Solution of Optimal Inventory Equation
Mamata Kuila · IOSR Journal of Mathematics · 2013
This paper deals with a functional equation of an optimal inventory equation having unbounded time period and one period lag in supply .The existence of the solution for this equation is proved through a dynamic programming approach. I. Introduction In this paper, we wish to study an analytic problem arising from an interesting stochastic allocation process occurring in the study of inventory. The successive approximation is a powerful analytic tool to prove the existence of the optimal policy of functional equation arising in optimization problems .Bellman(1 ) explored this area of optimization by investigating the existence and the behavior of the solution of various types of functional equations in game theory, inventory control problems ,dynamic programming ,bottleneck problems etc. Bhakta and Mitra (2) have established some existence theorems for functional equation arising in dynamic programming approach by contraction principle . Senapati and Panda (3) have established some existence theorems of a functional equation arising in continuous games. Panda (4) established existence results of functional equation of inventory equation, under the assumption of various costs associated with over supply and under supply when the total time period is unlimited. In this paper, the optimal inventory equation is considered which is concerned with the problem of stocking of supply items to meet and uncertain demand under the assumptions of various costs associated with over and under supply when the total time period is unlimited and the supply is one period in lag and the existence of the optimal policy is established by fixed point theorem. The problem we shall discuss here is one very particular case of the general problem of decision- making in the face of an uncertain future .The version we shall consider is concerned with the problem of stocking a supply of items to meet an uncertain demand, under the assumption that there are various costs associated with over supply and under supply. A certain set of items is to be ordered at various specified times, where the cost of ordering depends upon the number ordered of each item .There may or may not be some fixed costs which are independent of the number ordered .At various other times, demands are made upon the stocks of these items .The incentive for ordering lies in a penalty which is assessed whenever the demand for an item exceeds the supply .Here we want to determine the ordering policy at each stage which will minimize some average function of the overall cost of the process.