A tale of two linear programming formulations for crashing project networks

Abstract

Problems associated with time-cost trade-offs in project networks, which are commonly referred to as crashing problems, date back nearly 60 years. Many prominent management science textbooks provide a traditional linear programming (LP) formulation for a classic project crashing problem, in which the time-cost trade-off for each activity is continuous (and linear) over a range of possible completion times. We have found that, for students who are being introduced to time-cost trade-offs and the principles of project crashing, an alternative LP formulation facilitates a greater conceptual understanding. Moreover, the alternative formulation uses only half of the decision variables in the traditional formulation and has fewer constraints for many problems encountered in management science textbooks. Results from an MBA section of operations management suggest that students prefer the alternative formulation. Additionally, we have developed an Excel workbook that generates all possible paths for a network, allows students to manually evaluate crashing decisions, and generates the alternative LP formulation. We demonstrate the workbook using a small synthetic example and a larger, real-world network from the literature. We also show that the alternative formulation can be adapted easily to accommodate discrete project crashing problems for which the time-cost tradeoffs for activities are not necessarily linear.