The prize collecting traveling salesman problem.

*(English)*Zbl 0676.90089Summary: The following is a valid model for an important class of scheduling and routing problems. A salesman who travels between pairs of cities at a cost depending only on the pair, gets a prize in every city that he visits and pays a penalty to every city that he fails to visit, wishes to minimize his travel costs and net penalties, while visiting enough cities to collect a prescribed amount of prize money. We call this problem the Prize Collecting Traveling Salesman Problem (PCTSP). This paper discusses structural properties of the PCTS polytope, the convex hull of solutions to the PCTSP. In particular, it identifies several families of facet defining inequalities for this polytope. Some of these inequalities are related to facets of the ordinary TS polytope, others to facets of the knapsack polytope. They can be used in algorithms for the PCTSP either as cutting planes or as ingredients of a Lagrangean optimand.

##### MSC:

90C35 | Programming involving graphs or networks |

90B35 | Deterministic scheduling theory in operations research |

52Bxx | Polytopes and polyhedra |

90C27 | Combinatorial optimization |

##### Keywords:

scheduling; routing problems; Prize Collecting Traveling Salesman Problem; polytope; facet defining inequalities; knapsack polytope; cutting planes
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##### References:

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