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Convergence of estimated optimal inventory levels in models with probabilistic demands

The behavior of estimations of the optimal inventory level is analyzed. Two models are studied. The demands follow unknown probability distribution function. The included density functions are estimated and a plug-in rule is suggested for computing estimates of the optimal levels. Two search algorithms are proposed and compared using Monte Carlo experiments. | Yugoslav Journal of Operations Research 13 (2003), Number 2, 217-227 CONVERGENCE OF ESTIMATED OPTIMAL INVENTORY LEVELS IN MODELS WITH PROBABILISTIC DEMANDS Carlos N. BOUZA Departamento de Matem‹tica Aplicada Facultad de Matem‹tica y Computaci†n Universidad de La Haban bouza@matcom.uh.cu Abstract: The behavior of estimations of the optimal inventory level is analyzed. Two models are studied. The demands follow unknown probability distribution function. The included density functions are estimated and a plug-in rule is suggested for computing estimates of the optimal levels. Two search algorithms are proposed and compared using Monte Carlo experiments. Keywords: Backorders, simulated annealing, density function estimation. 1. INTRODUCTION The inventory problems to be analyzed can be formulated as follows: "given set of demands of a certain number of periods, determine the parameters that implement a policy that ensures minimum costs at a long run". We start with an inventory and at every time period t we examine the inventory position. The set-up cost c( s) is associated to each placed order. A holding cost c( h) is incurred per unit-time in the inventory stock. The backordered cost c(b) is incurred per unit-time per backordered unit of demand. The cost of using a policy is a linear combination of set-up, holding and backordered costs. We can consider this problem deterministic or we can assume that we face probabilistic demands. The later case is more realistic though a percent of the demands can be considered non-random. For example, the demands of a large buyer can be similar in any period. The level of the product in the firm should be set to reserve sufficient inventory in order to meet the deterministic demand. It is known before the next replenish. This problem is analyzed in Section 3 following the results of Haussmann-Thomas (1972). Section 4 is devoted to the analysis of an inventory model where all the demands during the stock out period are backordered