Variable neighborhood search for the travelling deliveryman problem

A travelling deliveryman needs to find a tour such that the total waiting time of all the customers he has to visit is minimum. The deliveryman starts his tour at a depot, travelling at constant velocity. In this paper we suggest a general variable neighborhood search based heuristic to solve this NP-hard combinatorial optimization problem. We combine several classical neighborhood structures and design data structure to store and update the incumbent solution efficiently. In this way, we are able to explore neighborhoods as efficiently as when solving the travelling salesman problem. Computational results obtained on usual test instances show that our approach outperforms recent heuristics from the literature.     

A general variable neighborhood search for solving the uncapacitated single allocation p-hub median problem

We present a new general variable neighborhood search approach for the uncapacitated single allocation p-hub median problem in networks. This NP hard problem is concerned with locating hub facilities in order to minimize the traffic between all origin-destination pairs. We use three neighborhoods and efficiently update data structures for calculating new total flow in the network. In addition to the usual sequential strategy, a new nested strategy is proposed in designing a deterministic variable neighborhood descent local search. Our experimentation shows that general variable neighborhood search based heuristics outperform the best-known heuristics in terms of solution quality and computational effort. Moreover, we improve the best-known objective values for some large Australia Post and PlanetLab instances. Results with the new nested variable neighborhood descent show the best performance in solving very large test instances.     

A heuristic for vehicle fleet mix problem using tabu search and set partitioning

The vehicle fleet mix problem is a special case of the vehicle routing problem where customers are served by a heterogeneous fleet of vehicles with various capacities. An efficient heuristic for determining the composition of a vehicle fleet and travelling routes was developed using tabu search and by solving set partitioning problems. Two kinds of problems have appeared in the literature, concerning fixed cost and variable cost, and these were tested for evaluation. Initial solutions were found using the modified sweeping method. Whenever a new solution in an iteration of the tabu search was obtained, optimal vehicle allocation was performed for the set of routes, which are constructed from the current solution by making a giant tour. Experiments were performed for the benchmark problems that appeared in the literature and new best-known solutions were found.     

A general variable neighborhood search variants for the travelling salesman problem with draft limits

In this paper, we present two general variable neighborhood search (GVNS) based variants for solving the traveling salesman problem with draft limits (TSPDL), a recent extension of the traveling salesman problem. TSPDL arises in the context of maritime transportation. It consists of finding optimal Hamiltonian tour for a given ship which has to visit and deliver products to a set of ports while respecting the draft limit constraints. The proposed methods combine ideas in sequential variable neighborhood descent within GVNS. They are tested on a set of benchmarks from the literature as well as on a new one generated by us. Computational experiments show remarkable efficiency and effectiveness of our new approach. Moreover, new set of benchmarks instances is generated.     

Tabu search for solving the black-and-white travelling salesman problem

The black-and-white travelling salesman problem (BWTSP) is an extension to the well-known TSP by partitioning the set of vertices into black and white vertices, and imposing cardinality and length constraints between two consecutive black vertices in a Hamiltonian tour. BWTSP has various applications in aircraft routing, telecommunication network design and logistics. In this paper, we develop several tabu search (TS) heuristics for solving the BWTSP. Our TS is built upon a new efficient neighbourhood structure, which exploits both the permutation and knapsack features of BWTSP. We also embed our TS as a heuristic procedure to improve the upper bound in a mixed-integer linear programming method. Extensive computational experiment on both benchmark and randomly generated instances shows effectiveness and efficiency of our algorithms. Our algorithms are able to obtain optimal and near optimal solutions to small instances in seconds, and find feasible solutions to large instances that have not been solved by the existing methods in the literature.     

Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.     

Formulations and Benders decomposition algorithms for multidepot salesmen problems with load balancing

This paper describes new models and exact solution algorithms for the fixed destination multidepot salesmen problem defined on a graph with n nodes where the number of nodes each salesman is to visit is restricted to be in a predefined range. Such problems arise when the time to visit a node takes considerably longer as compared to the time of travel between nodes, in which case the number of nodes visited in a salesman’s tour is the determinant of their ‘load’. The new models are novel multicommodity flow formulations with O(n²) binary variables, which is contrary to the existing formulations for the same (and similar) problems that typically include O(n³) binary variables. The paper also describes Benders decomposition algorithms based on the new formulations for solving the problem exactly. Results of the computational experiments on instances derived from TSPLIB show that some of the proposed algorithms perform remarkably well in cases where formulations solved by a state-of-the-art optimization code fail to yield optimal solutions within reasonable computation time.     

The Vehicle Routing-Allocation Problem: A unifying framework

Summary In this paper the Vehicle Routing-Allocation Problem (VRAP) is presented. In VRAP not all customers need be visited by the vehicles. However customers not visited either have to be allocated to some customer on one of the vehicle tours or left isolated. We concentrate our discussion on the Single Vehicle Routing-Allocation Problem (SVRAP). An integer linear programming formulation of SVRAP is presented and we show how SVRAP provides a unifying framework for understanding a number of the papers and problems presented in the literature. Specifically the covering tour problem, the covering salesman problem, the median tour problem, the maximal covering tour problem, the travelling salesman problem, the generalised travelling salesman problem, the selective travelling salesman problem, the prize collecting travelling salesman problem, the maximum covering/shortest path problem, the maximum population/shortest path problem, the shortest covering path problem, the median shortest path problem, the minimum covering/shortest path problem and the hierarchical network design problem are special cases/variants of SVRAP.     

A location-routing problem in glass recycling

In this study, a location-routing problem encountered in glass recycling is addressed. We formulate a combined maximal covering location problem in the presence of partial coverage and selective traveling salesman problem to determine the location of bottle banks and the route of a collecting vehicle that will daily visit a number of customers and the bottle banks. We propose a nested heuristic procedure to solve the problem. The outer loop of the heuristic is based on variable neighborhood search while the inner loop solves the traveling salesman problem on the locations defined. The performance of the heuristic procedure is demonstrated with computational experimentation on instances that are both randomly generated and are taken from the literature. An application of the procedure on a case study using a geographical information system is also reported.     

Creating very large scale neighborhoods out of smaller ones by compounding moves

This paper discusses neighborhood search algorithms where the size of the neighborhood is “very large” with respect to the size of the input data. We concentrate on such a very large scale neighborhood (VLSN) search technique based on compounding independent moves (CIM) such as 2-opts, swaps, and insertions. We present a systematic way of creating and searching CIM neighborhoods for routing problems with side constraints. For such problems, the exact search of the CIM neighborhood becomes NP-hard. We introduce a multi-label shortest path algorithm for searching these neighborhoods heuristically. Results of a computational study on the vehicle routing problem with capacity and distance restrictions shows that CIM algorithms are very competitive approaches for solving vehicle routing problems. Overall, the solutions generated by the CIM algorithm have the best performance among the current solution methodologies in terms of percentage deviation from the best-known solutions for large-scale capacitated VRP instances.     

A priori orienteering with time windows and stochastic wait times at customers

In the pharmaceutical industry, sales representatives visit doctors to inform them of their products and encourage them to become an active prescriber. On a daily basis, pharmaceutical sales representatives must decide which doctors to visit and the order to visit them. This situation motivates a problem we more generally refer to as a stochastic orienteering problem with time windows (SOPTW), in which a time window is associated with each customer and an uncertain wait time at a customer results from a queue of competing sales representatives. We develop a priori routes with the objective of maximizing expected sales. We operationalize the sales representative’s execution of the a priori route with relevant recourse actions and derive an analytical formula to compute the expected sales from an a priori tour. We tailor a variable neighborhood search heuristic to solve the problem. We demonstrate the value of modeling uncertainty by comparing the solutions to our model to solutions of a deterministic version using expected values of the associated random variables. We also compute an empirical upper bound on our solutions by solving deterministic instances corresponding to perfect information.     

A multi-space sampling heuristic for the vehicle routing problem with stochastic demands

The vehicle routing problem with stochastic demands consists in designing transportation routes of minimal expected cost to satisfy a set of customers with random demands of known probability distributions. This paper proposes a simple yet effective heuristic approach that uses randomized heuristics for the traveling salesman problem, a tour partitioning procedure, and a set partitioning formulation to sample the solution space and find high-quality solutions for the problem. Computational experiments on benchmark instances from the literature show that the proposed approach is competitive with the state-of-the-art algorithm for the problem in terms of both accuracy and efficiency. In experiments conducted on a set of 40 instances, the proposed approach unveiled four new best-known solutions (BKSs) and matched another 24. For the remaining 12 instances, the heuristic reported average gaps with respect to the BKS ranging from 0.69 to 0.15 % depending on its configuration.     

A tour construction heuristic for the travelling salesman problem

The tour construction heuristic that generates initial tours for the tour improvement heuristics plays an important role in solving the travelling salesman problem (TSP). With the help of an effective tour construction heuristic, the performance of a heuristic can be improved. In this study we present a new tour construction algorithm, the construction priority (CP). We incorporate the performance of the CP into metaheuristics such as tabu search, genetic algorithm, space smoothing, and noising methods. The performance of the CP is empirically compared with the nearest neighbour (NN) approach. Extensive computational comparison shows that the CP is considerably more effective compared to NN.     

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.     

Maximizing the profit per unit of time for the TSP with convex resource-dependent travelling times

This paper introduces a new problem that is an extension of the travelling salesman problem (TSP) in which the travelling times are resource dependent and the objective is to maximize the profit per unit of time. We present an optimal solution approach comprised of three main steps: (1) calculating the optimal amount of total resource required (regardless of the selected tour); (2) constructing the tour; and (3) assigning the optimal resource to each connection between vertices using the equivalent load method. This solution approach finds the optimal solution with the same computational complexity for solving the classic TSP.     

General variable neighborhood search for the uncapacitated single allocation <Emphasis Type="Italic">p</Emphasis>-hub center problem

In this paper we propose a general variable neighborhood search heuristic for solving the uncapacitated single allocation p-hub center problem (USApHCP). For the local search step we develop a nested variable neighborhood descent strategy. The proposed approach is tested on benchmark instances from the literature and found to outperform the state-of-the-art heuristic based on ant colony optimization. We also test our heuristic on large scale instances that were not previously considered as test instances for the USApHCP. Moreover, exact solutions were reached by our GVNS for all instances where optimal solutions are known.     

Random Tours in the Traveling Salesman Problem: Analysis and Application

Random solutions to the traveling salesman problem (TSP) exhibit statistical regularities across problem instances. These patterns can assist heuristic search for good solutions by providing easy estimates of the length of the optimal tour.     

A variable neighborhood search for the network design problem with relays

Given a set of commodities to be routed over a network, the network design problem with relays involves selecting a route for each commodity and determining the location of relays where the commodities must be reprocessed at certain distance intervals. We propose a hybrid approach based on variable neighborhood search. The variable neighborhood algorithm searches for the route for each commodity and the optimal relay locations for a given set of routes are determined by an implicit enumeration algorithm. We show that dynamic programming can be used to determine the optimal relay locations for a single commodity. Dynamic programming is embedded into the implicit enumeration algorithm to solve the relay location problem optimally for multiple commodities. The special structure of the problem is leveraged for computational efficiency. In the variable neighborhood search algorithm, the routes of the current solution are perturbed and reconstructed to generate neighbor solutions using random and greedy construction heuristics. Computational experiments on three sets of problems (80 instances) show that the variable neighborhood search algorithm with optimal relay allocations outperforms all existing algorithms in the literature.     

Local search for the probabilistic traveling salesman problem: Correction to the 2-p-opt and 1-shift algorithms

The probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space, where each customer requires a visit only with some known probability. A solution to this problem is an a priori tour which visits all customers, and the objective is to minimize the expected length of the a priori tour over all customer subsets, assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour. This problem belongs to the class of stochastic vehicle routing problems, a class which has received increasing attention in recent years, and which is of major importance in real world applications.Several heuristics have been proposed and tested for the probabilistic traveling salesman problem, many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem. In particular, two local search algorithms (2-p-opt and 1-shift) were introduced by Bertsimas.In a previous report we have shown that the expressions for the cost evaluation of 2-p-opt and 1-shift moves, as proposed by Bertsimas, are not correct. In this paper we derive the correct versions of these expressions, and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions.     

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.