Solving the Project Time/Cost Tradeoff Problem Using the Minimal Cut Concept

Abstract

This paper introduces a solution procedure for solving the project time/cost tradeoff problem of reducing a project duration at a minimum cost. The solution to the time/cost problem is achieved by locating a minimal cut in a flow network derived from the original project network. This minimal cut is then utilized to identify the project activities which should experience a duration modification in order to achieve the total project reduction. The paper will document this cut-based procedure and provide a practical application to a project situation.

BonnTools: Mathematical Innovation for Layout and Timing Closure of Systems on a Chip

“…If we have found such a cut of finite cost, we can improve the timing at the lowest possible power cost per time unit by speeding up the arcs from S toS and slowing down (if possible) the arcs fromS to S. This optimality statement is proved in [32] subject to the simplifying assumptions that the delay/power dependence is linear and that we can realize arbitrary V t -values within a given interval, which today's libraries typically do not allow. Nevertheless, the linearity of the delay/power dependence approximately holds locally and the discrete choosable values are close enough.…”

Decision Support and Optimization in Shutdown and Turnaround Scheduling

“…Later, Phillips and Dessouky (1977) gave an improved version of the original algorithms in which iterative cut computations in a graph of critical jobs yield the piecewise linear time-cost trade-off curve that describes the trade-off between project duration t and associated cost for all t. The running time is polynomial in the number of breakpoints of the optimal time-cost trade-off curve, which may, however, be exponential in the input size (see Skutella 1998b). (A breakpoint of such a piecewise linear function is a point in which the function is continuous but not differentiable.)…”

“…This problem without time windows can be solved optimally in polynomial time in the input and the number of breakpoints of the curve (Fulkerson 1961, Kelley 1961, Phillips and Dessouky 1977. In fact, most of our real-world instances do not require the more sophisticated and time-consuming algorithm that respects also time windows by Elmaghraby and Kamburowski (1992).…”

On project scheduling with irregular starting time costs

“…Fulkerson and Kelley computed the project cost curve through parametric solution of a minimum cost ow problem obtained as a dual of the linear programming formulation of the problem. Phillips and Dessouky [26] subsequently showed that the problem may be solved as a sequence of minimum cut problems.…”