Simultaneous Optimization of Several Response Variables

’ Simultaneous Optimization ofSeveral Response Variables GEORGE DERRINGER Batelle Columbus Laboratories, 505 King Avenue, Columbus, Ohio 43201 RONALD SUICH California State University, Fullerton, California 92634 A problem facing the product development community is the selection of a set of conditions which will result in a product with a desirable combination fjf properties. This essentially is a problem involving the simultaneous optimization of several response variables (the desirable combination of properties) which depend upon a number of independent variables or sets of conditions. Harrington, among others, has addressed this problem and has presented a desirability function approach. This paper will modify his approach and illustrate how several response variables can be transformed into a desirability function, which can be optimized by univariate techniques. Its usage will be illustrated in the development of a rubber compound for tire treads. Introduction programming model. However, a major disadvantage of these schemes is the philosophy upon which they are based. These methods involve optimization of one response variable subject to constraints on the remaining response variables. Often, however, the goal is the attainment of the best balance among several different response variables. In developing a compound for radiator hose, for example, it is more realistic to give water absorption, heat resistance, and tensile strength equal weights in the optimization than to optimize tensile strength while keeping the other properties within specified limits. A volves the selection of a set of conditions, the common problem in product development in- X’s, which will result in a produce with a desirable combination of properties, the Y ‘s. Essentially, this becomes a problem in the simultaneous optimization of the Y’s, or response variables, each of which depends upon a .set of independent variables, ‘X1, XZ , . . , , X,. As an example from the rubber industry, consider the problem of a tire tread compound. Here we have a number of response variables, such as PICO Abrasion Index, 200 percent modulus, elongation at break, and hardness. Each of these -response variables depends upon the ingredient variables, the X’s, such as hydrated silica level, silane coupling level, and sulfur level. We wish to select the levels for the X’s which will maximize the Y ‘s. Unfortunately, levels of the X’s which maximize Y1 might not even come close to maximizing YZ. Harrington [2] presented an optimization scheme utilizing what he termed the desirability function. Gatza and McMillan [l] gave a slight modification of Harrington’s function. We will employ a different form of this function and illustrate its use in the example of the development of a rubber compound for tire treads. In maximizing this function‘we will use a pattern search method similar to that presented by Hooke and Jeeves [4]. In addition, we will also plot this desirability function against two independent variables with the third held at its optimum level. One approach to this problem has been through the use of linear programming. Hartmann and Beaumont [3] and Nicholson and Pullen [5] described optimization schemes based upon the linear Dr. Derringer is Principal Research Scientist at BatelIe Memorial Institute. Dr. Suich is an associate profesor in the Department of Management Science. Development Suppose each of the K response variables is related to the p independent variables by Yij = fi(X1, XZ, e e n 3 Xp) + Eij KEY WORDS: Desirability, Multivariate, Optimization, Regression Journal of Quality Technology 214 i = 1,2, . . . , k j = 1, 2, . . . , ni Vol. 12, No. 4, October IS80 . Again using the radiator hose example. Pi must be greatly increased over Yi*. It is for these reasons that the geometric mean. D also has the property that if any dj = 0 (that is. which indicates an unacceptable product. We note that this function may differ for each Yj and that f.. . there is no highest value of Pi. Journal of Quality Technology . We then estimate qj by Pi. fi typically is unknown. xd#‘k (1) This single value of D gives the overall assessment of the desirability of the combined response levels. since we are considering a one-sided transformation here. 2. The individual de&abilities are then combined using the geometric mean. Yi* might be the tensile strength such that higher values of tensile strength would add little to the quality of the hose. For example. October 1980 iI i = 1. Therefore. X. .‘:z”. if Yi is the tensile strength of a radiator hose. Therefore.‘value of r = 0. 72. . The value Yj* gives the highest value of Pi. management might find values considerably above 1500 psi highly desirable and so choose a large value of r. The value Yi* gives the minimum acceptable value of Pi. then we can relate the average or expected responses vi to the p independent variables by ?)i=fi(X1. Clearly the range of D will fall in the interval [0. Graph of Transformation (2) for Various Values of r would make di = 0.. A . If we make the usual assumption that E(ejj) = 0 for each i. for example. say r = 10. On the other hand. dj would remain at 1.. rather than some other function of the di’s such as the arithmetic mean. As can be seen. WLIS just about as desirable as any other value of Pi above Yj*. a value of Yj below Yi* = 1500 psi would result in a product that could be unacceptable in the judgment of the management regardless of how desirable the other response variables might be. where 0 5 di 5 1. X2. The desirability function involves transformation of each estimated response variable Yj to a desirability value dj.Xp) D=(dlxdzx . from a practical viewpoint.. I0 di 215 “i* .. Yj* < Pi < Yi* C-3 1 1 Pi 2 Yi* and graphed in Figure 1.1. Figure 1 indicates a large value of r would be specified if it were very desirable for the value of Pi to increase rapidly above Y. .. since Pi 4 Yi* Vol. and thus D = 0. One-Sided Transformations In transforming Yj to di two cases arise: one-sided and two-sided desirability transformations.) dj= Yj 5 Yj* I-I r . . The user specifies this value of ‘Yi*. one can think of Yj* as the value for Pi such that higher values of Pi have little additional merit. even though any tensile strength above Yi* = 1500 psi would be acceptable. No. The value of r used in the transformation would again be specified by the user. In other words. For example. ) K. the desirability dj then increases slowly as Yi increases. The usual procedure is to approximate f. 4. . was used. knowing that any lower value of Pi would result in an overall unacceptable product.:. However. the overall product is unacceptable). . (Minimization of Pi is equivalent to maximization of Many transformations are possible-we shall consider the transformations given by -Pj. dj increases as Yj increases and is employed when Pi is to be maximized. l] and D will increase as the balance of the properties becomes more favorable. a small value of r would be specified if having values of Pi considerably above Yj* were not of critical importance.*. “i “i FIGURE 1. Actually. even though Yi+ is an acceptable value the desirability of the product would be greatly increased by having Pi considerably greater than Yj*.SIMULTANEOUS OPTlMlZATlON OF SEVERAL RESPONSE VARIABLES where fi denotes the functional relationship between Yi and Xl.X*. often (but not necessarily) by a polynomial function. the estimator obtained through regression techniques. . The value of di increases as the “desirability” of the corresponding response increases. would mean that any value of Yj above Yi. if one of the response variables is unacceptable) then D = 0 (that is. to maximize dj and thereby Dj. represents this relationship except for an error term l jj. In practice. For the one-sided case. the desirability function condenses a multivariate optimization problem into a univariate one. then small values of s and t would be chosen. October 1980 . < Yi* or Pi > Yi*. The value selected for Ci would be that value of Pi which was most desirable and could be selected anywhere between Yi* and Yi’. YZ Elongation at Break. One could also select a large value of s and a small value of t if it were desirable for Pi to increase rapidly to ci while almost any value of Pi above ci but below Yi* WCS also desirable. and sulfur level X3-was sought. The transformations presented in this paper may be viewed as a type of generalization of those above. of course.+ As a result. The values of s and t in the two-sided transformations play much the same role as r does in the one-sided transformation. above Yi. We shall consider the transformations given by I[ 1 Pi- di = [I Yi* Ci . the use of ci in (3) allows the user to set the most deisrable value of Yi anywhere between the lower and upper boundaries (Yi.Yi* t Ci .1 Yi ] “) for the two-sided transformation. The original transformation proposed by Harrington [2] is of the form di = eip(-exp(-Yi)) for the one-sided transformation and di = exp(. Graph of Transformation (3) for Various Values of 8 and f JOUfn8l of Ouality Technology PICO Abrasion Index. In this case the desirability di would not get large until Pi got close to ci.25 of the distance between Yi* and Yi*. it follows that D is a continuous function of the X. existing univariate search techniques can be used to maximize D over the independent variable domain. would be similar to a linear programming approach. Yl 200% Modulus. As an example.exp(-1)]. s. if almost any value of Y. Y3 Hardness. This figure also shows that large values for s and t would be selected if it were very desirable for the value of Pi to be close to ci. and t) and permits Yi* and Yi* to act as the boundary values. 4. Moderate values for s and t (near 1) would represent a compromise between the two extremes. For those Yls that are subject to constraints one uses extremely small values of the exponents (r. In this situation Yi* is the minimum acceptable value of Pi and Yi* is the maximum acceptable value. Yi* 7 t Yi i FIGURE 2.Yi* ’ Pi . silane coupling agent level X2. From (2) and (3) we see that the di’s are a continuous function of Yi’s and from (1) that D is ’ a continuous function of the d.‘s. Gatza and McMillan [l] used di = {exp[-exp(-Yi)]-exp(-l))/ [l . it may be noted that ci was chosen to lie at 0. For illustration. Values of Pi outside these limits would make the entire product unacceptable. This. Example In the development of a tire tread compound. and below Yi* were acceptable. di The procedure outlined can be used to maximize some of the di’s (corresponding to certain Pi’s) while in essence putting constraints on the other Yis. Therefore. and t) in (2) and (3) and may be viewed as special cases. In Figure 2 several different values of t and s are plotted. the optimal combination of three ingredient (independent) variables-hydrated silica level X1. In essence.GEORGE DERRINGER AND RONALD SUICH 216 Two-Sided Transformations The two-sided transformation arises when the response variable Yi has both a minimum and a maximum constraint. An added benefit of the method is the ability to plot D as a function of one or more independent variables. The properties to be optimized and constraint levels were as follows. a modification of Harrington’s which produces negative values of di for unacceptable properties. s. . Method of Optimization We have assumed that Yi is a continuous function of the Xh. We no longer restrict ourselves to particular members of the exponential family but consider transformations that offer the user greater flexibility in the setting of de&abilities.Yi* 0 I Yi* I Ci Pi 5 Ci < Pi I Yi* (3) P. 12. On the other hand. and Yi*) rather than exactly in the middle. Harrington’s and Gatza and McMillan’s transformations may be closely approximated by selection of the parameters (r. No. Y4 120 < Yl looo<Yz 4OO<Y~<600 60 < Y4 < 75 Vol. value less than 120 resulted in an unacceptable tire tread compound. As can be seen. we considered any PICO Abrasion Index above 170 to be only as desirable as one at 170. x2.5 62. a D value was obtained. This was done because we felt that the desirability increased in a linear manner. That is. A three-variable. Graph of Transformation (3) used for y. The resultant fitted coefficients are given in Table 2. along with experience. since we felt that a linear transformation expressed our evaluation of the desirability. however.: 1.5 75 x3 vol. we set Y1* = 170 and Yz* = 1300. we set Y1* = 120 and Yz. Here Ys* = 400 and Y3* = 600 while Y4* = 60 and Yq* = 75. and y. Each ?i was then transformed into a di. one could certainly utilize standard procedures in design and regression (including stepwise regression) in obtaining estimators Pi.5 1145 1090 1260 1344 430 390 390 :: :: . The four di’s were combined into a single D using (1).633 : 0 A33 tl. = 1000. 12. 2. Experimental Design Compound No.: .2.: 1300 380 ii.633 : 0 0 0 0 0 0 0 0 600 x2 67. Graph of Transformation (2) used for YI and k’. for each level of Xl. October 1980 Yl 1: +1 +1 500 FIGURE 4. along with the standard errors for each Yi.5 50)/10 and x3 are design levels iphr = parts per hundred) Journal of Quality Technology . It was felt from past experience that at least a second degree polynomial would be required to provide an adequate fit to the data. central composite design with six center points (shown in Table 1) was employed to generate the data which was then fitted to the second degree polynomials 3 3 Pi = L-l m .633 -1. 19 20 "4 400 60 1. A central composite response surface design was employed because of favorable past experience with such designs. and YZ the one-sided transformations given by (2) were used and are shown in Figure 3. +1 102 120 117 198 103 132 132 +1 : -1 0 0 139 102 154 t1. 3.5 74. Again constants of s = 1 and t = 1 were used.633 1:: 116 153 133 133 140 142 145 142 -1?33 t1. using (2) and (3) as illustrated in Figures 3 and 4.217 SIMULTANEOUS OPTlMlZATlON OF SEVERAL RESPONSE VARIABLES of Y3 and Y4.3)/0!5 = (phr (phr silica sulfur where x1.L L-l Vi* “i+ 120 170 1000 1300 "1 "2 FIGURE 3. for Tire Tread Example ’ X3 +1 -1 -1 +1 -1 +1 +1 2 xl "3 3 bo + C btXL + C z bLmxL%m IY 0 : : 0 : 0 0 = (phr silane - = .5 65 77. 4. With less previous experience. No. we selected midpoints cg = 500 and c4 = 67. For each of these. for Tire Tread Example For Y. For Y3 and Yq the two-sided transformations given by (3) were used and are shown in Figure 4. XZ. and X3 to obtain the yls.5 as the most desirable values di 1 (4) i = 1.2)/0. rotatable. y2 y3 Y4 900 a60 a00 2234 490 1289 470 410 570 240 640 270 67. x1 1 2 -1 +1 -1 t1 -1 +1 -1 4" 5 6 7 a 1: 11 12 13 14 1': 1. Any P. Hence. From a practical standpoint. Since it is important to have a good estimator Pi of qi for this optimization technique care should be taken to use good regression and design techniques.5 67 1270 1090 770 1690 700 1540 410 380 590 260 520 380 :: 2184 1784 520 290 . we set r = 1 in the transformation given by (2) for both Y1 and YZ. The next step was to use the coefficients given in Table 2 along with various values of Xl. 4. We then searched through TABLE 1. For this example. and X3. XZ. 0 Maximum composite desirability.88 10. do small changes in the independent variables result in sharp decreases in D? Since D is a function of the X variables. ^o Yl* Y2* Y3* Y4* d2 = 1.61 (Y.) 400. except to indicate the level of the X’s where the maximum D occurs.583 = 120 = -1 1000 I = 400 Y.69 (Y.17 69.57 5.13 -0.49 17.12 16.56 0. October 1980 . 5 FIGURE 5.43 a.11 268. Contour Plots of D for X1 and Xp for Tire Tread Example -.38 328.25 1. Figures 5. No.45 -1.06 -0.145 x3 = -0. The value of 0. the experimenter is generally interested in how stable the optimum is. 48 -83.40 -73.75 6. The maximum composite desirability was 0. (PICO) = 129.93 17.63 1.) 1261.88 5.868 Y.55 -124.32 1.- -1 I- -1 iI +.27 the levels of Xl.13 104.50 i 39‘.7 Y4 (Hardness) = 68. for Tire Tread Example Vol.583 has little numerical meaning.92 7. of course.) 139.656 d4 = 0.31 0. = 600 = 60 Y.79 199.91 -1.932 = 0. and 7 show the contour plots (sketched from a grid of D values) of D for two independent variables with the other held at its TABLE 3.38 -99. Contour Plots of D for X1 and X.25 20.55 (Y.41 4.583 and all of the constraints have clearly been met. The algorithm we employed generally converged in fewer than 250 iterations. = 75 Journal of Quality Technology = 0.91 -4.38 94.000 d3 = 0. however. 12.5 dl Y2 (Modulus) = 1300 Y3 (Elongation) = 465. 6.189 -1 1 0 I t1 x3 FIGURE 6. X2.67 -31.13 7. All of this was. done on a computer. 4.) 68. = X2 l- t1 a. For example.13 7.63 0. and X3 to find the optimum value for D.15 246. Optimum Compound and Predicted Properties x. Error b0 bl bz b3 bll b22 b33 b12 b13 b23 (Y.25 1.01 -3. it can be plotted to answer such questions.32 -1.21’8 GEORGE DERRINGER AND RONALD SUICH TABLE 2. The resulting optimum formulation is shown in Table 3.050 x2 = 0. Aside from finding the maximum D. Regresston Coefficients and Standard Error for Responses Std. A. pp. 1962. J.” Division of Rubber Chemistry. the advantage of being able to plot the desirability surface to determine its sensitivity to small changes in the independent variables is significant.. E.. Vol. Figure 5 shows the plot of X1 versus X2 with X3 held at its optimum. NICHOLSON. meaning that small departures from optimality of the X values would not appreciably decrease the desirability. 21. the optimum reached in Table 3 did prove to be satisfactory from a production standpoint. Summary -1 0 FIGURE 7. ‘HARRINCTON. This program also enables one to generate a response surface of D as a function of two of the independent variables. However. although slight deviations from the optimum levels of the X’s were instituted for other reasons. and B EAUMONT . 8. D. P. Contour Plots of D for Xz and X3 forTire Tread Example optimum. pp. 1965. Vol. since the surface is relatively flat near the optimum. No. 4. October 3-6. 10. “Optimum Compounding by Computer. A. 2. and JEEVES . 212. T. 1972. and PULLEN. a copy of the FORTRAN computer program used to maximize D in terms of the X. For example. C..” Industrial Quality Control. Vol. No.SIMULTANEOUS OPTlMlZATlON OF SEVERAL RESPONSE VARIABLES 219 Computer Program We have available. No. 12. References 1. x3 = -0. This proved no great problem in this example. numerous formulations are evaluated until one is found which is within all constraints. The desirability function approach is a considerable improvement over this method and usually not only requires fewer formulations to be evaluated but also results in more desirable property levels. 3. This becomes the “optimum” formulation. 4. Ohio.868. 272-275.. J R . No. R. American Chemical Society Fall Meeting. 2. No.” Computer Aided Design. Vol. October 1980 The simultaneous optimization of several responses has often been accomplished by a hit-ormiss approach. and MCMILLAN. “Statistical and Optimization Techniques in the Design of Rubber Compounds. N. It should be noted that any good optimization program may be used. pp. In such a procedure. This can then be used to obtain contour plots. “The Use of Experimental Design and Computerized Data Analysis in Elastomer Development Studies. R. GATZA. HARTMANN . Furthermore.. that is. the approach utilized in this example is not the only possible approach. C. and will provide upon request. p. 6..” Journal of the Institute of the Rubber Industry. E. T. R. 1. Cincinnati. Journal of the Association of Computing Machinery. 5. 1968. Obviously. E. 39-47. 1.. Vol. R. “The Desirability Function. Journal of Quality Technology . 494498. A. holding the other independent variables constant. 2. All three of these plots show the surface to be relatively flat near the maximum. Another feasible method would involve studying the coefficients in the fitted equations and overlaying contour plots. Paper No. HOOKE . 1969. 6. 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