- Email: [email protected]
Policies for lot sizing with setup learning
produhion economics ELSEVIER
Int. J. Production
Economics
48 (1997) 157-165
Policies for lot sizing with setup learning Ram Rachamadugu”,
Chong Leng Tan
ISOM Department, College of Business Administration, University of Toledo, Toledo, OH 43606, USA Received 23 June 1995; accepted
21 June 1996
Abstract
We address the issue of determining lot sizes in finite horizon environments when learning effects and/or emphasis on continuous improvement result in decreasing setup costs. Estimation of optimal lot sizes under these circumstances requires exact cost information on all future setup costs. In practice, this information is difficult to forecast. We analyze and evaluate a myopic policy which is intuitively appealing, easy to implement, and requires no information on future setup costs. We derive analytical measures for its effectiveness. Our analytical results, and computational experiments
show that the policy is a good choice for machine intensive environments which are characterized by high learning rates. Using an optimal lot sizing method developed by us, we further show that the myopic policy yields good results even when setup cost changes cannot be completely modeled by stylized learning curves used in earlier research studies. Managerial implications Keywords:
are discussed, and future research directions are indicated.
Lot sizing; Setup learning; Myopic policy
1. Introduction With the current emphasis on continuous improvement, managers have been striving to reduce setup costs (and times) in batch manufacturing operations. The fact that such efforts result in dramatic reductions in setup costs is evident from the single minute die exchange (SMDE) concept. Shingo [l] illustrates how such reductions take place in the automotive industry over long periods of time. Since setup costs are varying, more sophisticated methods which take into consideration decreasing nature of setup costs are required to determine the setup frequency, and optimal lot
* Corresponding
author.
E-mail: [email protected].
0925-5273/97/$17.00 Copyright PII SO925-5273(95)00084-9
0
sizes. Earlier papers proposed several methods to determine optimal lot sizes for finite horizons with decreasing setup costs [2,3]. They require a forecast of the learning rate, or how fast setup costs will decrease in future. This is difficult, primarily due to empirical nature of the learning curve theory [4]. Also, inappropriate aggregation in empirical data can lead to large errors in estimating the learning curve parameters [S]. Hence lot sizes found using optimal procedures are at best suspect due to these reasons. In this study, we analyze, and evaluate a myopic lot sizing policy for finite horizons. It does not require any information about future setup costs. Also, it can be used in any environment with decreasing setup costs while the policies suggested by earlier researchers [2,3] require that setup costs
1997 Elsevier Science B.V. All rights reserved
158
R. Rachamadugu,
C. L. TanlInt. J. Production
decrease as per the classical power function, Since the myopic policy analyzed in this paper is similar to the classical square root formula, it is likely to find easy acceptance among practitioners. We derive approximate analytical measures for its effectiveness. We also develop an enumerative method for determining optimal lot sizes when the setup costs change nonmonotonically. This procedure is used to validate the effectiveness of the myopic policy when setup cost reductions do not occur exactly as suggested by stylized learning curves. Our paper is organized as follows. In the next section, we provide a detailed description of the myopic policy, and illustrate it with a numerical example. Section 3 provides analytical results on its effectiveness. Section 4 provides computational results on the exact performance of the myopic policy in finite horizons. Section 5 describes the optimal procedure for determining lot sizes when setup cost trends are nonmonotonic. In the same section, we also present computational results on the effectiveness of the myopic policy when setup cost reductions are characterized by some degree of randomness, and do not exactly conform to the learning curve. Finally, Section 6 provides conclusions and future research directions.
2. Myopic policy Under the myopic lot sizing policy analyzed in this paper, number of units in the lotfor any setup is set so that the inventory carrying cost incurred during its depletion equals the setup cost for that lot. Hence the
lot size for any setup will equal the economic order quantity corresponding to that setup cost, i.e., lot size for the ith setup is given by the formula [6] Qi =
J5os,/c.
Though the myopic policy is analyzed here for finite horizons, it is very similar to the classical square root formula used in infinite horizon environments, Notation used in the paper is shown in Table 1. When the setup costs are large, myopic policy results in large lot sizes. As setup costs decrease, so do lot sizes. Earlier, Chand [7] and
Economics 48 (1997) 157-165
Table I Notation TIC* TIC,,, %I? N* R
s, C :!
Q’ Y Tg.
HG b
Ti WA, 4
Total cost for the optimal policy Total cost for the myopic policy Number of setups in the myopic policy Optimal number of setups Learning rate (also known as the progress ratio) Cost of ith setup Holding cost per unit per period Demand rate, units/period Number of units produced in ith setup Optimum lot size Length of the planning horizon Cumulative sum of setup costs, x1=, Si Total holding cost in a policy with n equal lot sizes (or setups) = DYZC/2n Learning index (= In R/in 2) Depletion time for the units produced in the ith batch (= Q,/o) Uniform random variate with minimum A, and maximum B
Replogle [3] alluded to similar policy, but did not explore it in detail. Rachamadugu [S, 91 discussed the worst case performance of the myopic policy. It was shown that the policy could result in costs as high as 100% above the optimum. However, in order to make an informed decision, managers should be concerned with not only extreme (worst case) performance, but also the average performance of decision rules. Results derived in this paper suggest that our myopic policy yields good results across a wide range of learning rates encountered in practice. A weakness of the myopic policy is that it results in unequal lot sizes. However, this is more than offset by its beneficial features. It does not require any information regarding future setup costs. For new production processes, comparable processes may not exist. Even when future setup cost patterns can be extracted from past experience, learning parameter estimation is difficult due to uncertainty and empirical nature of data [S]. Myopic policy needs only the current setup cost, which is easy to determine. Also, it results in large lot sizes when setup costs are high. Hence it results in low variances during the product life cycle. This would also be favored by most manufacturing managers whose incentives tend to be linked to cost variances. An
R. Rachamadugu,
C. L. Tan JInt. J. Production Economics
interesting feature of the myopic policy is its similarity to the classical economic order quantity, and also the well known part period balancing heuristic rule [lo]. In both these rules, inventory carrying cost for any lot either equals or is just about the same as the setup cost incurred for that lot. The myopic policy also possesses this characteristic.
2.1. Numerical
illustration
Consider the following numerical example [3]. A product has demand rate of 10 000 units per year. The initial setup cost is $12. Inventory carrying cost is $1 per unit per year. The demand lasts for 0.373 yr. A learning rate (R) of 80% is applicable i.e., when the cumulative number of setups doubles, setup cost reduces to 80% of the median setup cost. For example, setup cost for the second setup will be $9.60, and the setup cost for the fourth setup will be $7.68. Setup cost for the ith setup (Si) is given by the following power function [ 111: Si = Slih,
(2)
where b is known as the learning index. This characterization of learning has been very widely used in industry over several decades [ 111. It can easily be verified that the learning rate (R), and b are related by the following expression: b = In R/in 2.
Table 2 Lot sizes for the illustrative Setup i cost Si
Setup number
12.00 9.60 8.43 7.68 7.15 6.14 6.41 6.14 5.92 5.12
1 2 3 4 5 6 7 8 9 IO
problem
Production lot size (units)
Replenishment interval, Qi/D
Cumulative time, c Qi/D
489.898 438.178 410.492 391.918 378.091 367.156 358.158 350.542 343.959 200.000
0.049 0.044 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.020
0.049 0.093 0.134 0.173 0.211 0.248 0.284 0.319 0.353 0.373
This illustration shows that the myopic policy performs well even though it does not require any information on future setup costs. Next, we analytically show that it provides such good results in most practical situations.
3. Analytical evaluation of the myopic policy Myopic policy requires that the inventory carrying cost incurred should equal the setup cost in all setups (except possibly the last setup). Hence, Si = DTZ C/2,
(3)
Substituting the values, b in our problem equals - 0.3219. Lot size (Qi) for any setup in the myopic policy is simply found by using the economic order quantity formula given in (1). As shown in Table 2, we use lot sizes suggested in (1) until the end of the finite planning period. Since the horizon ends at 0.373 yr, the last batch (i = 10) consists of only 200 units, instead of a lot size (using a setup cost of $5.72) of 338.175 as suggested by (1). Total cost of the myopic policy for the above example is $147.93 ($75.79 setup costs and $72.14 inventory carrying costs). Optimal policy [2] results in 11 setups, a constant lot size of 332 units, a cost of $143.25 ($81.33 setup costs and $61.92 inventory carrying costs). Thus, the total cost of the myopic policy is only 3.3% more than the optimal value.
159
48 (I 997) 157-165
(4)
where Ti is the depletion time for the ith batch. Rewriting (4) Ti = J2SiIDC.
(5)
However, it is well known that learning effects can be characterized by the power function [l 11. Cheng [2] and Replogle [3] used the power function to characterize setup cost reductions. Using (2) in (5). T,
=
’
J ~
DC
iO.5h,
(6)
Let n, be the number of setups in the myopic policy. Using (6) j’=
5 i=l
Ti=
160
R. Rachamadugu,
C. L. TanlInt. J. Production
Rewriting (7)
J
i$li”.5b= Y $.1
(8)
However, Cheng [2] suggested using the following approximation:
Economics
Performance of the myopic policy relative to the optimum is given by the ratio TIC,/TIC*. If this ratio is 1, then the myopic policy is optimal. Clearly, it cannot be less than 1. Using (12) and (14), [2/(b + l)] [(b + 2)/21ztb+ i)lcb+‘)
TIC,,, TIC*=
1 + Cl/@ +
m
r,f(x) = jf(4dx
=
(9)
0
Using (9) in (8),
(10) Eq. (10) can be rewritten as
(11) Note that in the myopic policy, setup cost equals the inventory carrying cost for each lot, except possibly the last lot. Hence total cost of the myopic policy (TIC,) is approximately given by “In TIC,,, x 2
s
S1ibdi
0
Z(b+
1)
b+l
_b:I(b:2)s+i(DY21C)~~:“b+“.
(12) Next, we determine the cost of optimal policy. Using the approximation shown in (9), Q* is given by PI
Q* = (2SlD;1Yb)‘i’b+2~.
(13)
Using (2), (9) and (13), it can be seen that the optimal cost (TIC*), is given by TIC*
=
F
+ DySi F x
i=l
+ ‘TSi & 0
48 (1997) 157-165
111
f, + 2 bl@+2) H
b
(15)
’
Eq. (15) implies that the effectiveness of the myopic policy is dependent on the learning rate. When b = 0 (no learning at all), the myopic policy is nearly optimal, except for the “end-of-horizon” effects. In fact, when b = 0 and Y --f co, myopic policy reduces to the classical economic order quantity formula. Table 3 shows the performance of the myopic policy for different learning rates as predicted by (15). Empirical evidence [ 1l] suggests that learning rates fall in the range of 0.65-0.95. While machine intensive environments have very high learning rates, most operations with manual content range around 0.8 [11-131. Results in Table 3 show that the myopic policy performs well, particularly at high learning rates. For example, when learning rate is 0.9, our analytical results suggest that the myopic policy results in only 0.65% excess cost. Hence myopic policy is a good choice when learning rates are high. However, when learning rates are low (say 0.65), myopic policy results in costs as high as 18.27% above optimum. However, empirical evidence suggests that such low learning rates are very rare [l 1, Fig. 21. Table 3 Analytical estimates of myopic lot sizing policy performance Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65
b = In R/in 2
-
0.0740 0.1520 0.2345 0.3219 0.4150 0.5146 0.6215
TIC,/TIC*
1.0014 1.0065 1.0167 1.0342 1.0628 1.1085 1.1827
% Excess cost = (TIC,/TIC* xl00
- 1)
0.14 0.65 1.67 3.42 6.28 10.85 18.27
Note: Bold entries indicate results for very low learning rates.
R. Rachamadugu, C. L. Tan/M
J. Production Economics 48 (1997) 157-165
4. Computational experiments It should be noted that our analytical expression for the effectiveness of myopic policy (Eq. (15)) is an approximation due to two reasons. First, we use a continuous approximation for a discrete variable (Eq. (9)). Second, it disregards the fact that the last batch in the myopic policy could possibly be smaller than the economic lot size for that setup. In order to further validate our results, we conducted computational experiments to determine the exact performance of the myopic policy. The parameters chosen for our study are given below: R (learning rate) = 0.65-0.95 (step 0.05)
S1 (initial setup cost) = $100 C = $1 per unit per period D (demand) = 10000; 100000; 1000000 Y (horizon length) = 1 period These parameter combinations are consistent with similar studies on the effects of learning on lot sizes 171. In the simulation study, myopic policy was implemented as described in Section 2. For each setup, the lot size was determined as per the EOQ formula using its setup cost. However, the lot size for the last lot was reduced, if necessary, to the quantity required till the end of horizon. The total cost of implementing this myopic policy (TIC,,,) is made up of two components: (1) sum of all setup costs, and (2) sum of the inventory carrying costs for each lot. Optimum cost in each instance was found using Cheng’s [2] policy. However, the total number of
161
setups may not be an integer (since DY/Q*, where Q* is given by (13)). Therefore, the optimal total cost (TIC*) was found by taking the minimum of two possible total costs - rounding up or rounding down the number of setups (DY/Q*) to the nearest integer values. Both myopic and optimal policies were coded in Fortran 77 and run on VAX6420 minicomputer. Table 4 provides results of our computational experiments. For each learning rate, we calculated b (using Eq. (3)), and used it to derive the analytical estimate of the performance of myopic policy. This is shown in the second column of Table 4. Actual performance of the myopic policy, based on computational experiments, is shown in columns 3-5 of the table. It is clear from the table that at low learning rates, analytical estimates are biased against the myopic policy. Fig. 1 provides a graphical perspective on our results. It compares theoretical estimates of the myopic policy performance against its true performance (from simulation). It shows the percent excess cost resulting from the use of myopic policy against learning rate. The line graph shows the analytical estimates (Eq. (15)), which are independent of D. The bar graphs show the actual percent excess cost resulting from the use of myopic policy. It is clear from Fig. 1 that at low learning rates, actual performance of the myopic policy is significantly better than the analytical prediction. Though its true performance is slightly worse than the analytical forecast at high learning rates (0.8-0.95), actual performance is good (about 3%
Table 4 Analytical and exact evaluation of the myopic policy performance Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65 Note:
Analytical estimate (% excess cost) = (Eq. (15) - 1) x 100
% Excess cost due to use of myopic policy = (TIC,/TIC*
- 1) x 100
0=10ooo
100000
1000000
0.14 0.65 1.67 3.42 6.28 10.85 18.27
1.92 0.07 3.18 1.35 3.50 5.37 7.57
0.63 0.15 0.76 1.93 3.61 6.15 9.95
Bold entries cells indicate results for low learning rates.
0.15 0.27 0.85 2.09 4.00 6.54 10.60
162
R. Rachamadugu,
0.66
C. L. Tan/M.
0.7
.I Production
0.72
0.0
Economics
48 (1997) 157-165
0.26
0.0
0.22
R (Learning rate)
q D=
10,ooO
q
D = 100,000
q D = 1,OOO,OOO
-
hcrlytical6huW
Note: Bar graphs show the actual performance of the myopic policy.
Linegraph denotes the analytical&mate of myopic performance given by equation (15). Fig. 1. Performance of myopic policy.
above the optimum). Since machine intensive operations are characterized by high learning rates [14], myopic policy is an excellent choice for such environments.
5. Effects of random variations
There is significant amount of empirical evidence [4, 15-171 which suggests that costs do not decrease exactly as forecast by stylized learning curve formulations such as the well-known power function. Hence it is important to verify if the myopic policy will continue to yield good results even when setup cost decreases are somewhat random.
When there is some randomness in setup reduction patterns, evaluation of the myopic policy creates two difficulties. First, determination of optimal solution is not easy. We can use neither Cheng’s [2] approach (which is valid only when setup costs follow the power function) nor Chand’s [7] method which assumes nonincreasing setup costs. Second, there does not appear to be an analytically tractable solution. Hence we empirically evaluated the performance of the myopic policy. We developed a generic procedure to determine the optimal number of setups when setup costs changes are nonmonotonic. Clearly, this procedure is applicable to any given sequence of setup costs (S,, SZ, S3, . . ). Details of this procedure are provided in Appendix A.
R. Rachamadugu,
C. L. TanJlnt. J. Production
In order to study the effectiveness of the myopic policy in the presence of randomness in setup cost reductions, we conducted additional experiments. We generated setup costs for each setup as follows: s; = S,ih + rY(- O.lSi i*, O.lSr i”).
Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65
163
48 (1997) 157-165
Table 5 Exact evaluation
(16)
Sl represents the randomly generated ith setup cost. U(A, B) represents a random variate generated with a minimum value of A and a maximum value of B. Eq. (16) allows 20% variation around the expected cost. Clearly, the realized sequence of setup costs (Si, S;, S;, . . . ) in any particular instance do not necessarily follow the power function, nor can it be expected to decrease monotonicafly. Methodology provided in Appendix A was used to find the optimal solution. In this additional set of experiments, we used the same experimental settings as shown in Section 4. However, for each setting, we replicated 20 instances with setup costs generated as per (16). In each instance, myopic policy was evaluated using the setup cost generated from (16). Computational results are shown in Table 5. It shows the percentage excess cost (over the optimum) due to using myopic policy. Since each cell consists of 20 independent replications, we report in Table 5 the mean, and 95% confidence width for the mean (using t-statistic). These results show that the myopic policy continues to yield excellent results even when setup costs decreases as characterized by the learning curves, and also some degree of randomness. An interesting question worthy of investigation is the effect of random variations on the performance of the policy vis-a-vis its performance if the setup costs decrease exactly as predicted by the widely used power function. This analysis is shown in Fig. 2. It shows the 95% confidence interval (as vertical I), and the mean value for percentage excess cost due to the use of myopic policy when setup cost decreases are arbitrary at various learning rates. The same graph also shows the performance of the myopic policy when setup costs decrease exactly as per power function. It is clear from Fig. 2 (reported for D = 100000) that at high learning rate levels, deterioration in the performance of myopic policy is marginal. Results for the cases D = 10000and 100000 are similar. Hence it is
Economics
of the myopic
policy performance
Average % excess cost due to use of myopic policy = (TIC,/TIC* - 1) x 100 D=lOOOO
100000
1.951 0.430 3.083 1.610 3.382 5.744 7.570
0.860 0.888 1.724 2.427 4.486 7.261 11.462
(0.0280) (0.0625) (0.2645) (0.3920) (0.2325) (0.2465) (0.2105)
1000000 (0.1395) (0.2280) (0.2660) (0.1005) (0.1245) (0.0730) (0.1415)
0.442 0.737 1.519 2.972 5.309 8.837 14.276
(0.0815) (0.0710) (0.0715) (0.0545) (0.0580) (0.0550) (0.0740)
Nore: Entry in each cell represents the average of twenty random replications. The term in brackets indicates 95% confidence width.
reasonable to conclude that the myopic policy yields excellent results even when there are random variations in setup cost patterns for high learning rates.
6. Conclusions Japanese manufacturing methods, learning effects, current emphasis on total quality management (continuous improvement) and shorter product life cycles result in decreasing setup costs in a finite period of time. We discussed and analyzed a policy for lot sizing under these environments. Though optimal policies can help to reduce the costs in such situations, their use is difficult because of the need for information regarding future setup cost reduction patterns, and also length of the finite planning horizon. Continuing changes in technology and increasing rate of new product introductions make estimating both these factors very difficult, if not impossible. The myopic policy analyzed in this paper requires no such information for its implementation. It is also intuitively appealing, and easy to understand. Our analysis shows that it performs well, yielding solutions about 3% above the optimum in average sense for high learning rate environments such as machine intensive operations. Further, the myopic policy has an interesting feature - it results in low cost variance during the
R. Rachamadugu, C. L. TanlInt. J. Production Economics 48 (1997) 157-165
164
a A
m A
3
1.0 a0
A
I
I
I
I
I
0.70
a75
0.80
0.85
A a9o
I a65 I
95% C.I.
0
Mean
A
2 a95
Detemkistic
random
Fig. 2. Performance of myopic policy under random variation (% above optimum for D = 100000).
product life cycle. Hence it is more likely to be acceptable to practitioners such as production managers whose incentives are generally linked to cost variances. We also developed an optimization procedure to determine optimal lot sizes when setup costs may decrease nonmonotonically. This procedure was used to determine how well the myopic policy performs when stylized mathematical formulations used in earlier literature cannot fully characterize setup cost reductions. Our computational results show that the myopic policy continues to yield good results even when there are random variations in setup cost reductions.
Acknowledgements We thank the two anonymous reviewers and the editor for their constructive and helpful comments on the earlier version of our paper.
Appendix A. Procedure for determining optimal lot sizes for nonmonotonic setup costs Here we present a procedure for determining the optimal number of setups when setup costs vary nonmonotonically. It is based on a stopping rule. Let n be the number of setups used in a policy. Total cost of the policy is given by DY2C/2n
+ i
SC.
(Al)
i=l
Clearly, the minimum increase in setup costs due to additional setups is S,+ i. Also, the maximum reduction in the inventory carrying costs is D Y ‘C/2n. Hence, the search for the optimal number of setups can be terminated if Sn+l 2 DY2C/2n.
642)
Fig. 3 below provides a flowchart for determining the optimal number of setups using the above stopping rule.
R. Rachamadugu, C. L. Tan/M J Production Economics 48 (1997) 157-165
n c
1,
TS, c
S,,
HC!, c DYaC/2n,
N’ c 1,
TIC’
F TS,
165
+ HC,
ncn+l
No
N'
c
n,
TIC’
c TS,
+ HC,
STOP Note:
Notation is explained in Table 1.
Fig. 3. Flowchart for determining optimal policy under arbitrary setup costs
Cl1 Shingo, S., 1989. A Study of the Toyota Production System from an Industrial Engineering Viewpoint, revised version (English translation). Productivity Press, Cambridge, MA. VI Cheng, T.C.E., 1991. An EOQ model with learning effects on setup. Prod. Inv. Mgmt., 1st Quart., 83-84. c31Replogle, S.H., 1988. The strategic use of smaller lot sizes through a new EOQ model. Prod. Inv. Mgmt. 3rd Quart., 41-44. c41Vigil, D.P. and Sarper, H., 1994. Estimating the effects of parameter variability on learning curve models predictions. Int. J. Prod. Econom., 34: 185-200. c51Day, G. and Montgomery, D., 1985. Experience Curve: Evidence, Empirical Issues, and Applications. Management Science Institute, Cambridge, MA. pp. 213-238. CC1Nahmias, S., 1989. Production and Operations Analysis. Irwin (Richard D.) Homewood, IL. [71 Chand, SC., 1989. Lot sizes and setup frequency with learning in setups and process quality. Eur. J. Oper. Res., 42: 190-202. PI Rachamadugu, R., 1994. Performance analysis of a myopic lot sizing policy. Inst. Indust. Engrs. Trans., 26: 85-91.
c91Rachamadugu, R. and Schreiber, T.J., 1995. Optimal and heuristic policies for lot sizing with learning in setups. J. Oper. Mgmt., 13: 229-245. Cl01Vollman, T.E., Berry, W.L. and Whybark, D.C., 1992. Manufacturing Planning and Control Systems. 3rd ed. Irwin (Richard D.), Homewood, IL. Cl11Argote, L. and Epple, D., 1990. Learning curves in manufacturing. Science, 247: 920-924. Cl21Yelle, L.E. 1979. The learning curve: historical review and comprehensive survey. Dec. Sci., 10: 302-332. Cl31Hirschmann, W.B. 1964. Profit from the learning curve. Har. Bus. Rev., 42(l): 125-139. Cl41Meredith, J.R. 1987. The Management of Operations. Wiley, New York. Cl51Badiru, A.B., 1992. Computational study of univariate and multivariate learning curve models. IEEE Trans. Eng. Mgmt., 39: 176-188. Cl61Buck, J.R. and Cheng, S.W.J., 1993. Instructions and feedback effects on speed and accuracy with different learning curve models. IIE Trans., 25: 34-47. Cl71Reis, D. A., 1991. Learning curves in food services. J. Oper. Res. Sot., 42: 623-629.
Int. J. Production
Economics
48 (1997) 157-165
Policies for lot sizing with setup learning Ram Rachamadugu”,
Chong Leng Tan
ISOM Department, College of Business Administration, University of Toledo, Toledo, OH 43606, USA Received 23 June 1995; accepted
21 June 1996
Abstract
We address the issue of determining lot sizes in finite horizon environments when learning effects and/or emphasis on continuous improvement result in decreasing setup costs. Estimation of optimal lot sizes under these circumstances requires exact cost information on all future setup costs. In practice, this information is difficult to forecast. We analyze and evaluate a myopic policy which is intuitively appealing, easy to implement, and requires no information on future setup costs. We derive analytical measures for its effectiveness. Our analytical results, and computational experiments
show that the policy is a good choice for machine intensive environments which are characterized by high learning rates. Using an optimal lot sizing method developed by us, we further show that the myopic policy yields good results even when setup cost changes cannot be completely modeled by stylized learning curves used in earlier research studies. Managerial implications Keywords:
are discussed, and future research directions are indicated.
Lot sizing; Setup learning; Myopic policy
1. Introduction With the current emphasis on continuous improvement, managers have been striving to reduce setup costs (and times) in batch manufacturing operations. The fact that such efforts result in dramatic reductions in setup costs is evident from the single minute die exchange (SMDE) concept. Shingo [l] illustrates how such reductions take place in the automotive industry over long periods of time. Since setup costs are varying, more sophisticated methods which take into consideration decreasing nature of setup costs are required to determine the setup frequency, and optimal lot
* Corresponding
author.
E-mail: [email protected].
0925-5273/97/$17.00 Copyright PII SO925-5273(95)00084-9
0
sizes. Earlier papers proposed several methods to determine optimal lot sizes for finite horizons with decreasing setup costs [2,3]. They require a forecast of the learning rate, or how fast setup costs will decrease in future. This is difficult, primarily due to empirical nature of the learning curve theory [4]. Also, inappropriate aggregation in empirical data can lead to large errors in estimating the learning curve parameters [S]. Hence lot sizes found using optimal procedures are at best suspect due to these reasons. In this study, we analyze, and evaluate a myopic lot sizing policy for finite horizons. It does not require any information about future setup costs. Also, it can be used in any environment with decreasing setup costs while the policies suggested by earlier researchers [2,3] require that setup costs
1997 Elsevier Science B.V. All rights reserved
158
R. Rachamadugu,
C. L. TanlInt. J. Production
decrease as per the classical power function, Since the myopic policy analyzed in this paper is similar to the classical square root formula, it is likely to find easy acceptance among practitioners. We derive approximate analytical measures for its effectiveness. We also develop an enumerative method for determining optimal lot sizes when the setup costs change nonmonotonically. This procedure is used to validate the effectiveness of the myopic policy when setup cost reductions do not occur exactly as suggested by stylized learning curves. Our paper is organized as follows. In the next section, we provide a detailed description of the myopic policy, and illustrate it with a numerical example. Section 3 provides analytical results on its effectiveness. Section 4 provides computational results on the exact performance of the myopic policy in finite horizons. Section 5 describes the optimal procedure for determining lot sizes when setup cost trends are nonmonotonic. In the same section, we also present computational results on the effectiveness of the myopic policy when setup cost reductions are characterized by some degree of randomness, and do not exactly conform to the learning curve. Finally, Section 6 provides conclusions and future research directions.
2. Myopic policy Under the myopic lot sizing policy analyzed in this paper, number of units in the lotfor any setup is set so that the inventory carrying cost incurred during its depletion equals the setup cost for that lot. Hence the
lot size for any setup will equal the economic order quantity corresponding to that setup cost, i.e., lot size for the ith setup is given by the formula [6] Qi =
J5os,/c.
Though the myopic policy is analyzed here for finite horizons, it is very similar to the classical square root formula used in infinite horizon environments, Notation used in the paper is shown in Table 1. When the setup costs are large, myopic policy results in large lot sizes. As setup costs decrease, so do lot sizes. Earlier, Chand [7] and
Economics 48 (1997) 157-165
Table I Notation TIC* TIC,,, %I? N* R
s, C :!
Q’ Y Tg.
HG b
Ti WA, 4
Total cost for the optimal policy Total cost for the myopic policy Number of setups in the myopic policy Optimal number of setups Learning rate (also known as the progress ratio) Cost of ith setup Holding cost per unit per period Demand rate, units/period Number of units produced in ith setup Optimum lot size Length of the planning horizon Cumulative sum of setup costs, x1=, Si Total holding cost in a policy with n equal lot sizes (or setups) = DYZC/2n Learning index (= In R/in 2) Depletion time for the units produced in the ith batch (= Q,/o) Uniform random variate with minimum A, and maximum B
Replogle [3] alluded to similar policy, but did not explore it in detail. Rachamadugu [S, 91 discussed the worst case performance of the myopic policy. It was shown that the policy could result in costs as high as 100% above the optimum. However, in order to make an informed decision, managers should be concerned with not only extreme (worst case) performance, but also the average performance of decision rules. Results derived in this paper suggest that our myopic policy yields good results across a wide range of learning rates encountered in practice. A weakness of the myopic policy is that it results in unequal lot sizes. However, this is more than offset by its beneficial features. It does not require any information regarding future setup costs. For new production processes, comparable processes may not exist. Even when future setup cost patterns can be extracted from past experience, learning parameter estimation is difficult due to uncertainty and empirical nature of data [S]. Myopic policy needs only the current setup cost, which is easy to determine. Also, it results in large lot sizes when setup costs are high. Hence it results in low variances during the product life cycle. This would also be favored by most manufacturing managers whose incentives tend to be linked to cost variances. An
R. Rachamadugu,
C. L. Tan JInt. J. Production Economics
interesting feature of the myopic policy is its similarity to the classical economic order quantity, and also the well known part period balancing heuristic rule [lo]. In both these rules, inventory carrying cost for any lot either equals or is just about the same as the setup cost incurred for that lot. The myopic policy also possesses this characteristic.
2.1. Numerical
illustration
Consider the following numerical example [3]. A product has demand rate of 10 000 units per year. The initial setup cost is $12. Inventory carrying cost is $1 per unit per year. The demand lasts for 0.373 yr. A learning rate (R) of 80% is applicable i.e., when the cumulative number of setups doubles, setup cost reduces to 80% of the median setup cost. For example, setup cost for the second setup will be $9.60, and the setup cost for the fourth setup will be $7.68. Setup cost for the ith setup (Si) is given by the following power function [ 111: Si = Slih,
(2)
where b is known as the learning index. This characterization of learning has been very widely used in industry over several decades [ 111. It can easily be verified that the learning rate (R), and b are related by the following expression: b = In R/in 2.
Table 2 Lot sizes for the illustrative Setup i cost Si
Setup number
12.00 9.60 8.43 7.68 7.15 6.14 6.41 6.14 5.92 5.12
1 2 3 4 5 6 7 8 9 IO
problem
Production lot size (units)
Replenishment interval, Qi/D
Cumulative time, c Qi/D
489.898 438.178 410.492 391.918 378.091 367.156 358.158 350.542 343.959 200.000
0.049 0.044 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.020
0.049 0.093 0.134 0.173 0.211 0.248 0.284 0.319 0.353 0.373
This illustration shows that the myopic policy performs well even though it does not require any information on future setup costs. Next, we analytically show that it provides such good results in most practical situations.
3. Analytical evaluation of the myopic policy Myopic policy requires that the inventory carrying cost incurred should equal the setup cost in all setups (except possibly the last setup). Hence, Si = DTZ C/2,
(3)
Substituting the values, b in our problem equals - 0.3219. Lot size (Qi) for any setup in the myopic policy is simply found by using the economic order quantity formula given in (1). As shown in Table 2, we use lot sizes suggested in (1) until the end of the finite planning period. Since the horizon ends at 0.373 yr, the last batch (i = 10) consists of only 200 units, instead of a lot size (using a setup cost of $5.72) of 338.175 as suggested by (1). Total cost of the myopic policy for the above example is $147.93 ($75.79 setup costs and $72.14 inventory carrying costs). Optimal policy [2] results in 11 setups, a constant lot size of 332 units, a cost of $143.25 ($81.33 setup costs and $61.92 inventory carrying costs). Thus, the total cost of the myopic policy is only 3.3% more than the optimal value.
159
48 (I 997) 157-165
(4)
where Ti is the depletion time for the ith batch. Rewriting (4) Ti = J2SiIDC.
(5)
However, it is well known that learning effects can be characterized by the power function [l 11. Cheng [2] and Replogle [3] used the power function to characterize setup cost reductions. Using (2) in (5). T,
=
’
J ~
DC
iO.5h,
(6)
Let n, be the number of setups in the myopic policy. Using (6) j’=
5 i=l
Ti=
160
R. Rachamadugu,
C. L. TanlInt. J. Production
Rewriting (7)
J
i$li”.5b= Y $.1
(8)
However, Cheng [2] suggested using the following approximation:
Economics
Performance of the myopic policy relative to the optimum is given by the ratio TIC,/TIC*. If this ratio is 1, then the myopic policy is optimal. Clearly, it cannot be less than 1. Using (12) and (14), [2/(b + l)] [(b + 2)/21ztb+ i)lcb+‘)
TIC,,, TIC*=
1 + Cl/@ +
m
r,f(x) = jf(4dx
=
(9)
0
Using (9) in (8),
(10) Eq. (10) can be rewritten as
(11) Note that in the myopic policy, setup cost equals the inventory carrying cost for each lot, except possibly the last lot. Hence total cost of the myopic policy (TIC,) is approximately given by “In TIC,,, x 2
s
S1ibdi
0
Z(b+
1)
b+l
_b:I(b:2)s+i(DY21C)~~:“b+“.
(12) Next, we determine the cost of optimal policy. Using the approximation shown in (9), Q* is given by PI
Q* = (2SlD;1Yb)‘i’b+2~.
(13)
Using (2), (9) and (13), it can be seen that the optimal cost (TIC*), is given by TIC*
=
F
+ DySi F x
i=l
+ ‘TSi & 0
48 (1997) 157-165
111
f, + 2 bl@+2) H
b
(15)
’
Eq. (15) implies that the effectiveness of the myopic policy is dependent on the learning rate. When b = 0 (no learning at all), the myopic policy is nearly optimal, except for the “end-of-horizon” effects. In fact, when b = 0 and Y --f co, myopic policy reduces to the classical economic order quantity formula. Table 3 shows the performance of the myopic policy for different learning rates as predicted by (15). Empirical evidence [ 1l] suggests that learning rates fall in the range of 0.65-0.95. While machine intensive environments have very high learning rates, most operations with manual content range around 0.8 [11-131. Results in Table 3 show that the myopic policy performs well, particularly at high learning rates. For example, when learning rate is 0.9, our analytical results suggest that the myopic policy results in only 0.65% excess cost. Hence myopic policy is a good choice when learning rates are high. However, when learning rates are low (say 0.65), myopic policy results in costs as high as 18.27% above optimum. However, empirical evidence suggests that such low learning rates are very rare [l 1, Fig. 21. Table 3 Analytical estimates of myopic lot sizing policy performance Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65
b = In R/in 2
-
0.0740 0.1520 0.2345 0.3219 0.4150 0.5146 0.6215
TIC,/TIC*
1.0014 1.0065 1.0167 1.0342 1.0628 1.1085 1.1827
% Excess cost = (TIC,/TIC* xl00
- 1)
0.14 0.65 1.67 3.42 6.28 10.85 18.27
Note: Bold entries indicate results for very low learning rates.
R. Rachamadugu, C. L. Tan/M
J. Production Economics 48 (1997) 157-165
4. Computational experiments It should be noted that our analytical expression for the effectiveness of myopic policy (Eq. (15)) is an approximation due to two reasons. First, we use a continuous approximation for a discrete variable (Eq. (9)). Second, it disregards the fact that the last batch in the myopic policy could possibly be smaller than the economic lot size for that setup. In order to further validate our results, we conducted computational experiments to determine the exact performance of the myopic policy. The parameters chosen for our study are given below: R (learning rate) = 0.65-0.95 (step 0.05)
S1 (initial setup cost) = $100 C = $1 per unit per period D (demand) = 10000; 100000; 1000000 Y (horizon length) = 1 period These parameter combinations are consistent with similar studies on the effects of learning on lot sizes 171. In the simulation study, myopic policy was implemented as described in Section 2. For each setup, the lot size was determined as per the EOQ formula using its setup cost. However, the lot size for the last lot was reduced, if necessary, to the quantity required till the end of horizon. The total cost of implementing this myopic policy (TIC,,,) is made up of two components: (1) sum of all setup costs, and (2) sum of the inventory carrying costs for each lot. Optimum cost in each instance was found using Cheng’s [2] policy. However, the total number of
161
setups may not be an integer (since DY/Q*, where Q* is given by (13)). Therefore, the optimal total cost (TIC*) was found by taking the minimum of two possible total costs - rounding up or rounding down the number of setups (DY/Q*) to the nearest integer values. Both myopic and optimal policies were coded in Fortran 77 and run on VAX6420 minicomputer. Table 4 provides results of our computational experiments. For each learning rate, we calculated b (using Eq. (3)), and used it to derive the analytical estimate of the performance of myopic policy. This is shown in the second column of Table 4. Actual performance of the myopic policy, based on computational experiments, is shown in columns 3-5 of the table. It is clear from the table that at low learning rates, analytical estimates are biased against the myopic policy. Fig. 1 provides a graphical perspective on our results. It compares theoretical estimates of the myopic policy performance against its true performance (from simulation). It shows the percent excess cost resulting from the use of myopic policy against learning rate. The line graph shows the analytical estimates (Eq. (15)), which are independent of D. The bar graphs show the actual percent excess cost resulting from the use of myopic policy. It is clear from Fig. 1 that at low learning rates, actual performance of the myopic policy is significantly better than the analytical prediction. Though its true performance is slightly worse than the analytical forecast at high learning rates (0.8-0.95), actual performance is good (about 3%
Table 4 Analytical and exact evaluation of the myopic policy performance Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65 Note:
Analytical estimate (% excess cost) = (Eq. (15) - 1) x 100
% Excess cost due to use of myopic policy = (TIC,/TIC*
- 1) x 100
0=10ooo
100000
1000000
0.14 0.65 1.67 3.42 6.28 10.85 18.27
1.92 0.07 3.18 1.35 3.50 5.37 7.57
0.63 0.15 0.76 1.93 3.61 6.15 9.95
Bold entries cells indicate results for low learning rates.
0.15 0.27 0.85 2.09 4.00 6.54 10.60
162
R. Rachamadugu,
0.66
C. L. Tan/M.
0.7
.I Production
0.72
0.0
Economics
48 (1997) 157-165
0.26
0.0
0.22
R (Learning rate)
q D=
10,ooO
q
D = 100,000
q D = 1,OOO,OOO
-
hcrlytical6huW
Note: Bar graphs show the actual performance of the myopic policy.
Linegraph denotes the analytical&mate of myopic performance given by equation (15). Fig. 1. Performance of myopic policy.
above the optimum). Since machine intensive operations are characterized by high learning rates [14], myopic policy is an excellent choice for such environments.
5. Effects of random variations
There is significant amount of empirical evidence [4, 15-171 which suggests that costs do not decrease exactly as forecast by stylized learning curve formulations such as the well-known power function. Hence it is important to verify if the myopic policy will continue to yield good results even when setup cost decreases are somewhat random.
When there is some randomness in setup reduction patterns, evaluation of the myopic policy creates two difficulties. First, determination of optimal solution is not easy. We can use neither Cheng’s [2] approach (which is valid only when setup costs follow the power function) nor Chand’s [7] method which assumes nonincreasing setup costs. Second, there does not appear to be an analytically tractable solution. Hence we empirically evaluated the performance of the myopic policy. We developed a generic procedure to determine the optimal number of setups when setup costs changes are nonmonotonic. Clearly, this procedure is applicable to any given sequence of setup costs (S,, SZ, S3, . . ). Details of this procedure are provided in Appendix A.
R. Rachamadugu,
C. L. TanJlnt. J. Production
In order to study the effectiveness of the myopic policy in the presence of randomness in setup cost reductions, we conducted additional experiments. We generated setup costs for each setup as follows: s; = S,ih + rY(- O.lSi i*, O.lSr i”).
Learning rate (R)
0.95 0.90 0.85 0.80 0.75 0.70 0.65
163
48 (1997) 157-165
Table 5 Exact evaluation
(16)
Sl represents the randomly generated ith setup cost. U(A, B) represents a random variate generated with a minimum value of A and a maximum value of B. Eq. (16) allows 20% variation around the expected cost. Clearly, the realized sequence of setup costs (Si, S;, S;, . . . ) in any particular instance do not necessarily follow the power function, nor can it be expected to decrease monotonicafly. Methodology provided in Appendix A was used to find the optimal solution. In this additional set of experiments, we used the same experimental settings as shown in Section 4. However, for each setting, we replicated 20 instances with setup costs generated as per (16). In each instance, myopic policy was evaluated using the setup cost generated from (16). Computational results are shown in Table 5. It shows the percentage excess cost (over the optimum) due to using myopic policy. Since each cell consists of 20 independent replications, we report in Table 5 the mean, and 95% confidence width for the mean (using t-statistic). These results show that the myopic policy continues to yield excellent results even when setup costs decreases as characterized by the learning curves, and also some degree of randomness. An interesting question worthy of investigation is the effect of random variations on the performance of the policy vis-a-vis its performance if the setup costs decrease exactly as predicted by the widely used power function. This analysis is shown in Fig. 2. It shows the 95% confidence interval (as vertical I), and the mean value for percentage excess cost due to the use of myopic policy when setup cost decreases are arbitrary at various learning rates. The same graph also shows the performance of the myopic policy when setup costs decrease exactly as per power function. It is clear from Fig. 2 (reported for D = 100000) that at high learning rate levels, deterioration in the performance of myopic policy is marginal. Results for the cases D = 10000and 100000 are similar. Hence it is
Economics
of the myopic
policy performance
Average % excess cost due to use of myopic policy = (TIC,/TIC* - 1) x 100 D=lOOOO
100000
1.951 0.430 3.083 1.610 3.382 5.744 7.570
0.860 0.888 1.724 2.427 4.486 7.261 11.462
(0.0280) (0.0625) (0.2645) (0.3920) (0.2325) (0.2465) (0.2105)
1000000 (0.1395) (0.2280) (0.2660) (0.1005) (0.1245) (0.0730) (0.1415)
0.442 0.737 1.519 2.972 5.309 8.837 14.276
(0.0815) (0.0710) (0.0715) (0.0545) (0.0580) (0.0550) (0.0740)
Nore: Entry in each cell represents the average of twenty random replications. The term in brackets indicates 95% confidence width.
reasonable to conclude that the myopic policy yields excellent results even when there are random variations in setup cost patterns for high learning rates.
6. Conclusions Japanese manufacturing methods, learning effects, current emphasis on total quality management (continuous improvement) and shorter product life cycles result in decreasing setup costs in a finite period of time. We discussed and analyzed a policy for lot sizing under these environments. Though optimal policies can help to reduce the costs in such situations, their use is difficult because of the need for information regarding future setup cost reduction patterns, and also length of the finite planning horizon. Continuing changes in technology and increasing rate of new product introductions make estimating both these factors very difficult, if not impossible. The myopic policy analyzed in this paper requires no such information for its implementation. It is also intuitively appealing, and easy to understand. Our analysis shows that it performs well, yielding solutions about 3% above the optimum in average sense for high learning rate environments such as machine intensive operations. Further, the myopic policy has an interesting feature - it results in low cost variance during the
R. Rachamadugu, C. L. TanlInt. J. Production Economics 48 (1997) 157-165
164
a A
m A
3
1.0 a0
A
I
I
I
I
I
0.70
a75
0.80
0.85
A a9o
I a65 I
95% C.I.
0
Mean
A
2 a95
Detemkistic
random
Fig. 2. Performance of myopic policy under random variation (% above optimum for D = 100000).
product life cycle. Hence it is more likely to be acceptable to practitioners such as production managers whose incentives are generally linked to cost variances. We also developed an optimization procedure to determine optimal lot sizes when setup costs may decrease nonmonotonically. This procedure was used to determine how well the myopic policy performs when stylized mathematical formulations used in earlier literature cannot fully characterize setup cost reductions. Our computational results show that the myopic policy continues to yield good results even when there are random variations in setup cost reductions.
Acknowledgements We thank the two anonymous reviewers and the editor for their constructive and helpful comments on the earlier version of our paper.
Appendix A. Procedure for determining optimal lot sizes for nonmonotonic setup costs Here we present a procedure for determining the optimal number of setups when setup costs vary nonmonotonically. It is based on a stopping rule. Let n be the number of setups used in a policy. Total cost of the policy is given by DY2C/2n
+ i
SC.
(Al)
i=l
Clearly, the minimum increase in setup costs due to additional setups is S,+ i. Also, the maximum reduction in the inventory carrying costs is D Y ‘C/2n. Hence, the search for the optimal number of setups can be terminated if Sn+l 2 DY2C/2n.
642)
Fig. 3 below provides a flowchart for determining the optimal number of setups using the above stopping rule.
R. Rachamadugu, C. L. Tan/M J Production Economics 48 (1997) 157-165
n c
1,
TS, c
S,,
HC!, c DYaC/2n,
N’ c 1,
TIC’
F TS,
165
+ HC,
ncn+l
No
N'
c
n,
TIC’
c TS,
+ HC,
STOP Note:
Notation is explained in Table 1.
Fig. 3. Flowchart for determining optimal policy under arbitrary setup costs
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c91Rachamadugu, R. and Schreiber, T.J., 1995. Optimal and heuristic policies for lot sizing with learning in setups. J. Oper. Mgmt., 13: 229-245. Cl01Vollman, T.E., Berry, W.L. and Whybark, D.C., 1992. Manufacturing Planning and Control Systems. 3rd ed. Irwin (Richard D.), Homewood, IL. Cl11Argote, L. and Epple, D., 1990. Learning curves in manufacturing. Science, 247: 920-924. Cl21Yelle, L.E. 1979. The learning curve: historical review and comprehensive survey. Dec. Sci., 10: 302-332. Cl31Hirschmann, W.B. 1964. Profit from the learning curve. Har. Bus. Rev., 42(l): 125-139. Cl41Meredith, J.R. 1987. The Management of Operations. Wiley, New York. Cl51Badiru, A.B., 1992. Computational study of univariate and multivariate learning curve models. IEEE Trans. Eng. Mgmt., 39: 176-188. Cl61Buck, J.R. and Cheng, S.W.J., 1993. Instructions and feedback effects on speed and accuracy with different learning curve models. IIE Trans., 25: 34-47. Cl71Reis, D. A., 1991. Learning curves in food services. J. Oper. Res. Sot., 42: 623-629.