Tolerance optimization using the Lambert W function - Clemson ...

int. j. prod. res., 15 august 2004, vol. 42, no. 16, 3235–3251

**Tolerance** **optimization** **using** **the** **Lambert** W **function**: an

empirical approach

M. S. GOVINDALURI, S. SHIN and B. R. CHO*

This paper explores **the** integration of **the** **Lambert** W **function** to a tolerance

**optimization** problem with two unique features. First, **the** Taguchi loss **function**

has been a popular tool for quantifying a quality loss incurred by **the** customer.

This paper utilizes an empirical approach based on a well-established regression

analysis, which may be more appealing to engineers and better capture **the** customer’s

perception of product performance. Second, by trading off manufacturing

and rejection costs incurred by a manufacturer and quality loss incurred by **the**

customer, this paper shows how **the** **Lambert** W **function**, widely used in physics,

can be efficiently applied, which is perhaps **the** first attempt in **the** literature

related to tolerance **optimization** and syn**the**sis. Using **the** concept of **the**

**Lambert** W **function**, this paper derives a closed-form solution, which may

serve as a means for quality practitioners to make a quick decision on **the**ir

optimal tolerances without resorting to rigorous **optimization** procedures **using**

numerical methods. A numerical example is illustrated and a sensitivity analysis

is performed.

1. Introduction

Functional performance and economic considerations are **the** two primary factors

affecting **the** design of tolerances. A tight tolerance usually implies high manufacturing

cost due to additional manufacturing operations, slow processing rates,

additional care on **the** part of **the** operator, and a need for expensive measuring

and processing equipment. The **function**al performance, however, can be improved

by specifying a tight tolerance on a quality characteristic. On **the** o**the**r hand, a wide

tolerance reduces **the** manufacturing cost but may considerably lower **the** product

quality level. Thus, determining optimal tolerance involves a trade-off between **the**

level of quality based on **function**al performance and **the** costs associated with **the**

tolerance. While **the** issue of tolerance **optimization** is clearly important, it is one

of **the** least understood engineering problems. Note that **the** problem of determining

an optimal tolerance is equivalent to **the** determination of optimal lower and upper

specification limits.

The analysis in this paper differs from previous studies of **the** tolerance **optimization**

problem in two ways. First, in order to facilitate **the** economic trade-off,

researchers often express quality in monetary terms **using** a quality loss **function**.

Typically, **the** exact loss **function** representing **the** loss due to poor quality performance

is not known. Therefore, several ma**the**matical **function**s have been used

to approximate an actual loss. Among **the**m, **the** Taguchi loss **function** is one of

Revision received November 2003.

Advanced Quality Engineering Laboratory, Department of Industrial Engineering,

**Clemson** University, **Clemson**, South Carolina 29634, USA.

*To whom correspondence should be addressed. e-mail: bcho@ces.clemson.edu

International Journal of Production Research ISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd

http://www.tandf.co.uk/journals

DOI: 10.1080/00207540410001696311

3236 M. S. Govindaluri et al.

**the** most commonly used approximation tools for quantifying quality performance.

Although this loss **function** is considered to be a reasonable approximation for

many situations, an empirical loss **function** based on statistical regression of quality

loss data may be more appealing to engineers, particularly when **the** data regarding

losses due to quality performance are available. The tolerance **optimization** model

proposed in this paper considers **the** quality loss incurred by **the** customer, and

manufacturing and rejection costs incurred by **the** manufacturer. Second, this

paper shows how **the** **Lambert** W **function**, widely used in physics (Corless et al.

1996), can be efficiently applied to **the** tolerance **optimization** problem, which is

perhaps **the** first attempt in **the** literature related to tolerance **optimization** and

syn**the**sis. There are two significant benefits from **using** **the** **Lambert** W **function** in

**the** context of tolerance **optimization**. Most tolerance **optimization** models require

rigorous **optimization** procedures **using** numerical methods since closed-form solutions

are rarely found. By **using** **the** **Lambert** W **function**, quality practitioners can

not only express **the**ir solutions in a closed form but also quickly determine **the**ir

optimal tolerances without resorting to numerical methods, since a number of popular

ma**the**matical software packages, including Maple and Matlab, contain **the**

**Lambert** W **function** as an **optimization** component.

The remainder of this paper is organized as follows. A literature review is

presented in Section 2. The **Lambert** W **function** is briefly introduced in Section 3.

Section 4 includes a discussion pertaining to **the** determination of various tolerancerelated

costs. The proposed tolerance **optimization** model is presented in Section 5.

A numerical example is presented and a sensitivity analysis is conducted to demonstrate

**the** application of **the** proposed methodology in Section 6. Finally, conclusions

are presented in Section 7.

2. Literature review

A number of researchers have considered **the** problem of determining optimal

specification limits. Speckhart (1972), Spotts (1973), Chase et al. (1990), and Kim

and Cho (2000a) investigated **the** effect of tolerances on manufacturing cost, and

proposed models to determine tolerances for **the** minimization of manufacturing

cost. Tang (1988, 1991), Fathi (1990), Phillips and Cho (1999), and Kim and Cho

(2000b) studied **the** issue of tolerance design from **the** viewpoint of **function**al performance,

where **function**al performance is expressed in monetary terms by **using** **the**

Taguchi quality loss concept. Tang (1988) discussed an economic model for selecting

**the** most profitable tolerance in a complete inspection plan for **the** case where inspection

cost is a linear **function** of **the** tolerance. Fathi (1990) developed a graphical

approach for determining tolerances to minimize **the** quality loss and rejection costs

for a single quality characteristic. Tang and Tang (1989) and Tang (1991) fur**the**r

investigated screening inspection for multiple performance variables in a serial production

process. Tang and Tang (1994) presented a comprehensive literature review

related to **the** design of screening procedures. Kapur (1988), Kapur and Cho (1994),

and Kapur and Cho (1996) considered a tolerance **optimization** problem **using** truncated

normal, Weibull, and multivariate normal distributions, respectively. Jeang

(1997) proposed an **optimization** model for **the** simultaneous **optimization** of manufacturing

costs, rejection costs, and a quality loss **using** a process capability index

to establish a relationship between tolerance and standard deviation. The model

assumes a zero process bias condition; that is, **the** mean is equal to **the** target

value for **the** quality characteristic. Fur**the**r, **the** ratio of tolerance to **the** standard

Cost components

Closed-form

Author(s) Decision variable CM CR EQL solution

Spotts (1973),

Speckhart (1972),

Chase et al. (1990)

Kapur and

Cho (1994, 1996)

**Tolerance** Yes No No No

Number of standard

deviations from mean

No Yes Yes No

Jeang (1997) **Tolerance** Yes Yes Yes Yes

Jeang (1999) **Tolerance** Yes Yes Yes No

Phillips and

Cho (1999), Kim

and Cho (2000b)

Kim and

Cho (2000a)

Moskowitz and

Plante (2001),

Plante (2002)

**Tolerance** design **using** **the** **Lambert** W **function**

Number of standard

deviations from mean

No Yes Yes Yes

**Tolerance** Yes No No No

Number of standard

deviations from mean

Proposed model Number of standard

deviations from mean

No Yes Yes Yes

Yes Yes Yes Yes

Table 1. Summary of **the** literature and proposed model.

3237

deviation is assumed constant. Jeang (1999) demonstrated how response surface

methodology can be employed to determine tolerances of components in an assembly.

Phillips and Cho (1999) considered quality loss and rejection costs, where **the**

quality loss is determined empirically by applying regression analysis to historical

data of losses. They presented **optimization** models for **the** first-order and secondorder

empirical loss **function**s. Moskowitz et al. (2001) and Plante (2002) conducted

parametric and non-parametric studies for allocating **the** tolerances on design parameters

affecting a response. Table 1 provides a summary and comparison of **the**

literature with respect to **the** decision variable, manufacturing cost (CM), rejection

cost (CR), and expected quality loss (EQL), and **the** attainment of a closed-form

solution.

3. The **Lambert** W **function**

**Lambert** considered **the** trinomial equation x ¼ a þ x b and developed a series for

x in powers of a. This series can be transformed into a more symmetrical form

x x ¼ð Þ x þ

ð1Þ

by substituting x for x and setting x ¼ and a ¼ ( ) . Dividing equation (1)

by ( ) and letting converge to , equation (1) becomes

log x ¼ vx :

**Lambert**’s generalized series solution for x

ð2Þ

n is **the**n given by

x n ¼ 1 þ nv þ 1

2 nðn þ þ Þv2 þ 1

nðn þ

6

þ 2 Þðn þ 2 þ Þv3

þ 1

24 nðn þ þ 3 Þðn þ 2 þ 3 Þðn þ 3 þ Þv4 þ ð3Þ

3238 M. S. Govindaluri et al.

Using **the** series given by equation (3), Corless et al. (1996) shows that equation

(2) can be written as

log x ¼ v þ 21

2! v2 þ 32

3! v3 þ 43

4! v4 þ 54

5! v5 þ ð4Þ

The series shown in equation (4) converges to **Lambert** W (z), which is defined as

**Lambert** WðzÞ e **Lambert** WðzÞ ¼ z: ð5Þ

The **Lambert** W **function** is defined to be **the** multivalued inverse of **the** **function**

f (z) ¼ ze z . That is, **Lambert** W(z) can be any **function** that satisfies equation (5)

for all z. This **function** allows such **function**al equations as g(z)e g(z) ¼ z and

g(z) ¼ e **Lambert** W ln(z) , and ze z ¼ x where z ¼ **Lambert** W(x) to be solved. Readers are

referred to Corless et al. (1996) for detailed discussions. As shown in section 5, **the**

application of **the** **Lambert** W **function** to obtain a closed-form solution for **the**

optimal tolerance shown in equation (29) constitutes one of **the** principal contributions

in this paper.

4. Cost assessment in tolerance **optimization**

Until recently, it has often been assumed that a product is judged acceptable and

**the** customer loss is zero, if product performance falls within **the** specification limits.

However, it seems more reasonable to assume that a quality loss is incurred by

**the** customer, even when product performance deviates from **the** target within **the**

specification limits. An empirical loss **function** is used to evaluate **the** quality loss

within **the** specification limits in **the** proposed model. In addition to **the** loss incurred

by **the** customer, costs incurred by **the** manufacturer which are often ignored in **the**

tolerance **optimization** problems, such as rejection and manufacturing costs, are also

included in **the** proposed model.

Consider **the** quality characteristic Y that is normally distributed with mean

and variance

2 . Let f ( y) and F ( y) denote **the** probability density **function** and

cumulative distribution **function** of Y, respectively. The lower and upper specification

limits, LSL and USL, are defined as and þ , respectively, where >0.

Here, represents **the** number of standard deviations at which each specification

limit is located from **the** process mean.

4.1. Assessment of a quality loss

One of **the** most important issues encountered in **the** area of quality engineering

is **the** selection of a proper quality loss **function** (QLF) in order to relate a key quality

characteristic of a product to its quality performance. The QLF is a means to

quantify **the** quality loss of a product on a monetary scale, when a product performance

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

characteristics. This quality loss includes long-term losses related to poor reliability

and **the** cost of warranties, excess inventory, customer dissatisfaction, and eventual

loss of market share. Among **the** QLFs reported in **the** literature, **the** Taguchi loss

**function** has been identified as **the** most popular QLF in quantifying a quality loss.

However, we do not know **the** exact form of QLF in many real-world industrial

settings. It is often **the** case that empirical data on **the** costs associated with a quality

loss are available. Developing an empirical loss **function** based on regression analysis

ensures **the** best estimate of **the** QLF, since regression analysis is a well-established

statistical method for investigating cause-and-effect relationships.

**Tolerance** design **using** **the** **Lambert** W **function**

The linear regression form of **the** empirical loss **function**, ELF( y), for **the** quality

characteristic y is given by

ELF ð yÞ ¼b0 þ b1y þ b2 y 2 þ , ð6Þ

where **the** b’s are unknown parameters which need to be estimated **using** sample

data. The regression concept is concerned with analysing and establishing **the** prediction

for **the** response variable **using** **the** relationship between **the** ELF( y) and

**the** quality characteristic y. The principle of **the** least-squares method allows us to

determine **the** b’s in such a way as to minimize **the** sum of squares of **the** prediction

errors and to establish **the** QLF which best fits **the** actual data. Fur**the**rmore,

**the** application of regression analysis attempts to gain a better understanding of

**the** characteristics of **the** response of a system under study. Using **the** information

collected from a set of experimental trials, regression analysis helps establish a relationship

between **the** response variable and its influencing factors. Moreover, this

well-established statistical procedure allows us to calculate parameter estimations,

confidence intervals, and hypo**the**sis tests to analyse **the** data.

Because every manufacturing process experiences some inherent variability, Y can

be considered a random variable; that is, if f ( y) is **the** probability density **function**

for **the** random variable Y, **the** expected loss is evaluated by

Z

E½ELFð yÞŠ

¼ ELFð yÞ f ð yÞ dy: ð7Þ

y

Under a 100% inspection scheme, a product is rejected when y is not within **the**

specification limits. The expected quality loss under **the** 100% inspection scheme is

**the**n given by

E½ELFð yÞŠ ¼

Z USL

LSL

ELFð yÞ f ð yÞ dy: ð8Þ

If t denotes **the** tolerance on Y, **the**n t can be defined in terms of and as

t ¼ 2 . Therefore, LSL and USL become and þ , respectively, where

0 represents **the** number of standard deviations, at which **the** specification limits

are located from **the** mean. Therefore, **the** expected loss after implementing **the**

specification limits in terms of and can be written as

E½ELFð yÞŠ ¼

Z þ

ELFð yÞ f ð yÞ dy: ð9Þ

4.2. Assessment of rejection cost

Product performance that falls outside **the** specification limits is rejected and

**the** rejection cost is incurred by **the** manufacturer. Let CR be **the** unit rejection cost

incurred when a product falls below a lower specification limit or above an upper

specification limit. The expected rejection cost E [CR] is **the**n defined as

EC ½ RŠ

¼ CR Z LSL

1

f ð yÞ dy þ

Z 1

USL

Assuming USL ¼ LSL, equation (10) is expressed as

EC ½ RŠ

¼ 2CR Z 1

USL

3239

f ð yÞ dy : ð10Þ

f ð yÞ dy: ð11Þ

3240 M. S. Govindaluri et al.

If Y is a normally distributed random variable, **the**n **using** **the** transformation of

z ¼ (y ) , equation (11) simplifies to

EC ½ RŠ

¼ 2CR Z 1

ðzÞ dz, ð12Þ

and can also be written as

E½CRŠ¼2CR½1ðÞŠ, ð13Þ

where /( ), ( ), and z denote a standard normal density **function**, a cumulative

normal distribution, and a standard normal random variable, respectively.

4.3. Assessment of manufacturing cost

Additional manufacturing operations, slow processing rates, and additional care

on **the** part of **the** operator can increase **the** manufacturing cost. The manufacturing

cost usually contributes to a significant portion of **the** unit cost for many products

and its exclusion from **the** tolerance **optimization** model may result in a suboptimal

tolerance. Enforcing a 3 restriction (Krishnaswami and Mayne 1994, Speckhart

1972, Spotts 1973, Wilde and Prentice 1975, Su**the**rland and Roth 1975, Chase

et al. 1990, and Kim and Cho 2000a) may require selecting an expensive process

when tolerance selection is performed for **the** purposes of process selection. The

manufacturing cost-tolerance relationship proposed in this paper is free of this

ad hoc 3 assumption. **Tolerance** is defined in terms of , and as

t ¼ USL LSL ¼ð þ Þ ð Þ¼2 , ð14Þ

where for **the** process is known beforehand and is **the** decision variable (see

Section 5). The unit manufacturing cost CM is described by **the** following first-order

model:

CM ¼ a0 þ a1t þ ",

where " represents **the** least-squares regression error. The expected manufacturing

cost can **the**n be written as

E½CMŠ¼a0 þ a1t: ð15Þ

This linear manufacturing cost-tolerance modelling is often applied to many

industrial settings (see Patel 1980, Bjorke 1990, and Chase and Parkinson 1991).

Substituting t ¼ 2 in equation (15), E [CM] becomes

EC ½ MŠ

¼ a0 þ 2a1 : ð16Þ

5. Optimization model

The expected total cost, E [TC], is now given by

E½TCŠ ¼ E½ELF ð yÞŠþEC

½ RŠþEC

½ MŠ

¼ E½ELF ð yÞŠþ½PðYtÞþP

ðY tÞŠCRþEC

½ MŠ:

ð17Þ

Using equations (8), (11), and (15), **the** proposed **optimization** model is formulated

as

Minimize E½TCŠ ¼

Z USL

LSL

ELF ð yÞ f ð yÞ dy þ 2C R

Z 1

USL

f ð yÞ dy þ a 0 þ a 1t: ð18Þ

Note that **the** expected quality loss within specification limits can be written as

E½ELFð yÞŠ

¼

Z USL

LSL

Letting z ¼ ( y )/ , equation (19) becomes

Z USL

ðb 0 þ b 1y þ b 2 y 2 Þ f ð yÞ dy: ð19Þ

E½ELFðyÞŠ ¼

LSL

ðb0 þ b1y þ b2y 2 Z

Þ f ðyÞ dy

¼ b0 þ b1ð þ z Þþb2ð þ z Þ 2

ðzÞ dz

Z

Z

Z

¼ b0 ðzÞ dz þ b1 ð þ z Þ ðzÞ dz þ b2 ð þ z Þ 2 ðzÞ dz

Z

Z

Z

¼ b0 ðzÞ dz þ b1 ðzÞ dz þ z ðzÞ dz

Z

Z

þ b2 2

ðzÞ dz þ 2 z ðzÞ dz þ 2

Z

z 2 ðzÞ dz , ð20Þ

where ¼ (USL )/ (>0). In order to simplify equation (20), **the** following

derivations associated with **the** normal probability density **function** are utilized:

Z 1

r

ðzÞ dz ¼ 1 ðrÞ,

**Tolerance** design **using** **the** **Lambert** W **function**

Z 1

r

z ðzÞ dz ¼ ðrÞ, and

Z 1

r

z 2 ðzÞ dz ¼ 1 ðrÞþr ðrÞ:

ð21Þ

The detailed proofs related to equation (21) are given in Appendix A. Using

equation (21), E [ELF( y)] can be written as follows:

E½ELFð yÞŠ ¼ b0½ ð Þ ð ÞŠ þ b1 ½ ð Þ ð ÞŠ þ ½ ð Þ ð ÞŠ

þ b2 2

½ ð Þ ð ÞŠ þ

2

½½ ð Þþ ð ÞŠ ½ ð Þþð Þ ð ÞŠŠ

¼½2 ð Þ 1Šðb0 þ b1 þ b2 2

b2 2

Þþb2

2

½2ðÞŠ: ð22Þ

Similarly, E [CR] can be written as

EC ½ RŠ

¼ 2CR½PðYtÞþPðY tÞŠ

¼ 2CR Z 1

USL

f ð yÞ dy

¼ 2CR½1ðÞŠ: ð23Þ

Fur**the**r, E [CM] is defined as

EC ½ MŠ

¼ a0 þ 2a1 : ð24Þ

Substituting equations (22), (23), and (24) into equation (17), E [TC] can now be

written as

E½TCŠ ¼½2ðÞ1Šðb0þb1þb2 þ CR þ a0 þ 2a1 :

2

2 2

b2 CRÞþb2 ½2ðÞŠ 3241

To investigate **the** optimum, **the** first derivative with respect to

follows:

is calculated as

dE½TCŠ

¼ 2 ð Þ b0 þ b1 d

þ b2 2

b2 2 2

ð 1Þ CR þ 2a1 ¼ 0: ð26Þ

ð25Þ

3242 M. S. Govindaluri et al.

Equating @E [TC]/@ to zero and substituting

equation (26) becomes

2 1 pffiffiffiffiffi e

2

ð 2 =2Þ

b0 þ b1 þ b2 ð Þ¼ 1 pffiffiffiffiffi e

2

2

ð1=2Þ 2

,

b 2 2 ð 2

1Þ C R þ 2a 1 ¼ 0: ð27Þ

for

The **Lambert** W **function** is effectively employed to obtain a closed-form solution

, since **the** **function** is useful to solve expressions involving exponential terms

such as e . Details are presented **using** Lemma 1, and Propositions 1 and 2 in

Appendix B.

Proposition 2 states that if n5 ¼ n1e n2 ðn3 þ n4 Þ where 1, 2, 3, and 4 are not

**function**s of , **the**n **the** solution for can be given by

**Lambert** W

¼

n2n5 e ðn " #

2n3=n4Þ n4 þ n

n

2n3 1n4 :

n4n2 ð28Þ

Equation (27) can be expressed in **the** form n5 ¼ n1e n2 ðn3 þ n4 Þ shown in

Proposition 2, **using** such substitutions as ¼ 2 , n1 ¼ 2= ffiffiffiffiffi p

2 , n2 ¼ 1/2,

2 2 2

n3 ¼ b0 þ b1 þ b2 b2 C, n4 ¼ b2 and n5 ¼ 2a1 .

The closed-form solution for is now obtained by substituting 1, 2, 3, and 4

into equation (28). Thus, **the** optimal value of is

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffi 1

2 a1e

2b2 **Lambert** W

ðb v

u

" #

0þb1 þb 2

2 þCRÞ=2b 2

u

2

t

b 2

þ b 2ðb 0 þ b 1 þ b 2

2

b 2

b 2 2

b 2 2 CRÞ: ð29Þ

Consequently, **the** optimal LSL and USL are obtained as LSL* ¼ and

USL* ¼ þ . This **Lambert** W **function** is available in a number of standard

**optimization** software packages, such as Maple and Matlab.

5.1. Second derivative and **the** conditions for convexity

In this section, **the** second derivative is computed and **the** conditions for obtaining

**the** minimum value of E [TC] are investigated. The second derivative of E [TC] with

respect to is

@ 2 E½TCŠ @2 2

2 2 3

¼ 2 ð Þ 3b2 b0 b1 b2 þ b2 þ CR : ð30Þ

In order to obtain **the** minimum value of E [TC] at **the** stationary point(s) given in

equation (29), **the** second derivative must be greater than or equal to zero. Therefore,

3b 2 2

which fur**the**r implies that

ð 3b 2 2

b 0 b 1 b 2

b 0 b 1 b 2

2 þ b2 2 3 þ CR 0, ð31Þ

2 þ b2 2 2 þ CRÞ 0: ð32Þ

**Tolerance** design **using** **the** **Lambert** W **function**

In order to investigate **the** nature of **the** **function**, **the** inflection points need to be

considered. The inflection points can be determined by equating @ 2 E [TC]/@ 2 ¼ 0and

**the**n solving for . The inflection points are

¼ 0,

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b 2

2ð3b2 þ b0 þ b1 þ b 2

2 CRÞ

: ð33Þ

Using **the** inflection points, it can be concluded that

@ 2 E½TCŠ @2 ¼ 0 for ¼ 0, ð34Þ

@ 2 E½TCŠ @2 > 0 for 0 <

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b 2

<

2ð3b2 þ b0 þ b1 þ b 2

2 CRÞ

,

b2 ð35Þ

and

@ 2 E½TCŠ @2 < 0 for

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b 2

2ð3b2 þ b0 þ b1 þ b 2

2 CRÞ

<

b2

3244 M. S. Govindaluri et al.

E[TC]

E[TC]=153.32

δ*=0.9352

Figure 1. Relationship between E [TC] and .

Expected cost category Cost

0.9352. This optimal value can be verified from **the** plot of **the** expected total cost

shown in figure 1, where **the** minimum value of E [TC] at * ¼ 0.9352 is 153.32.

For * ¼ 0.9352, **the** different expected costs are shown in table 2.

It can be seen from table 2 that **the** manufacturing cost contributes to 62% of

E [TC]. Although manufacturing cost is commonly ignored in many tolerance **optimization**

models, it is evident from this particular example that a failure to consider

**the** manufacturing cost in tolerance **optimization** is likely to result in a suboptimal

tolerance (see table 2). Figure 2 shows how E [TC], E [CM], E [CR] and E [ELF( y)]

vary as changes.

Figure 3 shows that **the** first derivative of **the** expected total cost becomes zero

at **the** optimal value of * ¼ 0.9352. However, **the** negative value is ignored since

is always greater than zero. The conditions of convexity of **the** E [TC] **function**

are shown in figure 4, and it is concluded that @ 2 E [TC]/@ 2

is greater than zero in

**the** interval 0 1.667. This implies that E [TC] is a convex **function** and **the** local

minimum becomes **the** global minimum in **the** specified interval.

The sensitivity of **the** optimal specification limit to **the** changes in **the** mean and

standard deviation are shown in tables 3 and 4, respectively. It can be observed that

* gradually decreases as increases. The optimal USL and LSL occur when ¼ 2.0

and ¼ 5.0, and * ¼ 1.42 and 0.57, respectively. When ¼ 2.0, E [ELF(y)] becomes

larger, since an increase in tolerance implies a lower outgoing product quality.

Note that E [CR] is dependent only on *, and **the**refore **the** lowest value of E [CR]

is obtained when * ¼ 1.42, while **the** highest value is obtained when * ¼ 0.57.

δ

Percentage of

E [TC]

E [ELF( y)] 23.96 16

E [CR] 31.79 23

E [CM] 94.39 62

E [TC] 153.32 100

Table 2. Break-up of expected total cost at * ¼ 0.9352.

costs

[ ]

2

∂ E TC

2

∂ δ

E[C M ]

δ

E[TC]

Figure 4. Plot of @ 2 E [TC]/@ 2 with respect to .

δ

E[ELF(y)]

E[C R ]

Figure 2. Plots of E [CR], E [ELF (y)], and E [CM] with respect to .

[ TC]

dE

dδ

**Tolerance** design **using** **the** **Lambert** W **function**

Figure 3. Plot of dE[TC]/d with respect to .

δ

3245

3246 M. S. Govindaluri et al.

* E[CM] E [CR] E [ELF( y)] E [TC]

2.0 0.5 1.42 91.50 15.67 56.62 163.80

2.2 0.5 1.28 92.32 20.04 46.77 159.13

2.4 0.5 1.17 92.98 24.17 39.01 156.16

2.6 0.5 1.08 93.53 28.05 32.85 154.42

2.8 0.5 1.00 93.99 31.64 27.93 153.56

3.0 0.5 0.94 94.39 34.97 23.96 153.32

3.2 0.5 0.88 94.74 38.03 20.74 153.51

3.4 0.5 0.83 95.04 40.85 18.11 154.00

3.6 0.5 0.78 95.31 43.45 15.93 154.68

3.8 0.5 0.74 95.55 45.84 14.11 155.50

4.0 0.5 0.71 95.77 48.06 12.58 156.41

4.2 0.5 0.67 95.96 50.10 11.29 157.36

4.4 0.5 0.64 96.14 52.00 10.20 158.33

4.6 0.5 0.62 96.30 53.76 9.25 159.31

4.8 0.5 0.59 96.45 55.39 8.44 160.28

5.0 0.5 0.57 96.58 56.92 7.73 161.23

Table 3. Effect of on *.

* E [C M] E [C R] E [ELF( y)] E [TC]

0.0 5.0 0.58 94.21 56.23 16.46 166.90

0.1 5.0 0.58 94.21 56.27 16.52 167.00

0.2 5.0 0.58 94.22 56.36 16.64 167.22

0.3 5.0 0.58 94.25 56.50 16.82 167.56

0.4 5.0 0.57 94.27 56.68 17.06 168.01

0.5 5.0 0.57 94.31 56.92 17.35 168.58

0.6 5.0 0.57 94.35 57.21 17.70 169.26

0.7 5.0 0.56 94.40 57.56 18.10 170.05

0.8 5.0 0.55 94.46 57.94 18.55 170.95

0.9 5.0 0.55 94.52 58.39 19.03 171.95

1.0 5.0 0.54 94.60 58.90 19.55 173.05

1.2 5.0 0.52 94.77 60.11 20.65 175.54

1.4 5.0 0.50 94.99 61.61 21.77 178.37

1.6 5.0 0.48 95.24 63.43 22.81 181.48

1.8 5.0 0.44 95.55 65.64 23.61 184.80

2.0 5.0 0.41 95.92 68.34 23.99 188.26

Table 4. Sensitivity of * to changes in while holding at ¼ 5.0.

In o**the**r words, E [C R] is higher for lower values of * and vice versa. The effect

of | | (i.e. process bias) on * is shown in tables 4 and 5. Finally, table 6 shows

**the** effect of E [CM], E [CR], E [ELF(y)], and E [TC] on **the** changes in and .

The Maple program for **the** numerical example and sensitivity analysis is presented

in Appendix C.

7. Conclusions and fur**the**r study

Quality engineers are often faced with **the** problem of determining **the** optimal

tolerance in many industrial settings. When determining **the** tolerance, it is important

to consider a trade-off between costs incurred by a manufacturer and **the** customer.

In this paper, we showed how **the**se costs are quantified and integrated, and also

showed how **the** **Lambert** W **function** is incorporated into this tolerance optimi-

**Tolerance** design **using** **the** **Lambert** W **function**

* E [CM] E [CR] E [ELF( y)] E [TC]

0.0 2.0 1.44 94.22 14.85 25.34 134.41

0.1 2.0 1.44 94.23 14.90 25.48 134.61

0.2 2.0 1.44 94.24 15.00 25.79 135.04

0.3 2.0 1.43 94.27 15.17 26.27 135.70

0.4 2.0 1.43 94.30 15.39 26.90 136.58

0.5 2.0 1.42 94.34 15.67 27.68 137.69

0.6 2.0 1.40 94.38 16.03 28.62 139.03

0.7 2.0 1.39 94.44 16.45 29.69 140.58

0.8 2.0 1.37 94.50 16.94 30.90 142.35

0.9 2.0 1.36 94.58 17.52 32.23 144.33

1.0 2.0 1.33 94.66 18.19 33.67 146.52

1.2 2.0 1.29 94.86 19.84 36.80 151.50

1.4 2.0 1.23 95.09 21.99 40.13 157.21

1.6 2.0 1.16 95.38 24.78 43.41 163.57

1.8 2.0 1.07 95.72 28.45 46.28 170.46

2.0 2.0 0.97 96.13 33.35 48.21 177.69

Table 5. Sensitivity of * to changes in while holding at ¼ 2.0.

E [C M] E [C R] E [ELF( y)] E [TC]

0.2 3.0 0.5 98.80 84.15 1.11 184.06

0.4 3.0 0.5 97.60 68.92 3.64 170.16

0.6 3.0 0.5 96.40 54.85 8.70 159.95

0.8 3.0 0.5 95.20 42.37 16.80 154.37

0.9352 3.0 0.5 94.39 34.97 23.96 153.32

1.0 3.0 0.5 94.00 31.73 27.82 153.55

1.2 3.0 0.5 92.80 23.01 41.05 156.87

1.4 3.0 0.5 91.60 16.15 55.46 163.20

1.6 3.0 0.5 90.40 10.96 69.84 171.20

1.8 3.0 0.5 89.20 7.19 83.22 179.61

2.0 3.0 0.5 88.00 4.55 94.86 187.41

2.2 3.0 0.5 86.80 2.78 104.38 193.96

2.4 3.0 0.5 85.60 1.64 111.73 198.97

2.6 3.0 0.5 84.40 0.93 117.11 202.44

2.8 3.0 0.5 83.20 0.51 120.84 204.55

3.0 3.0 0.5 82.00 0.27 123.30 205.57

Table 6. Relationship between costs and .

3247

zation problem to derive a closed-form solution. This proposed model and solution

may be appealing to quality practitioners since **the** **Lambert** **function** is found

in a number of standard **optimization** software packages. The extension of **the**

proposed model to **the** case of more than one performance measure within

a multivariate context may be a good future study.

Appendix A: The integration of z/ (z)dz and z 2 / (z)dz

First, we consider **the** indefinite integral of **the** integrand zf (z)dz. Following

**the** definition of a standard normal random variable, **the** integral can be written as

Z

z ðzÞ dz ¼ 1 Z

pffiffiffiffiffi

ze

2

z2 =2

dz:

3248 M. S. Govindaluri et al.

Substituting u into z 2 , it follows that du ¼ 2z dz and z dz ¼ du/2. Thus **the** integral

can now be written as follows:

Z

z ðzÞ dz ¼ 1

2 ffiffiffiffiffi

Z

p

2

e u=2 As

du:

R e u=2 du ¼ e u=2 , we get

Z

z ðzÞ dz ¼ 1

2 ffiffiffiffiffi

Z

p e

2

u=2 du ¼

1

pffiffiffiffiffi e

2

u=2 ¼ ðzÞ:

Next, we consider **the** indefinite integral of z 2 /(z)dz. This integration can be

performed **using** integration by parts as follows:

Z Z

du

uvdx ¼ v

dx

Z Z

du

dx

vdx dx:

If z 2 /(z)dz is separated into two parts, z and z/(z), **the**n **the** integration by parts

formula can be applied as follows:

Z

z 2 Z

ðzÞ dz ¼ zz ð ðzÞÞ

dz

Z

Z

¼ z z ðzÞ dz ðzÞ dz

¼ z ðzÞ ðzÞ:

Appendix B: The application of **the** **Lambert** W for solving

g2 ¼ðvþ g1Þex and n5 ¼ n1en2vðn3 þ n4vÞ

Lemma 1: Suppose 2R 1 and **the** mapping : ! R be ¼ e . The solution for

is **the**n given by ¼ **Lambert** W ( ).

Proof: See Corless et al. (1996).

Proposition 1: If 2 ¼ ( þ 1)e x where 1 and 2 are not **function**s of , **the**n

¼ **Lambert** Wð 2 e 1Þ 1:

Proof: Consider **the** following equation:

2 ¼ð þ 1Þe : ðB-1Þ

Letting þ 1 ¼ C, equation (B-1) becomes

2 ¼ Ce C 1 : ðB-2Þ

To convert equation (B-2) in **the** standard form ¼ e , we first modify equation

(B-2) as follows:

2 e 1 ¼ Ce C 1 : ðB-3Þ

Next, substituting 2 e 1 ¼ ! into **the** right-hand side of equation (B-3), we

obtain ! ¼ Ce C 1 : Recalling Lemma 1, C is given by

C ¼ **Lambert** Wð!Þ: ðB-4Þ

Therefore, is obtained as

¼ **Lambert** Wð 2 e 1 Þ 1:

Proposition 2: If n5 ¼ n1 e n2 ðn3 þ n4 Þ where 1, 2, 3, and 4 are not **function**s of ,

**the**n

**Lambert** W

¼

n2n5 e ðn !

2n3=n4Þ n4 þ n

n

2n3 1n4 :

n4n2 Proof: In order to prove Proposition 2, we first consider **the** equation

n5 ¼ n1e n2 ðn3 þ n4 Þ: ðB-5Þ

Multiplying both sides by 2/ 1 4,, equation (B-5) becomes

n5n2 ¼

n1n4 n2 e

n4 n2 ðn3 þ n4 Þ: ðB-6Þ

Letting C ¼ ( 2/ 4)( 3 þ 4 ) equation (B-6) can be written as

Fur**the**r, **using**

**Tolerance** design **using** **the** **Lambert** W **function**

5 2e ð 2 3= 4Þ

1 4

5 2e ð 2 3= 4Þ

1 4

¼ Ce C : ðB-7Þ

¼ !,

equation (B-7) is now in **the** standard form ¼ e . That is,

C ¼ **Lambert** Wð!Þ: ðB-8Þ

Substituting

! ¼ 5 2 e ð 2 3= 4Þ

back into equation (B-8), **the** standard form in equation (B-8) can be written as

C ¼ **Lambert** W 5 2e ð !

2 3= 4Þ

: ðB-9Þ

Equation (B-9) can be expanded to express **the** solution in terms of by replacing

C with ( 2/ 4)( 3 þ 4 ). Thus, is given by

**Lambert** W

¼

n2n5e ðn !

2n3=n4Þ n4 þ n

n

2n3 1n4 :

n4n2 1 4

1 4

Appendix C: Maple program for **the** numerical example

3249

>NORMALPDF:¼ exp( (y 0)^2/2/1^2 )/sqrt(2*3.141592*1^2);

>NORMALCDF:¼ Int(NORMALPDF, y ¼ infinity .. x);

>EMPIRICAL LOSS:¼ (b0 þ b1*(u y) þ b2*(u þ y*s)^2);

>EXPECTED EMPIRICAL LOSS:¼ Int(EMPIRICAL LOSS * NORMALPDF,y¼ x..x);

>REJECTION COST:¼ 2*c*(1 NORMALCDF);

>MANUFACTURING COST:¼ a0 þ 2*a1*s*x;

>EXPECTED TOTAL COST:¼ EXPECTED EMPIRICAL LOSS þ REJECTION COST

þ MANUFACTURING COST;

3250 M. S. Govindaluri et al.

>plot([EXPECTED EMPIRICAL LOSS, REJECTION COST, MANUFACTURING

COST, EXPECTED TOTAL COST], x¼0..4);

>FIRST DERIVATIVE:¼ diff(EXPECTED TOTAL COST, x);

>n1:¼ 2*k/sqrt(2*pi);

>n2:¼ (u þ t)^2 þ s^2 1 c/k;

>n3:¼ 1/2;

>n4:¼ 2*a1*s;

>FIRST DERIVATIVE 1:¼ n1*(x^2 þ n2)*exp(n3*x^2) n4¼0;

>SOLVE1:¼ solve(FIRST DERIVATIVE 1, x);

>plot(SOLVE1, x¼0..4);

>SECOND DERIVATIVE:¼ diff(FIRST DERIVATIVE 1, x);

>SOLVE2:¼ solve(SECOND DERIVATIVE, x);

>plot(SOLVE2, x¼0..4);

References

BJORKE, O., 1990, Computer-Aided Tolerancing (New York, ASME).

CHASE, K. W., GREENWOOD, W. H., LOOSLI, B. G. and HAUGHLUND, L. F., 1990, Least-cost

tolerances allocation for mechanical assemblies with automated process selection.

Manufacturing Review, 3, 49–59.

CHASE, K. W. and PARKINSON, A. R., 1991, A survey of research in **the** application of tolerance

analysis to **the** design of mechanical assemblies. Engineering Design, 3, 23–27.

CORLESS, R. M., GONNET, G. H., HARE, D. E. G., JEFFREY, D. J. and KNUTH, D. E., 1996,

On **the** **Lambert** w **function**. Advances in Computational Ma**the**matics, 5, 329–359.

FATHI, Y., 1990, Producer-consumer tolerances. Journal of Quality Technology, 22(2),

138–145.

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., 1999, Robust tolerance design by response surface methodology. International

Journal of Production Research, 37(14), 3275–3288.

KAPUR, K. C., 1988, An approach for development of specifications for quality improvement.

Quality Engineering, 1, 63–77.

KAPUR, K. C. and CHO, B. R., 1994, Economic design and development of specifications.

Quality Engineering, 6, 401–417.

KAPUR, K. C. and CHO, B. R., 1996, Economic design of **the** specification region for multiple

quality characteristics. IIE Transactions, 28, 401–417.

KIM, Y. J. and CHO, B. R., 2000a, The use of response surface designs in **the** selection of

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

KIM, Y. J. and CHO, B. R., 2000b, Economic integration of design **optimization**. Quality

Engineering, 12(4), 561–567.

KRISHNASWAMI, M. and MAYNE, R. W., 1994, Optimizing tolerance allocation for minimum

cost and maximum quality. Proceedings of **the** 20th ASME Design Automation

Conference, DE, 69(1), 211–217.

MOSKOWITZ, H., PLANTE, R. and DUFFY, J., 2001, Multivariate tolerance design **using** quality

loss. IIE Transactions, 33, 437–448.

PATEL, A. M., 1980, Computer-aided assignment of tolerances. Proceedings of **the** 17th Design

and Automation Conference, ASME, Minneapolis, MN, 129–133.

PHILLIPS, M. D. and CHO, B. R., 1999, 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.

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

Journal of Engineering for Industry, 94, 447–453.

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

Engineering for Industry, 95, 762–764.

SUTHERLAND, G. H. and ROTH, B., 1975, Mechanism design: accounting for manufacturing

tolerances and costs in **function** generating problems. Journal of Engineering for

Industry, 98, 283–286.

**Tolerance** design **using** **the** **Lambert** W **function**

3251

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

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

TANG, K., 1991, Design of multi-stage screening procedures for a serial production system.

European Journal of Operational Research, 52(3), 280–290.

TANG, K. and TANG, J., 1989, Design of product specifications for multi-characteristic inspection.

Management Science, 35(6), 743–756.

TANG, K. and TANG, J., 1994, Design of screening procedures: a review. Journal of Quality

Technology, 26(3), 209–226.

WILDE, D. and PRENTICE, E., 1975, Minimum exponential cost allocation of sure-fit tolerances.

Journal of Engineering for Industry, 97, 1395–1398.