Polynomial-time algorithms to solve the single-item capacitated lot sizing problem with a 1-breakpoint all-units quantity discount

Abstract

This study investigates the single-item capacitated lot sizing problem considering a 1-breakpoint all-units quantity discount. Assuming the fixed ordering cost and the undiscounted unit purchase price to be non-increasing over the periods, the lot sizing problem under study is a special case of that with piecewise concave production cost function investigated by Koca et al. (2014). Koca’s DP algorithm solves our problem with a time complexity of O(T7), where T denotes the number of periods. It, however, fails in the case of large-sized instances in acceptable runtimes due to its higher complexity. Hence, the present study is a response to the need for more efficient algorithms to overcome this shortcoming. First, some properties of the optimal solution are proved. Second, these properties are exploited in an implicit enumeration exact algorithm. Third, certain speed-up techniques are employed to reduce the time complexity of the proposed algorithm from O(T5) to O(T4). Finally, a heuristic algorithm is presented for the problem. The proposed algorithms are compared with Koca’s DP algorithm and the commercial solver used to solve some test problems. Based on the runtimes observed, the exact algorithms proposed in this study are found to outperform both Koca’s DP one and the commercial solver. Moreover, the heuristic algorithm developed herein is found to be faster than both the exact algorithms in finding optimal solutions to most problem instances.