A Tabu Search Approach for the Prize Collecting Traveling Salesman Problem (original) (raw)
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Hybrid Metaheuristic for the Prize Collecting Travelling Salesman Problem
Lecture Notes in Computer Science, 2008
The Prize Collecting Traveling Salesman Problem (PCTSP) can be associated to a salesman that collects a prize in each city visited and pays a penalty for each city not visited, with travel costs among the cities. The objective is to minimize the sum of travel costs and penalties, while including in the tour enough cities to collect a minimum prize. This paper presents one solution procedure for the PCTSP, using a hybrid metaheuristic known as Clustering Search (CS), whose main idea is to identify promising areas of the search space by generating solutions and clustering them into groups that are them explored further. The validation of the obtained solutions was through the comparison with the results found by CPLEX.
Fifth International Conference on Hybrid Intelligent Systems (HIS'05), 2005
Problem. It can be associated to a salesman that collects a prize in each city visited and pays a penalty for each city not visited, with travel costs among the cities. The objective is to minimize the sum of the costs of the trip and penalties, including in the tour an enough number of cities that allow collecting a minimum prize. This paper approaches new heuristics to solve the PCTSP, using a hybrid evolutionary algorithm, called Evolutionary Clustering Search (ECS) and an adaptation of this, called * CS, where the evolutionary component will be substituted by the metaheuristics GRASP and VNS. The validation of the obtained solutions will be through the comparison with the results found by a commercial solver that was able to solve only small size problems.
Prize Collecting Travelling Salesman Problem
Proceedings of 5th the International Conference on Operations Research and Enterprise Systems, 2016
The Prize Collecting Travelling Salesman Problem (PCTSP) is an important generalization of the famous Travelling Salesman Problem. It also arises as a sub problem in many variants of the Vehicle Routing Problem. In this paper, we provide efficient methods to solve the linear programming relaxation of the PCTSP. We provide efficient heuristics to obtain the Generalized Subtour Elimination Constraints (GSECs) for the PCTSP, and compare its performance with an optimal separation procedure. Furthermore, we show that a heuristic to separate the primitive comb inequalities for the TSP can be applied to separate the primitive comb inequalities introduced for the PCTSP. We evaluate the effectiveness of these inequalities in reducing the integrality gap for the PCTSP.
A branch-and-cut and MIP-based heuristics for the Prize-Collecting Travelling Salesman Problem
RAIRO - Operations Research
The Prize Collecting Traveling Salesman Problem (PCTSP) represents a generalization of the well-known Traveling Salesman Problem. The PCTSP can be associated with a salesman that collects a prize in each visited city and pays a penalty for each unvisited city, with travel costs among the cities. The objective is to minimize the sum of the costs of the tour and penalties, while collecting a minimum amount of prize. This paper suggests MIP-based heuristics and a branch-and-cut algorithm to solve the PCTSP. Experiments were conducted with instances of the literature, and the results of our methods turned out to be quite satisfactory.
Modeling and Solving the Traveling Salesman Problem with Priority Prizes
Pesquisa Operacional, 2018
This paper addresses the Traveling Salesman Problem with Priority Prizes (TSPPP), an extension of the classical TSP in which the order of the node visits is taken into account in the objective function. A prize p ki is received by the traveling salesman when node i is visited in the k-th order of the route, while a travel cost c i j is incurred when the salesman travels from node i to node j. The aim of the TSPPP is to find the maximum profit n-node tour. The problem can be seen as a TSP variant with a more general objective function, aiming at solutions that in some way consider the quality of customer service and the delivery priorities and costs. A natural representation for the TSPPP is here grounded in the point of view of Koopmans and Beckmann approach, according to which the problem is seem as a special case of the quadratic assignment problem (QAP). Given the novelty of this TSP variant, we propose different mixed integer programming models to appropriately represent the TSPPP, some of them based on the QAP. Computational experiments are also presented when solving the MIP models with a well-known optimization software, as well as with a tabu search algorithm.
The online Prize-Collecting Traveling Salesman Problem
Information Processing Letters, 2008
We study the online version of the Prize-Collecting Traveling Salesman Problem (PCTSP), a generalization of the Traveling Salesman Problem (TSP). In the TSP, the salesman has to visit a set of cities while minimizing the length of the overall tour. In the PCTSP, each city has a given weight and penalty, and the goal is to collect a given quota of the weights of the cities while minimizing the length of the tour plus the penalties of the cities not in the tour. In the online version, cities are disclosed over time. We give a 7/3-competitive algorithm for the problem, which compares with a lower bound of 2 on the competitive ratio of any deterministic algorithm. We also show how our approach can be combined with an approximation algorithm in order to obtain an O(1)-competitive algorithm that runs in polynomial time.
On Prize-collecting Tours and the Asymmetric Travelling Salesman Problem
International Transactions in Operational Research, 1995
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 psnalities 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 (PC'I~P). 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.
An Hybrid GRASP+VNS Metaheuristic for the Prize-Collecting Traveling Salesman Problem
In the Prize-Collecting Traveling Salesman Problem (PCTSP) we have to determine a tour visiting each vertex in the graph at most one time. If a given vertex is selected then an associated prize is collected, if a vertex is unrouted a penalty must be paid. We want to minimize an objective function balancing between the travel cost and the total penalties in a such way that a sufficiently large prize is collected. In this paper we present an hybrid metaheuristic that combines Greedy Randomized Adaptive Search Procedure (GRASP) and Variable Neigboorhood Search procedure as a local search. Experimental results show that it is potentially a powerful heuristic device, since it greatly enhace different features of these two approaches.
Traveling Salesman Problems with Profits
Transportation Science, 2005
Traveling salesman problems with profits (TSPs with profits) are a generalization of the traveling salesman problem (TSP), where it is not necessary to visit all vertices. A profit is associated with each vertex. The overall goal is the simultaneous optimization of the collected profit and the travel costs. These two optimization criteria appear either in the objective function or as a constraint. In this paper, a classification of TSPs with profits is proposed, and the existing literature is surveyed. Different classes of applications, modeling approaches, and exact or heuristic solution techniques are identified and compared. Conclusions emphasize the interest of this class of problems, with respect to applications as well as theoretical results.
Budgeted Prize-Collecting Traveling Salesman and Minimum Spanning Tree Problems
Mathematics of Operations Research, 2019
We consider constrained versions of the prize-collecting traveling salesman and the prize-collecting minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. Rooted variants of the problems have the additional constraint that a given vertex, the root, must be contained in the tour/tree. We present a 2-approximation algorithm for the rooted and unrooted versions of both the tree and tour variants. The algorithm is based on a parameterized primal–dual approach. It relies on first finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is, in a precise sense, just within budget. We improve upon the best-known guarantee of 2 + ε for the rooted and unrooted tour versions and 3 + ε for the rooted and unrooted tree versions. Our analysis extends to the setting with weighted vertices, in which we want to maxim...