In this paper, the optimal production for an inventory control system of multi-items where items are either complementary and/or substitute is formulated with an uncertain resource constraint. Here, the production function is unknown and considered as a control variable. Also, the defective rates of the items are reliability dependent. The demand is stock dependent, shortages are not allowed. Unit production cost is reliability parameter and production rate dependent. It also includes the cost due to the environment protection measures. Set up cost is also reliability parameter dependent. The total profit, which consists of the sales proceeds, production cost, inventory holding cost and set up cost is formulated as a Pontryagin’s Optimal Control problem and evaluated using Taylor’s theorem, generalized reduced gradient technique and optimal control theory satisfying the Generalized Legendre conditions. Here, the inventory costs are fuzzy rough in nature and the resource constraint is imprecise. Fuzzy rough expectation is introduced and taken over the inventory costs and the constraint. The model is formulated in general form for n-items and in particular, is illustrated with three items for some numerical data. The optimum results are presented both in tabular form and graphically.
Key words: Multi-item inventory; Pontryagin’s maximum principle; Complementary and substitute item; Fuzzy rough expectation; Reliability.
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