Engineering Costs and Production Economics, 15 (1988) 381-386 Elsevier Science Publishers B.V., Amsterdam - Printed in Hungary zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM 381 A PRACTICAL NEAR-OPTIMAL ORDER QUANTITY METHOD zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ Victoria J. Mabin Scientific and Industrial Research Institute, Wellington. New Zealand INTRODUCTION Inventory control aims to balance investment in stocks with service to customers, by controlling the purchasing operations. An inventory control system is basically a set of rules to determine for each item when and how much to order, as well as how frequently stock should be reviewed. One of the commonest systems is the (Q, r) system, which places an order of size Q whenever the stock position falls to a re-order level, r. The determination of the optimal Q and r, to satisfy the required service level at the least cost, has not been straightforward, as Q and r are not independent. They are in fact connected by a non-linear (and even non-analytic) probability function. The traditional approach has been to assume an order quantity, solve for the re-order level, then substitute that value and recompute the order quantity. This process is repeated until convergence is obtained. The iteration converges rapidly, but is fairly taxing on computer resources. Several heuristic procedures have been proposed which attempt to provide approximations to the optimal solution (see e.g. Herron [5], Lewis [7]). Commercial packages also use ad hoc pro- cedures, e.g. IMPACT calculates the economic order quantity, EOQ, and then adjusts it using ad hoc rules based on empirical observations (see Brown [3, pp. 219-2211, Kleijnen and Rens [6], and Gardner [4]). In practice however, inventory controllers prefer to set the order quantities and safety stocks directly, using a known rule or formula, rather than rely on a computer iterative method. Formulae such as the Economic Order Quantity, EOQ (see [9]), allow direct calculation, but are frequently far from optimal. The purpose of this research was to find simple near-optimal formulae for the order quantity and safety stocks, which can be used by inventory controllers. This paper presents a method of deriving a closed form solution of the total cost function. This gives a simple formula for Q, which can be calculated directly. It replaces the need to compute an EOQ or use ad hoc procedures. The formula is simple enough to be calculated on a hand-calculator, although we are implementing it on an existing computerised inventory system. It appeals to inven- tory controllers and management because they can obtain an order quantity (or order cycle) and re- order level directly. Because they can understand it, they trust it more than ad hoc procedures. In addition, the formula saves considerably on com- puter time. The approach can be applied to service de- finitions, lead time demand distributions and inven- tory systems other than those assumed here. DEFINITION OF THE PROBLEM The total cost function, T, can be written as a function of the order quantity, Q. and re-order level, r, as follows: T (Q, r) = Holding Cost + Order Cost = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC (Q/2 + W VG + WQ>C,, with r = p+SS, (1) (2)