Optimal material ordering policy and allocation rule for a manufacturer making multiple products

Material ordering and allocation are important decisions for manufactures making multiple products, because those firms usually possess flexible production systems which can produce different products based on the same raw material. In this paper, we investigate the ordering policy (OP) and allocation rule (AR) of the raw materials for a manufacturer selling multiple products. The manufacturer's decision-making problem is analyzed under three scenarios: (1) joint decisions on OP and AR, (2) fixed AR, and (3) predetermined OP. We show that the latter two are not special cases of the first scenario, and they require different solution methods. Our objective is to derive the optimal solutions analytically. For the first scenario, we obtain the closed-form solution that is indeed optimal for the nonconcave profit function. For the fixed AR scenario, the products with twice-differentiable demands are studied, and the exact optimal OP for the raw material is achieved. Finally, if the OP is predetermined, we prove that the profit function is concave in AR and provide the associated optimality conditions, for which the optimal AR can be reached numerically. Different from the pervious heuristic approaches, these mathematically tractable solutions are easy to be applied by the practitioners.