Inventory of Deteriorating Complementary and Substitute Items with Stock Dependent Demand

SYNOPTIC ABSTRACT

The optimal production for an inventory control system of deteriorating multi-items where items are of complementary or substitute types is formulated with a resource constraint. Here, it is assumed that substitute products have linearly negative effect on one another's demands and this effect is positive for the complementary items. The effect of an item on the demand of the other item depends on the inventory level of the first item. The rate of deterioration of the items is stock dependent. An item's own demand is also stock dependent, shortages are not allowed, and deteriorated items are salvaged. The total profit which consists of the sales proceeds, production cost, inventory holding cost, salvage value is formulated as a Pontryagin Optimal Control problem for both steady and transient states and evaluated using optimal control theory satisfying the Generalized Legendre condition. The model is formulated in general form for n-items and in particular, is illustrated with two items for some numerical data. The optimum results are presented both in tabular form and graphically.

Key Words and Phrases:

• multi-item inventory
• deteriorating item
• Pontryagin maximum principle
• Generalized Legendre condition
• complement product
• substitute product