Development of the parametric tolerance modeling and optimization ...

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Development of the parametric tolerance modeling and optimization ...

Innovative Applications of O.R.

Development of the parametric tolerance modeling and optimization schemes

and cost-effective solutions

Sangmun Shin a, *, Pauline Kongsuwon a , Byung Rae Cho b

a Department of Systems Management and Engineering, Inje University, Gimhae 621749, South Korea

b Department of Industrial Engineering, Clemson University, Clemson, SC 29634, USA

article info

Article history:

Received 5 December 2008

Accepted 7 July 2010

Available online 17 July 2010

Keywords:

Process parameters

Tolerance

Quality loss

Lambert W function

Optimization

1. Introduction

abstract

The continuous improvement of the quality of products has

become an integral part of the business strategy of most enterprises.

Designing for quality, in particular, has proven to be a key

concept, helping many companies not only to improve product

quality but also to reduce costs. It is widely accepted that approximately

70% of production expenses are incurred by tolerance-related

design efforts (Caleb Li and Chen, 2001). A tight tolerance

usually implies high manufacturing cost due to the additional

manufacturing operations, slow processing rates, additional care

required on the part of the operator, and the expensive measuring

and processing equipment. As a result, the functional performance

is improved. On the other hand, a loose tolerance reduces the manufacturing

cost but may, at the same time, lower the product quality

level considerably. Thus, determining the optimal tolerance

involves a trade-off between the level of quality based on functional

performance and the associated costs. To facilitate this economic

trade-off, researchers typically express quality in monetary

terms using a quality loss function. This function is widely used in

the literature as a reasonable approximation of the actual loss to

the customer due to the deviation of the product performance from

its target value. By expressing the level of quality in monetary

* Corresponding author. Tel.: +82 55 320 3670; fax: +82 55 320 3632.

E-mail addresses: sshin@inje.ac.kr (S. Shin), p_kongsuwan@yahoo.com (P.

Kongsuwon), bcho@clemson.edu (B.R. Cho).

European Journal of Operational Research 207 (2010) 1728–1741

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

0377-2217/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2010.07.009

Most of previous research on tolerance optimization seeks the optimal tolerance allocation with process

parameters such as fixed process mean and variance. This research, however, differs from the previous

studies in two ways. First, an integrated optimization scheme is proposed to determine both the optimal

settings of those process parameters and the optimal tolerance simultaneously which is called a parametric

tolerance optimization problem in this paper. Second, most tolerance optimization models require rigorous

optimization processes using numerical methods, since closed-form solutions are rarely found. This

paper shows how the Lambert W function, which is often used in physics, can be applied efficiently to this

parametric tolerance optimization problem. By using the Lambert W function, one can express the optimal

solutions to the parametric tolerance optimization problem in a closed-form without resorting to

numerical methods. For verification purposes, numerical examples for three cases are conducted and sensitivity

analyses are performed.

Crown Copyright Ó 2010 Published by Elsevier B.V. All rights reserved.

terms, the problem of product trade-off with costs is converted

into a problem of minimizing the total expense, which is the

sum of the quality loss and the costs, i.e., those associated with

the tolerance, including manufacturing, inspection, and rejection

costs. Determining the optimal tolerance is equivalent to determining

the optimal specification limits because the term tolerance

refers to the distance between the lower and upper specification

limits of a product.

Most previous research addresses the tolerance optimization

problem through models seeking the optimal tolerance allocation

by using fixed settings of process parameters (i.e., the process

mean and variance). This research, however, differs from previous

studies of this problem in three ways. First, in this paper, an integrated

optimization scheme is proposed in order to determine the

optimal settings of those process parameters and the optimal tolerance

by simultaneously considering the quality loss incurred

by the customer, and the manufacturing and rejection costs

incurred by the producer. Second, this paper proposes three cases

in order to conduct a sequential optimization procedure by optimizing

process parameters and tolerance. Third, this paper then

shows how the Lambert W function, widely used in physics

(Corless et al., 1996), can be applied efficiently to the tolerance

optimization problem. There are two significant benefits from

using the Lambert W function in this context. Most current tolerance

optimization models require rigorous optimization processes

using complicated numerical methods since closed-form solutions

are rarely found. By using the Lambert W function, however,


quality practitioners can express their solutions in a closed-form,

in addition to being able to determine optimal tolerances quickly

without resorting to numerical methods, since a number of popular

mathematical software packages contain this function. Finally,

numerical examples for three cases are conducted and sensitivity

analyses are performed for verification purposes. An overview of

the proposed procedure is illustrated in Fig. 1.

2. Literature review

Determining the optimal process mean and tolerance is usually

considered separately in the literature. The following review investigates

the current methodologies used in both areas.

2.1. Process parameter optimization

The initial work in determining an optimal process mean probably

was began by Springer (1951), who considered the problem

with specified upper and lower specification limits under the

assumption of constant net income functions. However, when

the minimum content was dictated by government legislation, as

is often the situation, the lower limit was fixed while the upper

was arbitrary. In such a case, underfilled cans, for example, need

to be either reprocessed or sold at a lower price on a secondary

market. Golhar (1987) and Golhar and Pollock (1988) developed

models for the optimal process mean problem under the assumption

that overfilled cans could be sold in a regular market, while

underfilled ones require reprocessing. A more generalized treatment

of the optimal process mean problem, involving the determi-

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1729

Fig. 1. The overview of the proposed procedure.

nation of both the optimal process mean and the upper

specification limit when a filling amount follows an arbitrary continuous

distribution, was suggested by Liu and Raghavachari

(1997). In addition, Pakkala and Rahim (1999) presented a model

for the most economical process mean and production run, while

Al-Sultan and Pulak (2000) proposed an algorithm by considering

a manufacturing system for a two-stage series to find an optimal

mean based on a product lower specification limit.

While most researchers considered the process variance as a given

value, Rahim and Al-Sultan (2000) and Rahim et al. (2002)

studied the problem of jointly determining the process mean and

variance. In their research, the concept of variance reduction in

quality improvement and cost reduction is emphasized. Similarly,

Al-Fawzan and Rahim (2001) applied a Taguchi loss function to

determine the optimal process mean and variance jointly. Shao

et al. (2000) examined several methods for process mean optimization

when several levels of customer specifications are needed

within the same market, while Kim et al. (2000) achieved this joint

determination by integrating the variance reduction principles into

process mean optimization modeling using a process capability

concept.

Recent research has been concentrated in two areas. The first

area involves the approach used to determine product performance.

In this situation, where empirical data concerning the costs

associated with product performance are available, regression

analysis is typically used as exemplified by Teeravaraprug et al.

(2001), who showed a model for determining a cost-effective process

mean. In the absence of such data, however, Teeravaraprug

and Cho (2002) applied a quadratic loss function concept, based

on a multivariate normal distribution, to a process mean problem


1730 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

with multiple quality characteristics. The second area considers

more complex manufacturing situations involving a sequential

production system. Using a multi-stage production system, Bowling

et al. (2004) proposed a method for determining the optimum

process mean for each stage by integrating a Markovian model to

represent the production system, while Kang et al. (2004) applied

robust economic optimization concepts to optimize the process

mean and the robustness measure for a chemical process within

a multiobjective optimization framework. Jeong and Kim (2009)

also proposed a multi-response optimization method by using an

interactive desirability function incorporating decision maker’s

preference information.

2.2. Tolerance optimization

Tolerance, the second important focus in this research area, involves

the problem of determining optimal specification limits

from the viewpoint of cost reduction and functional performance.

For example, early work by Speckhart (1972), Spotts (1973), Chase

et al. (1990) and Kim and Cho (2000b) considered the reduction of

manufacturing cost in a tolerance allocation problem whereas

Fathi (1990), Phillips and Cho (1998) and Kim and Cho (2000a)

studied the issue of tolerance design from the viewpoint of functional

performance, expressing it in monetary terms using the

Taguchi quality loss concept. In an integrated study considering

the effect of both cost reduction and functional performance together,

Tang (1988) developed an economic model for selecting

the most profitable tolerance in situations where inspection cost

is a linear function. Tang and Tang (1989) then extended this investigation

to include screening inspection for multiple performance

variables in a serial production process. Also, in 1994, they comprehensively

addressed the concerns inherent in the design of such

screening procedures (Tang and Tang, 1994).

When integrating both viewpoints, i.e., cost and functional performance,

a trade-off is often necessary. Along this line, Jeang

(1997) considered the simultaneous optimization of manufacturing

cost, rejection cost, and quality loss using a process capability

index to establish a relationship between tolerance and standard

deviation. In an attempt to achieve a more realistic basis for use

in industrial settings, Kapur and Cho (1996) investigated tolerance

optimization problems using truncated normal, Weibull, and multivariate

normal distributions, respectively. However, in situations

where the historical data of losses are available, regression analysis

can be applied, as exemplified by Phillips and Cho (1998), who

studied the minimization of the quality loss and the rejection costs.

They developed optimization models using the first-order and second-order

empirical loss functions.

Research in the past several years has focused on an increasingly

more complex manufacturing environment. By Plante

(2002), trade-offs are considered for multivariate cases when conducting

parametric and nonparametric tolerance allocation. A genetic

algorithm is utilized to determine complex optimization

problems encountered by the selection of design and manufacturing

tolerances under different stack-up conditions (Singh et al.,

2003). Manarvi and Juster (2004) developed an integrated tolerance

synthesis model for tolerance allocation in assembly design.

Wang and Liang (2005) studied the tolerance optimization problem

in the context of machining processes, such as milling, turning,

drilling, reaming, boring, and grinding. Shin and Cho (2007) studied

two separate process parameters, such as process mean and

variability, in the bi-objective optimization framework. Recently,

Prabhaharan et al. (2007) and Peng et al. (2008) considered optimal

process tolerances for mechanical assemblies as a combinatorial

optimization problem by considering stack-up conditions. Lee

et al. (2007) proposed a method to find the process mean maximizing

the expected profit for a multi-product production process.

Hong and Cho (2007) suggested a joint optimization method associated

with the process target and tolerance limits based on measurement

errors. Jeang and Chung (2008) developed an

optimization model for product quality performance to determine

optimal use time, initial settings, a process mean, and a process tolerance

that simultaneously minimize total cost, quality loss, failure

cost, and tolerance cost. Chen and Kao (2009) proposed a process

control model for the canning/filling industry to find the process

mean and screening limits that minimize the expected total costs

by utilizing the concept of the surrogate variable which is assumed

to be highly correlated by the performance variable.

3. Cost structure for process parameter optimization

Loss to customers due to quality performance is zero when the

performance of all key quality characteristics occurs at their customer-identified

target values. However, due to the inherent variability

associated with the characteristics, the loss is always

incurred by the customer and it can be reduced by minimizing

the deviation from the target value of each quality characteristic.

However, it is often the case that process costs often increase as

we try to achieve a smaller deviation from the process target value.

An optimization of process parameter settings involves determining

the optimal process mean and standard deviation which simultaneously

minimize the loss incurred by the customer and the

process cost incurred by the manufacturer.

When modeling a problem to optimize the process parameter

settings, the customer loss and the process cost need to be expressed

in terms of the decision variables (process mean and

standard deviation). In the following sections, the loss incurred

by the customer is written in terms of these decision variables

by using a quadratic loss function, whereas the relationship between

the process cost and the decision variables is empirically

determined.

3.1. Loss due to process variability

A quality loss function (QLF) can be employed to quantify the

product quality on a monetary scale when its performance deviates

from customer-identified target value(s) in terms of one or more

key characteristics. The quality loss includes long-term losses related

to reliability and the cost of warranties, excess inventory,

customer dissatisfaction, and eventual loss in market share. A popular

QLF is a quadratic QLF which is a quasi-convex function with

many desirable mathematical properties including unimodality,

non-negative loss, zero loss due to target values, and accommodation

of discontinuities (Cho and Leonard, 1997). The quadratic QLF

can be applied to the situations either where there is little or no

information about the functional relationship between quality

and cost, or where there is no direct evidence to refute a quadratic

representation. Compared to other QLFs, such as step-loss or piecewise

linear loss functions, the quadratic QLF may be a good approximation

of measuring the quality of a product, particularly over the

range of characteristic values in the neighborhood of the target

values.

Assuming L(y) to be a measure of losses associated with the

quality characteristic y whose target value is T, the quadratic loss

function is given by

LðyÞ ¼kðy TÞ 2 ; ð1Þ

where k is a positive coefficient, which can be determined from the

information on losses relating to exceeding the tolerance given by

the customer. The expected loss is then defined as

E½L1ðyÞŠ ¼

Z 1

1

LðyÞf ðyÞdy ¼

Z 1

1

kðy TÞ 2 f ðyÞdy; ð2Þ


where L1(y) denotes the quality loss when inspection is not implemented,

the quality characteristic Y is normally distributed with

mean l and variance r 2 , and f(y) denotes the probability density

function of Y.

3.2. Process cost

To minimize the process bias (the distance between the process

mean and the process target) and the process variance, the

process parameter settings need to be optimized by achieving

the minimum process cost and quality loss. The process cost

can be empirically represented using the decision variables. The

following empirical linear model is often (Chase and Parkinson,

1991):

CM1 ¼ c0 þ c1l þ c2r þ c3lr: ð3Þ

3.3. Total cost

The expected total cost for determining the optimal process

parameters is then given as

E½TC1Š ¼E½L1ðyÞŠ þ E½CM1Š: ð4Þ

Using Eqs. (2) and (3), Eq. (4) can be expanded to

E½TC1Š ¼

Z 1

1

kðy TÞ 2 f ðyÞdy þðc0 þ c1l þ c2r þ c3lrÞ: ð5Þ

Eq. (5) can be written in terms of a standard normal random

variable (z) using the transformation z =(y l)/r. Thus, Eq. (5)

becomes

E½TC1Š ¼

Z 1

1

kfðl þ zrÞ Tg 2 /ðzÞdz þðc0 þ c1l þ c2r þ c3lrÞ:

Using the following properties of standard normal random

variables:

Z 1

1

/ðzÞdz ¼ 1;

Z 1

1

z/ðzÞdz ¼ 0; and

Z 1

then the expected total cost can be rewritten as

1

ð6Þ

z 2 /ðzÞdz ¼ 1; ð7Þ

E½TC1Š ¼kfðl TÞ 2 þ r 2 gþc0 þ c1l þ c2r þ c3lr: ð8Þ

The optimum values, l* and r*, can be determined by simultaneously

equating dE[TC 1]/dl and dE[TC 1]/dr to zero. Then, the

closed-form solutions of the optimal process mean and the variance

are obtained. The stationary points can be easily verified by

satisfying the following condition:

d 2 E½TC1Š

dl 2

!

d 2 E½TC1Š

dr 2

!

> d2 ! 2

E½TC1Š

dldr

4. Cost structure for tolerance optimization

: ð9Þ

A quadratic loss function is used to evaluate a quality loss

when an inspection with specification limits is implemented. Besides

the loss incurred by the customer, the costs incurred by the

manufacturer, such as the rejection and the manufacturing costs,

are also included. Depending on customer requirements, lower

and upper specification limits (LSL and USL) can be defined

respectively as l dr and l + dr, orT dr and T + dr where d

represents the number of standard deviations from the middle

of a tolerance range to each specification limit and it is always

greater than zero.

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1731

4.1. Quality loss

A quality loss is incurred by the customer when the product

performance determined by y falls inside the specification limits.

By modifying Eq. (1), the expected quality loss E[L2(y)], incurred

when the process inspection is implemented at the LSL and USL,

can be given as

E½L2ðyÞŠ ¼

Z USL

LSL

LðyÞf ðyÞdy ¼

Z USL

LSL

kðy TÞ 2 f ðyÞdy: ð10Þ

Letting z =(y l*)/r* where l* and r* denote the optimal process

mean and standard deviation resulting from the process

parameter optimization phase, Eq. (10) can be rewritten as

E½L2ðyÞŠ ¼

Z ðUSL l Þ=r

ðLSL l Þ=r

4.2. Rejection cost

kðzr þ l TÞ 2 /ðzÞdz: ð11Þ

A rejection cost is incurred by the manufacturer when the product

performance determined by y falls outside the specification

limits. Denoting CRas the rejection unit cost incurred when the

quality characteristic of a product falls below a lower specification

limit or above an upper specification limit, the expected rejection

cost E[CR] is defined as

E½CRŠ ¼CR

Z LSL

1

f ðyÞdy þ

Z 1

USL

f ðyÞdy : ð12Þ

Using z =(y l*)/r*, Eq. (12) can be modified as

Z ðUSL l Þ=r

!

E½CRŠ ¼CR 1

/ðzÞdz : ð13Þ

ðLSL l Þ=r

4.3. Manufacturing cost

Additional manufacturing operations, slow processing rates,

and improved care on the part of operators in achieving a tight tolerance

may increase manufacturing cost. The manufacturing cost

usually constitutes a significant portion of the unit production cost,

and its exclusion from the tolerance optimization model may result

in a suboptimal tolerance. The tolerance allocation models

(Speckhart, 1972; Spotts, 1973; Chase et al., 1990; Kim and Cho,

2000b) found in the literature enforce the 3r assumption in order

to minimize the manufacturing cost. The manufacturing cost-tolerance

relationship proposed in this paper is free of the ad hoc 3r

assumption. Even though the middle of a tolerance range is either

a process mean (l) or a process target (T), the tolerance range t can

be defined in terms of d, l, T and r as

t ¼ USL LSL ¼ðlþ drÞ ðl drÞ ¼ðT þ drÞ ðT drÞ ¼2dr:

ð14Þ

The manufacturing cost (CM2) is often described in literature

(Patel, 1980; Bjorke, 1989) as a first-order model

CM2 ¼ a0 þ a1t þ e; ð15Þ

where e represents the least-squares regression error. The expected

manufacturing cost E[CM2], after t is replaced with 2dr from Eq.

(14), can be written as

E½CM2Š ¼a0 þ 2a1dr: ð16Þ

4.4. Total cost

The expected total cost when the process inspection is implemented

can be given by

E½TC2Š ¼E½L2ðyÞŠ þ E½CRŠþE½CM2Š: ð17Þ


1732 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

Using the optimal process mean l* and standard deviation r*

determined in the process parameter optimization together with

Eqs. (11), (13) and (16), the proposed tolerance optimization model

is formulated as

Minimize E½TC2Š ¼

5. Proposed model

Z ðUSL l Þ=r

kðzr þ l TÞ

ðLSL l Þ=r

2 /ðzÞdz

Z ðUSL l Þ=r

!

þ CR 1

ðLSL l Þ=r

/ðzÞdz

þ a0 þ 2a1dr:

ð18Þ

When determining optimal tolerances, most research reported

in the literature assumes that the levels of the two process parameters,

i.e., mean and variance, are typically treated as given constants.

However, the arbitrary settings of these parameters in

connection with the tolerance optimization often affect the defective

rate, material cost, scrap or rework cost, and possibly cause a

loss to customers due to the deviation of the product performance

from its target value. Once the minimum process variability is

achieved by determining the optimal process parameter settings,

tolerance optimization can then be implemented. There are three

possible cases as shown in Fig. 1. First, when an arbitrary target

is considered in a new process design, a typical tolerance optimization

scheme focuses on the process mean by specifying

USL = l + dr and LSL = l dr. Second, when the process mean is

fixed at an arbitrary target, USL = T + dr and LSL = T dr are specified.

This case is usually applied to existing processes. The last

possible case is when the process can absorb some bias, thereby

allowing the process mean to be slightly off the target. The categorized

optimization strategy for each case is proposed.

Case I: Minimize E[TC1] for the process parameter optimization

Minimize E[TC 2] where USL = l + dr and LSL = l dr for the

tolerance optimization

Case II: Minimize E[TC 1] where l = T for the process parameter

optimization

Minimize E[TC 2] where USL = l + dr and LSL = l dr for the

tolerance optimization

Case III: Minimize E[TC 1] for the process parameter optimization

Minimize E[TC2] where USL = T + dr and LSL = T dr for the

tolerance optimization

The models for finding optimal process parameter settings and

tolerance range, together with an investigation of the conditions

for convexity are presented in each case.

5.1. Case I

5.1.1. Process parameter optimization

From Eq. (8), by equating @E[TC1]/@l and @E[TC1]/@r to zero, the

closed-form solutions for the optimal process mean l* and deviation

r* are obtained as

l ¼ 4k2T 2kc1 þ c2c3

4k 2

c2 ; ð19Þ

3

and

r ¼ 2kc2 þ 2kc3T c1c3

4k 2 þ c2 : ð20Þ

3

To verify that l* and r* represent the global minima of the

E[TC1], the following condition developed from Eq. (9) must be

satisfied:

ð2kÞð2kÞ > c 2

3 or 4k2 > c 2

3 :

5.1.2. Tolerance optimization

Replacing USL and LSL in Eq. (11) with l + dr and l dr,

respectively, and using the following results associated with the

normal probability density function:

Z r

1

Z r

1

/ðzÞdz ¼ UðrÞ;

Z r

1

z/ðzÞdz ¼ /ðrÞ; and

z 2 /ðzÞdz ¼ UðrÞ r/ðrÞ; ð21Þ

where r is a value of the standard normal random variable, the expected

quality loss can be determined as

E½L2ðyÞŠ ¼ 2k r 2 þðl TÞ 2

n o

UðdÞ 2kr 2 d/ðdÞ

n o

; ð22Þ

k r 2 þðl TÞ 2

and the expected rejection cost determined in Eq. (13) can be written

as:

Z d

E½CRŠ ¼CR 1 /ðzÞdz ¼ 2CR½1 UðdÞŠ: ð23Þ

d

Substituting Eqs. (22), (23) and (16) into Eq. (17), the expected

total cost is calculated as

E½TC2Š ¼2k r 2 þðl TÞ 2

n o

UðdÞ 2kr 2 d/ðdÞ

k r 2 þðl TÞ 2

n o

þ 2CRf1 UðdÞg þ a0

þ 2a1dr : ð24Þ

The optimum total cost can be derived by minimizing Eq. (24).

In order to determine the optimum value d*, the first derivative of

E[TC2] with respect to d is calculated as

@E½TC2Š

@d ¼ 2k/ðdÞ ðl TÞ2 þ d 2 r 2 CR

k þ 2a1r : ð25Þ

Then, equating Eq. (25) to zero and substituting /(d) with

e 1

2 d2=

ffiffiffiffiffiffi p

2p,

the result can be identified as follows:

1

e 2

2k d2

!

pffiffiffiffiffiffi

or

r 2

2p

ðl TÞ 2 þ d 2 r 2 CR

k þ 2a1r ¼ 0;

2k

pffiffiffiffiffiffi d 2 " #

kðl TÞ2 CR

þ

2p

kr 2

e 1

2 d2

¼ 2a1r : ð26Þ

Since it is nearly impossible to obtain a closed-form solution to

this complex equation, the Lambert W function is employed to obtain

such a solution of d effectively. Details are presented in Lemma

1, and in Propositions 1 and 2 of Appendix B. According to Proposition

2, ifg1 v2 þ g2 eg3v2 ¼ g4 where g1, g2, g3, and g4 are not

functions of v, then the solution for v can be given by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Lambert W

v ¼

g3g4 e g2g v

u

3

u

t

g1 : ð27Þ

g 3

g 2

Therefore, v, g1, g2, g3, and g4 in Proposition 2 can be respectively

substituted by d; 2k=r 2

p ffiffiffiffiffiffi

2

2p;

kðl TÞ CR =kr 2 ;

1=2, and 2a1r* from Eq. (26). Then, the closed-form solution

of d is defined as follows:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d ¼ 2Lambert W a1

pffiffiffi ð

pe

1 2 Þ

kðl TÞ2 CR kr 2

r k ffiffiffi

0

1

B

C

B

@ p C

kðl TÞ

2

A

2 CR

kr 2

v

u

!

u

;

t

ð28Þ


where l*, and d 2 , are the optimal process mean and variance obtained

from the process parameter optimization phase. Although

the closed-from solution presented in Eq. (28) consists of the Lambert

W function component, its computation is very simple. A negative

value of d* is ignored because d is always greater than zero, so

the negative sign from Eq. (28) is removed. After computing d*, the

optimal LSL and USL can be respectively calculated from l* d*r*

and l* + d*r*.

5.1.3. Investigation of the second derivative and the conditions for

convexity

To verify the validity of d*, the second derivative is computed

and the conditions for obtaining the minimum value of E[TC2] are

investigated. The second derivative of E[TC2] with respect to d is

@ 2 E½TC2Š

@ 2 d

CR 2

¼ 2kd/ðdÞ þ 2r

k

ðl TÞ 2

d 2 r 2

: ð29Þ

d*obtained from Eq. (28) is the global minimum of E[TC2] if the value

of Eq. (29) at d* is greater than zero. Therefore, the following

conditions must be satisfied:

2kd/ðdÞ CR 2

þ 2r

k

ðl TÞ 2

d 2 r 2 > 0;

or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

CR þ 2kr

0 < d <

2 kðl TÞ 2

s

; ð30Þ

and

kr 2

CR þ 2kr 2 kðl TÞ 2

kr 2 > 0: ð31Þ

In many industrial situations, the validity of this solution is further

suggested by setting the value of CR to a very large number

compared to the values of l* T and r 2 (Phillips and Cho, 1998).

5.2. Case II

5.2.1. Process parameter optimization

Applying the condition l = T to Eq. (8) and then equating

oE[TC 1]/@r to zero, the closed-form solutions for the optimal process

mean l* and deviation r* are obtained as

l ¼ T; ð32Þ

and

r ¼ c3l þ c2

2k

or r ¼ c3T þ c2

: ð33Þ

2k

5.2.2. Tolerance optimization

Replacing l, USL, and LSL in Eqs. (11) and (13) with T, l + dr,

and l dr, respectively, and exploiting the properties defined in

Eq. (21), the expected quality loss E[L 2(y)] and the expected rejection

cost E[CR] are respectively presented as follows:

E½L2ðyÞŠ ¼ kr 2 ½2UðdÞ 2d/ðdÞ 1Š; ð34Þ

and

E½CRŠ ¼CR 1

Z d

/ðzÞdz ¼ 2CRf1 UðdÞg: ð35Þ

d

Substituting Eqs. (34), (35) and (16) into Eq. (17), the expected

total cost can be defined as

E½TC2Š ¼kr 2 f2UðdÞ 2d/ðdÞ 1gþ2CRf1 UðdÞg þ a0 þ 2a1dr :

ð36Þ

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1733

The minimum total cost and d* can be obtained by minimizing

Eq. (36). The first derivative of E[TC 2] with respect to d is calculated

as follows:

@E½TC2Š

@d ¼ 2a1r 2CR/ðdÞþ2kd 2 r 2 /ðdÞ: ð37Þ

e 1

2 d2

Then, equating Eq. (37) to zero and substituting /(d) with

= ffiffiffiffiffiffi p

2p,

the result is given as

kr d 2 CR 1

e 2

kr 2 d2

¼ a1

p : ð38Þ

ffiffiffiffiffiffi

2p

d,

According to Proposition 2, substituting v, g1, g2, g3, and g4 with

kr*, CR=kr 2 pffiffiffiffiffiffi , 1/2, anda1 2p into Eq. (27), the closed-form

solution of d* is given by

d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Lambert W a1

C pffiffiffiffiffiffi

R

2kr 2p e

2

0

1

B

C CR

B

C

@ 2r k A þ

kr 2

v

u

;

t

ð39Þ

where l* and r 2 are the optimal process mean and variance obtained

from the process parameter optimization phase. After computing

d*, the optimal LSL and USL can be calculated from

l* d*r* and l* + d*r*, respectively.

5.2.3. Investigation of the second derivative and the conditions for

convexity

The validity of d* can be verified by the second derivative of

E[TC2] with respect to d. The closed-form solution presented in

Eq. (39) of d* is the global minimum if the following conditions

are satisfied:

@ 2 E½TC2Š

@ 2 d

¼ 2kd/ðdÞ CR

k þ 2r 2 ð1 d 2 Þ > 0;

or

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0 < d < 1 þ CR

r

; ð40Þ

and

2kr 2

1 þ CR

> 0: ð41Þ

2kr 2

5.3. Case III

5.3.1. Process parameter optimization

After applying the conditions for this case to Eq. (8), the expected

total cost is the same as that of Case I. Thus, the optimal

process mean l* and deviation r* are determined by Eqs. (19)

and (20), respectively. For more details, please see Section 5.1.

5.3.2. Tolerance optimization

Replacing USL, and LSL in Eqs. (11) and (13) with T + dr, and

T dr, respectively, and exploiting the properties defined in Eq.

(21), the expected quality loss E[L2(y)] and the expected rejection

cost E[CR] are respectively presented as follows:

Z sþd

E½L2ðyÞŠ ¼ kðzr þ l TÞ 2 /ðzÞdz;

or

s d

n o

Uðs þ dÞ

k 2r ðl TÞþr 2

n o

ðs þ dÞ /ðs þ dÞ

k ðl TÞ 2 þ r 2

n o

Uðs dÞ

n o

/ðs dÞ; ð42Þ

E½L2ðyÞŠ ¼ k ðl TÞ 2 þ r 2

þ k 2r ðl TÞþr 2

ðs dÞ


1734 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

and

E½CRŠ ¼CR

Z s d

/ðzÞdz þ

1

Z 1

sþd

/ðzÞdz

¼ CRf1 Uðs þ dÞþUðs dÞg; ð43Þ

where s =(T l*)/r*. Substituting Eqs. (42), (43) and (16) into Eq.

(17), the expected total cost can be written as

E½TC2Š ¼k ðl TÞ 2 þ r 2

n o

Uðs þ dÞ k ðl TÞ 2 þ r 2

n o

Uðs

k 2r ðl TÞþr


2

n o

ðs þ dÞ /ðs þ dÞ

þ k 2r ðl TÞþr 2

n

ðs

o

dÞ /ðs dÞ

þ CRf1 Uðs þ dÞþUðs dÞg þ a0 þ 2a1dr : ð44Þ

Even though the closed-form solution of d* for this case cannot

be provided, the optimum expected total cost and d* can be found

by minimizing Eq. (44). After d* is generated, the optimal LSL and

USL can be computed from T d*r* and T + d*r*, respectively.

6. Numerical example

To illustrate the application of the proposed models, an electronic

chip manufacturing company experiencing high warranty

costs and customer dissatisfaction associated with component failures

in a prime product will be used. This company is considering a

new process for manufacturing electronic chips. In view of the high

Fig. 2. Plots of E[TC1], E[L1], andE[CM1] with respect to r and l.

Table 1

Effect of r on E[TC 1] when l = l*.

r l E[L1(y)] E[CM1] E[TC1] 0.50 49.99 25.01 307.16 332.17

0.60 49.99 36.01 287.60 323.61

0.70 49.99 49.01 268.05 317.06

0.80 49.99 64.01 248.49 312.50

0.90 49.99 81.01 228.93 309.94

1.00 49.99 100.01 209.38 309.39

1.10 49.99 121.01 189.82 310.83

1.20 49.99 144.01 170.27 314.28

1.30 49.99 169.01 150.71 319.72

1.40 49.99 196.01 131.15 327.16

1.50 49.99 225.01 111.60 336.61

1.60 49.99 256.01 92.04 348.05

1.70 49.99 289.01 72.48 361.49

1.80 49.99 324.01 52.93 376.94

1.90 49.99 361.01 33.37 394.38

2.00 49.99 400.01 13.82 413.83

costs associated with the failure of the components and the low

inspection costs, the company follows a 100% inspection policy

on a key quality characteristic Y which is normally distributed with

mean l and standard deviation r.

Fig. 3. Plots of E[TC2], E[CR], E[L2], and E[CM2] with respect to d.

Fig. 4. Plot of @E[TC2]/@d with respect to d.

Fig. 5. Plot of @ 2 E[TC 2]/@d 2 with respect to d.


A quality loss incurred due to the deviation from the target value

of quality characteristic Y(T) is described as the quadratic function:

k(y T) 2 , where k and T are 100 and 50, respectively. The

quality loss coefficient k is a coefficient representing the magni-

Fig. 6. Plots of E[TC1], E[L1], andE[CM1] with respect to r and l.

Fig. 7. Plots of E[TC2], E[CR], E[L2], and E[CM2] with respect to d.

Fig. 8. Plot of @E[TC 2]/@d with respect to d.

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1735

tude of the loss incurred by the deviation of y from the target value.

As shown in Fig. 2, the relationship between the expected total

cost, the quality loss, and the manufacturing cost is defined in

Eq. (4) (i.e., E[TC 1]=E[L 1(y)] + E[CM 1]). If the loss coefficient k

Fig. 9. Plot of @ 2 E[TC2]/@d 2 with respect to d.

Fig. 10. Plots of E[TC 2], E[C R], E[L 2], and E[C M2] with respect to d.

Fig. 11. Plot of @E[TC 2]/@dwith respect to d.


1736 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

increases, the expected quality loss E[L1(y)] and the expected total

cost E[TC 1] also increase.

When inspection is not implemented, the empirical model associated

with manufacturing costs is described by the polynomial

presented in Eq. (3) where its regression coefficients c0, c1, c2,

and c 3 are 100, 6.1, 3.1, and 3.85, respectively. When inspection

is implemented, the manufacturing cost (CM2) is defined as the

polynomial model presented in Eq. (16) where a 0 and a 1 are 100

Fig. 12. Plot of @ 2 E[TC2]/@d 2 with respect to d.

Fig. 13. Plots of the effect of r on E[TC1], E[L1], and E[CM1] when l = l*.

Table 2

Effect of l on E[TC 1] when r = r*.

l r E[L1(y)] E[CM1] E[TC1]

49.00 0.98 196.04 210.99 407.03

49.20 0.98 160.04 211.45 371.49

49.40 0.98 132.04 211.92 343.96

49.60 0.98 112.04 212.38 324.42

49.70 0.98 105.04 212.61 317.65

49.80 0.98 100.04 212.85 312.89

49.90 0.98 97.04 213.08 310.12

50.00 0.98 96.04 213.31 309.35

50.10 0.98 97.04 213.54 310.58

50.20 0.98 100.04 213.78 313.82

50.40 0.98 112.04 214.24 326.28

50.50 0.98 121.04 214.48 335.52

50.60 0.98 132.04 214.71 346.75

50.80 0.98 160.04 215.17 375.21

51.00 0.98 196.04 215.64 411.68

and 0.2, respectively. When the product performance determined

by y falls outside the product specification limits, the rejection unit

cost (CR) is set to an arbitrary value such as 100 in this numerical

example.

6.1. Case I

6.1.1. Determining optimal settings of process parameters

Using the closed-form solution given in Eqs. (19) and (20), l*

and r* become 49.9883 and 0.9778, respectively. Fig. 2 plots

Fig. 14. Plots of the effect of l on E[TC1], E[L1], and E[CM1] when r = r*.

Table 3

Effect of r on d* when l = l*.

r l jl Tj d* E[CM2] E[CR] E[L2(y)] E[TC2] 0.80 49.99 0.01 1.2522 99.5993 21.0511 21.3398 141.9903

0.85 49.99 0.01 1.1790 99.5991 23.8395 21.1143 144.5529

0.90 49.99 0.01 1.1141 99.5989 26.5241 20.8129 146.9359

0.95 49.99 0.01 1.0561 99.5987 29.0913 20.4613 149.1513

1.00 49.99 0.01 1.0041 99.5984 31.5343 20.0792 151.2119

1.05 49.99 0.01 0.9571 99.5980 33.8506 19.6812 153.1298

1.10 49.99 0.01 0.9146 99.5976 36.0408 19.2786 154.9170

1.15 49.99 0.01 0.8759 99.5971 38.1075 18.8798 156.5845

1.20 49.99 0.01 0.8406 99.5965 40.0547 18.4912 158.1424

1.25 49.99 0.01 0.8084 99.5958 41.8868 18.1174 159.6001

1.30 49.99 0.01 0.7788 99.5950 43.6089 17.7623 160.9662

Fig. 15. Plots of the effect of r on E[TC 2], E[C R], E[L 2], E[C M2], and d* when l = l*.


E[TC1] versus l and r, where the minimum value of E[TC1] obtained

at l* and r* is 309.3380.

The sensitivity analysis of E[TC1] to the variation in process

standard deviation when the process mean is fixed at l* is presented

in Table 1 and Fig. 13. A similar analysis is conducted when

the process standard deviation is fixed at r* and it is shown in Table

2 and Fig. 14.

6.1.2. Determining optimal tolerance

Using the closed-form solution defined in Eq. (28), d* becomes

1.0269. This optimal value generates the minimum E[TC 2], optimal

LSL, and optimal USL at 150.3168, 48.9842, and 50.9924, respec-

Table 4

Effect of jl Tj on d* when r = r*.

l jl Tj r d* E[CM2] E[CR] E[L2(y)] E[TC2] 49.0 1.00 0.98 0.0685 99.9731 94.5356 5.4726 199.9813

49.1 0.90 0.98 0.4516 99.8230 65.1579 30.4354 195.4162

49.2 0.80 0.98 0.6182 99.7577 53.6425 35.0573 188.4574

49.3 0.70 0.98 0.7345 99.7121 46.2616 34.9641 180.9377

49.4 0.60 0.98 0.8222 99.6777 41.0981 32.8406 173.6164

49.5 0.50 0.98 0.8896 99.6513 37.3676 29.9179 166.9368

49.6 0.40 0.98 0.9412 99.6310 34.6598 26.8931 161.1839

49.7 0.30 0.98 0.9795 99.6160 32.7348 24.1974 156.5482

49.8 0.20 0.98 1.0059 99.6057 31.4458 22.1055 153.1570

49.9 0.10 0.98 1.0215 99.5996 30.7041 20.7882 151.0919

50.0 0.00 0.98 1.0266 99.5976 30.4619 20.3391 150.3987

Fig. 16. Plots of the effect of jl Tj on E[TC2], E[CR], E[L2], E[CM2], and d* when

r = r*.

Table 5

Effect of r on E[TC 1] when l = l*.

r l E[L1(y)] E[CM1] E[TC1] 0.5 50.00 25.00 307.20 332.20

0.6 50.00 36.00 287.64 323.64

0.7 50.00 49.00 268.08 317.08

0.8 50.00 64.00 248.52 312.52

0.9 50.00 81.00 228.96 309.96

1.0 50.00 100.00 209.40 309.40

1.1 50.00 121.00 189.84 310.84

1.2 50.00 144.00 170.28 314.28

1.3 50.00 169.00 150.72 319.72

1.4 50.00 196.00 131.16 327.16

1.5 50.00 225.00 111.60 336.60

1.6 50.00 256.00 92.04 348.04

1.7 50.00 289.00 72.48 361.48

1.8 50.00 324.00 52.92 376.92

1.9 50.00 361.00 33.36 394.36

2.0 50.00 400.00 13.80 413.80

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1737

tively. Plots of E[TC2], E[CM2], E[CR] and E[L2] with respect to d are

presented in Fig. 3.

Figs. 4 and 5 show that the first derivative of E[TC2] with respect

to d at d* is zero and the second derivative of E[TC 2] with respect to

d at d* is greater than zero, respectively. The latter figure also validates

the conditions for convexity presented in Eqs. (30) and (31).

Table 6

Effect of l on E[TC 1] when r = r*.

l r E[L1(y)] E[CM1] E[TC1] 49.0 0.98 96.04 210.99 307.03

49.2 0.98 96.04 211.45 307.49

49.4 0.98 96.04 211.92 307.96

49.6 0.98 96.04 212.38 308.42

49.7 0.98 96.04 212.61 308.65

49.8 0.98 96.04 212.85 308.89

49.9 0.98 96.04 213.08 309.12

50.0 0.98 96.04 213.31 309.35

50.1 0.98 96.04 213.54 309.58

50.2 0.98 96.04 213.78 309.82

50.3 0.98 96.04 214.01 310.05

50.4 0.98 96.04 214.24 310.28

50.5 0.98 96.04 214.48 310.52

50.6 0.98 96.04 214.71 310.75

50.8 0.98 96.04 215.17 311.21

51.0 0.98 96.04 215.64 311.68

Fig. 17. Plots of the effect of r on E[TC 1], E[L 1], and E[C M1] when l = l*.

Fig. 18. Plots of the effect of l on E[TC 1], E[L 1], and E[C M1] when r = r*.


1738 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

This is a solid proof that E[TC2] is a convex function and d* is the

global minimum because its value is between 0 and 1.7452.

The sensitivity analysis of d* to the change in r can be seen in

Table 3 or Fig. 15. It can be observed that d* gradually decreases

as r increases. Additionally, E[CM2] depends on d* and r and its value

is the lowest when r is 1.30. Similarly, the sensitivity analysis

of the effect of the process bias (or jl Tj)ond* is shown in Table 4

or Fig. 16. It is shown that, as the process bias increases, d* decreases

while E[CR] increases. For the 100% inspection of Y, E[L2]

is more affected by the process bias than by r, whereas E[C R]is

sensitive to both.

Table 7

Effect of r on d* when l = l*.

r l l Tj d* E[CM2] E[CR] E[L2(y)] E[TC2] 0.80 50.00 0.00 1.2555 99.5982 20.9297 21.4541 141.9821

0.85 50.00 0.00 1.1815 99.5983 23.7405 21.2063 144.5451

0.90 50.00 0.00 1.1158 99.5983 26.4520 20.8781 146.9284

0.95 50.00 0.00 1.0570 99.5983 29.0509 20.4949 149.1442

1.00 50.00 0.00 1.0041 99.5983 31.5310 20.0756 151.2050

1.05 50.00 0.00 0.9563 99.5983 33.8904 19.6344 153.1232

1.10 50.00 0.00 0.9129 99.5983 36.1303 19.1818 154.9105

1.15 50.00 0.00 0.8732 99.5983 38.2539 18.7256 156.5778

1.20 50.00 0.00 0.8369 99.5983 40.2658 18.2713 158.1354

1.25 50.00 0.00 0.8035 99.5983 42.1712 17.8230 159.5925

1.30 50.00 0.00 0.7726 99.5982 43.9758 17.3836 160.9576

Fig. 19. Plots of the effect of r on E[TC 2], E[C R], E[L 2], E[C M2], and d* when l = l*.

Table 8

Effect of r on d* when l = l*.

r l jl Tj d* E[CM2] E[CR] E[L2(y)] E[TC2] 0.80 49.99 0.01 1.2555 99.5982 20.9346 21.4538 141.9866

0.85 49.99 0.01 1.1815 99.5983 23.7450 21.2058 144.5491

0.90 49.99 0.01 1.1158 99.5983 26.4560 20.8776 146.9319

0.95 49.99 0.01 1.0570 99.5983 29.0546 20.4944 149.1473

1.00 49.99 0.01 1.0041 99.5983 31.5344 20.0750 151.2077

1.05 49.99 0.01 0.9563 99.5983 33.8935 19.6338 153.1256

1.10 49.99 0.01 0.9129 99.5983 36.1330 19.1813 154.9126

1.15 49.99 0.01 0.8732 99.5983 38.2564 18.7250 156.5797

1.20 49.99 0.01 0.8369 99.5983 40.2680 18.2708 158.1371

1.25 49.99 0.01 0.8035 99.5983 42.1732 17.8226 159.5941

1.30 49.99 0.01 0.7726 99.5982 43.9776 17.3831 160.9590

6.2. Case II

6.2.1. Determining optimal settings of process parameters

Similarly to case I, the minimum value of E[TC 1] obtained at the

optimal process mean and standard deviation (which are determined

by Eqs. (32) and (33), i.e., l* = 50.0000 and r* = 0.9780) is

309.3516 as illustrated in Fig. 6. In addition, the results of the sensitivity

analysis of E[TC 1] are given in Tables 5 and 6, and Figs. 17

and 18.

6.2.2. Determining optimal tolerance

Following the closed-form solution presented in Eq. (39), d* becomes

1.0267. This optimal value generates the minimum E[TC2],

the optimal LSL, and the optimal USL at 150.3166, 48.9959, and

51.0041, respectively. Plots of E[TC2], E[CM2], E[CR] and E[L2(y)] with

respect to d are presented in Fig. 7.

Fig. 8 shows that the first derivative of E[TC2] with respect to d

becomes zero when d is at the optimum point. The conditions of

convexity for E[TC2] discussed in Section 5.2 are presented in

Fig. 9, which shows that when d is at the optimum point (d*), the

value of @ 2 E[TC2]/@d 2 is greater than zero. It also shows the range

of d, which can be derived from Eq. (40). The result is the range between

0 and 1.2340, which contains the value of d*.

The effect of r on d* can be seen in Table 7 or Fig. 19. It can be

observed that d* decreases as r increases, and E[CR] is inversely

proportional to d*. For this case, the effect of the process bias (or

jl Tj) ond* is not provided because no l in the closed-form solution

of d* defined in Eq. (39). It can be concluded that l has no effect

to d*. For the 100% inspection of Y, to decrease the total cost,

E[L2] and r should be controlled because they are directly proportional

to the total cost.

Fig. 20. Plots of the effect of r on E[TC2], E[CR], E[L2], E[CM2], and d* when l = l*.

Table 9

Effect of jl Tj on d* when r = r*.

l jl Tj r d* E[CM2] E[CR] E[L2(y)] E[TC2] 49.0 1.00 0.98 1.0272 99.5974 51.7607 16.0869 167.4449

49.1 0.90 0.98 1.0271 99.5974 48.2578 16.8773 164.7325

49.2 0.80 0.98 1.0270 99.5974 44.9205 17.5967 162.1146

49.3 0.70 0.98 1.0270 99.5974 41.8078 18.2389 159.6441

49.4 0.60 0.98 1.0269 99.5974 38.9766 18.7994 157.3734

49.5 0.50 0.98 1.0269 99.5974 36.4801 19.2753 155.3528

49.6 0.40 0.98 1.0269 99.5974 34.3663 19.6651 153.6289

49.7 0.30 0.98 1.0269 99.5974 32.6769 19.9680 152.2423

49.8 0.20 0.98 1.0269 99.5974 31.4452 20.1840 151.2267

49.9 0.10 0.98 1.0269 99.5974 30.6961 20.3135 150.6070

50.0 0.00 0.98 1.0269 99.5974 30.4448 20.3565 150.3987


Fig. 21. Plots of the effect of jl Tj on E[TC 2], E[C R], E[L 2], E[C M2], and d* when

r = r*.

6.3. Case III

6.3.1. Determining optimal settings of process parameters

Since the closed-form solutions of l* and r* for this case are the

same as those of Case I, see the first part of Section 6.1 for more

details.

6.3.2. Determining optimal tolerance

Although the closed-form solution of d* for this case cannot be

formulated, d* can be generated by minimizing E[TC 2] given in Eq.

(44). By using MATLAB software package, the minimum E[TC2] is

obtained at 150.4008 when d is 1.0269, defined as d*. This optimal

value can be verified from the plots shown in Figs. 11 and 12. The

former shows that the first derivative of E[TC2] is zero at d* and the

latter shows that the second derivative of E[TC2] atd* is greater

than zero. The change of d to E[TC2], E[CM2], E[CR] and E[L2] is depicted

in Fig. 10. From d*, the optimum LSL and USL are set at

48.9959 and 51.0041, respectively.

The sensitivity analysis of the effect of r on d* is provided in Table

8 and Fig. 20. It can be observed that r has inverse variation in

d* but it has direct variation in E[C R]. Similarly, the sensitivity analysis

of the effect of the process bias (or jl Tj) ond* is provided in

Table 9 and Fig. 21. It shows that the process bias directly affect d*

and E[CR]. For the 100% inspection of Y, E[L2] is slightly affected by

r and jl Tj, whereas E[C R] is greatly affected by both r and

jl Tj.

7. Conclusions and further study

Very often, engineers face the problem of determining the optimal

tolerance in many industrial settings. This paper proposed an

integrated scheme for determining the optimal process parameters

and tolerances, the algorithm to quantify the cost of achieving optimal

process parameters and tolerances, together with a procedure

to incorporate Lambert W function to generate closed-form solutions

for tolerance optimization problems. The proposed methodologies

are superior to earlier models that determine optimal

tolerance assuming fixed settings for the process parameters.

Moreover, the proposed models and solutions might be appealing

to engineers because the function is found in most standard optimization

software packages. Continuous research efforts over the

past few decades in the area of tolerance optimization has led to

the formulation of effective optimization models that are free of

the restrictive assumptions imposed by earlier ones. Along these

lines, a fruitful area for future research is to extend the modeling

S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741 1739

methodology presented in this paper to a generalized tolerance

optimization problem involving multiple quality characteristics.

Further, the consideration of interactions between quality characteristics

and the use of a surrogate variable may be potential topics

for future investigations.

Acknowledgements

This work was supported by the Korea Science and Engineering

Foundation (KOSEF) grant funded by the Korea Government

(MOST) (No. R01-2007-000-21070-0). This work was supported

by the Korea Research Foundation Grant funded by the Korean

Government (MOEHRD, Basic Research Promotion Fund) (KRF-

2008-331-D00686).

Appendix A

Key notations, used in the model developments, are summarized

and explained as follows:

Y the quality characteristic Y of the product

y the observed value of Y

f(y) the density function of Y

l the process mean of Y

l* the optimum process mean of Y

r the standard deviation of Y

r* the optimum process standard deviation of Y

r2 the variance of Y

T the target value of Y

s the target value of Y in the standard normal distribution

domain, s =(y l)/r

k the quality loss coefficient

L(y) the quality loss associated with Y

L1(y) the quality loss associated with Y and incurred when process

inspection is not implemented

L2(y) the quality loss associated with Y and incurred when process

inspection is implemented

QLF quality loss function

CM1 the manufacturing unit cost incurred when process inspection

is not implemented

c0 the constant in the CM1 regression function

c1 the coefficient of l in the CM1 regression function

c2 the coefficient of r in the CM1 regression function

c3 the coefficient of lr in the CM1 regression function

d the number of standard deviations where the specification

limit is located from the middle of its tolerance range

LSL the lower specification limit associated with Y

USL the upper specification limit associated with Y

E[ ] the expected value

CR the rejection unit cost

/( ) the standard normal probability density function

U(.) the standard normal cumulative distribution function

z the standard normal random variable

t the tolerance range

CM2 the manufacturing unit cost incurred when process inspection

is implemented

a0 the constant in the CM2 regression function

a1 the coefficient of t in the CM2 regression function

e the least-squares regression error

TC1 the process total cost incurred when the process inspection

is not implemented

TC2 the total cost incurred when the process inspection is implemented

Appendix B

Lemma 1. Suppose v 2 R 1 and a mapping g:v ? Risg = ve v , then

the solution for v is given by v = LambertW(g).


1740 S. Shin et al. / European Journal of Operational Research 207 (2010) 1728–1741

Proof. See Corless et al. (1996). h

Proposition 1. If g2 =(v + g1)e v where g1 and g2 are not functions of

v, then v ¼ Lambert W g2eg ð 1Þ

g1 .

Proof. Consider the following equation:

g 2 ¼ðv þ g 1 Þe v : ðA-1Þ

Let v + g1 = w so Eq. (A-1) becomes

g 2 ¼ we w g 1: ðA-2Þ

To convert Eq. (A-2) in the standard form g = ve v , we first modify

Eq. (A-2) as follows:

g 2 e g 1 ¼ we w : ðA-3Þ

Next, substitute g 2 e g 1 ¼ x to the right hand side of Eq. (A-3)

then we obtain x = we w . From Lemma 1, w is given by:

w ¼ Lambert WðxÞ: ðA-4Þ

Therefore, v is obtained as

v ¼ Lambert W g2e g ð 1Þ

g1 :

Proposition 2. If g4 ¼ g1 v2 þ g2 eg3v2 , where g1, g2, g3, and g4 are

not functions of v, then

v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Lambert W g3g4 e g2g v

u

3

u

t

g1 :

g 3

g 2

Proof. In order to prove Proposition 2 using Lemma 1, we first consider

the equation

g4 ¼ g1 v2 þ g2 e g3v2 : ðA-5Þ

g 3g 4

g 1

Multiplied both sides with g3/g1, Eq. (A-5) becomes

¼ g3 v2 þ g2 e g3v2 : ðA-6Þ

Let w = g 3(v 2 + g 2), Eq. (A-6) can be written as

g 3g 4 e g 2g 3

g 1

¼ we w : ðA-7Þ

Further, substituted g3g4 e g2g3 g1 dard form g = ve v .

¼ x, Eq. (A-7) is now in the stan-

w ¼ Lambert WðxÞ: ðA-8Þ

Replaced x ¼ g3g4 e g2g3 g back into Eq. (A-8), the standard form in

1

Eq. (A-8) can be written as w ¼ Lambert W g3g4 e g2g3 g . Eq. (A-9)

1

can be expanded to express the solution in terms of v by replacing

w with g3(v + g2). Thus, the solution for v is given by

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Lambert W

v ¼

g3g4 e g2g v

u

3

u

t

g1 : ðA-9Þ

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