A polynomial time repeated cuts algorithm for the time cost tradeoff problem: The linear and convex crashing cost deadline problem

Abstract

The time cost tradeoff problem in project management, TCTP, is to achieve a given deadline on the project completion time by expediting the normal durations of activities at the cheapest cost possible. The linear TCTP, in which the expediting costs of each activity are linear, as a function of the number of time periods reduced, can be solved using linear programming. We present here an algorithm that solves the linear or convex costs TCTP that runs in polynomial time and calls only for a minimum s,t-cut routine at each iteration. This repeated cuts algorithm is related to the non-polynomial time algorithm by Phillips and Dessouky (1977), the PD-algorithm. The PD-algorithm reduces the project duration, at each iteration, by one time unit, at a minimum cost. The choice of the activities to expedite, in order to reduce the project duration by one unit, is determined by a solution to a minimum s,t-cut in a respective graph.We present here previously unknown properties of the PD-algorithm, and a new concept of cut-decomposition. These properties are used in devising the repeated cuts algorithm based on scaling. The repeated cuts algorithm solves in polynomial time, the linear as well as the convex TCTP. The algorithm solves the TCTP problem in polynomial time even when the durations and/or the target deadline are not necessarily integers.