ELSEVIER European Journal of Operational Research 103 (1997) 255-272 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH Theory and Methodology Models for 1-out-of Q systems with stochastic lead times and expedited ordering options for spares inventory Tadashi Dohi, Takeshi Shibuya, Shunji Osaki * Department of Industrial and Systems Engineering, Faculty of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima 739, Japan Received 1 March 1995; accepted 1 August 1996 Abstract The purpose of this article is to develop two kinds of continuous-time cyclic inventory models with stochastic lead times and expedited ordering options. The order form under consideration is similar to that arising in the spare part inventory management. Especially, the control problem to determine the timing for the regular order is considered, and the optimal policy which minimizes the long-run average cost is analytically characterized. Finally, numerical examples are presented to evaluate the uncertainty of stochastic lead times, and the optimal inventory policy, which is composed of both the ordering time and the order quantity, is numerically calculated. @ 1997 Elsevier Science B.V. Keywords: Spare part inventory model; Continuous time; Expedited order; Stochastic lead times; Optimal policies 1. Introduction Many kinds of inventory control policies have been extensively developed in the literature (see Hadley and Whitin [ 1 ] ). Among them, it is of great importance to analyze inventory control models with a delivery time lag by an expedited provision. In fact, complex order forms and various supply modes are required in real inventory systems, and efforts to cut down total shortage periods and to reduce operating costs are carried out by practition- ers. Since the recent development of sophisticated computer integrated manufacturing techniques, the inventory control under such a complex environment can be easily realized on computer. In most cases, however, it should be noted that the order and/or delivery schedule generated on computer can provide the best policy among possible candidates, but does not always guarantee the globally optimal policy. Thus, the mathematical modeling of complex inventory scheme under uncertainty is needed to analyze accurately the inventory mechanism. The inventory control models with an expedited order have been studied by many authors. Allen and D'Esopo [2] presented a simple inventory model with an expedited order. Related, but somewhat different models, have been considered by Barankin [3] and Rosenshine and Obee [4]. Especially, Barankin [3] discussed a model for a single planning period in which the two options for expedited orders are available. Recently, Moinzadeh and Nahmias [5] considered an inventory control model with two supply modes and formulated approximated * Corresponding author. 0377-2217/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0377-2217(96)00256- 1