Probabilistic multi-item inventory model with varying order cost under two restrictions: A geometric programming approach

A probabilistic multi-item inventory with varying order cost and zero lead time under two restrictions is treated in this paper under the following assumptions: (1) the maximum inventory level of each item is a constant multiple of the average quantity ordered; (2) the order cost is a continuous increasing function of the replenishment quantity, which itself is proportional to some number of periods covered by the replenishment quantity. The constant of proportionality is the average demand per period. The expected total cost of inventory management is composed of three components: the average purchase cost, which is a constant that does not enter into the optimization consideration; the expected ordering cost, and the expected holding cost. No shortages are to be allowed. An analytical solution of the optimal number of periods Nr^* (rounded integer) and the optimal maximum inventory level is derived using a geometric programming approach. There are four special cases corresponding to the three possible relaxations of the constraints plus the case of the classical probabilistic model of constant procurement cost combined with the absence of the constraints. Also, an illustrative numerical example is added with some graphs.