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A Priori Optimization of the Probabilistic Traveling Salesman Problem

Author

Listed:
  • Gilbert Laporte

    (École des Houtes Etudes Commerciales, Montreal, Quebec, Canada)

  • François V. Louveaux

    (Notre-Dame de la Paix, Namur, Belgium)

  • Hélène Mercure

    (École des Hautes Études Commerciales, Montreal, Quebec, Canada)

Abstract

The probabilistic traveling salesman problem (PTSP) is defined on a graph G = ( V , E ), where V is the vertex set and E is the edge set. Each vertex v i has a probability p i of being present. With each edge ( v i , v j ) is associated a distance or cost c ij . In a first stage, an a priori Hamiltonian tour on G is designed. The list of present vertices is then revealed. In a second stage, the a priori tour is followed by skipping the absent vertices. The PTSP consists of determining a first-stage solution that minimizes the expected cost of the second-stage tour. The problem is formulated as an integer linear stochastic program, and solved by means of a branch-and-cut approach which relaxes some of the constraints and uses lower bounding functionals on the objective function. Problems involving up to 50 vertices are solved to optimality.

Suggested Citation

  • Gilbert Laporte & François V. Louveaux & Hélène Mercure, 1994. "A Priori Optimization of the Probabilistic Traveling Salesman Problem," Operations Research, INFORMS, vol. 42(3), pages 543-549, June.
    • Handle: RePEc:inm:oropre:v:42:y:1994:i:3:p:543-549
    DOI: 10.1287/opre.42.3.543
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