Abstract
We propose in this paper a new formulation for the stability of the Traveling Salesman Problem (TSP) compared with its probabilistic version, the Probabilistic Traveling Salesman Problem (PTSP). It is a real extension of the TSP, where the number of customers to be served each time is a random variable. That is only a subset of customers will need its services, moreover this subset varies from day to day. From the literature, several methods of resolution of the TSP have been proposed. In order to use these methods as they are for the PTSP came the idea of the study of stability. It is interested in finding cases where the solution for the TSP is also for the PTSP. First we survey and comment a number of easy TSPs. We also present via the notion of “Master tour” the stable problems in the TSP-PTSP context. An exact Branch and Bound is used for recognizing the TSPs that are not stable. Finally, we propose a new modification method, -different from the usual method of PTSP-, called the method of Taxi Driver, it takes the structure of the tour into consideration.
Highlights
The probabilistic traveling salesman problem Probabilistic Traveling Salesman Problem (PTSP) is a generalization of the classical traveling salesman problem Traveling Salesman Problem (TSP) and one of the most important stochastic routing problems, whose formulation explicitly contains probabilistic elements
The exact resolution algorithm gives the results as shown in Table 4: We observe that PTSP = TSP, which breaks the equality of the random variables Long(,μ) and Longopt, which clearly gives, the non-stability of TSP on the Monge matrices
Remark: For the uniqueness of the induced tour, we opt for the following convention, when we find the taxi driver is obliged to rectify the trajectory of the tour, we choose the node that not yet visited, the smallest in its governorate
Summary
The probabilistic traveling salesman problem PTSP is a generalization of the classical traveling salesman problem TSP and one of the most important stochastic routing problems, whose formulation explicitly contains probabilistic elements. Khaznaji: Stability Analysis for the Modification Method where Longopt (ξ ) is the cost of optimal solution of problem through the subset of customers ξ , V is the set of all customers and P(ξ ) is the probability of presence of the ξ. This strategy is certainly optimal but it presents several disadvantages. Consider an a priori tour across the n nodes of V and for each subset of customers ξ , the method of modification μ to generate a solution through ξ.
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