On a periodic vehicle routing problem

This paper deals with a study on a variant of the Periodic Vehicle Routing Problem (PVRP). As in the traditional Vehicle Routing Problem, customer locations each with a certain daily demand are given, as well as a set of capacitated vehicles. In addition, the PVRP has a horizon, say T days, and there is a frequency for each customer stating how often within this T-day period this customer must be visited. A solution to the PVRP consists of T sets of routes that jointly satisfy the demand constraints and the frequency constraints. The objective is to minimize the sum of the costs of all routes over the planning horizon. We develop different algorithms solving the instances of the case studied. Using these algorithms we are able to realize considerable cost reductions compared to the current situation.     

A hybrid evolutionary algorithm for the periodic location-routing problem

The well-known vehicle routing problem (VRP) has been studied in depth over the last decades. Nowadays, generalizations of VRP have been developed for tactical or strategic decision levels of companies but not both. The tactical extension or periodic VRP (PVRP) plans a set of trips over a multiperiod horizon, subject to frequency constraints. The strategic extension is motivated by interdependent depot location and routing decisions in most distribution systems. Low-quality solutions are obtained if depots are located first, regardless of the future routes. In the location-routing problem (LRP), location and routing decisions are tackled simultaneously. Here for the first time, except for some conference papers, the goal is to combine the PVRP and LRP into an even more realistic problem covering all decision levels: the periodic LRP or PLRP. A hybrid evolutionary algorithm is proposed to solve large size instances of the PLRP. First, an individual representing an assignment of customers to combinations of visit days is randomly generated. The evolution operates through an Evolutionary Local Search (ELS) on visit day assignments. The algorithm is hybridized with a heuristic based on the Randomized Extended Clarke and Wright Algorithm (RECWA) to create feasible solutions and stops when a given number of iterations is reached. The method is evaluated over three sets of instances, and solutions are compared to the literature on particular cases such as one-day horizon (LRP) or one depot (PVRP). This metaheuristic outperforms the previous methods for the PLRP.     

Optimizing the periodic pick-up of raw materials for a manufacturer of auto parts

We describe a solution procedure for a special case of the periodic vehicle routing problem (PVRP). Operation managers at an auto parts manufacturer in the north of Spain described the optimization problem to the authors. The manufacturer must pick up parts (raw material) from geographically dispersed locations. The parts are picked up periodically at scheduled times. The problem consists of assigning a pickup schedule to each of its supplier’s locations and also establishing daily routes in order to minimize total transportation costs. The time horizon under consideration may be as long as 90 days. The resulting PVRP is such that the critical decision is the assignment of locations to schedules, because once this is done, the daily routing of vehicles is relatively straightforward. Through extensive computational experiments, we show that the metaheuristic procedure described in this paper is capable of finding high-quality solutions within a reasonable amount of computer time. Our main contribution is the development of a procedure that is more effective at handling PVRP instances with long planning horizons when compared to those proposed in the literature.     

Column generation based heuristic for tactical planning in multi-period vehicle routing

The periodic vehicle routing problem (PVRP) consists in establishing a planning of visits to clients over a given time horizon so as to satisfy some service level while optimizing the routes used in each time period. The tactical planning model considered here restricts its attention to scheduling visits and assigning them to vehicles while leaving sequencing decisions for an underlying operational model. The objective is twofold: to optimize regional compactness of the routes in a desire to specialize routes to restricted geographical area and to balance the workload evenly between vehicles. Approximate solutions are constructed using a truncated column generation procedure followed by a rounding heuristic. This mathematical programming based procedure can deal with problems with 50–80 customers over five working days which is the range of size of most PVRP instances treated in the literature with meta-heuristics. The paper highlights the importance of alternative optimization criteria not accounted for in standard operational models and provides insights on the implementation of a column generation based rounding heuristic.     

改进蚁群算法优化周期性车辆路径问题

周期性车辆路径问题(PVRP)是标准车辆路径问题(VRP)的扩展,PVRP将配送期由单一配送期延伸到T(T＞1)期,因此,PVRP需要优化每个配送期的顾客组合和配送路径。由于PVRP是一个内嵌VRP的问题,其比标准VRP问题更加复杂,难于求解。本文采用蚁群算法对PVRP进行求解,并提出采用两种改进措施——多维信息素的运用和基于扫描法的局部优化方法来提高算法的性能。最后,通过9个经典PVRP算例对该算法进行了数据实验,结果表明本文提出的改进蚁群算法求解PVRP问题是可行有效的,同时也表明两种改进措施可以显著提高算法的性能。     

A variable neighborhood search heuristic for periodic routing problems

The aim of this paper is to propose a new heuristic for the Periodic Vehicle Routing Problem (PVRP) without time windows. The PVRP extends the classical Vehicle Routing Problem (VRP) to a planning horizon of several days. Each customer requires a certain number of visits within this time horizon while there is some flexibility on the exact days of the visits. Hence, one has to choose the visit days for each customer and to solve a VRP for each day. Our method is based on Variable Neighborhood Search (VNS). Computational results are presented, that show that our approach is competitive and even outperforms existing solution procedures proposed in the literature. Also considered is the special case of a single vehicle, i.e. the Periodic Traveling Salesman Problem (PTSP). It is shown that slight changes of the proposed VNS procedure is also competitive for the PTSP.     

Generating hard instances for robust combinatorial optimization

While research in robust optimization has attracted considerable interest over the last decades, its algorithmic development has been hindered by several factors. One of them is a missing set of benchmark instances that make algorithm performance better comparable, and makes reproducing instances unnecessary. Such a benchmark set should contain hard instances in particular, but so far, the standard approach to produce instances has been to sample values randomly from a uniform distribution.In this paper we introduce a new method to produce hard instances for min-max combinatorial optimization problems, which is based on an optimization model itself. Our approach does not make any assumptions on the problem structure and can thus be applied to any combinatorial problem. Using the Selection and Traveling Salesman problems as examples, we show that it is possible to produce instances which are up to 500 times harder to solve for a mixed-integer programming solver than the current state-of-the-art instances.     

Core instances for testing: A case study

Data generation for computational testing of optimization algorithms is a key topic in experimental algorithmics. Recently, concern has arisen that many published computational experiments are inadequate with respect to the way test instances are generated. In this paper we suggest a new research direction that might be useful to cope with the possible limitations of data generation. The basic idea is to select a finite set of instances which ‘represent’ the whole set of instances. We propose a measure of the representativeness of an instance, which we call ε-representativeness: for a minimization problem, an instance x_ε is ε-representative of another instance x if a (1 + ε)-approximate solution to x can be obtained by solving x_ε. Focusing on a strongly NP-hard single machine scheduling problem, we show how to map the infinite set of all instances into a finite set of ε-representative core instances. We propose to use this finite set of ε-representative core instances to test heuristics.     

The balanced academic curriculum problem revisited

The Balanced Academic Curriculum Problem (BACP) consists in assigning courses to teaching terms satisfying prerequisites and balancing the credit course load within each term. The BACP is part of the CSPLib with three benchmark instances, but its formulation is simpler than the problem solved in practice by universities. In this article, we introduce a generalized version of the problem that takes different curricula and professor preferences into account, and we provide a set of real-life problem instances arisen at University of Udine. Since the existing formulation based on a min–max objective function does not balance effectively the credit load for the new instances, we also propose alternative objective functions. Whereas all the CSPLib instances are efficiently solved with Integer Linear Programming (ILP) state-of-the-art solvers, our new set of real-life instances turns out to be much more challenging and still intractable for ILP solvers. Therefore, we have designed, implemented, and analyzed heuristics based on local search. We have collected computational results on all the new instances with the proposed approaches and assessed the quality of solutions with respect to the lower bounds found by ILP on a relaxed and decomposed problem. Results show that a selected heuristic finds solutions of quality at 9%–60% distance from the lower bound. We make all data publicly available, in order to stimulate further research on this problem.     

A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem

The multi-activity shift scheduling problem involves assigning a sequence of activities to a set of employees. In this paper, we consider the variant where the employees have different qualifications and each activity must be performed in a specified time window; i.e., we specify the earliest start period and the latest finish period. We propose a matheuristic in which Lagrangian relaxation is used to identify a subset of promising shifts, and a restricted set covering problem is solved to find a feasible solution. Each shift is represented by a context-free grammar. Computational tests are carried out on two sets of instances from the literature. For the first set, the matheuristic finds a solution with an optimality gap less than 0.01% for 70% of the instances and improves the best-known solution for 16% of them; for the second set, the matheuristic reaches the best-known solutions for 55% of the instances and finds better solutions for 37.5% of them.     

A matheuristic for the cell formation problem

In this paper we propose a GRASP matheuristic coupled with an Integer Programming refinement based on Set Partitioning to solve the Cell Formation Problem. We use the grouping efficacy measure to evaluate the solutions. As this measure is nonlinear, we propose a fractional Set Partitioning approach and its linearization. Our method is validated on a set of 35 instances from the literature. The experiments found four unknown solutions. For all instances with known optima, our method is able to determine the optimum solutions.     

An automatic algorithm selection approach for the multi-mode resource-constrained project scheduling problem

This paper investigates the construction of an automatic algorithm selection tool for the multi-mode resource-constrained project scheduling problem (MRCPSP). The research described relies on the notion of empirical hardness models. These models map problem instance features onto the performance of an algorithm. Using such models, the performance of a set of algorithms can be predicted. Based on these predictions, one can automatically select the algorithm that is expected to perform best given the available computing resources. The idea is to combine different algorithms in a super-algorithm that performs better than any of the components individually. We apply this strategy to the classic problem of project scheduling with multiple execution modes. We show that we can indeed significantly improve on the performance of state-of-the-art algorithms when evaluated on a set of unseen instances. This becomes important when lots of instances have to be solved consecutively. Many state-of-the-art algorithms perform very well on a majority of benchmark instances, while performing worse on a smaller set of instances. The performance of one algorithm can be very different on a set of instances while another algorithm sees no difference in performance at all. Knowing in advance, without using scarce computational resources, which algorithm to run on a certain problem instance, can significantly improve the total overall performance.     

A metaheuristic approach to the dominating tree problem

This paper considers a recently introduced NP-hard problem on graphs, called the dominating tree problem. In order to solve this problem, we develop a variable neighborhood search (VNS) based heuristic. Feasible solutions are obtained by using the set of vertex permutations that allow us to implement standard neighborhood structures and the appropriate local search procedure. Computational experiments include two classes of randomly generated test instances and benchmark test instances from the literature. Optimality of VNS solutions on small size instances is verified with CPLEX.     

A new model for the asymmetric vehicle routing problem with simultaneous pickup and deliveries

The asymmetric vehicle routing problem with simultaneous pickup and deliveries is considered. This paper develops four new classes of valid inequalities for the problem. We generalize the idea of a no-good cut. Together, these help us solve 45-node randomly generated problem instances more efficiently. We report results on a set of benchmark instances in literature. In this set, we are able to show an order of magnitude improvement in computational times over currently published results in literature.     

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.     

A metaheuristic for the min–max windy rural postman problem with K vehicles

In this paper we deal with the min–max version of the windy rural postman problem with K vehicles. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here a metaheuristic that produces very good feasible solutions in reasonable computing times. It is based on the combination of a multi-start procedure with an Iterated Local Search. Extensive computational results on a large set of instances with up to 50 vertices, 184 edges and 5 vehicles are presented. The results are very good, the average gaps with respect to a known lower bound are less than 0.40% for instances with 2 or 3 vehicles and up to 1.60% when 4 or 5 vehicles are considered.     

A case study of algorithm selection for the traveling thief problem

Many real-world problems are composed of several interacting components. In order to facilitate research on such interactions, the Traveling Thief Problem (TTP) was created in 2013 as the combination of two well-understood combinatorial optimization problems. With this article, we contribute in four ways. First, we create a comprehensive dataset that comprises the performance data of 21 TTP algorithms on the full original set of 9720 TTP instances. Second, we define 55 characteristics for all TPP instances that can be used to select the best algorithm on a per-instance basis. Third, we use these algorithms and features to construct the first algorithm portfolios for TTP, clearly outperforming the single best algorithm. Finally, we study which algorithms contribute most to this portfolio.     

Computing with multi-row Gomory cuts

Recent advances on the understanding of valid inequalities from the infinite group relaxation has opened the possibility of finding a computationally effective extension to GMI cuts. In this paper, we investigate the computational impact of using a subclass of minimally valid inequalities from this relaxation on a wide set of instances.     

A backbone based TSP heuristic for large instances

We introduce a reduction technique for large instances of the traveling salesman problem (TSP). This approach is based on the observation that tours with good quality are likely to share many edges. We exploit this observation by neglecting the less important tour space defined by the shared edges, and searching the important tour subspace in more depth. More precisely, by using a basic TSP heuristic, we obtain a set of starting tours. We call the set of edges which are contained in each of these starting tours as pseudo-backbone edges. Then we compute the maximal paths consisting only of pseudo-backbone edges, and transform the TSP instance to another one with smaller size by contracting each such path to a single edge. This reduced TSP instance can be investigated more intensively, and each tour of the reduced instance can be expanded to a tour of the original instance. Combining our reduction technique with the currently leading TSP heuristic of Helsgaun, we experimentally investigate 32 difficult VLSI instances from the well-known TSP homepage. In our experimental results we set world records for seven VLSI instances, i.e., find better tours than the best tours known so far (two of these world records have since been improved upon by Keld Helsgaun and Yuichi Nagata, respectively). For the remaining instances we find tours that are equally good or only slightly worse than the world record tours.     

A biased random-key genetic algorithm for the Steiner triple covering problem

We present a biased random-key genetic algorithm (BRKGA) for finding small covers of computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems. Using a parallel implementation of the BRKGA, we compute improved covers for the two largest instances in a standard set of test problems used to evaluate solution procedures for this problem. The new covers for instances A ₄₀₅ and A ₇₂₉ have sizes 335 and 617, respectively. On all other smaller instances our algorithm consistently produces covers of optimal size.