The Optimal (s, S) Inventory Model with Dynamic and Deterministic Demands

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1 n ÒJ 6Ó The Optimal (s, S) Inventory Model with Dynamic and Deterministic Demands ƒf 6,''!# % 1 %! ˆ v o ÒJ 6Ó $ ÒJÓ Ò6Ó r nz d š j g +>@ FIABI FJKJ> B y j z z n ž n ÒJ 6Ó m v v v ÒJ 6Ó n z g n ÒJ 6Ó j n qv 1 ÒJ 6Ó j j g ˆ ÒJ 6Ó nz Abstract We consider the (s, S, i.e., reorder point s, order-up-to level S) inventory policies in a dynamic environment where forecasting is used to project the future demand of customers in a fixed planning horizon. The demand rates are finite, but not necessarily stationary; and the unfulfilled demands are lost in the system as lost sale. The considered total cost includes: procurement or setup cost, inventory carrying cost, and the cost of lost sale. The expected total cost per unit time is to be minimized by determining the optimal (s, S) policies. A dynamic programming model is used to derive the expected cost function. The analysis is based on optimality principle, conditioning on the state of (s, S) and the stage of period in the planning horizon. Close-formed analytic formula will be derived for the expected cost

2 function. Dominance theorems are established to develop polynomial time algorithms for finding the optimal (s, S) parameters, and the expected minimal total cost per period. We also studied the classical (s, S) approaches (demands are assumed stationary and follow some probability distributions) and the dynamic approach. The effects of length of planning horizon on the optimal policies is analyzed and contrasted. To aid decision making, parametric studies regarding the length of planning horizon, forms of demand distribution, and various parameters are conducted. Keywords: (s, S) Inventory model, Dynamic Demand, Lost Sale ƒ ÒJ 6Ó Veinott and Wagner %$ $ Ehrhardt & # Tijm and Groenvelt '# # Fu # $ z z nonstationary g Backorder n Blyth and Silver ' Sivazlian &! y z nonstationary z n Š Lost sale m v ÒJ 6Ó y g y k y Š n ÒJ 6Ó j y ± { n Dynamic lot sizing Wagner and Whitin $' % g ' n z r g ƒ -53 Brown '! Donald et al. " Silver and Peterson '$ William '! ' Miller '% Federgruen and Micheal " % Thomas and David $ " n ÒJ 6Ó Iglehart %" & j Iyer and Schrage! ' g +>@ FIABI n 2

3 qv ÒJ 6Ó y m ÒJ 6Ó j Š Lost salb n n ÒJ 6Ó qv J 6 n 5B@LIJ MB J 6 n z J 6 qv J 6 j nj 6 d d 1,d 2,...,d N z i - z? Š lost sales/unit/period E holding purchasing cost/unit setup/ordering cost constant lead time It K Net Inventory I t u d It m y It m I t vš jv y m Pt K Inventory Position δ t δ t =1, if Pt-1 s > Pt δ t = 0, if Pt s f t (P t, I t) K Φ t (P t,i t ) K Φ t (P t,i t ) K+(c (S- P t )]+h max(0, I t )+b max(0,- I t ) Flow Conservation K K! Pt It P = P + δ [ S P ] d t t 1 t 1 t 1 t 1 I = I + δ [ S P ] d t t 1 t L t L t 1 Pt = It + Inventory_on_order I nventory_on_order K f t (P t, I t) n x Forward Recursion v J 6 K v J 6 qvqv 2 O(N D ) v J 6 zqv jv 3

4 Theorem 1. There exists an optimal solution with at least one period t with Pt equal to the reorder point s. Theorem 2. There exists at least one optimal (s, S) policy such that Pt It qv qvqv O(N 4 ) Poisson, discrete uniform, geometric z qv k { 1. Blyth, C. and Silver, E. (1981), Continuous review (s, S) policies with lost sales. Management Science, 27, Brown, R.(1981), The new push for DRP. Inventory and Production Management, Donald, W. John, H. Thomas, R.(1991), Production and Inventory Management, 2 nd Edition, South-Western Publishing Co. 4. Ehrhardt, R.(1979), The power approximation for computing (s, S) policies. Management Science, 25, Fu, M.C.(1994), Sample path derivatives for (s, S) inventory systems. Operations Research, 42, Federgruen, A. Michael, T.(1993), The dynamic lot-sizing model with backlogging: a simple O(n log n) algorithm and minimal forecast horizon procedure. Naval Research Logistics, 40, Iglehart, D.(1963), Optimality of (s, S) policies in the finite horizon dynamic inventory problem. Management Science, 9, Iyer, A. Schrage, L.(1992), Analysis of the dynamic (s, S) inventory problem. Management Science, 38, Miller, B.(1986), Scarf s state reduction method, flexibility, and a dependent demand inventory model. Operations Research, 34, Ray, W.D.(1982), Computation of reorder levels when the demands are correlated and the lead time random. Journal of Operational Research Society, 32, Silver, E. Peterson, R.(1985), Decision systems for inventory management and production planning, Wiley. 12. Silvazlian, B.(1971), Dimensional and computational analysis in (s, S) inventory problem with Gamma distributed demand. Management Science, 17, Thomas, E., David, W.(1995), The finite horizon nonstationary stochastic inventory problem: Near myopic bounds, heuristics, testing. Management Science, 41, Tijms, H. Groenevelt, H.(1984), Approximations for (s, S) inventory systems with stochastic lead times and a service level constraint. European Journal of Operational Research, 17, Veinott, A. Wagner, H.(1965), Computing optimal (s, S) inventory policies,.management Science, 11, Wagner, H. Whitin, T.(1958), Dynamic version of the lot size model. Management Science, 5, Ward, J.(1978), Determining reorder point when demand is lumpy. Management Science, 24,

5 18. William, T.(1982), Reorder levels for lumpy demand. Journal of Operational Research Society, 33,