An electromagnetism metaheuristic for solving the Maximum Betweenness Problem

In this paper we present an electromagnetism (EM) metaheuristic for solving NP hard Maximum Betweenness Problem (MBP). A new encoding scheme with appropriate objective functions is implemented. Specific representation of the individuals enables the EM operators to explore the searching space in a way that achieves high quality solutions. An effective 1-swap based local search procedure improved by the specific caching technique is performed on each EM point. The algorithm is tested both on real and artificial instances from the literature. Experimental results show that the proposed EM approach achieves all previously known optimal solutions, except one, and achieves the best-known solutions or outperforms other approaches on all large-scale instances, except two. Provided statistical analysis indicates that the EM approach is significantly better than other approaches.     

Solving large <Emphasis Type="Italic">p</Emphasis>-median clustering problems by primal–dual variable neighborhood search

Data clustering methods are used extensively in the data mining literature to detect important patterns in large datasets in the form of densely populated regions in a multi-dimensional Euclidean space. Due to the complexity of the problem and the size of the dataset, obtaining quality solutions within reasonable CPU time and memory requirements becomes the central challenge. In this paper, we solve the clustering problem as a large scale p-median model, using a new approach based on the variable neighborhood search (VNS) metaheuristic. Using a highly efficient data structure and local updating procedure taken from the OR literature, our VNS procedure is able to tackle large datasets directly without the need for data reduction or sampling as employed in certain popular methods. Computational results demonstrate that our VNS heuristic outperforms other local search based methods such as CLARA and CLARANS even after upgrading these procedures with the same efficient data structures and local search. We also obtain a bound on the quality of the solutions by solving heuristically a dual relaxation of the problem, thus introducing an important capability to the solution process.     

The Multiple Trip Vehicle Routing Problem with Backhauls: Formulation and a Two-Level Variable Neighbourhood Search

In this paper a new VRP variant the Multiple Trip Vehicle Routing Problem with Backhauls (MT-VRPB) is investigated. The classical MT-VRP model is extended by including the backhauling aspect. An ILP formulation of the MT-VRPB is first presented and CPLEX results for small and medium size instances are reported. For large instances of the MT-VRPB a Two-Level VNS algorithm is developed. To gain a continuous balanced intensification and diversification during the search process VNS is embedded with the sequential VND and a multi-layer local search approach. The algorithm is tested on a set of new MT-VRPB data instances which we generated. Interesting computational results are presented. The Two-Level VNS produced excellent results when tested on the special variant of the VRPB.     

Variable neighborhood search for multi-objective resource allocation problems

The Resource Allocation Problem (RAP) is a classical problem in the field of operations management that has been broadly applied to real problems such as product allocation, project budgeting, resource distribution, and weapon-target assignment. In addition to focusing on a single objective, the RAP may seek to simultaneously optimize several expected but conflicting goals under conditions of resources scarcity. Thus, the single-objective RAP can be intuitively extended to become a Multi-Objective Resource Allocation Problem (MORAP) that also falls in the category of NP-Hard. Due to the complexity of the problem, metaheuristics have been proposed as a practical alternative in the selection of techniques for finding a solution. This study uses Variable Neighborhood Search (VNS) algorithms, one of the extensively used metaheuristic approaches, to solve the MORAP with two important but conflicting objectives—minimization of cost and maximization of efficiency. VNS searches the solution space by systematically changing the neighborhoods. Therefore, proper design of neighborhood structures, base solution selection strategy, and perturbation operators are used to help build a well-balanced set of non-dominated solutions. Two test instances from the literature are used to compare the performance of the competing algorithms including a hybrid genetic algorithm and an ant colony optimization algorithm. Moreover, two large instances are generated to further verify the performance of the proposed VNS algorithms. The approximated Pareto front obtained from the competing algorithms is compared with a reference Pareto front by the exhaustive search method. Three measures are considered to evaluate algorithm performance: D1_R, the Accuracy Ratio, and the number of non-dominated solutions. The results demonstrate the practicability and promise of VNS for solving multi-objective resource allocation problems.     

Bi-objective parallel machines scheduling with sequence-dependent setup times using hybrid metaheuristics and weighted min–max technique

In the literature of multi-objective problem, there are different algorithms to solve different optimization problems. This paper presents a min–max multi-objective procedure for a dual-objective, namely make span, and sum of the earliness and tardiness of jobs in due window machine scheduling problems, simultaneously. In formulation of min–max method when this method is combined with the weighting method, the decision maker can have the flexibility of mixed use of weights and distance parameter to yield a set of Pareto-efficient solutions. This research extends the new hybrid metaheuristic (HMH) to solve parallel machines scheduling problems with sequence-dependent setup time that comprises three components: an initial population generation method based on an ant colony optimization (ACO), a simulated annealing (SA) as an evolutionary algorithm employs certain probability to avoid becoming trapped in a local optimum, and a variable neighborhood search (VNS) which involves three local search procedures to improve the population. In addition, two VNS-based HMHs, which are a combination of two methods, SA/VNS and ACO/VNS, are also proposed to solve the addressed scheduling problems. A design of experiments approach is employed to calibrate the parameters. The non-dominated sets obtained from HMH and two best existing bi-criteria scheduling algorithms are compared in terms of various indices and the computational results show that the proposed algorithm is capable of producing a number of high-quality Pareto optimal scheduling plans. Aside, an extensive computational experience is carried out to analyze the different parameters of the algorithm.     

Variable neighborhood search for the Vertex Separation Problem

The Vertex Separation Problem belongs to a family of optimization problems in which the objective is to find the best separator of vertices or edges in a generic graph. This optimization problem is strongly related to other well-known graph problems; such as the Path-Width, the Node Search Number or the Interval Thickness, among others. All of these optimization problems are NP-hard and have practical applications in VLSI (Very Large Scale Integration), computer language compiler design or graph drawing. Up to know, they have been generally tackled with exact approaches, presenting polynomial-time algorithms to obtain the optimal solution for specific types of graphs. However, in spite of their practical applications, these problems have been ignored from a heuristic perspective, as far as we know. In this paper we propose a pure 0-1 optimization model and a metaheuristic algorithm based on the variable neighborhood search methodology for the Vertex Separation Problem on general graphs. Computational results show that small instances can be optimally solved with this optimization model and the proposed metaheuristic is able to find high-quality solutions with a moderate computing time for large-scale instances.     

Clustering search algorithm for the capacitated centered clustering problem

The capacitated centered clustering problem (CCCP) consists in partitioning a set of n points into p disjoint clusters with a known capacity. Each cluster is specified by a centroid. The objective is to minimize the total dissimilarity within each cluster, such that a given capacity limit of the cluster is not exceeded. This paper presents a solution procedure for the CCCP, using the hybrid metaheuristic clustering search (CS), whose main idea is to identify promising areas of the search space by generating solutions through a metaheuristic and clustering them into groups that are then further explored with local search heuristics. Computational results in test problems of the literature show that the CS found a significant number of new best-known solutions in reasonable computational times.     

Hybrid evolutionary algorithm for the Capacitated Centered Clustering Problem

The Capacitated Centered Clustering Problem (CCCP) consists of defining a set of p groups with minimum dissimilarity on a network with n points. Demand values are associated with each point and each group has a demand capacity. The problem is well known to be NP-hard and has many practical applications. In this paper, the hybrid method Clustering Search (CS) is implemented to solve the CCCP. This method identifies promising regions of the search space by generating solutions with a metaheuristic, such as Genetic Algorithm, and clustering them into clusters that are then explored further with local search heuristics. Computational results considering instances available in the literature are presented to demonstrate the efficacy of CS.     

Variable neighborhood search for the cost constrained minimum label spanning tree and label constrained minimum spanning tree problems

Given an undirected graph whose edges are labeled or colored, edge weights indicating the cost of an edge, and a positive budget B, the goal of the cost constrained minimum label spanning tree (CCMLST) problem is to find a spanning tree that uses the minimum number of labels while ensuring its cost does not exceed B. The label constrained minimum spanning tree (LCMST) problem is closely related to the CCMLST problem. Here, we are given a threshold K on the number of labels. The goal is to find a minimum weight spanning tree that uses at most K distinct labels. Both of these problems are motivated from the design of telecommunication networks and are known to be NP-complete [15].In this paper, we present a variable neighborhood search (VNS) algorithm for the CCMLST problem. The VNS algorithm uses neighborhoods defined on the labels. We also adapt the VNS algorithm to the LCMST problem. We then test the VNS algorithm on existing data sets as well as a large-scale dataset based on TSPLIB [12] instances ranging in size from 500 to 1000 nodes. For the LCMST problem, we compare the VNS procedure to a genetic algorithm (GA) and two local search procedures suggested in [15]. For the CCMLST problem, the procedures suggested in [15] can be applied by means of a binary search procedure. Consequently, we compared our VNS algorithm to the GA and two local search procedures suggested in [15]. The overall results demonstrate that the proposed VNS algorithm is of high quality and computes solutions rapidly. On our test datasets, it obtains the optimal solution in all instances for which the optimal solution is known. Further, it significantly outperforms the GA and two local search procedures described in [15].     

Parallel-machine scheduling problems with sequence-dependent setup times using an ACO,SA and VNS hybrid algorithm

This paper proposes a hybrid metaheuristic for the minimization of makespan in scheduling problems with parallel machines and sequence-dependent setup times. The solution approach is robust, fast, and simply structured, and comprises three components: an initial population generation method based on an ant colony optimization (ACO), a simulated annealing (SA) for solution evolution, and a variable neighborhood search (VNS) which involves three local search procedures to improve the population. The hybridization of an ACO, SA with VNS, combining the advantages of these three individual components, is the key innovative aspect of the approach. Two algorithms of a hybrid VNS-based algorithm, SA/VNS and ACO/VNS, and the VNS algorithm presented previously are used to compare with the proposed hybrid algorithm to highlight its advantages in terms of generality and quality for large instances.     

An ant colony system empowered variable neighborhood search algorithm for the vehicle routing problem with simultaneous pickup and delivery

Along with the progress in computer hardware architecture and computational power, in order to overcome technological bottlenecks, software applications that make use of expert and intelligent systems must race against time where nanoseconds matter in the long-awaited future. This is possible with the integration of excellent solvers to software engineering methodologies that provide optimization-based decision support for planning. Since the logistics market is growing rapidly, the optimization of routing systems is of primary concern that motivates the use of vehicle routing problem (VRP) solvers as software components integrated as an optimization engine. A critical success factor of routing optimization is quality vs. response time performance. Less time-consuming and more efficient automated processes can be achieved by employing stronger solution algorithms. This study aims to solve the Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD) which is a popular extension of the basic Vehicle Routing Problem arising in real world applications where pickup and delivery operations are simultaneously taken into account to satisfy the vehicle capacity constraint with the objective of total travelled distance minimization. Since the problem is known to be NP-hard, a hybrid metaheuristic algorithm based on an ant colony system (ACS) and a variable neighborhood search (VNS) is developed for its solution. VNS is a powerful optimization algorithm that provides intensive local search. However, it lacks a memory structure. This weakness can be minimized by utilizing long term memory structure of ACS and hence the overall performance of the algorithm can be boosted. In the proposed algorithm, instead of ants, VNS releases pheromones on the edges while ants provide a perturbation mechanism for the integrated algorithm using the pheromone information in order to explore search space further and jump from local optima. The performance of the proposed ACS empowered VNS algorithm is studied on well-known benchmarks test problems taken from the open literature of VRPSPD for comparison purposes. Numerical results confirm that the developed approach is robust and very efficient in terms of both solution quality and CPU time since better results provided in a shorter time on benchmark data sets is a good performance indicator.     

An integration of Lagrangian split and VNS: The case of the capacitated vehicle routing problem

In this paper, we propose an efficient and novel Lagrangian relaxation method which incorporates a new integer linear programming (ILP) formulation to optimally partition a giant tour in the context of a capacitated vehicle routing problem (CVRP). This approach, which we call Lagrangian split (Ls), is more versatile than the ILP which, in most cases, can be intractable using a conventional solver. An effective repair mechanism followed by a local search are also embedded into the process. The mathematical validity of the repair mechanism and its time complexity are also provided. An integration of Ls into a powerful variable neighbourhood search (VNS) is also presented. Computational experiments are conducted to demonstrate that Ls provides encouraging results when applied on benchmark instances and that the integration of Ls into a metaheuristic scheme produces good results when compared to those found by state-of-the-art methods.     

A GRASP with evolutionary path relinking for the truck and trailer routing problem

In the truck and trailer routing problem (TTRP) a heterogeneous fleet composed of trucks and trailers has to serve a set of customers, some only accessible by truck and others accessible with a truck pulling a trailer. This problem is solved using a route-first, cluster-second procedure embedded within a hybrid metaheuristic based on a greedy randomized adaptive search procedure (GRASP), a variable neighborhood search (VNS) and a path relinking (PR). We test PR as a post-optimization procedure, as an intensification mechanism, and within evolutionary path relinking (EvPR). Numerical experiments show that all the variants of the proposed GRASP with path relinking outperform all previously published methods. Remarkably, GRASP with EvPR obtains average gaps to best-known solutions of less than 1% and provides several new best solutions.     

Iterated local search heuristics for the Vehicle Routing Problem with Cross-Docking

This work addresses the Vehicle Routing Problem with Cross-Docking (VRPCD). The problem consists in defining a minimum cost set of routes for a fleet of vehicles that meets the demands of products for a set of suppliers and customers. The vehicles leave a single Cross-Dock (CD) towards the suppliers, pick up products and return to the CD, where products can be exchanged before being delivered to their customers. The vehicle routes must respect the vehicle capacity constraints, as well as the time window constraints. We adapted a constructive heuristic and six local search procedures from the literature of VRP, and made them efficient in the presence of the synchronization constraints of VRPCD. Besides, we propose three Iterated Local Search (Lourenço et al., 2010) heuristics for VRPCD. The first heuristic is a standard implementation of ILS, while the second extends the classic ILS framework by keeping a set of elite solutions, instead of a single current solution. The latter set is used in a restart procedure. As far as we can tell, this is the first ILS heuristic in the literature that keeps a population of current elite solutions. The third heuristic is an extension of the second that relies on an intensification procedure based on an Integer Programming formulation for the Set Partitioning problem. The latter allows a neighborhood with an exponential number of neighbors to be efficiently evaluated. We report computational results and comparisons with the best heuristics in the literature. Besides, we also present a new set with the largest instances in the literature of VRPCD, in order to demonstrate that the improvements we propose for the ILS metaheuristic are efficient even for large size instances. Results show that the best of our heuristics is competitive with the best heuristics in the literature of VRPCD. Besides, it improved the best solution known for half of the benchmark instances in the literature.     

Reactive Search strategies using Reinforcement Learning,local search algorithms and Variable Neighborhood Search

Optimization techniques known as metaheuristics have been applied successfully to solve different problems, in which their development is characterized by the appropriate selection of parameters (values) for its execution. Where the adjustment of a parameter is required, this parameter will be tested until viable results are obtained. Normally, such adjustments are made by the developer deploying the metaheuristic. The quality of the results of a test instance [The term instance is used to refer to the assignment of values to the input variables of a problem.] will not be transferred to the instances that were not tested yet and its feedback may require a slow process of “trial and error” where the algorithm has to be adjusted for a specific application. Within this context of metaheuristics the Reactive Search emerged defending the integration of machine learning within heuristic searches for solving complex optimization problems. Based in the integration that the Reactive Search proposes between machine learning and metaheuristics, emerged the idea of putting Reinforcement Learning, more specifically the Q-learning algorithm with a reactive behavior, to select which local search is the most appropriate in a given time of a search, to succeed another local search that can not improve the current solution in the VNS metaheuristic. In this work we propose a reactive implementation using Reinforcement Learning for the self-tuning of the implemented algorithm, applied to the Symmetric Travelling Salesman Problem.     

Two effective hybrid metaheuristic algorithms for minimization of multimodal functions

Global optimization problems naturally arise from many applications. We propose two hybrid metaheuristic algorithms for finding a global optimum of a continuous function. Our proposed algorithms are hybridizations of genetic algorithm (GA) and variable neighbourhood search (VNS). To increase the efficiency of our algorithms, for smooth functions we present an effective locally improving line search procedure, and for non-smooth functions, we use the simplex method proposed by Nelder and Mead. By use of the recently adopted non-parametric statistical tests of Kruskal–Wallis and Mann–Whitney for analysing the behaviour of evolutionary algorithms, we compare both the efficiency and the effectiveness of our proposed algorithms with efficiently representative metaheuristic algorithms such as the multiagent GA proposed by Liang et al., the ant colony algorithm proposed by Toksari, and the VNS of Toksari and Güner on a variety of standard test functions. Computational experiments demonstrate that our proposed algorithms are efficiently effective.     

Variable neighborhood search based approaches to a vehicle scheduling problem in agriculture

A vehicle scheduling problem (VSP) that arises from sugar beet transportation within minimum working time under the set of constraints reflecting a real‐life situation is considered. A mixed integer quadratically constrained programming (MIQCP) model of the considered VSP and reformulation to a mixed integer linear program (MILP) are proposed and used within the framework of Lingo 17 solver, producing optimal solutions only for small‐sized problem instances. Two variants of the variable neighborhood search (VNS) metaheuristic—basic VNS (BVNS) and skewed VNS (SVNS) are designed to efficiently deal with large‐sized problem instances. The proposed VNS approaches are evaluated and compared against Lingo 17 and each other on the set of real‐life and generated problem instances. Computational results show that both BVNS and SVNS reach all known optimal solutions on small‐sized instances and are comparable on medium‐ and large‐sized instances. In general, SVNS significantly outperforms BVNS in terms of running times.     

A variable neighborhood search for minimizing single machine weighted earliness and tardiness with common due date

Although the concept of just-in-time (JIT) production systems has been proposed for over two decades, it is still important in real-world production systems. In this paper, we consider minimizing the total weighted earliness and tardiness with a restrictive common due date in a single machine environment, which has been proved as an NP-hard problem. Due to the complexity of the problem, metaheuristics, including simulated annealing, genetic algorithm, tabu search, among others, have been proposed for searching good solutions in reasonable computation times. In this paper, we propose a hybrid metaheuristic that uses tabu search within variable neighborhood search (VNS/TS). There are several distinctive features in the VNS/TS algorithm, including different ratio of the two neighborhoods, generating five points simultaneously in a neighborhood, implementation of the B/F local search, and combination of TS with VNS. By examining the 280 benchmark problem instances, the algorithm shows an excellent performance in not only the solution quality but also the computation time. The results obtained are better than those reported previously in the literature.     

变邻域搜索算法综述

变邻域搜索算法(Variable Neighborhood Search,VNS)作为一种新的元启发式算法,已初步成功地用于解决优化问题,尤其是对于大规模组合优化问题效果良好。对VNS的扩展研究层出不穷,并将其成功地应用到旅行商问题、车辆路径问题、调度、图着色等问题中。简述了经典的元启发式算法,并依次论述了优化问题,VNS算法起源,VNS算法原理,VNS算法分析,扩展的VNS分析,VNS在初始解构造、邻域结构构造、局部搜索和停止准则几个方面的改进方法,针对不同版本的VNS归纳了其在各种优化问题应用情况。基于对改进的VNS的分类,从算法自身研究角度和实际应用角度提出了未来研究方向。