Multivariate tolerance design for a quadratic design parameter model

IIE Transactions (2002) 34, 565–571

Multivariate tolerance design for a quadratic design parameter model ROBERT PLANTE Krannert Graduate School of Management, Purdue University, 1310 Krannert Building, West Lafayette, IN 47907-1310, USA E-mail: [email protected] Received January 2000 and accepted November 2001

The determination of tolerance allocations among design parameters is an integral phase of product/process design. Such allocations are often necessary to achieve desired levels of product performance. Parametric and nonparametric methods have recently been developed for allocating multivariate tolerances. Parametric methods assume full information about the probability distribution of design parameter processes, whereas, nonparametric methods assume that only partial information is available, which consists of only design parameter process variances. These methods currently assume that the relationship between the design parameters and each of the performance measures is linear. However, quadratic response functions are increasingly being used to provide better approximations of the relationships between performance measures and design parameters. This is especially prevalent where there is a multivariate set of performance measures that are functions of a common set of design parameters. In this research we propose both parametric and nonparametric multivariate tolerance allocation procedures which consider the more general case where these relationships can be represented by quadratic functions of the design parameters. We develop the corresponding methodology and nonlinear optimization models to accommodate and take advantage of the presence of interactions and other nonlinearities among suppliers.

1. Introduction A number of research studies have been conducted on tolerance allocation (Chase and Parkinson, 1991). Many of these can be classified into full information (statistical) and partial information models (Moskowitz et al., 2001). In the full information model, the designer knows the process distribution for each design parameter, whereas, in the partial information case, the designer only possesses knowledge about the variance of each process. The partial information case is not uncommon, since designers seldom have sufficient data by which to specify the distribution of design parameter processes (Chase and Parkinson, 1991). In this paper we generalize the work of Moskowitz et al. (2001) by modeling the relationship between each performance measure and a set of design parameters as quadratic functions, which consist of interaction and squared terms, as well as linear terms. Such extensions are important and necessary given the prevailing use of quadratic response functions when estimating the influence of design parameters on performance measures (Myers and Montgomery, 1995). Such functions permit a more accurate second-order Taylor series approximation of the relationship between performance measures and design parameters in lieu of a first-order series approxi0740-817X

Ó 2002 ‘‘IIE’’

mation via the linear model. These methodological extensions are further required since, unlike a linear model, the presence of interaction terms in the nonlinear model prevents the independent allocation of design parameter tolerances to each supplier. For a given application, these extensions would also permit an assessment of the sensitivity or robustness of optimal tolerance allocations with respect to the type of response function estimated. In the next section, we develop an expected total loss model which will be used to assess tolerance allocations. We develop full information models and optimal parametric tolerance allocation schemes in Section 3. For the partial information case, upper bounds for two different expected total loss models are then developed in Section 4. Using these upper bounds, we then develop nonlinear optimization models for optimal tolerance allocations that minimize the maximum expected total loss. We then provide a summary of our research in Section 5.

2. Tolerance design In this section we develop a model for expected total loss which consists of: (i) loss realized by the manufacturer (expected external loss modeled via a Taguchi (1986) quadratic loss function); and (ii) loss experienced by

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suppliers for scrapping or reworking nonconforming product (expected internal loss). 2.1. Expected external loss A designer, via parameter design, has determined that each of M performance measures Yj ( j ¼ 1; . . . ; M) is a quadratic function of a common set of N design parameters as follows, Yj ¼

N X

aij Xi þ

N X

i¼1

aiij Xi2 þ

i¼1

N X N X

r02 Xi = the variance of the ith truncated design parameter distribution following screening on a specified tolerance; E½ = the expected value. Using Equation (1), Equation (3) can be expanded as follows, 2 !2 N N X X 2 02 4 rYj ¼ rXi aij þ 2aiij lXi þ aikj lXk i¼1

aikj Xi Xk ;

ð1Þ

i¼1 k>i

þ

where,

Note that while the Xi ’s are independent of each other, the Yj ’s are potentially dependent due to their joint dependence on the Xi ’s. Using a quadratic loss function (a second-order Taylor series expansion of a general loss function evaluated at a nominal or target performance), the designer specifies expected external loss (loss realized by the manufacturer/ customer) as follows (Taguchi, 1986), EEL ¼

M h i X kj r2Yj þ ðlYj  sYj Þ2 ;

ð2Þ

j¼1

where, EEL = expected external loss;

r2Yj = the variance of the jth performance measure; = the nominal or target for the jth performance measure.

The design parameters come from various supplier processes that are plausibly assumed to be independent, such that, for a quadratic response function (Fathi, 1991; Plante, 2001), "  # N X @Yj 2 02 2 rYj ¼ rXi ; E ð3Þ @Xi i¼1 where,

ð4Þ

Let us further assume that the designer has already established the required locations for the process means, lXi , such that li ¼ si . Thus, using (4) and (2), the expected external loss is, EEL ¼

N X

r02 Xi mi ;

ð5aÞ

i¼1

where, mi ¼

M X

2 kj 4 aij þ 2aiij lXi þ

j¼1

þ

N X

# a2ikj r02 Xk

þ

4a2iij r02 Xi

:

N X

!2 aikj lXk

k6¼i

ð5bÞ

k6¼i

The expression for mi in (5b) is of particular importance and is used throughout this paper. This expression contains the influential impact of tolerance allocations of all design parameter processes on the expected cost of the ith process. As expected, these influences are directly related to the estimated interaction effects aikj , and are realized via design parameter process means lXk and truncated variances r02 Xi .

= quadratic cost coefficient;

lYj = the mean of the jth performance measure, which equals the nominal or target value following parameter design; sYj

2 02 a2ikj r02 Xk þ 4aiij rXi :

k6¼i

Xi ¼ the ith design parameter, i ¼ 1; . . . ; N ; aij ¼ estimated linear effect of the ith design parameter on the jth performance; aiij ¼ estimated square effect of the ith design parameter on the jth performance; aikj ¼ estimated interaction effect of the ith design parameter on the jth performance.

kj

N X

k6¼i

#

2.2. Expected internal loss Since the proposed tolerance allocation procedure considers internal as well as external losses, the designer wishes to further consider two possible scenarios for internal costs. These scenarios are: (i) suppliers scrap process output that does not conform to tolerance; and (ii) suppliers rework nonconforming process output. For the scrap model, the designer specifies the expected internal loss model as an increasing linear function of the odds ratio between the proportion of product produced outside tolerance and the proportion of product within tolerance (Chase and Parkinson, 1991) as follows, X

EILS ¼ SXi 1  qXi =qXi ; ð6Þ i2XS

where,

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Multivariate tolerance design EILS = expected internal loss due to scrap; XS = set of suppliers that scrap nonconforming output; SXi = the scrap cost coefficient for the ith design parameter,

ETLR ¼

i Xh

RXi 1  qXi þ r02 Xi mi :

ð10Þ

i2XR

3. Parametric tolerance allocation and,

qXi ¼ Prob lXi  0:5TX Xi lXi þ 0:5TXi ; where, qXi = the yield (proportion of product within specifications) of the supplier process producing the ith design parameter; TXi = the tolerance allocated for the ith design parameter. For the rework model, the designer specifies the expected internal loss model as an increasing linear function of the proportion of product produced outside tolerance (Chase and Parkinson, 1991) as follows, X

EILR ¼ ð7Þ RXi 1  qXi ;

In this section we develop the parametric or full information expected cost models for the allocation of tolerances to design parameter processes. Under full information, a designer knows the process distributions of the design parameters. Although designers seldom have full information, the assumption of a normal process distribution is often used. For this reason alone, it is useful to develop the parametric approach. However, this development also serves an additional purpose by providing a comparative measure relative to the partial information case. Under full information, the values for process yield in (6) and (7) can be assessed. In addition, the variance of the truncated process distribution can be assessed in general as follows, TXi =2

r02 Xi

i2XR

where, EILR = the expected internal loss due to rework; XR = set of suppliers that rework nonconforming output; RXi = the per unit rework cost for the ith design parameter.

2.3. Expected total loss Using (5), (6), and (7), the expected total loss corresponding to the allocation of tolerances to design parameters can be expressed as follows,

 X   1  qX  02 i ETL ¼ þ rXi mi SXi qXi i2XS i Xh

þ : ð8Þ RXi 1  qXi þ r02 m i Xi i2XR

From (8), that portion of the total expected costs influenced by the scrap model is,

 X   1  qX  02 i ETLS ¼ þ rXi mi : ð9Þ SXi qXi i2X S

Similarly, we can express that component of the expected total loss for the rework model as,

¼

Z



2 xi  lXi f ðxi Þdxi ;

i ¼ 1; . . . ; N ;

ð11Þ

TXi =2

where, f ðxi Þ = the probability density associated with the ith design parameter. When a design parameter follows a normal distribution, the yield of the ith design parameter process can be expressed as,   TXi qXi ¼ 2U  1; ð12Þ 2rXi with truncated process variance assessed via (11) (Tang and Tang, 1989),       TXi TXi 2 r02  1  r ; ð13Þ ¼ r 2U T / Xi Xi Xi Xi 2rXi 2rXi where, UðÞ = the standard normal distribution function; /ðÞ = the standard normal density function. Using (6), (12) and (13) for the scrap model, that portion of the expected total loss due to the ith process, where the process distribution is normal, can be expressed as,   2ð1  UðTXi =2rXi ÞÞ ETLSi ¼ SXi 2UðTX =2rXi Þ  1   i     TXi TXi þ r2Xi 2U  1  rX i T X i / mi ; ð14aÞ 2rXi 2rXi where,

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Plante 2 mi ¼

6 M X 6 kj 6 6 4 j¼1

     3 T T X X k k 7 aij þ 2aiij lXi þ aikj lXk þ a2ikj r2Xk 2U  1  rXk TXk / 7 2r 2r X X k k 7: k6¼i k6¼i 7        5 T T Xi Xi 2 2  1  rXi TXi / þ 4aiij rXi 2U 2rXi 2rXi N X

!2

N X

Using (7), (12) and (13) for the rework model, that portion of the expected total loss due to the ith process, can be expressed as,          TXi TXi 2 ETLRi ¼ RXi 2 1  U þ rXi 2U 1 2rXi 2rXi   TXi  rXi TXi / mi : ð15Þ 2rXi Using (8), (14) and (15), the expected total loss when design parameter processes are described by a normal distribution is then,





ð14bÞ

collection for estimating response surface functions in the parameter design phase. Under partial information, the process yield and the variance resulting from truncated process distributions cannot be directly assessed. Consequently, we develop upper bound models for expected total loss that only require estimates of the variances of design parameter processes. The nonparametric method for the tolerance design problem can now be stated as determining tolerance allocations, TX0 i ’s, that minimize the maximum expected total loss as follows,

    X 2ð1  UðTX =2rX ÞÞ    TX  TXi i i i 2 þ rXi 2U  1  rXi TXi / mi ETL ¼ SXi 2UðTXi =2rXi Þ  1 2rXi 2rXi i2XS           X   TXi TXi TXi 2 þ RXi 2 1  U þ rXi 2U  1  rXi TXi / mi : 2rXi 2rXi 2rXi i2X

ð16Þ

R

The optimal tolerance allocations for minimizing the expected total loss in (16) are obtained by a constrained nonlinear optimization algorithm, such as the generalized reduced gradient (Lasdon et al., 1978). The required constraints for a solution to (16) are TXi 0, i ¼ 1; . . . ; N . Given the widespread use of Excel spreadsheets and the availability of optimization add-ins, such as SOLVER (based on the generalized reduced gradient method), we recommend that such nonlinear mathematical programming packages be employed to solve for each of the optimal tolerance allocations in (16).

4. Nonparametric tolerance allocation In this section, we develop nonparametric or partial information methods for the allocation of tolerances to design parameter processes. This is necessary since designers rarely have full information on design parameter processes. Under partial information, a designer possesses knowledge only about the variances of design parameter processes. Such information may be available from historical data or may be acquired from an actual assessment of supplier process variations. Such assessments may also be concurrently obtained during data

Minimize ½Maximum ð ETLS ¼ EILS þ EILR þ EELÞ : ð17Þ To obtain the maximum, or upper bound, on the expected total costs, with respect to tolerance allocation, it is evident from Equations (9) and (10) that a lower bound on qXi and an upper bound on r02 Xi are required. To obtain these bounds, it is assumed that the design parameter process distributions are symmetric and unimodal. The lower bound on process yield, using the Chebychev–Gauss inequality for unimodal distributions, is,

4 r2Xi Prob lXi  0:5TXi Xi lXi þ 0:5TXi 1  ; 9 ð0:5TXi Þ2 or, qXi 1 

16 r2Xi ; 9 TX2i

ð18Þ

where, r2Xi = the variance of the process which produces the ith design parameter.

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Multivariate tolerance design 4.1. Upper bound on suppliers’ expected loss Using (6) and (18) the upper bound on a suppliers’ expected loss and suppliers scrap nonconforming output is, " , !# X 16 r2Xi 16 r2Xi EILS ; SXi 1 9 TX2i 9 TX2i i2XS ð19Þ

¼ EILSB

If suppliers rework nonconforming output (7), the upper bound on a suppliers’ expected loss for an infinite range process is, ! X 16 r2Xi EILR ; RXi 9 TX2i i2XR ð20Þ

¼ EILRB : 4.2. Upper bound on external loss

To complete the derivation of upper bounds on total loss, an upper bound on expected external loss is required. As mentioned previously, an upper bound on r02 Xi is required to obtain an upper bound on expected external loss. For a symmetric, unimodal distribution Gray and Odell (1967) showed that the variance on interval [a; b] is bounded above by ðb  aÞ2 =12 which is the variance of a uniform distribution. Thus, given a tolerance TXi , such that the process distribution on the ith design   parameter is on the interval lXi  0:5TXi ; lXi þ 0:5TXi , the upper bound on the process variance is, r02 Xi

TX2i : 12

ð21Þ

Using (5), the upper bound on expected external loss is then, EEL

N X TX2

i

i¼1

12

Scrap nonconforming output ( " , !# ) X TX2i 16 r2Xi 16 r2Xi ETLS g ; SXi 1 þ 9 TX2i 9 TX2i 12 i i2XS ¼ ETLSB : ð23Þ Rework nonconforming output " ! # X TX2i 16 rXi g ; þ RXi ETLR 9 TX2i 12 i i2XR ¼ ETLRB : 4.3. Optimal tolerance allocations Since Equation (17) is separable into internal and external expected losses, then the tolerance allocations that minimize the maximum total expected losses can be determined by the following, @EILSB @EELB þ ¼ 0; @TXi @TXi @EILRB @EELB þ ¼ 0; @TXi @TXi

kj 4 aij þ 2aiij lXi þ

þ

N X k6¼i

ð26Þ

for suppliers that rework nonconforming product. Using (25) and (26), the optimal tolerance allocations, TXi ’s, that minimize each of the loss function upper bounds (23) and (24) can be determined by solving the following system of N nonlinear equations in N unknowns. 8 9 "  1=2 #1=2 =1=2