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** o**the**r h**and**, 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-**of**f between **the** level **of** quality based on functional

performance **and** **the** associated costs. To facilitate this economic

trade-**of**f, 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 **of**ten 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-**of**f 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 **the**n

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 **the**ir solutions in a closed-form,

in addition to being able to determine optimal **tolerance**s quickly

without resorting to numerical methods, since a number **of** popular

ma**the**matical s**of**tware 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 **of**ten **the** situation, **the** lower limit was fixed while **the** upper

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

to be ei**the**r 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 **the**ir 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 toge**the**r,

Tang (1988) developed an economic model for selecting

**the** most pr**of**itable **tolerance** in situations where inspection cost

is a linear function. Tang **and** Tang (1989) **the**n extended this investigation

to include screening inspection for multiple performance

variables in a serial production process. Also, in 1994, **the**y 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-**of**f is **of**ten 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** st**and**ard

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-**of**fs are considered for multivariate cases when conducting

**parametric** **and** non**parametric** **tolerance** allocation. A genetic

algorithm is utilized to determine complex **optimization**

problems encountered by **the** selection **of** design **and** manufacturing

**tolerance**s under different stack-up conditions (Singh et al.,

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

syn**the**sis 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 **tolerance**s 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 pr**of**it 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 **the**ir 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 **of**ten **the** case that process costs **of**ten 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** st**and**ard 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**

st**and**ard deviation). In **the** following sections, **the** loss incurred

by **the** customer is written in terms **of** **the**se 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 ma**the**matical 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 ei**the**r where **the**re is little or no

information about **the** functional relationship between quality

**and** cost, or where **the**re is no direct evidence to refute a quadratic

representation. Compared to o**the**r 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 **the**n 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 **of**ten (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 **the**n given as

E½TC1Š ¼E½L1ðyÞŠ þ E½CM1Š: ð4Þ

Using Eqs. (2) **and** (3), Eq. (4) can be exp**and**ed 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 st**and**ard normal r**and**om

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** st**and**ard normal r**and**om

variables:

Z 1

1

/ðzÞdz ¼ 1;

Z 1

1

z/ðzÞdz ¼ 0; **and**

Z 1

**the**n **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** st**and**ard 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** st**and**ard 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 ei**the**r

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 **of**ten 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** st**and**ard deviation r*

determined in **the** process parameter **optimization** toge**the**r 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 **tolerance**s, 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** **the**se parameters in

connection with **the** **tolerance** **optimization** **of**ten 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 **the**n 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, **the**reby

allowing **the** process mean to be slightly **of**f **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, toge**the**r 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** st**and**ard normal r**and**om 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, **the**n **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 fur**the**r

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** **the**n 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, **and**a1 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

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], **and**E[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** st**and**ard 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], **and**E[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

st**and**ard 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 st**and**ard 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 pro**of** 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** st**and**ard 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 **the**y 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 s**of**tware 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** fur**the**r study

Very **of**ten, 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** **tolerance**s, **the** algorithm to quantify **the** cost **of** achieving optimal

process parameters **and** **tolerance**s, toge**the**r 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 st**and**ard **optimization**

s**of**tware 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 **the**se

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.

Fur**the**r, **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** st**and**ard deviation **of** Y

r* **the** optimum process st**and**ard deviation **of** Y

r2 **the** variance **of** Y

T **the** target value **of** Y

s **the** target value **of** Y in **the** st**and**ard 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** st**and**ard 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** st**and**ard normal probability density function

U(.) **the** st**and**ard normal cumulative distribution function

z **the** st**and**ard normal r**and**om 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 , **the**n

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

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

Pro**of**. See Corless et al. (1996). h

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

v, **the**n v ¼ Lambert W g2eg ð 1Þ

g1 .

Pro**of**. 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** st**and**ard 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 h**and** side **of** Eq. (A-3)

**the**n 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, **the**n

v ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Lambert W g3g4 e g2g v

u

3

u

t

g1 :

g 3

g 2

Pro**of**. 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

Fur**the**r, 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** st**and**ard form in

1

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

1

can be exp**and**ed 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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