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USE OF MATHEMATICAL PROGRAMMING IN THE ANALYSIS OFCONSTRAINED AND UNCONSTRAINED INDUSTRIAL EXPERIMENTSPeter D. Kraus a; James C. Benneyan b; Neal A. Mackertich c

a Department of Mechanical and Industrial Engineering, University of Massachusetts, Amherst,Massachusetts b Mechanical, Industrial, and Manufacturing Engineering Department, Northeastern University,Boston, Massachusetts c Raytheon Systems Company, Sudbury, Massachusetts

Online Publication Date: 01 January 2000

To cite this Article Kraus, Peter D., Benneyan, James C. and Mackertich, Neal A.(2000)'USE OF MATHEMATICAL PROGRAMMINGIN THE ANALYSIS OF CONSTRAINED AND UNCONSTRAINED INDUSTRIAL EXPERIMENTS',Quality Engineering,12:3,395 —406

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Quality Engineering, 12(3). 395 -406 (2000)

USE OF MATHEMATICAL PROGRAMMING IN THE ANALYSIS OF CONSTRAINED AND

UNCONSTRAINED INDUSTRIAL EXPE-NTS

Peter D. Kraus Department of Mechanical and Industrial Engineering

Engineering Lab Building Box 32210 University of Massachusetts

Amherst, Massachusetts 0 1003

James C. Benneyan Mechanical, Industrial, and Manufacturing Engineering Department

334 Snell Engineering Center Northeastern University

Boston, Massachusetts 02 155

Neal A. Mackertich Raytheon Systems Company

528'Boston Post Road, M I S 1-1-824 Sudbury, Massachusetts 01 776

Key Words

Experimental design; Optimization; Mathematical pro- gramming; Cost minimization.

Introduction

In the analysis of industrial experimental results, trade- offs between the optimal settings for the response mean, the response variance, and their associated costs often are inevi- table in practice. In such situations, any of several methods typically are used to set the mean response equal to its target value with the response variance minimized to the extent

395

Copyright 8 2000 by Marcel Dekker, Inc.

possible. Typically, however, trade-offs between the costs and nonconforming rate associated with the resultant factor settings are not rigorously or explicitly explored. In order to better understand and address these trade-offs, mathematical optimization methods can complement other approaches by helping to explore alternate solutions more direcdy and de- termine the optimal factor settings that minimize poor qual- ity or total costs.

For example, in practice, a large percentage of industrial experiments are of the "nominal-the-best" type for which the studied response characteristic Y is continuous with a target value'T,, upper specification limit USL,, and lower specification limit LSL,. Three general objectives in this type of problem are to identify the best process design that

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KRAUS, BENNEYAN, AND MACKERTICH

either minimizes the nonconformance rate, minimizes the factor setting costs, or minimizes the total costs of noncon- formance and production. This article illustrates approaches in such cases for integrating mathematical programming with experimental design in order to help identify the over- all optimal settings, possibly subject to upper bounds on the nonconforming rate or total cost, and to help conduct sensitivity analysis in cases where accurate cost estimates are not easily obtainable. These approaches are compared to a more conventional method using an example adapted from practice.

A Conventional Approach: Minimize Variability and Adjust Mean to ~ a r g e t

1.n experimental situations for which both the response mean py and standard deviation u, are of concern, a conven- tional approach for selecting factor settings often is based on the following interrelated steps (e.g., see Refs. 1-3):

I . Design and conduct screening and response surface experiment(s).

2. Analyze results, identify significant mean and vari- ance effects, and develop appropriate prediction equations for fi,, and CY.

3. Minimize the predicted response standard deviation eY via appropriate settings for significant variance

. effects. 4. Set or adjust the predicted response mean f iy to the

target value T,,, such as by using one or more "ad- justment" factors or via optimization methods.

5. Set any insignificant factors at their lowest cost settings.

In the simplest case, assuming the first two activities have been conducted following either conventional screen- ing andlor response surface experimental methods, factor settings then can be chosen heuristically in the remaining steps first by identifying those factors and interactions that significantly impact the response variance and setting these in such a way so as to achieve the absolute minimum pos- sible estimated variance (within the experimental region). The response mean estimate often then is centered on target by appropriately setting some other "adjustment" factor (ideally one with a statistically significant result with respect to the mean but not with respect to variability). If such an adjustment factor (with no variance effect) is not available, then that with the largest estimated mean effect might be used for this purpose (alternatives are to use that with the

smallest estimated variance effect or with the largest ratio of mean-to-variance estimated effects).

In either case, if after setting the first factor all the way to one or the other end points of its experimental range (i.e., the coded + I or - 1 settings) still more adjustment is needed in order to move the response average to target, then a second factor is used for further adjustment (e.g., that with either the second largest estimated mean effect, second smallest estimated variance effect, or second largest ratio), and so on until the predicted mean response is adjusted such that f iy = T,. By following this general approach, the predicted mean iteratively can be set to its target value with the pre- dicted variability usually minimized as much as possible (or close to it). Although not the focus here, other ap- proaches include treating the mean and variance as multiple responses, using optimization or search algorithms to mini- mize the variance subject to the equality constraint that fiy = 5, or maximizing a desirability function; for example, see Refs. 4 and 5. Also, note that confirmation experiments are recommended at and about the selecled settings in order to verify their optimality.

An Example: Laver Sfock Removal

As an example, consider a laser stock removal process in which the response of interest Y is a depth of cut (mea- sured in millimeters), with approximately 160,000 units pro- duced per month and the following three factors available for adjustment:

Beam speed ( x , ) Frequency ( x 2 ) Power level (x , )

In order to determine the best settings for speed, frequency, and power that meet a customer depth-of-cut requirement of Y = 800 zh 40 p m most consistently and at minimum cost, each parameter can be set anywhere between the following minimum and maximum values:

Beam speed must be between 4 in./min (-1) and 16 in./min (+ 1). Frequency must be between 2000 Hz (-1) and 3000 Hz (+ 1) Power must be between 5 W (- 1) and 10 W (+ I)

with - 1 and -t 1 indicating the coded settings of each pro- cess parameter. These experimental conditions are summa- rized in Table 1, along with estimated relative monthly costs of operating at each factor setting, based on historical ex- perience and assumed here to be linear. Note that conduct- ing sensitivity analysis is especially important if such costs

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MATHEMATICAL PROGRAMMING IN INDUSTRIAL EXPERIMENTS

Table 1. Relative Operating Costs at Each Factor Setting

CODED FACTOR SPEED (x,) FREQUENCY (x2) POWER (x,)

SETTTNG SETTING REL. COST SETTING REL. COST SETTING REL. COST

Low (-1) 4 in./min $200 2000 Hz $200 5W $100 High ( + I ) 16 in./min $400 3000 Hz $800 10 W $200

cannot be estimated very precisely. For illustration purposes, Thus, the final recommended factor settings using this ap- assume that a series of experiments resulted in the regression proach are coefficie~~ts (or half-effects) for mean and standard deviation

Speed ( x , ) = 5 .= 0.92 (15.48 in./rnin) summarized in Table 2, with no evidence of nonlinearity, and

Frequency (x,) = 1 .OO (3000 Hz) the following prediction equations for the response mean fi,

Power (x,) = - 1.00 (5 W), and standard deviation &":

and

Using these results, the heuristic approach outlined earlier first produces x, = - 1, x, = + 1, and x,x, = -1 and thus x, = - 1 in order to minimize Sy. Because these settings re- sult in an estimated mean of

by = 750 + 40(- 1 ) +. 25(+1) + lo(- 1) - 2(-I)(+ 1)

= 725 < T, = 800.

with x , = - 1 making the largest negative contribution to the mean, then setting

and solving for x , yields

Table 2. Laser Stock Removal Mean and Variance Half-Effects

ESTIMATED COEFFICIENTS

FACTOR OR RESPONSE STANDARD INTERACTION MEAN DEVIATION

which in this case also agree with results obtained via opti- mization as shown later (i.e., minimizing variance subject to by = Ty).

Some Comments

As stated earlier, note that this approach results in factor settings that yield mean and standard deviation estimates of

= 800 mm,

which is the desired target value, and

which is only moderately greater than the minimum possible predicted standard deviation of

min &,, = 10.5 + (-I) - 2(1) + 0.5(1)(-1) = 7 mm.

Thus, the minimum possible estimated variance has been achieved subject to the requirement that the estimated pro- cess mean be centered equal to its target value. Note that in this example, these same factor settings and prediction equations would be produced by the largest ratio of effects (LRE) approach, whereas the smallest variance effect (SVE) approach here produces the different results of x , = = 0.3947, x2 = 1,x, = 1, fi, = 800 mm, and +y = 9.35 mm, which is larger than the minimum possible standard devia- tion of 8.92 mm found earlier given ji, = 800 rnrn. More- over, in any case, the nonconforming rate and costs asso- ciated with these results are typically only considered in passing and without much formal quantitative exploration of alternatives. It also is important to comment that none of these approaches necessarily result in the minimum expected

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KRAUS, BENNEYAN, AND MACKERTICH

number of nonconforming parts per million (NCPPM) or the overall minimum cost, unless all factors used for adjustment- to-target do not at all affect the response variance and all significant factors have negligible cost differences across their experimental ranges; both of these conditions often not being the case i n practice implicitly results in a quality-cost compromise.

For example, in the above case and assuming normality of the response Y, the estimated mean NCPPM correspond- ing to the identified factor settings under the largest-mean- effect (LME) and smallest-variance-effect (SVE) approaches are approximately 7.34 and 20.47, respectively (with process capability index estimates of approximately e,,, = 1.49 and 1.42, respcctively), which we will see can be slightly im- proved. Also, note that using the costs shown i n Table 1 and assuming linear costs and linear factor effects, the total factor

. operating cost corresponding to these factor settings can be seen to be

FC = Total production factor costs

= [(Cost at factori midpoint) Puctor i

+ (Cost half-range for Factor i)

X (Coded setting for factor i)]

$1292.1 1 per month,

where CT and C; represent the relative costs of each factor i at its high and low setting, respectively. As an aside, note that a little algebra simplifies this cost expression to

= {Sum of maximum costs + Sum of minimum costs 2

+ 2 [(Full cosl range,) (Setling,)] Pucior i

as before, which is significantly greater than the minimum possible total factor cost of

FC = C ; = Sum of minimum costs i

= $200 -t $200 -t $100 = $500 per month.

Therefore, this may not be necessarily the best approach in order to achieve the best compromise between the dual ob- jectives of the highest possible quality at the lowest possible cost (in this case achieving neither). As an alternate ap- proach to address these considerations more explicitly, after response mean and variance equations have been developed, mathematical programming models then can be used to help identify optimal factor settings and to explore trade-offs between operating costs and nonconformance rates. Several such possible uses are illustrated in the following sections, dependent on the primary experimental objective and any external cost or quality constraints. (Also, see Refs. 6-8 for examples of other uses of mathematical optimization in ex- perimental design scenarios.)

Model 1: Minimize Nonconformance Rate

If the primary objective is to minimize the nonconform- ing rate, possibly subject to some specified upper bound on acceptable production costs (i.e., the total operating costs of all factor settings), then the following general nonlinear programming (NLP) formulation can be used:

Minimize: Estimated nonconformance rate subject to:

1. Response mean and standard deviation prediction equations

2. Response upper and lower specification limits 3. 'Possible upper bound on total factor setting costs 4. Factor costs at upper and lower settings 5. All factor settings between their coded - 1 and + 1

settings

This model will identify the combination of factor settings that together minimize the predicted nonconforming rate subject to the above five sets of constraints. For example, using the following notation and assuming the response Y is normally distributed,

CT = the relative cost of factor i at its upper setting C; = the relative cost of factor i at its lower setting USL, = upper specification limit for the response Y

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MATHEMATICAL PROGRAMMING IN INDUSTRIAL EXPERIMENTS 399

LSL, = lower specification limit for the response Y @[z] = P(Z < z), where Z - N(0, 1) N^R~ = estimated nonconforming rate for the response Y

USL, - f i LSL , - p =I-@( , Y ) + @ ( t y)

FC = the total factor cost of all factors at their given settings

then in the laser removal example, if the total operating cost cannot be greater than, for example, $1 100, the above model becomes

USL, - ji LSL, - p Minimize: I - @

subject to:

FC 5 $1100

LSL, = 760, USL, = 840 C; = 200, C: = 400 C; = 200. C; = 800 C, = 100, Cf = 200

xi 2 -1 for i = 1, 2, 3

x; 5 + 1 for i = 1 . 2, 3.

Note that although several of the above constraints could be combined to make the formulation slightly more efficient computationally (e.g., all inequality and equality cost con- straints could be combined into the simpler requirement that $950 + $100x, + $ 3 0 0 ~ ~ + $SOX, 5 $1 loo), it can be left in the above form with negligible loss in efficiency for easier readability, comprehension, revision, and sensitivity analy- sis. Also note that prediction equations for the mean ,k, and standard deviation eY are included as equality constraints (rather than integrated into the objective function) as a matter of preference for the same reason, although this must not necessarily be the case..

The above optimization problem can be solved using any standard NLP software (e.g., LINDO, GAMS, CPLEX, and others), as well as optimization modules included in many spreadsheets (such as Microsoft Excel's, Lotus 1-2-3's, or Quattro Pro's Solvers). These programs use any of several optimization algorithms (9-11) to identify the optimal pa- rameter settings that will result in the minimum value for

the objective function (here, the nonconformance rate) while still satisfying all stated constraints. In the present example, solution of the above model results in a minimum objective function value (nonconformance rate) of 0.000498 (or an estimated 498 mean NCPPM) at the following optimal set- tings for the decision variables (factors):

Speed (x:) = 0.9834 (15.90 in./min) Frequency (xt) - 0.0055 (2502.77 Hz) Power ( x f ) = 1.00 (10 W)

These optimal factor settings result in an estimated response mean and standard deviation of

fi; = 750 + 40t0.9834) + '25(0.0055) + lO(1) - 2(0.9834)(0.0055) = 799.46 mm

and

8: .=: 10.5 + 0.9834 - 2(0.0055) + 0.5(0.0055)(1) = 1 1.475 mm,

which equates to a process capability estimate of e;k = 1.15 and a total operating factor cost equal to its upper bound

.FC* = $950 + $100x, + $300~~ + $SOX, - $950 + $100(0.9834) + $300(0.0055) + $SO(]) = $1 100 per month.

Note that, in practice, the total factor cost FC almost always will increase to its permitted upper bound in order to reduce NCPPM as close as possible to its absolute minimum (unless the total factor cost of all settings that together yield the ab- solute minimum possible NCPPM is less-than this constraint value). Alternatively, if an upper limit on the total factor cost does not exist or is not well defined, then this constraint can be set arbitrarily high, removed altogether. or iteratively adjusted via sensitivity analysis in order to explore a range of cost-quality trade-offs, as illustrated later in the Discus- sion section. In the present scenario, for exirmple, setting tbe right-hand side of the factor cost constraint equal to or greater than the sum of the maximum individual factor costs,

essentially results in an unconstrained minimum possible estimated mean NCPPM of 7.29 at a total factor cost of $1291.49. Also note that in the current example, this op- timization formulation happens to result .in the predicted response mean being nearly equal to the target value half- way between the lower and upper specification limits (here, ,k; = 799.46 versus T, = 800), although this must not al- ways be the case if one or more factors that impact the re-

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400 - KRAUS, BENNEYAN, AND MACKERTlCH

sponse mean have considerably higher costs relative to those impacting the response variance.

As an implementation aside, note that because the predic- tion equations and objective function are nonlinear, some caution should be taken to avoid local suboptimization (such as via the historical practice of verifying results from several different starting condilions), especially if using spreadsheet solvers. Additionally, most commercial spreadsheets have built-in normal cumulative density function statements that can be used in the above objective function, whereas if tra- ditional optilnization software is used, then a closed-form CDF approximation can be substituted in the objective func- tion or added as an equality constraint.

Model 2: Minimize Factor Costs

As a converse approach to the same problem, if the pri- mary objective instead is to minimize the total factor costs, now possibly while holding the nonconforming rate or mean NCPPM below some specified level (e.g., 0.1 % defective or 1000 NCPPM), then the following general NLP formulation can be used:

Minimize: Total factor costs subject to:

I. Response mean and standard deviation prediction equations

2. Response upper and lower specification limits 3. Possible upper bound on nonconformance rate 4. Factor costs at upper and lower settings 5. All coded factor settings between - 1 and -I- 1

.Using the same notation and assumptions as in Model 1, for the laser removal example this problem then becomes

(c:; c; + C; - crX i ) Minimize:

i - I 2

subject to :

fi), = 750 + 40x, + 25x2 + l o x , - 2x,x2

~3~ = 10.5 + x, - 2x, + 0 . 5 ~ ~ ~ ~

LSL, = 760, USL, = 840

USL, - ' LSL, - p,

Is, s 0.001

C ; = 200, c; = 400

c; = 200, c: = 800

c; = loo, c,' = 200

xi 2 - I for i = 1 , 2, 3 . . xi 5 + I fori = 1 , 2, 3

Solution of this mathematical optimization problem results in the following optimal factor settings (decision variables):

Speed (x : ) = I .OO (16 in./min) Frequency (x:) = -0.1473 (2426.37 Hz) Power (x;) = l .OO (10 W),

which result in the minimum possible total factor setting cost of

FC $950 + $100(1) + $300(-0.1473) + $50(1)

= $1055.82 month.

These settings, however, now result in an estimated response mean and standard deviation of

and

B,* 10.5 + ( 1 ) - 2(-0.1473) + 0.5(-0.1473)(1)

= 11.72 mm,

which increases the estimated nonconformance rate to 0.00 1 - (or 1000 NCPPM) and decreases C;k to approximately I .M. Note that the implied resultant est~mated nonconformance rate has been forced equal to its constraint upper bound, N^R, = 0.001, in order to reduce the total factor costs as much as possible. (Generally, this will be the case unless the factor settings that minimize total operating costs result in a nonconformance rate less than that specified in the con- straint.) Again by relaxing or tightening the FC or NCPPM constraint in either model, the trade-offs between these two performance criteria can be explored, as discussed and illus- trated further in the Discussion section.

Model 3: Minimize Expected Total Costs (Factor Setting Costs and

Expected Nonconformance Costs)

As an alternative to either of the above models, if the cost per nonconforming item is known or can be estimated rea- sonably, then the total cost due to both the factor settings and the nonconformance rate can be combined into a more global optimization model with the single objective being to mini- mize total costs as follows:

Minimize: Estimated total costs subject to:

1. Response mean and standard deviation prediction .equations

2. Response upper and lower specification limits 3. Defect cost and factor cost equations

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MATHEMATICAL PROGRAMMING IN INDUSTRIAL EXPERIMENTS 40 1

4. Factor costs at upper and lower settings 5. All coded factor settings between - 1 and + I

By adding the predicted expected nonconformance cost to the factor setting costs in the objective function of Model 2, in our example this becomes

Minimize:

5 (c: + c; + c: - c;~;) ; - I 2 2

mbject to:

$, = 750 + 40x, + 25x, + lox, - 2x,x,

eY = 10.5 + x, - 2x, + 0 . 5 ~ ~ ~ ~

LSL,, = 760, USL, = 840

c, = $1

V = 160,000

C; = 200, C; = 400

C; = 200, C: = 800

c; = 100, c; = 200

x , h -1 fori = 1,2 , 3

xis + 1 f o r i = 1.2, 3

where C, and V denote the relative cost per defective item (i.e., the average cost of not being within specifications) and the volume of items produced per time period, respectively. Again, note that this model can be thought of as a more global optimization model, as it captures both the costs of production (factor settings) and the associated costs of poor quality. Solution of this global model results in the following optimal settings for the decision variables:

Speed (x:) = 1 .OO (16 in./min) Frequency (x;) = -0.03496 (2482.52 Hz) Power ( x ; ) = 1.00 (10 W)

and, thus, a total factor cost of

FC* = $950 + $100(1) + $300(-0.03496) + $50(1) - $1089.51 per month.

These factor settings also produce an estimated response mean and standard deviation of

fij = 750 + 40(1) + 25(-0.03496)

+ lO(1) - 2(1)(-0.03496) = 799.20 mm

and

B,* = 10.5 + ( I ) - 2(-0.03496) + 0.5(-0.03496)(1)

= 1 1.552 mm,

which result in an estimated nonconfor~ning rate of 0.00055207 (or 552.07 NCPPM), an estimated process ca- pability of e;k = 1.13, and a minimum predicted expected total cost of

T^C = Total factor cost + Predicted expected nonconformance cost

+ P(Y C LSL, or Y > USL,)CDV

USLY - p,. LSL, - $, + I - @ [ ( &, ) + a ( q . )]c,v

x ($1)(160.000)

5 $1 177.84 per monlh.

Of course, although not included in the earlier two optimi- zation models, the associated total expected costs in each of these cases similarly can be computed for comparison pur- poses, as illustrated in the next section. The effect of the rela- tive cost per defect on the expected cost also is explored in the following section.

Discussion

Table 3 summarizes the results of each of the above ap- proaches, also including the total estimated cost for each approach in order to facilitate comparison with the global model. As shown, each approach can produce important dif- ferences in factor settings, nonconforming rates, and costs. For example, the global minimum total cost model (Model 3) results in an overall total cost that is approximately 8.9% and 12.3% smaller than the conventional LME and SVE ap- proaches, respectively. It also is important to note that the results of the first two (constrained) models are highly de- pendent on the value placed on the factor cost or noncon- forming rate constraint (as can be seen by considering their unconstrained results shown in parentheses in Table 3).

Sensitivity Analysis and Impact of Cost per Defect

The impact of the cost per defect relative to the factor costs can be explored by executing the total cost model (Model 3) for a variety of C, values, which also can be

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Table 3. Summary of Results of Each Approach

OE'TIMAL FACTOR SETTINGS ESTIMATED RESULTS OF FACTOR SETTINGS

METHOD FOR I 1 X 2 1 3 FACTOR DEFECT ESTIMATED SETTING FACTORS (SPEED) (FREQ.) (POWER) P, cV N ~ M e p k COSTS COST TOTAL COST

Conventional SVE 0.3947 1 1 800 9.39 20.67 1.42 $1,339.47 $3.31 $1,342.78 Conventional LME 0.92 1 1 1 - 1 800 8.92 7.34 1.49 $1.292.1 1 $1.17 $1,29;.28 Conventional LRE 0.92 1 1 1 - 1 800 8.92 7.34 1.49 $1,292.11 $1.17 $1.293.28

Min. NCPPM 0.9834 0.0055 1 799.46 11.48 497.74 1.15 $1,100.00 $79.64 $1,179.64 (unconstrained) (0.9148) (1) (- 1) (799.76) (8.91) (7.29) (1.49) ($1,29 1.49) ($1.17) ($1,292.65)

Min. facior cost 1 -0.1473 1 796.61 11.72 1,000 1.04 $1,055.82 $160 $1,215.82 (unconstrained) (- 1) (- 1) (- 1) (673) (12) ( 1,000,000) (-2.42) ($500) ($160,000) ($1 60,500)

Min. total costs I 0.0349 1 799.20 11.55 552.07 1.13 $1,089.51 $88.33 61,177.84

Nore: SVE: Smallest variance effect; LME: largesl mean effect; LRE: largest ratio of effects.

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MATHEMATICAL PROGRAMMING IN INDUSTRIAL EXPEIUMENTS

Table 4. Impact of Cost per Defect on Optimal Policy

RELATIVE OPTIMAL FACTOR SETTINGSa DECISION CRITERIA

COST PER X? : I? ESTIMATED TOTAL FACTOR ESTM ATED DEFECT (C,) (SPEED) (FREQ.) (POWER) NCPPM SETTING COST TOTAL COST

0 - 1 - 1 - 1 1,000.000 $500.00 $500.00 0.025 1 -0.5573 1 13,786.95 $932.82 $978.97 0.05 I -0.4155 1 6,038.15 $975.35 $1,023.66 0. I25 1 -0.2605 1 2,172.61 $1,021.85 $1,065.30 0.25 1 -0.1641 1 1,117.91 $1,050.78 S 1,095.49 0.5 I -0.0878 1 698.6 1 $1,073.65 $1,129.54 I I -0.0350 1 522.07 $1,089.51 $1,177.84 2 1 0.4709 -0.1863 152.88 $1.181.95 $1,230.87 4 1 0.7119 -0.7663 52.00 $1,225.25 $1,258.53 8 1 0.8263 - 1 24.76 $1 -247.89 $1,279.59

16 0.9693 0.9657 - 1 14.22 $1.268.15 $1,304.54 25 0.9163 0.9838 - 1 8.30 $1.286.78 $1,319.98 50 0.9103 1 - 1 7.32 $1,291.03 $1,349.55

100 0.9 1 26 I - 1 7.29 $1,291.26 $1,407.96

"Minimum total cost.

important for sensitivity analysis if this cost is not known with reasonable accuracy. As shown in Table 4, for example, the optimal policy in the laser removal scenario is fairly depen- dent on this value, with increases in CD resulting in settings that yield fewer defects but at the (lesser) expense of higher factor costs, as is intuitive. Also, note the pivoting of factors from their high or low settings to intermediate values. For ex- ample, increasing from CD < 4 to C, 2 16 in this case ex- changes the optimal settings from x, = 1 and - 1 < x, < 1 now to - 1 < x, < 1 and x, = - 1. It also is interesting that beyond this point, increases above CD 2 16 in this ex- ample only impact the optimal solution marginally, again useful knowledge if costs can only be estimated roughly to be within some general range. Similar sensitivity analysis obviously also could be conducted for any uncertain factor operating costs.

Optimal Trade-ofiFrontier

If defect or factor costs cannot be reasonably esti- mated even to be within general ranges, however, then eilher Model 1 or Model 2 still can be used effectively to explore trade-offs between factor setting costs and nonconforming rates (with both models producing identical information in opposite manners). For example, by iteratively solving Model 1 for a range of values on the maximum factor cost constraint (or Model 2 for a range of values for the maxi-

mum nonconforming rate constraint), the optimal possible relationship between factor costs and estimated NCPPM can be determined (i.e., the minimum possible factor cost at each NCPPM level or the minimum possible NCPPM at each factor cost level). Figure 1 illustrates this optimal painvise frontier for the present example, which can help a process manager to understand the minimum possible pairwise val- ues for each objective more clearly and then rnake informed decisions to move up or down this trade-off curve. To im- prove resolution, Figure 2 illustrates the same information but for the range from 0 to 5000 estimated NCPPM.

Figures 3 and 4 illustrate similar information but with the vertical axis based on the estimated rota1 cost T̂ C (rather than just FC) plotted versus estimated NCPPM on the hori- zontal axis. Figure 3 (plotted on a log scale) simply adds the cost of nonconforming items to the results in Figure 1 for four different cost per defect assumptions. Note that this figure represents the true (and in some cases very consider- able) impact of any given decision based on factor costs alone, rather than minimizing the total cost at each NCPPM level. Figure 4 then illustrates the minimum estimated total cost 'f? to achieve a specified NCPPM level (or minimum NCPPM to achieve a specified total cost), obtained by add- ing an artificial equality constraint on NCPPM to Model 3 (similar to that in Model 2). Each of the four curves in Figures 3 and 4 correspond to different assumptions for the relative per defect cost, with the case for which C, = 0 cor- responding to only the factor costs curve in Figure 1. As can

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KRAUS, BENNEYAN, AND MACKERTICH

Maximum FC (@ 7.3 NCPPM, Factors @ (.915.1,-1))

51.250

Minimum Estimated Non-Conforming park per Million

tn

Figure I . Optimal painvise frontier for minimum factor setting costs versus minimum estimated NCPPM (0 - 100,000).

5750 -- 0 m U.

E $500 .-

e .- Z

Maximum FC (@ 7.3 NCPPM. Factors @ (.Q15.1.-1)) - - - - _ _ - _ _ - - --------- _-.

51.250 - C

8 U

$1,000 -- .-

Minimum FC (@ 1,000,000 NCPPM, Factors @ (-1,-1,-I))

Minimum FC (@ 1,000,000 NCPPM. Factors @ (-1,-I,-!))

.- c

0 500 1,000 1,500 2,000 2,500 3,000 3.500 4.000 4,500 5,000

Minimum Estimated Nan-Conforming Parts per Million

Figure 2. Optimal painvise frontier for minimum factor setting costs versus minimum estimated NCPPM (only for 0-5060 NCPPM).

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MATHEMATICAL PROGRAMMING IN INDUSTRIAL EXPERIMENTS

Cost per Defect (CD) = $4 c-1 Cost per Defect (CD) = $1

...... Cost per Defect (CD) = S.5

Cost per Defect (CD) = $.I

- CD = 0 (Factor costs only)

Minimum Estimated Non-Conforming Parts per Million

Figure 3. Effect of estimated total cost (based on optimal NCPPM versus factor cost alone) (note log scale).

Cost per Defecl (CD) = 01 ...... Cost per Defect (CD) = 5.5

Cost per Defect (CD) = 0.1

Minimum Estimated Non-Conforming Parts per Million

Figure 4. Optimal relationship between estimated NCPPM and total costs.

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KRAUS, BENNEYAN, AND MACKERTICH

be seen in Figure 3, if the process is optimized based only on cstimated factor costs I? (i.e., Model 1 or 2), then increases in C, significantly increase the true estimated total cost YC. By optimizing on the total cost, however, the impact of in- creases or estimation errors in CD is reduced quite consider- ably, as shown in Figure 4. Thus, there clearly is significant value in using Model 3 even in cases for which the cost per defect cannot be estimated exactly, which is an interesting result.

More generally, integration of design of experiments and mathematical programming methods can complement tradi- tional postexperiment approaches for setting factors, par- ticularly when factor and nonconformance costs are of con- ccrn, as illustrated earlier, with spreadsheet implementation allowing sensitivity analysis to be conducted easily for dif- ferent cost, factor settings, response distribution, and other assumptions. These approaches also can easily be adapted to allow for several extensions, includihg nonlinear factor costs, smaller-the-better or larger-the-better type problems, one-sided specifications or asymmetric targets, multiple re- sponses, and the use of loss functions in place of specifica- tion-based nonconforming costs. Related multicriteria and goal programming optimization methods also could be used as alternatives to nonlinear programming models.

References

1 . Schmidt, S . R. and Launsby, R. G., Understanding /ndusfrial Designed Experiments, 4th ed., Air Academy Press, Colorado Springs, CO, 1995.

2. Ross. P. J . . Toguchi Techniques for Quality Engineering, McGraw-Hill, New York, 1988.

3. Phadke, M. S., Qualiry Engineering Using Robust Design, Prentice-Hall, Englewood Cliffs, NJ, 1989.

4. Vining, G. G.. Storistical Methodsfor Engineers, Duxbury Press, Pacific Grove, CA, 1998.

5 . Harrington, E. C., The Desirability Function, Ind. Qual. Con- trol, 2 1 ( 1 O), 494 -498 ( 1965).

6. Hartmann, N. E. and Beaumont, R. A., Optimum Cornpound- ing by Computer, J. Inst. Rubber I d . , 2(6), 272-275 (1968).

7. Nicholson, T. A. J. and Pullen, R. D., Statistical and Opti- misation Techniques of Rubber Compounds, Compurer Aided Design, 1 (I), 39-47 (I 969).

8. Heller, N. B, and Staats, G. E., Response Surface Optimiza- tion When Experimental Factors Are Subject to Costs and Constraints, Technometrics, 15(1), 1 13-123 (1 973).

9. Nash, S. G. and Sofer, A., Linear and Nonlinear Program- ming, McGraw-Hill. New York, 1996.

10. Bazarra, M. S., Sherali, H. D., and Shetty, C . M., Nonlinear Programming: Theory and Algorithms, 2nd ed., John Wiley & Sons, New York. 1993.

I 1 . Winston, W. L., Operations Research: Applications and Al- gorithms, 3rd ed., Duxbury Press, Boston, 1997.

About the Authors: peter Kraus is an instructor at the Uni- versity of Massachusetts, where he teaches courses in statis- tics and quality control. He also works as an independent consultant specializing in experimental design and is a mem- ber of ASQ.

James Benneyan is a professor in the College of Engi- neering at Northeastern University in Boston, MA, where he teaches and researches in statistical process control, quality engineering, and operations research. He is a member of ASQ, INFORMS, and 11E and has published in the J o u r n ~ l of Quality Technology, Annals of Operations Research, Pro- duction and Inventory Management Journal, Quality Prog- ress, and others.

Neal Mackertich is a principal engineer responsible for the application of statistical methods within the Sensors and Electronics Surface Radar business unit of the Raytheon Systems Company. He holds a Ph.D. in industrial engineer- ing and operations research' from the University of Massachusetts-Amherst and is a Senior Member of ASQ.

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