(1986)Optimal lot sizing, process quality improvement and setup cost reduction

Source: Operations Research, Vol. 34, No. 1 (Jan. - Feb., 1986), pp. 137-144Published by: INFORMS

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OPTIMAL LOT SIZING, PROCESS QUALITY IMPROVEMENT AND SETUP COST REDUCTION EVAN L. PORTEUS Stanford University, Stanford, California accepted March 1985) (Received June 1984; control. It costs can benefit production systems by improving quality This paper seeks to demonstrate that lower setup a simple model that captures a significant relationship between quality and lot size: while does so by introducing item. Once producing a lot, the process can go \"out of control\" with a given probability each time it produces another the process units throughout its production of the current lot. The system incurs out of control, an produces defective for each defective Thus, there is an incentive to and related piece that it produces. extra cost for rework operations units. The paper also introduces three a smaller fraction of defective for investing options produce smaller lots, and have the probability that the process moves out of control fewer yields defects, in quality (which improvements: (i) reducing lot sizes, lower (ii) reducing setup costs (which yields smaller larger lot sizes, fewer setups, and larger holding costs); the two previous options. and (iii) simultaneously using a specific form of By assuming holding costs, and fewer defects); we explicitly obtain the optimal investment each for the investment cost function strategy. We also briefly discuss option, A numerical to changes in underlying parameter values. illustrates the results. the sensitivity of these solutions example Recently, Porteus (1985, 1986) has contributed to our understanding of the economic trade-offs associated with production systems with very small lot sizes. For example, very small lot sizes are an important part of the Japanese Just-In-Time (JIT) (1982) provide systems. Hall (1983) and Schonberger Por- useful details about such systems. In particular, teus (1985) introduces the option of investing in re- in the classical undis- ducing the setup cost parameter an optimal setup counted EOQ model and determines cost level. (Porteus 1986 deals with the discounted those (1985), however, case.) As pointed out in Porteus papers address only some of the benefits associated with reducing the setup cost: reduced setup and hold- benefits ing costs. They do not deal with the additional of improved quality control, flexibility, and increased effective capacity. This paper begins to address the benefits of improved quality control. With the model that is postulated, improved output quality (percent- can be age of units produced that meet specifications) achieved simply by reducing the lot size. Reducing setup cost can also improve output quality, because it further reduces the optimal lot size. The novel aspect of the model is that it is the first, to the author's relationship to explicitly show a significant knowledge, between quality and lot size. The quality control lit- erature focuses on many aspects of quality, such as inspection; process control charts; machine mainte- quality improvement; and replacement; nance, repair and designing for quality. See Girschick and Rubin (1974), Pier- (1952), Fetter (1967), Ross (1971), Juran skalla and Voelker (1976), Crosby (1979), Juran and Gryna (1980), Monahan (1982), andl Feigenbaum (1983) for a sampling. Recently, Fine (1983) devel- oped a model that considers improved quality achieved through learning. However, these references do not address the possible relationship between lot size and quality. This paper models this relationship in a very simple way: while producing a lot, a process each can go \"out of control\" with a given probability is assumed The process time another unit is produced. to be in control before beginning production of the produces defective lot. Once out of control, the process units and continues to do so until the entire lot is produced. Although simple, this assumption is sup- ported by Hall (p. 157): out of the Once most production processes start to drift are on basic variance capable, they keep range of which they can be determined and a reason which doing it, usually for and various corrected. If production runs are short, wear factors which normally cause a process to drift out of control of limited duration. are We assume that the system incurs an extra cost, for operations for each defective piece rework and related produced. Thus, there is an incentive to produce smaller lots with this model, because of the smaller of units that are defective. This paper fraction expected therefore first tackles the problem of specifying the optimal lot size that accounts for the consequences of 728 lot sizing production, 335 quality effect, interaction. improvement and setup Subject classification: 197 quality Operations Research Vol. 34, No. 1, January-February 1986 137 0030-364X/86/3401-0 l37 $01.25 ? 1986 Operations Research Society of America 138 / PORTEUS defects. Section 3 then introduces the option of in- vesting in process quality improvement by means of reducing the process quality parameter, the probability of the process moving out of control. Such an invest- ment yields better output quality (fewer defects), a larger lot size, fewer setups, and larger holding costs. For a specific form for the investment cost, we explic- itly derive the optimal solution. Section 4 extends the results of Porteus (1985) to this model. That is, the option of investing in reducing the setup cost param- eter is allowed, without allowing for the simultaneous investment in process quality improvement. Com- pared to the basic model, such investment yields a smaller lot size, lower holding costs, and better output quality. Again, by assuming a specific form of the investment cost function, we explicitly obtain the optimal solution. Section 5 combines the options of the previous two sections and determines the optimal setup cost and process quality level simultaneously. The results break into four cases, depending on whether or not investment is made for process quality improvement and/or setup cost reduction. A single numerical example is used throughout the paper to illustrate the various problems posed. Table I in Section 5 summarizes the results from this exam- ple. (Some readers may wish to examine the table before reading further.) The last section summarizes the paper. 1. The Basic Model and Preliminaries The basic model is the classical undiscounted EOQ model, with m the deterministic, continuous sales (demand) rate, K the setup cost, c the unit production cost, i the fractional per unit time opportunity cost of capital, and h the nonfinancial per unit time holding cost. Each of these parameters is assumed to be strictly positive. The basic model is modified as follows. First, while producing a single unit of the product, the production process (machine) becomes \"out of con- trol\" and begins to produce defective products, with probability q. We assume q is strictly positive. Once the process is out of control, it remains that way until the remainder of the lot has been produced. That is, the process follows a two-state Markov chain during production of the lot, with a transition occurring with each unit produced. (The transition probabilities are self-evident.) One interpretation of these assumptions is that the firm uses the inspection policy suggested by Hall that inspects only the first and last pieces of the lot. (If the last piece is good, then the entire lot is judged to be good. If not, further inspection is needed to determine the defective pieces.) Another interpre- tation is that the type of defect cannot be identified until the piece is processed at the next work station and that the entire lot is produced at the previous work station before any are transferred to the next station. As a final interpretation, the process is contin- uously monitored but a delay is required before the operators can determine that the process is out of control. If that delay is long compared to the produc- tion rate, then it may be reasonable to assume that the entire lot is produced before knowing whether the process is out of control. (The monitoring is still useful, since it can help identify which, if any, pieces need rework.) To finish specification of the model, we assume that each defective unit costs an additional CR to correct (for rework and related operations). We assume CR is strictly positive. For simplicity, we as- sume that detectives are corrected instantaneously, once discovered, so that shortages in satisfying de- mand do not occur. Let Q denote the lot size and q :=1 - q. We first develop the expected number of detectives in a lot of size Q. Lemma 1. The expected number of defectives in a lot of size Q is Q- 0(1 - q----, OQ) which is a strictly increasing, strictly convex function of Q. The probability of a randomly selected the lot being defective is 1 - - piecefrom 1 4Q)/qQ, which is a strictly increasing, strictly concave 4( function of Q. Proof. For purposes of the proof, let en denote the expected number of detectives in a lot of size n, given that the process is in control before beginning the lot. Then, en= qn + qen1 n-I n-I -qn - q i=O i=I i =qn (1 - qn) 4[1 + (n - -) q -q n nq] q =n - q(1- q which verifies the form of the functions. That they are strictly convex and concave, respectively, follows di- rectly from the s cond derivatives. That they are strictly increasing is intuitively clear but requires some inequality logic to be applied to the first derivatives. For example, -if q + 0.01, so there is 1 chance in 100 of the process going out of control with each unit Lot Sizing, Process Quality Improvement and Setup Cost Reduction / 139 produced, then the expected percentage of defective units in a lot of 100 is 37%, whereas the percentage drops to 22% for a lot size of 50. If q is decreased to 0.001, the percentage drops to 4.9% for a lot of 100 and to 2.5% for a lot of 50. These figures illustrate how a seemingly small probability of the process going out of control can lead to a relatively high fraction of defective units in large lots, and how lowering the lot size can significantly reduce the defective fraction. Thus, the stage is set for determining the optimal lot size, while accounting for the quality effects of this model. 2. The Optimal Lot Size The total cost per unit time consists of the usual setup and holding costs included in the EOQ model plus the rework costs, which equal the rework cost per unit times the expected number of detectives per lot times the number of lots per unit time: mK +iQ Q 2 + c m -CRq(l q Working with this expression leads to the following results. Theorem 1. If q is close to zero, then the following statements are valid. (a) The total cost per unit time is approximately f(Q) = m + Q (h + mcRq). (b) The optimal lot size is approximately Q* Q + .mK mcRq (c) The induced optimal total cost per unit time is approximately f(Q*) = V2mK(h + mcRq). Proof. (a) Since 4 = 1 - q is close to one, we use a Taylor series expansion of qQ and obtain 4Q = e (In)Q - 1 + (ln 4)Q + [(In 4)Q]2 The result follows upon using In q -q/q, and mcRq/q- mcRq. (b) Two approaches lead to this result. The first minimizes the expression given in (a). The second finds the partial derivative of the exact total cost expression and then uses a Taylor series expansion of it. Both approaches lead to the same result. Further details are omitted. (c) This statement follows directly from (a) and (b). An interesting aspect of these results is that to adjust for the quality effects of this model, one need only increase the original holding cost parameter of the classical EOQ model by mcRq. This increase in the holding cost may be significant. For example, consider the numerical example given in Porteus (1985): K = 100, c = 50, h = 0.5, m = 1000, and i = 0.15. We assume that CR= 25 and q = 0.0004. If the quality effects of this model are ignored, the lot size is 158, an average of 3.1 % are defective, and the (exact) total cost (including setups, holding, and rework) is $2044. If the formula from Theorem 1(b) is used, the lot size becomes 105, the average percentage defective drops to 2.1 %, and the total cost drops to $1895, which is a 7% reduction. Thus, even though the probability of the process going out of control during production of a unit seems to be quite small, a substantial reduction in the lot size can reduce the average fraction defective and the total cost. Hereafter we refer tof(Q) and Q* (of Theorem 1 (a) and (b)) as the total cost and optimal lot size, respec- tively, without mention of their being approximations. In effect, we are assuming they are exact in the anal- yses that follow. It is useful to determine how the optimal lot size and the total cost vary as functions of the parameters of the model. Let Qq(q) denote the optimal lot size, Q*, of Theorem 1(b), and fq(q) = f(Qq(q)), the optimal total cost, as functions of the parameter q. Define analogous functions for the parameters K, m, c, and h as well. Theorem 2. If q is close to zero, then the following statements are valid. (a) Qq(q) is strictly decreasing and strictly convex (b) fq(q) (in q). is strictly increasing and strictly concave. (c) qf,(q) is strictly positive and strictly increasing. (d) QK(K) and fK(K) are strictly increasing and strictly concave. (e) KfK(K) is strictly positive and strictly increasing. (f) Qm(m) and fm(m) are strictly concave. strictly increasing and (g) Qc(c) and Qh(h) are strictly decreasing and strictly convex. (h) fc(c) and fJ(h) are strictly increasing and strictly concave. 140 / PORTEUS Proof. (a) Direct computation leads to Qq~q) = - mcRK Q(h + mcRq)2 Further direct computation reveals that Qq'(q) is strictly positive. (b) Use of the chain rule shows that > 0. f,(q) = mcRQ/2 Therefore, fq'(q) = mcRQ5(q)/2 < 0. (c) The derivative of this expression has the same sign as Q2(h + mcRq) - m2KcRq, which is strictly positive. (d)-(h) These conclusions follow from differentia- tion and substitution. For example, f\"(m) has the same sign as CRqQ' - 2K, which is negative. Part (a) shows that as the probability q decreases (the process quality is improved), the lot size increases at an increasingly faster rate. Part (b) shows that the total cost decreases at an increasingly faster rate, at the same time. Of course, if q decreases to zero, then the classical EOQ results hold. Thus, the optimal lot size for the limiting case of this model is no more than the classical EOQ. The expression qf,(q) in part (c) is similar to point elasticity in economics, except that instead of measuring percentage change for each per- centage change, it measures actual change (in total cost) for each percentage change (in the probability q). Thus, it shows that each time a successive reduction of, say, 10% is made in q, the total cost will decrease but by a smaller amount than for the previous reduc- tion. Part (e) says the same thing for reductions in the setup cost. The remaining parts of the theorem merely confirm that the parametric properties of the classical EOQ model are preserved in this model. 3. The Optimal Process Quality Level In this section, we consider the option of investing in improving the process quality, namely in reducing the probability q. The idea is simply an application of those presented in Porteus (1985, 1986), so explana- tory details are omitted here. Let aq(q) denote the investment cost of changing the probability to the level q. In general, we seek to minimize w(q) := iaq(q) +fq(q). We assume that ag is convex and decreasing, so- we seek to minimize the sum of a convex and a concave function. Porteus (1986) discusses algorithms that can be used to solve such problems. Our approach in this paper is to assume that fderive explicit results. The particular a, has a particular orm and form we choose is the same one exploited in Porteus (1985), namely the logarithmic form: aq(q) = a - b ln(q), for 0 q < qO, where a, b and qo are given positive constants. We interpret qO as the original process quality level, and we therefore seek to determine whether any reduction is called for. In this case, it costs a fixed amount to reduce q by a fixed percentage. For example, if qO is 0.0004, it may cost $20 to reduce the probability by ten percent to 0.00036 and another $20 to reduce it to 0.000324, and so on. As in Porteus (1985), without loss in generality, if we assume that aq(qo) = 0, as we do, it is straightforward to derive the unique values of a and b, given a cost of making each ten percent reduction. (When that cost is 20, then a = 1485 and b= 190.) Theorem 3. If b is strictly positive, then the following statements are valid. (a) w has a unique local minimum on [0, qO]. (b) The optimal process quality probability, q*, satis- fies q* minqo, (ib)2 + ib((ib)2 + 2mKih)1/2) (c) q* is a (strictly) decreasing function of m (holding other parameters constant). Similarly, q* is an increasing function of b and h, and a decreasing function of K and cR. Proof. (a) Here, w'(q) = -ib/q + sign as f,(q), which has the same qfJ(q) - ib, which is a strictly increasing function, by Theorem 2(c). We consider the latter function to be defined for all q , 0, and since it is negative when q = 0, and positive when q = oo, part (a) follows. (b) The equation w'(q) = 0 can be rewritten as (m3cRK)q2 - (2(ib)2mcR)q - 2(ib)2h = 0. The result then follows from the quadratic formula. (c) These results follow directly from writing out the partial derivatives. Part (a) shows that even though w is the sum of a convex and a concave function, it can be minimized without concern. Indeed, part (b) gives the optimal solution. Part (c) shows how that solution varies as a function of the parameters. In particular, there is a critical sales rate level such that if the sales rate is below that level, no improvement in process quality should be carried out, and, above that level, the higher the sales rate, the more should be invested in improv- ing process quality. Analogous statements apply for the other parameters. and Setup Cost Reduction Lot Sizing, Process Quality Improvement / 141 Consider the numerical example discussed earlier, with the addition of the option of investing in im- proving process quality. Suppose that q0 = 0.0004, SO that it is possible to reduce q below 0.0004, and that each ten percent reduction in q costs $20. Thus, a = -1485 and b = 190. The theorem suggests that q be reduced to 0.000015. The resulting lot size is 155, which is about the same as the classical EOQ, and the total cost is 1388, a 32% reduction on the original 2044. The average fraction defective is 0.1 1%, com- pared with the original 3.1%. These reductions are substantial. However, they are done without consid- ering the possibility of reducing the setup cost, as discussed in Porteus (1985, 1986). The next section focuses on this option alone, and the following section combines the two options. 4. The Optimal Setup Cost In this section, as in Porteus (1985, 1986), we allow the option of investing in reducing the setup cost. Again, we assume the logarithmic form: an investment of aK(K) = A - B ln(K) is required to change the setup cost to K from the original level of K0. Thus, we seek to minimize v(K) := iaKK) + fK(K) over K E [0, Ko]. Theorem 4. If B is strictly positive, then the following statements are valid. (a) v has a unique local minimum on [0, Ko]. (b) The optimal setup cost, K*, satisfies K* = min(Ko, 'm(h 2(iB)2 + mcRq) ) (c) K* is a (strictly) decreasing function of q (holding other parameters constant). Similarly, K* is an increasing function of B and a decreasing function of m, cR, and h. Proof. Here, v '(K) = -iB/K + m/Q = 0 has only one positive root, which yields a relative minimum of v. The root is easily seen to be as given in (b). Part (c) follows directly. Part (b) can be analyzed to show when it is optimal to invest in reducing setups, as was done in Porteus (1985). For example, holding other parameters con- stant, there is a critical sales rate such that setup reduction should be carried out if and only if demand exceeds that rate. Part (c) (or a direct analysis of (b)) shows that there will always be at least as much setup reduction when the effect of quality is taken into account as there is when the effect of quality is ignored (when, implicitly, q = 0 is used). Consider again the numerical example discussed earlier, with the addition of the option of investing in lowering the setup cost, but not (yet) in improving process quality. Suppose that it costs $200 to reduce the setup by ten percent and that Ko = 100. Thus, A = 8742 and B = 1898. Then, if the results of Porteus (1985) are used, in which the effect of the process quality is ignored in the optimization (but not in the evaluation of costs), then the setup cost is reduced to 20, the lot size is 71, the number of defective parts per hundred is 1.43, and the total cost is reduced to 1382, which is a 32% reduction on the original 2044. If the effect of quality is incorporated into the optimization over the setup cost, so that the results of Theorem 4 are followed, the setup cost is reduced to 9, the lot size to 32, the number of defective parts per 100 to 0.65, and the total cost to 1259, a 38% reduction on the original total. Thus, in this example, if we ignore the effects of quality while considering reducing the setup cost, a substantial reduction in total costs can be realized. In Porteus (1985), the estimated savings from this action amounted to 20%, whereas the actual savings were much higher, 32%. In this example, the latter savings percentage is the same as that obtained from investing solely in improving process quality (without the op- tion of reducing the setup cost), as seen in the previous section. If the effect of quality is explictly taken into account, the setup cost should be lowered further, with a corresponding decrease in the lot size and in the percentage of defective units. The process config- uration is very different from that suggested in the numerical example of the previous section. Now, the process quality must remain at its current level, but the setup cost is dramatically lowered, so there are small lot sizes and a relatively low defect rate. In the previous section, the process quality was dramatically improved by decreasing q by about 96%. The lot size was quite large, similar to the classical EOQ, and the defect rate was substantially smaller than that given in this section (0.11 vs. 0.65). We shall see in the next section that when we allow simultaneous investment in improving process quality and reducing setups, the optimal configuration will lie somewhere between these two extremes. 5. The Simultaneously Optimal Process Quality and Setup Cost We now combine the options of investing in improved process quality and reduced setup costs: we seek to 142 / PORTEUS minimize iaq(q) + iaK(K) + mK + Q 2(h + mcRq) over Q, q, and K. For purposes of the theorem to follow, let b m~c~qO~o m 2cRqOK0 ifo and B =- JoA 2i' where fo:= /2mKo(h + mcRqo), which is the sum of the optimal setup, holding, and rework costs per unit time when using the optimal lot size and making no investment in process quality improvement or setup reduction. We can now identify four different cases that arise. Let C1 lb band B > ij, C2 = {_ B h + cj qO andB Bb + mKoh/2i2 and b > BmcRqO/(h + mcRqo) im- ply that B2 > mKo(h + mcRqo)/2i2, which contradicts B < B. C1 and C4 are seen to be disjoint by a similar argument, using the complements, The remaining cases are seen to be disjoint by inspection. (b)-(e) The approach taken here is to partition the results into the four cases, determined by whether or not investment is made to reduce q and/or K. The cases are derived as necessary conditions. That they are sufficient follows from their being mutually exclu- sive, verified in (a). The properties of Q* and f* are consequences of q* and K*. For example, consider the second case, when q* = qO and K* < Ko. For K* to be optimal, it is necessary that K* = K*(qo), as given in Theorem 4(b). That is, K* = 2(ib)2/m(h + mcRqo) < KO, which implies that B < B. Similarly, for q* to be optimal, it is necessary that q* = q*(K*), as given in Theorem 3(b). That is, the solution of w'(q) = 0 must be greater than or equal to qO when using K = K*. This condition is equivalent to 2ib(h + mcRqo)1/2 ? mcR(2mK*)1/2 which, after substituting for K*, is equivalent to b mcRqo B h+mcRqO' which completes specification of C2. The other cases follow similarly. Lot Sizing, Process Quality Improvement and Setup Cost Reduction / 143 TABLE I Results of Numerical Example Characteristic q Classical EOQ 0.0004 Quality Adjusted EOQ 0.0004 Optimal Quality 0.000015 Optimal Unadjusted 0.0004 Optimal Adjusted ~~~~~Setup Setup 0.0004 Optimal and Setup Quality 0.000036 K Q Defective % Cost % Savings 100 158 3.1 2044 - 100 105 2.1 1895 7 100 155 0.11 1388 32 20 71 1.43 1382 32 9 32 0.65 1259 38 18.2 64 0.12 1123 45 of the cases C1, C2, C3, and C4 cannot Specification be readily simplified. For example, the set b < b and B < B contains all of C4 and parts of C2 and C3. We now return to the numerical example consid- ered previously. We allow both of the options for in Sections q and Kas specified investment in reducing 3 and 4. Recall that for this example, Ko = 100, = c = 50, h = 0.5, m = 1000, i = 0.15, CR = 25, qo 0.0004, b= 190, B = 1898, a = -1485, andA = 8742. are based on the assump- (The latter four parameters tions that no reduction in the process quality param- or in the setup cost K below Ko implies eter q below qo no investment cost need be incurred, and each ten percent reduction in q costs 20 and each such reduc- tion in K costs 200.) The results are given in Table I, along with those for the cases discussed in previous sections. The results from all columns in Table I except the last have already been discussed. For instance, the column entitled \"Optimal Unadjusted Setup\" gives the results after optimizing over the setup cost while assuming that q is zero even though it is not. When a out in this case, simultaneous optimization is carried some investment in both process quality improvement and setup reduction is made. However, as is true in general, the optimal investment in process quality improvement is less than that given by the \"Optimal Quality\" case in which reductions in setup are not allowed. Similarly, as is also true in general, the opti- mal investment in setup cost reduction is less than case. The that given by the \"Optimal Adjusted Setup\" optimal fraction defective of 0.12% is substantially level and almost as low as that given below the original quality by the case in which only investment in process improvement is allowed. A 45% reduction in total costs is achieved. In this example, case C4 applies, and it is optimal to invest in both quality improvement and in setup reduction. Theorem 5(e) shows that once this case applies, if the sales rate is increased, this case will continue to apply. It shows that if the sales rate is doubled, then both q and K are cut in two. As a result, the optimal lot size does not depend on the sales rate (in this region). Indeed, neither does the optimal level of operating costs (total costs less amortized invest- ment costs). Thus, once a firm exceeds a certain sales rate, its holding, setup, and rework costs should not increase as the market for that product expands. In- deed, that total depends solely on i, the opportunity cost of capital, and B, the parameter indicating how costly it is to make reductions in the setup cost. Note that q* depends on B, and K* depends on b. That is, the optimal process quality depends on how costly it is to make setup cost reductions and the optimal setup cost depends on how costly it is to make process quality improvements. Thus, these decisions cannot be made independently. Note also the dominant role of i, the opportunity cost of capital. It has a squared effect on the optimal setup cost, a linear effect on the optimal lot size and the optimal operating costs, and a less than linear effect (through h = h + ic) on the optimal process quality. Thus, accurate measurement of this parameter is critical. For instance, using the nominal opportunity cost of capital rather than the real opportunity cost of capital will result in process quality, setup cost, and lot size all being too large. If two firms accurately measure their opportunity costs of capital, but have differing figures, perhaps due to differences in financial strength, the one with the smaller opportunity cost of capital will have better process quality, smaller setup costs, smaller lot sizes, and lower total cost. These observations all are based on assuming the case C4 applies and, of course, that the investment cost functions are logarithmic as spec- ified. Analyses of the other cases are similar and therefore omitted. 6. Summary This paper has introduced a model that shows a significant relationship between quality and lot size. in which this relationship is valid, taking For situations it into account results in reducing the lot size and decreasing the fraction of defective units. The paper 144 / PORTEUS has also introduced the option of investing in process quality improvement and setup cost reduction. Each option is taken up alone and then the two are consid- ered simultaneously. In general, algorithms such as those proposed in Porteus (1986) must be used to solve the stated optimization problems. However, if we assume that the investment cost function has a logarithmic form, explicit optimal solutions can be derived. The sensitivity of these solutions to changes in the parameter values is discussed briefly. A numer- ial example illustrates the results. The fractions of defective units for the different cases vary from 3.1 to 0.1 , which is similar to the range found by Garvin (1983) when comparing U.S. and Japanese manu- facturers' electrical assembly-line defects in air conditioners. has pointed out that one unreal- One of the referees istic aspect of this model is the assumed static nature of the quality, setup, and lot size optimization. The a dynamic process of qual- stories from Japan suggest ity improvement and setup cost reduction whereby, slowly but surely, the firm works through alternating steps of small improvements in quality and setup costs. A more realistic model would incorporate an experimental/informational dimension within a dy- namic setting. However, the current static model does suggest that since the relevant cost functions are not convex but do have unique local minima, a small cost reduction from an incremental change (in either qual- ity of setup cost) does not rule out the possibility of a larger cost reduction from the next incremental change. That is, marginal analysis works, but does not necessarily result in decreasing marginal returns. Acknowledgment The author is very grateful to Professors Marvin Lieberman and James Patell for suggesting ways of examining the relationship between process quality and lot sizing, and to the two anonymous referees for their helpful suggestions. References CROSBY, P. 1979. Quality Is Free. McGraw-Hill, New York. FEIGENBAUM, A. 1983. Total Quality Control. McGraw- Hill, San Francisco. FETTER, R. 1967. The Quality Control System. Richard D. Irwin, Inc., Homewood, Ill. FINE, C. 1983. Quality Control and Learning in Produc- tive systems. 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