A heuristic approach for multiple item and location ordering problem with quantity discount and capacity constraint

This paper addresses a geographically distributed vending machine inventory control problem where the multiple product choices in every vending machine and the varied demand for each product in every location's vending machine exist. The ordering quantity for each product and the desirable number of each product type in a vending machine must be simultaneously decided to maximize the total expected profit. Considering that the ordering costs are in piece-wise function and only one type of product is allocated in a slot of a vending machine, the proposed problem correlates to the piece-wise constrained nonlinear integer-programming problem. There is difficulty in deriving the exact optimal solutions to this problem since its computational complexity appears to be a nondeterministic polynomial problem. This paper presents a novel heuristic approach in finding (1) the ordering quantity for each product, (2) the product types, and (3) the corresponding quantity allocated in each vending machine. The numerical results show the effectiveness of the proposed methodology in solving the aforementioned problem. In this paper, the solutions found by this study's approach are as well as those found by the brute force method, and if not equivalent, perform better than those by the Lingo software.