A matheuristic for the Team Orienteering Arc Routing Problem

Computers & Operations Research, 2016

The Team Orienteering Problem (TOP) is an attractive variant of the Vehicle Routing Problem (VRP). The aim is to select customers and at the same time organize the visits for a vehicle fleet so as to maximize the collected profits and subject to a travel time restriction on each vehicle. In this paper, we investigate the effective use of a linear formulation with polynomial number of variables to solve TOP. Cutting planes are the core components of our solving algorithm. It is first used to solve smaller and intermediate models of the original problem by considering fewer vehicles. Useful information are then retrieved to solve larger models, and eventually reaching the original problem. Relatively new and dedicated methods for TOP, such as identification of irrelevant arcs and mandatory customers, clique and independent-set cuts based on the incompatibilities, and profit/customer restriction on subsets of vehicles, are introduced. We evaluated our algorithm on the standard benchmark of TOP. The results show that the algorithm is competitive and is able to prove the optimality for 12 instances previously unsolved.

2019

This paper focuses on a recent variant of the Orienteering Problem (OP), namely the Capacitated Team OP (CTOP) which arises in the logistics industry. In this problem, each node is associated with a demand that needs to be satisfied and a score that need to be collected. Given a set of homogeneous fleet of vehicles, the objective is to find a path for each vehicle in order to maximize the total collected score, without violating the capacity and time budget. We propose an Iterated Local Search (ILS) algorithm for solving the CTOP. Two strategies, either accepting a new solution as long as it improves the quality of the solutions or accepting a new solution as long as there is no constraint violation, are implemented. For solving difficult instances, we simplify the move operator of local search in order to reduce the computational time. Instead of exploring all possible nodes in all paths to be moved, we only focus on nodes in the path with the least remaining amount of time. Comput...

Routing problems have many practical applications in distribution and logistics management. The Traveling Salesman Problem (TSP) and its variants lie at the heart of routing problems. The Orienteering Problem (OP) is a subset selection version of well-known TSP which comes from an outdoor sport played on mountains. In the OP, the traveller must finish its journey within a predetermined time (cost, distance), and gets a gain (profit, reward) from the visited nodes. The objective is to maximize the total gain that the traveller collects during the predetermined time. The OP is also named as the selective TSP since not all cities have to be visited. The Team Orienteering Problem (TOP) is the extension of OP by multiple-traveller. As far as we know, there exist a few formulations for the TOP. In this paper we present two new integer linear programming formulations (ILPFs) for the TOP with O(n 2) binary variables and O(n 2) constraints, where n is the number of nodes on the underlying graph. The proposed formulations can be directly used for the OP when we take the number of traveller as one. We demonstrate that, additional restrictions and/or side conditions can be easily imported for both of the formulations. The performance of our formulations is tested on the benchmark instances from the literature. The benchmark instances are solved via CPLEX 12.6 by using the proposed and existing formulations. The computational experiments demonstrate that both of the new formulations outperform the existing one. The new formulations are capable of solving optimally most of the benchmark instances, which have solved by using special heuristics so far. As a result, the proposed formulations can be used to find the optimal solution of small-and moderate-size real life OP and TOP by using an optimizer.

RAIRO - Operations Research, 2021

In this paper, a new version of the location-routing problem (LRP), named orienteering location-routing problem (OLRP) is investigated. The problem is composed of two-well known problems: team orienteering problem (TOP) and LRP. There are some challenging practical applications in logistics, tourism, military operations, and other fields, which can be modeled by OLRP. The problem is to consider the location and routing with a special objective function. In the OLRP, a set of nodes with specific scores is given, and some stations among candidate stations should be established. Moreover, there are some routes limited in length, which start from a station, visit some nodes and then return to the same station. Maximizing the sum of the collected scores is the goal of OLRP. To model the problem, an integer linear programming model is proposed. Against a commercial solver, a heuristic GRASP is developed for solving the standard test problems. Most test problems are found difficult to solv...

2015

The Team Orienteering Problem with Time Windows (TOPTW) is a well-known variant of the Vehicle Routing Problem (VRP) whose aim is to maximize the total amount of profit collected from the visited customers while taking into consideration some resource limitations. In this variant, customers must be served at most once in their time windows while respecting the predefined maximum travel time limit of each vehicle. In this paper we propose a column generation based algorithm to solve the linear relaxation of TOPTW. A dynamic programming algorithm is used to solve the subproblems in order to generate additional columns. Experiments conducted on the benchmark of TOPTW show the effectiveness of our algorithm and the strengthen of our formulation since we were able to prove the optimality of several instances by finding their integer solutions at the root node while solving the linear relaxation of the model.

2020

The Team Orienteering Problem (TOP) is an NP-hard routing problem in which a fleet of identical vehicles aims at collecting rewards (prizes) available at given locations, while satisfying restrictions on the travel times. In TOP, each location can be visited by at most one vehicle, and the goal is to maximize the total sum of rewards collected by the vehicles within a given time limit. In this paper, we propose a generalization of TOP, namely the Steiner Team Orienteering Problem (STOP). In STOP, we provide, additionally, a subset of mandatory locations. In this sense, STOP also aims at maximizing the total sum of rewards collected within the time limit, but, now, every mandatory location must be visited. In this work, we propose a new commodity-based formulation for STOP and use it within a cutting-plane scheme. The algorithm benefits from the compactness and strength of the proposed formulation and works by separating three families of valid inequalities, which consist of some gen...

Engineering Optimization

The orienteering problem (OP) is a routing problem that has numerous applications in various domains such as logistics and tourism. The objective is to determine a subset of vertices to visit for a vehicle so that the total collected score is maximized and a given time budget is not exceeded. The extensive application of the OP has led to many different variants, including the team orienteering problem (TOP) and the team orienteering problem with time windows. The TOP extends the OP by considering multiple vehicles. In this article, the team orienteering problem with variable profits (TOPVP) is studied. The main characteristic of the TOPVP is that the amount of score collected from a visited vertex depends on the duration of stay on that vertex. A mathematical programming model for the TOPVP is first presented and an algorithm based on iterated local search (ILS) that is able to solve modified benchmark instances is then proposed. It is concluded that ILS produces solutions which are comparable to those obtained by the commercial solver CPLEX for smaller instances. For the larger instances, ILS obtains good-quality solutions that have significantly better objective value than those found by CPLEX under reasonable computational times.

Transportation Science, 2005

In this article, we introduce a new arc routing problem that we call the profitable arc tour problem. This problem is defined on a graph in which profits and travel costs are associated with the arcs. The objective is to find a set of cycles in the graph that maximizes the collection of profit minus travel costs, subject to constraints limiting the number of times that profit is available on arcs and the maximal length of cycles. The problem is related both to constrained flow problems and to vehicle-routing problems. We tackle it from this standpoint and propose a branch-and-price algorithm for its solution. In the column-generation phase, the issue of the collection decisions while traveling through the arcs is addressed. In the branching phase, the fact that viewing solutions in terms of flow variables regularly induces an integer flow matrix leads us to introduce a branching method called the flow-splitting method. Finally, the relationships of this problem with constrained flow...

Information Processing Letters, 2015

In this article we present approximation algorithms for the Arc Orienteering Problem. Specifically, we give a O(log 2 m)−approximation algorithm in directed graphs, where m is the number of arcs, while in undirected graphs, we obtain a (6 + + o(1))−approximation algorithm for the general case and a (4 + )−approximation algorithm for instances with unit profits. Moreover, we obtain approximation algorithms for the Mixed Orienteering Problem.

Computers & Operations Research, 2015

This paper introduces a new arc routing problem for the optimization of a collaboration scheme among carriers. This yields to the study of a protable uncapacitated arc routing problem with multiple depots, where carriers collaborate to improve the prot gained. In the rst model the goal is the maximization of the total prot of the coalition of carriers, independently of the individual prot of each carrier. Then, a lower bound on the individual prot of each carrier is included. This lower bound may represent the prot of the carrier in the case no collaboration is implemented. The models are formulated as integer linear programs and solved through a branch-and-cut algorithm. Theoretical results, concerning the computational complexity, the impact of collaboration on prot and a game theoretical perspective, are provided. The models are tested on a set of 971 instances generated from 118 benchmark instances for the Privatized Rural Postman problem, with up to 102 vertices. All the 971 instances are solved to optimality within few seconds.