Characterization of the probabilistic traveling salesman problem

We show that stochastic annealing can be successfully applied to gain new results on the probabilistic traveling salesman problem. The probabilistic "traveling salesman" must decide on an a priori order in which to visit n cities (randomly distributed over a unit square) before learning that some cities can be omitted. We find the optimized average length of the pruned tour follows E((L) over bar (pruned))=rootnp(0.872-0.105p)f(np), where p is the probability of a city needing to be visited, and f(np)-->1 as np-->infinity. The average length of the a priori tour (before omitting any cities) is found to follow E(L-a priori)=rootn/pbeta(p), where beta(p)=1/[1.25-0.82 ln(p)] is measured for 0.05less than or equal topless than or equal to0.6. Scaling arguments and indirect measurements suggest that beta(p) tends towards a constant for p<0.03. Our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths, exploiting the sampling error to provide the analog of thermal fluctuations in simulated (thermal) annealing. The method has general application to the optimization of functions whose cost to evaluate rises with the precision required

S-ACO: An Ant-Based Approach to Combinatorial Optimization Under Uncertainty

Abstract. A general-purpose, simulation-based algorithm S-ACO for solving stochastic combinatorial optimization problems by means of the ant colony optimization (ACO) paradigm is investigated. Whereas in a prior publication, theoretical convergence of S-ACO to the globally opti-mal solution has been demonstrated, the present article is concerned with an experimental study of S-ACO on two stochastic problems of fixed-routes type: First, a pre-test is carried out on the probabilistic traveling salesman problem. Then, more comprehensive tests are performed for a traveling salesman problem with time windows (TSPTW) in the case of stochastic service times. As a yardstick, a stochastic simulated annealing (SSA) algorithm has been implemented for comparison. Both approaches are tested at randomly generated problem instances of different size. It turns out that S-ACO outperforms the SSA approach on the considered test instances. Some conclusions for fine-tuning S-ACO are drawn.

A multiobjective metaheuristic for a mean-risk static stochastic knapsack problem

Stochastic knapsack problem, Stochastic combinatorial optimisation, Mean-risk objectives, Multiobjective combinatorial optimisation, Multiobjective metaheuristics,