Prize Collecting Traveling Salesman Problem Research Articles

Metaheurísticas híbridas para resolução do problema do caixeiro viajante com coleta de prêmios

O Problema do Caixeiro Viajante com Coleta de Prêmios (PCVCP) pode ser associado a um caixeiro que coleta um prêmio em cada cidade visitada e paga uma penalidade para cada cidade não visitada, com um custo de deslocamento entre as cidades. O problema encontra-se em minimizar o somatório dos custos da viagem e penalidades, enquanto inclui na sua rota um número suficiente de cidades que lhe permita coletar um prêmio mínimo preestabelecido. Este trabalho contribui com o desenvolvimento de metaheurísticas híbridas para o PCVCP, baseadas em GRASP e métodos de busca em vizinhança variável (VNS/VND) para solucionar aproximadamente o PCVCP. De forma a validar as soluções obtidas, propõe-se uma formulação matemática a ser resolvida por um solver comercial, objetivando encontrar a solução ótima para o problema, sendo este solver aplicado a problemas de pequeno porte. Resultados computacionais demonstram a eficiência da abordagem híbrida proposta, tanto em relação à qualidade da solução final obtida quanto em relação ao tempo de execução.

Upper bounds for revenue maximization in a satellite scheduling problem

This paper presents upper bounds for the Satellite Revenue Selection and Schedulingproblem (SRSS). A compact model of this generalized Prize Collecting Traveling Salesman Problem with Time Windows is defined and enriched with valid inequalities based on task interval reasoning. The non-concavity of the objective function to be maximized is also studied. Finally a Russian Dolls approach combines bounds on nested sub-problems. These first upper bounds for the SRSS problem are compared to best known solutions of the benchmark of the optimization challenge organized by the French OR society.

On approximating a geometric prize-collecting traveling salesman problem with time windows

We study a scheduling problem in which jobs have locations. For example, consider a repairman that is supposed to visit customers at their homes. Each customer is given a time window during which the repairman is allowed to arrive. The goal is to find a schedule that visits as many homes as possible. We refer to this problem as the prize-collecting traveling salesman problem with time windows (TW-TSP). We consider two versions of TW-TSP. In the first version, jobs are located on a line, have release times and deadlines but no processing times. We present a geometric interpretation of TW-TSP on a line that generalizes the longest monotone subsequence problem. We present an O(log n) approximation algorithm for this case, where n denotes the number of jobs. This algorithm can be extended to deal with non-unit job profits. The second version deals with a general case of asymmetric distances between locations. We define a density parameter that, loosely speaking, bounds the number of zig-zags between locations within a time window. We present a dynamic programming algorithm that finds a tour that visits at least OPT/ density locations during their time windows. This algorithm can be extended to deal with non-unit job profits and processing times.

New classes of efficiently solvable generalized Traveling Salesman Problems

We consider the n‐city traveling salesman problem (TSP), symmetric or asymmetric,with the following attributes. In one case, a positive integer k and an ordering (1,..., n) ofthe cities is given, and an optimal tour is sought subject to the condition that for any pairi, j ∈ (1..., n), if j ≥ i + k, then i precedes j in the tour. In another case, position i in the tourhas to be assigned to some city within k positions from i in the above ordering. This case isclosely related to the TSP with time windows. In a third case, an optimal tour visiting m outof n cities is sought subject to constraints of the above two types. This is a special case ofthe Prize Collecting TSP (PCTSP). In any of the three cases, k may be replaced by city‐specificintegers k(i), i = 1,..., n. These problems arise in practice. For each class, we reducethe problem to that of finding a shortest source‐sink path in a layered network with a numberof arcs linear in n and exponential in the parameter k (which is independent of the problemsize). Besides providing linear time algorithms for the solution of these problems, the reductionto a shortest path problem also provides a compact linear programming formulation.Finally, for TSPs or PCTSPs that do not have the required attributes, these algorithms canbe used as heuristics that find in linear time a local optimum over an exponential‐sizeneighborhood.

The hot strip mill production scheduling problem: A tabu search approach

The Hot Strip Mill Production Scheduling Problem is a hard problem that arises in the steel industry when scheduling steel coil production. It can be modeled as a generalization of the Prize Collecting Traveling Salesman Problem with multiple and conflicting objectives and constraints. In this paper we formulate this problem as a mathematical program and propose a heuristic method to determine good approximate solutions. The heuristic is based on Tabu Search and a new idea called “Cannibalization”. Computational results on production data from Dofasco are presented and analyzed. Comparison with actual production schedules indicates that the proposed method could produce significantly better schedules.

A Lagrangian heuristic for the Prize CollectingTravelling Salesman Problem

In this paper, we consider the Prize Collecting Travelling Salesman Problem (PCTSP),which is a variant of the Travelling Salesman Problem (TSP), where a tour visiting eachnode at most once in a given graph has to be computed such that a prize is associated witheach node and a penalty has to be paid for every unvisited node: moreover, a knapsackconstraint guarantees that a sufficiently large prize is collected. We develop a Lagrangianheuristic and obtain an upper bound in the form of a feasible solution starting from a lowerbound to the problem recently proposed in the literature. We evaluate these bounds utilizingboth randomly generated instances and real ones with very satisfactory results.

On Prize‐collecting Tours and the Asymmetric Travelling Salesman Problem

We consider a variant of the Travelling Salesman Problem which is to determine a tour visiting each vertex in the graph at most at one time; if a vertex is left unrouted a given penalty has to be paid. The objective function is to find a balance between these penalties and the cost of the tour. We call this problem the Profitable Tour Problem(PTP). If, in addition, each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize is collected, we have the well‐known Prize‐collecting Travelling Salesman Problem (PCTSP). In this paper we summarize the main results presented in the literature, then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreover, we show, through computational experiments, that large size instances of the asymmetric PTP can be solved exactly.

On prize-collecting tours and the asymmetric travelling salesman problem

We consider a variant of the Travelling Salesman Problem which is to determine a tour visiting each vertex in the graph at most at one time; if a vertex is left unrouted a given penalty has to be paid. The objective function is to find a balance between these penalities and the cost of the tour. We call this problem the Profitable Tour Problem (PTP). If, in addition, each vertex is associated with a prize and there is a knapsack constraint which guarantees that a sufficiently large prize is collected, we have the well-known Prize-collecting Travelling Salesman Problem (PCTSP). In this paper we summarize the main results presented in the literature, then we give lower bounds for the asymmetric version of PTP and PCTSP. Moreover, we show, through computational experiments, that large size instances of the asymmetric PTP can be solved exactly.

The prize collecting traveling salesman problem: II. Polyhedral results

AbstractThe task of developing daily schedules for a steel rolling mill has been formulated as a Prize Collecting Traveling Salesman (PCTS) Problem, in which a salesman who gets a prize for every city he visits seeks a minimum‐cost tour including enough cities to collect a required amount of prize money. This paper addresses the facial structure of the PCTS polytope, the convex hull of solutions to the PCTS problem. In an earlier paper, we generalized to the PCTS polytope the subtour elimination inequalities for the Asymmetric Traveling Salesman (ATS) polytope. Here, we give a general method for deriving a facet defining inequality for the PCTS polytope from any facet defining inequality for the ATS polytope. We apply the procedure to several well‐known families of facet inducing inequalities for the ATS polytope: comb, odd CAT, SD, clique tree, and lifted cycle inequalities. We also extend the cloning and clique lifting procedure for the ATS polytope to the PCTS polytope.

The prize collecting traveling salesman problem

AbstractThe 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 vitis 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.