A new probability model for insuring critical path problem with heuristic algorithm

In order to obtain an adequate description of risk aversion for insuring critical path problem, this paper develops a new class of two-stage minimum risk problems. The first-stage objective function is to minimize the probability of total costs exceeding a predetermined threshold value, while the second-stage objective function is to maximize the insured task durations. For general task duration distributions, we adapt sample average approximation (SAA) method to probability objective function. The resulting SAA problem is a two-stage integer programming model, in which the analytical expression of second-stage value function is unavailable, we cannot solve it by conventional optimization algorithms. To avoid this difficulty, we design a new hybrid algorithm by combining dynamic programming method (DPM) and genotype-phenotype-neighborhood based binary particle swarm optimization (GPN-BPSO), where the DPM is employed to find the critical path in the second-stage programming problem. We conduct some numerical experiments via a critical path problem with 30 nodes and 42 arcs, and discuss the proposed risk averse model and the experimental results obtained by hybrid GPN-BPSO, hybrid genetic algorithm (GA) and hybrid BPSO. The computational results show that hybrid GPN-BPSO achieves the better performance than hybrid GA and hybrid BPSO, and the proposed critical path model is important for risk averse decision makers.