Dynamic Lot Size Model

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THE DYNAMIC LOT SIZE MODEL WITHSTOCHASTIC DEMANDS:

A DECISION trORIZON STUDY*SITA BHASKARAN

Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 15260and

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 firstfew periods that are of immediate importance to the manager. Decision/Forecasthorizon (DH/FH) research deals with the question of whether optimal decisions inthe first or first few periods (known as DH) can be made without regards to thedata from some future period (known as FH) onwards.

In this paper the Dynamic Lot Size Model (DLSM) with stochastic demands isstudied. The model has a setup cost and hence does not have a convex cost strvictureusually assumed in the earlier DH/FH research involving stochastic problems. It isshown that in the stochastic DLSM, unlike in the convex cost problems, there areexamples without DH/FH. However, randomly generated problems seem univer-sally to have decision horizons. Sufficient conditions for the existence of DH/FHare obtained in terms of salvage functions, in terms of states of the system, andin terms of decision strategies. A comparison of these approaches concludes thepaper.

Keywords : Decision horizon, forecast horizon, stochastic demands, concave cost,dynamic lot size model (DLSM), Wagner Whitin algorithm.

RESUMEDans un probleme de decision multi-periode, ce sont normalement les decisionsdans la ou les premiere(s) periodes qui sont d'une importance immediate pour lemanager. La recherche de l'horizon de decision / de prevision (DH/FH) a pourbut de savoir si les decisions (connues comme DH) peuvent etre faites sans con-sideration de donnees des periodes futures (connues comme FH).

Le modele dynamique de la taille des lots (DLSM) en cas de demande stochas-tique est etudie dans cet article. Le modele a un cout de commande et de la, n'apas la structure convexe des couts normalement supposee dans la recherche DH/FHanterieure impliquant des problemes stochastiques. II est demontre que dans leDLSM stochastique, contrairement aux problemes de cout convexe, il existe des

Received Dec 1983, Revised May 1987 INFOR vol. 26, no. 3, 1988

213

Electronic copy available at: http://ssrn.com/abstract=1422108

214 S. BHASKARAN AND S. SETHI

exemples sans DH/FH. Cependant, des problemes generes au hasard semblent uni-versellement avoir des horizons de decision. On obtient des conditions suffisantespour l'existence de DH/FH en termes de fonction de cout de remplacement, entermes d'etats du systeme et en termes de strategies de decision. Une comparaisonde ces approches conclut cet article.

1. INTRODUCTIONIn a multiperiod decision problem it is usually the decisions in the first or firstfew periods that are of immediate importance to the manager. Decision/Forecasthorizon research deals with decision making in these periods. To find the optimalinitial decision, finite horizon problems of increasingly longer horizon are solvedtill the initial decision converges. A decision horizon procedure is a stopping rulespecifying when further increasing of the problem horizon will have negligible or noeffect on the initial decision. The horizon {T) when the stopping rule is satisfied iscalled the forecast horizon (FH) and the number of initial periods (t < T) for whichthe optimality of the solution is unaffected by demand beyond period T is calledthe decision horizon (DH). Bes and Sethi (1987) provide a more elaborate definitionof DH/FH.

The decision horizon concept has been studied mainly in problems with de-terministic demands. Wagner and Whitin (1958) were the first to introduce thedecision horizon concept. They used it in the area of production planning prob-lems and solved the Dynamic Lot Size Model. The work on this problem was latersurveyed and extended by Lundin and Morton (1975).

Decision horizon research in problems with stochastic demands has been re-stricted 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 otherstochastic decision horizon research because the DLSM has a setup cost and hencedoes not have convex cost structure. Thus unlike all DH research till now the twincomplexities of stochastic demand and non-convex costs are considered simultane-ously.

In Section 2 we formulate the stochastic DLSM. A backward algorithm is de-rived for its solution and an illustrative example presented. Section 3 illustrates thatunlike 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 stochasticDLSM.

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 no-tions of regeneration points, production points and regeneration sets (Lundin andMorton (1975)). These notions do not generalize to the case of stochastic demandsbecause 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

DYNAMIC LOT SIZE MODEL 215

case in two ways the state regeneration framework and the value function regen-eration framework. Stochastic decision horizon research for undiscounted problemstill now has used certain monotonicity properties of convex cost problems arisingfrom the value function regeneration framework to get decision horizon results andto prove, under mild conditions, that DH/FH exist. These monotonicity propertiesdo not hold for non-convex cost problems and hence the methods of convex costproblems will not work for non-convex cost problems. Bes and Sethi (1987) studydiscounted stochastic problems.

2.2 The Stochastic DLSMConsider 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 inven-tory and average shortage, respectively, in a period. Demands A are independentrandom variables. The problem with horizon T (or T-period problem) is

C(T) = min'{P,}

subject toIt = It-i + Pt - Dt, lo = 0.

Pt>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 policyThe optimal policy (see e.g. Hillier and Lieberman (1980)) at period t, t e (1, T) isdetermined by two critical numbers St and St{>St) as follows:

St L-l if I 1 < s0 otherwise.

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

216 S. BHASKARAN AND S. SETHI

Ht < H" and 0 < /, < /",for every t e (1, T).

This leads to an optimal policy based on a single critical number St for everyperiod:

' \ 0, otherwise.Also, because it is never optimal to hold inventory for more than {K/h) periods, thefollowing stronger bounds on inventory are obtained:

/", V)where V = {K/h)H". Thus, /, G {0,1,..., W}.

2.5 Solution methods: Forward vs. Backward AlgorithmsThe Wagner and Whitin forward algorithm and the Lundin and Morton results solvethe deterministic DLSM efficiently but are not applicable to stochastic problemsbecause the notions of regeneration set and regeneration monotonicity (Lundin andMorton (1975)) are not applicable to stochastic problems. Backward procedures arereadily applicable to stochastic problems since they derive an optimal action forevery possible inventory level. We will derive such an algorithm. This algorithm iscomputationally more demanding than the Wagner and Whitin algorithm as it doesnot use previously obtained solutions.Backward algorithm. Since actions depend only on the cost function modulo anadditive 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 ofperiod t to the end of period T. The cost is relative because a largeenough constant is subtracted to make iv,r(^*) = 0-

Thus 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 ExampleWe now present a typical example to illustrate the nature of the solution to thestochastic DLSM: Let K = 1, h = 0.05, I" = oo and the demand sequence bePeriod (/): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Ht: 2 4 8 3 1 5 4 7 3 2 6 8 9 1 1 4 3 6 5 5Lt-. 1 3 4 2 1 4 4 5 1 2 5 3 8 1 1 2 3 4 2 4

DYNAMIC LOT SIZE MODEL 217

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

10-;

i 1 -^1 2 3 4 5 6 7t

T = 8

Figure la. A Typical Example (Part 1)

218 S. BHASKARAN AND S. SETHI

1 2 3 4 5 6 7 8 9 10

T = 10

15-

10^

I' ^

1 1 1

= 11

1 2 3 4 5 6 7 8 9 10 Ut

9 10 11 12

Figure 1b. A Typical Example (Part 2)

3. NONEXISTENCE OF DECISION HORIZONCases where decision/forecast horizons do not exist have not been studied. Exis-tence 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 empir-ical study. But there are problems where DH/FH do not exist, as we will show. Inthe DLSM with almost stationary (but not stationary) demand it sometimes hap-pens, in both the deterministic and stochastic cases, that as problems of increasinglylonger horizon are solved the optimal initial production keeps oscillating infinitely

DYNAMIC LOT SIZE MODEL 219

proving that decision horizons do not always exist. This does not happen in convexcost problems. Below are such examples.

3.1 Deterministic ExampleLet K = I, h = .Qh and consider the following three demand sequencesPeriod (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... T5 5 5 5 5 5 5 5 5 5 5 5 5 5... 55 4 5 4 5 5 5 5 5 5 5 5 5 5 . . . 54 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 slightlychanging the demand structure in the first few periods it is possible to constructexamples where the optimal initial solution is unique and the initial productionoscillates. This is shown in Case 2.

Case 2. The solution, in this case, is always unique and the optimal initial productionis 14, 18 or 9 according asT = 3, 3n -I-1, or 3 -f- 2 respectively. Hence no DH. Thecost penalty (i.e., excess over minimum cost) on using the three initial productionsPi = 14, 18 and 9 is given in Table 1.

Table 1.Cost Penalty Table: Deterministic Example (Case 2)

Pi14189

T3n

0.1.2

3/7-1-1

.150.1

3n + 2.05.15

Thus, for e < .15, there is no production Pi with cost guaranteed to be within eof optimal. Oscillation of the initial decision can also occur when the demand hasa 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.

220 S. BHASKARAN AND S. SETHI

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

7 7 7 7 7 7 7 7 77 7 6 6 6 6 6 6 6

T16

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

Table 2.Cost Penalty Table: Stochastic Example

Pi1421

TEven

0.05

Odd.075

0

For e < .05, there is no production Pi with cost guaranteed to be within e ofoptimal. It should be noted that the demand is nearly stationary and the dispersionof 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/FHSufficient conditions for the existence of DH/FH in the stochastic DLSM are ob-tained using three different approaches: States, Decision strategies, and Salvagefunctions.

4.1 StatesIn problems with stochastic demands, decisions only determine the expected prob-ability 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 aprobability distribution of inventory levels. This done, it is possible to formulatea forward dynamic programming algorithm to solve the stochastic problem in thefollowing 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 thecheapest path (from t = O)hy which it can be reached.

DYNAMIC LOT SIZE MODEL 221

Step 3. If the optimal initial decision corresponding to all "states" of thesystem at period / + 1 is the same, stop, f + 1 is a FH. If not, setf = r + 1 and go to Step 2.

Example 4.1. Let AT = 1, A = .1, / " = 10 and as in previous examples let A have atwo point distribution:

Period (0 : 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2122Ht\ 5 4 5 5 3 5 5 4 5 5 5 4 5 5 3 5 4 5 4 5 5 5Lt: 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 ateach period. In the forward algorithm the number of states at end of period t canincrease 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 reducedto a few possibilities (in this example the number varies between 2 and 5) then theforward algorithm is applicable.

4.2 Decision StrategiesIt is a common property for the product of the transition matrices of a Markovchain 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 outafter sufficient time. This leads to the following result:

Theorem 4.1 : If there exists a T > 2, such that for all possible decisions betweenperiods 2 and T, the "state" at the end of period T is independent of the "state" at theend of period 1, then T is a forecast horizon.

The above theorem provides conditions for the existence of FH which are indepen-dent 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 10Ht: 5 4 6 5 7 3 5 6 4 6Lt: 1 2 2 2 1 2 2 2 2 2

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.

222 S. BHASKARAN AND S. SETHI

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

If I* = 0 the above inequality reduces to

-K < G{I) 1, the critical number Si corresponding to the solutionto the T-period problem with salvage function G, is the same for every function Gwithin 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 allgive 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 solvingincreasingly longer horizon problems and randomly generating 50 possible salvagefunctions for every horizon length, an apparent FH of 6 was obtained with a DHof 2.

DYNAMIC LOT SIZE MODEL 223

4.5 Comparison of the three approachesNo disposal vs. Disposal. In most of the paper the model considered is the stan-dard DLSM in which disposal of inventory is not allowed. This keeps the form ofpolicy simple. This is suitable for the state and decision strategy approaches whichare based on enumeration of possible decision strategies. In the salvage functionapproach, 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 theefficiency of the salvage function approach.

While in the state and salvage function approaches, the T-period problem issolved 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 underconsideration at every period. Otherwise, one runs into computer storage problemsin the state approach and into CPU time problems in the decision strategy approach.The combinatorial complexity of the salvage function approach exceeds that of thefirst two approaches. However, trying out a number of possible salvage functions isone way of determining apparent forecast horizons.

ACKNOVt'LEDGEMENTThis 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: Applicationsto Dynamic Stochastic Optimization Problems" Math, of O.R. forthcoming.

2. Bhaskaran, S. and S.P. Sethi (1987) "Decision and Forecast Horizons in a StochasticEnvironment: 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 ,pp 113-118.

4. Chand, S. and T.E. Morton (1986) "Minimal Forecast Horizon Procedures for DynamicLot 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 DayInc.

6. Kleindorfer, P. and H. Kunreuther (1978) "Stochastic Horizons for the Aggregate Prob-lem" Management Science Vol. 24, No. 5, pp 485- 497.

1. Lundin, R.A. and T.E. Morton (1975) "Planning Horizons for the Dynamic Lot SizeModel: Zabel vs. Protective Procedures and Computational Results" Operational Re-search Vol. 23, No. 4, pp 711-734.

8. Morton, T.E. (1978a) "Universal Planning Horizons for Generalized Convex ProductionScheduling" Operations Research Vol. 26, No. 6, pp 1046-1057.

9. Morton, T.E. (1978b) "The Nonstationary Infinite Horizon Inventory Problem" Man-agement Science Vol 24, No. 14, pp 1474-1482.

10. Morton, T.E. (1979) "Infinite Horizon Dynamic Programming Models A PlanningHorizon Formulation" Operations Research Vol. 27, No. 4, pp 730-742.

224 S. BHASKARAN AND S. SETHI

11. Morton, T.E. and W.E. Wecker (1977) "Discounting, Ergodicity and Convergence forMarkov 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 SizeModel" 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 GeneralMotors Research Laboratories. She works in the Production andLogistics Group of the Operating Sciences Departemnt. Her re-search interests are in applied Operations Research and her cur-rent work is in the areas of facility location and supplier ratio-nalization. She received her Ph.D. at the University of Adelaide,Australia and prior to joining GM taught at the Graduate Schoolof Business, University of Pittsburgh.

Suresh P, Sethi is General Motors Research Professor of Opera-tions Management at the University of Toronto. He has an MBAfrom Washington University and an M.S. and a Ph.D. in Indus-trial Administration from Carnegie-Mellon University. He wasa Connaught Senior Research Fellow in 1984-85. He is currently aPrincipal Investigator for the Manufacturing Research Corpora-tion of Ontario, a Center of Excellence funded by the ProvincialGovernment. His research interests are in production planning

problems under uncertainty, problems of scheduling in manufacturing, and stochas-tic dynamic optimization problems. His articles on these and other topics haveappeared in a variety of journals including Management Science, Operations Research,Mathematics of Operations Research, SIAMJ. of Control and Optimization, SIAM Review, Ad-vances in Applied Probability, J. ofEcon. Theory, Naval Research Logistics Quarterly, Journalof Optimization Theory and Application, IEEE Transactions on Automatic Control, EuropeanJournal of Operational Research, International Journal of Production Research, and The FMSMagazine. He has also co-authored a book and co-edited a special issue for INFORon the subject of optimal control theory and applications.