Constrained Inventory Allocation PHILLIP G. BRADFORD University of Alabama Department of Computer Science Box 870290, Tuscaloosa, AL 35487-0290. USA pgb@cs.ua.edu MICHAEL N. KATEHAKIS Rutgers University Department of MSIS 180 University Avenue, Newark NJ 07102 USA mnk@andromeda.rutgers.edu Abstract: This paper shows how to allocate a limited supply of inventory in a centralized distribution system with multiple retailers subject to minimal supply commitments and maximum delivery limits for each retailer. The objective is to maximize the number of units sold by all retailers in one time period. The analysis is done by building on a classical result of Derman. The demand at each retailer is described by a probability density function and unsold portions of the supply lose their value at the end of the period. Key–Words: Inventory Control, Nonlinear Optimization 1 Introduction We study the following allocation problem. Suppose at the beginning of a given time period there exists T units of some product (or service) which are to be allocated among k retailers. The demand at each re- tailer is a random variable with a distribution which depends on the retailer. At the end of the time period the unused product (or service) loses its value. The single supplier has contractual obligations to provide a minimum amount of l i ≥ 0 and a maximum amount of u i ≥ l i to retailer i, i =1,...,k. The problem of interest here is that of allocating the T units among the k retailers so that the amount a i that retailer i re- ceives satisfies the commitments, i.e., l i ≤ a i ≤ u i , i =1,...,k and the expected number of units sold is maximized. This model is a generalization of [2] where the case l i =0, and u i = ∞ was considered. This paper’s central motivation follows [11], in that there is a clear desire to have simple and effective models to enhance management reasoning. How may we balance expected supply and expected demand of a retailer given contract constraints? What if the sup- plier has a different expected supply function than the retailer’s expected demand function? The paper [10] gives a generalized framework for interpreting and solving nonlinear-additive functions constrained by the sum of their inputs. The focus is on the algorithmic solution to such problems. Variations on these problems are very close to the one consid- ered by the current paper. Furthermore, [1] character- izes and extends probabilistically constrained convex programming on discrete distributions. Single source supplies to several retailers or locations are considered in [3] with finite time periods. The focus is on min- imizing cost for the system. An infinite horizon case is considered in [4]. In [4] it is shown it may be ex- pressed as single location problems. The paper [12] considers the balancing of inventory with a horizon constraint. Other related work includes [6], [5], [11], [8] and [7]. 2 Main Theorem Let the D i be the random demand for retailer i. We assume that it is a nonnegative continuous random variable with continuous density f i (·) and c.d.f. F (·). We will assume that F is absolutely continuous for all positive values, and that P (D i = 0) = 0. Let a i de- note the amount of product allocated to retailer i. Un- der an allocation a =(a 1 ,...,a k ), the total expected sales at retailer i are: S i (a i )=  a i 0 xf (x)dx + a i  ∞ a i f (x)dx. (1) The problem then is to maximize total sales of the sys- tem: G(a 1 ,...,a k )= k  i=1 S i (a i ) (2) l i ≤ a i ≤ u i ,i =1,...,k, (3) k  i=1 a i ≤ T. (4) Here we allow some l i to be equal to 0 and some u i to be equal to ∞. Proceedings of the 10th WSEAS Interbational Conference on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 504