Prize Collecting Traveling Salesman Problem Research Articles

A Novel Algorithm for Solving the Prize Collecting Traveling Salesman Problem Based on DNA Computing.

DNA computing is a new pattern of computing that combines biotechnology and information technology. As a new technology born in less than three decades, it has developed at an extremely rapid rate, which can be attributed to its advantages, including high parallelism, powerful data storage capacity, and low power consumption. Nowadays, DNA computing has become one of the most popular research fields worldwide and has been effective in solving certain combinatorial optimization problems. In this study, we use the Adleman-Lipton model based on DNA computing for solving the Prize Collecting Traveling Salesman Problem (PCTSP) and demonstrate the feasibility of this model. Then, we design a simulation experiment of the model to solve some open instances of PCTSP. The results illustrate that the model can satisfactorily solve these instances. Finally, the comparison with the results of the Clustering Search algorithm and the Greedy Stochastic Adaptive Search Procedure/Variable Neighborhood Search method reveals that the optimal solutions obtained by this simulation experiment are significantly superior to those of the other two algorithms in all instances. This research also provides a method for proficiently solving additional combinatorial optimization problems.

APPROXIMATION ALGORITHMS WITH CONSTANT FACTORS FOR A SERIES OF ASYMMETRIC ROUTING PROBLEMS

In this paper, the first fixed-ratio approximation algorithms are proposed for the series of asymmetric settings of the well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all the proposed algorithms rely on original reductions of the considered problems to auxiliary instances of the Asymmetric Traveling Salesman Problem and employ the breakthrough approximation results for this problem obtained recently by O. Svensson, J. Tarnawski, L. Végh, V. Traub and J. Vygen. On the other hand, approximation of the cycle cover problem was proved by more deep extension of their approach.

Routing Replenishment Workers: The Prize Collecting Traveling Salesman Problem in Scattered Storage Warehouses

Many online retailers apply scattered (or mixed-shelves) storage in the picking areas of their warehouses. Instead of keeping unit loads together, individual pieces of stock keeping units (SKUs) are stored on various shelves throughout the warehouse. This storage strategy increases the probability that—whatever it is that customers order jointly—somewhere in the warehouse these products will be located in close proximity. Hence, a picker can retrieve them without excessive unproductive walking. The price for this advantage on the picking side, however, is additional effort for the replenishment workers (also denoted as stowers) when restocking the shelves. Instead of moving only a single homogeneous unit load toward an SKU’s designated storage position, each stower has to travel along multiple open shelf spaces until all products on the cart are stored on shelves. The resulting stower routing problem is equivalent to the well-known prize collecting traveling salesman problem (PCTSP). While the PCTSP for general graphs is known to be strongly [Formula: see text]-hard, we show that in a warehousing environment, where all open storage positions are located along parallel aisles, it is only binary [Formula: see text]-hard. The special parallel-aisle structure allows us to derive an exact solution algorithm with pseudo-polynomial runtime, which solves even instances with hundreds of open storage positions to proven optimality in just a few seconds. Our computational tests show that the performance gains of an optimized stowing process over the status quo, where stowers operate without decision support, are significant. Especially, when the fill level of a warehouse is high, directing stowers on optimized routes promises huge improvements. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by the German Science Foundation/Deutsche Forschungsgemeinschaft (DFG) by the grant “Routing of human and automated order pickers in modern warehouses” (BO 3148/14-1 and BO 1972/2-1). Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0173 .

Hybrid genetic algorithm for undirected traveling salesman problems with profits

AbstractThe orienteering problem (OP) and prize‐collecting traveling salesman problem (PCTSP) are two typical TSPs with profits, in which each vertex has a profit and the goal is to visit several vertices to optimize the collected profit and travel costs. The OP aims to collect the maximum profit without exceeding the given travel cost. The PCTSP seeks to minimize the travel costs while ensuring a minimum profit threshold. This study introduces a hybrid genetic algorithm that addresses both the OP and PCTSP under a unified framework. The algorithm combines an extended edge‐assembly crossover operator to produce promising offspring solutions, and an effective local search to ameliorate each offspring solution. The algorithm is further enforced by diversification‐oriented mutation and population‐diversity management. Extensive experiments demonstrate that the method competes favorably with the best existing methods in terms of both the solution quality and computational efficiency. Additional experiments provide insights into the roles of the key components of the proposed method.

Reinforcement Learning With Multiple Relational Attention for Solving Vehicle Routing Problems.

In this article, we study the reinforcement learning (RL) for vehicle routing problems (VRPs). Recent works have shown that attention-based RL models outperform recurrent neural network-based methods on these problems in terms of both effectiveness and efficiency. However, existing RL models simply aggregate node embeddings to generate the context embedding without taking into account the dynamic network structures, making them incapable of modeling the state transition and action selection dynamics. In this work, we develop a new attention-based RL model that provides enhanced node embeddings via batch normalization reordering and gate aggregation, as well as dynamic-aware context embedding through an attentive aggregation module on multiple relational structures. We conduct experiments on five types of VRPs: 1) travelling salesman problem (TSP); 2) capacitated VRP (CVRP); 3) split delivery VRP (SDVRP); 4) orienteering problem (OP); and 5) prize collecting TSP (PCTSP). The results show that our model not only outperforms the learning-based baselines but also solves the problems much faster than the traditional baselines. In addition, our model shows improved generalizability when being evaluated in large-scale problems, as well as problems with different data distributions.

Formulations and a Lagrangian relaxation approach for the prize collecting traveling salesman problem

AbstractThe prize collecting traveling salesman problem (PCTSP) is a generalization of the traveling salesman problem (TSP) in which a tour starting at a root node must visit a subset of nodes to collect a prescribed amount of total prize minimizing the summation of travel costs and penalty costs associated with nonvisited nodes. We propose new formulations and compare them with adaptations to the PCTSP of strong formulations from the literature for the asymmetric TSP. One of the formulations proposed in this paper provided the tighter linear relaxation bounds on computational experiments based on benchmark instances for the asymmetric TSP. We then develop a Lagrangian relaxation approach embedding heuristic schemes to obtain both lower and upper bounds. The proposed heuristic has shown to be successful in obtaining optimal or near‐optimal solutions with information from the Lagrangian subproblems. Our Lagrangian approach can thereupon provide small optimality gaps, being an efficient alternative to bound optimal values of large instances.

A Branch-and-Cut and MIP-based heuristics for the Prize-Collecting Travelling Salesman Problem

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.

Multi-Dimensional User Preference Path Scheme Based on the Prize Collecting Traveling Salesman Problem

Multi-Dimensional User Preference Path Scheme Based on the Prize Collecting Traveling Salesman Problem

Prize Collecting Traveling Salesman Problem with Ridesharing

The Prize Collecting Traveling Salesman Problem with Ridesharing is a model that joins elements from the Prize Collecting Traveling Salesman and the collaborative transport. The salesman is the driver of a capacitated vehicle and uses a ridesharing system to minimize travel costs. There are a penalty and a bonus associated with each vertex of a graph, G, that represents the problem. There is also a cost associated with each edge of G. The salesman must choose a subset of vertices to be visited so that the total bonus collection is at least a given a parameter. The length of the tour plus the sum of penalties of all vertices not visited is as small as possible. There is a set of persons demanding rides. The ride request consists of a pickup and a drop off location, a maximum travel duration, and the maximum amount the person agrees to pay. The driver shares the cost associated with each arc in the tour with the passengers in the vehicle. Constraints from ride requests, as well as the capacity of the car, must be satisfied. We present a mathematical formulation for the problem investigated in this study and solve it in an optimization tool. We also present three heuristics that hybridize exact and heuristic methods. These algorithms use a decomposition strategy that other enriched vehicle routing problems can utilize.

A Unified PTAS for Prize Collecting TSP and Steiner Tree Problem in Doubling Metrics

We present a unified (randomized) polynomial-time approximation scheme (PTAS) for the prize collecting traveling salesman problem (PCTSP) and the prize collecting Steiner tree problem (PCSTP) in doubling metrics. Given a metric space and a penalty function on a subset of points known as terminals, a solution is a subgraph on points in the metric space whose cost is the weight of its edges plus the penalty due to terminals not covered by the subgraph. Under our unified framework, the solution subgraph needs to be Eulerian for PCTSP, while it needs to be a tree for PCSTP. Before our work, even a QPTAS for the problems in doubling metrics is not known. Our unified PTAS is based on the previous dynamic programming frameworks proposed in Talwar (STOC 2004) and Bartal, Gottlieb, Krauthgamer (STOC 2012). However, since it is unknown which part of the optimal cost is due to edge lengths and which part is due to penalties of uncovered terminals, we need to develop new techniques to apply previous divide-and-conquer strategies and sparse instance decompositions.

Traveling Salesman Problem with Optional Bonus Collection, Pickup Time and Passengers

This study introduces a variant of the Traveling Salesman Problem, named Traveling Salesman Problem with Optional Bonus Collection, Pickup Time and Passengers (PCVP-BoTc). It is a variant that incorporates elements of the Prize Collecting Traveling Salesman Problem and Ridesharing into the PCV. The objective is to optimize the revenue of the driver, which selectively defines which delivery or collection tasks to perform along the route. The economic effect of the collection is modeled by a bonus. The model can be applied to the solution of hybrid routing systems with route tasks and solidary transport. The driver, while performing the selected tasks, can give rides to persons who share route costs with him. Passengers are protected by restrictions concerning the maximum value they agree to pay for a ride and maximum travel duration. The activity of collecting the bonus in each locality demands a specific amount of time, affects the route duration, and is interconnected with the embarkment of passengers. Two mathematical formulations are presented for the problem and validated by a computational experiment using a solver. We propose four heuristic algorithms; three of them are hybrid metaheuristics. We tested the mathematical formulation implementations for 24 instances and the heuristic algorithms for 48.

Weighted matching with pair restrictions

The weighted matroid parity problems for the matching matroid and gammoids are among the very few cases for which the weighted matroid parity problem is polynomial time solvable. In this work we extend these problems to a general revenue function for each pair, and show that the resulting problem is still solvable in polynomial time via a standard weighted matching algorithm. We show that in many other directions, extending our results further is impossible (unless P = NP). One consequence of the new polynomial time algorithm is that it demonstrates, for the first time, that a prize-collecting assignment problem with “pair restriction” is solved in polynomial time. The prize collecting assignment problem is a relaxation of the prize-collecting traveling salesman problem which requires, for any prescribed pair of nodes, either both nodes of the pair are matched or none of them are. It is shown that the prize collecting assignment problem is equivalent to the prize collecting cycle cover problem which is hence solvable in polynomial time as well.

Robust optimization for routing problems on trees

The prize-collecting travelling salesman problem (pc-tsp) is a known variant of the classical travelling salesman problem. In the pc-tsp, we are given a weighted graph $$G=(V,E)$$ with edge weights $$\ell :E\rightarrow {\mathbb {R}}_+$$ , a special vertex $$r\in V$$ , penalties $$\pi :V\rightarrow {\mathbb {R}}_+$$ and the goal is to find a closed tour K such that $$r\in V(K)$$ and such that the cost $$\ell (K)+\pi (V{\setminus } V(K))$$ , which is the sum of the weights of the edges in the tour and the cost of the vertices not spanned by K, is minimized. In this paper, we study the pc-tsp from a viewpoint of robust optimization. In our setting, an operator must find a closed tour in a (known) edge-weighted tree $$T=(V,E)$$ starting and ending in the depot r while some of the edges may be blocked by “avalanches” defining the scenario $$\xi $$ . The cost $$f(K,\xi )$$ of a tour K in scenario $$\xi $$ is the cost resulting from “shortcutting” K, i.e. from restricting K to the nodes which are reachable from r in scenario $$\xi $$ , i.e. in the graph $$T {\setminus } \xi =(V,E{\setminus }\xi )$$ . In the absolute robust version of the problem one searches for a tour which minimizes the worst-case cost over all scenarios, while in the deviation robust counterpart, the regret, that is, the deviation from an optimum solution for a particular scenario, is sought to be minimized. We show that both versions of the problem can be solved in polynomial time on trees.

The Canadian Tour Operator Problem on paths: tight bounds and resource augmentation

In the prize-collecting travelling salesman problem, we are given a weighted graph $$G=(V,E)$$G=(V,E) with edge weights $$\ell :E\rightarrow \mathbb {R}_+$$l:EźR+, a special vertex $$r\in V$$rźV, penalties $$\pi :V\rightarrow \mathbb {R}_+$$ź:VźR+ and the goal is to find a closed tour $$T$$T such that $$r\in V(T)$$rźV(T) and such that the cost $$\ell (T)+\pi (V\setminus V(T))$$l(T)+ź(V\V(T)), which is the sum of the edges in the tour and the cost of the vertices not spanned by $$T$$T, is minimized. We consider an online variant of the prize-collecting travelling salesman problem related to graph exploration. In the Canadian Tour Operator Problem the task is to find a closed route for a tourist bus in a given network $$G=(V,E)$$G=(V,E) in which some edges are blocked by avalanches. An online algorithm learns from a blocked edge only when reaching one of its endpoints. The bus operator has the option to avoid visiting each node $$v\in V$$vźV by paying a refund of $$\pi (v)$$ź(v) to the tourists. The goal consists of minimizing the sum of the travel costs and the refunds. We study the problem on a simple (weighted) path and prove tight bounds on the competitiveness of deterministic algorithms. Specifically, we give an algorithm with competitive ratio equal to the golden ratio $$\phi =(1+\sqrt{5})/2$$ź=(1+5)/2. We also study the effect of resource augmentation, where the online algorithm either pays a discounted cost for traversing edges or for the penalties.

A Tabu Search Approach for the Prize Collecting Traveling Salesman Problem

Abstract The Prize Collecting Traveling Salesman Problem is a generalization of the Traveling Salesman Problem. A salesman collects a prize for each visited city and pays a penalty for each non visited city. The objective is to minimize the sum of the travel costs and penalties, but collecting a minimum pre-established amount of prizes. This problem is here addressed by a simple, but efficient tabu search approach which had improved several upper bounds of the considered instances.

Tourist Activity Simulation Model for Assessing Real-Time Tour Information Systems

The purpose of this article is to develop a framework for assessing the effects of tour information systems and information dissemination strategies by incorporating tourists’ decision processes into an activity-based model. There are two important features in tourists’ decision-making processes: (1) They generally have limited information and experiences on tour sites, and (2) their value of time is higher than normal travelers' because of their time limitations. These two features are, however, the main reasons that make tour information services attractive to tourists and thus should also be incorporated into a modeling framework for investigating the effects of the real-time tour information system. This study develops an agent-based simulation model that incorporates activity chaining behavior into tour scheduling systems. The underlying assumption used in the model is that tourists schedule their activities within their time and budget constraints to maximize their utilities. The model seeks individuals’ optimal tour schedules by solving a prize-collecting multiple-day traveling salesman problem (PC-MD-TSP). The simulation model also includes real-time and end-of-day rescheduling capabilities by taking tourists’ unpredicted delays into consideration. From a numerical test, the pros and cons of tour information services are found and the optimal strategy for a tour information service is also discussed. The results show that tour information services increase tourists’ satisfaction with less expenditure by allowing tourists to visit more places in an effective way.

A branch‐and‐cut algorithm for the undirected prize collecting traveling salesman problem

AbstractGiven an undirected graph with edge costs and vertex prizes, the aim of the Prize Collecting Traveling Salesman Problem (PCTSP) is to find a simple cycle minimizing the total edge cost while collecting at least a minimum amount of prizes. In this article, we present a branch‐and‐cut algorithm to solve the PCTSP. We have adapted and implemented some classical polyhedral results for the PCTSP and derived inequalities from cuts designed for the Orienteering Problem. Computational results on instances with more than 500 vertices are reported. © 2009 Wiley Periodicals, Inc. NETWORKS, 2009

Approximation algorithms for group prize-collecting and location-routing problems

In this paper we develop approximation algorithms for generalizations of the following three known combinatorial optimization problems, the Prize-Collecting Steiner Tree problem, the Prize-Collecting Travelling Salesman Problem and a Location-Routing problem. Given a graph G = ( V , E ) on n vertices and a length function on its edges, in the grouped versions of the above mentioned problems we assume that V is partitioned into k + 1 groups, { V 0 , V 1 , … , V k } , with a penalty function on the groups. In the Group Prize-Collecting Steiner Tree problem the aim is to find S , a collection of groups of V and a tree spanning the rest of the groups not in S , so as to minimize the sum of the costs of the edges in the tree and the costs of the groups in S . The Group Prize-Collecting Travelling Salesman Problem, is defined analogously. In the Group Location-Routing problem the customer vertices are partitioned into groups and one has to select simultaneously a subset of depots to be opened and a collection of tours that covers the customer groups. The goal is to minimize the costs of the tours plus the fixed costs of the opened depots. We give a ( 2 − 1 n − 1 ) I -approximation algorithm for each of the three problems, where I is the cardinality of the largest group.

An iterated local search heuristic for the capacitated prize-collecting travelling salesman problem

This paper considers a variant of the travelling salesman problem named the capacitated prize-collecting travelling salesman problem (CPCTSP), which is derived from the colour-coating production scheduling in a cold rolling mill. The objective of the CPCTSP is to minimize the travel cost and the penalties paid for unvisited customers in such a way that a sufficiently large prize is collected and the demand of the visited customers does not exceed the salesman's capacity. For this problem, we propose an iterated local search (ILS) heuristic adopting guided kick and enhanced dynasearch. The experimental results on randomly generated instances show that the proposed heuristic outperforms the improved tabu search algorithm using frequency-based memory, and the further experimental results on instances collected from real colour-coating production also show that the proposed ILS algorithm is more effective and efficient than the currently adopted manual scheduling method.

The online Prize-Collecting Traveling Salesman Problem

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.