THE DYNAMIC LOT SIZE MODEL WITH

THE DYNAMIC LOT SIZE MODEL WITH STOCHASTIC DEMANDS: A DECISION trORIZON STUDY* SITA BHASKARAN Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260 and Operating Sciences Department, General Motors Research Laboratories, Warren, MI 48090 SURESH P. SETHI Faculty of Management Studies, University of Toronto, Toronto, M5S 1V4 ABSTRACT In a multiperiod decision problem it is usually the decisions in the first or first few periods that are of immediate importance to the manager. Decision/Forecast horizon (DH/FH) research deals with the question of whether optimal decisions in the first or first few periods (known as DH) can be made without regards to the data from some future period (known as FH) onwards. In this paper the Dynamic Lot Size Model (DLSM) with stochastic demands is studied. The model has a setup cost and hence does not have a convex cost strvicture usually assumed in the earlier DH/FH research involving stochastic problems. It is shown that in the stochastic DLSM, unlike in the convex cost problems, there are examples without DH/FH. However, randomly generated problems seem universally to have decision horizons. Sufficient conditions for the existence of DH/FH are obtained in terms of salvage functions, in terms of states of the system, and in terms of decision strategies. A comparison of these approaches concludes the paper. Keywords : Decision horizon, forecast horizon, stochastic demands, concave cost, dynamic lot size model (DLSM), Wagner Whitin algorithm.

RESUME Dans un probleme de decision multi-periode, ce sont normalement les decisions dans la ou les premiere(s) periodes qui sont d'une importance immediate pour le manager. La recherche de l'horizon de decision / de prevision (DH/FH) a pour but de savoir si les decisions (connues comme DH) peuvent etre faites sans consideration de donnees des periodes futures (connues comme FH). Le modele dynamique de la taille des lots (DLSM) en cas de demande stochastique est etudie dans cet article. Le modele a un cout de commande et de la, n'a pas la structure convexe des couts normalement supposee dans la recherche DH/FH anterieure impliquant des problemes stochastiques. II est demontre que dans le DLSM stochastique, contrairement aux problemes de cout convexe, il existe des

Received Dec 1983, Revised May 1987

INFOR vol. 26, no. 3, 1988

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S. BHASKARAN AND S. SETHI exemples sans DH/FH. Cependant, des problemes generes au hasard semblent universellement avoir des horizons de decision. On obtient des conditions suffisantes pour l'existence de DH/FH en termes de fonction de cout de remplacement, en termes d'etats du systeme et en termes de strategies de decision. Une comparaison de ces approches conclut cet article.

1. INTRODUCTION In a multiperiod decision problem it is usually the decisions in the first or first few periods that are of immediate importance to the manager. Decision/Forecast horizon research deals with decision making in these periods. To find the optimal initial decision, finite horizon problems of increasingly longer horizon are solved till the initial decision converges. A decision horizon procedure is a stopping rule specifying when further increasing of the problem horizon will have negligible or no effect on the initial decision. The horizon {T) when the stopping rule is satisfied is called the forecast horizon (FH) and the number of initial periods (t < T) for which the optimality of the solution is unaffected by demand beyond period T is called the decision horizon (DH). Bes and Sethi (1987) provide a more elaborate definition of DH/FH. The decision horizon concept has been studied mainly in problems with deterministic demands. Wagner and Whitin (1958) were the first to introduce the decision horizon concept. They used it in the area of production planning problems and solved the Dynamic Lot Size Model. The work on this problem was later surveyed and extended by Lundin and Morton (1975). Decision horizon research in problems with stochastic demands has been restricted to cases with convex or proportional cost structure. Bhaskaran and Sethi (1987) survey this literature. In this paper the Dynamic Lot Size Model (DLSM) with stochastic demands is studied. This study differs fundamentally from other stochastic decision horizon research because the DLSM has a setup cost and hence does not have convex cost structure. Thus unlike all DH research till now the twin complexities of stochastic demand and non-convex costs are considered simultaneously. In Section 2 we formulate the stochastic DLSM. A backward algorithm is derived for its solution and an illustrative example presented. Section 3 illustrates that unlike convex cost problems, DH/FH may not exist in non-convex cost problems. Section 4 derives three sufficient conditions for the existence of FH in the stochastic DLSM. 2. STOCHASTIC DYNAMIC LOT SIZE MODEL (DLSM) 2.1 Decision Horizon Research — Undiscounted Problems with Stochastic Demands. In deterministic undiscounted problems, decision horizon results depend on the notions of regeneration points, production points and regeneration sets (Lundin and Morton (1975)). These notions do not generalize to the case of stochastic demands because the inventory levels to be visited at future periods are not known beforehand. The notion of regeneration set was generalized by Morton (1979) for the stochastic

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case in two ways — the state regeneration framework and the value function regeneration framework. Stochastic decision horizon research for undiscounted problems till now has used certain monotonicity properties of convex cost problems arising from the value function regeneration framework to get decision horizon results and to prove, under mild conditions, that DH/FH exist. These monotonicity properties do not hold for non-convex cost problems and hence the methods of convex cost problems will not work for non-convex cost problems. Bes and Sethi (1987) study discounted stochastic problems. 2.2 The Stochastic DLSM Consider a multiperiod undiscounted inventory problem with Dt, Pt, It '• f = 1,2,... denoting the nonnegative demand, production, and ending inventory respectively, in period t. The production cost is a fixed cost and is K, if Pt > 0, and 0 otherwise. Holding and penalty costs of h and p per unit are charged on the average inventory and average shortage, respectively, in a period. Demands A are independent random variables. The problem with horizon T (or T-period problem) is C(T) = min£' {P,}

subject to

It = It-i + Pt - Dt, Pt>0,

lo = 0.

te (1, T)

where

i^^^^^\K, ifx>0 and riyx — Utj ^)

^ X ^ Ut

^'^•"•^ ^ \^ hx/2 -h p{Dt - x)/2

if X < A-

2.3 Form of optimal policy The optimal policy (see e.g. Hillier and Lieberman (1980)) at period t, t e (1, T) is determined by two critical numbers St and St{>St) as follows: St — L-l 0

if I 1 < s otherwise.

2.4 Two point demand distribution In numerical examples, for computational simplicity, we assume that demand Dt takes on either a high value Ht or a low value Lt, both being equally probable integers. We also assume that backlogging is not allowed and that demands and inventory levels are bounded, i.e., there exist H" and / " such that

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for every t e (1, T). This leads to an optimal policy based on a single critical number St for every period: ' \ 0, otherwise. Also, because it is never optimal to hold inventory for more than {K/h) periods, the following stronger bounds on inventory are obtained: /", V) where V = {K/h)H". Thus, /, G {0,1,..., W}. 2.5 Solution methods: Forward vs. Backward Algorithms The Wagner and Whitin forward algorithm and the Lundin and Morton results solve the deterministic DLSM efficiently but are not applicable to stochastic problems because the notions of regeneration set and regeneration monotonicity (Lundin and Morton (1975)) are not applicable to stochastic problems. Backward procedures are readily applicable to stochastic problems since they derive an optimal action for every possible inventory level. We will derive such an algorithm. This algorithm is computationally more demanding than the Wagner and Whitin algorithm as it does not use previously obtained solutions. Backward algorithm. Since actions depend only on the cost function modulo an additive constant, the cost relative to a base state /* is defined to always be zero. Let Ft,T{I) '• Minimum relative cost of going from inventory level / at the end of period t to the end of period T. The cost is relative because a large enough constant is subtracted to make iv,r(^*) = 0Thus Ftj{I) satisfies these backward relations for every f e (0, T - 1) : FtAJ) = ftAi) - ftAn

FTAI)

=0

The above equations are solved iteratively to compute i^o, 2.6 A Typical Example We now present a typical example to illustrate the nature of the solution to the stochastic DLSM: Let K = 1, h = 0.05, I" = oo and the demand sequence be Period (/): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Ht: 2 4 8 3 1 5 4 7 3 2 6 8 9 1 1 4 3 6 5 5 Lt-. 1 3 4 2 1 4 4 5 1 2 5 3 8 1 1 2 3 4 2 4

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As increasingly longer horizon problems are solved, the nature of the solution and the inventory level in the various periods are sho^wi in Figure 1. The numbers Ht, St are marked as heavy dots. We note that for 7" > 6 we have Si = 6. Thus the first period decision converges. As problem horizon increases, the number of initial periods, for which the decision converges, also increases. This indicates the existence of DH/FH but there is no way of rigorously identifying them.

10-;

i

1 2

3

4

5

6

1

7

-^

t

T= 8

Figure la. A Typical Example (Part 1)

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T = 10 1

2

3

4

5

6

7

8

9

10

•

15-

•

10^

'

^

I

1

2

3

4

5

= 11

6

7

8

9

1

1

10

1

U

t

9

10

11

12

Figure 1b. A Typical Example (Part 2)

3. NONEXISTENCE OF DECISION HORIZON Cases where decision/forecast horizons do not exist have not been studied. Existence issues in DH/FH research are not yet settled. Lundin and Morton (1975) remark that decision horizons or near decision horizons existed in their vast empirical study. But there are problems where DH/FH do not exist, as we will show. In the DLSM with almost stationary (but not stationary) demand it sometimes happens, in both the deterministic and stochastic cases, that as problems of increasingly longer horizon are solved the optimal initial production keeps oscillating infinitely

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proving that decision horizons do not always exist. This does not happen in convex cost problems. Below are such examples. 3.1 Deterministic Example Let K = I, h = .Qh and consider the following three demand sequences Period (0 ". Demand (A) Case 1: Demand (A) Case 2: Demand (A) Case 3:

1 2 3 4 5 6 7 8 9 10 11 12 13 14... T 5 5 5 5 5 5 5 5 5 5 5 5 5 5... 5 5 4 5 4 5 5 5 5 5 5 5 5 55... 5 4 5 5 4 4 4 5 4 5 4 5 4 5 4 ...

Case 1. The T-period solution is as follows: If, for some positive integer n (n > 2) • T = Zn, the optimal solution is unique and the initial production is 15, • r = 3n -I-1, the optimal solution is not unique and initial productions of 15 and 20 are both optimal, • T = Zrt -\- 2, the optimal solution is not unique and initial productions of 10 and 15 are both optimal. Thus, in this case the initial decision does converge (to 15) and there is a DH (= 3) even though there may be other optimal initial decisions as well. By slightly changing the demand structure in the first few periods it is possible to construct examples where the optimal initial solution is unique and the initial production oscillates. This is shown in Case 2. Case 2. The solution, in this case, is always unique and the optimal initial production is 14, 18 or 9 according asT = 3«, 3n -I-1, or 3« -f- 2 respectively. Hence no DH. The cost penalty (i.e., excess over minimum cost) on using the three initial productions Pi = 14, 18 and 9 is given in Table 1. Table 1. Cost Penalty Table: Deterministic Example (Case 2)

T Pi

14 18 9

3n 0 .1 .2

3/7-1-1

3n + 2

.15 0 .1

.05 .15

Thus, for e < .15, there is no production Pi with cost guaranteed to be within e of optimal. Oscillation of the initial decision can also occur when the demand has a repetitive pattern as shown in Case 3. Case 3. Here, except for the initial six periods, the demand is alternately 5 and 4. The optimal initial decision is again unique and is 14, 18 or 9 according as T ~ 3n, 3« -h 1 or 3« -f- 2 respectively. Again, no DH.

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3.2 Stochastic Example Consider the previous example with the following stochastic demand sequence (having a two point distribution): Period (t) : 1 2 3 4 5 6 7 8 9

T

777777777 776666666

1 6

For every T > 2, the solution of the T-period problem is unique. The optimal initial production (Pi) is 14 or 21 according as T is even or odd respectively. The cost penalty (i.e., excess over minimum cost) on using the two initial productions Pi = 14 and 21 are given in Table 2: Table 2. Cost Penalty Table: Stochastic Example

T Pi

Even

14 21

0 .05

Odd .075 0

For e < .05, there is no production Pi with cost guaranteed to be within e of optimal. It should be noted that the demand is nearly stationary and the dispersion of demand is small (i.e., difference between high and low demand is small) — thus the demand is only marginally stochastic. Only in such cases, to the authors' knowledge and experience, DH/FH does not exist. 4. THREE APPROACHES FOR DETERMINING DH/FH Sufficient conditions for the existence of DH/FH in the stochastic DLSM are obtained using three different approaches: States, Decision strategies, and Salvage functions. 4.1 States In problems with stochastic demands, decisions only determine the expected probability distribution of inventory levels. Thus, it helps to generalize the notion of "inventory level" to that of "state" of the system, where a "state" is defined as a probability distribution of inventory levels. This done, it is possible to formulate a forward dynamic programming algorithm to solve the stochastic problem in the following manner: Step 1. Set^ = O. Step 2. Enumerate all "states" that can be reached at the end of period t-i-1 from the end of period t. For every such "state" consider only the cheapest path (from t = O)hy which it can be reached.

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Step 3. If the optimal initial decision corresponding to all "states" of the system at period / + 1 is the same, stop, f + 1 is a FH. If not, set f = r + 1 and go to Step 2. Example 4.1. Let AT = 1, A = .1, / " = 10 and as in previous examples let A have a two point distribution: Period (0 : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2122 Ht\ 5 4 5 5 3 5 5 4 5 5 5 4 5 5 3 5 4 5 4 5 5 5 Lt: 2 3 3 5 2 4 2 4 3 4 4 2 5 4 2 5 4 3 4 4 3 4 On applying the above forward algorithm FH = 22, 5i = 6 and DH ~ 1. Let IX^I : be the number of states at end of period t. Then \Xt\ is shown below: t: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 \Xt\: 5 13 11 13 26 83 29 55 85 184 148 109 156 97 113 97 97 65 161 129 209 161 In the backward algorithm of Section 2 there are only {W + 1) inventor}^ levels at each period. In the forward algorithm the number of states at end of period t can increase rapidly with t and make the algorithm computationally complex. However, if as in Example 4.1, the number of possible decisions at any period can be reduced to a few possibilities (in this example the number varies between 2 and 5) then the forward algorithm is applicable. 4.2 Decision Strategies It is a common property for the product of the transition matrices of a Markov chain to tend to a constant matrix (i.e., a matrix with identical rows). This property, known as "loss of memory" implies that the effect of the initial decision wears out after sufficient time. This leads to the following result: Theorem 4.1 : If there exists a T > 2, such that for all possible decisions between periods 2 and T, the "state" at the end of period T is independent of the "state" at the end of period 1, then T is a forecast horizon. The above theorem provides conditions for the existence of FH which are independent of K and h. They are strong conditions which are sufficient but not necessary. In particular, these conditions will never be satisfied by deterministic problems. Example 4.2. Let / " = 10. Consider the two point demand sequence

Period {t)\ 1 2 3 4 5 6 7 8 9 10 Ht: 5465735646 Lt: 1222122222 On applying Theorem 4.1, FH = 10. DH depends on K and h. Morton and Wecker (1977) deal with infinite horizon optimality for Markov Decision Processes.

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4.3 Salvage Functions Incorporating a salvage function for terminal inventory leads to the following observation: An A'^-period problem is equivalent for any T T is equivalent to a problem with horizon length T and appropriate salvage function G{I) for terminal inventory. G{I) is subject to the bounds (proof omitted)

If I* = 0 the above inequality reduces to

-K < G{I)
1, the critical number Si corresponding to the solution to the T-period problem with salvage function G, is the same for every function G within the bounds given above, then T is an FH. The number of possible salvage functions between the given bounds is very large. However, one could try out an arbitrary number of salvage functions and if they all give the same first period decision, conclude that T is an apparent FH. Example 4.3 In the typical example of Section 2, let disposal be free. On solving increasingly longer horizon problems and randomly generating 50 possible salvage functions for every horizon length, an apparent FH of 6 was obtained with a DH of 2.

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4.5 Comparison of the three approaches No disposal vs. Disposal. In most of the paper the model considered is the standard DLSM in which disposal of inventory is not allowed. This keeps the form of policy simple. This is suitable for the state and decision strategy approaches which are based on enumeration of possible decision strategies. In the salvage function approach, the model is altered to allow disposal as this bounds salvage functions. Disposal complicates the form of policy but a complicated policy does not affect the efficiency of the salvage function approach. While in the state and salvage function approaches, the T-period problem is solved for increasing values of T, this is not so in the decision strategy approach. The first two approaches are feasible only if a few alternative decisions are under consideration at every period. Otherwise, one runs into computer storage problems in the state approach and into CPU time problems in the decision strategy approach. The combinatorial complexity of the salvage function approach exceeds that of the first two approaches. However, trying out a number of possible salvage functions is one way of determining apparent forecast horizons. ACKNOVt'LEDGEMENT This research is supported in part by grant A4619 from NSERC of Canada. 5. REFERENCES 1. Bes, C. and S.P. Sethi (1987) "Concepts of Forecast and Decision Horizons: Applications to Dynamic Stochastic Optimization Problems" Math, of O.R. forthcoming. 2. Bhaskaran, S. and S.P. Sethi (1987) "Decision and Forecast Horizons in a Stochastic Environment: A Survey" Optimal Control Applications and Methods Vol.8, pp 201-217. 3. Chand, S. (1982) "A note on Dynamic Lot Sizing in a Rolling Horizon Environment" Decision Sciences No\. 13, No. l , p p 113-118. 4. Chand, S. and T.E. Morton (1986) "Minimal Forecast Horizon Procedures for Dynamic Lot Size Models" Naval Research Logistics Quarterly Vol. 33, No. 1, pp 111-122. 5. Hillier, F.S. and G.J. Lieberman (1980) Introduction to Operations Research Holden Day Inc. 6. Kleindorfer, P. and H. Kunreuther (1978) "Stochastic Horizons for the Aggregate Problem" Management Science Vol. 24, No. 5, pp 485- 497. 1. Lundin, R.A. and T.E. Morton (1975) "Planning Horizons for the Dynamic Lot Size Model: Zabel vs. Protective Procedures and Computational Results" Operational Research Vol. 23, No. 4, pp 711-734. 8. Morton, T.E. (1978a) "Universal Planning Horizons for Generalized Convex Production Scheduling" Operations Research Vol. 26, No. 6, pp 1046-1057. 9. Morton, T.E. (1978b) "The Nonstationary Infinite Horizon Inventory Problem" Management Science Vol 24, No. 14, pp 1474-1482. 10. Morton, T.E. (1979) "Infinite Horizon Dynamic Programming Models — A Planning Horizon Formulation" Operations Research Vol. 27, No. 4, pp 730-742.

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11. Morton, T.E. and W.E. Wecker (1977) "Discounting, Ergodicity and Convergence for Markov Decision Processes" Management Science Vol 23, No. 8, pp 890-900. 12. Wagner, H.M. and T.M. Whitin (1958) "Dynamic Version of the Economic Lot Size Model" Management Science Vol 5, pp 89-96. 13. Zabel, E. (1964) "Some Generalizations of an Inventory Planning Horizon Theorem" Management Science Vol. 10, pp 465-471.

Sita Bhaskaran is a Senior Research Scientist at the General Motors Research Laboratories. She works in the Production and Logistics Group of the Operating Sciences Departemnt. Her research interests are in applied Operations Research and her current work is in the areas of facility location and supplier rationalization. She received her Ph.D. at the University of Adelaide, Australia and prior to joining GM taught at the Graduate School of Business, University of Pittsburgh. Suresh P, Sethi is General Motors Research Professor of Operations Management at the University of Toronto. He has an MBA from Washington University and an M.S. and a Ph.D. in Industrial Administration from Carnegie-Mellon University. He was a Connaught Senior Research Fellow in 1984-85. He is currently a Principal Investigator for the Manufacturing Research Corporation of Ontario, a Center of Excellence funded by the Provincial Government. His research interests are in production planning problems under uncertainty, problems of scheduling in manufacturing, and stochastic dynamic optimization problems. His articles on these and other topics have appeared in a variety of journals including Management Science, Operations Research, Mathematics of Operations Research, SIAMJ. of Control and Optimization, SIAM Review, Advances in Applied Probability, J. ofEcon. Theory, Naval Research Logistics Quarterly, Journal of Optimization Theory and Application, IEEE Transactions on Automatic Control, European Journal of Operational Research, International Journal of Production Research, and The FMS Magazine. He has also co-authored a book and co-edited a special issue for INFOR on the subject of optimal control theory and applications.