EXISTENCE AND DERIVATION OF FORECAST HORIZONS IN A DYNAMIC LOT SIZE MODEL WITH NONDECREASING HOLDING COSTS

Abstract

We are concerned with a discrete‐time undiscounted dynamic lot size model in which demand and the production setup cost are constant for an initial few periods and the holding cost of inventory is an arbitrary nondecreasing function assumed to be stationary (i.e., explicitly independent of time) in the same initial few periods. We show that there exists a finite forecast horizon in our model and obtain an explicit formula for it. In addition, we obtain fairly general conditions under which the existence of a solution horizon in the model implies the existence of a forecast horizon. We also derive an explicit formula for the minimal solution horizon. These results extend the earlier ones obtained for the dynamic lot size model with linearly increasing holding costs.

Keywords

FORECAST HORIZONS SOLUTION HORIZONS DECISION HORIZONS DYNAMIC LOT SIZE MODEL GRAPH THEORY

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References

Aronson

and Thompson

G. L.

(1984), “Forward Methods in Mathematical Programming,” J. Large Scale Systems, 7, 1–16.

Bensoussan

Crouhy

, and Proth

J. M.

(1983), Mathematical Theory of Production Planning, Advanced Series in Management, North‐Holland, Amsterdam.

Bes

and Sethi

S. P.

(1988), “Concepts of Forecast and Decision Horizons: Applications to Dynamic Stochastic Optimization Problems.” Mathematics of Operations Research, 13, 295–310.

Bhaskaran

and Sethi

(1987), “Decision and Forecast Horizons in a Stochastic Environment: A Survey,” Optimal Control Applications and Methods, 8, 201–217.

Bhaskaran

(1988), “The Dynamic Lot Size Model with Stochastic Demands: A Decision Horizon Study,” INFOR, 26, 213–224.

Bylka

(1974), “Horizon of Optimization Problems on Multigraphs,” Polish Academy of Sciences, P.W.N., Warsaw.

Bylka

(1982), “Horizon Theorems for the Solution of the Dynamic Lot‐Size Model,” Proceedings of the Second International Symposium on Inventories, Budapest, Hungary, 649–659.

Chand

and Morton

T. E.

(1986), “Minimal Forecast Horizon Procedures for dynamic Lot‐Size Model,” Naval Research Logistics Quarterly, 33, 111–122.

Chand

Sethi

S. P.

, and Proth

J. M.

(1990), “Existence of Forecast Horizons in Undiscounted Discrete‐Time Lot Size Models,” Operations Research, 38, 884–892.

10.

Chand

Sethi

, and Sorger

(1992), “Forecast Horizons in the Discounted Dynamic Lot Size Model,” Management Science, 38, 1034–1048.

11.

Los

(1967), “Horizons in Dynamic Programs,” Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, California University Press, Berkeley, CA., 479–490.

12.

Lundin

and Morton

(1975), “Planning Horizon for the Dynamic Lot Size Model: Zabel vs. Protective Procedures and Computational Results,” Operations Research, 23, 711–735.

13.

Morton

(1979), “Infinite Horizon Dynamic Programming Model: A Planning Horizon Formulation,” Operations Research, 27, 730–742.

14.

Morton

(1981), Forward Algorithms for Forward‐Thinking Managers. Applications of Management Science, 1, JAI Press, Greenwich, CT. 1–55.

15.

Sethi

(1990), “Dynamic Optimization: Decision and Forecast Horizons,” In Systems and Control Encyclopedia, Supplementary Vol. 1, edited by Singh

M. G.

, Pergamon Press, Oxford, UK 192–198.

16.

Sethi

and Sorger

(1991), “A Theory of Rolling Horizon Decision Making,” Annals of Operations Research, 29, 387–416.

17.

Wagner

(1969), Principles of Operations Research, Englewood Cliffs, NJ, Prentice Hall.

18.

Wagner

H. M.

and Whitin

T. M.

(1958), “Dynamic Version of the Economic Lot Size Model,” Management Science, 5, 86–96.

19.

Zangwill

W. I.

(1968), “Minimum Concave Cost Flows in Certain Networks,” Management Science, 14, 429–450.