DETERMINATION OF THE OPTIMAL PROCESS MEAN AND THE UPPER LIMIT FOR A CANNING PROBLEM Damodar Y. Golhar Management Department Western Michigan University Kalamazoo, Michigan 49008 Stephen M. Pollock Department of Industrial & Operations Engineering The University of Michigan Ann Arbor, Michigan 48109 Technical Report #86-28 July 1986

DETERMINATION OF THE OPTIMAL PROCESS MEAN AND THE UPPER LIMIT FOR A CANNING PROBLEM Damodar Y. Golhar Management Department Western Michigan University Kalamazoo Stephen M. Pollock Industrial & Operations Engineering Department The University of Michigan Ann Arbor Abstract We deal with a canning problem where the amount of an expensive ingredient put into a can is a random variable whose mean can be set by the canner. Underfilled (below a specified lower limit) and overfilled (above a controllable upper limit) cans are emptied and refilled. We assume that the fill is normally distributed with known variance, and assuming a reasonable cost structure obtain optimal values for the process mean and the upper limit. Simple approximate analytical expressions relating these optimal values to fundamental parameters are also established. An explicit measure is given for the value of being able to impose an upper limit on the fill.

I NTRODUCTI ON There is a vast literature on quality control that focuses on defining economic upper and lower control limits for a process. A common assumption is that the process is set optimally and the problem is usually to detect deviation from the "normal" performance of the process at an early stage [see Montgomery (1980)]. In this paper we, instead, address the issue of finding the optimal process settings in conjunction with a pre-determined control limit. In particular, we consider a canning problem where cans are filled with an expensive ingredient called "fill". The amount put in a can is a random variable, with a mean (the "process mean") set by the canner. Filled cans are weighed: underfilled cans (those weighing below a specified limit) are emptied and refilled at the expense of a reprocessing cost (this might include, among other costs, the cost of production time lost); cans weighing above the specified limit are sold in a regular market for a fixed price. If the process mean is set very high then the probability of underfilled cans becomes small, and the canner saves reprocessing cost at the expense of sending out too much ingredient at no benefit to him. (The cost of the excess fill is sometimes referred to as "give-away" cost). On the other hand, if the canner sets the process mean too low then he will save on the give-away cost but the reprocessing cost will increase because 1

more cans will be underfilled. An immediate problem then is to set the process mean at the most economic level. In addition, overfilled cans bring a fixed price in a regular market. Hence, it is not profitable to sell cans that are filled too excessively. The canner may have an option of reprocessing cans that are filled over a controllable upper limit, in which case it is desirable to know the most economic upper limit. Thus, the canner faces the combined problem of finding optimal values for both the process mean and an upper limit. Bettes (1962) studied the problem of finding optimal values for both the process mean and the upper limit. However, his procedure is based on trial and error, is computationally tedious, and does not give accurate values. Hunter and Kartha (1977) found the optimal process mean only, with the assumption that underfilled cans can be sold in a secondary market for a fixed price. Bisgaard, Hunter and Pallesen (1984) later suggested that this assumption is unrealistic because it implies that empty cans can be sold for as much as almost acceptably full ones, and instead looked at the situation where underfilled cans are sold in the secondary market at a price proportional to the amount of ingredient in a can. However, in some cases (such as pharmaceuticals) the product may be sold only in the regular market. The secondary market may also be so far away that transportation and other related costs may make is prohibitive to sell a substandard product. In such 2

cases if a can is underfilled then the only available alternative is to empty and refill it with an associated reprocessing cost. Golhar (1986) formulated such a problem and found the optimal process mean setting. But his assumption that the overfilled cans, no matter the amount of fill, be sold for a fixed price is unrealistic. At times, it may not be profitable to sell overfilled cans at a fixed price and there can be a controllable upper limit such that overfilled cans (i.e., weighing above this limit) can be reprocessed. Here, we extend Golhar's (1986) model to a process where both the process mean and the upper limit can be controlled and show the superiority of such a policy over controlling the process mean only. THE PROBLEM Let g(X;g,a2) represent a normal density function for the random fill X, and f(X) and F(X) be the standard normal density and distribution functions respectively. L is a pre-specified minimum weight limit and U a controllable upper limit. Thus, a can weighing between L and U is sold in a regular market, at a price A. If a can weighs below L or above U, it is emptied and refilled at the (reprocessing) cost R. C denotes the cost of the contents/unit. The objective of a canner is to find the optimal process setting I* and the upper limit U* that will maximize the expected profit P per can sold. 3

THE SOLUTION Let P(X;A,U) denote the profit for a can sold with contents X and P(g,U) its expected value. If the amount of fill X is such that L < x < U then the can is sold for A and the net profit is A - Cx. On the other hand, if the can weighs below L or above U then the can is emptied and is refilled at cost R. The refilled can will then realize the expected profit P(O,U). Hence, for a refilled can the net expected profit is P(L,U) - R. The profit per can sold is therefore: A - CX for L < X < U P(X;r,U) = P(p,U) - R otherwise Hence, the expected profit is: U L P(0,U) = (A-Cx) g(x;,Ta2) dx + {P(A,U) -R} g(x;J,a2)dx L X 0 + J {P(I,U) - R} g(x;,a2=)dx (1) U Using the well known result that: U f x g(x;A,oa2) dx = LF[U-1]- Ff[fU]-F[LfJ-]+ +f[L-M] (2) L a O J aaLoCrI and letting ti = U-U and ta = L-U we get: R+Ca [f(t2)-f(ti)] P(H,U) = A - C + R - F(t2)-F(t2) (3) 4

Note that without an upper control limit, i.e., U = a, equation (3) becomes: R+Ccf (t2) P(j1,) = A - C) + R -- (4) 1-F(ta) which is essentially the same relationship obtained by Golhar (1986). We will show later the degree to which the process with upper control (given by equation (3)) is more profitable than without upper control (equation (4)). In order to find H* and U* (the most economic levels of and U), we first show that P(Q,U) in equation (3) is a concave function of H and U. Since the last term of equation (3) has f(ti) and f(ta) in the numerator, and F(ta) and F(t2) in the denominator, it is difficult to show the desired concavity analytically. However, the appendix numerically establishes the concavity of P(0,U) over a wide range of values of ) and U. Taking the first derivative of P(I,U) with respect to U, and equating to zero, gives: a P(H,U) _ ---- = 0 = t [F(t)-F(t2) + f(ti)-f(t2) - M (5) a U where M S R/(Co), a constant for any given process. Similarly, equating the first derivative of P(o,U) with respect to A to zero, and combining the result with equation (5), we get: d P2(,U) =0 = 0 = F(t)F(t)-f(t2) ) ti-t1 (6) a u L 5

Optimal values of ti and t= can be obtained numerically by solving equations (5) and (6) simultaneously. RESULTS Table 1 gives t~* and ta*, the optimal values for t~ and tsolving equations (5) and (6) as a function of M. The optimal process setting H* is obtained through the relation )* = L - eta* and the optimal upper limit U* is obtained via U* = ~ + ot*,. For M between 0.1 and 2, these optimal values are plotted (in units of o) in figure 1. A convenient way of examining the resulting expected profit is to look at the excess over what would be obtained if each can could be filled to exactly L (achievable only when -a-*0), in which case the profit would be A - CL. We can define the minimum expected excess cost per can as E = A - CL - P(l*,Uw) For different M, values of E are computed (in units of cc) and are given in table 2. For M between 0.1 and 3, Figure 2 shows the behavior of E. To see how the parameters affect AU U* and the resulting costs, consider the following example: suppose initially C = $0.5 per ounce, R = $0.2, o =.4 ounces, and L = 3 ounces. The constant M-= 1, and from table 1, ti* = 1.657 and t* = -.75. Hence, H* = L +.75 a = 3.30 ounces and U* = L + 2.425 a = 3.97 ounces. From table 2, this results in a cost per can of E = (.5)(.4)(1.409) = $0.28 per can. Now, suppose due to process 6

Table 1. Optimal values of tl and t2 for a given M. M ti^ t a 0.1 0.478 -0.236 0.2 0.682 -0.334 0.3 0.843 -0.410 0.4 0.983 -0.474 0.5 1.111 -0.530 0.6 1.230 -0.581 0.7 1.342 -0.628 0.8 1.450 -0.671 0.9 1.555 -0.711 1.0 1.657 -0.750 1.1 1.757 -0.786 1.2 1.855 -0.820 1.3 1.952 -0.853 1.4 2.049 -0.884 1.5 2.145 -0.913 1.6 2.240 -0.942 1.7 2.335 -0.969 1.8 2.430 -0.995 1.9 2.524 -1.020 2.0 2.619 -1.044 2.2 2.809 -1.088 2.4 2.998 -1.130 2.6 3.189 -1.168 2.8 3.380 -1.204 3.0 3.572 -1.237 3.2 3.764 -1.268 3.4 3.957 -1.298 3.6 4.151 -1.325 3.8 4.344 -1.351 4.0 4.539 -1.375 4.5 5.026 -1.432 5.0 5.515 -1.482 5.5 6.006 -1.526 6.0 6.498 -1.567 7.0 7.483 -1.639 8.0 8.472 -1.700 9.0 9.462 -1.754 10.0 10.454 -1.801

1. - 1.41.1 L /a 1.''"' 1 1, - 4-... n 0.6 -o C( CllL 0.8 ~. 0_, -..."r'" 0.5 -l 0.4 - 0.3" 0 —-B —- --- 0.1 0.5 0.9 1.3 Process ccnstant M Figure 1: Optimal U and p values as functions of M Figure 1' Optimal U and p values as functions of M 1.7

innovations the process standard deviation is halved. Therefore, M = 2 which gives ti* = 2.619 and ta* = -1.044, with corresponding = = 3.10 and U* = 3.36, and E = (.5)(.2)(1.662) = $0.16 per can, a 43% saving. PROFITABILITY OF A PROCESS WITH UPPER CONTROL LIMIT It is of interest to compare the expected profit obtained above to the case where there is no upper control limit, in which case the objective is to find the optimal value of ta = (L-W)/o that will maximize the expected profit given by equation (4). Golhar (1986) has solved this problem, with results given in table 2 and plotted in figure 3. Notice that having an upper limit gives a higher expected profit, with the advantage decreasing as M increases. An upper limit allows a tighter control on the fill, resulting in higher profit. However, controlling both parameters (I and U) might be expensive and/or time consuming. These results allow the canner to determine the value of seeking to control both parameters, as a function of M. APPROXIMATIONS TO OBTAIN OPTIMAL VALUES FOR THE PARAMETERS In this section we develop simple approximate relationship between to* and ta* as a function of M, that can be used as an alternative to table 1 for M < 1. It is reasonable to assume that the process constant M for real processes would be small. (For values of M greater than 7

Table 2 Comparative evaluation of the advantage of being able to have an upper central limit. E = excess cost per can. No Upper Limit Upper Limit Available t=2 E | tlg tat E (units of ccr) (units of ca) 0.1 0.364 0.858 0.478 -0.236 0.613 0.2 0.059 0.998 0.682 -0.334 0.816 0.3 -0.126 1.091 0.843 -0.410 0.954 0.4 -0.261 1.165 0.983 -0.474 1.058 0.5 -0.366 1.224 1.111 -0.530 1.141 1.0 -0.701 1.433 1.657 -0.750 1.409 1.5 -0.899 1.565 2.145 -0.913 1.559 2.0 -1.040 1.664 2.619 -1.044 1.663 2.5 -1.149 1.742 3.094 -1.149 1.742 3.0 -1.237 1.808 3.572 -1.237 1.808 3.5 -1.311 1.865 4.054 -1.311 1.865 4.0 -1.375 1.914 4.539 -1.375 1.913 5.0 -1.482 1.996 5.515 -1.482 1.998 6.0 -1.567 2.065 6.498 -1.567 2.065 7.0 -1.639 2.121 7.483 -1.639 2.121 8.0 -1.700 2.172 8.472 -1.700 2.172 9.0 -1.754 2.215 9.462 -1.754 2.215

1, - ^.-a-~~~_1.6 ILIJ I F_~- l 4-' 1.5-'-" V) C 1 4-~ ~ 1.13 -./ 0,9 q, f 1,.3- / 41 / "-'~ 1 1- / 0.9 0.8 - / 0.7 - 0,6 I... —-,,-,,............. 0. 0.1 0,5 1 1.5 2 2.5 3 P roce.s s constai. nt M Figure 2: Expected excess cost as a function of M when upper limit is available

Iw 4-' V>r I? o. 0 U --.2) rU)L CL Uc 1.7 -- 1 6 - 1,51.4 - 1.3 - 1.21.1 - 1 - 0.90.8 - 0.70.6 - 0.1 0.5 1 1.5 Figure 3: Process cotnstant M Expected excess costs for a process with and without upper limit

about 2, the optimal process setting is more than one standard deviation above the minimum weight limit L, a situation that would not be readily tolerated in most practical situations). It can be seen from table 1 that as M approaches 0 the optimal values of ti and t2 also approach 0. Using Taylor series expansions, the standard normal density and distribution functions can be approximated for y —O, as f(y) 2 T 1 - a1 (Y - Ya/6) and F(y) = 2 + -(2x) Let yi and y2 represent the values of ti* and ta* respectively, obtained using this approximation. Equation (5) can then be rewritten as (2% Yl(Y1-YY) + vz7 Y-Y-12 M (7) Solving equation (7) we get: Y1 - Y2 = f(2M') (8) where M' = f(21) M Similarly, equation (6) becomes 1(2) [ Y - Y3/6 - Y2 + Y23/6] - - )[1-Y22/2](Y - Y2)=0, (9) which reduces to: 3 Y~ Y=2 = 2Y=3 + y13 (10) Simultaneously solving equations (8) and (10) gives: Ya = -0.746 fM (11) 8

and Y = -2 Y. (12) For values of M c 2, the values of yi and ya are plotted against actual to* and t2* (obtained from table 1) in figure 4. It is seen that the approximations fit very well. In particular, approximation (11) gives values of t2* that are within 1% of the actual, for M c 2. DISCUSSION The canning problem is in general subject to ever-changing values of the process constants R, C and a. With changes in technology the reprocessing cost R and fill precision a should change. The cost of ingredients, C, is a volatile function of market forces and should change often. Finally, with aging of the process itself the precision a will change. Industry should, in turn, respond by adjusting the optimal process setting t* and the upper limit U*. Table 1 and relations (11) and (12) should prove useful for finding these settings without carrying out the detailed calculations. However, adjusting these two parameters simultaneously could prove to be expensive and/or time consuming; in which case industry might want to cost-out the value of having the capability of controlling the upper limit. Table 2 can then be used to determine the cost effectiveness of the upper control limit. 9

3 2.5 1.5 1 0,5 0 -0.5 -1 -1,5 NctJaA t i t1 Actual t2 2 0.1 0,4 0.7 1,0 1.3 1.6 1.9 Figure 4: Process conrstant M Actual and approximate t1, t2 functions of M values as

ADDendix We show numerically that the function P(,U) = A- CA + R-R + Ca[f(ta)-f(t1)] (Al) P(,U) A - - F(t1) - F(t) (A is concave over parameters U and A, where ti = (U-g)/a and to = (L-g)/a. Hence, we show that the function: R )R C+ f(tz)-f(tz) S1(U,U) = -P(X U) - — a + + (t ) Co= -- F(ti)-F(ta) (A2) is convex over U and H. This is equivalent of showing: L M + f(t2)-f(tl) S2 = SI- - -t 2 + M. F(to)-F(ti) (A3) is convex with respect to t, and to2 To show that equation (A3) is convex with respect to to, we fix t2 and show that: S (ti) = M + f(t2)-f(tl) (A) S3(t1) F(t)-F(t2) (A4) is convex over tz. For different tX, SsCti) was computed for 0 S M S 10 and fixed -0.1 s ta S -2.1 ( for a fixed t2 - -0.6, sample computations are given in table Al). It was seen that S3(tz) was convex over t1 for a wide range of M and to values. To show that equation (A3) is convex with respect to t2, we first express tz in terms of to: to= UL + to = K + t2 (A5) a.

Hence equation (A3) becomes: S2(t:) = -t + M + f(t2) - f(ta + K) (6) F(t2 + K)-F(t2) (A6) For different to, S2(t2) values were computed for 0.1 < M S 10 and 0.1 I K c 3. ( Table A2 gives a sample of these computations for a fixed K = 0.5 ). It was seen that equation (A6) was convex over tz for a wide range of M and K values.

Table Al Sample computations of S3(tl) for a fixed t2 = -0.6 0 M Thy 0.5 1.0 2.0 3.0 4.0 5.0 7.5 10.0 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 1.92416 1.22003 1.05787 1.04657 1.07662 1.1086 1.13008 1.14135 1.14599 1.14764 1.14798 4.13949 2.53202 2.03101 1.86544 1.82189 1.81994 1.82694 1.83282 1.83564 1.83682 1.83697 8.57015 5.15601 3.97729 3.50317 3.3124 3.24261 3.22067 3.21575 3.21495 3.21518 3.21495 13.0008 7.77999 5.92358 5.14091 4.80296 4.66529 4.6144 4.59869 4.59426 4.59354 4.59293 17.4315 10.404 7.86986 6.77864 6.29349 6.08797 6.00813 5.98162 5.97357 5.9719 5.97091 21.8621 13.028 9.81614 8.41637 7.78403 7.51065 7.40185 7.36455 7.35288 7.35026 7.34889 32.9388 19.5879 14.6818 12.5107 11.5104 11.0673 10.8862 10.8219 10.8012 10.7962 10.7938 44.0154 26.1479 19.5476 16.605 15.2367 14.624 14.3705 14.2792 14.2494 14.2421 14.2388

Table A2 Sample computations of S2(t2) for a fixed K = 0.5 M T1\ 0.5 1.0 2.0 3.0 4.0 5.0 7.5 10.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 454.81 102.368 30.4239 11.6471 5.71705 3.60284 2.86618 2.85575 3.35193 4.57205 7.31258 13.7642 30.317 79.575 238.244 909.355 204.409 60.5444 23.0108 11.1577 6.94062 5.47714 5.46672 6.4672 8.9161 14.4048 27.3143 60.4375 158.94 476.34 1818.45 408.491 120.785 45.738 22.0391 13.6162 10.6991 10.6887 12.6977 17.6042 28.5892 54.4146 120.678 317.67 952.531 2727.54 612.572 181.026 68.4653 32.9205 20.2918 15.921 15.9106 18.9283 26.2923 42.7736 81.5148 180.919 476.401 1428.72 3636.63 816.654 241.267 91.1926 43.8019 26.9673 21.1429 21.1325 25.1588 34.9804 56.958 108.615 241.16 635.131 1904.91 4545.72 1020.74 301.508 113.92 54.6833 33.6429 26.3649 26.3544 31.3893 43.6685 71.1424 135.715 301.401 793.861 2381.1 6818.45 1530.94 452.111 170.738 81.8868 50.3318 39.4197 39.4093 46.9656 65.3887 106.603 203.466 452.004 1190.69 3571.58 9091.17 2041.14 602.713 227.55 109.09 67.0207 52.4745 52.4641 62.542 87.109 142.064 271.217 602.606 1587.51 4762.06 3.2 1000.73 2000.74 4000.77 6000.8 8000.82 10000.8 15000.9 20001 I

REFERENCES: Bettes, D.C.(1962) "Finding an Optimal Target Value in Relation to a Fixed Lower Limit and an Arbitrary Upper Limit", Applied Statictics, Vol.11, pp. 202-210. Bisgaard, S., Hunter, W.G., and Pallesen, L.(1984) "Economic Selection of Quality of Manufactured Product", Technometrics, Vol.26, No.l, pp. 9-18. Burr, I.W.(1949) "A New Method of Approving a Machine or Process Setting: Part I", Industrial Quality Control, Vol.5, No.4, pp. 12-18. Golhar, D.Y.(1986) "Determination of the Best Mean Contents for a Canning Problem", to appear in the Journal of Quality Technology. Hunter, W.G. and Kartha, C.P.(1977) "Determining the Most Profitable Target Value for a Production Process", Journal of Quality Technology, Vol.9, No.4, pp. 176-181. Montgomery, D.C.(1980) "The Economic Design of Control Charts: A Review and Literature Survey", Journal of Quality Technology, Vol. 12, No.2, pp. 75-87. Nelson, L.S.(1978) "Best Target Value for a Production Process", Journal of Quality Technology, Vol.10, No.2, pp. 88-89. Springer, C.H.(1951) "A Method of Determining the Most Economic Position of a Process Mean", Industrial Quality Control, Vol.8, pp. 36-39.