A Polynomial Time Branch and Bound Algorithm for the Single Item Economic Lot Sizing Problem with All Units Discount and Resale

The purpose of this paper is to present a polynomial time algorithm which determines the lot sizes for purchase component in Material Requirement Planning (MRP) environments with deterministic time-phased demand with zero lead time. In this model, backlog is not permitted, the unit purchasing price is based on the all-units discount system and resale of the excess units is possible at the ordering time. The properties of an optimal order policy are argued and on the basis of them, a branch and bound algorithm is presented to construct an optimal sequence of order policies. In the proposed B&B algorithm, some useful fathoming rules have been proven to make the algorithm very efficient. By defining a rooted tree graph, it has been shown that the worst-case time complexity function of the presented algorithm is polynomial. Finally, some test problems which are randomly generated in various environments are solved to show the efficiency of the algorithm. Received: 5 March 2012; Accepted: 20 May 2012