Variable Neighborhood Decomposition Search

The recent Variable Neighborhood Search (VNS) metaheuristic combines local search with systematic changes of neighborhood in the descent and escape from local optimum phases. When solving large instances of various problems, its efficiency may be enhanced through decomposition. The resulting two level VNS, called Variable Neighborhood Decomposition Search (VNDS), is presented and illustrated on the p-median problem. Results on 1400, 3038 and 5934 node instances from the TSP library show VNDS improves notably upon VNS in less computing time, and gives much better results than Fast Interchange (FI), in the same time that FI takes for a single descent. Moreover, Reduced VNS (RVNS), which does not use a descent phase, gives results similar to those of FI in much less computing time.     

A variable neighborhood search for the capacitated arc routing problem with intermediate facilities

The capacitated arc routing problem (CARP) focuses on servicing edges of an undirected network graph. A wide spectrum of applications like mail delivery, waste collection or street maintenance outlines the relevance of this problem. A realistic variant of the CARP arises from the need of intermediate facilities (IFs) to load up or unload the service vehicle and from tour length restrictions. The proposed Variable Neighborhood Search (VNS) is a simple and robust solution technique which tackles the basic problem as well as its extensions. The VNS shows excellent results on four different benchmark sets. Particularly, for all 120 instances the best known solution could be found and in 71 cases a new best solution was achieved.     

Cooperative Parallel Variable Neighborhood Search for the p-Median

We propose a cooperative multi-search method for the Variable Neighborhood Search (VNS) meta-heuristic based on the central-memory mechanism that has been successfully applied to a number of difficult combinatorial problems. In this approach, several independent VNS meta-heuristics cooperate by asynchronously exchanging information about the best solutions identified so far, thus conserving the simplicity of the original, sequential VNS ideas. The p-median problem (PM) serves as test case. Extensive experimentations have been conducted on the classical TSPLIB benchmark problem instances with up to 11948 customers and 1000 medians, without any particular calibration of the parallel method. The results indicate that, compared to sequential VNS, the cooperative strategy yields significant gains in terms of computation time without a loss in solution quality.     

Neighborhood structures for the container loading problem: a VNS implementation

This paper presents a Variable Neighborhood Search (VNS) algorithm for the container loading problem. The algorithm combines a constructive procedure based on the concept of maximal-space, with five new movements defined directly on the physical layout of the packed boxes, which involve insertion and deletion strategies. The new algorithm is tested on the complete set of Bischoff and Ratcliff problems, ranging from weakly to strongly heterogeneous instances, and outperforms all the reported algorithms which have used those test instances.     

Exact and heuristic solutions of the global supply chain problem with transfer pricing

We examine the example of a multinational corporation that attempts to maximize its global after tax profits by determining the flow of goods, the transfer prices, and the transportation cost allocation between each of its subsidiaries. Vidal and Goetschalckx [Vidal, C.J., Goetschalckx, M., 2001. A global supply chain model with transfer pricing and transportation cost allocation. European Journal of Operational Research 129 (1), 134–158] proposed a bilinear model of this problem and solved it by an Alternate heuristic. We propose a reformulation of this model reducing the number of bilinear terms and accelerating considerably the exact solution. We also present three other solution methods: an implementation of Variable Neighborhood Search (VNS) designed for any bilinear model, an implementation of VNS specifically designed for the problem considered here and an exact method based on a branch and cut algorithm. The solution methods are tested on artificial instances. These results show that our implementation of VNS outperforms the two other heuristics. The exact method found the optimal solution of all small instances and of 26% of medium instances.     

A variable neighborhood search with an effective local search for uncapacitated multilevel lot-sizing problems

In this study, we improved the variable neighborhood search (VNS) algorithm for solving uncapacitated multilevel lot-sizing (MLLS) problems. The improvement is twofold. First, we developed an effective local search method known as the Ancestors Depth-first Traversal Search (ADTS), which can be embedded in the VNS to significantly improve the solution quality. Second, we proposed a common and efficient approach for the rapid calculation of the cost change for the VNS and other generate-and-test algorithms. The new VNS algorithm was tested against 176 benchmark problems of different scales (small, medium, and large). The experimental results show that the new VNS algorithm outperforms all of the existing algorithms in the literature for solving uncapacitated MLLS problems because it was able to find all optimal solutions (100%) for 96 small-sized problems and new best-known solutions for 5 of 40 medium-sized problems and for 30 of 40 large-sized problems.     

Bringing order into the neighborhoods: relaxation guided variable neighborhood search

In this article we investigate a new variant of Variable Neighborhood Search (VNS): Relaxation Guided Variable Neighborhood Search. It is based on the general VNS scheme and a new Variable Neighborhood Descent (VND) algorithm. The ordering of the neighborhood structures in this VND is determined dynamically by solving relaxations of them. The objective values of these relaxations are used as indicators for the potential gains of searching the corresponding neighborhoods. We tested this new approach on the well-studied multidimensional knapsack problem. Computational experiments show that our approach is beneficial to the search, improving the obtained results. The concept is, in principle, more generally applicable and seems to be promising for many other combinatorial optimization problems approached by VNS. NICTA is funded by the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.The Institute of Computer Graphics and Algorithms is supported by the European RTN ADONET under grant 504438.     

A modified variable neighborhood search for the discrete ordered median problem

This paper presents a modified Variable Neighborhood Search (VNS) heuristic algorithm for solving the Discrete Ordered Median Problem (DOMP). This heuristic is based on new neighborhoods’ structures that allow an efficient encoding of the solutions of the DOMP avoiding sorting in the evaluation of the objective function at each considered solution. The algorithm is based on a data structure, computed in preprocessing, that organizes the minimal necessary information to update and evaluate solutions in linear time without sorting. In order to investigate the performance, the new algorithm is compared with other heuristic algorithms previously available in the literature for solving DOMP. We report on some computational experiments based on the well-known N-median instances of the ORLIB with up to 900 nodes. The obtained results are comparable or superior to existing algorithms in the literature, both in running times and number of best solutions found.     

Variable neighborhood search: Principles and applications

Systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization, called variable neighborhood search (VNS). We present a basic scheme for this purpose, which can easily be implemented using any local search algorithm as a subroutine. Its effectiveness is illustrated by solving several classical combinatorial or global optimization problems. Moreover, several extensions are proposed for solving large problem instances: using VNS within the successive approximation method yields a two-level VNS, called variable neighborhood decomposition search (VNDS); modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed VNS (SVNS) heuristic. Finally, we show how to stabilize column generation algorithms with help of VNS and discuss various ways to use VNS in graph theory, i.e., to suggest, disprove or give hints on how to prove conjectures, an area where metaheuristics do not appear to have been applied before.     

A Metaheuristic to Solve a Location-Routing Problem with Non-Linear Costs

The paper deals with a location-routing problem with non-linear cost functions. To the best of our knowledge, a mixed integer linear programming formulation for the addressed problem is proposed here for the first time. Since the problem is NP-hard exact algorithms are able to solve only particular cases, thus to solve more general versions heuristics are needed. The algorithm proposed in this paper is a combination of a p-median approach to find an initial feasible solution and a metaheuristic to improve the solution. It is a hybrid metaheuristic merging Variable Neighborhood Search (VNS) and Tabu Search (TS) principles and exploiting the synergies between the two. Computational results and conclusions close the paper.     

Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. This work is supported by the RTN ADONET under grant 504438.     

Local search and suffix tree for car-sequencing problem with colors

This paper describes an approximation solution method for the car sequencing problem with colors. Firstly, we study the optimality of problems with a single ratio constraint. This study also introduces a data structure for efficient calculation of the penalties related to ratio constraints. We describe the constructive greedy algorithm and variable neighborhood search adjusted for the problem with colors. Tabu metaheuristic is used to improve the results obtained by VNS. We then represent the cars with their constraints as letters over an alphabet and apply the algorithm to spell the motifs in order to improve the number of batch colors without decreasing the costs associated to the set of ratio constraints. The algorithm achieves 19 out of the 64 best results for instance sets A and B. These instances are the reference instances for Challenge ROADEF.     

Skewed VNS enclosing second order algorithm for the degree constrained minimum spanning tree problem

We develop ideas to enhance the performance of the variable neighborhood search (VNS) by guiding the search in the shaking phase, and by employing the Skewed strategy. For this purpose, Second Order algorithms and Skewed functions expressing structural differences are embedded in an efficient VNS proposed in the literature for the degree constrained minimum spanning tree problem. Given an undirected graph with weights associated with its edges, the degree constrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. Computational experiments are conducted on the largest and hardest benchmark instances found in the literature to date. We report results showing that the VNS with the proposed strategies improved the best known solutions for all the hardest benchmark instances. Moreover, these new best solutions significantly reduced the gaps with respect to tight lower bounds reported in the literature.     

求解带有时间窗和提前/拖期惩罚的飞机着陆问题的遗传算法

研究了带有时间窗、飞机着陆的总提前/拖期惩罚最小为目标函数的飞机着陆问题。针对此问题设计了一种遗传算法进行求解。染色体表示为飞机着陆次序和着陆跑道两个向量,一个新的解码算法来计算飞机的着陆时间。采用数据库OR-Library中的实例进行数值实验,实验结果表明：设计的算法是有效的, 主要原因是解码算法能大大提高解的质量。该算法对于求解带有时间窗、目标函数为提前/拖期惩罚最小的调度问题具有借鉴意义。     

A memetic algorithm approach for solving the multidimensional multi-way number partitioning problem

In this paper, we describe a generalization of the multidimensional two-way number partitioning problem (MDTWNPP) where a set of vectors has to be partitioned into p sets (parts) such that the sums per every coordinate should be exactly or approximately equal. We will call this generalization the multidimensional multi-way number partitioning problem (MDMWNPP). Also, an efficient memetic algorithm (MA) heuristic is developed to solve the multidimensional multi-way number partitioning problem obtained by combining a genetic algorithm (GA) with a powerful local search (LS) procedure. The performances of our memetic algorithm have been compared with the existing numerical results obtained by CPLEX based on an integer linear programming formulation of the problem. The solution reveals that our proposed methodology performs very well in terms of both quality of the solutions obtained and the computational time compared with the previous method of solving the multidimensional two-way number partitioning problem.     

Model and algorithm for inventory/routing decision in a three-echelon logistics system

This paper addresses an integrated inventory and routing problem in a three-echelon logistics system, which consists of a supplier, a central warehouse and a group of retailers. The inventory decision of each member and the routing decision among members of the system are made simultaneously, with the objective of minimizing the overall average cost of the system. A strategy named fixed partition and power-of-two (FP–POT) is proposed for the considered problem and a variable large neighborhood search (VLNS) algorithm, which is a special case of variable neighborhood search (VNS) algorithm, is developed. The efficiency of the strategy as well as the algorithm is illustrated by comparing computational results with a lower bound. The advantage of the proposed VLNS algorithm is further shown by getting better results for the problems in a two-echelon logistics system, which have been solved by a Tabu Search algorithm recently.     

The Parallel Variable Neighborhood Search for the p-Median Problem

The Variable Neighborhood Search (VNS) is a recent metaheuristic that combines series of random and improving local searches based on systematically changed neighborhoods. When a local minimum is reached, a shake procedure performs a random search. This determines a new starting point for running an improving search. The use of interchange moves provides a simple implementation of the VNS algorithm for the p-Median Problem. Several strategies for the parallelization of the VNS are considered and coded in C using OpenMP. They are compared in a shared memory machine with large instances.     

The Team Orienteering Problem with Time Windows: An LP-based Granular Variable Neighborhood Search

The Team Orienteering Problem (TOP) is a known NP-hard problem that typically arises in vehicle routing and production scheduling contexts. In this paper we introduce a new solution method to solve the TOP with hard Time Window constraints (TOPTW). We propose a Variable Neighborhood Search (VNS) procedure based on the idea of exploring, most of the time, granular instead of complete neighborhoods in order to improve the algorithm’s efficiency without loosing effectiveness. The method provides a general way to deal with granularity for those routing problems based on profits and complicated by time constraints. Extensive computational results are reported on standard benchmark instances. Performance of the proposed algorithm is compared to optimal solution values, when available, or to best known solution values obtained by state-of-the-art algorithms. The method comes out to be, on average, quite effective allowing to improve the best know values for 25 test instances.     

Serial-batching group scheduling with release times and the combined effects of deterioration and truncated job-dependent learning

This paper investigates a single machine serial-batching scheduling problem considering release times, setup time, and group scheduling, with the combined effects of deterioration and truncated job-dependent learning. The objective of the studied problem is to minimize the makespan. Firstly, we analyze the special case where all groups have the same arrival time, and propose the optimal structural properties on jobs sequencing, jobs batching, batches sequencing, and groups sequencing. Next, the corresponding batching rule and algorithm are developed. Based on these properties and the scheduling algorithm, we develop a hybrid VNS–ASHLO algorithm incorporating variable neighborhood search (VNS) and adaptive simplified human learning optimization (ASHLO) algorithms to solve the general case of the studied problem. Computational experiments on randomly generated instances are conducted to compare the proposed VNS–ASHLO with the algorithms of VNS, ASHLO, Simulated Annealing (SA), and Particle Swarm Optimization (PSO). The results based on instances of different scales show the effectiveness and efficiency of the proposed algorithm.     

Hybrid variable neighbourhood approaches to university exam timetabling

In this paper, we investigate variable neighbourhood search (VNS) approaches for the university examination timetabling problem. In addition to a basic VNS method, we introduce variants of the technique with different initialisation methods including a biased VNS and its hybridisation with a Genetic Algorithm. A number of different neighbourhood structures are analysed. It is demonstrated that the proposed technique is able to produce high quality solutions across a wide range of benchmark problem instances. In particular, we demonstrate that the Genetic Algorithm, which intelligently selects appropriate neighbourhoods to use within the biased VNS, produces the best known results in the literature, in terms of solution quality, on some of the benchmark instances. However, it requires relatively large amount of computational time. Possible extensions to this overall approach are also discussed.