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. 


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 


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 


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 


! 

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


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 


kðzr þ l TÞ 

ðLSL l Þ=r 

2 /ðzÞdz 


! 

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 


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

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


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 


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 


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 


r ¼ c3l þ c2 

2k 

or r ¼ c3T þ c2 

: ð33Þ 

2k 


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 


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 


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 


2kr 2 

1 þ CR 

> 0: ð41Þ 

2kr 2 

5.3. Case III 


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. 


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



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 

dÞ 

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. 


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.


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 


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*.


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 


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 


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 


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. 


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 


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. 


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 


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).


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 


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 


Lambert W 

v ¼ 

g3g4 e g2g v 

u 

3 

u 

t 

g1 : ðA-9Þ 

References 

g 3 

g 2 

Al-Fawzan, M.A., Rahim, M.A., 2001. Optimal control of deteriorating process with a 

quadratic loss function. Quality and Reliability Engineering International 17 (6), 

459–466. 

Al-Sultan, K.S., Pulak, M.F.S., 2000. Optimum target values for two machines in 

series with 100% inspection. European Journal of Operational Research 120 (1), 

181–189. 

Bjorke, O., 1989. Computer-Aided Tolerancing. American Society of Mechanical 

Engineers, USA. 

Bowling, S.R., Khasawneh, M.T., Kaewkuekool, S., Cho, B.R., 2004. A Markovian 

approach to determining optimum process target levels for a multi-stage 

serial production system. European Journal of Operational Research 159 (3), 

636–650. 

Caleb Li, M.H., Chen, J.C., 2001. Determining process mean for machining while 

unbalanced tolerance design occurs. Journal of Industrial Technology 17 (1), 1– 

6. 

Chase, K.W., Parkinson, A.R., 1991. A survey of research in the application of 

tolerance analysis to the design of mechanical assemblies. Research in 

Engineering Design 3 (1), 23–27. 

Chase, K.W., Greenwood, W.H., Loosli, B.G., Haughlund, L.F., 1990. Least-cost 

tolerances allocation for mechanical assemblies with automated process 

selection. Manufacturing Review 3 (1), 49–59. 

Chen, C.H., Kao, H.S., 2009. The determination of optimum process mean and 

screening limits based on quality loss function. Expert Systems with 

Applications 36 (3-2), 7332–7335. 

Cho, B.R., Leonard, M.S., 1997. Identification and extensions of quasiconvex quality 

loss functions. International Journal of Reliability, Quality and Safety 

Engineering 4 (2), 191–204. 

Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E., 1996. On the 

Lambert W function. Advances in Computational Mathematics 5, 329–359. 

Fathi, Y., 1990. Producer–consumer tolerances. Journal of Quality Technology 22 (2), 

138–145. 

Golhar, D.Y., 1987. Determination of the best mean contents for a canning problem. 

Journal of Quality Technology 19 (2), 82–84. 

Golhar, D.Y., Pollock, S.M., 1988. Determination of the optimal process mean and 

the upper limit for a canning problem. Journal of Quality Technology 20 (3), 

188–192. 

Hong, S.H., Cho, B.R., 2007. Joint optimization of process target mean and tolerance 

limits with measurement errors under multi-decision alternatives. European 

Journal of Operational Research 183, 327–335. 

Jeang, A., 1997. An approach of tolerance design for quality improvement and 

cost reduction. International Journal of Production Research 35 (5), 1193– 

1211. 

Jeang, A., Chung, C.P., 2008. Process mean, process tolerance, and use time 

determination for product life application under deteriorating process. The 

International Journal of Advanced Manufacturing Technology 36 (1–2), 97– 

113. 

Jeong, I.J., Kim, K.J., 2009. An interactive desirability function method to 

multiresponse optimization. European Journal of Operational Research 195, 

412–426. 

Kang, J.S., Suh, M.H., Lee, T.Y., 2004. Robust economic optimization of process design 

under uncertainty. Engineering Optimization 36 (1), 51–75. 

Kapur, K.C., Cho, B.R., 1996. Economic design of the specification region for multiple 

quality characteristics. IIE Transactions 28 (3), 401–417. 

Kim, Y.J., Cho, B.R., 2000a. Economic integration of design optimization. Quality 

Engineering 12 (4), 561–567. 

Kim, Y.J., Cho, B.R., 2000b. The use of response surface designs in the selection of 

optimum tolerance allocation. Quality Engineering 13 (1), 35–42. 

Kim, Y.J., Cho, B.R., Phillips, M.D., 2000. Determination of the optimum process 

mean with the consideration of variance reduction and process capability. 

Quality Engineering 13 (2), 251–260. 

Lee, M.K., Kwon, H.M., Hong, S.H., Kim, Y.J., 2007. Determination of the optimum 

target value for a production process with multiple products. International 

Journal of Production Economics 107 (1), 173–178. 

Liu, W., Raghavachari, M., 1997. The target mean problem for an arbitrary quality 

characteristic distribution. International Journal of Production Research 35 (6), 

1713–1728. 

Manarvi, I.A., Juster, N.P., 2004. Framework of an integrated tolerance synthesis 

model and using FE simulation as a virtual tool for tolerance allocation in 

assembly design. Journal of Materials Processing Technology 150 (1–2), 182– 

193. 

MATLAB. . 

Pakkala, T.P.M., Rahim, M.A., 1999. Determination of an optimal setting and 

production run using Taguchi’s loss function. International Journal of Reliability, 

Quality and Safety Engineering 6 (4), 335–346. 

Patel, A.M., 1980. Computer-aided assignment of tolerances. In: Proceedings of the 

17th Design and Automation Conference, Minneapolis, USA. 

Peng, H.P., Jiang, X.Q., Liu, X.J., 2008. Concurrent optimal allocation of design and 

process tolerances for mechanical assemblies with interrelated dimension 

chains. International Journal of Production Research 46 (24), 6963–6979. 

Phillips, M.D., Cho, B.R., 1998. An empirical approach to designing product 

specifications: A case study. Quality Engineering 11 (1), 91–100. 

Plante, R., 2002. Multivariate tolerance design for a quadratic design parameter 

model. IIE Transactions 34 (6), 565–571. 

Prabhaharan, G., Ramesh, R., Asokan, P., 2007. Concurrent optimization of assembly 

tolerances for quality with position control using scatter search approach. 

International Journal of Production Research 45 (21), 4959–4988. 

Rahim, M.A., Al-Sultan, K.S., 2000. Joint determination of the target mean and 

variance of a process. Journal of Quality Maintenance Engineering 6 (3), 192– 

199. 

Rahim, M.A., Bhadury, J., Al-Sultan, K.S., 2002. Joint economic selection of target 

mean and variance. Engineering Optimization 34 (1), 1–14. 

Shao, R.L., Fowler, J.W., Runger, G.C., 2000. Determining the optimal target for a 

process with multiple markets and variable holding cost. International Journal 

of Production Economics 65 (3), 229–242.

Shin, S., Cho, B.R., 2007. Integrating a bi-objective paradigm to tolerance 

optimization. International Journal of Production Research 45 (23), 5509–5525. 

Singh, P.K., Jain, P.K., Jain, S.C., 2003. Simultaneous optimal selection of design and 

manufacturing tolerances with different stack-up conditions using genetic 

algorithms. International Journal of Production Research 41 (11), 2411–2429. 

Speckhart, F.H., 1972. Calculation of tolerance based on a minimum cost approach. 

Transactions of the ASME, Journal of Engineering for Industry 94, 447–453. 

Spotts, M.F., 1973. Allocation of tolerances to minimize cost of assembly. 

Transactions of the ASME, Journal of Engineering for Industry 95, 762–764. 

Springer, C.H., 1951. A method for determining the most economic position of a 

process mean. Industrial Quality Control 8 (1), 36–39. 

Tang, K., 1988. Economic design of product specifications for a complete inspection 

plan. International Journal of Production Research 26 (2), 203–217. 


Tang, K., Tang, J., 1989. Design of product specifications for multi-characteristic 

inspection. Management Science 35 (6), 743–756. 

Tang, K., Tang, J., 1994. Design of screening procedures: A review. Journal of Quality 

Technology 26 (3), 209–226. 

Teeravaraprug, J., Cho, B.R., 2002. Designing the optimal target levels for multiple 

quality characteristics. International Journal of Production Research 40 (1), 37– 

54. 

Teeravaraprug, J., Cho, B.R., Kennedy, W.J., 2001. Designing the most cost effective 

process target using regression analysis: A case study. Process Control and 

Quality 11 (6), 469–477. 

Wang, P., Liang, M., 2005. An integrated approach to tolerance synthesis, process 

selection and machining parameter optimization problems. International 

Journal of Production Research 43 (11), 2237–2262.

Development of the parametric tolerance modeling and optimization ...

Create successful ePaper yourself

Delete template?

Save as template?