHILLERS 111
4.1 INTRODUCTION TO CASE STUDY 115
4.1.1 Motivation for Case Study: Benefits of an Integrated Product and
Process Design in Make-to-Order Systems 115
4.1.2 A Methodology for Modeling Production Systems for Integrated
Product and Process Design 117
4.2 DESCRIPTION OF THE DESIGN PROBLEM AND ANCILLARY
x

INFORMATION 120
4.2.1 Absorption Refrigeration 120
4.2.2 Objective and Ancillary Information 124
4.3 BASELINE FORMULATION AND SOLUTION 128
4.3.1 Formulation of Decision for Product Design 130
4.3.2 Formulation of Decision for Production System Design 135
4.3.3 Results of Baseline Approach 146
4.4 COOPERATIVE FORMULATION AND SOLUTION 152
4.4.1 Task 1: Expand Scope of Decision 154
4.4.2 Task 2: Determine Input Needed for Analysis and Identify
Modeling Techniques 154
4.4.3 Task 3: Define Boundaries of Decision 156
4.4.4 Results of Cooperative Model 162
4.5 NON-COOPERATIVE FORMULATION AND SOLUTION 163
4.5.1 Task 1: Expand Scope of Decision 165
4.5.2 Task 2: Determine Input Needed for Analysis and Identify
Modeling Techniques 166
4.5.3 Task 3: Define Boundaries of Decision 169
4.5.4 Results of Non-Cooperative Model 178
4.6 NON-COOPERATIVE FORMULATION AND SOLUTION WITH
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PROBABILISTIC DESIGN MODEL 180
4.7 COMPARISON OF SOLUTIONS 189
4.6 CLOSURE OF CASE STUDY AND OPEN ISSUES 198
CHAPTER 5
4.6.1 Review of Case Study 198
4.6.2 Open Issues 199
CLOSURE 201
5.1 CRITICAL REVIEW OF THE THESIS: DISCUSSION OF RESEARCH
QUESTIONS AND SHORTCOMINGS 202
5.2 CONTRIBUTIONS AND RELEVANCE OF THE THESIS 208
5.3 RECOMMENDATIONS FOR FUTURE WORK 210
APPENDIX A
THE PARAMETRIC DECOMPOSITION APPROACH 212
APPENDIX B
FORTRAN FILES 220
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B.1 SAMPLE OF FORTRAN FILES FOR OptdesX 221
B.2 SUBROUTINES TO FIND THE MINIMUM NUMBER OF TUBES
GIVEN LENGTH AND TUBES SELECTION 246
B.3 RESPONSE SURFACE NETWORK MODEL 251
REFERENCES 260
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LIST OF TABLES
Table 1.1 Research Questions and Hypotheses 25
Table 2.1 Strategy, Tactics and Control Decisions 38
Table 2.2 Contents of Static and Extensive-Form Games 49
Table 3.1 Explicit and Implicit Coordination Regarding Processes Underlying
Coordination 64
Table 4.1 Estimated Distribution of Demand 124
Table 4.2 Alternatives for Selection of the Evaporator Tube 131
Table 4.3 Alternatives for Selection of the Absorber Tube 131
Table 4.4 Parameters for Absorber-Evaporator Module Synthesis 132
Table 4.5 Material Cost Goal for each Refrigeration Capacity 132
Table 4.6 Constraints Values for Product Design 132
Table 4.7 Cost of Raw Material at each Subassembly Station 133
Table 4.8 Parameters for Production System Design 140
Table 4.9 Goals and Constraint Values for Production System Design 140
Table 4.10 Results for Each Capacity using Simulated Annealing 146
Table 4.11 Results for Each Capacity using a Clustering-based Algorithm 147
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Table 4.12 Results for Each Capacity using FALP 148
Table 4.13 Results for Minimum Production Cost and Cycle Time 150
Table 4.14 Baseline Solution for Production System Design 151
Table 4.15 Cycle Times Estimated with Response Surface Network Model and
Discrete Event Simulation 151
Table 4.16 Cycle Times for 20 Replicates of Simulation 152
Table 4.17 Implementation of Cooperative Formulation and Solution 153
Table 4.18 Initial Value of Design Variables for Optimization Process 162
Table 4.19 Results of Cooperative Model 162
Table 4.20 Results of Cooperative Model for Each Capacity 163
Table 4.21 ANOVA Table for MfgC Regression 174
Table 4.22 Summary of Fit for MfgC 174
Table 4.23 ANOVA Table for CT Regression 175
Table 4.24 Summary of Fit for CT 175
Table 4.25 Initial Value of Design Variables for Optimization Process 178
Table 4.26 Results of Product Design: MC and “Predicted” Values of MfgC,
COST and CT 178
Table 4.27 Results of Non-Cooperative Model for Each Capacity 179
Table 4.28 Production System Design Results with Non-Cooperative Model 179
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Table 4.29 Results of Product Design: MC and Predicted Values of MfgC ,
COST and CT 188
Table 4.30 Results of Non-Cooperative Probabilistic Model for Each Capacity 188
Table 4.31 Production System Design with Probabilistic Non-Cooperative Model 188
Table 4.32 Safety Stock of Evaporator and Absorber Tubes in Number of Tubes 194
xvi

LIST OF FIGURES
Figure 1.1 A Hybrid Paradigm for Decision Support 15
Figure 1.2 A Decision-Based Method for Enterprise Design 17
Figure 1.3 Implementation Strategy 26
Figure 2.1 Various Formulations in Optimization Theory 42
Figure 2.2 Example of an Extensive Game 51
Figure 2.3 Enterprise Design as a Game 58
Figure 3.1 Implicit and Explicit Coordination of Decisions 63
Figure 3.2 Integrating Implicit and Explicit Coordination 65
Figure 3.3 A Framework for Enterprise Design 67
Figure 3.4 A Single Objective Optimization Problem and the Multigoal
Compromise DSP 76
Figure 3.5 Mathematical Form of a Compromise DSP 77
Figure 3.6 Full Cooperation: Pareto Solutions 85
Figure 3.7 Approximate-Cooperation using Compromise DSPs 86
Figure 3.8 Solution of a Two-Players Approximate-Cooperative Model 87
Figure 3.9 Conceptual Outline of RRS Construction 89
Figure 3.10 Non-Cooperative Compromise DSPs 90
xvii

Figure 3.11 Solution of a Two-Players Non-Cooperative Model 90
xviii

Figure 3.12 Leader/Follower Solution Process 92
Figure 3.13 Compromise DSPs for a Two-Players Leader/Follower Game 93
Figure 3.14 Combining Cooperative Protocols in Enterprise Design 92
Figure 3.15 Coordination of Decisions with Ranged Solutions 95
Figure 3.16 A Flexible Preference Function 97
Figure 3.17 A Comparison of Preference Structure 98
Figure 3.18 Mapping between Design Variables and Design Performance 99
Figure 3.19 Design Preference Index 101
Figure 3.20 Statistical Constraint Feasibility 104
Figure 3.21 The Probabilistic-Based Multobjective Design Model 108
Figure 4.1 Hierarchical Objectives in a Manufacturing Organization 116
Figure 4.2 Modeling the Production System as a Network of Response Surfaces 119
Figure 4.3 Absorption Refrigeration Cycle 121
Figure 4.4 Layout of Absorption Chillers 123
Figure 4.5 Estimated Distribution of Demand 124
Figure 4.6 Production line of the Absorber-Evaporator Module 125
Figure 4.7 Roadmap to Case Study 127
Figure 4.8 Agents Involved in the Design Process 129
Figure 4.9 Baseline Compromise DSP for Product Designers 134
Figure 4.10 Compromise DSP for Building a Payoff Matrix for Production
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System Design 144
Figure 4.11 Baseline Compromise DSP for Production System Design 145
Figure 4.12 Comparison of the Value of the Final Deviation Function Z for each
Capacity using San, GLOBAL and FALP 149
Figure 4.13 Implementation of Design Strategy for Cooperative Model 159
Figure 4.14 Compromise DSP for Cooperative Model 161
Figure 4.15 Modified Deviation Function Z mp 164
Figure 4.16 Stackelberg Protocol between Product Design and Production System
Design 166
Figure 4.17 Implementation of Design Strategy for Non-Cooperative Model 171
Figure 4.18 Fit of MfgC Regression 173
Figure 4.19 Fit of CT Regression 176
Figure 4.20 Stackelberg Compromise DSP for Product Design 177
Figure 4.21 Preference Functions for COST and CT 181
Figure 4.22 Stackelberg Probabilistic Formulation for Product Design 186
Figure 4.23 Stackelberg Probabilistic Formulation for Production System Design 187
Figure 4.24 Deviation from Material Cost Target for Each Capacity 190
Figure 4.25 Average Material Cost of the Various Solutions 190
Figure 4.26 Average Production Cost of the Various Solutions 191
Figure 4.27 Average Total Cost of the Various Solutions 192
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Figure 4.28 Average Production Time of the Various Solutions 192
Figure 4.29 Deviation from Targets of Total Cost and Production Time of the
Various Solutions 193
Figure 4.30 Sensitivity of Cooperative Solution to Changes in the Length 195
Figure 4.31 Sensitivity of Probabilistic Solution to Changes in the Length 195
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a An action in a game
A Set of feasible actions in a game
B Design region with a local minimum
CAD Computer Aided Design
CCD Central Composite Design
CE Concurrent Engineering
CIM Computer Integrated Manufacturing
NOMENCLATURE
d Deviation variable in a Compromise DSP
DFM Design for Manufacturability
DBD Decision Based Design
DOE Design of Experiments
DPI Design Preference Index
DSP Decision Support Problem
f Cost (objective) function
F Function or model
g Equality vector constraint
h Inequality vector constraint, history of a game
xxii

k Stage of a game
xxiii

K Total number of stages in a game
N Number of players
P Preference function
Po
Payoff matrix
r Dimensions of a cost function
RRS Rational Reaction Set
s A pure strategy of a player in a game
S Design space set
SC Supercriterion
T Target values
W Weight of an objective in an Archimedean formulation
x Design variable
X Design vector
y Level of performance of a system
ˆ
z Normalized deviation function of a Compromise DSP
Z Deviation function of a Compromise DSP
Subscripts
p Pareto solution
n Nash solution
xxiv

Superscripts
* Optimum value
m Dimensions of a design space S
mp modified Pareto solution
w Worst value
xxv

SUMMARY
xxvi
Enterprise design is embodied by the integration of product design, manufacturing process
design and organization design to enhance competitiveness by achieving manufacturing
enterprises that are designed in an integrated and internally consistent manner. However, due to
the complexity and size of an enterprise, it is likely that its design must be accomplished by a
group of functional departments that deal with different aspects of its activity. Thus, it is
necessary to provide a rigorous framework that enhances an efficacious collaboration of the
various decision-makers that comprise an enterprise. In this thesis such a framework is built
with the following elements:
?? A representation of the enterprise activity as an extensive game.
?? An enterprise design method based on a common model for decision support.
?? A multiobjective decision model, the Compromise DSP.
?? A probabilistic design model to attain flexible and robust decisions and handle uncertainty
and imprecision.
The merging of these elements provides a consistent mathematical framework for coordination
of decisions based on design formulations, which fosters its implementation in a computer-
supported environment. This framework is exercised in a case study, namely, the integrated

xxvii
product and process design of an absorber/evaporator module for a family of absorption
chillers.

CHAPTER 1
FOUNDATIONS FOR DEVELOPING A FRAMEWORK FOR ENTERPRISE
DESIGN
Enterprise Design is a term coined to characterize the integration of the processes of
product design, manufacturing process design and organization design to enhance the
competitiveness of manufacturing organizations by achieving enterprises that are designed in a
unified, integrated and internally consistent manner. In this work it is asserted that a practical
and efficacious integration can be achieved by formulating and solving appropriately the
individuals’ design problems under a unified decision-support framework. In this first chapter,
the foundations for developing such a framework are examined, along with its motivations and
research questions to be explored. The core notion is to envision design and manufacturing as
disciplines that can be scientifically carried out with a decision-based perspective.
1

1.1 MOTIVATION
In recent years there has been a steady rise in customers’ demands. Customers expect variety,
low prices, high quality, and responsive service. Due to these requirements, manufacturing
organizations have to more efficaciously coordinate their activities to maintain their
competitiveness. The notion of integrating the processes of product design, manufacturing
process design, and organization design has been characterized with the term of “enterprise
design” (Peplinski, 1997). It is postulated that such integration fosters competitiveness by
achieving manufacturing enterprises that are designed in an internally consistent manner.
However, due to the overwhelming complexity and size of the problem of designing an
enterprise in its entirety, it is likely that the problem must be accomplished by a group of
functional departments that deal with different aspects of the enterprise activity. In other words,
an enterprise is a system whose scale and complexity necessitates decomposition into a set of
smaller interdependent activities, undertaken by distinct groups making individual decisions.
In this thesis a perspective in which this decomposition is not only necessary, but also desirable,
is embraced, as voiced by Little (Little, 1992):
Building complete models of a system and setting all the control variables misses the
major opportunities for system improvement that are possible by finding new ways to
empower the people on the front lines of the organization by giving them information,
training and tools, with which to improve their own performance.
Herbert Simon (Simon, 1996, p. 41) also supports this belief:
2

As a single decision may be influenced by a large number of facts and criteria of choice,
some fraction of these premises may be specified by superiors without implying complete
centralization. Organizations can localize and minimize information demands by
decentralizing decisions. Matters of fact can be determined wherever the most skill and
information is located to determine them.
Organizations operating in a complete centralized way would exceed the limits of human
procedural rationality and lose many of the advantages attainable from the use of hierarchical
authority and employee empowerment. But, if decentralization of decisions is desirable, how
can an effective integration of the enterprise activities be achieved? Peplinski and Mistree
(1997) offer an answer to this dilemma:
The alternative approach of enterprise design is that of promoting all of the domaindependent
system modeling schemes across the enterprise and then integrating all of these
varied models into a decision based design... The key shift in thinking is from visualizing a
design process as initiated by a single designer to recognizing the actual enterprise design
process as the combined actions of the empowered group of decision makers that
comprise the organization.
The essential feature of this enterprise design vision is that enterprise design should not be
viewed as an attempt to design products, manufacturing processes, and organizations all at once
in a huge design effort. Instead, it is asserted that enterprise design should be based on an
approach in which support is provided for identifying interdependencies, and tools are provided
for modeling across boundaries.
Decomposing a problem, in this case enterprise design, into smaller problems is a
common approach in design. However, the decomposition of the enterprise in functional
groups, with a variety of characteristics, needs, requirements and constraints, poses the problem
3

of coordinating the activities of the various groups, as each one typically resorts to optimize its
own subsystem. Although this approach is advantageous locally, the individually motivated
decisions are usually not advantageous for the system as a whole. Even more, the objectives of
the various design processes are frequently in conflict with each other. Thereupon, it is
necessary to establish an effective cooperative framework for enterprise design, and to
formulate and solve the resulting local design problems with an appropriate approach, consistent
with the aforementioned framework. According to the preceding discussion, the objective of
this work is twofold:
1. To establish a mathematically supported cooperative framework that enhances a
practical, effective and efficient integration of the enterprise design processes.
2. To provide an appropriate approach for the formulation and solution of design
problems, consistent with such a cooperative framework.
In working out this objective, the focus is on manufacturing enterprises that produce discrete
products. Such systems are normal in mechanical, electrical and electronics industries.
However, the framework that is constructed in this thesis is generic and can be readily applied
to such systems.
1.2 INTELLECTUAL FOUNDATIONS
1.2.1 Systems Thinking in Design and Manufacturing
4

Designers, engineers and production managers carry a heavy responsibility since their ideas,
knowledge and skills determine in a decisive way the competitiveness of their enterprises. This
responsibility places the onus on them to carry out their activities in a more rigorous way.
Manufacturing enterprises cannot afford anymore a trial-and-error approach to design of
products, its processes and organization.
The traditional approach to design and planning has its roots in the reductionist method:
the work is decomposed into its distinct subsystems and then each subsystem is “optimized.”
But alternatively, design of subsystems can be viewed much more in terms of the interactions
between them and in the light of the performance of the system as a whole. The main reason
that motivates adopting such a systems thinking is that a system is usually more than the sum of
its constitutive parts, because the parts are dynamically interrelated and interdependent. In
other words, in complex systems it is both the organization of components and their physical
properties that largely determine behavior. Hence, the major opportunities for improvements
of the manufacturing enterprises lie at the interfaces (e.g., between sales and manufacturing, or
between product development and manufacturing). The major benchmarks behind a systems-
thinking mindset are (Wright, 1989):
?? Goal seeking. A system’s inner actions result in some position of dynamic equilibrium
where activities can be directed to goal attainment.
5

?? Holism. A system is an inseparable entity. Hence, each system should be studied as a
whole, i.e., holistically. Holism views a whole entity with all its parts in natural motion and
does not force divisions of the indivisible.
?? Hierarchy. A system has subparts or subsystems that are organized in a hierarchy, from
those of major scope down to those subsets of minimal influence on goal attainment.
?? Equifinality. An open system can reach its goals in a number of ways. This flexibility on the
means to goals is made possible because, unlike a closed system with a pre-set alternative,
open systems can change inputs and some can also change goals. Hence, the tendency
toward inertia can be thwarted by open-systems when they can be managed.
With such a systems-thinking perspective, subsystems are designed with emphasis in the overall
system performance, and the process of design is forced into a total systems context.
1.2.2 Design as a Science of the Artificial
Increasing manufacturing competitiveness is demanding organizations to develop products and
their manufacturing processes more efficaciously. Designers should not anymore rely solely in
experience and intuition. Such experience and intuition will always be desirable, and probably
necessary, but designers should be more systematic and support their activity with rigorous
analysis in order to be effective in the new marketplace. Systematization and rigor brings
design into the realm of science.
6

It should be apparent that natural science and design are essentially different, the
fundamental difference being that the former is concerned with analysis of phenomena, whereas
design is concerned with the synthesis of artificial systems to function. However, if the general
structure of design as a discipline and most of its elements can be understood, and it can be
carried out within a systematic methodology and supported with rigorous analysis and synthesis,
design can be envisioned as a science of the artificial (Simon, 1996).
Ordinary systems of logic serve natural sciences well because the concern of standard
logic is with declarative statements, suited for assertions about the world and far for inferences
from those assertions. On the other hand, design is concerned with how things ought to be.
Even when there have been developed a number of constructions of modal logic for handling
“shoulds”, and “oughts” of various kinds, none of these systems has been sufficiently developed
to be adequate to handle the requirements of design. However, as affirmed by Herbert Simon
(Simon, 1996, p. 115), “the requirements of design can be met fully by a modest
adaptation of ordinary declarative logic. Thus a special logic of imperatives is
unnecessary.”
There exists a considerable area of design practice where standards on rigor in inference
are high, in particular the so-called optimization methods. Since the optimization problem is a
standard mathematical problem --to find an admissible combination of parameters that maximize
a utility function subject to constraints-- it is evident that the logic used to deduce the answer is
the standard logic of the predicate calculus on which mathematics rests. This formalism avoids
7

making use of a special logic of imperatives by dealing with sets of possible worlds: we simply
ask what values the command variables would have to maximize the utility function while
meeting the stated constraints, and we conclude that these are the values the command variables
should have. Hence, a scientific design practice should incorporate (Simon, 1996):
1. A utility theory as a logical framework for rational choice among alternatives, and tools for
actually making the computations.
2. The body of techniques for actually deducing which of the available alternatives is the
optimum, which include mathematical programming, simulation, queuing theory, and control
theory, to name a few.
3. Algorithms (that can be of a heuristic nature) to search for alternatives.
4. A theory of structure and design organization.
5. Representation of design problems.
The subject of computational techniques need not be limited to optimization.
Traditional engineering design methods make use of “figures of merit” to compare between
designs in terms of “better” and “worse” without actually providing a judgment of “best.” In
design, we usually satisfy because we are unable to optimize, as the set of all alternatives cannot
be always evaluated due to computational capacity, nor we can recognize the best alternative,
even if we are fortunate enough to generate it. This is so because in real-world systems there is
always present a certain amount of incompleteness and uncertainty in the information available.
Herbert Simon coined the term “satisficing” to describe those solutions that are “good enough”
8

ather than optimal. By using this term to describe the results of a “scientific” design effort it is
recognized that in building a science of design there are segments of such an effort that can be
organized with a systematic and formal theory, while other segments have to be more pragmatic,
more empirical. This is the fundamental tenet of a decision-based design paradigm, which is
described in the next section.
1.3 FRAME OF REFERENCE: DECISION-BASED DESIGN
Ideally, engineering design involves two steps: (1) determine all possible design options,
and (2) choose the best one. Simple as this may sound, these two steps are extremely difficult
or impossible to perform for all but the most simple systems. Some of the difficulties
encountered are (Hazelrigg, 1996):
?? For most systems, the range of possible design options is virtually limitless.
?? It is not possible to know exactly how a particular design will perform until it is built. Thus,
design is a matter of decision making under conditions of uncertainty and risk. Engineers
frequently use models to assist in predicting the behavior of a design. But models, which are
abstractions of reality, are never precise and never predict the future with certainty.
Moreover, models cannot be fully validated until after the design has been decided upon and
the system built.
9

?? In order to compare and rank order design options, it is necessary to have a valid measure
of value. It is not a trivial task to identify a valid value measure, particularly under conditions
of uncertainty and risk.
?? The dimensionality of a typical design is so large that it is computationally infeasible to
search all possible options for the best one.
As a consequence of these difficulties, engineering design can never be reduced to a prescriptive
procedure exclusive of human input. Hazelrigg (1996) asserts that human input will always be
needed for the following:
?? assessment of probabilities (or their equivalent) describing random events,
?? determination of an appropriate value measure, and
?? judgment on which options to include in consideration of alternative design, and which to
neglect.
Thus, design is a decision-making process, rather than a problem solving process. “Decision-
based design” is a term coined to emphasize such a perspective from which to develop
methods for design.
In decision-based design it is asserted that the principal role of a designer is to make
decisions (Mistree, et al., 1990). Decisions help bridge the gap between an idea and reality.
In general, decisions are characterized by information from many sources and disciplines and
may have wide-ranging repercussions. In decision-based design they represent a unit of
communication that has both domain-dependent and domain-independent features. This notion
10

is consistent with the definition of a decision as, “a choice from among a set, either closed or
open, of options, an irrevocable allocation of resources” (Hazelrigg, 1996). Such a
conception of design brings the rationality of decision analysis to the field of engineering design.
The discipline of decision analysis is based on a normative approach in which decision
making has three main elements (Hazelrigg, 1996): identification of options, determination of
expectations of each option, and an expression of values. The resulting rule is, “the preferred
decision is the option whose expectation has the highest value.” An expectation comprises the
range of possible outcomes of a decision paired with probabilities of occurrence. Determination
of expectations on each option is the realm of modeling. Such expectations are practically
always probabilistic because expectations relate to the future, and it is very difficult to predict
the future with precision and certainty.
There is a difference between solving problems and making decisions. The solution of a
problem depends only on the statement of the problem, the data or boundary conditions, laws
of nature and axioms of mathematics. Decisions however, involve options, expectations and
values. And, whereas problems can be solved in the absence of options and values, decisions
cannot be made without their consideration. Thus, decision making cannot be done in the
absence of human values, which are an expression of human needs and desires. There is no
clear connection between the laws of nature and human values, and therefore, there are no
known physical laws that govern the determination of values. The purpose of values in decision
making is to enable the rank ordering of alternatives. It is convenient to use a quantitative
11

function to automate the process of rank ordering options because it is too cumbersome to
make an assessment of relative merits of every design alternative versus every other design
alternative. Such a comparison is generally too complex to make it accurately and consistently
without the use of a mathematical value model, particularly in the presence of uncertainty.
Therefore, in decision-based design, a design decision is a decision that is intended to maximize
value.
By adopting decision-based design as the frame of reference for achieving enterprise
integration it is possible to develop a comprehensive and practical approach for enterprise
design, based on a common paradigm for decision support. This is discussed in Section 1.4.
1.4 A DECISION-BASED ENTERPRISE DESIGN
1.4.1 The Concept of Enterprise Design
Given the focus of this work in manufacturing enterprises, a fitting description of an enterprise is
offered by Gale and Eldred (1996):
An enterprise is a system of business objects that are in functional symbiosis. Symbiosis
means that business objects work together to accomplish mutually beneficial goals. The
business objects of the enterprise include people, machines, buildings, processes, events,
and information that combine to produce the products of the enterprise.
In other words, an enterprise is the sum of its people and the jobs and tasks they perform, its
products, processes and facilities, its information systems and flows, and so on. Clearly all of
these aspects come into being at some time, and then remain dynamic, changing over time.
12

Hence, as affirmed by Peplinski (1997), “an enterprise as a whole is designed, either implicitly
or explicitly.”
In practice, because of the complexity of dealing with enterprises in their entirety, the
design of each aspect is treated separately by functional departments such as research and
development, product design, production, distribution, human resources, etc. However, in this
thesis all of these aspects are grouped in three major different design processes: product design,
manufacturing process design, and organization design. In this work, these processes are
defined as follows:
?? Product design is the conception of a physical artifact to fulfill a set of requirements.
?? Manufacturing process design includes microprocess design and macroprocess design.
Microprocess design is the field of process engineering, which deals with the specification of
equipment, methods, fixtures and process conditions of individual operations for the
manufacture of a product. On the other hand, macroprocess design is the scope of
operations management, which can be defined as the design of the production system that
create the firm’s products, and includes facility location and layout, and the planning,
scheduling and operation of manufacturing facilities.
?? Organization design refers to the design of the organization itself, through the setting of
variables under management control. These variables include the organization structure
(boundaries, levels), technologies (both product and process), people (hiring, training), tasks
(work design), reward systems and information systems.
13

Enterprise design is embodied by the integration of these three design processes
(Peplinski, 1997). It is clear that enterprise design encompasses nearly all aspects of a
manufacturing organization. Designing a manufacturing enterprise is a huge task, and clearly it is
beyond the scope of one single person. It is necessary to carry out the enterprise activity
through decomposition of tasks and responsibilities, assigned to various individuals. Almost all
of these tasks involve making decisions (which usually require problem solving) rather than
merely solving problems. Consequently, the enterprise activity becomes a decision-making
process, in which such decisions are made in face of particular aspects of the enterprise
requirements.
The view of an enterprise activity as a decision-making process suggests pursuing
enterprise integration through coordination of decisions. There are two options to coordinate
decisions: either a centralized and prescriptive process can be specified to enforce coordination,
or an empowered decision-based approach in which support is provided for identifying
interdependencies and tools are provided for modeling across boundaries. Within this last
vision, enterprise design becomes (Peplinski, and Mistree, 1997):
?? identifying and assessing the enterprise-wide effects of each local decision,
?? identifying and assessing the effects of other external decisions on the local decision at hand,
and
?? making each decision to satisfy as many of the enterprise-wide goals as possible, while being
as robust as possible to external decisions that are beyond the decision-maker’s control.
14

Based on this perspective of empowerment through decision-making, a method for enterprise
design has been proposed by Peplinski and Mistree (1997), based on a common model for
decision support. This method is examined in the next section.
1.4.2 A Decision-Based Method for Enterprise Design
In Figure 1.1 a model for decision support spanning operations research, engineering design,
and systems theory, proposed by Peplinski and Mistree (1997), is illustrated. In this model for
decision support, a decision is represented in terms of a set of design variables ? X ? and a set of
n goals and constraints captured by target values {T1, T2, ..., Tn} and models {F1(X), F2(X), ...,
Fn(X)}. Regions of interest are specified for each design variable a priori, and these regions
define the realm of potential solutions for the decision. Goals and constraints represent the
motives for making the decision, and may encompass measures of system cost, performance,
quality, and so on. The target values Ti represent the “aspiration space” or what ideally will be
achieved with the decision, and the models Fi(X) quantify to what extent these aspirations are
achieved by the actual values of the design variables. Analysis is then the process of computing
the values of Fi(X) for given values of {X}, and synthesis is the process of using these computed
values to find the best settings for the design values. A specific value is identified from the
region of interest for each design variable, and this set of values represents the solution that
comes closest to achieving all of the goal targets while not violating any constraint.
15

Synthesis
Analysis
Objectives & Constraints
(Requirements)
T 1 T 2 T n
F 1(x)
Models
F 2 (x) F n (x)
Design Variables
{x}
Figure 1.1 A Hybrid Paradigm for Decision Support (Peplinski, 1997)
This representation builds upon the strengths of operations research, engineering design
and systems theory. It follows the operations research structure of decision variables,
constraints, and an objective function. It is built upon the virtues of engineering design as it
capitalizes on and integrates the basic activities in engineering design, namely modeling, analysis
and synthesis, where modeling encompass the universe of techniques that exist to predict system
performance or behavior, analysis represents the process of performance prediction, and
synthesis represents the prescriptive process of finding the values of the design variables. This
model for decision support has the following characteristics:
?? It is domain-independent.
?? The partitioning of activities into modeling, analysis and synthesis fosters the creation of
methods and tools for decision support.
?? It sets the stage for concrete and specific definitions of interdependence and integration in
terms of decisions.
Synthesi
s
16

Based on this model for decision support, Peplinski and Mistree proposed an enterprise design
method, which is described in broad strokes as a series of tasks in Figure 1.2. These tasks
correspond to the development, formulation and solution of an enterprise design decision
according to the decision support model. The method is represented as a relatively straight
sequence of tasks, but in actuality iteration will likely occur within and across tasks. Each of the
tasks in Figure 1.2 are expanded into more detailed diagrams of tasks, information flows, and
decisions, but due to space constraints these diagrams are not given here.
In Figure 1.2, a designer begins with a baseline decision, represented in terms of design
variables, goals and constraints, and models for analysis. In product design the variables may
represent system parameters or characteristics, while in organization design the variables may
represent different policies or courses of action. This decision is likely geared toward local
measures of merit and may not capture its enterprise-wide implications.
17

Baseline Decision
Task 1 Expand Scope of Decision
Task 2a Identify Modeling Techniques
Task 2b Determine Input Needed for
Analysis
Task 3 Define Boundaries of Decision
Task 4 Transform and Integrate Models
Task 5 Generate Potential Solutions
Recommendation
Figure 1.2 A Decision-Based Method for Enterprise Design
(Peplinski, 1997)
18

Task 1 of the method is employed to explore the potential impact of the baseline
decision. A wide range of enterprise-wide effects are generated by a process of ideation
(brainstorming is a specific example of this) both with the design variables and with an
awareness of overall customer requirements. From this process, additional goals are identified
to expand the scope of the decision.
During Tasks 2a and 2b modeling schemes are identified to quantify the relationships
between the baseline design variables and the additional goals from Task 1. These modeling
schemes are selected from the diverse range of existing system modeling techniques in the
enterprise (which include finite element analysis, system dynamics, discrete event simulation,
stochastic models, regression models, physical prototypes, to name a few); in this fashion
integration is fostered. The two tasks are represented separately to imply a process of iteration;
modeling techniques are identified both by examining the goals and by examining the design
variables. Iteration may be required to resolve inconsistencies. At the conclusion of these tasks
a modeling scheme is selected for each goal and constraint in the decision. These modeling
schemes are tentative and require further hardening. Also, these modeling schemes may require
additional input information, represented by additional clouds. The additional goals and input
information from Tasks 2a and 2b may overlap with other decisions made by different people or
at different times throughout the enterprise. Thus the interdependencies between decisions
become explicit. Because of empowerment it not intended to gather all decisions under some
centralized control, and because there are limitations on the size of any one decision, so at some
19

point the boundaries of the decision must be drawn. This is represented in Task 3. These
boundaries may be forced to slice through interdependencies, so they may be resolved by
formulating a decision that is robust to external issues. The information that remains inside the
decision boundary is then classified into constraints, design variables and noise factors.
As Task 4 is initiated, all of the elements of the decision are potentially in place. Modeling
techniques have been identified for each goal and constraint, and the input needed for each
model is captured by the sets of design variables, noise factors, and constants that have been
identified. The issues that remain are whether the models are efficient enough, and whether
models are in a format that allows integration into a decision formulation. Peplinski and Mistree
(1997) suggest the use of statistical metamodeling techniques during Task 4 to resolve these
issues.
During Task 5 an enterprise design decision is formulated mathematically and solved.
“Solution” occurs through an in-depth process of exercising the formulation - changing
parameters, testing assumptions, and exploring “what-if” scenarios - thereby generating sets of
potential solutions for consideration. In the end one solution is selected, one that satisfies as
many of the enterprise-wide goals as possible and one that is as robust as possible to the effects
of external decisions outside the designer’s control. This solution is the recommendation
submitted for implementation. This method is discussed in detail in Peplinski (1997).
20

1.5 RESEARCH QUESTIONS AND HYPOTHESES
As indicated in Section 1.1, the objective of this work is to establish a mathematical framework
for enterprise design within a decision-based mindset. There are various issues related to such
an aim. The first is the need of modeling the enterprise design process itself. Accordingly, a
first research question is:
Research question 1: How can an enterprise design process be modeled
mathematically in the context of decision-based design?
The fundamental tenet behind this question is that the enterprise design processes, namely,
product process design, manufacturing process design, and organization design, are decision-
making processes carried out by several employees who have to coordinate their decisions.
Mathematical analysis of strategic behavior, where a decision-maker’s action depends on
decisions by others is typical in game theory. Thus, game theory can provide the underpinning
for modeling mathematically an enterprise design process. Because all the decisions of an
enterprise cannot be made simultaneously, the enterprise design process has an important
dynamic structure. The games that are used to model and analyze such dynamic situations are
known as games in extensive form. Therefore, the enterprise design process can be abstracted
as an extensive game. This notion is posed as Hypothesis 1:
Hypothesis 1: An enterprise design process can be mathematically modeled as an
extensive game in a decision based context.
21

Application of game theory classically has had two goals. A first goal is a descriptive goal
of understanding why the players behave as they do which includes describing the results of a
certain game structure. A second goal is a more practical one of being able to advise the
players of the game the best strategy to take. In enterprise design, the later goal can be
expressed as how to appropriately formulate and make local decisions, i.e., decisions carried
out by an individual agent of the organization, while enhancing the convergence of the global
enterprise design process to a common satisfying solution. Here global is used to characterize
the activity carried out by all the decision-makers that comprise an enterprise. This last concern
is expressed as a research question as follows:
Research question 2: How can enterprise decisions be coordinated through a design
formulation based on a game theoretical representation of the enterprise design
process?
Game theory is divided into two branches: cooperative and non-cooperative game theory.
Essentially, in non-cooperative game theory the unit of analysis is the individual participant in the
game who is concerned with doing as well for himself/herself as possible subject to clearly
defined rules and possibilities. If individuals happen to undertake behavior that in common
parlance would be described as “cooperative,” then this is done because such behavior is in the
best interests of each individual. In comparison, in cooperative game theory the unit of analysis
is most often the group or the coalition; when a game is specified, part of the specification is
22

what each group or coalition of players can achieve without reference to how the coalition
would affect a particular outcome or result.
At first, it seems desirable to coordinate enterprise decisions using a cooperative
formulation. However, there are two major drawbacks in trying to attain such solutions: first, a
cooperative formulation is difficult to implement since it requires a single model to quantify all the
relevant aspects that might affect the payoff of each player. Second, some enterprise decisions
involve quite different time scope and have to be made at different times. It is not realistic to
suppose that a designer can incorporate all the various aspects of an enterprise in solving a local
design problem. Therefore, the focus on this thesis is in the coordination of decisions using non-
cooperative game theory. This perception is enunciated in Hypothesis 2.1:
Hypothesis 2.1: Enterprise decisions can be effectively coordinated with a non-
cooperative formulation of the individual design problems.
A fundamental assumption in coordinating decisions through a game theoretical
formulation, either cooperative or non-cooperative, is that the decision-makers that comprise
the enterprise behave rationally, that is, their decisions are unbiased results of their analyses and
estimations.
Multiple criteria and multiple objectives, sometimes in conflict with each other, usually
characterize the resulting design formulations. Therefore, a multi-objective design model is
necessary to embody the design formulation. In this thesis the Compromise Decision Support
Problem is used as such a model. The Compromise DSP is a multi-objective decision model
23

which is a hybrid formulation based on Mathematical Programming and Goal Programming
(Mistree et al., 1993). It is used to determine the values of design variables to satisfy a set of
constraints and to achieve as closely as possible a set of conflicting goals. This assertion is
captured in Hypothesis 2.2:
Hypothesis 2.2: The Compromise DSP is an adequate decision model to embody game
theoretical design formulations within an enterprise design framework.
The embodiment of the aforementioned protocols using the Compromise DSP is discussed in
Chapter 3.
When considering an enterprise design problem, a designer has to deal with uncertain and
incomplete information from other designers. This is so because of the use of non-local
information approximations, the need to forecast information, and the unavoidable randomness
of the manufacturing environment (e.g., demand fluctuations). Therefore, the design problem is
no longer choosing the right course of action but to find a way of calculating, very
approximately, a good course of action. With this shift, the design solution becomes estimation
under uncertainty. The real goal of a local action, within a systems-thinking perspective, is to
establish a satisficing local solution that is also satisficing to the other members of the system.
The question is what are satisficing conditions for other members of the system?
The use of approximations of non-local relevant information helps setting expectations of
other members of the organization (assuming rational behavior). Therefore, it is possible to
provide them with conditions that are reasonably close to their expectations. However, because
24

each one wants to optimize their own performance, it is expected (furthermore, it is desirable)
that there will be a certain number of negotiation and iteration. One way to improve
convergence of the process is to provide each designer with the flexibility to find a satisficing
solution in a region instead of trying to meet a single point. One design approach that is
consistent with this notion is the probabilistic-based design model of Chen and Yuan (1997). In
this work, it is suggested the use of Chen’s and Yuan’s probabilistic-based model in formulating
and solving Compromise DSPs as an effective means to enhance coordination of decisions.
This is expressed in the following research question and its corresponding hypothesis:
Research question 3: How can the design problems be solved to enhance the
coordination of enterprise design decisions?
Hypothesis 3: A probabilistic-based design approach enhances the coordination of
enterprise design decisions.
In this probabilistic approach, the distribution of performance is collected through
statistical analysis and the design parameters are expressed by probabilistic distributions. There
are two main reasons behind Hypothesis 3:
?? As argued in Section 1.3, when dealing with uncertain and incomplete information, only an
expected value (i.e., a probabilistic solution) is a valid measure to rank alternatives.
?? By providing design solutions as ranges, instead of single point values, the possibility to meet
a common satisfying solution is enhanced.
The probabilistic-based design model is explained in Chapter 3.
25

1.6 ORGANIZATION OF THE THESIS
In this chapter a series of research questions and their respective hypothesis are elaborated.
These questions and hypotheses are summarized in Table 1.1.
Table 1.1 Research Questions and Hypothesis
RESEARCH QUESTIONS AND HYPOTHESES
Research question 1
How can an enterprise design process be modeled mathematically in the context of decisionbased
design?
?? Hypothesis 1: An enterprise design process can be mathematically modeled as an
extensive game in a decision-based design context.
Research question 2
How can enterprise decisions be coordinated through a design formulation based on a game
theoretical representation of the enterprise design process?
?? Hypothesis 2.1: Enterprise decisions can be effectively coordinated with a noncooperative
formulation of the individual design problems.
?? Hypothesis 2.2: The Compromise DSP is an adequate decision model to embody game
theoretical design formulations within an enterprise design framework.
Research question 3
How can the design problems be solved to enhance the coordination of enterprise design
decisions?
?? Hypothesis 3: A probabilistic-based design approach enhances the coordination of
enterprise design decisions.
26

These hypotheses embody the framework for enterprise design that is posed as the
objective of this work in Section 1.1. A scheme of the strategy followed in developing this
framework is illustrated in Figure 1.3.
A common
paradigm for
decision
support
The
Compromise
DSP
Enterprise Design as an Extensive Game
Game Theory
Decision-Based Design
Design as a Scientific Discipline
Systems Thinking
Closure
Case Study
A
Probabilistic
Design
Model
Figure 1.3 Implementation Strategy
Chapter 5
Chapter 4
Chapter 3
Chapter 2
Chapter 1
In this chapter the foundations of a framework for enterprise design are established,
namely, systems thinking, decision-based design, and design as a scientific discipline. In
Chapter 2 the enterprise design process is abstracted as an extensive game. The abstraction is
27

ased on the fact that an enterprise is composed of decision-makers who have to negotiate the
value of design parameters to find mutually satisfying solutions. An overview of existing
enterprise modeling tools and methods is presented in Section 2.1. In Section 2.2 the use of
game theory to model and support design processes is examined. Based on such analysis, the
extensive-form game is discussed as a means of modeling the enterprise design process in
Section 2.3. The implications of the resulting model are discussed in Section 2.4.
In Chapter 3 the formulation of design problems is discussed, based on the perception of
enterprise integration as coordination of decisions. In Section 3.1, two different coordination
mechanisms, namely implicit coordination and explicit coordination, are examined. A way to
merge them and capitalize their different advantages with a decision-based perspective using a
common model of decision support and the Compromise DSP is presented in Sections 3.2 and
3.3. Section 3.4 contains a discussion of how coordination based on the use of compromise
DSPs can be translated into specific design formulations, either cooperative or non-cooperative,
as well as methods to implement both. In Section 3.6 a probabilistic approach for solving the
resulting compromise DSPs is proposed, ideally facilitating the cooperative process.
In Chapter 4, the framework developed in the first three chapters is exercised in a case
study: the integrated product and process design of an absorber/evaporator module for a family
of absorption chillers. Therefore, the approach is applied to both product design and process
design. The synthesis of the system is performed with four different approaches:
1. The various individuals involved in the design process acting in isolation (Section 4.3).
28

2. The individuals coordinating their decisions through a cooperative formulation of the design
problem (Section 4.4).
3. The individuals coordinating their decisions through a non-cooperative formulation of the
design problem and using a non-probabilistic design model (Section 4.5).
4. The individuals coordinating their decisions through a non-cooperative formulation of the
design problem and using a probabilistic design model (Section 4.6).
In Section 4.7 the results obtained with the four approaches are compared and discussed, and
in Section 4.8 the case study is summarized and open issues are examined.
Finally, in Chapter 5 the thesis is surveyed, its relevance for the engineering and design
practice is discussed and possible directions of future work are identified.
29

CHAPTER 2
ENTERPRISE DESIGN AS AN EXTENSIVE GAME
In this chapter, the first element of the enterprise design framework proposed in Chapter
1 is examined, namely, the abstraction of an enterprise design process as an extensive game.
The motivation for using a game theoretical representation of the enterprise is to capture the
dynamics of interaction between decision makers who are trying to coordinate their activities
through the appropriate formulation of their decisions. First, a review of previous enterprise
modeling representations, along with a discussion on their use as a decision support tool, is
presented. Then, the use of game theory in engineering design is reviewed in order to introduce
the basic concepts that are used to abstract an enterprise design process as a game. The core
of this chapter is Section 2.3 where it is addressed how an enterprise design process can be
mathematically modeled as an extensive game. Attention is paid to a particular type of solution
of extensive games, Stackelberg equilibrium. The assumptions and definitions used in
abstracting design processes as games are examined. Finally, the chapter ends with a
discussion of the relevance of modeling an enterprise activity as a game.
29

2.1 OVERVIEW OF ENTERPRISE MODELING: ARCHITECTURES,
TOOLS AND METHODS
An imperative for achieving a framework for enterprise design and integration is enterprise
modeling. Designers need models of the enterprise to allow them to “tune” the formulation of
their individual decisions with the performance of the system.
Most of the efforts in enterprise modeling have been undertaken in Computer Integrated
Manufacturing (CIM), and they have primarily focused on creating architectures, modeling tools
and modeling methods (Kateel, et al., 1996). This section presents a brief overview of some
selected architectures, tools and methods to exemplify the enterprise models that are being
currently used in CIM.
2.1.1 Modeling Architectures
In CIM, an architecture is defined as a “body of rules that define those system features which
directly affect the manufacturing environment into which the system is placed. These features
include system configuration, component locations, interfaces between the system and its
environment, and modes of operation” (O’Sullivan, 1994). Several enterprise modeling
architectures have been proposed in the literature, and surveys can be found in (Kateel, et al.,
1996) and (O’Sullivan, 1994). A brief overview of selected architectures follows:
CIM-OSA: This CIM-Open Systems Architecture has three levels of model derivation:
requirements definition, design specification and implementation description. It also has four
30

views (functional, informational, resource, and organizational) and three levels of model
instantiation: generic, partial, and particular (Jorysz and Vernadat, 1990).
NBS: This architecture, developed by the National Bureau of Standards, is based on the
concept of hierarchical control. The NBS architecture was developed to consist of five levels of
levels of hierarchy: facility, shop, cell, work station, and machine. The decomposition is based
on procedures, functions, and rules (O’Sullivan, 1994).
ICAM: The Air Force’s Integrated Computer Aided Manufacturing project offers a suite
of IDEF (ICAM DEFinition) modeling methods, from functional modeling (IDEF0), information
modeling (IDEF1), data modeling (IDEF3), object oriented design (IDEF4), and ontology
capture (IDEF5), among others (O’Sullivan, 1994).
CAM-I: The Computer Aided Manufacturing-International architecture is a general
representation of manufacturing enterprises. The functional decomposition method used to
create this architecture is prescribed to indicate the details such as company policies and
procedures, organizational structure and standards (Doumeingts, et al., 1995).
IMPACS: The IMPACS architecture uses IDEF0, Data Flow Diagrams (DFD),
“Graphe a Resultatis er Activities Intercies” (GRAI) Grids and nets, IDEF1x and group
technology. IMPACS outlines a cellular architecture. Software modules such as dispatcher,
scheduler, mover, producer and monitor control the production cells. These software modules
are designed to be compatible with each other even if different vendors develop them.
O’Sullivan (1994) states that the IMPACS architecture has been widely accepted among
31

manufacturing software vendors as a useful and practical interpretation of the production
management system.
2.1.2 Modeling Tools
CIM models are built using the modeling tools prescribed by modeling methodologies. These
tools refer to techniques used for diagrammatically representing functions or activities. Some of
the most widely known modeling tools are:
IDEF Modeling Tools: IDEF0, IDEF1, IDEF1x, and IDEF2 are modeling tools
developed by the ICAM project of the US Air Force. These four main modeling tools,
complement each other to provide functional, informational and dynamic models of the system.
a) IDEF0 Models: IDEF0 is a comprehensive and expressive functional modeling
language capable of graphically representing a wide variety of business, manufacturing
and other types of enterprise operations to any level of detail. The basic construction
block of IDEF0 is a function block linked to other function blocks by inputs, outputs,
mechanisms and controls. Links between the blocks may be either physical objects
(material flow) or information flow. These models have three important features:
“context” which indicates the position that the subject model takes up in the systems
hierarchy, “viewpoint’ which refers to the perspective which the model adopts, and the
“purpose” which indicates the reason for existence of the model (Colquhoun, et al.,
1993).
32

) IDEF1, and IDEF1x Models: IDEF1 is a technique for modeling information
requirements of a function, in terms of the structure of information. IDEF1x is an
extension of IDEF1 and deals with the flow of information (Pandya, 1995).
c) IDEF2 Models: IDEF2 models have the capability to describe as well analyze a
system. However, IDEF2 is an unsupported simulation language. Hence, other
simulation models such as ARENA (Drevna and Kasales, 1994), or PROMODEL
(Baird and Leavy, 1994) are more commonly used.
Structured System Analysis (SSA): SSA is a data flow approach to systems design. It
is also quite effective in representing the flow of physical entities. This technique prompts the
user to think in terms of what to accomplish rather than how. It is more detailed and software
oriented than IDEF0 and is the most preferred modeling tool for data flows (Gane and Sarson,
1979).
GRAI Grids and GRAI Nets: GRAI grids and nets are the tools developed by the
GRAI Laboratories of France to model decision-flow processes in manufacturing environments.
In the GRAI grid, various activities are modeled with respect to the decisions and information
flows between them while the GRAI net delineates the decision-making process itself.
2.1.3 Modeling Methods
33

Modeling methods in CIM are guidelines to combine the modeling tools described in the
previous section to model a particular system. Some of the methods used for modeling
enterprise systems include IDEF, SSADM, SADT, GIM and CIM-OSA cube:
IDEF Method: This methodology prescribes the integration of IDEF0, IDEF1 and
IDEF2 and models to describe functional, informational and dynamic models of an enterprise.
Structured Systems Analysis and Design Methodology (SSADM): SSADM is a
procedural framework developed specifically for use in system development projects. It uses
three modeling tools: data flow diagrams, logical data structures, and entity life histories to
provide function, data, and event views of the systems (Pandya, 1995).
Structured Analysis and Design Technique (SADT): A system analysis and design
methodology to represent the structure of the system diagrammatically. This method makes use
of a number of tools such as activity diagrams, data diagrams, node lists and data dictionaries to
represent the structure of the system. The analysts are required to consider activity or function
and data views of the system being modeled, thereby encouraging the creation of integrated
enterprises.
CIM-OSA Method: Defines an integrated methodology to design, implement, operate,
and maintain an enterprise. The CIM-OSA method deploys many well-accepted ideas and
principles to design a CIM enterprise.
Object-Oriented Approach: This methodology proposed by Kim, et al. (1993) consists
of an analysis phase and a design phase. In the analysis phase, functional decomposition is
34

employed to define the information flow among the manufacturing functions and their
infrastructures. Decomposed functions are represented by functional diagrams, which in turn
are transformed into an object-oriented information model (an object oriented database
management system).
2.1.4 Enterprise Modeling and Enterprise Design
The enterprise modeling schemes discussed so far are integrated modeling schemes, i.e., they
are intended to be applied to model an enterprise at nearly any level of abstraction. Enterprise
integration in CIM is thus pursued by constructing and applying an integrated modeling scheme
or by integrating existing models. The argument in favor of the former approach (a single
model) is that local models built with the big picture in mind do not search for local optima,
rather they contribute to the system effectiveness. On the other hand, the argument in favor of
the second approach (of many small linked models) is that a unified model that captures all
aspects and variables of an enterprise is extremely cumbersome for practical use.
In order to determine how to model an enterprise it is necessary to consider what is the
model for and how is it going to be used. In a framework for decision-based enterprise design,
an enterprise model is needed to support and improve design decisions. This need motivates the
following research question:
35

How can an enterprise design process be modeled mathematically in a decision-based
design context?
The hypothesis posed in Section 1.5 to answer this question is that an enterprise design
process, in the context of decision-based design, can be mathematically modeled using game-
theoretical constructs. Emphasis is given to the need of a mathematical model because
mathematical models support rational choice among design alternatives. Without mathematical
models, the comparison of alternatives and the tradeoff between competing objectives lacks the
rigor that is demanded in decision-based design. This game theoretical representation of the
enterprise is the first floor of the enterprise design framework that is being constructed in this
thesis.
Before proceeding with a discussion on how to model the enterprise activity using game
theory, it is worth examining why the CIM architectures previously described do not to fulfill the
need that originated the preceding research question. For a decision-based enterprise design,
we require a representation of the enterprise that provides insight in how the individuals or
agents of an enterprise should coordinate their decisions in carrying out their respective tasks.
Even if a single model of an enterprise were affordable, computationally and economically, to
support any design decision within an enterprise (which is very unlikely), there would still be the
problem that such a model would not provide any insight in how to formulate those decisions,
which in a decision-based design paradigm have to be formulated, at least the important ones.
On the other hand, mathematically modeling a system’s behavior, where one decision-maker’s
36

action depends on decisions by others, is well established in game theory. Examples of modeling
a design process using game theoretical representations can be found in (Rao, et al., 1997),
(Lewis and Mistree, 1997) and (Lewis, 1997).
2.2 A GAME THEORETICAL REPRESENTATION OF THE
ENTERPRISE DESIGN PROCESS
2.2.1 Decisions in a Manufacturing Enterprise
Manufacturing decisions differ greatly with regard to the length of time over which their
consequences persist. This makes it essential to use different planning horizons in the
decision-making process. For example, the decision to construct a new plant influences
operations for years; thus, we must forecast these effects years into the future in order to make
a reasonable decision. On the other hand, the effect of selecting a particular part to work on a
particular workstation affects the system only within hours or minutes. Within a given company,
longer time horizons are generally used at the corporate office, which is responsible for long-
range business planning, than at the plant where day-to-day execution decisions are made. In
37

Table 2.1 various manufacturing decisions that are made over long, intermediate, and short
planning horizons are listed according to Hopp and Spearman (1996).
In general, long-range decisions address strategy, by considering such questions as
what to make, how to make it, how to finance it, how to sell it, where to make it, where to get
materials, and general principles for operating the system. Intermediate-range decisions address
tactics for determining what to work on, who will work on it, what actions will be taken to
maintain the equipment, and so on. These tactical decisions must be made within the physical
and logical constraints established by the strategic long-range decisions. Finally, short-range
decisions address control by moving materials and workers, adjusting processes and
equipment, and taking whatever actions are required to ensure that the system continues to
function towards its goal. Therefore, when trying to integrate the various activities of the
enterprise it is necessary to formulate decisions that involve quite different scope and time
horizon, which complicates their integration into joint formulations.
Table 2.1 Strategy, Tactics, and Control Decisions
(Hopp and Spearman, 1996)
Time Horizon Length Representative Decisions
Long-term (strategy) Year-decades Financial decisions
Marketing strategies
Product designs
Process technology decisions
Capacity decisions
Facility locations
Supplier contracts
38

Personnel development programs
Plant control policies
Quality assurance policies
Intermediate-term (tactics) Week-year Work scheduling
Staffing assignments
Preventive maintenance
Sales promotions
Purchasing decisions
Short-term (control) Hour-week Material flow control
Worker assignments
Machine setup decisions
Process control
Quality compliance decisions
Emergency equipment repairs
Due to the different horizon and scope of decisions, manufacturing enterprises are
organized in a hierarchical structure, with the lowest level dealing with issues that are either
limited in scope or where, because of their frequency, a programmed response is appropriate.
At the middle level of the hierarchy, there is greater freedom for making decisions but still within
constraints of authority posed by the higher levels. At high levels in the hierarchy, the decision is
typically one that affects the whole system. Clearly, this hierarchical organization of the
enterprise requires defining what is the range of authority of its agents, and mechanisms to
coordinate and resolve their conflicts at any level. Game theory can provide a mathematical
representation of an organization that facilitates the coordination of individuals. This is discussed
in the next section.
39

2.2.2 Decisions and Game Theory
Rousseau, in his Discourse on the Origin and Basis of Equality Among Men, comments
(quoted from Fudenburg and Tirole, 1993, p. 3):
If a group of hunters set out to take a stag, they are fully aware that they would all have
to remain faithfully at their posts in order to succeed; but if a hare happens to pass near
one of them, there can be no doubt that he pursued it without qualm, and that once he
had caught his pray, he cared very little whether or not he had make his companions
miss theirs.
To make the situation described by Rousseau into a game, suppose that there are only two
hunters, and that they must decide simultaneously whether to hunt for stag or for hare. If both
hunt for stag, they will catch one stag and share it equally. If both hunt for hare, they each will
catch one hare. If one hunts for hare while the other tries to take a stag, the former will catch a
hare and the latter will catch nothing. Each hunter prefers half a stag to one hare. This is a
simple example of a game. The hunters are the players. Each player has the choice between
two strategies: hunt stag and hunt hare. The payoff to their choice is the prey. How should the
hunters act? What is the expected outcome of the game? These are the issues of interest in
game theory.
In a general sense, a “game” is a set of rules completely specifying a competition, including
the permissible actions of, and information available to each participant, the criteria for
termination of the competition, and the distribution of payoffs (Websters, 1984). From a
systems perspective, a “game” is defined as follows (Myerson, 1991):
40

A game consists of multiple decision-makers who each control a specified subset of
system variables and who each seek to minimize their own cost functions subject to their
individual constraints.
Game theory is the study of the strategic interactions of such games (Fudenburg and Tirole,
1995).
Now consider a more practical case than the aforementioned hunting game. Consider the
problem of a manufacturer -let’s call it P- that must choose between a high and a low price for
the seasonal batch of products. In making this choice P wants to think about what the prices of
other similar and substitute products are likely to be. P could simply define its pricing policy for
some given exogenous beliefs about the competitor’s prices, but it seems more satisfactory to
try to predict these prices from some knowledge of the industry. In particular, P knows that the
other firms choose their own prices on the basis of their own predictions of the market
environment, including P’s prices. The game-theoretic way for P to use this knowledge is to
build a model of the behavior of each individual competitor, and to look for a solution that forms
an equilibrium of the model. This relatively simple example of pricing is typical in game theory,
which has been extensively used in business administration, economics and disciplines alike, but
game theory has recently been applied to other fields such as engineering design.
2.2.3 Fundamentals of Game Theory in Design
41

Traditionally, “optimal” design of parametric systems is generally referred to the concept of a
minimum or maximum. However, there are many other optimization concepts defined for
parametric systems if we allow optimal design to include multiple criteria optimization with
multiple optimizers. Such an extended view, more in line with a systems perspective, brings
optimality in parametric systems into the realm of game theory along with its numerous
optimization concepts.
Generally during a design process, the variables in a system are constrained by a system
of equality and inequality equations of the form (Vincent, 1983):
42
g(X) = 0 (2.1)
h(X) ? 0 (2.2)
where X is a vector variable and g and h are vector functions, that is ? x ,..., x ?
? ? g ( X ),..., g ( X ) ? , and h( X ) ? h1(
X ),..., hq(
X ) ?
g( X ) 1
n
X 1
? ,
? . The equations given by (2.1)
and (2.2) may be thought of as defining the system under study. In mathematical terms the
“system” is a constraint set defined by
? X g(
X ) ? 0 and h(
X ? 0?
m X ? S ?
)
(2.3)
where S m designates an m-dimensional space of real numbers. Thus, X is the set of all points in
the m-dimensional variable space that satisfy all the constraints. For each choice of X, one or
more costs (or objectives) may be defined by means of a cost function (or objective function)
f(X):
? f ( X ),..., f ( X ) ?
f ( X ) ? 1
r
(2.4)
m

When r = 1 we have a scalar-valued cost function. For r >1 we have either a vector-valued
cost criterion or a game, depending whether one or more designers are involved in selecting
the point X and the relation of the various costs to the various players. This classification of
optimality problems is illustrated in Figure 2.1.
N = 1
N 1
?
Number of Objectives
r = 1 r 1
?
Scalar
Optimization
Problem
Scalar
Games
Vector
Optimization
Problem
Vector
Games
Figure 2.1 Various Formulations in Optimization Theory
(Mesterton-Gibbons, 1992)
Typical optimization processes focus on the upper-left quadrant, namely scalar
optimization problems with one objective and one decision-maker. With only one designer
selecting design parameters we have either a problem in scalar optimization or vector
optimization. In the former case, a point X ? is a minimum point if and only if there does not
exist a point such that f(X)

y replacing X? S m by X? B m ? S m , where B m is a region centered at X ? and B m ? S m is the set
of points common to B m and S m .
For vector costs a point Xp? S m is said to be a Pareto-minimum if and only if there does
not exist a point X? S m such that fi(X) ? f(Xp) for all i ? ? 1,...,r?
and fj(X) ? f(Xp) for at least
one j ? 1,...,
r?
? (Vincent, 1983). If Xp is Pareto-minimal then for any other X at least one of
the costs that compose the vector cost must increase or all costs remain the same. Under this
definition it is usually possible for many points to have the Pareto-minimum property. Indeed for
most problems in engineering design, the Pareto-minimal points are not isolated. Vincent
(1983) provides a thorough discussion of the mathematical conditions for achieving Pareto-
minimum points. Here, it is enough to note that one designer choosing parameters for a vector-
valued optimization problem should try to restrict choices to the Pareto-minimal set. This set
may be obtained by a number of methods such as the mini-max method (Osyczka, 1984)
multicriteria dynammic programming (Rosenman and Gero, 1983), and scalarization methods
such as compromise programming (Goicoechea, et al., 1982), goal programming (Ignizio,
1985), the global criterion method (Rao, 1984), the ratios method (Metwalli, et al., 1984), to
name some. Vanderpooten (1989) and Hwang (1980) provide a survey of various of these
methods.
Now suppose that a design project has been broken down into a number of subprojects
each with a set of design objectives and each with a design team in charge of optimizing those
objectives. Since communication is generally possible between the subgroups and since each
44

designer must consider the possible choices made by other design teams, it is envisioned that
there will be many tentative designs which will be passed back and forth between the various
design teams before some ultimate design is finally obtained. Operations researchers and
economists, in studying competitive systems, have developed theories for games that are
applicable to this kind of situation. This is the most general case of the various optimization
formulations that are shown in Figure 2.1.
Two game-theoretical models are of relevance for this thesis to describe the interaction of
players: the cooperative game model based on the concept of Pareto equilibrium, and the
non-cooperative model based on the concept of Nash equilibrium. In cooperative games,
each player compromises his/her objective in order to allocate the resources in such a way that
the performance of the system of players be as optimal as possible. On the other hand, in non-
cooperative games each player selects a share of resources with the view of optimizing his/her
performance. These models are described next using a simple 2-player terminology:
?? Pareto or cooperative model:
In this model both players have knowledge of the other players’ information and they work
together to find a Pareto solution. If the players cooperate, they can be expected to obtain
better solutions than when they do not. A pair (x1p, x2p) is Pareto optimal if no other pair (x1,
x2) exists such that:
45

f 1 (x 1, x 2 ) ? f 1 (x 1 p , x 2p ) & f 2 (x 1 , x 2 ) ? f 2 (x 1p , x 2p ) (2.5)
In the cooperative solution the group of players allocate resources with the intent that all
players be at an optimal point as far as possible. The group must decide how to distribute
resources so that a gain for one player does not result in an unacceptable loss for other players.
One method is to distribute the resources such that all players are as far from their worst cases
as possible. To find such a solution it is necessary to formulate a supercriterion as a measure
of compromise. This supercriterion, a function of the objectives, is essentially a penalty function
that penalizes solutions with objectives that are too close to predefined worst cases (Rao,
1991). Implementing cooperative game theory requires a two-step algorithm. First, the set of
Pareto solutions is generated. Then the Pareto solution that has the most optimal supercriterion
is selected as the best design. This method is difficult to implement in automated routine. Rao
and Freiheit (1991) suggest a method to perform both tasks simultaneously by combining the
Pareto optimal solution generation and the supercriterion into one objective by subtracting the
supercriterion from the Pareto optimal objective. A single-objective optimization algorithm is
used to minimize this new objective. This formulation change the characteristic of the
optimization problem such that it is neither truly minimizing the Pareto objective nor is it truly
maximizing the supercriterion, but according to Rao and Freihet (1991), “from an engineering
point of view, the computational savings warrant the acceptance of a solution which deviates
slightly from a true Pareto optimum.” They support their assert with numerical examples. The
algorithm to implement this approach is presented in Chapter 3.
46

?? Nash or non-cooperative model:
In this model players make decisions based on prediction of others players’ strategy. A
strategy pair (x1n, x2n) is a Nash solution if
where
f1( x1
n , x2
n ) min f1(
x1
, x2
n ), x1
? X1
47
? (2.6)
f 2(
x1
n , x 2n
) ? min f 2(
x1n
, x 2 ), x 2 ? X 2
(2.7)
? x1n
? X 1 : f1(
x1
, x2
) ? min f1(
x1
, x 2 ) ?
x1 n ( x2
) RRS 1(
x1
) ?
n
x1
? X 1
? (2.8)
? x 2n
? X 2 : f 2(
x1
, x 2 ) ? min f 2(
x1
, x2
) ? x2
X 2
x 2 n ( x1
) RRS 2(
x1
) ?
n
?
? (2.9)
are called the rational reaction sets (RRS) of the two players. The RRS is a set of solutions
that a player constructs based on public information from other players.
Non-cooperative game theory is based on the assumption that each player is looking
out for his/her own interests. Each players selects a share of the resources only with the view of
optimizing his/her own objective. The players then bargain with each other until equilibrium is
reached. The resultant solution is known as Nash equilibrium, and no player may improve his
objective by deviating from it as long as the other players maintain their choices. The solution
can be dependent on the order in which the resources are allocated and the order in which the
players choose the resources when more than one equilibrium point exists.
In complex problems, finding exact expressions for the RRS is difficult since it involves
finding functions relating the independent variables of one player as a function of the independent
variables of the other players. In this case, the RRS have to be approximated.

So far, the fundamental concepts on game theory required to build a game-theoretical
representation of the enterprise design process have been introduced. In the next section these
fundamentals are applied in building such a model.
2.3 ENTERPRISE DESIGN AS AN EXTENSIVE GAME
2.3.1 Extensive-Form Games
In the examples presented in Section 2.2, namely, the hunters’ game and the oligopoly-pricing
problem, the players choose actions simultaneously. However, in the design of an enterprise,
i.e., its product, its manufacturing process and its organization, all decisions cannot be made
simultaneously. A more general form of non-cooperative games, the so-called games in
extensive form, can be used to model such a dynamic situation. Attention is given in this work
to a particular kind of extensive-form games known as multi-stages games with observed
actions.
A multi-stage game with observed actions are games in which (1) all players (including
“Nature”) knew the actions chosen at all previous stages 0, 1, 2, ..., k-1 when choosing their
actions at stage k, and (2) all players move “simultaneously” in each stage (Fudenburg and
Tirole, 1995). Players move simultaneously in stage k if each player chooses an action at stage
k without knowing the stage-k action of any other player. However, “simultaneous” moves do
not exclude games where players move in alternation, as there is the possibility that some of the
players have the one-element choice “do nothing.”
An extensive form of a game contains the following information:
48

1. The set of players.
2. Number of stages and order of moves, i.e., who moves when.
3. The player’s payoffs as a function of the moves that were made.
4. What the players’ choices are when they move.
5. What each player knows when making choices.
6. Probability distributions over any exogenous events.
The difference in the content of information of extensive games and static games is shown in
Table 2.2. The fundamental difference is that in extensive games strategies correspond to
contingent plans, whereas in static games strategies correspond to uncontingent actions. That
is, in a game where sequential moves are to be performed (as in chess), one action is taken in
contingency of possible future actions, whereas in a static or one-stage game (like flipping once
a coin) there is no need to develop a strategy that considers future actions.
Table 2.2 Contents of Static- and Extensive-Form Games
STATIC GAME EXTENSIVE GAME
Set of players
? i?
i ? 1,
2 N
I ? ,...,
One stage game
K=1
Set of players
? i?
i ? 1,
2 N
I ? ,...,
Number of stages K ? 1
and order of Moves
Players’ payoff as function of design variables Players’ payoff as function of design variables and
history
49

k
f i (X )
( X , H )
Players’ choices
A ?
i
S
i
f k ? ? 1,
2,...,
K?
i
Players’ choices
k
A ? A ( H ) ? S k ? ? 1,
2,...,
K?
Players’ information when making a choice Players’ information when making a choice
Probability distributions over exogenous events Probability distributions over exogenous events
In a stage k of a multi-stage game, all players i ? ? choose simultaneously actions from
choice sets Ai(h k ) (some of the choice sets may be “do nothing”) where h k denotes the
information sets or “history” of the game at the stage k of the game. Note that Ai is a subset of
the design space of the player, Si. At the end of each stage, all players observe the action
profile of the stage.
Let a k ? a1 k , a2 k ,..., an k ? ? be the stage-k action profile (the actions performed by the n
players at stage k), where ai k is the value of the design vector Xi that is controlled by player i at
stage k. Let also h 0 ? ? be the history at the start of the play. At the beginning of stage 1,
players know history h 1 , which can be identified with a 0 given that h 0 is trivial. In general, the
available actions of the ith player in stage k depend on what has happened previously,
1 k?1
k
? a , a ,..., a ? f ? ?
a ?
i
i
50
k 0 ? f
h
(2.10)
In this setting, a pure strategy for player i is simply a plan of how to play in each stage k for
possible history h k .

Let K be the total number of stages in the game, and let H k denote the set of all stage-k
histories as follows:
A ( H
i
k
51
k
) ? ? A ( h )
(2.11)
hk
? H k
A pure strategy si for player i is a sequence of maps ? ? K k
of player i’s feasible actions Ai(H k ):
i
si k 0
? , where each s k k
i maps H to the set
si k : H k ? Ai(H k ) (2.12)
Hence, the stage-0 actions are a 0 =s 0 (h 0 ), the stage-1 actions are a 1 =s 1 (a 0 ), the stage-2 actions
are a 2 =s 2 (a 0 ,a 1 ), and so on. This is called the path of the strategy profile. Since the terminal
histories represent an entire sequence of play, we can represent each player i’s payoff as a
K ?1 r
function f : H ? R .
i
To illustrate this, consider a game with two stages and two players, depicted as a game
tree in Figure 2.2 where qi represents the selection of player i. Both Player 1 and Player 2 have
as choice options the set of actions {Ø,3,4,6} (Ø is the option “do nothing”) and Player 2 acts
only after Player 1 has acted. Thus,
A1(
H
A ( H
1
1
2
) ?
) ?
1
? 3,
4,
6?
A2(
H ) ? ? ? ?
2
? ? A2
( H ) ? ? 3,
4,
6?
The payoff vectors are displayed next to the corresponding terminal nodes. The information
that players have when choosing their actions is usually represented using the information set

h ? H that partitions the nodes of the tree; that is, every node is in exactly one information set h
and the action set at h is denoted as A(h). Finally, the probability distributions over exogenous
events are represented as moves by “Nature” that have some probability distribution. For
example, in the previous game, if Player 1 were “Nature”, a probability distribution on its
actions could be a vector [p(3)=0.3, p(4)=0.2, p(6)=0.5].
q2=3
q2=4
q1=3
q 2=6
q2=3
q 1=4
Player1
q2=4
q 2=6
q1=6
q 2=3
q2=4
q2=6
(18,18) (15,20) (9,18) (20,15) (16,16) (8,12) (18,9) (12,8) (0,0)
Payoff of player
1
Player 2
Figure 2.2 Example of an Extensive Game
2.3.2 Enterprise Design as an Extensive Game
Payoff of
player 2
The mathematics of extensive-form games, described in the previous section, can be readily
applied to model an enterprise design process. As a simple example, consider a
Leader/Follower relationship between two design teams, where the design teams are abstracted
52

as players in a game. The actions of the two players are to set the value of the design variables
X1 for player 1 and X2 for player 2.
Player 1, the leader, chooses the value of X1 first, and player 2 observes X1 before
choosing X2. Since player 2 observes player 1’s choice before choosing X2, player 2 can
condition his choice of X2 on the observed level of X1. Since player 1 moves first, she cannot
condition her output on player 2’s. Thus, it is natural to expect that player 2’s strategy in this
game be a map of the form:
s2: X1? X2
Given this strategy, the outcome is a vector X=[X1, s2(X1)] with payoffs f1(X) and f2(X).
In the Stackelberg equilibrium, which is the expected equilibrium in this game, player 2’s
strategy is to choose, for each X1 the level of X2 that solves min f2(X1, X2), so that X2 is equal to
the reaction function or rational reaction set RRS2(X1). Given that player 1 expects the
strategy of player 2 to be RRS2(X1) , her choice of X1 should be the solution to min f1(X1,
RRS2(X1)). Thus, a formulation of a strategy for player 1, the leader, would be as follows:
minimize f 1 ( X1,
X2
), ( X1,
X2
) S1
? S2
? (2.13)
satisfying ? RRS ( X ) ? X ( X )
(2.14)
X2 2 1 2n
1
Why is this Stackelberg equilibrium the expected one? An informal answer is that, as
observed by Fudenburg and Tirole (1995), this is the only “credible” equilibrium if both players
act “rationally.” A formal answer is that the Stackelberg equilibrium is consistent with the
notion of backward induction, so called because the idea is to start by solving the optimal
53

choice of the last mover for each possible situation he might face, and then work backward to
compute the optimal choice for the player before.
The ideas of credibility and backward induction have been generalized with the concept of
subgame-perfect equilibrium (see Fudenburg, and Tirole, 1995), which extends the idea of
backward induction to extensive games where players move simultaneously in several periods.
In this case, the backward-induction algorithm is not applicable because there are several “last
movers” and each of them must know the moves of the others to compute the optimal choice.
As an illustrative example of how a two-players game can be used in an actual design
scenario consider the collaboration between a stress engineer and a manufacturing engineer in
the design and production of a wing section of an aircraft who have to negotiate a solution. The
stress engineer and the manufacturing engineer interact using a CAD drawing (the public
information). The stress engineer needs a solution that transmits loads well through the
structure, and the manufacturing engineer needs a structure that is easy to fabricate, using, for
example, an automatic riveting machine. The criteria used by each specialist are private in that
they are complex and concerned with their particular technologies. In this example, the stress
engineer is concerned with arrangements such that the loads carried in the elements are well
formed, in that internal load is transmitted throughout the structure under a given external load
specification. The description of the stress engineer concerns loads, stresses, transmission, and
techniques for finding them. He/she works privately with a model to calculate load patterns that
must match the load transmission properties of the sized geometry. On the other hand, the
54

manufacturing engineer is concerned with a design that can be produced with the machines
currently available using techniques and tooling currently in use. For example, he/she would like
to keep rivet spacing constant, or at least to a small number of different rivet spacings, in order
to limit tooling set-up cost.
Aircraft design is one example of a large-scale design of a system, where there are
many specialists interacting, each with their own technology and language. For example, there
are aerodynamicists who use surface models and flow-field equations; there are maintainability
engineers concerned with access, disassembly, and replacement; there are hydraulic engineers,
and thermodynamic experts, to name just a few. They have to collaborate and negotiate to
produce a design that satisfies all of them. This means that each agent has a justification of the
design that he or she is satisfied with.
Clearly, a common and single model would not be practical to support each design
decision of the aircraft development. Thus, its design has to be partitioned into a series of
problems handled by disciplinary teams, and the overall design proceeds by the cooperation
and negotiation of individuals, teams and/or departments, each one with private information.
Moreover, each agent may use several different models or partial models for different purposes.
Because the criteria used by each agent is private to them in that they are complex and
concerned with their particular goals and technologies, each agent has limited ability to
understand other agents’ motivations and decisions. Therefore, the design proceeds by
successive coordination of decisions.
55

Aircraft design is also an example of a decision making process in which each agent has
public and private information during the design process. In order to take the aircraft design
process into an extensive-form game consider how the six points that define an extensive game
can be readily applied:
1. The set of players. They would vary depending on the level of hierarchy of the organization
and the extent of integration that is considered. At a high level of hierarchy the players
could be functional departments such as product engineering, manufacturing, marketing,
purchasing, etc. At a lower level they could be disciplinary teams within a department and
some “external” players to include relevant external factors over which the team does not
have control.
2. The order of moves. Who moves when depends on the policies of the organization.
Usually there is always an established plan for product development that is used to specify
when and how do different agents participate in the design process.
3. The players’ payoffs as a function of the moves (or history of the design process). This is
straightforward if there are clear goals defined in the beginning of each stage of the design
process (the payoff could be the deviation from a previously specified target).
4. The players’ choices when they move. This is a subtle point when considering design
processes, specially at early stages, because the alternatives may not be known in advance
(for example, new technologies might be developed during the process).
56

5. Player knowledge when making choices. This includes private and public information.
Public information is any information that is common to all the design teams, such as a CAD
model which serves as baseline model for all the players involved in the process between
updating meetings.
6. Probability distributions over exogenous events. Here, exogenous events may refer to
movements by “Nature” such as market demand, interest rates, suppliers’ deliveries, etc.
It should be apparent that an enterprise design process can be abstracted as a multi-stage
game, in which each of the design groups is a player who strives to act in the interest of the
organization, while at the same time tries to optimize his/her standing (e.g., minimize production
cost or maximize reliability).
The relevant assumptions and definitions that are used in this thesis to model an enterprise
design process as a game are:
Assumption 2.1: Decisions are supported with mathematical models.
Assumption 2.2: No agent of an enterprise has sufficient competence and resources to solve an
entire enterprise design problem. In other words, even if an individual involved in a design
problem strives consciously to solve it considering all the aspects and variables of the system,
he/she will do so in a way that reflects cognitive and computational limitations.
Assumption 2.3: The common link among each agent is the company for which they all work.
They all act respecting the social rules established by the organization.
57

Assumption 2.4: The design variables of different agents do not overlap. The local control of
each agent is exclusive. That is, if one agent controls X1? S1 and another agent controls X2? S2
then S S ? ? .
1 ? 2
Given these assumptions, the following definitions are used to formalize an enterprise
design process as a game:
Definition 2.1: In enterprise design, a player in a game is the decision maker or agent, which is
embodied by a designer, design team or department, and their associated analysis and synthesis
design tools.
Definition 2.2: A strategy of an agent is embodied by a design formulation.
Definition 2.3: A payoff of an agent is the value of the motivating function at a given move.
Definition 2.4: Public information is knowledge that is accessible to any agent of an
enterprise.
Definition 2.6: A stage of the game is demarcated by public events when one or more agents
of an enterprise make design decisions public.
The equivalency between design and games based on the discussion carried out in this
chapter is shown in Figure 2.3.
58

Figure 2.3 Enterprise Design as a Game
In the following section, the relevance of a game theoretical representation of the
enterprise for enterprise design is discussed.
2.4 RELEVANCE OF MODELING AN ENTERPRISE DESIGN PROCESS AS
A GAME
Enterprise Design
Enterprise Design Process
Agents of the Enterprise
Design Formulations
Value of Cost Function
Game
Extensive Game
Players
Players’ Strategy
Players’ Payoff
The first and perhaps most important contribution of a game theoretical representation of the
enterprise lies in the framing of issues. The language of extensive form games permits us to ask
questions about the dynamics of interaction between decision makers; and the language of other
parts of non-cooperative game theory are well suited to pose questions about cooperative
interactions in which parties have proprietary information. Indeed, a game theoretical model
focuses attention on issues of dynamics and proprietary information, framing the issues so that
the “physics” of interaction -who does what, when, with what information- come to the fore. If
59

the mechanics of interaction between decision-makers matter, as they do when trying to
integrate enterprise activities, then simply focusing attention and framing cooperation in this way
is a contribution. It has to be noted that a game theoretical representation does not exclude the
use of other enterprise models, such as CIM models. Moreover, CIM models could provide
knowledge on dependencies between the agents of an enterprise, thus facilitating the
construction of a game theoretical model useful for individual decision making. In Chapter 3 is
examined how agents can formulate and solve problems using a game theoretical representation
of an enterprise design process.
60

CHAPTER 3
FORMULATION AND SOLUTION OF ENTERPRISE DESIGN PROBLEMS
In Chapter 2 it is presented a game theoretical representation of the enterprise design
process. This representation provides the underpinning of a framework for enterprise design
that is based on three key elements:
?? An enterprise design method based on a common paradigm for decision support.
?? A multi-objective decision model, the Compromise DSP.
?? A probabilistic design model to attain flexible and robust decisions and handle uncertainty
and imprecision
In this chapter, these elements are described and integrated into a consistent construction for
supporting coordination of decisions through design formulations, which is the essential feature
of a decision-based perspective for enterprise integration.
60

3.1 ENTERPRISE INTEGRATION AS COORDINATION OF
DECISIONS: IMPLICIT AND EXPLICIT COORDINATION
Peplinski (1997) defines organization design as “the design of the organization itself through the
setting of variables under management control.” These variables include the organization
structure, technologies, people, tasks, reward systems, information systems, etc. These issues
are largely influenced by human behavior, which makes organization behavior only partially
amenable to mathematical modeling and analysis. However, there are basic patterns of human
behavior that can be recognized and used for improving the design of an organization. For
example, it is clear that individuals act according to their motivations and rewards. Therefore,
improving the quality of individual decisions does not necessarily improve the performance of
the system unless the coordination of such decisions is improved as well. Such coordination is
typically pursued through team meetings, notices, centralized decision making or information
sharing (Lizotte and Chaib-draa, 1997; Browning, 1997).
In the team-meetings approach, individuals from different disciplines or groups reason on
each other’s decisions as a team. Notices are recipes describing a decision sent by a
responsible party to all the groups of the enterprise that might be affected. In the centralized
approach, all the design decisions are gathered in one place and one decision- maker reviews
them to coordinate all tasks and resources and solve conflicts. These three approaches are
informal ways to pursue enterprise integration, and do not enhance the quality of individual
decision making.
61

In the information-sharing approach enterprise integration is pursued by constructing and
applying integrated architectures that encompass all the aspects of an enterprise such that the
agents that comprise it have more complete information to make decisions. In Section 2.1 it is
argued that the effectiveness of this approach is limited because no individual can handle such
amount of information efficaciously. This approach has also the inconvenient that it does not
provide any insight in how to formulate and make decisions consistently in an integrated
enterprise environment.
In the decision-based perspective embraced in this work, enterprise integration is pursued
by coordinating decisions, i.e., managing dependencies between the various agents through the
appropriate formulation and solution of their decisions. There are two kinds of coordination,
namely, implicit and explicit coordination. In implicit coordination the agents of an enterprise
establish social rules that they agree to respect in order to act in a coordinated manner. A
simple example of an implicit coordination mechanism, in the context of the integration of
product design and manufacturing, are DFM guidelines (e.g., Boothroyd, 1989), which
represent a set of rules that a designer should follow to avoid potential manufacturing problems.
Explicit coordination, on the other hand, asks agents to make explicit choices to ensure
coordination. In order to make such explicit choices, they have to identify potential interactions
between their activities and those of other agents. An example of this kind of coordination are
the intelligent CAD systems (e.g., Lu and Subramanyam, 1988), which utilize the state-of-the
art computer technology to explicitly anticipate manufacturing problems of a design.
62

IMPLICIT COORDINATION
Coordination achieved
by establishing social
rules that agents should respect
Coordination
achieved by
identifying
interactions
with other
agents
EXPLICIT COORDINATION
Figure 3.1 Implicit and Explicit Coordination of Decisions
In Table 3.1 it is shown a comparison of implicit and explicit coordination according to
Lizotte and Chaib-draa (1997). Implicit coordination is generally easier and less costly to
design and implement than explicit coordination because two of the processes required for
achieving coordination are reduced to a minimum. The group decision-making process is
determined in advance using conventions called “social rules.” However, implicit coordination
reduces the autonomy of agents because they should only consider alternatives that respect such
social rules. Explicit coordination, on the other hand, increases an agent’s autonomy since the
agent possesses a margin of authority that is not limited to alternatives respecting social rules.
Nevertheless, increasing decision-makers’ autonomy forces them to deal with incomplete and
uncertain information of other agents of the enterprise. An organization should be able to
63

alance the advantages of both kinds of coordination to achieve an effective integration of its
activities.
Table 3.1 Explicit and Implicit Coordination Regarding Processes Underlying
Coordination (Lizotte and Chaib-draa, 1997)
Processes Implicit Coordination Explicit Coordination
1. Coordination Less costly to design and more
effective, but can reduce agent’s
autonomy and it is domain
dependent
2. Group decision making Static, i.e., the decision is
determined in advance using
conventions
3. Communication Generally done by exchanging signs
and signals
4. Perception of
common
objects
Perceiving same agents and
objects (sharing data files, etc.)
More costly to design and less
effective but can improve agent’s
autonomy
64
Dynamic, i.e., implies evaluating
alternatives and making a common
decision between agents by
authority (decision power)
Generally done by exchanging
alternatives, evaluations and
choices
Knowing part of agents’ intentions,
goals and plans
For example, within explicit coordination, in order to determine the effect of setting the
value of variable x as a or b, a designer would have to specify the states and actions of all other
players as a function of x. An alternative approach would be to determine the possible
strategies of each player independently, and to make the transition function from one stage to
the next probabilistic. In this last approach, for example, if player 1 decides to set x = a, the
result would be a design with an expected payoff that depends on other players’ behavior. In
the first approach, complete information about other players must be supplied and the transition

function produces a specific prediction that supports a decision. In the second approach, the
transition function produces a highly uncertain prediction.
Instead of adopting either extreme view, a more practical approach is to use the social
rules of an implicit-coordination mechanism that constrain actions of agents to “frame” explicit
coordination. These rules should specify which of the actions that are in general available are in
fact allowed in a given stage. They do so by a predicate over that stage. The transition function
can now take as an argument not only the previous history of the process and the possible
actions taken by the agents, but also the system constraints in force, such that it produces a
better estimate of possible points of equilibrium between agents. Generally speaking, a
prediction based on constraints will be more general than a prediction that is based on the
precise states and actions of all other players, and more specific than a prediction based on pure
probabilistic estimates. This idea is illustrated in Figure 3.2.
Enterprise Design Space
Constraints
set by social
rules
Points of
Equilibrium
Figure 3.2 Integrating Implicit and Explicit Coordination
65

The goal of a coordination-mechanism design is to strike a good balance between
implicit and explicit coordination, allowing freedom to the individual decision-makers on the one
hand and ensuring a harmonious cooperative behavior among them on the other. Furthermore,
the integrative mechanism should be systematic to take advantage of a computer-supported
environment. A systematic coordination mechanism, that capitalizes the advantages of both
kinds of coordination, and is mathematically supported, can be established by promoting a
framework for enterprise design based on the following key elements:
?? A representation of the enterprise activity as an extensive game.
?? An enterprise design method based on a common paradigm for decision support.
?? A multi-objective decision model, the Compromise DSP.
?? A probabilistic design model to attain flexible and robust decisions and handle uncertainty
and imprecision.
The merging of these elements provides a consistent mathematical framework for coordination
of decisions based on design formulations, which is the objective of this work. This framework
is illustrated in Figure 3.3, where it is shown how this framework can support the enterprise
design method of Peplinski and Mistree (1997), described in Section 1.4.
Is in Task 1, when a decision-maker expands the scope of a baseline decision to address
needs, goals and constraints of other agents of the enterprise, that a representation of the design
process as a game is brought up. Depending on the number of objectives that are considered,
the game can be either a scalar game or a vector game.
66

Enterprise Design Method
Baseline Decision
Task 1
Expand Scope
of Decision
Task 2
Determine Input
Needed for
Analysis
Identify Modeling
Techniques
Task 3
Define Boundaries
of Decision
Task 4
Transform and
Integrate Models
Task 5
Generate Potential
Solutions
Recommendation
Enterprise Design as an Extensive Game
Identify Players
of the Game
Define Order of Moves
Define Players’ Choices
Identify Public and
Private Information
Probabilities on
Exogenous Events
Define Players’ Payoffs
Define Strategies
Given
Find
Satisfy
Minimize
Compromise DSP
Figure 3.3 A Framework for Enterprise Design
Scalar
Games
Vector
Games
Probabilistic
Design Model
During Tasks 2 and 3, where it is determined the input needed for analysis and modeling
techniques, and when the boundaries of decision are set, is when the basic information required
to define a strategy in this game is identified, i.e., the order of moves, players’ choices,
information available, probabilities on exogenous events, and the players’ payoffs (see Section
2.3.1). With this information it is possible to embody a strategy, using a compromise DSP, as
67

explained in Section 3.5. Finally, and depending on the characteristics of the problem, the
compromise DSP can be formulated as a probabilistic decision model, in order to obtain
interval solutions, and to capture uncertainty and imprecision. This framework is described in
detail in the rest of this chapter.
3.2 GAME THEORY AS A FOUNDATION FOR COORDINATION OF DESIGN
DECISIONS
In Section 3.1 it is argued that an organization must adopt a coordination mechanism that
balances the advantages of implicit and explicit coordination. Given the abstraction of an
enterprise activity as a game, as discussed in Chapter 2, it should not be difficult to consider the
social rules of an implicit coordination mechanism analog to the rules of a game. During an
enterprise design process, abstracted as an extensive-game, the actions of a player are taken
from a set of alternatives that might depend on the previous history of the game. The problem in
defining the strategies of players is that the state in which each player ends up after taking a
particular action at a particular state depends also on actions and states of other players. Thus,
in principle, the transition of the entire system can be thought of a mapping from the states and
actions of all players to the new states of all players. In other words, the agents or players in
this game have to think ahead and devise a strategy considering other agents’ goals, intentions
and motivations.
A player in a game that acts rationally seeks the best solution for his/her own performance
while respecting the rules of the game. In doing so he/she has to assume that all other players
68

will do so as well. These rules on the one hand constrain her autonomy and possible choices,
but on the other hand guarantee certain behaviors on the part of other agents, so that she can
formulate a specific design strategy. If all players involved in the game act in such a rational
way, the expected outcome is a point of equilibrium (if it exists), either a Pareto solution or a
Nash solution, depending on the cooperative protocol. Thus, the design of the organization, i.e.,
the design of its structure, communication and information systems, rewards systems, etc., has
to be carried out in such a way that motivates and facilitates agents to achieve this equilibrium.
Adopting game theory as a foundation for coordination of agents has important
implications. A Pareto or full cooperative solution requires centralization of decisions, which in
turn requires a model of the enterprise to quantify all the important effects of a decision. A non-
cooperative protocol, on the other hand, empowers employees by decentralizing decisions and
does not require the large information demands for decision making that are necessary in
attaining a Pareto solution. This discussion has settle the foundation for Hypothesis 2.1:
Hypothesis 2.1: Enterprise decisions can be effectively coordinated using a non-
cooperative formulation of the individual design problems.
However, it must be kept in mind that, given the same conditions, a cooperative solution is
usually better than a non-cooperative one.
According to the preceding discussion, the rules of an enterprise design game should
define at least, but are not restricted to:
69

1. Who are the players and which is their respective authority (organization structure).
2. How players interact (information, communication and negotiation systems).
3. Individual and systems goals which in turn define the payoffs of players (a reward system).
If these rules are clear it is easier for individuals to formulate strategies of action and
coordinate their decisions. Note that these “rules” already exist in almost every manufacturing
organization. The difference is that now these rules should be settle in such a way that they
promote the achievement of a particular kind of equilibrium.
In either a Pareto or a Nash solution, each player perceives the solution as optimal, or at
least “satisficing,” in the sense that they cannot do better by deviating from such solution. This is
a natural perspective to coordinate agents because it fosters the design of an organization with
controls that are consistent with, not contrary to, the behavior of individuals, who generally act
according to their own interests. This perspective has also the advantage that brings the insight,
constructs and tools of game theory to improve the individual decision-making.
How is one to derive appropriate rules for this game, i.e., to design the coordination
mechanism? One approach could be to handcraft laws for each organization given certain
existing conditions. An alternative approach could be to derive a general theory of social laws
(see Shoham and Tennenholtz, 1992) and derive from it appropriate laws for individual
organizations. These issues are beyond the scope of this thesis, but certainly game theory can
70

provide insightful contributions to organization design in general, and the design of coordination
mechanisms for enterprise integration in specific.
In Sections 3.3 and 3.4 it is shown how a game-theoretical coordination mechanism can
be implemented in practice by using a common model for decision support across the enterprise
and a multi-objective decision model, the Compromise DSP.
3.3 IMPLICIT COORDINATION THROUGH A COMMON PARADIGM FOR
DECISION SUPPORT
In the previous section it is mentioned that the basic elements that rule the behavior of the agents
of an organization are:
?? The structure of the organization, i.e., to settle which individuals and responses are
associated to each level of decision.
?? Information, communication and negotiation systems, i.e., the way in which information is to
be filtered and passed to other agents at the same or different level of authority.
?? Individual and systems goals.
In order to pursue a decision-based enterprise design, it must be added to the former
requirements another element, namely, “the way in which decisions are formulated.” This
element is important because it fosters enterprise integration through the creation and
implementation of a common paradigm for decision support and a unified decision model,
thus promoting coordination between agents by providing a common language for
71

communication and negotiation. These common paradigm for decision support and decision
model should be mathematically supported, because mathematics provides some of the basic
elements of a scientific design practice, discussed in Section 1.2.2, such as:
?? A valid framework for rational choice among alternatives (therefore, to solve conflicts) and
tools for actually making the computations.
?? A body of techniques for actually deducing which of the available alternatives is the
optimum.
?? Algorithms to search for alternatives and perform tradeoffs.
As mentioned in Section 1.3.1, Peplinski (1997) proposes a model for decision support in
enterprise design that fulfills this aim. In such a model, a decision is represented in terms of:
?? A set of design variables {X}
?? A set of goals and constraints and their targets Ti, and
?? Models Fi(X) that quantify the relationships between the variables and objectives.
This model sets the stage for concrete and specific definitions of interdependence and
integration in terms of decisions. Based on this model, Peplinski (1997) propose an enterprise
design method, through which diverse issues across product design, manufacturing process
design and organization design can be integrated (see Section 1.4). This enterprise design
method establishes a specific set of steps in making decisions, thus predicates over each agent
by constraining the decision making approach. In other words, it asks individuals to follow a
72

formal and rigorous approach in making decisions, thus guaranteeing at some extent the rational
outcome that is assumed in game theory.
In the following section it is examined the implementation of an explicit coordination
mechanism based on the use of a common decision model across the enterprise, the
Compromise DSP, which is the next element of the framework for enterprise design that is
being constructed.
3.4 EXPLICIT COORDINATION THROUGH A UNIFIED DECISION MODEL:
THE COMPROMISE DSP
As mentioned in Section 3.1, in explicit coordination the agents of an organization have to be
aware of the system-wide effects of their decisions and make choices based on such awareness.
Various approaches for explicit coordination have been considered in recent years in the
context of a concurrent product and manufacturing process development, such as design with
features (Dixon, et al., 1988), enhanced CAD systems (Lu and Subramanyam, 1988), and
multi-objective design evaluations (Hwang, et al., 1980). Agrell (1994) examines the
advantages of each one and concludes that a multi-objective design evaluation approach is the
best way to deal with an explicitly integrated design problem. In this thesis the Compromise
DSP is used as a rigorous and practical tool to solve multi-objective design problems.
The Compromise DSP is a multi-objective decision model which is a hybrid formulation
based on Mathematical Programming and Goal Programming Technique (Mistree and Muster,
73

1988). The Compromise DSP is used to determine the values of design variables to satisfy a
set of constraints and to achieve as closely as possible a set of conflicting goals. The
Compromise DSP is used to model such decisions since it is capable of handling constraints,
goals, and multiple objectives. In particular, the Compromise DSP offers the following
capabilities:
• Accurately represent single-objective or multi-objectives.
• Use either preemptive or Archimedean formulation to prioritize objectives.
• Have hard constraints or soft constraints (goals).
• Generate feasible solutions more frequently.
• Quickly generate results for several different weighting schemes.
The Compromise DSP has been successfully used in designing aircrafts (Lewis and Mistree,
1997), thermal energy systems (Bascaran, et al., 1987) damage tolerant structural systems
(Shupe et al., 1984), and material composite design (Karandikar and Mistree, 1990), to name
some applications. A compromise DSP is stated in terms of the following system descriptors:
?? Variables: system variables and deviation variables
?? Constraints: system constraints and goal constraints
?? Bounds: on system and deviation variables
?? Objective: in terms of deviation variables only
The resulting compromise DSP is stated in words as follows:
74

Given
An alternative that is to be improved through modification.
Assumptions used to model the domain of interest.
The system parameters.
The goals for the design.
Find
The values of the independent system variables that describe the attributes of an artifact.
The values of the deviation variables that indicate the extent to which the goals are achieved.
Satisfy
The system constraints that must be satisfied for the solution to be feasible.
The system goals that must achieve a specified target value as much as possible.
The upper and lower bounds on the system variables.
Minimize
The objective function Z, which is a measure of the deviation of the system performance
from that implied by the set of goals and their associated priority levels or relative weights.
The difference between the compromise DSP and a traditional single objective
formulation for a two dimensional problem is illustrated in Figure 3.4. In the case of the single
objective formulation, shown in Figure 3.4a, the objective is a function of the system variables.
The space representing all feasible solutions (the feasible design space) is surrounded by the
system constraints and bounds of the problem. The objective is to maximize the value of Z, and
as shown in the figure, the solution is at vertex A.
In the compromise DSP, the set of system constraints and bounds again define the
feasible design space, whereas the set of system goals defines the aspiration space (see Figure
3.4b). For feasibility, the system constraints and bounds must again be satisfied whereas the
system goals are to be achieved to the extent possible. The solution to this problem represents
a tradeoff between that which is desired (as modeled by the aspiration space) and that can be
achieved (as modeled by the design space).
75

X 2
Objective
Function
Z = W1 A1(X) + W2 A2(X) + W3 A3(X)
Feasible
Design
Space
A
Bounds
System constraints
(a)
Direction of increasing Z
X 1
X 2
Feasible
Design
Space
A1(X) + d1 - - d1 + = G1
A
Bounds
System constraints
System goals
(b)
Aspiration
Space
X 1
76
A2(X) + d2 - - d2 + = G2
A3(X) + d3- - d3 + = G3
Deviation
Function
Z = W3 (d1 -+ d1 +) + W3 (d2 -+ d2 +) + W3 (d3 -+ d3 +)
Figure 3.4 A Single Objective Optimization Problem and the Multi-goal
Compromise DSP (Mistree, et al., 1993)
The solution for the compromise DSP shown in Figure 3.4b is at vertex A. This is the
same solution as that obtained for the problem illustrated in Figure 3.6a except that in the
compromise DSP case (with the aspiration space modeled), the best possible solution can be
identified. This solution is referred to as a satisficing solution since it is a feasible point that
achieves the system goals to the extent that is possible. Given a solution, it is left to a
designer to accept this solution or to explore the problem further by modifying the aspirations
and/or the feasible design space and re-solving. The values of the deviation variables provide a
measure for assessing the degree to which each of the goals has not been achieved.
The system descriptors, namely, system and deviation variables, system constraints,
system goals, bounds and the deviation function are described in (Mistree, et al. 1993) and are
therefore not repeated here. The mathematical form of the compromise DSP is shown in Figure
3.5.

Given
An alternative to be improved through modification.
Assumptions used to model the domain of interest.
The system parameters:
n number of system variables
p+q number of system constraints
p equality constraints
q inequality constraints
m number of system goals
gi(X) system constraint function:
gi(X) = Ci(X) – Di(X)
fk(di ) function of deviation variables to be minimized at priority level k
for the preemptive case.
Find
Xi i = 1, …, n
? ?
i , di
d i = 1, …, m
Satisfy
System constraints (linear, nonlinear)
gi(X) = 0 ; i = 1, …, p
gi(X) = 0 ; i = p+1, …, p+q
System goals (linear, nonlinear)
Bounds
Ai(X) + d - i – d+ i = Gi ; i = 1, …, m
X i min = Xi = X i max ; i = 1, …, n
d - i , d+ i = 0 ; i = 1, …, m
d - i . d + i = 0 ; i = 1, …, m
Minimize
Archimedean formulation
? ?
Z ? Wi(di ? di )
?
Figure 3.5 Mathematical Form of a Compromise DSP
(Mistree, et al., 1993)
77

In the compromise DSP, each goal, Ai, has two associated deviation variables
?
d i , which indicate the extent of the deviation from the target. The deviation variables,
?
i
78
?
d i and
?
di and
d , are both positive, and the product constraint, ? ? 0
? ?
d d , ensures that at least one of the
deviation variables for a particular goal is always zero.
Usually, goals are not equally important. To determine a solution on the basis of
preference, the goals may be rank-ordered into priority levels. Customers rate certain product
qualities higher than other qualities. Designers usually seek a solution that minimizes all
unwanted deviations from the desired qualities. There are various methods for measuring the
effectiveness of the minimization of these unwanted deviations. In this work it is used the
Archimedean or weighted sum method, in which the deviation function Z is formulated as:
Z ? W i(d i
with Wi ? 0
?
?
W i ? 1
i
? ? di
The weighted sum method generates a Pareto solution, but is deficient in that it is difficult
to determine which weights will produce the best compromise solution. A systematic
redistribution of the weights on the objectives does not necessarily produce a continuous
Pareto-optimal solution set. To overcome this problem Rao and Freiheit (1991) proposed a
modified approach as follows (in terms of a compromise DSP deviation function Z):
1. From a constant initial design vector, minimize each of the objectives separately, recording
the values of the other objectives at each optimal design vector.
? )
i

Minimize Zi ? di ? ? di ? , i ? 1, 2, ...,m (3.1)
subject to system constraints. For each objective function Zi an “optimal” solution Xi *
is obtained.
2. With the values obtained in the previous step, construct a pay-off matrix as:
? Z1(
X
?
? ? Z1(
X
P o ?
?
? Z1
( X
*
1
*
2
*
m
)
)
)
Z
Z
Z
2
2
2
( X
( X
( X
Normalize the objectives so that no objective due to its magnitude will be favored as
follows:
*
1
*
2
*
m
Z ˆ
i ? Zi ? Zi *
w
Zi ? Z i
)
)
)
Z
Z
Z
m
m
m
( X
( X
( X
*
1
*
2
*
m
) ?
?
) ?
?
?
) ?
79
* (3.2)
where Zi w is the worst value and Zi * is the optimum value of the ith objective.
3. Formulate a super-criterion SC. A formulation for a super-criterion which has been used
effectively (Rao and Feiheit, 1991) is:
m
SC ? [1 ? ˆ ? Z i ]
(3.3)
Note that due to the normalization, SC will always have a value between zero and one,
which is the same magnitude of the normalized objectives.
4. Formulate a Pareto optimal objective Z using the normalized objectives:
i? 1
Z ? W ˆ ? iZ i i ? 1, ...,m (3.4)

5. Since Z has to be minimized and S has to be maximized, a new objective Z mp that has to be
minimized is constructed as:
subject to the system constraints and
80
Z mp ? Z ? SC (3.5)
Z ˆ
i ? ? Zi mp ? ˆ Z i w (3.6)
with the design vector X now including the priorities on objectives. This technique greatly
simplifies the search for a “satisficing” solution since it is not necessary to construct the entire
set of Pareto points. However, it must be remembered that it does so by changing the
characteristic of the original problem.
Karandikar and Mistree (1993) and Peplinski, et al. (1996) have already shown that the
Compromise DSP is an adequate tool to support the integration of enterprise design processes.
This method has proven to be efficient because of the expected reduction in the number of
iterations and it is effective because it can deal with the complexity of the problem in terms of
hierarchy and the existence of multiple objectives in design.
The discussion carried out in this section provides the foundation for Hypothesis 2.2:
Hypothesis 2.2: the Compromise DSP is an adequate decision model to embody game
theoretical design formulations within an enterprise design framework.
In the next section it is discussed how the common paradigm for decision support
discussed in Section 3.4 facilitates the creation of compromise DSPs, thus establishing a
consistent mechanism for coordination of decisions, which fosters its implementation in a

computer-supported environment, and which capitalize on both implicit and explicit
coordination.
3.5 FORMULATION OF DESIGN PROBLEMS IN ENTERPRISE
DESIGN
3.5.1 Approximation in Engineering Design
Explicit coordination requires individuals to make explicit choices based on awareness of the
system-wide effects of their decisions. To do so they have to be able to predict such effects. In
“The Sciences of the Artificial” Herbert Simon comments (Simon, 1996):
In real life business firm must choose product quality and the assortment of products it
will manufacture. It often has to invent and design some of these products. It must
schedule the factory to produce a profitable combination of them and devise marketing
procedures and structures to sell them. So we proceed step by step from the simple
caricature of the firm depicted in the textbooks, to the complexities of real firms in the
real world of business. At each step towards realism, the problem gradually changes
from choosing the right course of action (substantive rationally) to finding a way of
calculating, very approximately, where a good course of action lies (procedural
rationality).
In a perfect world, approximation and procedural rationality would not be needed, but
limited human beings as we are, we do not have the ability nor sufficient information to create
“exact” models that capture the real behavior of natural or artificial systems, even the most
simple ones. Therefore, we build models that approximate such behavior with different degrees
of accuracy. Usually, the boundaries, elements and internal relationships of these models are
selected to explore different parameters drawn form a single simple model and each based on
81

different simplifying and independence assumptions depending on our needs and interests. In
essence, a formulated design problem and its supporting models must be stated in sufficiently
simple terms that finding a solution is a manageable process and, at the same time, they must be
a close-enough approximation of the real-world that the solution is useful. In doing so, we have
two choices when faced with complex real-world problems in design (Muster and Mistree,
1988):
1. To develop an algorithm, based on a relatively simple model, by means of which an exact
optimal solution can be found, and provided the assumptions on which the model is based
can be satisfied exactly.
2. To develop an approximate algorithm or heuristic based on a relatively complex model that
represents the real world more closely than the simple model referred to earlier.
When dealing with multidisciplinary problems, like in enterprise design problems, the motivations
for approximation become even more critical because direct coupling of a design space search
(DSS) code to several multidisciplinary analysis models may be impractical due to
computational constraints. Consequently, most optimizations of complex engineering systems
couple a DSS code to easy-to-calculate approximations of the objective and constraints. An
“optimum” of the approximate problem is found; then, the approximation models are upgraded
and the search is performed again on a smaller design space that surrounds the “optimum” point
found on the previous step.
82

Approximation of complex models without simplifying assumptions may take many forms.
Global reduced-basis approximations (e.g., Kirsch, 1991) are occasionally used in engineering
system optimization. However, most often the approximations are either local, linear, or
quadratic, based on the derivatives. Occasionally, intermediate variables or intermediate
response quantities (e.g., Kodiyalam and Vanderplaats, 1989) are used to improve the
accuracy of the approximation. Another method is a variable-complexity modeling (VCM)
technique (e.g., Unger, et al., 1992) in which both simpler and more sophisticated models
alternate during the optimization procedure. The sophisticated model provides a scale factor,
which is periodically updated to correct the simpler model. Traditional derivative-based
approximations can be combined with global VCM approximations by using a derivative-based
linear approximation for the scale factor (Chang, et al., 1993). Toropov and Markine (1996)
generalize this approach, constructing an approximation based on both the design variables and
the prediction of the simpler model.
A different kind of approximation is statistical-based approximation. Response surface
techniques replace the objective and constraints functions with simple functions, often
polynomials, which are fitted to data computed at a set of carefully selected design points
(Chen, et al., 1996). These approximations may be regarded either as global or local
approximations depending on the design space that they span. Other approaches of this nature
are Neural Networks-based regression, kriging and inductive learning. Unfortunately, statistical-
based approximations do not scale well to a large number of variables (Sobieszcanski-Sobieski
83

and Haftka, 1997). Nevertheless, these techniques provide a convenient representation of data
from one discipline to other disciplines and to the system. Because design points are pre-
selected, rather than chosen by an optimization algorithm, planning and coordinating the solution
process “off line” is possible.
In summary, to permit practical solution of enterprise design problems, we require to
approximate the behavior of systems with relatively simple models, but as Simon (1996) notes:
“it is done at the cost of shaping and squeezing the real-world problem.”
3.5.2 The Cooperative Formulation using Compromise DSPs
Up to this point, it has been examined how game theory can provide a foundation for
coordination of decisions. In Section 3.5.1 it is asserted that the Compromise DSP is an
adequate decision model to embody game theoretical formulations of decisions. In this section
it is explained how to formulate a design problem to achieve a Pareto or cooperative solution
(defined mathematically in Section 2.2.3), using the Compromise DSP. The way in which this
solution can be achieved is as follows:
1. Combine the models of each player into one encompassing model.
2. Solve the model using an appropriate technique.
Here the strategy of a player is embodied by a design formulation, in this case using a
compromise DSP. A full cooperative model (using compromise DSPs) is illustrated in Figure
3.6. The compromise DSPs of Player 1 and Player 2 are combined into one compromise DSP
84

and solved using the cumulative design variables, constraints, goals, and deviation variables of
both players.
Player 1
Player 2
Find
x1, d +
1 ,d1 -
Satisfy
constraints
g 1(x1,s1,x2,s2 h 1 (x1 ,s1 ,x2 ,s 2 )=0.0
goals
f 1 (x1 ,s1 ,x2 ,s2 )+ d -
1 -d +
1 =1.0
Minimize
Z1 (x1 ,s1 ,x2 ,s2 )
Find
x2 , d +
2 ,d -
2
Satisfy
constraints
g2(x1,s1,x2,s2 h 2(x1,s 1,x2,s2)=0.0 goals
f -
2(x1,s1,x2,s 2)+d 2 -d +
2 =1.0
Minimize
Z2(x 1,s1,x2,s2) Find
x1 ,x2 , d +
1 ,d -
1 , d +
2 ,d2 -
Satisfy
constraints
g 1(x1,s1,x2,s 2) 0.0
h 1 (x1 ,s1 ,x2 ,s2 )=0.0
g 2(x1,s1,x2,s 2) 0.0
h 2(x1,s1,x2,s2)=0.0 goals
f -
1(x1,s1,x2,s2)+d 1 -d +
1 =1.0
f 2 (x1 ,s1 ,x2 ,s2 )+d -
2 -d +
2 =1.0
Minimize
Z(x1 ,s1 ,x2 ,s2 )
Figure 3.6 Full Cooperation: Pareto Solutions (Lewis, 1997)
Although conceptually obtaining a cooperative solution is simple and theoretically sound, it
is difficult to implement since the models that the different agents utilize are usually complex
analysis packages, and many times are solved at different points in a design process using
approximate and/or incomplete information. Combining these models is typically impractical due
to the sheer size of the models and analysis routines. Therefore, the definition of a cooperative
solution in complex systems design has to be extended to “approximate cooperation” (Lewis,
1997) where models remain separate but linked through approximations of the coupling
variables which are needed by more than one agent. Approximate cooperation is achieved
85

using approximations of the state variables, including constraints and goals needed from the
other players, obtained using any suitable technique. However, approximation of every
constraint, goal, and state variable is unrealistic. Hence, only those system variables that are
largely affected by the local design variables are approximated. Because of the non-linearity of
the resulting enterprise design problems it is difficult to know a priori which variables of other
players are relevant to the local design decision. CIM models, screening experiments and
sensitivity analysis can help a designer’s judgment to this aim. The approximation technique to
be used depends on the problem; for example, when dealing only with continuum variables a
derivative-based method, such as the Global Sensitivity Equations approach (see Lewis, 1997)
is appropriate. Using the Compromise DSP and approximations of non-local information, the
cooperative design problems are now formulated as “approximate-cooperative DSPs”
(Lewis, and Mistree, 1997). This concept is illustrated in Figure 3.7.
Player 1
Given
X2 Approximations of Player 2’s
information
Player 1’s Models
Find
X1 + -
d1i , d1i Satisfy
constraints
goals
Minimize
Z1 = [ f1 , f2 ]
Player 2
Given
X1 Approximations of Player 1’s
information
Player 2’s Models
Find
X2 + -
d2i , d2i Satisfy
constraints
goals
Minimize
Z2 = [ f1 , f2 ]
P1 P2
Figure 3.7 Approximate-Cooperation using Compromise DSPs
86

Note that due to the use of approximations the problem is not anymore a single problem,
and iterations between the players’ solutions are required. If the heuristic proposed by Rao and
Freihet (1991) previously described in Section 3.4 (or a similar one) is not used, in any single
iteration each player has to generate an individual set of Pareto solutions, each one requiring
solving a compromise DSP. From the individual’s set each player choose a solution and then
convergence is checked. Clearly, the modified method is advantageous in order to reduce the
number of compromise DSPs to be solved in an approximate-cooperative model. The
algorithm for the solution of the approximate-cooperative design model is illustrated in Figure
3.8 considering two players. The process starts with some initial values of their design
variables, say x10 and x20, and each player solves an approximate-cooperative DSP. Once the
solution of each player is obtained, the values of the resulting design variables, x1 and x2, are
compared, and if they differ they update the values of these variables and solve again their
respective compromise DSPs until convergence is achieved.
x 10
x 20
x 1
x 2
Approximate-Cooperative
DSP for Player 1
Approximate-Cooperative
DSP for Player 2
No
Convergence?
x 1
x 2
87

Figure 3.8 Solution of a Two-Players Approximate-Cooperative Model
3.5.3 The Non-Cooperative Formulation using Compromise DSPs
As discussed in Section 3.2, it might be more practical in enterprise design to pursue
coordination of design decisions using a non-cooperative protocol than using a cooperative one.
In this section it is shown how to achieve non-cooperative solutions using the Compromise
DSP.
Finding the rational reaction sets (RRS) of the players is the first step in formulating a
strategy for a non-cooperative game. Similar to the cooperative case where complete
cooperation is not realistic in the design of complex systems, finding exact mathematical RRS of
other players is not always possible. Each player might use models that consist of multiple,
nonlinear constraints and objectives that require nonlinear programming and/or heuristic
techniques in order to solve for the independent variables. To develop a closed form equation
for one or more variables as functions of other variables is difficult. Therefore, the RRS of
players might have to be found by using approximation techniques. In this work the RRS are
approximated using response surfaces.
The steps followed in this work to construct the RRS of each player is as follows (adapted
from Lewis, 1997):
1. Use statistical software as the design of experiments (DOE) driver.
88

1.a) Based on the characteristics of the input and response variables set up the appropriate
experimental design.
1.b) Set the variables of the players which are required constant in a compromise DSP
2. Solve the individual compromise DSP using the modified Pareto solution described in
Section 3.4.2.
3. Send the values of the design variables obtained in the previous step to the DOE driver.
4. Determine if the full experiment is finished.
?? If not, continue by moving to the next experiment point and repeat Step 2.
?? Otherwise, construct the required response surfaces.
This algorithm is illustrated in Figure 3.9.
sP1
1
P1’s Design Space
xP1
xP1, sP1
xP1, sP1
xP1, sP1
xP1, sP1
P2’s Compromise
DSP
Given
Find
Satisfy
Minimize
Given
Find
Satisfy
Minimize
Given
Find
Satisfy
Minimize
Given
Find
Satisfy
Minimize
xP2, sP2
xP2, sP2
xP2, sP2
xP2, sP2
Rational Reaction
Set xP2, sP2 = f(xP1, sP1)
y
Response Surface
Equations
Figure 3.9 Conceptual Outline of RRS Construction (Lewis, 1997)
2
3
89

In Figure 3.10 is shown the compromise DSPs of two players with a non-cooperative
formulation. The given information is information required from the other player that is
unknown. Therefore, a player’s solution must be found using unknown variables.
Player I Player II
Given
unknown s II, x II
Find
x I, d i + ,di -
Satisfy
constraints
g I(x I,s I,x II,s II
h I (x I ,s I ,x II ,s II )=0.0
goals
f i(x I,s I,x II,s II)+d i - -d i + =1.0
Minimize
Z I (x I ,s I ,x II ,s II )
Given
unknown s I , x I
Find
x II , d i + ,di -
Satisfy
constraints
g II (x I ,s I ,x II ,s II
h II (x I ,s I ,x II ,s II )=0.0
goals
f i (x I ,s I ,x II ,s II )+d i - -d i + =1.0
Minimize
Z II(x I,s I,x II,s II)
Figure 3.10 Non-cooperative Compromise DSPs (Lewis, 1997)
The steps to solve a non-cooperative formulation are as follows (Lewis, 1997):
1. Construct Rational Reaction Set of each player I, RRSi.
2. Using an appropriate technique, find the intersection points of the RRS of the players.
90
X RRS ? RRS (i ? j)
(3.7)
n ? i
j
3. Determine which solution fall in the ranges of the design variables.
These three steps are shown for a two-player case in Figure 3.11, where player 1 controls the
value X1 and player 2 controls the value of X2.

Construct RRS 1
Construct RRS 2
Figure 3.11 Solution of a Two-Player Non-cooperative Model
A particular case of non-cooperative games that is used in this work is the Stackelberg or
Leader/Follower game. This model arises in some dynamic situations (extensive games) where
one player dominates others. This is the case, for example, of a sequential design process
where a product designer hands the design over the production department. In the Stackelberg
formulation the leader may be able to use knowledge of the followers’ response to his
advantage in minimizing his own deviation function. The followers may also benefit from having
a leader in that they do not have to guess what the leader will do. This behavior of the follower
is dictated by his strategy and embodied by his/her RRS, which describes how the follower will
behave in response to any decision made by the leader (this is the principle of backward
induction).
1997):
1 2
The process for solving the Stackelberg formulation is as follows (adapted from Lewis,
1. Construct the RRS of the followers
Find X 1 and X 2 such that
RRS 1(
X 2)
? RRS 2(
X 1)
2. Allowing the leader access to the followers’ RRS solve the leader’s compromise DSP.
3. Solve the followers’ compromise DSPs
3a) If one of the followers is the leader of the remaining players, solve a
3
Determine which solutions
fall in the feasible design space
of players
91

leader/follower formulation for remaining players using steps 1 and 2.
3b) Otherwise, solve the appropriate design formulation (cooperative, non-
cooperative, or single player).
These steps to solve a Stackelberg formulation are shown schematically in Figure 3.12, and the
compromise DSPs of the leader and follower are shown in Figure 3.13, considering two
players. The leader’s given information includes the RRS of the follower, while the follower’s
given information includes the leader’s solution.
1st. Leader
Given
Compromise DSP
Construct RRS
of the followers Compromise DSP
Yes
Leader among
previous leader’s
followers?
Figure 3.12 Leader/Follower Solution Process
F , sF = f(xL ,sL )
Find
xL , d +
i ,di -
Satisfy
constraints
g(xL ,sL ,xF ,sF h(xL ,sL ,xF ,sF )=0.0
goals
fi (xL ,sL ,xF ,sF )+d -
i -di + =1.0
Minimize
ZL (xL ,sL ,xF ,sF )
No
Given
x L, s L
Find
x F, d j + ,d j -
Satisfy
constraints
g(x L,s L,x F,s F
h(x L,s L,x F,s F)=0.0
goals
f j (x L ,s L ,x F ,s F )+d j - -d j + =1.0
Minimize
Z F (x L ,s L ,x F ,s F )
92
Solve Follower’s
Compromise DSP
Figure 3.13 Compromise DSPs for a Two-Players Leader/Follower Game

(Lewis, 1997)
It is important to note that these cooperative protocols can be combined for coordination
of decisions in an enterprise. For example, some small groups can use a cooperative protocol
between them, and maintain a non-cooperative one with other groups. This idea, which is
illustrated in Figure 3.14, is in line with a hierarchical organization of manufacturing enterprises.
GROUP B
Full Cooperative
GROUP A
Leader/Follower
LEADER/FOLLOWER
NON-COOPERATIV E
GROUP C
Approximate-Cooperative
Figure 3.14 Combining Cooperative Protocols in Enterprise Design
93

In the next section is described how to enhance this framework with a probabilistic
design approach such that the players can obtain flexible solutions, ideally facilitating
coordination of decisions.
3.6 A PROBABILISTIC-BASED DESIGN APPROACH FOR ENHANCING
COORDINATION WITH FLEXIBLE SOLUTIONS
One of the most critical issues in design is making early decisions on a sound basis. When
addressing the interests of multiple groups and criteria in making a decision, a designer faces the
additional problem of finding a solution that is satisfying to all those groups. However, the early
stages of design are the most uncertain and obtaining precise information from other groups
upon which support decisions can be made is usually impossible. Thus, in the early stages it is
known that a solution that seems “optimal” at the present will probably have to be modified to
meet the requirements of those other groups that will suggest changes based on their private
information and concerns. Therefore, obtaining the “best” solution in an early stage has little
meaning. Instead, it is better to attain solutions that are flexible, yet robust to future changes. In
this section it is described a design approach that is intended to provide such a kind of solutions.
3.6.1 Why a Probabilistic-Based Design Approach?
94

Antonsson and Otto (1995) distinguish two kinds of uncertainty in design: uncertainty
associated with uncontrolled variations (e.g., manufacturing variations, material property
variations, demand fluctuations) and uncertainty in choosing among alternatives, that they called
“imprecision.” An imprecise variable is a variable that may potentially assume any value within a
range because the designer does not have all the information required for deciding the final
value. To represent an imprecise variable, x, a range of numbers can be used or, more
completely, a range and a function defined on the range to describe the preference for particular
values. In this way variables whose values are not known precisely can be formally
represented.
The need for a methodology to represent and manipulate uncertainty and imprecision have
led to a series of approaches such as Bayesian methods (Vadde, et al., 1991), fuzzy design
methods (Zhou, et al., 1992), and utility theory (Thurston and Liu, 1991), to name some.
Independently of the method chosen, the use of intervals encourages the passing of imprecise
design information between agents (Antonsson and Otto, 1995), and permits the early release
of design information from one group to others in advance of more precise information, thus
ideally facilitating the integration of the enterprise design activities. It also facilitates
coordination and group decision making because it is easier to find common satisfying solutions
when the “satisficing” values of the design variables are not fixed points but regions. This notion
is illustrated in Figure 3.15.
95

Point Solutions
X 1 ? A
X 2 ? B
* *
Ranged Solutions
X 1 ? A ? ? A
* *
X 2 ? B ? ? B
Figure 3.15 Coordination of Decisions with Ranged Solutions
In this work, the probabilistic-based design model of Chen and Yuan (1997) is used, because it
provides a consistent scheme. The following sections are mostly summarized from Chen and
Yuan (1997) and Parkinson, et al. (1993).
3.6.2 Preference Functions for Design Requirements
In conventional design optimization, design requirements have to be precisely satisfied under all
circumstances. For example, a design requirement on weight, W, can be stated as “it is desired
the weight to be less than 800 lbs.” This can be modeled by a constraint, W ? 800 lbs, plus
maybe a design objective on minimizing W. The modeling is slightly different if using goal
programming, where designers’ preferences are specified by the performance target value and
expressed using one of the following three notions: (1) smaller-the-beter (STB), (2) larger-the-
better (LTB), or (3) center-the-better (CTB). The previous requirement on W could be
modeled by a goal (STB) with the target value set at 800 lbs. The goal could be considered as
96

either rigid (constraint) or soft (objective). In all circumstances, 800 lbs is picked as a precise
number to distinguish designs that are desirable or undesirable. However, the target level of
performance that will be judged as desirable may be uncertain, especially in the early stages of
product development and when addressing the interests of various enterprise groups. Hence, it
is hard for a designer to make such a distinct evaluation.
In the previous example, designers may not want to absolutely reject the design when the
gross weight is slightly above 800 lbs. Instead, they would like to consider designs in the range
of 750 to 850 lbs, with a varying degree of desirability. To capture a varying degree of
preference on the levels of performance, a preference function might be included in a design
model. An example of a preference function is shown in Figure 3.16.
P(y)
1
0
750 800 850
W (lbs)
Figure 3.16 A Flexible Preference Function (Chen and Yuan, 1997)
In general, a preference function P(y) is a function defining the relationships between the
degree of desirability, P, and the level of performance, y. The function is usually defined in the
range of 0 and 1, where a value of 1 represents full preference. Any design with desirability 1 is
fully acceptable, whereas a design with a value of 0 is unacceptable. For the same problem
97

mentioned earlier, a linear function may be used as the preference function for W. In Figure
3.16 when W ? 800 lbs, a design is fully acceptable, and a design is unacceptable if W is
greater than 850 lbs. In the range of 800 to 850 lbs the degree of desirability decreases from 1
to 0. A comparison is made in Figure 3.17 to show that when the preference functions are used
for expressing STB, LTB and CTB, they are more flexible, compared those in conventional
optimizations. In the next sections is studied how to use this flexible specification of preference
for ranged design solutions. From now on, it is assumed that the ranges of design variables are
symmetrical with respect to their mean.
STB
LTB
CTB
Conventional
Optimization
P(y)
1.0
0
P(y)
1.0
0
P(y)
1.0
0
y 0
y 01
y0
y0
y02
y
y
y
Proposed Model
P(y)
1.0
0
P(y)
1.0
0
y 1
y 0 y 1
Figure 3.17 A Comparison of Preference Structure (Chen and Yuan, 1997)
3.6.3 Ranged Design Solutions
P(y)
1.0
0
y 1
y 01
y 0
y 0
y 02
y 2
y
y
y
98

To enhance coordination of decisions and integration since an early stage of the enterprise
design process it is desirable to have a design model that considers ranges of solutions for
design variables instead of point solutions. In this work, the uncertainty associated with such a
ranged design solution is modeled by probabilistic distributions of design and performance
variables. This is illustrated in Figure 3.18, where it is shown how variation in design variables,
shown on the left side, causes a variation in the dependent performance function, as shown on
the right side.
x1
x 2
Figure 3.18 Mapping between Design Variables and Design Performance
Monte Carlo simulations or other statistical techniques, such as DOE, can be used to
generate the probability density function of the performance attribute. However, when the
computation time is demanding, it is an accepted practice to assume the performance follows a
normal distribution (Chen and Yuan, 1997). This practice follows from the central limit
theorem, which states that if a variable depends on a large number of independently acting
random causes or variables, then its distribution closely follows a normal distribution, even
y
99

though the independent causes or variables are not normally distributed. In such a case, the
probability density function is as follows:
100
2
1 ?
?
? ?
? 1 ? y ? y ?
f y ? exp ? ? ? ?
(3.8)
? 2?
? 2 ? ? ?
y ? ? ? y ? ?
If this assumption holds, the standard deviation ? can be approximated as ? y / 3,
where ? y is
the half-width of the variable range. In this case, the probability of the performance falling within
the range y ? ? y is 99.8%.
3.6.4 Design Preference Index
To evaluate the goodness of a design when both the design performance and the preference
level of performance vary within ranges, Chen and Yuan suggest the use of a Design Preference
Index (DPI). Mathematically, the DPI is defined as the expected preference function value of
design performance within the range of design solutions. This is estimated as follows for a
continuous y:
? ?
y ? ? y
?
DPI ? E P( y) ? P( y) f ( y) dy
y ? ? y
(3.9)
This function can be used as a measure to evaluate the goodness of designs with varying
performance in successfully satisfying a ranged set of requirements.
Three different designs with different values of DPI are shown in Figure 3.19. In design
(a) the preference function value P(y) remains within the range y ? ? y , where y is the average

value of the performance measured and DPI=1. This means that this design can fully meet the
design requirements under the variation of performance. However, DPI=1 cannot be achieved
in some cases, as illustrated in design (b) and (c). In design (c) the whole range of performance
falls totally outside the preferred range, which results in DPI=0. Hence, a design with a value of
DPI=1 corresponds to the design with the highest expected value of performance with respect
to the target acceptance function. Thereupon, maximizing DPI as closely as possible to 1
becomes the design objective. To evaluate the DPI, a numerical method can be used to
integrate the product of P(y) and f(y) within the range of variation y ? ? y .
P(y)
1.0
0
(a) DPI = (b) 0

Equation (3.9) is used to measure the probability of designs being acceptable. The novelty in
the model proposed by Chen and Yuan is in applying this concept to evaluate ranged solutions
that are assumed to be picked from a range on a random basis.
3.6.5 Constraint Feasibility Evaluation for Ranged Solutions
When solving a constrained design problem where the design variables are expressed as ranges
there are two option to evaluate feasibility: (1) to formulate the problem such that the constraint
g ( x ) has to be rigidly satisfied, i.e., satisfied for the whole range of values of x, or (2) to
formulate the constraint as to be satisfied within a confidence interval using statistical analysis.
For the former case, known as worst-case specification, the constraint is formulated as
102
g ( x,
x ) b
w
? ?
(3.10)
where g ( x,
x )
w
? represents the worst possible value of the constraint in the interval of
solution, and b is the maximum value of the constraint function for the solution to be feasible. A
constraint formulated as in Equation (3.10) is used to guarantee that the whole range of the
design solution, x ? ? x , satisfies the constraint. If the worst value, g ( x,
x ),
w
? is unknown, it
can be approximated by two means: (1) evaluating the combinations of design solutions in the
range based on Monte Carlo simulation and then choosing the worst case, or (2) estimating it
from a first order Taylor series approximation as:
? g
? g ? ? ? x j
(3.11)
j ? x j

where ? g is the worst-case deviation of the constraint function:
g w
103
? g ? ? g
(3.12)
Worst-case specification is almost always overly conservative. There are some
conditions that must be treated as a worst-case, but generally it is unlikely that all variables will
simultaneously occur in the worst possible combination. More reasonable, for such cases, is to
consider variations in performance as random variables, and the low probability of a worst-case
combination can be taken into account through statistical analysis. By allowing some fixed
number of infeasible designs to occur, a designer can use larger ranges and/or back away from
the “optimum” design some amount smaller than for a worst-case specification. In this case,
assuming that all variables are independent and the fluctuations are small, the standard deviation
of a constraint can be estimated as follows:
?
g
?
? ? g ?
? ? ? x j ?
j ? ? x j ?
where ? g is the standard deviation of the constraint g.
2
(3.13)
Because constraint functions are now described by probabilistic distributions, we can
modify the conventional notion of constraint feasibility. If a constraint distribution straddles the
right hand side, b in Figure 3.20, only a certain fraction of designs will be feasible, depending on
the location of the distribution relative to b. Suppose that at the nominal value (no variation
considered), the constraint function is equal to b, then the constraint is binding. Introducing
variation, and if the induced constraint distribution is symmetric, for half of the possible designs

the constraint is violated, as shown in the right side of Figure 3.20. It is possible to control the
number of infeasible designs by shifting the distribution of the constraint into the feasible region.
How far the distribution is to be shifted depends on the particular distribution type, the values of
distribution parameters (such as standard deviation) and which percentage of possible designs
are allowed to be infeasible.
y=b
Infeasible designs
with no shift
y
Shift
Figure 3.20 Statistical Constraint Feasibility
b
Infeasible designs
with shift
Statistical specification of feasibility can be used to develop a method for achieving
“feasibility robustness,” which refers to a feasibility that is characterized by a definable
probability, set by a designer, to remain feasible relative to a nominal constraint, as it is
subjected to variations in the variables (Parkinson, 1993). Feasibility robustness can be
obtained by increasing the value of the constraint function during optimization by a suitable
amount (for a constraint g ? b ). This has the effect of reducing the size of the feasible region
with an accompanying degradation in the optimal value of the objective. In equation form,
104

where ? g is the shift of the constraint.
105
g ? ? g ? b
(3.14)
For a worst-case analysis the shift is as given in Equation (3.11). For a statistical
specification it is given as:
? g ? k ? g
(3.15)
The value of k is selected to obtain a particular confidence of feasibility. For example, if a
constraint is described by a normal distribution, then k=3 shifts the constraint distribution such
that 99.8% of the possible designs are feasible for that constraint.
The linear approximations, Equations (3.11) and (3.13), are accurate for normal
distributions. When this assumption does not hold it is possible to extend the analysis by using
second order Taylor series to approximate first to third moments of the distribution, i.e., the
mean, the standard variation and the skewness, and then fitting a three-parameter gamma
distribution to these three values (Lewis and Parkinson, 1998). This allows defining a whole
family of distributions. When the skewness is zero, the gamma distribution closely matches a
normal distribution. The analysis can be significantly more accurate when the functions involved
are highly non-linear, but it can also become computationally expensive.
3.6.6 A Probabilistic-Based Multiobjective Design Model
So far, four basic notions that are used to obtain flexible design solutions have been described,
namely:

(1) flexible preference functions to capture a varying degree of preference for a design,
(2) evaluation of system behavior within a range using probability density functions,
(3) the combination of the former two concepts in an index, the DPI, to evaluate interval design
solutions, and
(4) evaluation of feasibility for interval solutions.
With these four elements it is now possible to formulate a compromise DSP for solving design
problems that involve interval design variables, i.e., problems whose solution is a set of ranged
variables. This is the final block for constructing a mathematical framework for enterprise
design, as proposed in Section 1.1.
In general, the probabilistic-based design model, shown in Figure 3.21, is constructed by
modifying the conventional compromise DSP (Section 3.4) to include probabilistic
representations of performance variations and the multiple goals on maximizing the design
metrics, DPIs, as close as possible to 1. The required changes are as follows (from Chen and
Yuan, 1997):
(1) In Section “Given” of the compromise DSP include, in addition to the given elements of a
conventional compromise DSP, the following:
?? Preference functions, Pi ( yi
), for the ranged performance variables yi ? y ? ? y.
106
?? Probability density functions ( y ) of the functional relationship between design
fi i
variables xj and performance, yi ? F(
x j ) , obtained through probability analysis,
Monte Carlo simulations, DOE or any other suitable technique.

107
(2) In Section “Find,” substitute the design variables x j with the average value of the variable
x j and the variation of each variable, ? xj, as design variables.
(3) In Section “Satisfy”:
(3.1) For the system constraints:
??Include lower limiting values for the deviation of design variables, ? xi
min , to ensure a
minimum flexibility of the design solutions. Without this constraint, the range of a design
variable, ? xi, will otherwise tend to converge to zero.
??Add a limiting value for the deviation of performance, ? yi
max,
as a constraint, in order to
find “robust” design regions, i.e., ranged solutions for which the performance does not
deviate more than a specified amount.
??Formulate constraint feasibility, g k ( x j ), using either a worst-case specification (Equation
3.10) or a statistical specification (Equation 3.15).
(3.2) For the system goals:
??Instead of the original form for defining goals in a conventional compromise DSP,
? y ? d ? d ? G (where Gi is the value of the goal for variable yi), define the system
i
i
?
i
i
goals as to achieve a DPIi of 1 for each of the original objectives, as shown in Equation
(3.12), where the DPIs are defined as in Equation (3.8).
(3.3) For the bounds of the design variables:
??Define bounds for both the average value of the design variables, x j , and their
variation, ?
x j .

The probabilistic design model, embodied in the compromise DSP shown in Figure 3.20, is a
formal design model to achieve flexible, yet robust, design decisions which is domain-
independent, thus appropriate for the coordination of decisions of the different enterprise design
processes. Particularly, the Design Preference index, DPI, provides a valid measure of value for
evaluating imprecise designs, thus enabling rational comparison and choice between interval
solutions, which is one of the fundamental requirements for designing in a decision-based design
paradigm, as discussed in Section 1.3.
Given
System parameters:
n number of design variables
q number of system constraints
m number of system goals
gi (x) system constraint function
F i(x) functional relationship between performance and design variables
Pi (yi) preference functions of design requirements
fi (yi) probability density functions of design performance
Find
System variables
x i
nominal value of design variables, i = 1, ..., n
? xi range of design variables, i = 1, ..., n
? ?
d i , d i
deviation variables, i = 1, ..., m
Satisfy
System constraints
? xi ? ? ximin i = 1, ..., n
? yi ? ? yimax i = 1, ..., m
g ( x , x ) b
w
? ? (for worst-case) i = 1, ..., q (Eq. 3.10)
g ? k ? g ? b (for statistical feasibility) i = 1, ..., q (Eq. 3.15)
System goals
? ?
DPIi ? d i ? di
= 1 i = 1, ..., m (Eq. 3.9)
Bounds
x L ? x ? x U i = 1, ..., n
? xi L ? ? xi ? ? xi U i = 1, ..., n
? ?
d i , d i ? 0 i = 1, ..., m
108

? ?
d i ? di
= 0
Minimize
i = 1, ..., m
Deviation function in preemptive or archimedean formulation
? ?
Z = Z( d i , di
) i = 1, …, m
Figure 3.21 The Probabilistic-Based Multiobjective Design Model
(Chen and Yuan, 1997)
109

When early design decisions are expressed as ranges, designers have more freedom in
adapting their decisions to satisfy the needs and goals of other agents of the organization, which
is the paramount objective of any coordination mechanism. This is the final block of the
mathematical framework for enterprise design proposed in Section 1.1. In Section 3.7, all the
elements that comprise the framework are tied together into a consistent structure for supporting
coordination of decisions in enterprise design.
3.7 TYING IT ALL TOGETHER: A MATHEMATICAL FRAMEWORK
FOR ENTERPRISE DESIGN
Up to this point, the four basic elements that comprise a mathematical framework, which is
shown in Figure 3.3, for integration of the enterprise design processes have been presented.
The fundamental tenet that bears this framework, rooted in a decision-based perspective, is that
the integration of the enterprise achieved through the appropriate coordination of decisions is
generally more efficacious than integrating those decisions into joint formulations. These four
basic elements are, as mentioned in Section 3.1:
?? A representation of the enterprise activity as an extensive game.
?? A common paradigm for decision support.
?? A multi-objective decision model, the Compromise DSP.
?? A probabilistic design model to attain flexible and robust decisions and handle uncertainty
and imprecision.
110

The merging of these elements provides a consistent mathematical framework for coordination
of decisions based on design formulations. The partitioning of elements of a decision into design
variables, models, requirements (goals and constraints) and activities (modeling, analysis and
synthesis) of the unified decision support model facilitates the mathematical formulation of
decisions as compromise DSPs. Game theory provides the mathematics to appropriately
formulate those compromise DSPs in seeking specific kinds of equilibrium. Finally, these
compromise DSPs can be formulated to achieve flexible solutions and to design with uncertain
and incomplete information, using a probabilistic design model, thus facilitating the coordination
of decisions and, therefore, the integration of the enterprise design processes.
111
Observe that each of these elements is “independent” of the others, including the
enterprise design method. That is, each of them holds by itself as a tool/method for decision
support. However, it is their merging that creates a practical and comprehensive framework for
enterprise integration that is anchored on a solid mathematical foundation.
Another characteristic of this framework is its “openness.” It does not exclude any tool or
method that contributes to an improvement of individual or group decision-making. The better
the individual decision-making, the easier is to find points of equilibrium. Thus, this framework
contributes to the integration of the enterprise activities, while empowering the agents of an
organization by decentralizing decisions and encouraging a better individual decision making.
Finally, the systematic and mathematical nature of this framework fosters its
implementation on a computer-supported environment. This framework is exercised in Chapter

4 in a case study, namely, the integrated product and process design of an absorber/evaporator
module for a family of absorption chillers.
112

CHAPTER 4
CASE STUDY: INTEGRATED PRODUCT AND PROCESS DESIGN OF AN
ABSORBER/EVAPORATOR MODULE FOR A FAMILY OF ABSORPTION
CHILLERS
In this chapter the framework for enterprise design constructed in Chapters 2 and 3 is
exercised in the preliminary design of an absorber/evaporator module for a family of absorption
chillers, which are produced in a make-to-order system.
The approach is demonstrated step by step for this case study, in which the design
process is abstracted as an extensive game that involves nine players: eight product designers
who want to minimize cost of material and one production system designer who wants to
minimize manufacturing cost and production time. Thus, the benefits of the proposed enterprise
design framework for the coordination of enterprise design decisions are studied in this case
study, while considering the issue of design strategies for family design accounting for production
process.
111

NOMENCLATURE FOR CASE STUDY
A Absorber tube selection
BC Burden (overhead) cost
Ca
Ce
COST Total cost
Cost of absorber tube
Cost of evaporator tube
CT Cycle time (total production time)
D Expected demand per year
DPI Design Preference Index
E Evaporator tube selection
F Replenishment frequency
FALP Foraging-Adaptive Linear Programming algorithm
GLOBAL Global optimization algorithm based on clustering method
IC Inventory cost
ir Annual interest rate
k Proportional factor in burden cost estimation
LC Labor cost
LFC Leader/follower solution using a conventional design approach
LFP Leader/follower solution using a probabilistic design approach
LT Normalized tube length
112

MC Material cost
MfgC Production cost
MSi Capacity at station Si
Na
Ne
Number of tubes of the absorber
Number of tubes of the evaporator
NAT Normalized number of tubes of the absorber
NET Normalized number of tubes of the evaporator
n(r) expected number of backorders during a cycle
Ops Number of operators
PC Product cost
Q Replenishment quantity
r Reorder point
SA Safety stock of absorber tubes
SAi Subassembly station i
SAn Simmulated Annealing Algorithm
SE Safety stock of evaporator tubes
Si Station i
SCV Squared coefficient of variation
TH Throughput
UA Heat transfer coefficient multiplied by the area of heat exchange
113

? P Pressure drop?
?? Mean processing time
?? Expected demand during lead time?
? Standard deviation
Subindices
Abs Absorber
AP Aggregated product
Evap Evaporator
T Individual absorber/evaporator design
114

4.1 INTRODUCTION TO CASE STUDY
In Chapters 2 and 3 is presented a framework for enterprise design that is based on a game
theoretical representation of the enterprise design process. Following the implementation
strategy presented in Section 1.6, the next step is to exercise this framework in a case example,
namely the design of an absorber/evaporator module for a family of absorption chillers, which
are produced in a make-to-order system.
4.1.1 Motivation for Case Study: Benefits of an Integrated Product and
Process Design in Make-to-Order Systems
115
Make-to-order systems are “systems in which jobs, both the physical material and the job
order, arrive from outside the shop, and the process that generates the arrivals is independent of
whatever is happening in the shop, and its associated input and output stores” (Buzacott and
Shanthikumar, 1993). This kind of systems is commonly found in the production of complex
equipment with low demand, such as ships, aircrafts, turbines, etc. These systems, particularly
those that offer product variety, face a large amount of variability in manufacturing processes
time, supply deliveries and demand, which in turn cause long lead-times and large work-in-
process. How can product design improve the performance of make-to-order systems?
A hierarchy of objectives that links the fundamental objective of a manufacturing
organization to various supporting subordinate objectives according to Hopp and Spearman
(1996) is illustrated in Figure 4.1. Clearly, there are conflicts. High inventory is desirable for

fast response, but low inventory is attractive for keeping total assets low to return on investment
be high. High utilization is needed to keep unit costs down, and low utilization is needed for
good responsiveness. More variability is needed for greater product variety, but less variability
is needed to keep inventory low and throughput high. Ideally, these objectives should be
analyzed concurrently in order to obtain an appropriate compromise between them, particularly
between greater product customization and shorter production times, which are both critical in
customers’ preference for this kind of products. The motivation for this case study is, therefore,
to apply the framework for enterprise design and integration constructed in Chapters 2 and 3 to
explore its benefit in the integrated product and process design of a make-to-order product.
High
Throughput
Less
Variability
Low
Costs
Low Unit
Costs
High
Utilization
Low
Inventory
High
Profitability
Quality
Product
Short
Cycle Time
High
Sales
Fast
Response
Low
Utilization
High Customer
Service
High
Inventory
Many
Products
Figure 4.1 Hierarchical Objectives in a Manufacturing
Organization (Hopp and Spearman, 1996)
More
Variability
116

Before addressing the case study, the strategy followed in this work to model and analyze
a production system for an integrated product and process design is presented in Section 4.1.2.
4.1.2 A Methodology for Modeling Production Systems for Integrated
Product and Process Design
In order to assess the effect of product changes in the production system, it is necessary
to have some model that relates such changes with the performance of the production line. In
simulation-based manufacturing models, the probabilistic nature of many events, such as product
rework or arrival variability, is represented by sampling from a distribution representing the
pattern of occurrence of the event. Thus, to represent the typical behavior of a system it is
necessary to run the simulation model for a sufficiently long time, so that all events can occur a
reasonable number of times. Direct coupling of product and manufacturing processes models
(such as FEM models) with a discrete event simulation of the production system is
computationally too expensive and time consuming to support product design decisions.
117
To overcome this problem, Peplinski, et al., (1997) propose creating a “bank” of
response surfaces, one for each machine or processing center. These response surfaces
capture the relationship between a reduced number of local input variables and a small set of
design requirements such as flowtime, scrap rate, etc. These response surfaces could then be
“assembled” to represent the larger manufacturing system. They suggest two types of
constraints to assemble the response surfaces: (1) the exit flow from each upstream machine

must match the arrival flow of each downstream, and (2) input flows to a processing center must
not exceed its capacity. It is easy to see that a set of small response surface modules should
increase a designer’s flexibility in quickly modeling different shop floor configurations.
However, this approach fails because, in general, the variability of arrivals, and therefore of
departures, from a station is not independent of other stations of the system.
In this thesis a hybrid method is proposed, which combines the notion of a bank of
response surfaces, as proposed by Peplinski and co-authors, with the parametric
decomposition approach (described in Appendix A) to build models of manufacturing systems
as response surface networks, which do not have the aforementioned limitation.
118
In the parametric decomposition approach, each station is characterized by two
parameters: the mean (?? and the squared coefficient of variation (SCV) of the processing times.
Therefore, only two parameters for each station are needed. Thus, it is not necessary to fit a
response surface for the flowtime, yield, waiting time, machine utilization, etc. of each station as
proposed by Peplinski and co-authors, and it is possible to capture interaction between stations.
The proposed method of building models of the manufacturing system as response surface
networks consists of four steps, as illustrated in Figure 4.2:
1. Identify relevant factors and ranges for each process (machining, forging,
welding, assembly, etc).

2. Design appropriate experiments for fitting the mean processing times and its
SCV for each process as functions of the factors identified in Step 1, using the
appropriate modeling technique.
3. Fit the response surfaces.
4. Assemble the response surfaces in a network using the parametric
decomposition approach.
1. Identify Factors and Ranges
x
Control
Factors
Noise
Factors
z Service
Mean Time
?
Process
y
x2
x1
Welding
y
2 c
SCV
x2
x1
Cutting
y
2. Design Experiments
The Parametric
Decomposition Approach
4. Assemble the Response Surfaces in a Network
x2
x2
x1
x1
Machining Assembly
MANUFACTURING SYSTEM
MODEL
y
Models of Individual
Processes
3. Build Response Surface
Models of Individual
Processes
Figure 4.2 Modeling the Production System as a Network of Response Surfaces
y
x2
x1
119

In Step 1, the initial exploration space is defined, i.e., the control and noise factors which
mostly influence the mean process times and the SCVs are identified as well as their range of
value. In the second and third steps, response surfaces of the mean processing times and SCVs
as function of the control and noise variables are built and validated. Finally, in the fourth step,
the different stations are synthesized in a network using the response surfaces obtained in the
previous steps. It should be apparent that once a network of response surfaces is built, it is
possible to quickly observe the effect of any change in a product or process in the system
performance, thus allowing an efficient and systematic product-process exploration.
Furthermore, any change in the network is straightforward and requires a minimum remodeling
effort, which is not the case with simulation-based models. In the following section this method
is applied in solving the case study.
4.2 DESCRIPTION OF THE DESIGN PROBLEM AND ANCILLARY
INFORMATION
4.2.1 Absorption Refrigeration
Today there are several manufacturers of large absorption machines, which are used when the
operating and maintenance costs of electrically driven refrigeration equipment are not
economical or are restricted due to environmental regulations. This product is a typical example
of expensive and highly customized products with low production volumes which obliges
120
manufacturers to produce them in make-to-order systems. Manufacturers of absorption

equipment usually face large demand fluctuations, hard-to-forecast market status, complex
supply logistics, large work-in-process levels, and considerable lead-times. This case study
illustrates how the proposed enterprise design framework can help to improve the performance
of such systems via an integrated product and process design.
121
Absorption chillers consists of four basic components: (1) the evaporator, (2) the
absorber, (3) the generator, and (4) the condenser; as shown in the absorption cycle illustrated
in Figure 4.3. Usually, the absorbent in conventional absorption cycles is Lithium Bromide
(LiBr), a salt that has the ability to absorb water (the refrigerant).
Heat
Exchanger
GENERATOR CONDENSER
Strong Solution
Weak Solution
Steam
ABSORBER EVAPORATOR
Condensing
Water
Refrigerant Vapor
Refrigerant Vapor
Refrigerant
Refrigerant
Figure 4.3 Absorption Refrigeration Cycle
Condensing
Water
Cooling
load

A brief description of these components follows:
GENERATOR. The generator is the equivalent of the compressor in a conventional
compression refrigeration cycle. It is a vessel where heat is applied which causes the mixture of
refrigerant and absorbent to boil and provides the motive force for the process. The refrigerant
vapor (steam) and absorbent droplets (LiBr and water) leave the generator and enter a
condenser.
CONDENSER. The vapors pass from the separator to the condenser where cooling
water from a cooling tower circulates through the condenser removing the heat added at the
generator. The low temperature steam vapor is condensed into a liquid refrigerant where it
passes through an orifice into the evaporator which operates under a vacuum. This condenser
performs the same function as the condenser of a conventional compression refrigeration
system.
122
EVAPORATOR. The evaporator serves the same function as in a conventional
compression refrigeration system inasmuch as it absorbs the heat from the process load. The
expansion of the refrigerant (water) within a vacuum causes boiling and therefore absorbs heat
from the chilled water coils located within the evaporator. This chilled water is used to satisfy
the process load.
ABSORBER. The refrigerant (water) vapor will not condense under the vacuum
required for evaporation. Some means of converting the vapor back into a liquid must take

place before the refrigerant is returned back to the generator to repeat the process. This is
where the absorption cycle differs from the conventional compression refrigeration cycle. The
LiBr solution, which was concentrated within the low temperature generator when the
refrigerant (water) was boiled off, passes from the low temperature generator through a heat
exchanger to the absorber. Cooling coils within the absorber absorb the heat load from the
evaporator and, unavoidably, some of the residual heat from the concentrated solution, to be
dissipated through the cooling tower. The concentrated LiBr solution absorbs the refrigerant
vapor from the evaporator. This LiBr/water solution is now diluted and is pumped through the
heat exchanger, to recover heat from the concentrated solution, before returning to the
generator to repeat the process. When the refrigerant vapor is absorbed, a vacuum is created
which allows expansion to take place between the condenser and the evaporator.
123
Usually, the absorber and evaporator are combined in a single shell, the
absorber/evaporator module, illustrated in Figure 4.4, which is the focus of this case example.

Figure 4.4 Layout of Absorption Chillers
(Carrier Mexico)
4.2.2 Objective and Ancillary Information 1
A manufacturer of absorption refrigeration equipment offers a family of chillers with the
following refrigeration capacities: 600, 700, 800, 900, 1000, 1100, 1200 and 1300 tons. A
market study has estimated the demand for the next years as approximately 80 units/year, with a
distribution of such demand as shown in Table 4.1 and Figure 4.5.
Table 4.1 - Estimated Distribution of Demand
Tons, T 600 700 800 900 1000 1100 1200 1300
%
demand
DT
0.05
0.25
0.2
0.15
0.1
0.05
0
0.15
0.16
0.05
0.15
0.12
600 700 800 900 1000 1100 1200 1300
Capacity (tons of refrigeration)
1 The actual information of the product design and the production system has been disguised to protect the
competitive position of the manufacturer.
0.25
124
0.07

Figure 4.5 Estimated Distribution of Demand
Prior to this work, the absorber-evaporator module has been uniquely designed for each
capacity to minimize the material cost (the traditional approach in designing heat exchangers).
This in turn has created a large variety of components. Because of this variety, low production
volumes and large fluctuations in demand, it is not economical at the present to protect the line
against disruptions, delays and randomness by carrying safety stock. It also creates difficulties
for inventory management, material handling, storage and control of the floor shop. In this
example, the concern is with the preliminary design of the absorber/evaporator module (to
determine its length and number and type of tubes) for the family of chillers using the
mathematical framework established in previous chapters and applying the design principles of
standardization and robustness.
Consider a production line of the absorber/evaporator module as shown in Figure 4.6.
125

SA1
SA2
SA3
SUB-ASSEMBLIES
STATIONS
Absorber and
Evaporator Tubes
S1 S2
S3
STATION 1 STATION 2 STATION 3
Figure 4.6 Production Line of the Absorber/Evaporator Module
126
DEPART
There are 3 main subassemblies produced in the subassembly stations SA1, SA2, and
SA3, each with one server. These subassemblies are assembled in Station S1. After this
station, the assemblies go to a second station, S2, and finally to Station S3, where the
evaporator tubes and the absorber tubes are assembled. There are two working shifts of 8
hours each and the system operates only during weekdays.
The manufacturer wants to reduce the variety of tubes to one combination of tube type
and length for the absorber and one for the evaporator (to reduce tube variety from 16 to 2).
This will allow carrying safety stock to avoid disruptions at Station S3, and will facilitate material
handling and storage, thus fostering a better control of the shop floor, shorter lead times, lower
costs and better customer service. Simply stated, the objective is to determine one
combination of length and tubes selection (material, diameter, thickness, number of fins
per inch, etc.) for the absorber/evaporator modules.

The design of the absorber/evaporator module is carried out with four approaches. The
objective is to show the application of the framework for enterprise design described in Chapter
3 and test the hypothesis posed in Chapter 1. In Section 4.3 is described and implemented an
approach in which each of the agents involved in the design process acts in isolation (which is
called Baseline Solution). These agents are 8 product designers, each responsible of the design
of an absorber/evaporator module for a particular refrigeration capacity, and one production
system designer who is responsible of designing a production system that can meet the expected
demand with minima production cost and cycle time. The information and formulation of these
baseline decisions are the input of the enterprise design method.
Once the baseline decisions are formulated, the expansion of decisions into a game, in
order to coordinate decisions between product design and production system design, is
carried out with three different approaches, described in Sections 4.4 to 4.6. This plan of
action is illustrated in Figure 4.7.
127

Baseline Decisions
Product Design
Section 4.3
Production System
Design
Reformulate Decisions to
Integrate Product Design and
Production System Design
Using a Cooperative
Game Model
Section 4.4
Section 4.5
Using a Non-cooperative
(Stackelberg) Game Model
Reformulation of
Design Model
as a Probabilistic
Design Model
Section 4.6
Figure 4.7 Roadmap to Case Study
190000
187500
185000
182500
180000
177500
175000
Comparison of
Solutions
Baseline Coop Conv-NC Prob-NC
Section 4.7
In Section 4.4, it is studied a solution (called Cooperative Solution) obtained with a
cooperative protocol between the 9 agents. In Sections 4.5 is adopted a cooperative protocol
between the 8 product designers, and in Section 4.6 is adopted a non-cooperative protocol
between them as a group and the production system designer. This last case is formulated using
a “conventional” design approach in Section 4.5 (called Conventional Non-Cooperative
Solution), and a probabilistic design approach in Section 4.6 (called Probabilistic Non-
Cooperative Solution). Here “conventional” approach refers to a non-probabilistic one.
4.3 BASELINE FORMULATION AND SOLUTION
128

The objective of this section is threefold:
1. To present information regarding the design problem that is used in Sections 4.4 to 4.6.
2. To obtain a design solution for comparison with those solutions obtained by applying the
enterprise design framework developed in Chapters 2 and 3, and
3. To verify the accuracy of the manufacturing systems model and the effectiveness of the
optimization algorithm that is used in obtaining subsequent solutions.
129
Assume that there are 8 designers, each responsible of achieving a design of an
absorber/evaporator module for a particular refrigeration capacity that minimizes the material
cost. There is also a production system designer who, after the design of the family of products
is obtained by the group of product designers, has to design a production system that can meet
the expected demand of the absorber/evaporator modules, while seeking minima production
cost and cycle time. This situation is illustrated in Figure 4.8. In this section it is described a
solution achieved when each of these designers acts in isolation.

Goal
Minimize Material Cost
Product Designers
600 700 800 900 1000 1100 1200 1300
Capacities
DESIGN VARIABLES
DESIGN VARIABLES
Manufacturing Process
Designer
Length, Number of Tubes, Tube Selection
for each Capacity
Goal
Minimize Production Cost
Minimize Production Time
Capacity of Stations S1, S2 and S3
Tubes Replenishment Frequency
Figure 4.8 Agents Involved in the Design Process
Nine baseline decisions are identified, one for each agent. In Section 4.3.1 is described
the formulation of decisions for the product designers, and in Section 4.3.2 for the production
system designer. In Section 4.3.3 the solution obtained is presented, and in Section 4.3.4
validation and verification issues are studied.
130

4.3.1 Formulation of Decision for Product Design
The variables that a product designer controls are the value of the product design variables for a
particular capacity, i.e., the length of the tubes (LT), number of evaporator tubes (NeT), number
of absorber tubes (NaT), and the selection of tubes for the evaporator and the absorber, ET and
AT, where the subscript T is used to refer to individual products/designers (T=1,...,8). Each
product has to be carried out for the following standard specifications:
Entering temperature of chilled water 54 o F
Leaving temperature of chilled water 44 o F
Flow rate of chilled water 2.4 GPM/ton
Refrigerant fluid Fresh Water
Saturation temperature of refrigerant 41 o F
Flow rate of water in the absorber 4.5 GPM/ton
Entering (absorber) water temperature 85 o F
Solution fluid LiBr/H2O
Entering solution concentration 62%
Entering solution temperature 110 o F
Entering solution flow rate 222.5 lb / (hr ton)
Fouling factor for tube walls 0.00025 hrft 2 o F/BTU
Fouling factor for shell walls 0.00025 hrft 2 o F/BTU
131

4.2.
The commercial alternatives for selection of tubes for the evaporator are shown in Table
Table 4.2 Alternatives for Selection of the Evaporator Tubes
Evaporator Tube E 1 2 3 4
Outer Diameter (in) 0.625 0.75 0.875 1.0
Wall Thickness (in) 0.025 0.025 0.025 0.025
Fins/in 30 30 30 30
Fin Height (in) 0.05 0.05 0.05 0.05
Fin Thickness (in) 0.02 0.02 0.02 0.02
Material copper copper copper copper
Cost C e (dollars/in) 6.00 6.50 7.0 7.5
Similarly, the options for selection of tubes for the absorber are shown in Table 4.3.
Table 4.3 Alternatives for Selection of the Absorber Tubes
Absorber Tube A 1 2 3 4
Outer Diameter (in) 0.625 0.75 0.875 1.0
Wall Thickness (in) 0.03 0.03 0.03 0.03
Material copper copper copper copper
Cost C a (dollars/in) 3.25 3.50 3.75 4.0
The classification of parameters that define the synthesis of the products is shown in Table 4.4.
The only goal is to achieve some target of material cost and the constraints include a minimum
UA (the overall heat transfer coefficient U multiplied by the area of heat exchange A) in both the
absorber and the evaporator, and a maximum pressure drop of the chilled water. These
product cost goals and constraints values for each refrigeration capacity are shown in Tables
132

4.5 and 4.6. The cost of raw material (without considering tubes) for each station is given in
Table 4.7.
Table 4.4 Parameters for Absorber-Evaporator Module Synthesis
Parameter Name Symbol Category Range Units
1 Tube length L design variable 17.0 - 24.0 ft
2 Evaporator tube
selection
E design variable 1 - 4
3 Absorber tube
selection
A design variable 1 - 4
4 Number of tubes of
the evaporator
Ne design variable 440 - 900
5 Number of tubes of
the absorber
Na design variable 440 - 900
6 Material cost MC goal dollars
Table 4.5 Material Cost Goal for each Refrigeration Capacity
Refrigeration Capacity Material Cost Goal
T
MCT,goal (dollars)
600 tons 99000
700 tons 112500
800 tons 126000
900 tons 139500
1000 tons 153000
1100 tons 166500
1200 tons 180000
1300 tons 193500
133

Table 4.6 Constraint Values for Product Design
Capacity min UA Evaporator
(BTU/ hr o F)
min UA Absorber
(BTU/ hr o F)
Max Pressure Drop ? P
(ft of water)
600 tons 771100 575035 23
700 tons 905300 677235 23
800 tons 1039500 779435 23
900 tons 1173700 881635 23
1000 tons 1307900 983835 23
1200 tons 1576300 1188235 23
1400 tons 1844708 1392635 23
1600 tons 2113108 1597035 23
Table 4.7 Cost of Raw Material at Each Subassembly Station (in Dollars)
Capacity
T
MCSA1 MCSA2 MCSA3 600 20000 15000 4000
700 22000 16000 4500
800 24000 17000 5000
900 26000 18000 5500
1000 28000 19000 6000
1100 30000 20000 6500
1200 32000 21000 7000
1300 34000 22000 7500
Given length, tubes selection and number of tubes for each capacity, the cost of material
(MCT) is calculated as:
134
MC T ? L T( N eTC e ? N aTC a) ? MC T, SA1 ? MC T,SA2 ? MC T, SA3 (4.1)
where Ce and Ca are the cost of the evaporator and absorber tubes (Tables 4.2 and 4.3). For
each capacity, the goal for product design is formulated as follows:
MC T
MC T ,goal
? d ? ? d ? ? 1 (4.2)

135
? ?
where MCT,goal are the target values for cost of material given in Table 4.5, and d and d are
deviation variables used in a compromise DSP.
This information (design variables, bounds, goals and constraints) embody the baseline
decision for each product designer. A compromise DSP corresponding to these baseline
decisions is shown in Figure 4.9. This compromise DSP has to be solved by each product
designer for a specific refrigeration capacity.

Given
?? A refrigeration capacity
?? Heat-transfer models of the absorber-evaporator
?? Commercial tube alternatives (Table 4.2).
Find
?? Length LT of tubes
?? Selection of tube for the evaporator ET
?? Selection of tube for the absorber AT
?? Number of tubes of the evaporator, NeT
?? Number of tubes of the absorber NaT
Satisfy
System Constraints
?? UAEVAP ? UAEVAP min (Table 4.5)
?? UAABS ? UAABS min (Table 4.5)
?? ? P ? ? Pmax (Table 4.5)
System Goals for each capacity (minimize material cost):
??
MC T
MC T ,goal
? d ? ? d ? ? 1 (Equation 4.2)
Bounds for each capacity
?? Na min ? Na ? Na max (Table 4.3)
?? Ne min ? Ne ? Ne max
?? Lmin ? L ? Lmax
?? d ? , d ? ? 0
d ? ?d ? ? 0
Minimize
The deviation function
?? Z ? d ?
Figure 4.9 Baseline Compromise DSP for Product Designers
136

4.3.2 Formulation of Decision for Production System Design
The baseline decision of the production system designer is that of specifying the required
number of servers at station S1, S2 and S3 of the production line, and to set a safety stock level
of absorber tubes and evaporator tubes. The performance measurement is based on the
expected cycle time and production cost.
The approach described in Section 4.2.2 for modeling production systems is now applied
to the production line of the absorber/evaporator module. The steps are as follows:
1. Identify relevant factors that affect processing times and SCVs of the various
stations. In this case the product design variables that mainly affect the processing times
are the length and number of tubes of both the absorber and the evaporator. The SCVs are
all considered exponentially distributed, which gives a SCV of one for all the stations
regardless of the value of the design variables.
2. Design experiment. A full factorial CCD design is selected for fitting second order
response surfaces to the processing times at stations as functions of the product design
variables LT, Ne,T and Na,T.
3. Fit response surfaces of the mean processing time (???of the various stations. The
statistical analysis software JMP® (SAS, 1995) is used to this aim. The resulting response
surfaces for the mean processing times ? of each station are:
137
? SA1 ? 24.159 ? 0.063 LT ? 0.563 NET ? 0.688 NAT ? 0.045 LT 2 ? 0.125 NET ?LT ? 0.080 NET 2
? 0.125 NAT ?LT ? 0.125 NAT ?NET ? 0.205 NAT 2 (4.3)

138
? SA2 ? 29 .295 ? 0.312 LT ? 2.062 NET ? 1.397 NAT ? 0.023 LT 2 ? 0.125 NET ?LT ? 0.102 NET 2
? 0.125 NAT ?LT ? 0.125 NAT ?NET ? 0.227 NAT 2 (4.4)
? SA3 ? 8.636 ? 1.25 NET ? 1.25 NAT ? 0.056 NET 2 ? 0.25 NAT ?LT ? 0.25NAT ?NET ? 0.068NAT 2 (4.5)
? S1 ? 66.04 ? 1.312 LT ? 3.801 NET ? 4.062 NAT ? 0.147 LT 2 ? 0.125 NET ?LT ? 0.147 NET 2
? 0.125 NAT ?LT ? 0.875 NAT ?NET ? 0.397 NAT 2 (4.6)
? S2 ? 63.272 ? 1.05LT ? 4.375NET ? 4.875NAT ? 0.363 LT 2 ? 0.011NET 2 ? 0.378 NAT 2 ??????
? S3 ? 81.954 ? 2.397 LT ? 6.187 NET ? 5.937 NAT ? 0.102 LT 2 ? 0.125 NET ?LT ? 0.227 NET 2
? 0.125 NAT ?LT ? 0.125 NAT ?NET ? 0.522 NAT 2 (4.8)
where LT, NET, and NAT are normalized values of the length (LT), number of tubes in the
evaporator (Ne,T) and number of tubes of the absorber (Na,T), such that their values are
within the range [-2, 2]. They are calculated using the upper and lower bounds of the
variables (Table 2) as follows:
LT ?
L ? 17
1.75
NET ? N e ? 440
115
NAT ? N a ? 440
115
? 2 (4.9)
? 2 (4.10)
? 2 (4.11)
4. Link the response surfaces (Equations 4.3 to 4.8) in a network using the parametric
decomposition approach (Appendix A), considering a SCV of one at all the stations.
The response surface network model allows estimating the processing times (?), the
utilization (u) at each station, and the expected production time (CT), as functions of the
product design variables.

Once the mean processing times, ?, are obtained for a particular combination of design
variables using the response surface network model, the labor cost (LC) can be estimated as
follows:
LC ? ? SA1 Ops SA1 ? ? SA2 Ops SA2 ? ? SA3 Ops SA3 ? (MS1)? S1 Ops S1 ?
(MS2)? S2 Ops S2 ? (MS3)? S3 Ops S3
139
(4.12)
where Opsi is the number of operators at the station i, and MS1, MS2 and MS3 are the number
of servers (the capacity) at Stations S1, S2 and S3 respectively. In this case there are 2
operators at the subassemblies stations SA1, SA2 and SA3, and there is one operator for each
server at stations S1, S2 and S3. These values are fixed.
For simplicity, only inventory, labor and burden (overhead) costs are considered in
estimating the production cost (MfgC):
where BC is the burden cost and IC is the inventory cost.
follows:
MfgC = LC + BC + IC (4.13)
For this example, the burden cost is estimated as being proportional to the labor cost as
BC = k LC (4.14)
where k is a constant coefficient. With the raw material cost, MC (Table 4.7), and the number
of operators at each station the cost of the product at each station i (PCi) can now be

approximated. For example, for the station SA1, the product cost is simple the material cost
MCSA1, whereas for station S1 it is calculated as:
where ICi is estimated as follows:
?
PC S1 ? MC S1 ? MC i ? LC i ? BC i ? IC i
SA1
SA2
SA3
IC i ? ir(PC i )
TH year
140
(4.15)
(4.16)
where ir is the annual interest rate and THyear is the annual throughput of the product. Here is
considered an ir of 0.1. The cost at other stations is calculated similarly. At station S3 the cost
of the tubes is added to the raw material cost given in Table 4.7.
To estimate IC it is necessary to know the safety stock of evaporator and absorber
tubes. In setting a safety stock it is desired a confidence of at least 99% (a fill rate S ? 0.99)
that there will be no delays at S3 due to a stockout of tubes. Given this constraint, and
according to Hopp and Spearman (1996), the problem for specifying a safety stock is
formulated for each tonnage as follows (shown for the evaporator tubes):
Given Cost of tubes Ce (Table 4.2)
Expected demand of tubes per year Dtubes = D(Ne)
Find Safety stock SE
Satisfy Average replenishment frequency ? F
Average fill rate S ? 0.99
Minimize Inventory investment, ICe

The “Satisfy” and “Minimize” statements can be written mathematically as:
Satisfy D tubes
Q
Minimize C e
1 ? n(r)
Q
141
? F (4.18)
? 0.99 (4.19)
?? Q ??
?? ? r ? ? ?? (4.17)
?? 2 ??
where Q is the replenishment quantity (in units), r is the reorder point (in units), ? is the expected
demand during the replenishment lead time (in units), and n(r) is the expected number of
backorders during a cycle. It follows that Q is chosen as:
Q ? D tubes
F (4.19)
Given Q, r is chosen as the smallest value that satisfies the Equation (4.19), that is:
n(r) ? (1? 0.99) D tubes
F (4.20)
Thus, the problem is to find a value of F that minimizes inventory investment (Equation
4.17). In this case example, the replenishment lead-time is 7 weeks, and the possible F values
are 54, 27, 18, 12 and 10.8 replenishments/year.
Once F is chosen, and r is estimated using Equation (4.20), the safety stock of
evaporator tubes SE is estimated as:
SE = r - ? (4.21)
A similar approach is used to obtain the safety stock for the absorber tubes.

142
Based on these considerations, the classification of parameters that define the
production system synthesis is shown in Table 4.8. The goals include production cost and cycle
time. Two constraints are considered: a maximum utilization of 0.9 at any station and an
average fill rate at station S3 of 99% in setting the safety stock of tubes. In Table 4.9 the values
of the production targets for each refrigeration capacity are shown.
Table 4.8 Parameters for Production System Design
Parameter Name Symbol Category Range Units
1 Capacity at station SA1 MSA1 constant 1
2 Capacity at station SA2 MSA2 constant 1
3 Capacity at station SA3 MSA3 constant 1
4 Capacity at station S1 MS1 design variable 1 ,2,...,5
5 Capacity at station S2 MS2 design variable 1 ,2,...,5
6 Capacity at station S3 MS3 design variable 1,2,...,5
7 Replenishment
frequency
F design variable 10.8,12,
18,27,54
8 Production cost MfgC goal dollars
9 Expected cycle time CT goal hours

Table 4.9 Goals and Constraint Values for Production System Design
Production Cost/Unit Expected Cycle-Time Fill Rate
MfgC (dollars)
CT (hours)
S
Capacity GOAL GOAL CONSTRAINT
600 tons 6000 280 0.99
700 tons 7000 290 0.99
800 tons 8000 300 0.99
900 tons 9000 310 0.99
1000 tons 10000 320 0.99
1100 tons 11000 330 0.99
1200 tons 12000 340 0.99
1300 tons 13000 350 0.99
For the synthesis of the system, the value of the individual goals shown in Table 4.9 are
“aggregated” into two systems goals as follows:
143
GoalAP ? ? Goal T DT (4.22)
T
where T indicates individual products (T=1,2,...,8), and DT is the estimated fraction of the total
demand of each product (Table 4.1). Thus, for the aggregated product, the MfgC target value
is:
MfgCgoal=0.05(6000)+0.15(7000)+0.16(8000)+0.05(9000)+0.15(10000)+0.12(11000)+
0.25(12000)+0.07(13000) = 9810 (dollars) (4.23)
Similarly, for CT the target value is:

144
CTGOAL=0.05(280)+0.15(290)+0.16(300)+0.05(310)+0.15(320)+0.12(330)+
0.25(340)+0.07(350)=318 (hours) (4.24)
Using the target given by Equation (4.23), the production cost goal for a compromise
DSP is formulated as:
MfgC
MfgC goal
Similarly, the CT goal is formulated as follows:
CT
CT goal
? d 1 ? ? d1 ? ? 1 (4.25)
? d 2 ? ? d2 ? ? 1 (4.26)
An objective function Z ? W1(d1 ? ? d1 ? ) ? W2 (d2 ? ? d2 ? ) can now be formulated where Wi is the
weight given to the achievement of the ith goal in an Archimedean formulation of the deviation
function (Section 3.4).
As this is a two-objective problem, the modified approach of Rao and Freihet (1992)
described in Section 3.4.2, can be applied. First, two compromise DSPs are solved to
construct a payoff matrix P:
P ? Z ??
11 Z ??
12
?? ???
?? Z21 Z 22??
d ??
1 ??
?? d2 ? *
( X1 )
? *
d1 (X2 )
? *
( X1 )
? *
d2 (X2 )
??
??
?? (4.27)
where X1 * is the design vector X1 * =[MS1, MS2, MS3, F]1 obtained when minimizing only the
production cost (i.e., W2=0), and X2 * is the design vector obtained when minimizing only the
cycle time (W1=0). di ? ( X) is the value of the deviation variable associated with the achievement

of the ith goal when the design vector is X. The compromise DSP for obtaining the payoff
matrix is shown in Figure 4.10.
With the values of the payoff matrix (Equation 4.27), a modified deviation function is
formulated as:
145
Z mp = Z - SC+1 (4.28)
where 1 is added to make Z mp always positive, and the other terms are given by:
Z ? W1 ˆ
Z 1 ? W2 ˆ
Z 2 (4.29)
SC ? (1 ? ˆ
Z 1)(1 ? ˆ
Z 2) (4.30)
ˆ
Z 1 ?
ˆ
Z 2 ?
Z1 ? Z1( X1 * )
* * (4.31)
Z1 ( X2 ) ? Z1 (X1 )
Z2 ? Z2 (X2 * )
* * (4.32)
Z 2 ( X1 ) ? Z2 ( X2 )
In this case study, and to facilitate the comparison of solutions obtained with different
approaches, fixed values of W1 = W2 = 0.5 are used when solving compromise DSPs with
deviation function Z mp . The resulting baseline compromise DSP for the synthesis of the
production system is shown in Figure 4.11.

Given
?? Production model
?? Inventory model
?? Product design variables (LT, AT, ET, Ne,T, Na,T) T=1,2,...,8
Find
?? Capacity (number of servers) at station S1, MS1
?? Capacity at station S2, MS2
?? Capacity at station S3, MS3
?? Replenishment frequency, F
?? Deviation variables di ? ,di ? i=1,2
Satisfy
System Constraints
?? Fill Rate at S3 (for tubes safety stock)
S ? 0.99
?? Maximum utilization
u ? ???
(Equation 4.20)
System Goals for each capacity (minimize average production cost):
?? MfgC ? ?
? d1 ? d1 ? 1 (Equation 4.25)
??
MfgC goal
CT
CT goal
? d 2 ? ? d2 ? ? 1 (Equation 4.26)
Bounds
?? 1? MS1 ? 5
?? 1? MS2 ? 5
?? 1? MS3 ? 5
?? di ? , di ? ? 0
di ? ?di ? ? 0
Minimize
The deviation function
?? Z = Z ? W1d1 ? ? W2d2 ?
Figure 4.10 Compromise DSP for Building a Payoff Matrix for Production System
Design
146

Given
?? Production model
?? Inventory model
?? Product design variables (LT, AT, ET, Ne,T, Na,T) T=1,2,...,8
?? Payoff matrix P (Equation 4.27)
Find
?? Capacity (number of servers) at station S1, MS1
?? Capacity at station S2, MS2
?? Capacity at station S3, MS3
?? Replenishment frequency, F
?? Deviation variables di ? ,di ? i=1,2
Satisfy
System Constraints
?? Fill Rate at S3 (for tubes safety stock)
S ? 99%
?? Maximum station utilization
u ? ???
System Goals for each capacity
(Equation 4.20)
?? MfgC ? ?
? d1 ? d1 ? 1 (Equation 4.25)
??
MfgC goal
CT
CT goal
? d 2 ? ? d2 ? ? 1 (Equation 4.26)
Bounds
?? 1? MS1 ? 5
?? 1? MS2 ? 5
?? 1? MS3 ? 5
?? di ? , di ? ? 0
di ? ?di ? ? 0
Minimize
The deviation function
Z mp = Z-SC+1 (Equation 4.28)
Figure 4.11 Baseline Compromise DSP for Production System Design
147

4.3.3 Results of Baseline Approach
In the present case example the deviation function is highly non-linear and exhibits various
minima. In general, for this kind of function it is impossible to know if an optimum is a global
optimum or not. In order to verify at some extent the quality of the optimization algorithm that is
used in this work, three different algorithms are compared, namely, Simulated Annealing (SAn),
a clustering-based algorithm (GLOBAL), and the Foraging-Adaptive Linear Programming
(FALP) algorithm.
The first algorithm, SAn, is solved using the commercial optimization software OptdesX ®
(Balling, et al., 1998). The parameters used in the optimization process are: 100 optimization
cycles, a probability of accepting a worse design at the beginning of the optimization of 0.5, and
a probability of accepting a worse design at the end of the optimization of 1.e-6. For all the
capacities the initial starting point is LT =20.5, Ne,T=670, Na,T= 670, ET=3, and AT=3. The
final values of the design variables and material cost for each refrigeration capacity are shown in
Table 4.10.
Table 4.10 Results for Each Capacity using Simulated Annealing
Tons LT (ft) AT ET Na,T Ne,T MC T (dollars)
600 17.01 2 2 443 453 115415
700 18.46 2 2 441 478 128348
800 19.45 2 2 464 457 135363
900 17.06 2 2 498 697 156150
1000 22.51 2 3 448 544 167891
1100 19.47 3 2 637 538 176333
1200 23.82 2 3 489 615 195988
148

1300 23.95 2 3 527 687 214625
Clustering-based algorithms are a modified form of the standard multi-start procedure.
Simply explained, in a clustering-based algorithm a local search starting from several points
distributed over the entire search domain is performed. The three main steps of the algorithm
are: (1) sample points in the search domain, (2) transform the sampled point to group them
around local minima, and (3) apply a clustering technique to identify groups that represent
neighborhoods of local minima. Here is used the GLOBAL routine of Csendes (Csendes,
1988) to solve the compromise DSP of Figure 4.9 with this technique. The local search
procedure used in GLOBAL is a Quasi-Newton type, which uses the so-called DFP (Davidon-
Fletcher-Powell) update formula. Note that the solutions obtained with this algorithm are
independent of initial conditions. Because the GLOBAL routine is for continuous variables only,
at each iteration of the continuous variables an exhaustive search on possible combinations of
tubes selection is performed. A penalty function method is used for constraining the solution.
The results obtained with this method are shown in Table 4.11.
Table 4.11 Results for Each Capacity Using a Clustering-based Algorithm
Tons LT (ft) AT ET Na,T Ne,T MC T (dollars)
600 17.04 2 2 476 440 115942
700 18.11 2 2 523 440 127445
800 18.26 4 2 504 494 141445
900 19.13 2 2 617 522 155719
1000 23.05 3 3 535 440 170238
149

1100 22.285 4 3 528 501 181691
1200 23.70 4 3 530 493 192033
1300 23.90 2 3 747 528 214321
The third algorithm, FALP (Lewis and Mistree, 1996), is a solution scheme for mixed
discrete/continuous design problems, which uses a successive approximation programming
method to constrain the solution. The discrete portion of the scheme is based on the notion of
foraging of animals and includes dynamic memory structure and schema identification. The
continuous portion of the scheme is solved using the Adaptive Linear Programming (ALP)
algorithm. Both the FALP and the ALP algorithms are part of the DSIDES software (Mistree,
et al., 1993). The initial design point for each capacity is the same as with the SAn algorithm
and the optimization is performed with a maximum of 100 cycles. The results are shown in
Table 4.12, where the solutions marked with the symbol “*” are infeasible solutions considering
an allowable relative constraint violation of 0.001. With SAn and the clustering-based algorithm
all the solutions are feasible under this criterion.
Table 4.12 - Results for Each Capacity using FALP
Tons LT AT ET Na,T Ne,T MC (dollars)
600 17.30 2 2 440 440 115120
700 18.95 2 2 463 474 131593
800 22.16 2 2 445 488 150806
900 20.51 4 2 463 467 149743*
1000 23.25 3 3 440 440 162973*
150

1100 24.00 3 3 440 440 170020 *
1200 23.05 4 3 486 486 182958 *
1300 22.61 3 3 578 578 203987 *
Observing Table 4.11 to 4.12, it is clear that the value of the design variables do not
follow monotonic behavior with increasing capacity requirements. It is also observed that, in
general, each algorithm reaches a different minimum. In Figure 4.12 it is shown the value of the
deviation function Z for each capacity.
Z
0.25
0.2
0.15
0.1
0.05
0
* Infeasible
*
600 700 800 900 1000 1100 1200 1300
Capacity
Sim.Ann.
GLOBAL
Figure 4.12 Comparison of the Value of the Deviation Function Z for each Capacity
Using SAn, GLOBAL and FALP Algorithms
Clearly, the solutions obtained for this particular case with SAn algorithm are generally
better than those obtained with the clustering-based method and FALP. Particularly, with the
FALP algorithm the solution tends to be infeasible. The reason is that the continuous part of
*
FALP
*
*
*
151

FALP, the ALP algorithm is more suitable for highly constrained problems, which is not the
case of this problem. Based on this observation, the SAn algorithm is used to solve subsequent
compromise DSPs.
The resultants MfgC and CT obtained by solving the compromise DSP shown in Figure
4.11 with SAn are presented in Table 4.13.
Table 4.13 - Results for Minimum Production Cost and Cycle Time
MS1 MS2 MS3 F
MfgC
(dollars)
CT
(hours)
Min COST 2 1 2 54 46,762 703
Min CT 5 5 5 54 65,680 448
With these values of MfgC and CT, and the targets of MfgCgoal=9810 and CTgoal=318.1,
obtained with Equations (4.23) and (4.24), a payoff matrix (Equation 4.27) is constructed as:
and the modified deviation function is
where
152
Po ? Z11 Z ?? ?? ?? 3.76 5.69??
12
?? ??? ?? ?? (4.35)
?? Z21 Z22 ?? ?? 1.21 0.41??
Z mp = Z - SC+1 (4.36)
Z ? 1
( Z
ˆ
1 ?
2 ˆ
Z 2) (4.37)
SC ? (1 ? ˆ
Z 1)(1 ? ˆ
Z 2) (4.38)
ˆ
Z 1 ? Z 1 ? 3.76
5.69 ? 3.76 (4.39)

153
Z
ˆ
2 ? Z 2 ? 0.41
1.21? 0.41 (4.40)
With the modified deviation function given in Equation 4.36, the compromise DSP shown in
Figure 4.11 is solved, and the results are shown in Table 4.14.
Table 4.14 Baseline Solution for Production System Design
Design Variables Performance Variables
MS1 MS2 MS3 F
MfgC
(dollars)
CT
(hours)
Z mp
3 2 3 54 52430 476 0.652
The results of CT that are shown in Tables 4.13 and 4.14 are those predicted with the
response surface network. As a means to validate this model of the production system, in
Tables 4.15 it is shown a comparison of the values predicted with this model and the results of a
discrete-event simulation, based on the values of length and number of tubes of Table 4.5. The
simulation was carried out using the commercial simulation software ARENA® (Kelton, et al.,
1998), with 20 replicates and a simulation time equivalent to 500,000 hrs (57 years) of
operation. The results of CT for the 20 replicates are shown in Table 4.16.
Table 4.15 Cycle Times in Hours Estimated with Response Surface Network
Model (RSNM) and Discrete Event Simulation (DES)
600 700 800 900 1000 1100 1200 1300 MEAN

RSNM 395 409 415 469 448 466 478 501 448
DES 404 413 424 459 448 467 471 490 448
Difference
%
2.2 1.0 2.1 2.2 0 0.2 1.5 2.2 0
Table 4.16 Cycle Times in Hours for 20 Replicates of Simulation
Rep CT Rep CT Rep CT Rep CT
1 439 6 440 11 451 16 450
2 448 7 446 12 448 17 442
3 445 8 443 13 457 18 449
4 453 9 453 14 450 19 453
5 436 10 453 15 453 20 456
154
In Table 4.15 it is observed that there is no significant difference between the estimations
of the Response Surfaces Network Model and Discrete Event Simulation.
Once the accuracy of the production system model and the quality of the algorithm for
solving the compromise DSPs have been verified with the data presented in this section, the
cooperative and non-cooperative models are applied to the design of the absorber/evaporator
module in Sections 4.4 and 4.5.
4.4 COOPERATIVE FORMULATION AND SOLUTION

In this section a full-cooperative model with a conventional (non-probabilistic) solution scheme
is applied in the design of the absorber/evaporator module. The objective in doing so is
threefold:
1. To verify the applicability of game theoretical representations for modeling enterprise
design processes that involve product design and production system design
(Hypothesis 1).
2. To examine how the Compromise DSP is an adequate decision model to embody a
cooperative design formulation (Hypothesis 2.2).
3. To develop information to be used in the validation of a non-cooperative solution.
In doing so, the enterprise design method described in Section 1.4.4 is applied step by step
within the framework constructed in this work, which is shown in Figure 3.3. The
complementary tasks shown in the right side of Table 4.17 are required to model the integrated
product and process design as a cooperative game (see Section 2.3).
Table 4.17 Implementation of Cooperative Formulation and Solution
Tasks Enterprise Design Method Complementary Tasks
» Baseline decision
1 Expand scope of decision ?? Identify players of the game,
2 Determine Input needed for Analysis ?? Define number of stages, K, and order
and Identify Modeling Techniques of moves
?? Define players’ set of choices, A
?? Identify public and private information
?? Identify probabilities on exogenous
events
3 Define boundaries of decision ?? Define players’ payoffs, fi(X)
155

?? Define strategies
4 Transform and integrate models
5 Generate potential solutions ?? Embody strategy with a compromise
DSP
» Recommendation
Following Table 4.17, the input required is the baseline decision for each of the
individuals involved in the design process. These baseline decisions are described in Section
4.3, and they are not repeated here. The remaining tasks are described in sections 4.4.1 to
4.4.5.
4.4.1 Task 1: Expand Scope of Decision
The baseline decisions described in Section 4.3 are expanded by three means:
1. By abstracting the design process as a game with multiple players, i.e., identifying other
agents of the enterprise whose interests, needs or constraints should be considered in
making a local decision. In this case, as explained in Section 4.1, there are 9 players
involved in the game (N=9): 8 product designers who have to design an
absorber/evaporator module for each capacity and a production system designer.
2. Adopting a cooperative protocol among all the players that they agree to respect in order to
act in a coordinated manner (this is akin to an implicit coordination mechanism, as discussed
in Section 3.1). In this case, a “full” cooperative protocol is adopted, in which the 9 players
merge their goals and models such that the problem is formulated and solved as a single
156
one. In order to do this they have to adopt system goals, such that the goals of the players

as a group are now to reach some specific targets of total cost (MC+MfgC) and cycle
time.
3. By adopting standardization of tubes for the entire family of absorber/evaporator modules,
such that there will be only one length, L, and tubes selection, E and A, for the 8 individual
products that comprise the family.
4.4.2 Task 2: Determine Input Needed for Analysis and Identify Modeling
Techniques
In the cooperative model that is studied in this section, the 9 players act as a unit and formulate
a single compromise DSP. Thus, the merging of the information required by all of the individual
players in the baseline decision is the information required by the coalition. This information is
presented in Section 4.3. Following Table 4.17, the complementary tasks required at this step
are as follows:
?? Define the order of moves. A full cooperative game is, by definition, a static game (K=1).
Therefore, there is only one movement at this stage, which is performed by the whole
coalition of players.
?? Define the players’ choices: A player’s set of choices (Ai) is embodied by the combination
of values that the design variables that the player controls can take within their bounds. In
this case example, for the cooperative protocol the only design variable is a vector X:
? L,
E,
A,
N , N , MS1,
MS2,
MS3,
F ?
X e,
T a,
T
157
? T=1,2,...,8 (4.41)

The bounds and possible values that the elements of this vector can take are the same as in
the baseline decisions (Tables 4.2, 4.3, 4.4 and 4.8). Note that there is only one length, L,
and tubes selections, E and A, but there are 8 different variables Ne,T and Na,T, one for each
capacity.
?? Define what each player knows when making choices. Because all the players act as a
unit, no player has private information. All the information presented during the formulation
of the baseline decisions in the previous section is available to the group as a whole.
Mathematically, if Ii represents the information set that is private to player i, the information
available to each player in the cooperative game is given by:
158
I i ? I j j ? 1,..., N (4.42)
?? Identify probability distributions over exogenous events. The only exogenous event
considered in this case is demand, which is considered as a Poisson process with a mean
arrival rate of 80 units/year, and with a distribution as shown in Table 4.1. Finally, the
modeling techniques to be used in the full cooperative formulation are the same as in the
baseline decisions.
4.4.3 Task 3: Define Boundaries of Decision
The boundaries of decision are embodied by the variables and parameters controlled by a
decision maker that can be altered within the restrictions posed by the organization, which in the
game theoretical framework being applied are defined as the set of choices A, identified in Task

2. In the cooperative case, the boundaries of decision of the group as a coalition (referred here
as Acoop) are simply the union of the individual sets Ai:
159
A coop ? A i ? A j ? i, j (4.43)
During Task 3 it is necessary to define the players’ payoffs as a function of the design
variables, fi( X) . As explained in section 2.3, the payoffs are defined in the framework that is
being applied as the value of the deviation function. In order to design the entire family
simultaneously, the goals are redefined as “aggregated” goals, i.e., the individual goals of the
players are merged into system goals as follows:
GoalAP ? ? Goal T DT (4.44)
T
where DT is the fraction of the demand associated with each capacity of refrigeration, shown in
Table 4.1 (Section 4.3), and the goals for each capacity, GoalT, are shown in Tables 4.5 and
4.8 (Section 4.3).
In this case, the goals are redefined as to achieve some targets of total cost and cycle
time, given the targets of material cost, production cost and cycle time presented in Section 4.3.
In this case the total cost is simply calculated as the sum of the material cost and production
cost.
COST = MC + MfgC (4.45)
Using Equations (4.44) and (4.45) the goals for the system synthesis are formulated as follows:
COST
COST AP, goal
? d 1 ? ? d1 ? ? 1 (4.46)

where the target for COST is calculated as:
And the goal for CT is formulated as:
160
COSTAP, goal ? ? (MCgoal ? MfgCgoal) T DT ? 160245
(4.47)
T
CT
CT goal
? d 2 ? ? d2 ? ? 1 (4.48)
The target of cycle time, CTgoal, remains the same as in the formulation of the baseline decision
for the production system design, i.e., CTgoal = 318.
As this is a two-objective problem, the modified approach of Rao and Freihet is applied.
The procedure is the same as in the synthesis of the production system design described in
Section 4.3, except that the production cost goal is now substituted with total cost. Therefore,
the payoff of each player, fi(X), is the same for all of them, and equal to a modified deviation
function Z mp :
where:
fi(X) = Z mp = Z - SC+1 (4.49)
Z ? W1 ˆ
Z 1 ? W2 ˆ
Z 2 (4.50)
SC ? (1 ? ˆ
Z 1)(1 ? ˆ
Z 2) (4.51)
ˆ
Z 1 ?
ˆ
Z 2 ?
Again, W1 and W2 are chosen to be equal to 0.5.
Z1 ? Z1( X1 * )
* * (4.52)
Z1 ( X2 ) ? Z1 (X1 )
Z2 ? Z2 (X2 * )
* * (4.53)
Z 2 ( X1 ) ? Z2 ( X2 )

With the information described up to this point, it is possible to formulate a design strategy
s. According to the definition given in Section 2.2.3 of a cooperative solution, the strategy s can
be simply stated as “find a design vector X * such that Z mp has the minimum possible value,”
which can be mathematically written as:
161
s : X * ? Z mp (X * ) ? Z mp ( X) (4.54)
where X is defined in Equation 4.41. The algorithm for implementing this strategy is shown in
Figure 4.13.
3
HEAT TRANSFER
ANALYSIS
MODEL
L, E, A, Tons, NeT, NaT
UAEVAP, UAABS, DP
1
L, E, A
2 DETERMINE
MINIMUM
NUMBER OF
TUBES
Tons = 600,..., 1300
4
OPTIMIZATION DRIVER
L, Ne, Na, E, A
MS1
MS2
MS3
F
PRODUCTION MODEL
INVENTORY MODEL
UAEVAP, UAABS, DP
Zmp,
Value of Constraints
5
Evaluate Zmp
Evaluate
constraints
u
COST
CT
Figure 4.13 Implementation of Design Strategy for Cooperative Model

An optimization driver, OptdesX® (Module 1), passes values of L and a selection of
tubes E and A, along with values of the production system variables MS1, MS2, MS3 and F.
Then, the minimum number of tubes that can satisfy the heat transfer requirements for each
capacity are calculated (or the number for which the violation of the constraint is as small as
possible). The search of the number of tubes (module 2 of Figure 4.13) is performed using a
bisection search method. Once the number of tubes for each capacity is obtained, the value of
the dependent variables COST and CT are evaluated in module 4, then the value of the
deviation function and the constraints is calculated in module 5, and finally the results are passed
to the optimization driver.
Given a design strategy and an approach to implement it, the next tasks in the enterprise
design method are to transform and integrate models, embody the design strategy in a design
formulation and to generate potential solutions. In this case no transformation of models is
applied and the complete models of all the players are used in the solution process. The
strategy is embodied in a compromise DSP as shown in Figure 4.14. The solution achieved
with the cooperative model is described in the next section.
162

163

Given
?? Heat Transfer Models
?? Production model (Section 4.3.2)
?? Inventory model (Section 4.3.2)
?? Commercial tube alternatives (Table 4.2)
?? A payoff matrix
Find
?? Design vector X=[L, E, A, NeT, NaT, MS1, MS2, MS3, F]; T=1,2,..,8
?? Deviation variables di ? ,di ? i=1,2
Satisfy
System Constraints
?? UAEVAP ? UAEVAP min (Table 4.5)
?? UAABS ? UAABS min (Table 4.5)
?? ? P ? ? Pmax (Table 4.5)
?? Fill Rate at S3 (for tubes safety stock)
S ? 99% (Equation 4.20)
?? Maximum station utilization
u ? ???
System Goals for each capacity
??
COST ? ?
? d1 ? d1 ? 1 (Equation 4.46)
??
COST APgoal
CT
CT goal
Bounds
? d 2 ? ? d2 ? ? 1 (Equation 4.48)
?? NaT min ? Na ? NaT max (Table 4.3)
?? NeT min ? Ne ? NeT max
?? Lmin ? L ? Lmax
?? 1? MS1 ? 5 (Table 4.8)
?? 1? MS2 ? 5
?? 1? MS3 ? 5
?? di ? , di ? ? 0
di ? ?di ? ? 0
Minimize
The modified deviation function Z mp (Equation 4.49)
Figure 4.14 Compromise DSP for Cooperative Model
164

4.4.4 Results of Cooperative Model
The compromise DSP shown in Figure 4.14 is solved using a Simulated Annealing algorithm.
The parameters for the optimization process are: 100 cycles, a probability of accepting a worse
design of 0.5 at the beginning of the optimization and a probability of 1.e-6 at the end. The
optimization process is carried out starting from three different initial design vectors, as shown in
Table 4.18. The final value of the design variables and the aggregated cost and cycle time are
shown in Table 4.19. The final number of tubes and the material cost for each capacity are
shown in Table 4.20.
Table 4.18 Initial Value of Design Variables for Optimization Process
Case L(ft) E A MS1 MS2 MS3 F
1 17 1 1 1 1 1 10.8
2 20.5 2 3 3 3 3 18
3 24 4 4 5 5 5 54
Table 4.19 Results of Cooperative Model
Case L(ft) E A MS1 MS2 MS3 F
COST
(dollars)
CT
(hours)
Z mp
1 19.45 2 2 3 2 2 54 220050 530 0.63
2 19.45 2 2 3 2 2 54 220050 530 0.63
3 22.95 3 4 3 2 2 54 219484 466 0.51
165

Table 4.20 Results of Cooperative Model for each Capacity
Case 1 Case 2 Case 3
Capacity Ne Na MC Ne Na MC Ne Na MC
(dollars)
(dollars)
(dollars)
600 440 440 124580 440 440 124580 440 440 150078
700 440 446 128488 440 446 128488 440 440 153578
800 457 504 138086 457 504 138086 440 440 157078
900 510 575 153120 510 575 153120 440 440 160578
1000 568 647 168854 568 647 168854 440 457 165639
1100 626 647 183839 626 647 183839 475 503 178984
1200 687 708 199952 687 708 199952 522 554 194717
1300 745 780 215687 745 780 215687 568 600 209829
It is observed that the algorithm found two different minima, one for Case 1 and Case 2,
and one for Case 3. Clearly, the solution obtained in Case 3 is the best of the three. With an
increment of only 0.9% in the total cost with respect to the other solution cycle time is reduced
69 hours (12%). In Figure 4.14 it is shown a plot of contours of the deviation function, Z mp ,
against the variables length and evaporator tube selection for the absorber tube selection A=2,
and A=4, considering other design variables fixed at the values given in Table 4.19.
In Figure 4.15 we can observe a strong similarity between the deviation function with
absorber tube selection A=2 and A=4, where the two minima found are present. Interestingly,
the minima are in the middle of the design space for these particular values of A, MS1, MS2,
166
MS3 and F. Further discussion of this solution is carried out in Section 4.7, where these results

are compared with those achieved with other approaches, and the fulfillment of the objectives
posed at the beginning of this section is examined.
A = 2
MS1 = 3
MS2 = 2
MS3 = 2
F = 54
EVAPORATOR TUBE
Design Solution
A = 2
17.0
CONTOURS Z mp
17.0 18.75 20.5 22.25 24.0
LENGTH
LENGTH
20 .5 24.0
22.25
Zmp
1
2
3
EVAPORATOR TUBE
a) Deviation Function with Absorber Tube 2
4
A = 4
MS1 = 3
MS2 = 2
MS3 = 2
F = 54
EVAPORATOR TUBE
A = 4
17.0
CONTOURS Z mp
17.0 18.75 20.5 22.25 24.0
LENGTH
LENGTH
20 .5
22.25
24.0
Z mp
1
2
3
EVAPORATOR TUBE
b) Deviation Function with Absorber Tube 4
Figure 4.15 Modified Deviation Function Z mp
4.5 NON-COOPERATIVE FORMULATION AND SOLUTION
In this section, the absorber/evaporator design problem is solved using a Stackelberg or
Leader/Follower game model between the product design group and the production system
4
167

designer. As explained in Section 2.3, the Stackelberg game is a particular class of non-
cooperative games. The objectives of this section are as follows:
1. To continue the verification initiated in Section 4.4 of the applicability of game
theoretical representations for modeling enterprise design processes that involve
product design and production system design (Hypothesis 1).
2. To verify that product design and production operations management, which are two
core activities of an enterprise, can be effectively coordinated using non-cooperative
formulations of individual design problems (Hypothesis 2.1).
3. To verify that the Compromise DSP is an adequate decision model to embody a non-
cooperative design formulation (Hypothesis 2.2).
The implementation is akin to that carried out in the cooperative formulation described in the
previous section. Again, the baseline decisions are those described in Section 4.3. The
remaining tasks are described in Section 4.5.1 to 4.5.4.
4.5.1 Task 1: Expand Scope of Decision
Similarly as in the cooperative formulation, the baseline decisions are expanded by four means:
1. By abstracting the design process as a multi-player game.
2. Adopting a game theoretical protocol to coordinate decisions. In this case, the 8 product
designers adopt a full cooperative protocol among them, and they, as a group, act as
168

leaders of the production system designer in a Stackelberg (Leader/Follower) game. This
protocol is illustrated in Figure 4.16.
3. By adopting standardization of tubes for the entire family of absorber/evaporator modules,
as in the cooperative case.
4. By adopting system goals in making “local” decisions. In this case, both the product design
group and the production system designer adopt total cost and cycle time as performance
measures.
Goal
Minimize Material Cost
Product Designers
600 700 800 900 1000 1100 1200 1300
Capacities
RRS f
DESIGN VARIABLES
DESIGN VARIABLES
Manufacturing Process
Designer
X lg =[L, E, A, N eT , N aT ] T=1,2,…,8
Goal
Minimize Production Cost
Minimize Production Time
X f =[MS1, MS2, MS3, F]
Figure 4.16 Stackelberg Protocol between Product Design and
Production System Design
169

4.5.2. Task 2: Determine Input Needed for Analysis and Identify Modeling
Techniques
Because the product design group and the production system designer do not act as a single
player, two sets of information have to be distinguished. The information required by the group
of product designers and the information required by the production system designer. For
simplicity, in this section and in Section 4.6, the group of 8 product designers is referred as the
leader group, and the production system designer as the follower. Once the players are
identified, in order to determine the input needed for analysis, the game is set up by the following
tasks:
?? Define the order of moves. In this case the game has at least two stages ( K ? 2). In the
first stage the leader group initiates the process while the follower does “nothing.” After a
design is obtained by the leader group in the first stage, in the second move the follower
designs a production system to meet the required demand using the information generated by
the leader group in the previous stage. In this example only two stages are considered, but
the leader group could proceed to more detailed stages of the design process (if necessary)
or they could wait until the follower provides them with “updated” information of the
production system.
170
k
?? Define the players’ choices. The set of choices at stage k for the leader group, Alg, are
defined as follows:
A
k
lg
k
k k
? X lg ? Slg
: X f ( X ) ? RRS ( X ) ?
? (4.55)
lg
f
lg

171
k k
where Xlg is the design vector of the leader group, and RRSf (Xlg)
is the rational reaction set
of the follower at stage k. In this case, Xlg is as defined as follows:
lg
? L,
E,
A,
N eT , NaT
?
X ? T = 1,2, ...,8 (4.56)
Note that the leader group coordinates decisions with the follower via the RRSf. The
k
point Xlg ? RRSf is by definition a point of Nash equilibrium. Thus, the “leader” restricts the
set of alternatives, Alg, to those that form Nash equilibrium with the follower. This is a hybrid
of the explicit and implicit coordination mechanisms discussed in Section 3.1. It is implicit
because the leader group agrees on a set of rules defined by the Stackelberg protocol and
restricts choices accordingly. It is explicit because the decision is based on an explicit
evaluation of alternatives that satisfy Equation (4.55).
On the other hand, the follower’s options for this specific case are defined as:
Af k ?
??
??
? k ? 1
?? k k ?1
Af ? Xlg
?? ? k ? 1
(4.57)
Equation (4.57) means that at stage 1 the only alternative of the follower is “do nothing,”
whereas for later stages the set of choices depends only on the absorber/evaporator design,
k?1
Xlg , obtained by the leader group at the previous stage.
k
?? Define what each player knows when making choices. Let I lg be the set of information
that is private to the group of product designers. This set is the union of the information sets
of the players that comprise the group. The information available to the leader group at stage
k in the Stackelberg protocol is at least , but not restricted to:

172
k k
Ilg ? RRSf
(4.58)
In this work only two stages are considered, thus only the RRSf for stage k=1 is needed.
From this point forward the convention that RRS refers to
1 RRS f is adopted.
For the production system designer, the follower, the information available at the
beginning of stage k (for k>1) is as follows:
k? 1
If ? Xlg (4.59)
where If is the private information of the follower, described in Section 4.3.2.
?? Identify probability distributions over exogenous events. This is the same as in the
cooperative model.
Finally, the modeling techniques to be used in the non-cooperative model for each
group are the same as in the baseline decisions. The only addition is that, according to the
procedure described in Section 3.5.3, the RRS is modeled using response surfaces. The
construction of the RRS is described at the end of the present section.
4.5.3 Task 3: Define Boundaries of Decision
In the non-cooperative case, the boundaries of decision of the leader group (Alg) are simply the
union of the individual sets of the 8 different product designers:
A lg ? A i ? A j , i, j ? f (4.60)
where Alg is given in Equation (4.55). For the follower, the boundaries of decision are simply
the alternative choices Af given by Equation 4.57.

During Task 3 it is necessary to define the payoffs of the players as function of the
design variables. In this case, the goals and deviation functions for both the leader group
mp mp
( Zlg ) and the follower (Zf ) are the same as in the cooperative case (Equations 4.46, 4.48,
mp mp mp
and 4.49). Thus, Zlg ? Z f = Z .
Now it is possible to define a design strategy for the leader group, slg, and the follower, sf.
According to the definition of a Stackelberg solution, (Section 2.3.2), the strategy for the leader
is mathematically written as:
Similarly, the follower’s strategy is:
work.
173
* mp * * mp
slg : Xlg ? Z (Xlg,RRS
f ( Xlg )) ? Z ( Xlg, RRSf (Xlg)) (4.61)
s
f
: X
* f
mp * mp
? Z ( X , X ) ? Z ( X , X )
(4.62)
* f
In Figure 4.17 it is shown the implementation of these two strategies followed in this
lg
f
* lg

2
3
1
FOLLOWER
LEADER GROUP
FOLLOWER
CONSTRUCT RRS
RRS
DETERMINE Xlg
Xlg
DETERMINE Xf
1.1
1.2
1.3
2.1
2.2
L,E,A
2.3
3.1
3.2
DOE Driver
Xl g
Optimization Software
Evaluate Z
Xf
Production Model
mp
Zmp
1.4
Evaluate constraints
Value of Constraints
u
CT,
MfgC
Inventory Model
Optimization Software
Ne,N a
Determine Minimum
Number of Tubes
Tons = 600,..., 1300
Xlg constraints
Heat Transfer Model
Optimization Software
Xf
Production Model
Inventory Model
MfgC, CT
Xlg
constraints
Z mp
Value of Constraints
2.4
Zmp
Value of Constraints
3.3
Evaluate Z mp
Evaluate constraints
Evaluate Z mp
Evaluate constraints
u
CT,
MfgC
Figure 4.17 Implementation of Design Strategies for Non-Cooperative Model
174

In Module 1, the follower constructs the RRS, following the procedure given in Section
3.5.3, as follows:
?? Design an experiment for constructing response surfaces of MfgC and CT as functions of
Xlg using a DOE driver (Module 1.1).
?? For each level of the experiment find the “optimal values” of MfgC and CT using an
optimization algorithm (Module 1.2), and appropriate analyses routines (Modules 1.3 and
1.4).
?? Fit response surfaces to the values of MfgC and CT obtained in step 3 using the DOE
driver. The RRS is the vector [MfgC(Xlg), CT(Xlg)], where MfgC(Xlg) and CT(Xlg) are
the response surfaces.
175
Note than in this example, the RRS of the follower is not the design vector Xf(Xlg), i.e.,
the values of MS1, MS2, MS3 and F as functions of Xlg, but the resulting values of the
responses MfgC and CT of the follower. The reason to do so is that the leader group does not
actually need for its calculations to know the value of the follower’s design variables. They only
need from the follower the value of CT and MfgC in order to evaluate a payoff function. Using
MfgC and CT as the elements of the RRS vector greatly expedites computation during the
product synthesis.
The compromise DSP used in step one is the same as in the baseline decision for the
production system design, illustrated in Figure 4.10 (Section 4.3). The resultant response
surfaces are:

MfgC? 49263 ? 1215L C ? 442E C ? 412 A C ? 2451NE C ? 2208NA C
CT ? 505.
5 ? 11.
1L
- 365 LC EC ? 684 LC AC ? 479LC NAC ? 357EC AC
? 376E C NE C ? 798E C NA C ? 686A C NE C ? 824A C NA C
- 504NEC 2 - 392NEC NAC -345NAC 2
? 3.
63E
C
2
C
? 0.
1E
? 3.
8E
C
C
NA
? 26.
7 A
C
C
? 9.
4NE
? 23.
41NE
2
C
C
? 15.
4NE
? 3.
0L
where LC, EC, AC, NEC and NAC are normalized values obtained as follows:
NEC ?
NAC ?
LC ?
L ? 17
1.75
?? ?2, if E ? 1
??
?? ?1, if E ? 2
EC ? ??
?? 1, if E ? 3
??
?? 2, if E ? 4
?? ?2, if E ? 1
??
?? ?1, if E ? 2
EC ? ??
?? 1, if E ? 3
??
?? 2, if E ? 4
?
T
?
T
C
NA
C
C
E
C
? 4.
5NA
2
C
176
(4.60)
(4.61)
- 2 (4.62)
(4.63)
(4.64)
NeT DT ? 440
115
? 2 (4.65)
N aTD T
115
? 440
? 2 (4.66)
By using Equations (4.62) to (4.66) the value of all the variables for fitting a response
surface are in the same range of [-2,2]. With Equations (4.65) and (4.66), the number of
variables for fitting the response surfaces are reduced from 19 (considering the number of tubes
for both the absorber and evaporator for the 8 different capacities, i.e., 16 variables) to 5, with

the consequent reduction in the size of the experiment, and therefore, in the time and
computational effort required to build the response surfaces. In Figures 4.18 and 4.19, and in
Tables 4.21 to 4.24, it is verified that these response surfaces have an adequate accuracy for
practical purposes.
In Table 4.21 it is shown the ANOVA results for the MfgC fitting with a confidence level
of 99%. The hypothesis that the model adequately fits the data is accepted under this level of
confidence. The adequacy of the fit can also be observed in Table 4.22 where it is shown the
R 2 value, which can be loosely interpreted as the proportion of the variability in data that can be
“explained” by means of the response surface. Clearly, the R 2 value of 95% is appropriate for
product design purposes.
Table 4.21 ANOVA Table for MfgC Regression
Source DF Sum of Squares Mean Square F Ratio Prob>F
Model 17 384.5e+06 22.6e+06 11.6 0.0004
Error 9 17.6e+06 1.9e+06
Total 26 402.0e+06
Table 4.22 Summary of Fit for MfgC
R 2 0.956
Root Mean Square Error 139
Mean of Response 48426
Observations 27
177

MfgC
55000
52500
50000
47500
45000
42500
40000
40000 42500 45000 47500 50000
MfgC Predicted
52500 55000
Figure 4.18 Fit of MfgC Regression
The response surface for CT is also adequate for approximating the CT, as shown in
Figure 4.19, and Tables 4.23 and 4.24. Again, the R 2 value is around 0.95.
Table 4.23 ANOVA Table for CT Regression
Source DF Sum of Squares Mean Square F Ratio Prob>F
Model 10 40967 4097 42

Xlg:
CT
600
550
500
450
450 500 550 600
CT Predicted
Figure 4.19 Fit of CT Regression
179
Once the RRS is built and validated, the leader group uses the RRS to find a design vector
Xlg=[L,E,A,NeT,NaT], T=1,2,...,8 (4.67)
An optimization driver (Module 2.1) passes values of L, E, and A, to a subroutine (Module 2.2)
that searches the minimum number of tubes for each capacity that satisfies the constraints or that
violates them the least. Then, the values of Xlg are passed to Module 2.3 where the deviation
function, Zmp, is evaluated.
Once the design vector Xlg is obtained by the leader group using the previous
implementation, the follower follows a similar approach to find the values of Xf.
Given these design strategies, both the leader group and the follower embody their
strategies in a compromise DSP. The compromise DSP for the leader group is shown in

Figures 4.20. The follower’s compromise DSP is the same one shown in Figure 4.10 (Section
4.3).
Given
?? Heat-transfer models of the absorber-evaporator
?? Commercial tube alternatives (Table 4.2)
?? Rational reaction set of follower RRS (Equations 4.60 & 4.61)
?? A payoff matrix P
Find
Design Vector Xlg=[L,E,A,NeT,NaT]
?? Length of tubes, L
?? Selection of tube for the evaporator heat-exchanger, E
?? Selection of tube for the absorber heat-exchanger , A
?? Number of tubes of the evaporator for each capacity , NeT; T=1,2,...,8
?? Number of tubes of the absorber for each capacity , NaT; T=1,2,...,8
?? Deviation variables di ? ,di ? ; i=1,2
Satisfy
System Constraints
?? UAEVAP ? UAEVAP min (BTU/ hr o F) (Table 4.5)
?? UAABS ? UAABS min (BTU/ hr o F) (Table 4.5)
?? ? P ? ? Pmax (Table 4.5)
?? X f (X lg) ? RRS( X lg)
System Goals
COST
??
??
COST APgoal
CT
CT goal
Bounds
? d 1 ? ? d1 ? ? 1 (Equation 4.46)
? d 2 ? ? d2 ? ? 1 (Equation 4.48)
?? NaT min ? Na ? NaT max (Table 4.3)
?? NeT min ? Ne ? NeT max
?? Lmin ? L ? Lmax
?? di ? , di ? ? 0
?? di ? ?di ? ? 0
Minimize
?? The modified deviation function
Z mp (Equation 4.49)
180

Figure 4.20 Stackelberg Compromise DSP for Product Design
181

4.5.4 Results of Non-Cooperative Model
The compromise DSPs for the non-cooperative model are solved using a Simulated Annealing
algorithm, as in the baseline and cooperative solutions. The parameters for the optimization
process are the same as in the cooperative model to facilitate comparison of results in Section
4.7. The optimization process is initiated from three different points, shown in Table 4.25. In
Table 4.26 it is presented the material cost (MC), the total cost (COST) and cycle time (CT)
“predicted” using the RRS, and the value of the deviation function, Z mp , for each solution. The
number of tubes and the material cost for each capacity are shown in Table 4.27 (using the L
value obtained in Case 1).
Table 4.25 Initial Value of Design Variables for Optimization Process
Case L(ft) E A
1 17 1 1
2 20.5 2 3
3 24 4 4
Table 4.26 Results of Product Design: MC and “Predicted” Values of MfgC, COST
and CT
Case L(ft) E A
MC
(dollars)
MfgC
(dollars)
COST
(dollars)
CT
(hours)
Z mp
1 19.24 2 2 167241 49430 216413 461 0.380
2 19.25 2 2 167901 49512 216613 462 0.386
3 19.27 2 2 168201 49553 217754 462 0.396
182

Table 4.27 Results of Non-Cooperative Model for Each Capacity
Case 1
Capacity Ne Na MC
600 440 440 123216
700 440 446 127118
800 457 518 137556
900 518 590 153468
1000 579 647 168376
1100 636 719 183791
1200 697 791 199703
1300 758 863 215615
In Table 4.26 it is observed that the solution practically converges to the same point,
starting from completely different points of the design space. This is probably due to the use of
second order response surfaces to model the RRS of the production system designer, which
tends to “smooth” the deviation function.
In Table 4.28, it is shown the values of MfgC, COST and CT obtained by the production
system designer using the values of the product design variables of Case 1.
Table 4.28 Production System Design Results with Non-Cooperative Model
MS1 MS2 MS3 F MfgC CT
3 3 3 54 51238 468
183

It is observed that the difference between the values of production cost and cycle time
predicted by the leader group (Table 4.26) and the final values of MfgC and CT estimated by
the follower (Table 4.28) are very similar, with an error of 3.4% for MfgC. Given this error,
the design process can be carried out again, with an “updated” RRS approximated around the
product design vector, Xlg, obtained in the first stage, or, if the cost and/or time that this
redesign process requires are not justified considering this magnitude of error, the design
process continues to more detailed stages. Generally, differences in the estimate of different
agents are unavoidable if the exact RRS are not known and have to be approximated by
response surfaces or any other means. Thus, iteration and negotiation between agents is
expected and desirable, the difference being now that the negotiation is based on quantitative
estimates obtained in a consistently manner. In Section 4.6 the non-cooperative model is
modified with the probabilistic design model discussed in Section 3.6, such that differences on
estimations and other sources of uncertainty and imprecision are formally considered during the
product and manufacturing process design. A comparison of results with other approaches and
further discussion is carried out in Section 4.7.
4.6 NON-COOPERATIVE FORMULATION AND SOLUTION WITH
PROBABILISTIC DESIGN MODEL
The probabilistic design model presented in Section 3.5 is now applied to the development of a
flexible design solution of the absorber/evaporator module. The motivation is threefold:
184

1. To “capture” and deal with uncertainty and imprecision that results from the use of
approximations
2. To achieve an absorber/evaporator module design that is robust to external
variability, such as changes in the cost of raw material, and
3. To achieve a flexible design solution of the absorber/evaporator module to facilitate
coordination of product design and the production system design, thus verifying
Hypothesis 3.
The only difference with the process studied in Section 4.5 is in the formulation of the
compromise DSPs of the leader group (Figure 4.19) and the follower (Figure 4.10). The
necessary modifications are carried out according to the procedure described in Section 3.6.6,
and the probabilistic compromise DSPs are shown in Figures 4.22 and 4.23.
For the product design:
(1) In Section “Given” of the compromise DSP include:
?? Preference functions for the ranged performance variables.
The preference functions for this case study are defined as shown in Figure 4.21.
185

P(COST)
1.0
0.0
160245
COST (dollars)
320000
P(CT)
1.0
0.0
318
CT (hours)
Figure 4.21 Preference Functions for COST and CT
For the variable COST, a linear preference function is defined as follows:
618
186
? 1
if 160245 ? COST
? 320000 ? COST
P ( COST ) ? ?
? 320000 ? 160245
? 0
if 160245 ? COST ? 320000 (4.68)
otherwise
where “160245” is the value of the COST target and “320000” is some higher value,
such that a design that results in COST within this range is acceptable. Similarly, for CT
the preference function is defined as:
? 1
if 318
? COST
? 600 ? CT
P ( CT ) ? ?
? 600 ? 318
? 0
if 318
? COST ? 600
(4.69)
otherwise
?? Probability density functions for COST and CT. To simplify calculations, it is assumed a
normal distribution of the probability density functions of these two variables.
Accordingly, they are defined as follows:
2
1
?
?
? 1 ? COST ? COST ?
f ( COST ) ? exp ? ?
? ?
(4.70)
? 2?
? 2 ?
? ?
COST ? ? ? COST ? ?

(2) In Section “Find”:
187
2
1
?
?
? 1 ? CT ? CT ?
f ( CT ) ? exp ? ? ?
?
?
? ? ?
(4.71)
? 2?
2
? ? ?
CT
CT ? ?
?? Substitute the design variables L, Ne,T and Na,T, with the average values,
L , N
e,
T
and N
a T,
(3) In Section “Satisfy”:
(3.1) For system constraints:
, and their half-width variation, ? L, ? Ne,T and ? Na,T.
?? Include lower limiting values for the deviation of L:
? L ? 0.25 (ft) (4.72)
?? Add a limiting value for the deviation of performance of COST and CT. It is chosen to
limit the change of COST and CT as 10% of the average value within the design range
as follows:
? COST ? 0.10 COST (4.73)
? CT ? 0.10 CT (4.74)
?? Define the constraints using a statistical feasibility formulation as:
UA ? 3 ? ? UA
(4.75)
EVAP
ABS
UA,
EVAP
UA,
ABS
EVAP,
MIN
UA ? 3 ? ? UA
(4.76)
? ? ? ? P
ABS,
MIN
P 3 ? ? P
(4.77)
where the constraint values UAEVAP,MIN, UAABS,MIN, ? PMAX have the same values as in
the baseline decision formulation.
MAX

(3.2) For system goals:
?? Redefine the system goals as:
188
? ?
DPICOST ? d ? d ? 1
(4.78)
1
1
? ?
DPICT ? d ? d ? 1
(4.79)
where the DPI of the variables COST and CT is calculated as:
2
COST?
? COST
2
DPI COST ? ? P(
COST ) f ( COST ) d(
COST )
(4.80)
COST?
? COST
CT ? ? CT
DPI CT ? ? P(
CT ) f ( CT ) d(
CT )
(4.81)
CT ? ? CT
The integral of Equations (4.80) and (4.81) is evaluated numerically during the
optimization process.
(3.3) In Section “Bounds:”
?? Define the bounds for L, Ne,T and Na,T as follows:
e,
T MIN
LMIN MAX
? ? L ? L ? L ? ? L
(4.82)
N ? ? N ? N ? N ? ? N
(4.83)
a,
T MIN
e T,
a,
T
e,
T
a,
T
e,
T MAX
N ? ? N ? N ? N ? ? N
(4.84)
a , T MAX
The aforementioned changes are required for the transformation of the compromise DSP of
Figure 4.19 into a probabilistic formulation. Besides these changes, in the formulation of the
probabilistic design problem it is considered that there are two sources of uncertainty, which are
not considered in the conventional compromise DSP: (1) uncertainty regarding the cost of the
e,
T
a,
T

tubes, and (2) uncertainty that results from approximating the RRS with response surfaces. The
first one is included as a means to illustrate the effect of uncertainty caused by factors that are
external to the enterprise, whereas the second is included as a means to design considering that
there are errors in the values of MfgC and CT predicted during product design.
To include these considerations the following changes are done:
?? In section “Given” include a range for deviation of variables or factors due to
189
uncertainty. In this case example it is considered that the cost of evaporator tubes might
vary within a range of ? 0. 5dollars
around the nominal value, and the absorber tubes
within a range of ? 0. 25 dollars. Also, it is assumed that the relative error of MfgC
calculated with Equation (4.60) is around 10%, and the error for CT calculated with
Equation (4.61) is around 5%.
During the optimization process, for each combination of the design variables that is evaluated, a
CCD experiment is performed considering all the possible deviations of factors and variables,
and the standard deviation of any variable is approximated by one sixth of the maximum and
minimum values of the variable obtained in the experiment:
?
VARIABLE
?
1
6
? max ? VARIABLE CCD ? ? min?
VARIABLE CCD ?
And the deviation of constraints and performance variables is then approximated by:
(4.85)
? UAEVAP ?3? UA,EVAP (4.86)
? UAABS ?3? UA,ABS (4.87)
? P ?3? ? P (4.88)

190
? COST ?3? COST (4.89)
? CT ?3? CT (4.90)
These considerations apply also in the formulation and solution of a compromise DSP for
the production system design, shown in Figure 4.23. Besides the reformulation of the
compromise DSP the strategy and implementation is the same as that carried out in Section 4.5.

Given
?? Heat-transfer models of the absorber-evaporator
?? Commercial tube alternatives (Table 4.2)
?? Rational reaction set of follower RRS (Equations 4.60 & 4.61)
?? Preference function for COST (Equation 4.68)
?? Preference function for CT (Equation 4.69)
?? Probability density function for COST (Equation 4.70)
?? Probability density function for CT (Equation 4.71)
?? Half-width ranges for cost of tubes
? C e=0.5 (dollars)
? C a=0.25 (dollars)
?? Assumed half-width range of error of the RRS:
? MfgC=0.1MfgC
? CT=0.05CT
?? A payoff matrix P
Find
?? Design vector X lg=[L,E,A,N eT,N aT,? L,? N e,T,? N a,T]; T=1,2,...,8
?? Deviation variables di ? ,di ? ; i=1,2
Satisfy
System Constraints
?? ? L ? 0.25 (ft) (Equation 4.72)
?? ? COST ? 0.10 COST (Equation 4.73)
?? ? CT ? 0.10 CT (Equation 4.74)
?? UA EVAP ? 3 ? UA,
EVAP ? UAEVAP,
MIN
(Equation 4.75)
UA ? 3 ? ? UA
(Equation 4.76)
?? ABS UA,
ABS ABS,
MIN
?? ? P ? ? ? P ? ? PMAX
?? Xf (X lg) ? RRS( Xlg) 3 (Equation 4.77)
System Goals
??
? ?
DPI COST ? d1
? d ? 1 1
(Equation 4.78)
??
? ?
DPI CT ? d ? d ? 1
(Equation 4.79)
Bounds
2
?? ? L ? L ? L ? ? L
2
LMIN ? MAX
(Equation 4.82)
N eT,
MIN ? ? N eT,
? N eT,
? N e,
T MAX ? ? N e
(Equation 4.83)
N ? ? N ? N ? N ? ? N
(Equation 4.84)
?? T,
?? a T, MIN a T, a T, a,
T MAX a T,
?? di ? , di ? ? 0
di ? ?di ? ? 0
Minimize
?? The modified deviation function Z mp (Equation 4.49)
Figure 4.22 Stackelberg Probabilistic Formulation for Product Design
191

Given
?? Production System Model
?? Inventory Model
?? Replenishment frequency alternatives
?? Product Design Vector X lg=[L,E,A,N eT,N aT,? L,? N e,T,? N a,T]; T=1,2,...,8
?? Preference function for COST (Equation 4.68)
?? Preference function for CT (Equation 4.69)
?? Probability density function for COST (Equation 4.70)
?? Probability density function for CT (Equation 4.71)
?? Half-width ranges for cost of tubes
? C e=0.5
? C a=0.25
?? A payoff matrix P
Find
?? Production system design vector Xf=[MS1,MS2,MS3,F]
?? Deviation variables di ? ,di ? ; i=1,2
Satisfy
System Constraints
?? ? COST ? 0.10 COST (Equation 4.73)
?? ? CT ? 0.10 CT (Equation 4.74)
?? ? u ? 3 ? u ? 0.
9
System Goals
??
? ?
DPI COST ? d1
? d ? 1 1
(Equation 4.78)
??
? ?
DPI CT ? d ? d ? 1
(Equation 4.79)
Bounds
?? 1? MS1 ? 5
?? 1? MS2 ? 5
?? 1? MS3 ? 5
?? di ? , di ? ? 0
di ? ?di ? ? 0
Minimize
2
2
?? The modified deviation function Z mp (Equation 4.49)
Figure 4.23 Stackelberg Probabilistic Formulation for Production System Design
192

The solution of the compromise DSP of product design (Figure 4.22) is shown in Table
4.29, along with the average values of MC, and predicted MfgC , COST and CT . The range
of the number of tubes and material cost for each capacity are shown in Table 4.30. The
production system variables and the average values of MfgC, COST and CT obtained by the
production system designer are shown in Table 4.31.
Table 4.29 Results of Product Design: MC and
Predicted Values of MfgC , COST and CT
L (ft) E A MC
dollars
MfgC
dollars
COST
dollars
CT
hours
? COST
dollars
? CT
hours
193
mp Z
18.6-19.39 2 2 175043 46873 221916 460 21370 11.5 0.39
Table 4.30 Results of Non-Cooperative Probabilistic Model for Each Capacity
Capacity Ne Na MC(dollars)
600 440-440 440-440 120928-124272
700 440-440 446-468 126253-129179
800 453-468 518-540 138200-139528
900 511-529 575-611 152873-155522
1000 572-590 647-683 168941-171460
1100 629-651 719-755 184505-187520
1200 690-715 780-827 199827-203095
1300 752-777 852-899 216021-219154
Table 4.31 Production System Design with Probabilistic Non-Cooperative Model

MS1 MS2 MS3 F MfgC
dollars
COST
dollars
CT
hour
s
? COST
dollars
? CT
hours
mp Z
194
3 3 3 54 51367 220115 471 2335 2 0.91
In Tables 4.29 and 4.31 is observed that the difference between the cost and the cycle
times estimated during product design with the RRS is less than 1% for COST and around 2.5%
for CT. Note that the largest deviation of the COST from the average within the interval would
be of 21370 dollars or 10%. Similarly, the largest change for CT would be of 2.5%. That is, if
the variable length were not chosen to be 19 ft (the average) but some other value within the
interval [18.6-19.39] then the COST would not vary more than 10% and the CT more than
2.5%. This is further discussed in the next section where these results are compared with those
achieved in the previous sections.
4.7 COMPARISON OF SOLUTIONS
In Section 4.3 to 4.6 four different solutions are obtained for the design of the
absorber/evaporator module. In Figure 4.23 it is shown the deviation of the material cost from
the original targets for the various refrigeration capacities obtained with each solution
(Z=MC/MCgoal-1). The value shown for the probabilistic solution is that estimated with the
average length of 19 ft. Interestingly, in the cooperative solution there is a large increment of the
material cost for the smallest capacities. This effect is even more evident in the average or
“aggregated” cost, which is shown in Figure 4.24. As expected, the baseline solution in which
each product designer strikes at minimizing the cost of material for each capacity is the one with

the lowest material cost, but the solutions achieved with the standardization of tubes, either
cooperative or non-cooperative, are almost as good as the baseline one, the largest difference
being approximately of 8000 dollars (4%).
Z MC
0.6
0.5
0.4
0.3
0.2
0.1
0
600 700 800 900 1000 1100 1200 1300
Capacity
Baseline
Coop
Conv-NC
Prob-NC
Figure 4.23 Deviation from Material Cost Target for Each Capacity
190000
187500
185000
182500
180000
177500
175000
Baseline Coop Conv-NC Prob-NC
Figure 4.24 Average Material Cost (in Dollars) of the Various Solutions
195

In Figure 4.25 it is illustrated the average production cost. In this case, it is the
cooperative solution (Coop) the one with the lowest cost, and the baseline is the one with the
highest cost, the difference being approximately 6,000 dollars/unit. However, when considering
total cost the baseline solution is slightly better. This is shown in Figure 4.26, where it is
observed how both the baseline and the conventional non-cooperative (Conv-NC) solutions
achieve a lower average total cost than the cooperative one. The probabilistic non-cooperative
(Prob-NC) solution, for the mean length of 19 ft, has a cost as high as the one of the
cooperative solution. However, it has to be remembered that the mean value is not necessarily
the best value of the interval of solution. Actually, the solution obtained with the conventional
non-cooperative model (L=19.14) is within the interval of solution obtained with the
probabilistic model (18.6? L ? 19.39).
53000
52000
51000
50000
49000
48000
47000
46000
45000
44000
43000
42000
Baseline Coop Conv-NC Prob-NC
Figure 4.25 Average Production Cost (in Dollars) of the Various Solutions
196

221000
220000
219000
218000
217000
216000
215000
Baseline Coop Conv-NC Prob-NC
Figure 4.26 Average Total Cost (in Dollars) of the Various Solutions
The values of average production time estimated for the various solutions are shown in
Figure 4.27. The cooperative solution is in this case much better than the others. Particularly,
the time estimated in the non-cooperative solutions and the baseline one are 40 hours above.
480
470
460
450
440
430
420
410
400
Baseline Coop Conv-NC Prob-NC
Figure 4.27 Average Production Time (in Hours) of the Various Solutions
197

In Figure 4.28 is shown the deviation of total cost and cycle time from the original
targets (Z=Variable/Target –1 .0). It is observed that the difference between the cost
achieved between the various solutions is minimal but it is in the reduction in cycle time that the
cooperative solution overrides the others. The performance of the non-cooperative solutions is
very similar to the baseline one. However, the real benefit of the approach is in the reduction of
components’ variety. With the baseline solution there is a large variety of tubes. The
manufacturer would have to ask the supplier to provide 8 different tubes for the absorber and 8
different tubes for the evaporator, thus increasing the complexity of the system. On the other
hand, with the cooperative and non-cooperative solutions there is a single combination of length
and tubes selection for the evaporator and one for the absorber. This advantage is evident in
Table 4.32, where the required safety stock for each solution is presented.
Z
0.6
0.5
0.4
0.3
0.2
0.1
0
Baseline Coop Conv-NC Prob-NC
COST
Figure 4.28 Deviation from Targets of Total Cost and Production Time of the
Various Solutions
CT
198

Table 4.32 Safety Stock of Evaporator and Absorber Tubes in Number of Tubes
Baseline Cooperative Conv-NC Prob-NC
SE SA SE SA SE SA SE SA
600 1577 1542
700 2124 1959
800 1983 2013
900 2426 1733 4561 4715 5561 6187 5673 6318
1000 2417 1990
1100 2558 3029
1200 3325 2643
1300 2249 1725
Total 18659 16634 4561 4715 5561 6187 5673 6318
Investment
(dollars)
2892530 1343690 770800 455474 727781 436173 736957 442260
The number of tubes to keep as safety stock, in order to have the desired fill rate of
0.99 at station S3, is approximately three to four times the inventory required with the other
solutions. Clearly, the material handling and storage is facilitated with this reduction. The
difference in the investment in safety stock is also important. Due to the variety of tubes, the
safety stock in the baseline solution is as large as 3,300,000 dollars, whereas for the other
solutions is in the order of 1,100,000 dollars.
199
An important issue is the sensitivity of the solutions to future changes of the design
variables, particularly the length. The sensitivity of the total cost and cycle time for an interval of
? 0. 38 ft around the value of the length of 22.95 ft obtained with the cooperative model are
shown in Figure 4.29. Similarly, the sensitivity of the probabilistic solution is shown in Figure
4.30. The cooperative solution is, as expected, more sensitive to changes on the length,
whereas the probabilistic solution shows a smooth monotonic behavior within the range.

COST (dollars)
220600
220400
220200
220000
219800
219600
219400
219200
219000
218800
22.57 22.722 22.874 23.026 23.178
Length (ft)
Cost
CT (hours)
Figure 4.29 Sensitivity of Cooperative Solution to Changes in the Length
COST (dollars)
221500
221000
220500
220000
219500
219000
218500
218000
217500
Cost
CT
Cooperative
Solution
Range of Probabilistic Solution
18.62 18.772 18.924 19.076 19.228
Length (ft)
CT
Non-Probabilistic
Solution
455
450
445
440
435
430
425
420
CT (hours)
Figure 4.30 Sensitivity of Probabilistic Non-Cooperative Solution
to Changes in the Length
480
475
470
465
460
455
200

study:
In summary, the following observations are made regarding the results obtained in the case
?? On the game theoretical representation of the integrated product and process design
(Hypothesis 1).
The integrated design process of the absorber/evaporator module was successfully modeled
using either a cooperative or a non-cooperative model depending on the desired kind of
coordination. The mathematics of game theory provided an appropriate framework for
expanding decisions and identifying design strategies within the integrated design process.
?? On the use of a non-cooperative formulation to coordinate decisions between product
design and manufacturing process design (Hypothesis 2.1).
The solution achieved with a cooperative formulation is clearly better than the one achieved
by coordinating the integrated product and process design using a non-cooperative
formulation. Even when it is expected that the solution achieved by integrating all the
models into a joint solution be better, at least for small problems, the use of a second order
response surface to approximate the rational reaction set of the production system designer
contributes to this deviation of results because local minima are “lost” with the
approximation. However, it is also clear that with the non-cooperative formulation a
solution was achieved that was as good as the baseline and the cooperative solutions in
terms of total cost, and better than the baseline one in terms of reduction of cycle time and
components’ variety.
201

It has to be remarked that the cooperative solution is very good for the whole system
performance, but it is so by requiring a large “loss” for some of the individual products (see
Figure 4.23), whereas the non-cooperative solution obtained is more equitable for the
various product designers, thus achieving a better balance between their interests.
?? On the use of the Compromise DSP to embody the cooperative and non-cooperative
formulations (Hypothesis 2.2).
The Compromise DSP proved to be a very flexible decision model, which can be easily
used to formulate cooperative or non-cooperative design strategies. The goal-oriented
structure of this formulation provided an adequate model to embody the integrated product
and process design of the absorber/evaporator module.
?? On the use of a probabilistic design model to enhance the coordination of product
design and manufacturing process design of the absorber/evaporator module
(Hypothesis 3).
202
The interval representation of the design solution allows the group of product designers
to maintain their design freedom while enabling the production system designer to have a
preliminary design of the production system that is robust within the interval of solution.
There are various advantages of the probabilistic approach:
?? It was possible to capture uncertainty associated with factors that are external to the
enterprise.

203
?? It was possible to capture uncertainty associated with inaccuracy of the RRS
approximations.
?? It is possible to obtain robust solutions that are insensitive to the imprecision
associated with this early stage of the absorber/evaporator design process, thus
facilitating the integrated product and production system design.
The main disadvantage of the approach is in the need to evaluate the distribution of
performance at every iteration of the optimization process, which can become time
consuming.
4.8 CLOSURE OF CASE STUDY AND OPEN ISSUES
4.8.1 Review of Case Study
Make-to-order systems can be effectively improved by adopting a systems perspective in the
design of their products. This perspective is implemented in this chapter by two means. First,
by applying the enterprise design framework constructed in this thesis and second, by applying
the design principles of standardization and robustness.
The enterprise design framework is applied by:
?? Modeling the design process as a game between product designers and the production
system designer.
?? Expanding the scope of individual decisions by adopting system’s goals.

?? Coordinating the decisions of the players using either a cooperative or a non-cooperative
formulation of strategies.
?? Embodying the strategy of the players with a common decision model, the Compromise
DSP.
?? Providing ranged solutions using a probabilistic design model as a means to enhance the
coordination of the individual’s decisions.
Thus, the various hypotheses posed in Chapter 1 are being tested in this example. In applying
this framework a family of products is designed as efficiently as a single product while
incorporating production issues since a very early stage of the design process.
4.8.2 Open Issues
An important consideration that has to be done regarding the use of the non-cooperative
game theory in coordinating enterprise design decisions is the issue of building rational reaction
sets. In this work response surfaces are used to construct them, but generally speaking, when
the number of players and/or design variables become large, the construction of the RRS using
design of experiments might be extremely cumbersome. It is necessary to look for alternative
methods for building the RRS.
In the case on an integrated product-process development it is necessary to somehow
estimate the effect of changes in the design variables in the production system. In this thesis it is
proposed to do so by modeling the floor shop as a response surface network. This approach
204

proved to be effective for this case example in which the number of variables that largely affect
the processing times were small. However, when the functions are highly nonlinear, the space is
relatively unconstrained and there is large freedom in choosing design variables, response
surfaces might not provide appropriate fit.
Regarding the design of product families, in this case study it is proposed to do so using
the notion of an “aggregated” product that is used as a product platform. The design of the
family is carried out by applying adaptive design on this aggregated product to satisfy the
requirements of the individual products of the family. This greatly facilitates the family design
process. Here it is suggested to do the “aggregation” based on demand, because it facilitates the
analysis of production performance. It is difficult to assess at the present the effectiveness of
this approach but certainly it opens an interesting avenue for future research.
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CHAPTER 5
CLOSURE
201
This thesis is motivated by the need of developing a mathematical framework for
enterprise integration. In this chapter the thesis argument developed throughout this work to
answer the posed research questions, and applied to a case study in Chapter 4, is critically
reviewed, namely, the merging of game theory, an enterprise design method based on a
common model for decision support, the Compromise DSP and a probabilistic design model for
the appropriate coordination of design decisions. Emphasis is given to potential shortcomings of
using game theory as a mathematical foundation for enterprise design and integration.
The contributions and relevance of the work are listed and discussed in this chapter, and
areas of future work are identified.

5.1 CRITICAL REVIEW OF THE THESIS: DISCUSSION OF RESEARCH
QUESTIONS AND SHORTCOMINGS.
Presently, it is recognized that an efficacious integration of the various enterprise activities
improves the competitiveness of manufacturing organizations. The fundamental thesis developed
in this work, which is based on a decision-based perspective, is that the integration of the
enterprise design processes, namely, process design, manufacturing process design and
organization design, should be based on an appropriate formulation and solution of individual
design decisions. Accordingly, two key objectives are posed in this work:
1. To establish a mathematically supported cooperative framework that enhances a practical,
effective and efficient integration of the enterprise design processes, and
2. To provide an appropriate model to support formulation and solution of design problems,
consistent with the aforementioned cooperative framework.
These fundamental objectives are embodied by three central research questions:
1. How can an enterprise design process be modeled mathematically in the
context of decision-based design?
2. How can enterprise decisions be coordinated through a design formulation
based on a game theoretical representation of the enterprise design process?
3. How can the design problems be solved to enhance the coordination of
enterprise design decisions?
202

203
In answering Question 1, the enterprise activity is perceived as a decision making process,
and therefore it is necessary to use a model that captures the dynamics of decisions.
Mathematically modeling of a system behavior, where one decision-maker’s action depends on
decisions by others, as occur in any manufacturing organization, is well established in game
theory. Thus, it is proposed to model the enterprise design process as a game, particularly as
an extensive game. This assertion has been extensively discussed in Chapter 2 and applied in
the integrated product and process design of the absorber/evaporator module presented in
Chapter 4. Organization design problems are not explicitly integrated in this case study.
However, it is apparent that the proposed framework for coordinating decisions has important
implications for organization design, particularly in the setting of reward and information systems,
as discussed in Section 3.2. Such a design should promote the achievement of a particular kind
of equilibrium depending on the protocol that is used. In other words, it is necessary to prepare
the appropriate field for the specific game that is played by the various agents that comprise the
enterprise.
It is noted that game theory has some shortcomings as a foundation for enterprise design.
Particularly, a game theoretical model is only a descriptive model and not a normative one. That
is, game theory identifies optimal strategies for the players in a game but does not address the
issue of how to structure the game. This is akin to asking (quoted from Kreps, 1990): “if the
players in the game are so smart, why don’t they change the rules and play a game where they

can do better?” In this sense, it is descriptive and does not provide a normative theory for
enterprise management and organization design.
A second problem that poses the use of game theory is that it is based on the assumption
of rationality of individuals that have transitive preferences. Kenneth Arrow has shown that this
condition is not always sufficient for group decision making (see Hazelrigg, 1996). Although the
group consists of rational individuals, the group itself may be irrational. In general, the more
members that comprise the group and the more alternatives among which the group must
choose, the more likely it is that the group will be irrational. Most manufacturing organizations
consist of several individuals pursuing the selection of a preferred course of action from among a
set of alternatives. In such a case and in the absence of an externally imposed preference
ordering, it is a virtual certainty that the design team will exhibit intransitive preferences. This
condition will lead to a course of action that is guided by the intransitive preferences of the
group. In short, the decision might not be rational. Thus, Arrow’s Impossibility Theorem (AIT)
shows the suboptimality of many of the answers the proposed approach gets. Hence, why is
the proposed approach useful?
204
In answering this question it is necessary to examine the requirements that the AIT
imposes to an organization. First, it is necessary for the organization to state explicitly the utility
against which the design should be judged. Second, the organization must create a set of
incentives and rewards that is in the best interest of each individual to make use of the stated
utility measure. However, uncertainty and bounded rationality prevents the organization to have

a single utility function to be used by any individual. That is, the fundamental goal of any
organization is to profit; thus, it is the only utility measure that could be used to this aim.
However, because each individual has different information, what one would be perceive as an
optimal decision to maximize profit, might not be perceived as such by others. Also, the profit
that a decision causes depends on future events that cannot be predicted with certainty.
Consider the case example of Chapter 4. Two system goals are posed: to minimize cost and to
minimize cycle time of production. The reason to do so is that it is difficult to predict accurately
how does a reduction of the cycle time benefit sales and therefore it is difficult to forecast the
real value of cycle time reduction for the organization in units of dollars.
Therefore, because of uncertainty and bounded rationality, the problem changes from
choosing the right course of action which maximizes profit to finding a way of calculating, very
approximately, where a good course of action lies. The logic on using game theory for
coordinating decisions is that such a course of action usually lies on a point of equilibrium of the
different individuals’ preferences. Game theory provides the mathematical tools to estimate
such equilibrium. However, particularly at early stages, such equilibrium may not exist because
individuals begin with different expectations and are trying to learn what to expect of others.
Hence, it would seem clear that it is necessary to complement this approach with other means.
Thus, the proposed framework does not exclude other ways of enterprise integration, such as
team meetings or CIM, that help us to better estimate the preferences and motivations of other
members of the enterprise.
205

A third problem on the use of game theory for coordination of decisions is the need for
precise protocols and mathematical models. Because human behavior cannot be reduced to a
set of equations, manufacturing organizations are only partially amenable to mathematical
analysis and modeling. Moreover, intuition and experience drive the individual’s sense of what
is satisfactory. Thus, the equilibrium that arises with a particular group of individuals might be
different to the one achieved with a different group even if all other conditions remain the same.
However, by adopting a common model for decision support and formulating the resulting
design problem using a common decision model it is expected this problem to be ameliorated.
It has to be noted that the use of a rigorous decision model should not aim at suppressing
intuition and experience but to provide a foundation to support them and therefore to improve
the decision making process. Thus, the answer to the research question 2 that is explored in this
thesis is that enterprise integration can be effectively pursued by formulating the design problems
using a non-cooperative-game formulation of the design problems. It is recommended the use
of the Compromise Decision Support Problem (DSP) to embody the resulting design
formulations. The reason to do so is that, as previously argued, it is difficult to measure the
profitability of a decision using a single scalar measure. Thus, a multiobjective decision model is
needed. The Compromise DSP is a rigorous tool that fulfills this need.
206
The concern that motivates the third research question is that for most real-world
problems it is not possible to find the optimum solution. Even the prediction of points of non-
cooperative equilibrium is difficult in actuality. Therefore, it is characteristic of the enterprise

design activity that the final outcome is built from a search of a satisficing solution that is
gradually refined. Thus, the idea of a fixed goal is inconsistent with our limited ability to
determine the best point of equilibrium. The real results of our actions are to establish initial
conditions for the next succeeding stage of action. What are the goals for a given stage are in
fact criteria for choosing the initial conditions that we will leave to our successors. An attractive
alternative in conducting this search of a solution is to leave as many alternatives as possible for
future decision making, avoiding irreversible commitments that cannot be undone, through
flexible solutions. A probabilistic design model is proposed as a formal means to develop
flexible solutions to meet a ranged set of design requirements, ideally facilitating the convergence
of the group decision making process to an equilibrium that is satisfactory for the individuals that
comprise the group.
In summary, the proposed framework developed in this thesis for the integration of the
enterprise activities based on the formulation of individuals design problems is based on the
following elements:
?? A game theoretical representation of the enterprise activity
?? A common paradigm for decision support which builds upon the strengths of operations
research, engineering design and systems theory.
?? An enterprise design method based on a philosophy of empowerment through decision
making.
?? A common decision model, the Compromise DSP.
207

?? A probabilistic design model to attain flexible and robust decisions, and to handle
uncertainty and imprecision.
These elements are merged in this work into a consistent framework for enterprise design and
integration, as is explained in Section 3.6. This framework has been exercised in a case study in
Chapter 4: the integrated product and process design of an absorber/evaporator module for a
family of absorption chillers. Even when the results of the case study are encouraging, the
effectiveness of the approach for larger and more complex problems has yet to be proved.
5.2 CONTRIBUTIONS AND RELEVANCE OF THE THESIS
A fundamental foundation of this work is the tenet of enterprise design as a scientific discipline.
In Chapter 1 it is argued that a scientific design practice should incorporate:
1. Representation of design problems.
2. A theory of structure and design organization.
3. A utility theory as a logical framework for rational choice among alternatives, and tools for
actually making the computations.
4. The body of techniques for actually deducing which of the available alternatives is the
optimum.
5. Algorithms to search for solutions.
This work provides a contribution on all these elements. First, it provides a game theoretical
representation of the enterprise design process based on viewing the enterprise activity as a
208

group decision making process. This representation focuses attention on issues of cooperative
dynamics within an organization, so that the “physics” of interaction between the agents that
comprise it come to the fore. As commented in Chapter 2, if the mechanics of interaction
between individuals matter for achieving enterprise integration, this representation is by itself a
relevant contribution.
This work contributes to a theory of structure and enterprise design organization as it
poses a mathematically based theory for collaboration of individuals based on the appropriate
formulation of their design problems. This concept is built upon the constructs of game theory,
operations research and systems theory. By using a hybrid paradigm for decision support that is
domain independent, this theory capitalizes and integrates the structure of engineering design
based on modeling, analysis and synthesis.
On the need of a utility theory and the body of techniques for the synthesis of designs, the
application of game theory provides clear objectives based on the notion of Nash and Pareto
equilibrium. Depending on the kind of solution that is desired, game theory provides the basic
theory that is required for identifying such equilibrium.
The proposed framework provides also a solution technique that is based on theory of
optimization that translates into specific algorithms. Thus, it provides a foundation for rigorously
implementing enterprise design and integration.
209
A secondary, but also important contribution of this work, is the development of a method
for modeling production systems for product design purposes. Simulation based models are too

cumbersome to be used during product optimization at early design stages, thus they do not
facilitate an integrated product/process development. Instead, it is proposed to model the
system as a response surface network, in which the response surfaces of individual processing
stations relate product characteristics with production time and processing variability. These
response surfaces are the modules for the construction of a queuing model. This method allows
a quickly evaluation of the impact of changes of product variables in the production system, and
enhances the practical implementation of the integration of product design and manufacturing
process design using a non-cooperative protocol because it greatly facilitates building the
required rational reaction sets.
Finally, the case study provides specific contributions as well. It deals with the important
issue of developing design methods for products that are produced in make-to-order systems.
Low volume, uncertain demand and high product variety characterize these systems. These
characteristics result in long lead times, large work in process and high manufacturing costs.
The approach developed on this case study, which focus on standardization, modularity and
robustness, is a steppingstone for further progress in this area.
In summary, this thesis provides a formal means to coordinate decisions of the individuals
that comprise a manufacturing system based on an approach that aims at framing the tradeoffs
needed to coordinate decisions through controls that are consistent, not contrary to employee
empowerment. Because the major opportunities for improvement of manufacturing enterprises
210

lie at the interfaces (e.g., between product design and manufacturing), manufacturing
organizations can greatly benefit from this approach.
5.3 RECOMMENDATIONS FOR FUTURE WORK
Even when the proposed framework seems to be consistent there are several issues that have
arisen through the development of this work and remain open. Maybe the most challenging
issue is that of organization design. Unfortunately, organization design is, in general, not
amenable to mathematical support and analysis. The proposed approach for enterprise
integration requires the mathematical formulation and solution of problems. Therefore, it is
extremely important to develop mathematical-based methods for the modeling, analysis and
synthesis of organization design problems.
Another problem is the difficulty to deal with large and complex problems. Cooperative-
based integration based on large models is not practically due to computational and cognitive
limitations. Non-cooperative-based coordination seems to be more practical, but building
rational reaction sets when several agents are involved in making a decision is also very
cumbersome. This is a critical problem that has to be studied if non-cooperative game theory is
intended to be the foundation of enterprise integration.
In the specific issue of manufacturing systems modeling, the proposed response surface
network modeling method has important shortcomings, particularly at the use of response
surfaces to approximate behavior of process stations when mixed continuous/discrete product
211

variables are present. Alternative approximation techniques are required to tackle this problem.
Also, for large complex production systems, parametric decomposition-based models are
usually not accurate.
212

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Zhou, Q.-J., Allen, J. K. and Mistree, F., 1992, "Decisions under Uncertainty: The Fuzzy
Compromise Decision Support Problem," Engineering Optimization, Vol. 20, pp. 21-
43.
267

APPENDIX A
THE PARAMETRIC DECOMPOSITION APPROACH
Queuing networks have been used to model the performance of a variety of production
systems. Since exact results exist for only a limited class of networks, the decomposition
methodology has been used extensively to obtain approximate results. In this Appendix it is
given a brief overview of the parametric decomposition approach, which is a means to
approximating the steady-state performance measures of open queueing networks with general
arrival processes and general service-time distributions. This modeling approach is combined
with a response surface technique in order to develop efficacious models of production systems
in Chapter 4.
212

Queueing networks have been used to model the performance of a variety of complex systems
such as production job shops, flexible manufacturing systems, service sytems, etc. However,
exacts results exist only for the limited class of product-form type (Jackson type) networks.
The exact results encountered in the literature basically rely on the following assumptions (Whitt,
1983a):
?? Exponential distribution for service times.
?? Homogeneous service requirement at all stations (the distribution of service time is
independent of the product type).
?? Priority discipline independent of customer class.
?? Poisson arrivals.
These are overly restrictive in many practical situations. Since the exact results do not extend to
more general networks, it has led to the development of approximation schemes. These may be
broadly described under the following four categories:
?? Diffusion approximation.
?? Mean value analysis
?? Operational analysis
?? Decomposition methods
The first three approaches are not directly related to this work, and are mentioned here only for
completeness. Bitran and Tirupati (1993) provide a detailed list of references on these
schemes.
213

Decomposition Methods are essentially attempts to generalize the notion of independence
and product form results for Jackson networks to more general systems. This approach relies
on two notions:
1. The nodes can be treated as being stochastically independent, and
2. two parameters (typically mean and variance) approximations provide reasonable accurate
results.
This approach was first proposed by Reiser and Kobayashi (1974) and has been used by
several researchers in developing approximations. The work by Buzacott and Shanthikumar
(1993) and Whitt (1983a, 1995) are representative of this approach. Essentially the method
involves three steps:
Step 1. Analysis of interaction between stations.
Step 2. Decomposition of the network into subsystems of individual stations and their
analysis.
Step 3. Recomposition of the results to obtain the network performance.
The first step is critical to the decomposition approach. The interaction between stations is
analyzed by looking at the network as a composite of three basic processes:
(a) superposition or merging,
(b) flow through a queue or a station, and
(c) splitting or decomposition.
214

The superposition process (a) typically represents the arrivals at a station while (b) describes the
departures from a station. The splitting and the merging processes model the product routing in
the network.
(a) The merging process considers superposition of arrivals at a station (say station j). Let ? ij
and caij be the arrival rate and the square coefficient of variation 1 (scv) of interarrival time of the
flow from station i to j (i = 0 for external arrivals). The resulting mixture is approximately
described with an arrival rate ? j and scv caj as follows
?
j
?
N
N ij
? ? ij , ca j ? ?
i?
0
i? 0?
j
?
ca
ij
215
(A.1)
(b) The flow process describes approximately the scv of the departure stream from a station. If
caj and csj are the scv of the interarrival and service time at station j with a utilization ? j the scv
of the departures is approximately given by
cd
j
j
ij
2 ? ? j ? ca j
2 ? ? cs ? 1 ?
(A.2)
When there is more than one machine m at a station, the following is a reasonable way to
estimate cdj [2]:
2
2 ? j
? 1?
( 1?
? j )( ca j ? 1)
? ( cs ? 1)
(A.3)
m
cd j
j
where m is the number of machines at the station. Notice that this reduces to (A.2) when m=1.
(c) In the decomposition process a product stream with arrival rate ? j and scv of cdj is split
into a number of substreams, each stream representing the flow from station j to other stations in

the network. The routing is assumed to be Markovian. If pi is the probability that a job would
follow pat l , then the substream i is characterized by the following parameters:
216
? ij ? p i ? j , cd ji ? p i cd j ? 1? p i (A.4)
A rationale for these approximations can be found in Whitt [19]. He examines two methods -
the stationary interval method and the asymptotic method in approximating a point process by a
renewal process. These methods are used to determine the moments of the renewal process
that approximates the point process. For example, the asymptotic method to approximate the
superposition process leads to (A.1). Also shows that (A.2) can be interpreted as an
approximation resulting from the stationary interval method. This explanation involves the use of
an approximation for the mean waiting time in a GI/G/1 2 queue. Expression (A.4) implicitly
assumes that interdeparture intervals of the product stream are iid with mean 1/? j and scv of
cdj. (Note that caji = cdji) Whitt (1983a) explain that (A.3) is in fact exact under the
assumption of Markovian routing.
The interactions among stations are reflected by the scv of the arrivals at each station.
Combining the approximations (considering m=1) for the three basic processes described
above leads to the following linear systems:
N
? i ? ? r0i ? ? ? jr ji , i = 1, 2,..., N
(A.5)
j?1
1
The square coefficient of variation, scv, of a random variable is defined as the ratio of its variance to the
square of the mean
2
Kendall’s notation which characterizes a station. In this case a one-machine station, with general
exponentially distributed interarrival times, and generally dis tributed process times.

?
where
i
ca
i
N
N
2
2
? j ? ? j ? r jica
j ? ? ? r0i
? ? ? jr
ji ? j rjics
j ? 1?
r ji ? ,
217
2 ? ? 1 ?
i = 1,2,..., N (A.6)
j?
1
? = arrival rate into the system,
? j = net arrival rate at station j,
? j = utilization at station j
caj = scv of arrivals
csj = scv of service at station j,
j?
1
Notice that (A.5) consists of standard flow balance equations to determine the net arrivals at
each station. The internal flows ? ij are given by ? i?rij. (A.6) is the approximation to compute
the scv of arrivals at each station.
In Step 2, each station is analyzed based on the partial information obtained in Step 1.
Shanthikumar and Buzacott examine M/G/1 and GI/G/1 approximation to evaluate the station at
the performance. Whitt (1983a, 1983b) also examines the GI/G/m queue. In particular the
GI/G/1 and the G/G/m approximations require only the mean and scv of the interarrival and
service times.
Finally in Step 3 these results are synthesized and performance measures for the network
are estimated. This procedure is straightforward for mean queue lengths, and lead times.
In developing the queueing network analyzer (QNA) Whitt (1983a) essentially used an
approach similar to the one described above in analyzing the interactions between the nodes.
The superposition or the merging process has been modified as follows:

chj = wcaj + 1 -w
where chj is the modified scv of the merged stream, caj is determined by (A.1) and w is a
weighting function that depends on the station utilization ? j and ? ij. In the QNA the weight w is
determined as follows:
2
? 1 ? 4(
1 ? ? ) ( ? ) ?
m ?
? 1
where ? ? ?
? 1?
218
? 1 w ?
j ? 1
(A.7)
2
kj
2
k j
(A.8)
This modification is motivated by the observation that neither the asymptotic method (which
leads to Equation A.1) nor the stationary interval approach by itself perform well over a wide
range of ca for approximating the superposition process. (A.7) above is a hybrid approximation
of both approaches. In the QNA a modified form of (A.2), similar to (A.3), is used to account
for multiple servers at each station. The splitting process is based on Markovian routing. The
estimation of station performance measures in Step 2 is also modified to provide for multiple
servers at each station. However, the approximations continue to use the first two moments of
the arrival and service distributions.
In Witt (1995) it is proposed to replace the variability parameters of arrival processes
by variability functions that depend on the utilization ?. These variability functions provide an
improved framework for parametric-decomposition approximations.

Bitran and Tirupati (1988) propose a modification to the parametric decomposition
approach for the analysis of multiproduct queueing networks with deterministic routings, which
can be viewed as a generalization of the one used by Shanthikumar and Buzacott and Whitt. It
is based on the notion that the scv of product departures can be approximated as the sum of
two terms, one that represents the influence of station congestion and service time and one that
represents the interference due to the presence of multiple products. Since the estimation of the
interference effect is difficult to evaluate exactly, they propose three approximations that are
asymptotically exact. The approximation used in this work is based on the notion that the
aggregation of the arrival streams of a large number of products may be approximated by a
Poisson process, resulting in the following scv of departure for product i at any station:
where
i
i
i
? p ? ( p ) ca ?
i
i
i
219
cd ? p cd ? ( 1 ? p ) 1?
(A.9)
pi = ? i?? ?
? i = arrival rate of product I
????total arrival rate at the station = ? ? i
cdi = scv of departure for product i at station
cd = scv of departure of station
cai = scv of arrival of product i at station
Details on this approximation can be studied in Bitran and Tirupati (1988).
Buzacott and Shanthikumar (1991), Whitt (1995), and Bitran and Tirupati (1988)
report encouraging results based on experiments comparing their respective approximations
with simulation.

APPENDIX B
FORTRAN FILES
220

221

B.1 SAMPLE OF FORTRAN FILES FOR OptdesX
222
C FORTRAN FILE FOR OptdesX FOR PRODUCT DESIGN - BASELINE
C FORMULATION
c ANALYSIS SUBROUTINES FOR ABSORPTION CHILLER OPTIMIZATION WITH c
OptdesX
c Gabriel Hernandez
C June 05, 1998
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Absorber/Evaporator Design"
return
end
c-----------------------------------------------------------------------
--
C LIST OF VARIABLES
c A = Scaled value for number of absorber tubes [-2,2]
c CAPACITY = refrigeration capacity (tons)
c COST = Material Cost (dollars) of the Tubes of the
Absorber/Evaporator
c COSTA = Cost/ft of different Absorber Tubes (Vector)
c COSTE = Cost/ft of different Evaporator Tubes (Vector)
c DIE = Internal diameter (in) of evaporator tubes (vector)
c DOA = External diameter (in) of absorber tubes (vector)
c DOE = External diameter (in) of evaporator tubes (vector)
c DPE = pressure drop of chilled water at evaporator
c DPE_MIN = minimum DPE required
c DR = Root diameter (in) of evaporator tubes (vector)
c E = Scales value for number of tubes of the evaporator [-2,2]
c FH = Height of fin (in) of evaporator tubes (vector)
c FPI = Number of fins/in of evaporator tubes (vector)
c FW = Width of fin (in) of evaporator tubes (vector)
c I = Integer variable of evaporator tubes selection
c J = Integer variable of absorber tubes selection
c L = Length of tube in ft
c LT = Scaled value for length of tubes [2,2]
c NABSTA= Number of absorber tubes options
c NAT = Number of tubes of the Absorber
c NEVPTA= Number of evaporator tubes options
c NET = Number of tubes of the Evaporator
c NRA = Number of rows of the absorber
c NATR= Number of tubes/row of the absorber
c THKA= Thickness (in) of tubes of the absorber
c THKE= Thickness (in) of tubes of the evaporator
c TONS= Required capacity
c UAA = Heat Transfer Exchange Coefficient multiplied by
c area of heat exchange for the absorber

223
c UAANOM= UAA required
c UAE = Heat Transfer Exchange Coefficient multiplied by
c area of heat exchange for the evaporator
c UAENOM= UAE required
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c ... variables passed between OptdesX and these subroutines
DOUBLE PRECISION DLT,DE,DA,DCAP,DCOST,DUAE,DUAA,DDP,
+ DSELE,DSELA,DL,DE,DA,
+ DTONS,DUAENOM,DUAANOM
c ----------------------------------------------------------------------
-----
INTEGER I,J,NEVPTA,NABSTA,
+ NRA,NATR
PARAMETER (NEVPTA=4,NABSTA=4)
REAL A,
+ COST,COSTE(NEVPTA),COSTA(NABSTA),
+ DIE(NEVPTA),DOA(NABSTA),DOE(NEVPTA),DR(NEVPTA),
+ E,
+ FH(NEVPTA),FPI(NEVPTA),FW(NEVPTA),
+ L,LT,
+ NAT,NET,
+ THKA(NABSTA),THKE(NEVPTA),TONS,
+ UAA,UAANOM,UAE,UAENOM
C -----------------------------------------------------------
C TUBES ALTERNATIVES
DATA (DOE(I), I=1,NEVPTA) /0.625,0.75,0.875,1.0/,
& (THKE(I), I=1,NEVPTA) /0.025,0.025,0.02,0.025/,
& (FPI(I), I=1,NEVPTA) /30,30,30,30/,
& (DIE(I), I=1,NEVPTA) /0.444,0.569,0.694,0.819/,
& (DR(I), I=1,NEVPTA) /0.5,0.625,0.75,0.875/,
& (FW(I), I=1,NEVPTA) /0.02,0.02,0.02,0.02/,
& (FH(I), I=1,NEVPTA) /0.05,0.05,0.05,0.05/,
& (COSTE(I),I=1,NEVPTA) /6.0,6.5,7.0,7.5/,
& (DOA(I), I=1,NABSTA) /0.625,0.75,0.875,1.0/,
& (THKA(I), I=1,NABSTA) /0.03,0.03,0.03,0.03/,
& (COSTA(I),I=1,NABSTA) /3.25,3.50,3.75,4.0/
c...input scalar AV values
call avdsca(DLT,'length')
call avdsca(DE,'NET')
call avdsca(DA,'NAT')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
call avdsca(DTONS,'Tons')
c ... Scaled Length

LT=REAL(DLT)
c ... Scaled number of tubes
E=REAL(DE)
A=REAL(DA)
c
c...Selection of Tubes
c ... For the evaporator
I=INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
c ... For the absorber
J=INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
c
c ---------------------------------------------------------------
c... value of constraints
TONS=DTONS
c... linear relationship between required capacity and UA required
UAENOM=1372.183*TONS-34091.8
UAANOM=1021.806*TONS-38163.0
DPE_MIN=23.0
C ****************************************************************
c... Transform scaled variables to actual values
L=1.75*(LT+2)+17.0
NET=115*(E+2)+440
NAT=115*(A+2)+440
DL=L
DE=NET
DA=NAT
c -----------------------------------------------------------------
C...compute heat transfer functions
C...EVAPORATOR
c ... pass TONS,DOE,THKE,FPI,DIE,DR,FW,FH,L,NET
C ... get UAE and DPE
CALL EVAP(TONS,DOE(I),THKE(I),FPI(I),DIE(I),DR(I),
& FW(I),FH(I),L,NET,UAE,DPE)
C...ABSORBER
C NOTE: THE NUMBER OF ROWS IS FIXED AS 12
NRA=12
NATR = INT(NAT/NRA)
c ... pass TONS,DOA,THKA,L,NRA,NATR
c ... get UAA and CAPACITY
CALL ABSE(TONS,DOA(J),THKA(J),L,NRA,NATR,UAA,CAPACITY)
C ------------------------------------------------------------
c
c ... Compute cost of tubes
COST =COSTE(I)*(L*NET)+COSTA(J)*(L*NAT)
c
c ************************************************************
C Value of dependent variables for optimization driver
224

DUAA=UAA
DUAE=UAE
DDP=DPE
DCAP=CAPACITY
DCOST=COST
DUAENOM=UAENOM
DUAANOM=UAANOM
C ***************************************************************
c...output scalar AF values
call afdsca(DCAP,'capacity')
call afdsca(DUAE,'UAevp')
call afdsca(DUAA,'UAabs')
call afdsca(DDP,'DP')
call afdsca(DCOST,'Cost')
call afdsca(DL,'Length')
call afdsca(DE,'Evap')
call afdsca(DA,'Abs')
call afdsca(DUAENOM,'UAENOM')
call afdsca(DUAANOM,'UAANOM')
return
end
225
C FORTRAN FILE FOR PRODUCTION SYSTEM DESIGN WITH
C MODIFIED DEVIATION FUNCTION - BASELINE FORMULATION
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DCOST,
+ DMatCOST,DU,DZ,DFN1,DFN2
DOUBLE PRECISION DCT,DF,DMfgCOST,DMS1,DMS2,DMS3
c production variables

INTEGER I,J,K,NABSTA,NEVPTA,A(8),E(8)
PARAMETER (NEVPTA=4,NABSTA=4)
REAL COSTE(NEVPTA),COSTA(NABSTA),
+ FN1,FN2,
+ L(8),LT(8),
+ NAT(8),NET(8),NE(8),NA(8),
+ PROB(8),
+ TONS(8),TCOST,
+ Z
REAL CT,
+ F,
+ MS1,MS2,MS3,MfgCOST,
+ TCT,
+ U
c -----------------------------------------------------------
c ... PRODUCT DATA
DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,NEVPTA) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,NABSTA) /3.25,3.50,3.75,4.0/
c ... baseline solution
c & (L(I), I=1,8) /17.01,18.46,19.45,17.06,22.51,19.25,
c + 23.82,23.95/,
c & (E(I) ,I=1,8) /2,2,2,2,3,2,3,3/,
c & (A(I) ,I=1,8) /2,2,2,2,2,2,2,2/
c & (NE(I),I=1,8) /443,441,464,498,445,645,489,527/,
c & (NA(I),I=1,8) /453,478,457,697,549,727,615,687/
c Z11 = 46762.
c Z12 = 65680.
c Z21 = 703.6
c Z22 = 447.7
c
c ... regular cooperative solution
c DO 5 I=1,8
c L(I) = 22.06
c E(I) = 3
c A(I) = 2
c5 CONTINUE
c DATA (NE(I),I=1,8) /440,440,440,440,460,511,557,608/,
c & (NA(I),I=1,8) /440,440,446,504,554,612,662,719/
c Z11 = 219817
c Z12 = 239155.
c Z21 = 709.3
c Z22 = 458.3
c
C ... modifed cooperative solution
c DO 5 I=1,8
c L(I) = 19.25
c E(I) = 2
c A(I) = 2
226

c5 CONTINUE
c DATA (NE(I),I=1,8) /440,440,440,440,446,493,539,586/,
c & (NA(I),I=1,8) /440,440,440,493,540,601,647,708/
c Z11 = 219965.
c Z12 = 239301.
c Z21 = 697.
c Z22 = 455.
c
c ... non cooperative solution
DO 5 I=1,8
L(I) = 19.25
E(I) = 2
A(I) = 2
5 CONTINUE
DATA (NE(I),I=1,8) /440,440,457,518,579,636,697,758/,
& (NA(I),I=1,8) /440,446,518,590,647,719,791,863/
Z11 = 216630.
Z12 = 235456.
Z21 = 614.
Z22 = 494.
c
c...input scalar AV values
c
call avdsca(DMS1,'MS1')
call avdsca(DMS2,'MS2')
call avdsca(DMS3,'MS3')
call avdsca(DF,'F')
c
c product analysis
c
c...compute scalar AF values
c
c...intermediate constants
C **********************************************************
C...compute functions
MatCOST=0.0
DO 100 K=1,8
I=E(K)
J=A(K)
LT(K)=1.75*(L(K)-17.0)-2.0
NET(K)=(NE(K)-440)/115.0-2
NAT(K)=(NA(K)-440)/115.0-2
MatCOST=PROB(K)*(COSTE(I)*(L(k)*NE(K))+
+ COSTA(J)*(L(k)*NA(K)))+MatCOST
100 CONTINUE
MatCOST=MatCOST+52335.0
c
c
c production analysis
MS1=REAL(DMS1)
MS2=REAL(DMS2)
227

c
MS3=REAL(DMS3)
F =REAL(DF)
CALL QNA(MS1,MS2,MS3,F,LT,NET,NAT,I,J,U,CT,MfgCOST)
228
COST=MatCOST+MfgCOST
c ************************************************************
C modified Pareto objective
TCOST = 169810.0
TCT = 318.1
Z11 = Z11/TCOST-1.0
Z12 = Z12/TCOST-1.0
Z21 = Z21/TCT-1.0
Z22 = Z22/TCT-1.0
W1 = 0.5
W2 = 0.5
c
c ... COST GOAL
F1 = COST/TCOST-1.0
FN1 = (F1-Z11)/(Z12-Z11)
c ... CT Goal
F2 = CT/TCT-1.0
FN2 = (F2-Z22)/(Z21-Z22)
c
c ... Formulate Supercriterion
S = (1-FN1)*(1-FN2)
c ... Formulate a Pareto Optimal objective FC
FC = W1*FN1+W2*FN2
c ... Modified Objective Function
Z = FC-S+1
C EVALUATE DEVIATION FUNCTION USING AN ARCHIMEDEAN FORMULATION
C WEIGHT OF EACH GOAL AT EACH PRIORITY LEVEL
DCOST = COST
DMatCOST = MatCOST
DU = U
DMfgCOST = MfgCOST
DCT = CT
DFN1 = FN1
DFN2 = FN2
DZ = Z
C
C
************************************************************************
****
c...output scalar AF values
call afdsca(DU,'Utilization')
call afdsca(DCOST,'COST')
call afdsca(DMatCOST,'MatCOST')
call afdsca(DMfgCOST,'MfgCOST')
call afdsca(DCT,'CT')
call afdsca(DFN1,'FN1(COST)')

call afdsca(DFN2,'FN2(CT)')
call afdsca(DZ,'Z')
return
end
229
C FORTRAN FILE FOR OptdesX for COOPERATIVE SOLUTION WITH MODIFIED
C DEVIATION FUNCTION
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DLT,DE,DA,DCOST,DFEAS,
+ DSELE,DSELA,DL,DE,DA,
+ DMatCOST,DU
DOUBLE PRECISION DCT,DF,DMfgCOST,DMS1,DMS2,DMS3
DOUBLE PRECISION DFN1,DFN2,DZ
c production variables
INTEGER I,J,K,NABSTA,NEVPTA
PARAMETER (NEVPTA=4,NABSTA=4)
REAL A,
+ COSTE(NEVPTA),COSTA(NABSTA),
+ E,FEASE,FEASA,FEAS,
+ L,LT,
+ NAT(8),NET(8),NE,NA,
+ PROB(8),
+ TONS(8),TCOST
REAL CT,
+ F,
+ MS1,MS2,MS3,MfgCOST,
+ TCT,
+ U
REAL F1,F2,FN1,FN2,FC,

+ S,
+ W1,W2,
+ Z11,Z12,Z21,Z22,Z
c -----------------------------------------------------------------
DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,NEVPTA) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,NABSTA) /3.25,3.50,3.75,4.0/
c -----------------------------------------------------------------
c...input scalar AV values
call avdsca(DLT,'length')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
c
call avdsca(DMS1,'MS1')
call avdsca(DMS2,'MS2')
call avdsca(DMS3,'MS3')
call avdsca(DF,'F')
c
c...compute scalar AF values
c...intermediate constants
C **********************************************************
c ... product analysis
LT=REAL(DLT)
E=REAL(DE)
A=REAL(DA)
I =INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
J =INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
L = 1.75*(LT+2)+17.0
DL= L
C...compute functions
MatCOST= 0.0
FEAS = 16.0
DO 100 K=1,8
c ............. Find minimum number of tubes for EVAP
CALL MINEVP(L,I,TONS(K),NE,FEASE)
NET(K)=(NE-440)/115.0-2
c ............. Find minimum number of tubes for ABS
CALL MINABS(L,J,TONS(K),NA,FEASA)
NAT(K)=(NA-440)/115.0-2
FEAS = FEAS-FEASE-FEASA
MatCOST=PROB(K)*(COSTE(I)*(L*NE)+COSTA(J)*(L*NA))+MatCOST
100 CONTINUE
MatCOST=MatCOST+52335.0
c
230

231
c ----------------------------------------------------------------------
-------
c ... production analysis
MS1=REAL(DMS1)
MS2=REAL(DMS2)
MS3=REAL(DMS3)
F =REAL(DF)
CALL QNA(MS1,MS2,MS3,F,LT,NET,NAT,I,J,U,CT,MfgCOST)
c
COST=MatCOST+MfgCOST
c
************************************************************************
*****
C modified Pareto objective
TCOST = 160000.0
TCT = 318.1
Z11 = 214698/TCOST-1.0
Z12 = 233959/TCOST-1.0
Z21 = 769/TCT-1.0
Z22 = 394/TCT-1.0
W1 = 0.5
W2 = 0.5
c
c ... COST GOAL
F1 = COST/TCOST-1.0
FN1 = (F1-Z11)/(Z12-Z11)
c ... CT Goal
F2 = CT/TCT-1.0
FN2 = (F2-Z22)/(Z21-Z22)
c
c ... Formulate Supercriterion
S = (1-FN1)*(1-FN2)
c ... Formulate a Pareto Optimal objective FC
FC = W1*FN1+W2*FN2
c ... Modified Objective Function
Z = FC-S+1
c
c
************************************************************************
*****
DCOST = COST
DMatCOST = MatCOST
DFEAS = FEAS
DU = U
DMfgCOST = MfgCOST
DCT = CT
DFN1 = FN1
DFN2 = FN2
DZ = Z
c
C

232
C
************************************************************************
****
c...output scalar AF values
call afdsca(DL,'Length')
call afdsca(DFEAS,'PRODUCT_FEAS')
call afdsca(DU,'Utilization')
call afdsca(DCOST,'COST')
call afdsca(DMatCOST,'MatCOST')
call afdsca(DMfgCOST,'MfgCOST')
call afdsca(DCT,'CT')
call afdsca(DFN1,'FN1(COST)')
call afdsca(DFN2,'FN2(CT)')
call afdsca(DZ,'Z')
return
end
c FORTRAN FILE FOR OptdesX for BUILDING RATIONAL REACTION SETS
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Non-cooperative Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DLT,DNET,DNAT,
+ DSELE,DSELA
DOUBLE PRECISION DCT,DF,DMfgCOST,DMS1,DMS2,DMS3,DU
c production variables
INTEGER I,J,NABSTA,NEVPTA
PARAMETER (NEVPTA=4,NABSTA=4)
REAL LT,
+ NAT,NET

REAL CT,
+ F,
+ MS1,MS2,MS3,MfgCOST,
+ U
c -----------------------------------------------------------
c...input scalar AV values
c.....values of experiment
call avdsca(DLT,'length')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
call avdsca(DNET,'NET')
call avdsca(DNAT,'NAT')
c
c .... values of rational reaction set
call avdsca(DMS1,'MS1')
call avdsca(DMS2,'MS2')
call avdsca(DMS3,'MS3')
call avdsca(DF,'F')
c
c product analysis
LT=REAL(DLT)
NET=REAL(DNET)
NAT=REAL(DNAT)
I=INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
J=INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
c...compute scalar AF values
c
c...intermediate constants
c **********************************************************
C...compute functions
c.... production analysis
MS1=REAL(DMS1)
MS2=REAL(DMS2)
MS3=REAL(DMS3)
F =REAL(DF)
CALL QNA(MS1,MS2,MS3,F,LT,NET,NAT,I,J,U,CT,MfgCOST)
c
c ************************************************************
C EVALUATE DEVIATION FUNCTION USING AN ARCHIMEDEAN FORMULATION
C WEIGHT OF EACH GOAL AT EACH PRIORITY LEVEL
DU = U
DMfgCOST= MfgCOST
DCT = CT
c
C
233

234
C
************************************************************************
****
c...output scalar AF values
call afdsca(DU,'Utilization')
call afdsca(DMfgCOST,'MfgCOST')
call afdsca(DCT,'CT')
return
end
C FORTRAN FILE FOR OptdesX FOR FINDING PAYOFF MATRIX FOR
C PRODUCT DESIGN - NON COOPERATIVE
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DLT,DCOST,DFEAS,
+ DSELE,DSELA,DL,DNAT,DNET,DNE,DNA,
+ DMatCOST,DZ1
DOUBLE PRECISION DCT,DMfgCOST,DZ2
c production variables
INTEGER I,J,K,NABSTA,NEVPTA
PARAMETER (NEVPTA=4,NABSTA=4)
REAL A,
+ COSTE(NEVPTA),COSTA(NABSTA),
+ E,FEASE,FEASA,FEAS,
+ L,LT,
+ NAT,NET,NE,NA,
+ PROB(8),
+ TONS(8),TCOST
REAL CT,

+ MfgCOST,
+ TCT,
+ XE(4),XA(4)
c
c -----------------------------------------------------------
DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,NEVPTA) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,NABSTA) /3.25,3.50,3.75,4.0/
DATA (XA(I), I=1,4) /-2,-1,1,2/,
& (XE(I), I=1,4) /-2,-1,1,2/
c...input scalar AV values
call avdsca(DLT,'length')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
c
c product analysis
TCOST = 150435.0
TCT = 318.1
LT=REAL(DLT)
I=INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
J=INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
c...compute scalar AF values
c...intermediate constants
C **********************************************************
L=1.75*(LT+2)+17.0
DL=L
C...compute functions
MatCOST= 0.0
FEAS =16.0
NET = 0.0
NAT = 0.0
DO 100 K=1,8
c ............. Find minimum number of tubes for EVAP
CALL MINEVP(L,I,TONS(K),NE,FEASE)
NET=NET+PROB(K)*NE
c ............. Find minimum number of tubes for ABS
CALL MINABS(L,J,TONS(K),NA,FEASA)
NAT=NAT+PROB(K)*NA
FEAS = FEAS-FEASE-FEASA
MatCOST=PROB(K)*(COSTE(I)*(L*NE)+COSTA(J)*(L*NA))+MatCOST
100 CONTINUE
c ... code vars
NET=(NET-440)/115.0-2
NAT=(NAT-440)/115.0-2
235

236
E=XE(I)
A=XA(J)
DNET = NET
DNAT = NAT
DNE = NE
DNA = NA
c
MatCOST=MatCOST+52335.0
c
c ... Rational Reaction Sets of Production Designer
MfgCOST = 49263.35+1215.23*LT-442.57*E-
412.04*A+2451.33*NET+2208.98*NAT
+ -365.50*LT*E-684.65*LT*A-479.31*LT*NET-
768.31*LT*NAT+357.44*E*A
+ +376.55*E*NET+798.4*E*NAT+686.45*A*NET+823.85*A*NAT
+ -504.06*NET**2-392.44*NET*NAT-344.81*NAT**2
CT = 505.48+11.08*LT+0.09*E+26.67*A+23.41*NET-
3.0*LT*NET+3.63*E**2
+ -3.78*E*NAT+9.38*NET**2+15.38*NET*NAT+4.50*NAT**2
c
COST=MatCOST+MfgCOST
c ************************************************************
C EVALUATE DEVIATION FUNCTION USING AN ARCHIMEDEAN FORMULATION
C WEIGHT OF EACH GOAL AT EACH PRIORITY LEVEL
DCOST = COST
DMatCOST = MatCOST
DFEAS = FEAS
DMfgCOST = MfgCOST
DCT = CT
DZ1 = COST/TCOST-1.0
DZ2 = CT/TCT-1.0
c
C
C
************************************************************************
****
c...output scalar AF values
call afdsca(DL,'Length')
call afdsca(DNET,'NET')
call afdsca(DNAT,'NAT')
call afdsca(DNE,'NE')
call afdsca(DNA,'NA')
call afdsca(DFEAS,'PRODUCT_FEAS')
call afdsca(DCOST,'COST')
call afdsca(DMatCOST,'MatCOST')
call afdsca(DMfgCOST,'MfgCOST')
call afdsca(DCT,'CT')
call afdsca(DZ1,'Z1(COST)')
call afdsca(DZ2,'Z2(CT)')
return

end
237
C FORTRAN FILE FOR OptdesX FOR PRODUCT SYSTEM DESIGN WITH
C MODIFIED DEVIATION FUNCTION
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DLT,DCOST,DFEAS,
+ DSELE,DSELA,DL,
+ DMatCOST,DNET,DNAT,DNE,DNA
DOUBLE PRECISION DCT,DMfgCOST
DOUBLE PRECISION DFN1,DFN2,DZ
c production variables
INTEGER I,J,K,NABSTA,NEVPTA
PARAMETER (NEVPTA=4,NABSTA=4)
REAL A,
+ COSTE(NEVPTA),COSTA(NABSTA),
+ E,FEASE,FEASA,FEAS,
+ L,LT,
+ NAT,NET,NE,NA,
+ PROB(8),
+ TONS(8),TCOST
REAL CT,
+ MfgCOST,
+ TCT
REAL F1,F2,FN1,FN2,FC,
+ S,
+ W1,W2,
+ XE(4),XA(4),
+ Z11,Z12,Z21,Z22,Z
c -----------------------------------------------------------------

DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,NEVPTA) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,NABSTA) /3.25,3.50,3.75,4.0/
DATA (XA(I), I=1,4) /-2,-1,1,2/,
& (XE(I), I=1,4) /-2,-1,1,2/
c -----------------------------------------------------------------
c...input scalar AV values
call avdsca(DLT,'length')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
c
c...compute scalar AF values
c...intermediate constants
C **********************************************************
c ... product analysis
LT=REAL(DLT)
I =INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
J =INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
L = 1.75*(LT+2)+17.0
DL= L
C...compute functions
MatCOST= 0.0
FEAS =16.0
NET = 0.0
NAT = 0.0
DO 100 K=1,8
c ............. Find minimum number of tubes for EVAP
CALL MINEVP(L,I,TONS(K),NE,FEASE)
NET=NET+PROB(K)*NE
c ............. Find minimum number of tubes for ABS
CALL MINABS(L,J,TONS(K),NA,FEASA)
NAT=NAT+PROB(K)*NA
FEAS = FEAS-FEASE-FEASA
MatCOST=PROB(K)*(COSTE(I)*(L*NE)+COSTA(J)*(L*NA))+MatCOST
100 CONTINUE
c ... code vars
NET=(NET-440)/115.0-2
NAT=(NAT-440)/115.0-2
E=XE(I)
A=XA(J)
DNET = NET
DNAT = NAT
DNE = NE
DNA = NA
238

239
c
MatCOST=MatCOST+52335.0
c
c ... Rational Reaction Sets of Production Designer
MfgCOST = 49263.35+1215.23*LT-442.57*E-
412.04*A+2451.33*NET+2208.98*NAT
+ -365.50*LT*E-684.65*LT*A-479.31*LT*NET-
768.31*LT*NAT+357.44*E*A
+ +376.55*E*NET+798.4*E*NAT+686.45*A*NET+823.85*A*NAT
+ -504.06*NET**2-392.44*NET*NAT-344.81*NAT**2
CT = 505.48+11.08*LT+0.09*E+26.67*A+23.41*NET-
3.0*LT*NET+3.63*E**2
+ -3.78*E*NAT+9.38*NET**2+15.38*NET*NAT+4.50*NAT**2
c
COST=MatCOST+MfgCOST
c
************************************************************************
*****
C modified Pareto objective
TCOST = 169810.0
TCT = 318.1
Z11 = 211837/TCOST-1.0
Z12 = 233385/TCOST-1.0
Z21 = 631.6/TCT-1.0
Z22 = 448.0/TCT-1.0
W1 = 0.5
W2 = 0.5
c
c ... COST GOAL
F1 = COST/TCOST-1.0
FN1 = (F1-Z11)/(Z12-Z11)
c ... CT Goal
F2 = CT/TCT-1.0
FN2 = (F2-Z22)/(Z21-Z22)
c
c ... Formulate Supercriterion
S = (1-FN1)*(1-FN2)
c ... Formulate a Pareto Optimal objective FC
FC = W1*FN1+W2*FN2
c ... Modified Objective Function
Z = FC-S+1
c
c
************************************************************************
*****
DCOST = COST
DMatCOST = MatCOST
DFEAS = FEAS
DU = U
DMfgCOST = MfgCOST
DCT = CT

240
DFN1 = FN1
DFN2 = FN2
DZ = Z
c
C
C
************************************************************************
****
c...output scalar AF values
call afdsca(DL,'Length')
call afdsca(DNET,'NET')
call afdsca(DNAT,'NAT')
call afdsca(DNE,'NE')
call afdsca(DNA,'NA')
call afdsca(DFEAS,'PRODUCT_FEAS')
call afdsca(DCOST,'COST')
call afdsca(DMatCOST,'MatCOST')
call afdsca(DMfgCOST,'MfgCOST')
call afdsca(DCT,'CT')
call afdsca(DFN1,'FN1(COST)')
call afdsca(DFN2,'FN2(CT)')
call afdsca(DZ,'Z')
return
end
C FORTRAN FILE FOR OptdesX FOR PROBABILISTIC PRODUCT DESIGN
C WITH MODIFIED DEVIATION FUNCTION
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DCHL,DCHCOST,DCHCT,DCOST,DCT,

241
+ DFEAS,
+ DL,DLT,DPICOST,DPICT,
+ DMatCOST,DMfgCOST,
+ DSELE,DSELA
DOUBLE PRECISION DFN1,DFN2,DZ
c production variables
INTEGER I,J,NABSTA,NEVPTA
PARAMETER (NEVPTA=4,NABSTA=4)
REAL ACHCT,ACHCOST,
+ COST,CT,
+ CHL,CHMfgC,CHCOST,CHCT,CHDCT,
+ FEAS,
+ L,LT,
+ MfgCOST
REAL F1,F2,FN1,FN2,FC,
+ S,
+ W1,W2,
+ Z11,Z12,Z21,Z22,Z
c -----------------------------------------------------------
c...input scalar AV values
call avdsca(DLT,'Length')
call avdsca(DSELE,'EvapTube')
call avdsca(DSELA,'AbsTube')
call avdsca(DCHL,'DLength')
c ************************************************************
CHMfgC = 0.1
CHDCT = 0.05
ACHCOST= 0.1
ACHCT = 0.1
c ************************************************************
I=INT(DSELE)
IF(I.LT.1) I=1
IF(I.GT.NEVPTA) I=NEVPTA
J=INT(DSELA)
IF(J.LT.1) J=1
IF(J.GT.NABSTA) J=NABSTA
LT = REAL(DLT)
CHL = REAL(DCHL)
L = 1.75*(LT+2)+17.0
DL = L
C...compute functions
c ... Evaluate DPIs
CALL EVAL_DPI(CHL,LT,I,J,
+ COST,MfgCOST,CT,
+ CHCOST,CHMfgC,CHDCT,CHCT,
+ FEAS,
+ DPI_COST,DPI_CT)
c ----------------------------------------------------------------------
---
C EVALUATE DEVIATION FUNCTION USING AN ARCHIMEDEAN FORMULATION
C WEIGHT OF EACH GOAL AT EACH PRIORITY LEVEL

242
C modified Pareto objective
Z11 = 0.178
Z12 = 0.431
Z21 = 0.994
Z22 = 0.477
W1 = 0.5
W2 = 0.5
c
c ... COST GOAL
F1 = 1-DPI_COST
c WRITE(*,*) Z11,Z12,F1
FN1 = (F1-Z11)/(Z12-Z11)
c WRITE(*,*) FN1
c ... CT Goal
F2 = 1-DPI_CT
FN2 = (F2-Z22)/(Z21-Z22)
c
c ... Formulate Supercriterion
S = (1-FN1)*(1-FN2)
c ... Formulate a Pareto Optimal objective FC
FC = W1*FN1+W2*FN2
c ... Modified Objective Function
Z = FC-S+1
c
DCHCOST = 1.0-CHCOST/(ACHCOST*COST)
DCHCT = 1.0-CHCT/(ACHCT*CT)
DFN1 = FN1
DFN2 = FN2
DCOST = COST
DCT = CT
DFEAS = FEAS
DMatCOST = COST-MfgCOST
DMfgCOST = MfgCOST
DPICOST = DPI_COST
DPICT = DPI_CT
DZ = Z
C
************************************************************************
****
c...output scalar AF values
call afdsca(DFEAS ,'FEAS')
call afdsca(DCHCOST ,'DCOST')
call afdsca(DCHCT ,'DCT')
call afdsca(DFN1 ,'FN1')
call afdsca(DFN2 ,'FN2')
call afdsca(DPICOST ,'DPICOST')
call afdsca(DPICT ,'DPICT')
call afdsca(DZ ,'Z')
call afdsca(DCOST ,'COST')
call afdsca(DCT ,'CT')
call afdsca(DMatCOST,'MatCOST')

call afdsca(DMfgCOST,'MfgCOST')
return
end
SUBROUTINE EVAL_DPI(DL,LT,I,J,
+ AvgCOST,MfgC,AvgCT,
+ DCOST,DMfgC,DDCT,DCT,
+ FEASTOT,
+ DPI_COST,DPI_CT)
INTEGER I,J,K,M,N,P
PARAMETER(N=27)
INTEGER CCD(N,5)
REAL A,AvgCOST,AvgCT,
+ COSTA(8),COSTE(8),CA,CAT,CE,CET,COST,CT,
+ DCA,DCE,DCOST,DCT,DDC,DDCT,DL,DLT,DLTC,
+ DMC,DMfgC,DPI_COST,DPI_CT,
+ FEAS,FEASE,FEASA,FEASTOT,
+ E,
+ LC,LT,LTC,
+ MatCOST,MAXCOST,MAXCT,MINCOST,MINCT,MfgC,Mfg,
+ NA,NAT,NE,NET,
+ PROB(8),
+ TONS(8),
+ VAL(N),
+ XA(4),XE(4),
+ YCOST(N),YCT(N)
c ... Fractional Factorial Experiment w/resolution 5
DATA ((CCD(K,M),M=1,5),K=1,N)
+ /-1,-1,-1,-1, 1,
+ -1,-1,-1, 1,-1,
+ -1,-1, 1,-1,-1,
+ -1,-1, 1, 1, 1,
+ -1, 1,-1,-1,-1,
+ -1, 1,-1, 1, 1,
+ -1, 1, 1,-1, 1,
+ -1, 1, 1, 1,-1,
+ 1,-1,-1,-1,-1,
+ 1,-1,-1, 1, 1,
+ 1,-1, 1,-1, 1,
+ 1,-1, 1, 1,-1,
+ 1, 1,-1,-1, 1,
+ 1, 1,-1, 1,-1,
+ 1, 1, 1,-1,-1,
+ 1, 1, 1, 1, 1,
+ -2, 0, 0, 0, 0,
+ 2, 0, 0, 0, 0,
+ 0,-2, 0, 0, 0,
+ 0, 2, 0, 0, 0,
+ 0, 0,-2, 0, 0,
+ 0, 0, 2, 0, 0,
+ 0, 0, 0,-2, 0,
+ 0, 0, 0, 2, 0,
243

+ 0, 0, 0, 0,-2,
+ 0, 0, 0, 0, 2,
+ 0, 0, 0, 0, 0/
DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(K),K=1,4) /6.0,6.5,7.0,7.5/,
& (COSTA(K),K=1,4) /3.25,3.50,3.75,4.0/
DATA (XA(K), K=1,4) /-2,-1,1,2/,
& (XE(K), K=1,4) /-2,-1,1,2/
C -----------------------------------------------------------
c ... high and low values for preference functions
MINCOST =169180.0
MINCT =318.1
MAXCOST =320000.0
MAXCT =600.0
c -----------------------------------------------------------
DLTC=4.0/7.0*DL
CET =COSTE(I)
CAT =COSTA(J)
FEASTOT= 0.0
DO 200 K=1,N
IF (CCD(K,1).EQ.-2) THEN
DLT=-DLTC
ELSE IF(CCD(K,1).EQ.-1) THEN
DLT=-DLTC/2.0
ELSE IF(CCD(K,1).EQ.0) THEN
DLT=0.0
ELSE IF(CCD(K,1).EQ.1) THEN
DLT=DLTC/2.0
ELSE
DLT=DLTC
END IF
IF (CCD(K,2).EQ.-2) THEN
DCE=-0.50
ELSE IF(CCD(K,2).EQ.-1) THEN
DCE=-0.25
ELSE IF(CCD(K,2).EQ.0) THEN
DCE= 0.0
ELSE IF(CCD(K,2).EQ.1) THEN
DCE= 0.25
ELSE
DCE= 0.50
END IF
IF (CCD(K,3).EQ.-2) THEN
DCA=-0.25
ELSE IF(CCD(K,3).EQ.-1) THEN
DCA=-0.125
ELSE IF(CCD(K,3).EQ.0) THEN
DCA= 0.0
ELSE IF(CCD(K,3).EQ.1) THEN
244

245
DCA= 0.125
ELSE
DCA= 0.25
END IF
IF (CCD(K,4).EQ.-2) THEN
DM=-DMfgC
ELSE IF(CCD(K,4).EQ.-1) THEN
DM=-DMfgC/2.0
ELSE IF(CCD(K,4).EQ.0) THEN
DMC= 0.0
ELSE IF(CCD(K,4).EQ.1) THEN
DMC= DMfgC/2.0
ELSE
DMC= DMfgC
END IF
IF (CCD(K,5).EQ.-2) THEN
DDC=-DDCT
ELSE IF(CCD(K,5).EQ.-1) THEN
DDC=-DDCT/2.0
ELSE IF(CCD(K,5).EQ.0) THEN
DDC= 0.0
ELSE IF(CCD(K,5).EQ.1) THEN
DDC= DDCT/2.0
ELSE
DDC= DDCT
END IF
LTC=LT+DLT
LC =1.75*(LTC+2)+17.0
CE=(1+DCE)*CAT
CA=(1+DCA)*CET
C ... Calculate COST & MfgC
MatCOST= 0.0
FEAS =16.0
NET = 0.0
NAT = 0.0
DO 100 P=1,8
c ............. Find minimum number of tubes for EVAP (99.7% CI)
CALL MINEVP(LC,I,TONS(P),NE,FEASE)
NET=NET+PROB(P)*NE
c ............. Find minimum number of tubes for ABS (99.7% CI)
CALL MINABS(LC,J,TONS(P),NA,FEASA)
NAT=NAT+PROB(P)*NA
FEAS = FEAS-FEASE-FEASA
MatCOST=PROB(P)*(CE*(LC*NE)+CA*(LC*NA))+MatCOST
100 CONTINUE
FEASTOT=FEASTOT+FEAS
MatCOST=MatCOST+52335.0
c ----------------------------------------------------------------------
--c
... Rational Reaction Sets
c ... code vars

246
NET=(NET-440)/115.0-2
NAT=(NAT-440)/115.0-2
E=XE(I)
A=XA(J)
MfgC = 49263.35+1215.23*LTC-442.57*E-412.04*A+2451.33*NET
+ +2208.98*NAT
+ -365.50*LTC*E-684.65*LTC*A-479.31*LTC*NET-768.31*LTC*NAT
+ +357.44*E*A
+ +376.55*E*NET+798.4*E*NAT+686.45*A*NET+823.85*A*NAT
+ -504.06*NET**2-392.44*NET*NAT-344.81*NAT**2
CT = 505.48+11.08*LTC+0.09*E+26.67*A+23.41*NET-3.0*LTC*NET
+ +3.63*E**2-
3.78*E*NAT+9.38*NET**2+15.38*NET*NAT+4.50*NAT**2
c ----------------------------------------------------------------------
---
MfgC=(1+DMC)*MfgC
CT =(1+DDC)*CT
COST=MatCOST+MfgC
c
C ESTIMATE DEVIATION VARIABLES
C modified Pareto solution
C
YCOST(K)= COST
YCT(K) = CT
C
C ************************************************************
200 CONTINUE
c ... DPI for COST
CALL SORT(YCOST,VAL,AvgCOST,DCOST,MAXCOST,MINCOST,N)
CALL INTEGRAL(DPI_COST,YCOST,VAL,N)
C ... DPI for CT
CALL SORT(YCT,VAL,AvgCT,DCT,MAXCT,MINCT,N)
CALL INTEGRAL(DPI_CT,YCT,VAL,N)
c ------------------------------------------------------------
RETURN
END
SUBROUTINE SORT(Y,VAL,AVG,DY,TMAX,TMIN,N)
INTEGER N,FIRST, PASS, K
REAL AUX,AVG,
+ DY,
+ F,
+ PY,
+ STDDEV,
+ TEMP,TMAX,TMIN,
+ VAL(N),
+ Y(N)
LOGICAL SORTED
PASS=1
SORTED=.FALSE.
DO WHILE(.NOT.SORTED)
SORTED=.TRUE.

DO 10 FIRST=1,N-PASS
IF(Y(FIRST).GT.Y(FIRST+1)) THEN
TEMP=Y(FIRST)
Y(FIRST)=Y(FIRST+1)
Y(FIRST+1)=TEMP
SORTED=.FALSE.
END IF
10 CONTINUE
PASS=PASS+1
END DO
STDDEV=(Y(N)-Y(1))/6.0
STDDEV=ABS(STDDEV)
DY=3*STDDEV
AVG =(Y(N)+Y(1))/2.0
DO 20 K=1,N
IF(Y(K).LT.TMIN) THEN
PY=0.0
ELSE IF(Y(K).GT.TMAX) THEN
PY=0.0
ELSE
PY=(TMAX-Y(K))/(TMAX-TMIN)
END IF
F=-((Y(K)-AVG)/STDDEV)**2.0/2.0
F=EXP(F)
AUX=SQRT(6.2832)
F=1.0/(AUX*STDDEV)*F
VAL(K)=F*PY
20 CONTINUE
RETURN
END
SUBROUTINE INTEGRAL(DPI,X,Y,N)
INTEGER I,N
REAL X(N),Y(N),DPI
DPI=0.0
DO 10 I=1,N-1
DPI=DPI+((Y(I+1)-Y(I))/2.0+Y(I))*(X(I+1)-X(I))
10 CONTINUE
RETURN
END
247
c FORTRAN FILE FOR PRODUCT DESIGN WITH PROBABILISTIC MODEL
C AND MODIFIED DEVIATION FUNCTION
c=======================================================================
=======
c...subroutine ANAPRE
c preprocessing routine

c-----------------------------------------------------------------------
-------
subroutine anapre(modelN)
character*17 modelN
248
modelN = "Chiller Design"
return
end
c-----------------------------------------------------------------------
-------
c=======================================================================
=======
c...subroutine ANAFUN
c sample analysis routine for chiller design
c-----------------------------------------------------------------------
-------
subroutine anafun
c product variables
DOUBLE PRECISION DCHCOST,DCHCT,DCOST,DCT,
+ DF,
+ DMatCOST,DMfgCOST,DMS1,DMS2,DMS3,
+ DU
DOUBLE PRECISION DFN1,DFN2,DZ,DZCOST,DZCT
c production variables
INTEGER I,J
REAL ACHCT,ACHCOST,
+ COST,CT,
+ CHL,CHCOST,CHCT,
+ F,
+ L,
+ MS1,MS2,MS3,MfgCOST,
+ U
REAL F1,F2,FN1,FN2,FC,
+ S,
+ W1,W2,
+ Z11,Z12,Z21,Z22,Z
c
c ... product design solution
c ... Note: LT is the normalized value of L[17,24] -> LT[-2,2]
L = 19.09
c ... CHL is the half-width range of L
CHL = 0.26
c ... I is the evaporator tube, J is the absorber tube
I = 2
J = 4
5 CONTINUE
c -----------------------------------------------------------
c...input scalar AV values
call avdsca(DMS1,'MS1')
call avdsca(DMS2,'MS2')
call avdsca(DMS3,'MS3')

249
call avdsca(DF,'F')
c ************************************************************
ACHCOST= 0.1
ACHCT = 0.1
c ************************************************************
C...compute functions
c ... Evaluate DPIs
MS1 = DMS1
MS2 = DMS2
MS3 = DMS3
F = DF
CALL EVAL_DPI(CHL,L,I,J,
+ COST,MfgCOST,CT,
+ CHCOST,CHCT,
+ MS1,MS2,MS3,F,
+ U,
+ DPI_COST,DPI_CT)
c ----------------------------------------------------------------------
---
C EVALUATE DEVIATION FUNCTION USING AN ARCHIMEDEAN FORMULATION
C WEIGHT OF EACH GOAL AT EACH PRIORITY LEVEL
C modified Pareto objective
Z11 = 0.170
Z12 = 0.304
Z21 = 1.000
Z22 = 0.417
W1 = 0.5
W2 = 0.5
c
c ... COST GOAL
F1 = 1-DPI_COST
FN1 = (F1-Z11)/(Z12-Z11)
c ... CT Goal
F2 = 1-DPI_CT
FN2 = (F2-Z22)/(Z21-Z22)
c
c ... Formulate Supercriterion
S = (1-FN1)*(1-FN2)
c ... Formulate a Pareto Optimal objective FC
FC = W1*FN1+W2*FN2
c ... Modified Objective Function
Z = FC-S+1
c
DCHCOST = 1.0-CHCOST/(ACHCOST*COST)
DCHCT = 1.0-CHCT/(ACHCT*CT)
DCOST = COST
DCT = CT
DFN1 = FN1
DFN2 = FN2
DU = U
DMatCOST = COST-MfgCOST

250
DMfgCOST = MfgCOST
DZCOST = 1.0-DPI_COST
DZCT = 1.0-DPI_CT
DZ = Z
C
************************************************************************
****
c...output scalar AF values
call afdsca(DU ,'Utilization')
call afdsca(DCHCOST ,'DCOST')
call afdsca(DCHCT ,'DCT')
call afdsca(DFN1 ,'FN1')
call afdsca(DFN2 ,'FN2')
call afdsca(DZ ,'Z')
call afdsca(DZCOST ,'Z1(COST)')
call afdsca(DZCT ,'Z2(CT)')
call afdsca(DCOST ,'COST')
call afdsca(DCT ,'CT')
call afdsca(DMatCOST,'MatCOST')
call afdsca(DMfgCOST,'MfgCOST')
return
end
c=======================================================================
=======
c...subroutine ANAPOS
c postprocessing routine
c-----------------------------------------------------------------------
-------
subroutine anapos
return
end
SUBROUTINE EVAL_DPI(DL,L,I,J,
+ AvgCOST,MfgC,AvgCT,
+ DCOST,DCT,
+ MS1,MS2,MS3,F,
+ MAXU,
+ DPI_COST,DPI_CT)
INTEGER I,ISEED,J,K,M,N,LTABLE
PARAMETER(N=5)
REAL AvgCOST,AvgCT,
+ COST,CT,
+ DCOST,DCT,DL,
+ DPI_COST,DPI_CT,COSTE(4),COSTA(4),
+ F,
+ L,LC,LT(8),L_RAND,
+ MatCOST,MAXCOST,MAXCT,MINCOST,MINCT,MfgC,MAXU,

+ MS1,MS2,MS3,
+ NA(8),NAT(8),NE(8),NET(8),NETABLE(8,5),NATABLE(8,5),
+ PROB(8),
+ TONS(8),
+ VAL(N),
+ YCOST(N),YCT(N)
DATA (TONS(K), K=1,8) /600,700,800,900,1000,1100,1200,1300/,
& (PROB(K), K=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,4) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,4) /3.25,3.50,3.75,4.0/
ISEED=20029
DATA ((NETABLE(M,K),K=1,5),M=1,8)
+ /440,440,440,440,440,
+ 440,440,440,440,440,
+ 475,467,464,457,453,
+ 536,532,525,518,514,
+ 600,593,586,579,572,
+ 661,654,647,640,633,
+ 726,716,708,701,694,
+ 787,780,769,762,751/
DATA ((NATABLE(M,K),K=1,5),M=1,8)
+ /440,440,440,440,440,
+ 440,440,440,440,440,
+ 482,482,482,467,467,
+ 554,539,539,528,528,
+ 611,611,601,601,590,
+ 672,672,662,661,647,
+ 744,734,734,719,708,
+ 805,805,791,780,781/
C -----------------------------------------------------------
c ... high and low values for preference functions
MINCOST =169180.0
MINCT =318.1
MAXCOST =320000.0
MAXCT =600.0
c -----------------------------------------------------------
MAXU = 0.0
DO 200 K=1,N
c
c random length [L-DL,L+DL]
L_RAND=(L-DL)+2*DL*RANDOM(ISEED)
LC=(L_RAND-17.0)/1.75-2.0
MatCOST=0.0
DO 90 M=1,8
LT(M)=LC
IF(LC.LT.-1.0) THEN
LTABLE=1
ELSE IF(LC.LT.0) THEN
LTABLE=2
ELSE IF(LC.LT.1) THEN
251

c
c
c
LTABLE=3
ELSE IF(LC.LE.2) THEN
LTABLE=4
END IF
x=L-DL+LTABLE*DL/2.0
NE(M)=(NETABLE(M,LTABLE+1)-NETABLE(M,LTABLE))/DL
NE(M)=NE(M)*(L_RAND-x)+NETABLE(M,LTABLE+1)
NET(M)=(NE(M)-440)/115.0-2.0
NA(M)=(NATABLE(M,LTABLE+1)-NATABLE(M,LTABLE))/DL
NA(M)=NA(M)*(L_RAND-x)+NATABLE(M,LTABLE+1)
NAT(M)=(NA(M)-440)/115.0-2.0
252
MatCOST=PROB(M)*(COSTE(I)*L_RAND*NE(M)+
+ COSTA(J)*L_RAND*NA(M))+MatCOST
90 CONTINUE
MatCOST=MatCOST+52335.0
c
CALL QNA(MS1,MS2,MS3,F,LT,NET,NAT,I,J,U,CT,MfgC)
IF(U.GT.MAXU) MAXU=U
c
COST=MfgC+MatCOST
c ----------------------------------------------------------------------
---
C
YCOST(K)= COST
YCT(K) = CT
C
C ************************************************************
200 CONTINUE
c ... DPI for COST
CALL SORT(YCOST,VAL,AvgCOST,DCOST,MAXCOST,MINCOST,N)
CALL INTEGRAL(DPI_COST,YCOST,VAL,N)
C ... DPI for CT
CALL SORT(YCT,VAL,AvgCT,DCT,MAXCT,MINCT,N)
CALL INTEGRAL(DPI_CT,YCT,VAL,N)
c ------------------------------------------------------------
RETURN
END
B.2 SUBROUTINES TO FIND MINIMUM NUMBER OF TUBES GIVEN LENGTH AND TUBES
SELECTION
C SUBROUTINE TO FIND MINIMUM NUMBER OF TUBES OF EVAPORATOR
SUBROUTINE MINEVP(L,I,TONS,MinNET,FEAS)
c production variables
INTEGER I,NEVPTA
c product variables
PARAMETER (NEVPTA=4)

REAL DPE,
+ DIE(NEVPTA),DOE(NEVPTA),DR(NEVPTA),
+ FH(NEVPTA),FPI(NEVPTA),FW(NEVPTA),FEAS,
+ L,FLOOR,ROOF,
+ NET,MinNET,
+ THKE(NEVPTA),TONS,
+ UAE,UAENOM
C TUBES ALTERNATIVES
c evaporator
DATA (DOE(I), I=1,NEVPTA) /0.625,0.75,0.875,1.0/,
& (THKE(I), I=1,NEVPTA) /0.025,0.025,0.02,0.025/,
& (FPI(I), I=1,NEVPTA) /30,30,30,30/,
& (DIE(I), I=1,NEVPTA) /0.444,0.569,0.694,0.819/,
& (DR(I), I=1,NEVPTA) /0.5,0.625,0.75,0.875/,
& (FW(I), I=1,NEVPTA) /0.02,0.02,0.02,0.02/,
& (FH(I), I=1,NEVPTA) /0.05,0.05,0.05,0.05/
UAENOM=1372.183*TONS-34091.8
FLOOR=440
ROOF =900
MinNET=900
FEAS=0
CALL EVAP(TONS,DOE(I),THKE(I),FPI(I),DIE(I),DR(I),
& FW(I),FH(I),L,ROOF,UAE,DPE)
IF(UAE. GE. UAENOM. AND. DPE.LE.23.0) THEN
FEAS=1
CALL EVAP(TONS,DOE(I),THKE(I),FPI(I),DIE(I),DR(I),
& FW(I),FH(I),L,FLOOR,UAE,DPE)
IF(UAE. GE. UAENOM. AND. DPE.LE.23.0) MinNET=FLOOR
END IF
IF(FEAS.EQ.1 .AND. MinNET.GT.FLOOR) THEN
NET=FLOOR
DO 100 WHILE(ABS(MinNET-NET). GT. 1.0)
NET=(FLOOR+ROOF)/2.0
MinNET=(NET+ROOF)/2.0
CALL EVAP(TONS,DOE(I),THKE(I),FPI(I),DIE(I),DR(I),
& FW(I),FH(I),L,NET,UAE,DPE)
IF(UAE. GT. UAENOM. AND. DPE.LE.23.0) THEN
ROOF=NET
ELSE
FLOOR=NET
END IF
100 CONTINUE
ELSE
END IF
RETURN
END
C SUBROUTINE TO FIND MINIMUM NUMBER OF TUBES OF ABSORBER
SUBROUTINE MINABS(L,J,TONS,MinNAT,FEAS)
c production variables
INTEGER I,J,NABSTA,NRA,NATR
253

254
c product variables
PARAMETER (NABSTA=4)
REAL DOA(NABSTA),THKA(NABSTA),
+ L,FLOOR,ROOF,FEAS,
+ NAT,MinNAT,
+ TONS,
+ UAA, UAANOM
C TUBES ALTERNATIVES
DATA (DOA(I), I=1,NABSTA) /0.625,0.75,0.875,1.0/,
& (THKA(I), I=1,NABSTA) /0.03,0.03,0.03,0.03/
c evaporator
UAANOM=1021.806*TONS-38163.0
FLOOR=440
ROOF =900
MinNAT=900
FEAS=0
NRA=12
C CALCULATE EVAPORATOR UA
NATR = INT(ROOF/NRA)
CALL ABSE(TONS,DOA(J),THKA(J),L,NRA,NATR,UAA,CAPACITY)
IF(UAA. GE. UAANOM. AND. CAPACITY.GE.TONS) THEN
FEAS=1
NATR = INT(FLOOR/NRA)
CALL ABSE(TONS,DOA(J),THKA(J),L,NRA,NATR,UAA,CAPACITY)
IF(UAA. GE. UAANOM. AND. CAPACITY.GE.TONS)
MinNAT=FLOOR
END IF
IF(FEAS.EQ.1 .AND. MinNAT.GT.FLOOR) THEN
NAT=FLOOR
DO 100 WHILE(ABS(MinNAT-NAT). GT. 1.0)
NAT=(FLOOR+ROOF)/2.0
MinNAT=(NAT+ROOF)/2.0
NATR = INT(NAT/NRA)
CALL ABSE(TONS,DOA(J),THKA(J),L,NRA,NATR,UAA,CAPACITY)
IF(UAA. GT. UAANOM. AND. CAPACITY.GE.TONS) THEN
ROOF=NAT
ELSE
FLOOR=NAT
END IF
100 CONTINUE
ELSE
END IF
RETURN
END

B.3 RESPONSE SURFACE NETWORK MODEL
C RESPONSE SURFACE NETWORK MODEL QNA
SUBROUTINE QNA(MS1,MS2,MS3,F,LT,NTE,NTA,ET,AT,MAXU,CTTOT,COST)
C ***************************************************************
INTEGER I,J,K,L,NODES,R,ET,AT
PARAMETER (NODES=5,R=8)
REAL A(R),ALFA(NODES),AR(R),ARN(NODES),ARRIVAL,AVAL,
+ ASSYM,
+ B(NODES,NODES),BS(NODES),
+ CA2(R),CA2N(NODES),CT(R),CTtot,
+ CS2(R,10),CS2N(NODES),C02N(NODES),
+ DN(NODES),DRN(NODES),
+ EN(NODES),EW(NODES),
+ FR(NODES,NODES),FRN(NODES),F,
+ MATRIX(NODES,NODES),MF(NODES),MR(NODES),
+ MAXU,MS1,MS2,MS3,
+ NTE(R),NTA(R),
+ P(NODES,NODES),P0(NODES),PCT(R),
+ Q(NODES,NODES),QAUX(NODES),
+ TSN(NODES),TS(R,10),
+ U(NODES),U0(NODES),
+ V(NODES),VAVG(NODES),
+ W(NODES),WAVG(NODES),X(NODES),
+ D,S1,S2,S3,LT,NET,NAT
REAL CTQ,AUX,S1AUX,USA1,USA2,USA3,CTQSA1,CTQSA2,CTQSA3
REAL PCSA1,PCSA2,PCSA3,PCS1,PCS2,PCS3,PCWTM1,PCWTM2,PCWTM3,
+ MCSA1,MCSA2,MCSA3,
+ ICSA1,ICSA2,ICSA3,ICS1,ICS2,ICS3,
+ LCSA1,LCSA2,LCSA3,LCS1,LCS2,LCS3,
+ WIPSA1,WIPSA2,WIPSA3,WIPWTM1,WIPWTM2,WIPWTM3,
+ TSA1,TSA2,TSA3,
+ COST,PCTUBE
INTEGER M(NODES),NN(R),N(R,10),Z
C --------------------------------------------------------------
C INPUT DATA
DATA (BS(I),I=1,NODES) /1.0,1.0,1.0,1.0,1.0/,
& (M(I),I=1,NODES) /1,1,3,2,3/,
& (MF(I),I=1,NODES) /80.0,0.0,80.0,80.0,80.0/,
& (MR(I),I=1,NODES) /88.0,88.0,88.0,88.0,88.0/,
& (PCT(I),I=1,R) /0.05,0.15,0.16,0.05,0.15,
& 0.12,0.25,0.07/
C
C
C Specify route
M(3) = MS1
M(4) = MS2
M(5) = MS3
TSA1=0.0
TSA2=0.0
TSA3=0.0
255

ARRIVAL=80.0/365./24.
DO 10 Z=1,R
NN(Z)=5
N(Z,1)=1
N(Z,2)=2
N(Z,3)=3
N(Z,4)=4
N(Z,5)=5
AR(Z)=ARRIVAL*PCT(Z)
NET=NTE(Z)
NAT=NTA(Z)
AVAL=0.59
C --------------------------------------------------------
C RESPONSE SURFACES
TS(Z,1)=24.159+0.063*LT+0.563*NET+0.688*NAT-0.045*LT**2
+ +0.125*NET*LT+0.080*NET**2+0.125*NAT*LT
+ -0.125*NAT*NET+0.205*NAT**2
TS(Z,1)=TS(Z,1)/AVAL
C This function calculates the time before the TACK-UP
C It is considered that the fabrication of the subcomponents
C start at the same time and the processing time of each is
C exponentially distributed and independent of the others
C MEAN TIMES OF THE PRODUCT
S1=24.159+0.063*LT+0.563*NET+0.688*NAT-0.045*LT**2
+ +0.125*NET*LT+0.080*NET**2+0.125*NAT*LT
+ -0.125*NAT*NET+0.205*NAT**2
S1=S1/AVAL
S2=29.295-0.312*LT+2.062*NET+1.397*NAT-0.023*LT**2
+ -0.125*NET*LT-0.102*NET**2-0.125*NAT*LT
+ -0.125*NAT*NET-0.227*NAT**2
S2=S2/AVAL
S3=8.636+1.25*NET+1.25*NAT-0.056*LT**2
+ -0.056*NET**2-0.25*NAT*LT
+ +0.25*NAT*NET+0.068*NAT**2
S3=S3/AVAL
TSA1=TSA1+S1*PCT(Z)
TSA2=TSA2+S2*PCT(Z)
TSA3=TSA3+S3*PCT(Z)
S1AUX=S1
S1=1.0/S1-ARRIVAL
S2=1.0/S2-ARRIVAL
S3=1.0/S3-ARRIVAL
ASSYM=1.0/S1+1.0/S2+1.0/S3
+ -1.0/(S1+S2)-1.0/(S1+S3)
+ -1.0/(S2+S3)+1.0/(S1+S2+S3)
ASSYM=ASSYM-S1AUX
TS(Z,2)=ASSYM
TS(Z,3)=66.04+1.312*LT+3.801*NET+4.062*NAT+0.147*LT**2
+ -0.125*NET*LT+0.147*NET**2-0.125*NAT*LT
+ +0.875*NAT*NET+0.397*NAT**2
TS(Z,3)=TS(Z,3)/AVAL
256

TS(Z,4)=63.272+1.05*LT+4.375*NET+4.875*NAT-0.363*LT**2
+ +0.011*NET**2-0.378*NAT**2
TS(Z,4)=TS(Z,4)/AVAL
TS(Z,5)=81.954+2.937*LT+6.187*NET+5.937*NAT+0.102*LT**2
+ -0.125*NET*LT+0.227*NET**2-0.125*NAT*LT
+ +0.125*NAT*NET-0.522*NAT**2
TS(Z,5)=TS(Z,5)/AVAL
CS2(Z,1)=1.0
CS2(Z,2)=0.0
CS2(Z,3)=1.0
CS2(Z,4)=1.0
CS2(Z,5)=1.0
10 CONTINUE
CLOSE(15)
C Obtain external arrival rates to each node I
S1=0.0
DO 110 J=1,NODES
ARN(J)=0.0
C Check if the first node of customer j is i
DO 100 K=1,R
IF(N(K,1).EQ.J) ARN(J)=ARN(J)+AR(K)
100 CONTINUE
S1=S1+ARN(J)
110 CONTINUE
DO 115 I=1,NODES
P0(I)=ARN(I)/S1
115 CONTINUE
C Obtain FR(i,j)
DO 150 I=1,NODES
DO 140 J=1,NODES
FR(I,J)=0.0
DO 130 K=1,R
DO 120 L=1,NN(K)-1
IF(N(K,L).EQ.I. AND .N(K,L+1).EQ.J) THEN
FR(I,J)=FR(I,J)+AR(K)
END IF
120 CONTINUE
130 CONTINUE
140 CONTINUE
150 CONTINUE
C Obtain departure rates from node I out of network
DO 160 J=1,NODES
DRN(J)=0.0
C Check if the first node of customer j is i
DO 155 K=1,R
IF(N(K,NN(K)).EQ.J) DRN(J)=DRN(J)+AR(K)
155 CONTINUE
160 CONTINUE
C
C Obtain the routing matrix Q
DO 190 I=1,NODES
257

DO 180 J=1, NODES
S1=0.0
DO 170 K=1,NODES
S1=S1+FR(I,K)
170 CONTINUE
Q(I,J)=REAL(FR(I,J))/REAL(DRN(I)+S1)
180 CONTINUE
190 CONTINUE
C Obtain service-time for the nodes
USA1=ARRIVAL*TSA1
USA2=ARRIVAL*TSA2
USA3=ARRIVAL*TSA3
WIPSA1=USA1/(1-USA1)
WIPSA2=USA2/(1-USA2)
WIPSA3=USA3/(1-USA3)
CTQSA1=TSA1*WIPSA1
CTQSA2=TSA2*WIPSA2
CTQSA3=TSA3*WIPSA3
DO 220 J=1,NODES
S1=0.0
S2=0.0
S3=0.0
DO 210 K=1,R
DO 200 L=1,NN(K)
IF(N(K,L).EQ.J) THEN
S1=S1+AR(K)*TS(K,L)
S2=S2+AR(K)*(TS(K,L)**2)*(CS2(K,L)+1)
S3=S3+AR(K)
END IF
200 CONTINUE
210 CONTINUE
TSN(J)=S1/S3
CS2N(J)=S2/S3/TSN(J)**2-1.0
220 CONTINUE
C --------------------------------------------------------
C Eliminate immediate feedback
DO 225 I=1,NODES
QAUX(I)=Q(I,I)
TSN(I)=TSN(I)/(1.0-Q(I,I))
CS2N(I)=CS2N(I)*(1.0-Q(I,I))+Q(I,I)
Q(I,I)=0.0
DO 224 J=1,NODES
IF(J.NE.I) Q(I,J)=Q(I,J)/(1-QAUX(I))
224 CONTINUE
225 CONTINUE
C --------------------------------------------------------
C SOLVE SYSTEM OF TRAFFIC-RATE EQUATIONS
DO 240 I=1,NODES
DO 230 J=1,NODES
IF(I.EQ.J) THEN
MATRIX(I,J)=1.0-BS(I)*Q(I,I)
258

ELSE
MATRIX(I,J)=-Q(J,I)
END IF
230 CONTINUE
240 CONTINUE
OPEN(UNIT=15,FILE='matrix')
write(15,*) NODES
do 242,i=1,nodes
do 241,j=1,nodes
write(15,*) MATRIX(I,J)
241 CONTINUE
242 CONTINUE
DO 243 J=1,NODES
WRITE(15,*) ARN(J)
243 CONTINUE
CLOSE(15)
CALL SYSTEM('solveqs.out')
OPEN(UNIT=15,FILE='lambdas')
DO 260 I=1,NODES
C Calculate utilization and server load
READ(15,*) FRN(I)
U0(I)=FRN(I)*TSN(I)/REAL(M(I))
IF(U0(I).GT.0.999) U0(I)=0.999
ALFA(I)=FRN(I)*TSN(I)
260 CONTINUE
CLOSE(15)
C ---------------------------------------------------------
C
C Calculate arrival rate to node j from node i
C proportion of arrivals to j from i
C departure rate out of network from i
C total departure rate
C Calculate modified variability function U(i)
D=0.0
MAXU=0.0
DO 280 I=1,NODES
S1=0.0
IF (I.LT.NODES) THEN
AUX=(1-U0(I+1))**2/(1-U0(I))**2
U(I)=U0(I)*MIN(1.0,AUX)
ELSE
U(I)=U0(I)
END IF
DO 270 J=1, NODES
FR(I,J)=FRN(I)*BS(I)*Q(I,J)
P(I,J)=FR(I,J)/FRN(J)
S1=S1+Q(I,J)
270 END DO
DN(I)=FRN(I)*BS(I)*(1-S1)
D=D+DN(I)
IF(U(I).GT.MAXU) MAXU=U(I)
259

280 CONTINUE
DO 300 J=1,NODES
V(J)=0.0
DO 290 I=1,NODES
V(J)=V(J)+P(I,J)**2
290 CONTINUE
V(J)=1.0/(V(J)+P0(J)**2)
W(J)=1.0/(1+4*((1-U(J))**2)*(V(J)-1))
S1=REAL(M(J))
X(J)=1+1.0*(AMAX1(CS2N(J),0.2)-1.0)/SQRT(S1)
300 CONTINUE
C ----------------------------------------------------------
C Calculate VAVG(J) AND WAVG(J)
DO 330 J=1,NODES
VAVG(J)=0.0
S2=0.0
DO 320 K=1,R
S1=0.0
DO 310 L=1,R
IF(N(L,1).EQ.J) S1=S1+AR(L)
310 CONTINUE
IF(N(K,1).EQ.J) THEN
VAVG(J)=VAVG(J)+(AR(K)/S1)**2
S2=S2+(AR(K)/S1)*CA2(K)
END IF
320 CONTINUE
IF(VAVG(j).EQ.0) THEN
VAVG(J)=0.0
WAVG(J)=0.0
C02N(J)=0.0
ELSE
VAVG(J)=1/VAVG(J)
WAVG(J)=1.0/(1.0+4*(1-U(J))**2*(VAVG(J)-1))
C02N(J)=1-WAVG(J)+WAVG(J)*S2
END IF
330 CONTINUE
C ------------------------------------------------------------
C NOTE: MODIFICATION OF A(J) AND B(I,J) BASED ON
C GENERAL FORMULA OF BITRAN AND TIRUPATI (1988) FOR
C MULTIPRODUCT QUEUING NETWORKS
OPEN(UNIT=15,FILE='matrix')
write(15,*) NODES
DO 350 J=1,NODES
S1=0.0
DO 340 I=1, NODES
B(I,J)=W(J)*P(I,J)*Q(I,J)*BS(I)*(1-U(I)**2+
& (1-Q(I,J))*(1-P(I,J)))
S1=S1+P(I,J)*((1-Q(I,J))*P(I,J)+BS(I)*
& Q(I,J)*U(I)**2*X(I))
S1=S1
IF(I.EQ.J) THEN
260

MATRIX(I,J)=1.0-B(I,I)
ELSE
MATRIX(J,I)=-B(I,J)
END IF
write(15,*) MATRIX(I,J)
340 CONTINUE
A(J)=1.0+W(J)*((P0(J)*C02N(J)-1.0)+S1)
350 CONTINUE
C ---------------------------------------------------------------
C Solve system of traffic variability equations
DO 351 J=1,NODES
WRITE(15,*) A(J)
351 CONTINUE
CLOSE(15)
CALL SYSTEM('solveqs.out')
OPEN(UNIT=15,FILE='lambdas')
DO 355 I=1,NODES
C Calculate utilization and server load
READ(15,*) CA2N(I)
355 CONTINUE
CLOSE(15)
C --------------------------------------------------------------
C CALCULATE THE PERFORMANCE OF THE NETWORK (Congestion at the nodes)
CTQ=0.0
DO 800 I=1,NODES
S1=2*(M(I)+1)
S1=SQRT(S1)-1.0
EW(I)=(CA2N(I)+CS2N(I))/2.0*TSN(I)*
& U0(I)**S1/(M(I)*(1.0-U0(I)))
EN(I)=ALFA(I)+FRN(I)*EW(I)
C Return to the original measures (before eliminating feedback)
FRN(I)=FRN(I)/(1.0-QAUX(I))
TSN(I)=(1.0-QAUX(I))*TSN(I)
CS2N(I)=(CS2N(I)-QAUX(I))/(1.0-QAUX(I))
EW(I)=(1-QAUX(I))*EW(I)
CTQ=CTQ+EW(I)
800 CONTINUE
CTtot=0.0
DO 900 K=1,R
CT(K)=0.0
DO 850 I=1,NODES
IF(I.EQ.2) TS(K,I)=TSN(I)
CT(K)=TS(K,I)+CT(K)
850 CONTINUE
CT(K)=CT(K)+CTQ
CTtot=CTtot+CT(K)*PCT(K)
900 CONTINUE
910 FORMAT(1X,F12.3,F12.3)
920 FORMAT(1X,F12.3,F12.3,F12.3)
C
C -------------------------------------------------------------------
261

C Estimate cost of the product
MCSA1=27620.0
MCSA2=18810.0
MCSA3= 5905.0
PCSA1=MCSA1
PCSA2=MCSA2
PCSA3=MCSA3
LCSA1=17.0*2*TSA1
LCSA2=17.0*2*TSA2
LCSA3=17.0*2*TSA3
COST=LCSA1+LCSA2+LCSA3
ICSA1=WIPSA1*PCSA1*0.1/80.0
ICSA2=WIPSA2*PCSA2*0.1/80.0
ICSA3=WIPSA3*PCSA3*0.1/80.0
COST=COST+ICSA1+ICSA2+ICSA3
PCWTM1=PCSA1+LCSA1+ICSA1
PCWTM2=PCSA2+LCSA2+ICSA2
PCWTM3=PCSA3+LCSA3+ICSA3
WIPWTM1=TSN(2)*ARRIVAL
WIPWTM2=(TSN(2)+TSN(1)+CTQSA1-TSA2-CTQSA2)*ARRIVAL
WIPWTM3=(TSN(2)+TSN(1)+CTQSA1-TSA3-CTQSA3)*ARRIVAL
PCS1=PCWTM1+WIPWTM1*PCWTM1*0.1/80.0+
+ PCWTM2+WIPWTM2*PCWTM2*0.1/80.0+
+ PCWTM3+WIPWTM3*PCWTM3*0.1/80.0
COST=COST+WIPWTM1*PCWTM1*0.1/80.0+
+ PCWTM2+WIPWTM2*PCWTM2*0.1/80.0+
+ PCWTM3+WIPWTM3*PCWTM3*0.1/80.0
LCS1=17.0*MS1*TSN(3)
ICS1=PCS1*(ARRIVAL*EW(3))*0.1/80.0
PCS2=PCS1+LCS1+ICS1
LCS2=17.0*MS2*TSN(4)
ICS2=PCS2*ARRIVAL*EW(4)*0.1/80.0
PCS3=PCS2+ICS2+LCS2
LCS3=17.0*MS3*TSN(5)
ICS3=PCS2*ARRIVAL*EW(5)*0.1/80.0
CALL CTUBES(PCTUBE,F,LT,NTE,NTA,ET,AT)
COST=COST+LCS1+ICS1+LCS2+ICS2+LCS3+ICS3+PCTUBE
END
SUBROUTINE CTUBES(COST,F,LT,NET,NAT,ET,AT)
INTEGER R,I,ET,AT
REAL INVE,INVA,Q,F,S,TETA,CE(8),CA(8),D(8),NE(8),NRVAL,
+ NA(8),F1(8),PCT(8),L(8),DEM,AUX1,AUX2,CAVG,COST,
+ CE1(8),CA1(8),LT,NET(8),NAT(8),COSTE(4),COSTA(4)
DEM=80.0
DATA (PCT(I),I=1,8) /0.05,0.15,0.16,0.05,
+ 0.15,0.12,0.25,0.07/
DATA (COSTE(I),I=1,4) /6.0,6.5,7.0,7.5/,
& (COSTA(I),I=1,4) /3.25,3.50,3.75,4.0/
S=0.99
INVE=0.0
262

INVA=0.0
CAVG=0.0
DO 20 I=1,8
D(I)=DEM*PCT(I)
CE(I)=COSTE(ET)
CA(I)=COSTA(AT)
L(I)=(LT+2)*1.75+17
NE(I)=(NET(I)+2)*115+440
NA(I)=(NAT(I)+2)*115+440
CE1(I)=L(I)*NE(I)*CE(I)
CA1(I)=L(I)*NA(I)*CA(I)
CAVG=CAVG+PCT(I)*(CE1(I)+CA1(I))
F1(I)=F*PCT(I)
TETA=D(I)/54.0*7.0
AUX1=INT(D(I)/F1(I))
AUX2=NINT(D(I)/F1(I))
Q=MAX0(AUX1,AUX2)
R=1
CALL NR(R,TETA,NRVAL)
AUX1=(1.0-S)*Q
DO WHILE (NRVAL.GT.AUX1)
CALL NR(R,TETA,NRVAL)
R=R+1
END DO
INVE=INVE+CE1(I)*(Q/2.0+R-TETA)
INVA=INVA+CA1(I)*(Q/2.0+R-TETA)
WRITE(18,*) Q*NE(I), R*NE(I),Q*NA(I),R*NA(I),(R-TETA)*NE(I)
20 CONTINUE
COST=(INVE+INVA)*0.1/80.0
RETURN
END
SUBROUTINE NR(R,TETA,NRVAL)
INTEGER K,R
REAL GR,NRVAL,PR,TETA
GR=EXP(-TETA)
DO 20 K=1,R
CALL PROB(PR,K,TETA)
GR=GR+PR
20 CONTINUE
CALL PROB(PR,R,TETA)
NRVAL=TETA*PR+(TETA-R)*(1.0-GR)
RETURN
END
SUBROUTINE PROB(PR,K,LAMBDA)
INTEGER I,K
REAL LNPK,PR,LAMBDA,AUX
LNPK=-LAMBDA+K*LOG(LAMBDA)
DO 10 I=1,K
263

AUX=REAL(I)
LNPK=LNPK-LOG(AUX)
10 CONTINUE
PR=EXP(LNPK)
RETURN
END
264