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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 synthesis. 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 their

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 functional performance, however, can be improved

by specifying a tight tolerance on a quality characteristic. On the other 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 functional 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 mathematical functions have been used

to approximate an actual loss. Among them, 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

synthesis. 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 their solutions in a closed form but also quickly determine their

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

mathematical 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 functional performance,

where functional 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) further

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. Further, 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 functions. 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 then 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 functional 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. Furthermore,

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 hypothesis 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

then given by

E½ELFð yÞŠ ¼

Z USL

LSL

ELFð yÞ f ð yÞ dy: ð8Þ

If t denotes the tolerance on Y, then 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 then 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, then 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, Sutherland 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 then 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

Further, 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

functions of , then 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 further 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

then 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 therefore 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


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 other 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 further 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 these 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), then 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 then 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 functions of , then

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 functions of ,

then

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

Further, 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);

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