A lagrangian relaxation and ACO hybrid for resource constrained project scheduling with discounted cash flows

Abstract

We consider a project scheduling problem where a number of tasks need to be scheduled. The tasks share resources, satisfy precedences, and all tasks must be completed by a common deadline. Each task is associated with a cash flow (positive or negative value) from which a “net present value” is computed dependent upon its completion time. The objective is to maximize the cumulative net present value of all tasks. We investigate (1) Lagrangian relaxation methods based on list scheduling, (2) ant colony optimization and hybrids of (1) and (2) on benchmark datasets consisting of up to 120 tasks. Considering lower bounds, i.e., maximizing the net present value, the individual methods prove to be effective but are outperformed by the hybrid method. This difference is accentuated when the integrality gaps are large.

Appendix 1: ACO using sum rule

Table 8 shows ACO with the default model compared with ACO with the sum rule. The default model consists of a distribution for each position or variable from which a task is selected. In the sum rule, a task is selected by summing over the pheromone values of that task over the preceding positions:

$$\begin{aligned} \mathbf{P}(\pi _i = k) = \frac{[\sum _{l=1}^{i}\tau _{lk}] \eta _{k}^{\beta }}{\sum _{j\in \mathcal{J} {\setminus } \{\pi _{1},\ldots ,\pi _{i-1}\}} \left( [\sum _{l=1}^{i}\tau _{lj}] \eta _{j}^{\beta }\right) } \end{aligned}$$

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Table 8 ACO using default model and ACO using sum rule.