Tabu search and GRASP for the maximum diversity problem

In this paper, we develop new heuristic procedures for the maximum diversity problem (MDP). This NP-hard problem has a significant number of practical applications such as environmental balance, telecommunication services or genetic engineering. The proposed algorithm is based on the tabu search methodology and incorporates memory structures for both construction and improvement. Although proposed in seminal tabu search papers, memory-based constructions have often been implemented in naïve ways that disregard important elements of the fundamental tabu search proposals. We will compare our tabu search construction with a memory-less design and with previous algorithms recently developed for this problem. The constructive method can be coupled with a local search procedure or a short-term tabu search for improved outcomes. Extensive computational experiments with medium and large instances show that the proposed procedure outperforms the best heuristics reported in the literature within short computational times.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

A simple and effective algorithm for the MaxMin diversity problem

The challenge of maximizing the diversity of a collection of points arises in a variety of settings, including the setting of search methods for hard optimization problems. One version of this problem, called the Maximum Diversity Problem (MDP), produces a quadratic binary optimization problem subject to a cardinality constraint, and has been the subject of numerous studies. This study is focused on the Maximum Minimum Diversity Problem (MMDP) but we also introduce a new formulation using MDP as a secondary objective. We propose a fast local search based on separate add and drop operations and on simple tabu mechanisms. Compared to previous local search approaches, the complexity of searching for the best move at each iteration is reduced from quadratic to linear; only certain streamlining calculations might (rarely) require quadratic time per iteration. Furthermore, the strong tabu rules of the drop strategy ensure a powerful diversification capacity. Despite its simplicity, the approach proves superior to most of the more advanced methods from the literature, yielding optimally-proved solutions for many problems in a matter of seconds and even attaining a new lower bound.     

A hybrid metaheuristic method for the Maximum Diversity Problem

The Maximum Diversity Problem (MDP) consists in selecting a subset of m$m$ elements from a given set of n$n$ elements (n>m$n>m$) in such a way that the sum of the pairwise distances between the m$m$ chosen elements is maximized. We present a hybrid metaheuristic algorithm (denoted by MAMDP) for MDP. The algorithm uses a dedicated crossover operator to generate new solutions and a constrained neighborhood tabu search procedure for local optimization. MAMDP applies also a distance-and-quality based replacement strategy to maintain population diversity. Extensive evaluations on a large set of 120 benchmark instances show that the proposed approach competes very favorably with the current state-of-art methods for MDP. In particular, it consistently and easily attains all the best known lower bounds and yields improved lower bounds for 6 large MDP instances. The key components of MAMDP are analyzed to shed light on their influence on the performance of the algorithm.     

Reinforcement learning enhanced multi-neighborhood tabu search for the max-mean dispersion problem

This paper presents a highly effective reinforcement learning enhancement of multi-neighborhood tabu search for the max-mean dispersion problem. The reinforcement learning component uses the Q-learning mechanism that incorporates the accumulated feedback information collected from the actions performed during the search to guide the generation of diversified solutions. The tabu search component employs 1-flip and reduced 2-flip neighborhoods to collaboratively perform the neighborhood exploration for attaining high-quality local optima. A learning automata method is integrated in tabu search to adaptively determine the probability of selecting each neighborhood. Computational experiments on 80 challenging benchmark instances demonstrate that the proposed algorithm is favorably competitive with the state-of-the-art algorithms in the literature, by finding new lower bounds for 3 instances and matching the best known results for the other instances. Key elements and properties are also analyzed to disclose the source of the benefits of our integration of learning mechanisms and tabu search.     

Strategic oscillation for the capacitated hub location problem with modular links

The capacitated single assignment hub location problem with modular link capacities is a variant of the classical hub location problem in which the cost of using edges is not linear but stepwise, and the hubs are restricted in terms of transit capacity rather than in the incoming traffic. We propose a metaheuristic algorithm based on strategic oscillation, a methodology originally introduced in the context of tabu search. Our method incorporates several designs for constructive and destructive algorithms, together with associated local search procedures, to balance diversification and intensification for an efficient search. Computational results on a large set of instances show that, in contrast to exact methods that can only solve small instances optimally, our metaheuristic is able to find high-quality solutions on larger instances in short computing times. In addition, the new method, which joins tabu search strategies with strategic oscillation, outperforms the previous tabu search implementation.     

A hybrid metaheuristic approach to solving the UBQP problem

This paper presents a hybrid metaheuristic approach (HMA) for solving the unconstrained binary quadratic programming (UBQP) problem. By incorporating a tabu search procedure into the framework of evolutionary algorithms, the proposed approach exhibits several distinguishing features, including a diversification-based combination operator and a distance-and-quality based replacement criterion for pool updating. The proposed algorithm is able to easily obtain the best known solutions for 31 large random instances up to 7000 variables (which no previous algorithm has done) and find new best solutions for three of nine instances derived from the set-partitioning problem, demonstrating the efficacy of our proposed algorithm in terms of both solution quality and computational efficiency. Furthermore, some key elements and properties of the HMA algorithm are also analyzed.     

A Tabu Search Algorithm for the Quadratic Assignment Problem

Tabu search approach based algorithms are among the widest applied to various combinatorial optimization problems. In this paper, we propose a new version of the tabu search algorithm for the well-known problem, the quadratic assignment problem (QAP). One of the most important features of our tabu search implementation is an efficient use of mutations applied to the best solutions found so far. We tested this approach on a number of instances from the library of the QAP instances—QAPLIB. The results obtained from the experiments show that the proposed algorithm belongs to the most efficient heuristics for the QAP. The high efficiency of this algorithm is also demonstrated by the fact that the new best known solutions were found for several QAP instances.     

A note on heuristic approach based on UBQP formulation of the maximum diversity problem

The maximum diversity problem (MDP) is a challenging NP-hard problem with a wide range of real applications. Several researchers have pointed out close relationship between the MDP and unconstrained binary quadratic program (UBQP). In this paper, we provide procedures to solve MDP ideas from the UBQP formulation of the problem. We first give some local optimality results for r-flip improvement procedures on MDP. Then, a set of highly effective diversification approaches based on sequential improvement steps for MDP are presented. Four versions of the approaches are used within a simple tabu search and applied to 140 benchmark MDP problems available on the Internet. The procedures solve all 80 small- to medium-sized problems instantly to the best known solutions. For 22 of the 60 large problems, the procedures improved by significant amounts the best known solutions in reasonably short CPU time.     

Maximally diverse grouping: an iterated tabu search approach

The maximally diverse grouping problem (MDGP) consists of finding a partition of a set of elements into a given number of mutually disjoint groups, while respecting the requirements of group size constraints and diversity. In this paper, we propose an iterated tabu search (ITS) algorithm for solving this problem. We report computational results on three sets of benchmark MDGP instances of size up to 960 elements and provide comparisons of ITS to five state-of-the-art heuristic methods from the literature. The results demonstrate the superiority of the ITS algorithm over alternative approaches. The source code of the algorithm is available for free download via the internet.     

Heuristics and metaheuristics for the maximum diversity problem

This paper presents extensive computational experiments to compare 10 heuristics and 20 metaheuristics for the maximum diversity problem (MDP). This problem consists of selecting a subset of maximum diversity from a given set of elements. It arises in a wide range of real-world settings and we can find a large number of studies, in which heuristic and metaheuristic methods are proposed. However, probably due to the fact that this problem has been referenced under different names, we have only found limited comparisons with a few methods on some sets of instances. This paper reviews all the heuristics and metaheuristics for finding near-optimal solutions for the MDP. We present the new benchmark library MDPLIB, which includes most instances previously used for this problem, as well as new ones, giving a total of 315. We also present an exhaustive computational comparison of the 30 methods on the MDPLIB. Non-parametric statistical tests are reported in our study to draw significant conclusions.     

A divide and merge heuristic for the multiprocessor scheduling problem with sequence dependent setup times

This article proposes lower bounds, as well as a divide and merge heuristic for the multiprocessor scheduling problem with sequence dependent setup times (MSPS). The heuristic is tested on randomly generated instances and compared with a previously published tabu search algorithm. Results show that the proposed heuristic is much faster than tabu search while providing similar quality solutions.     

Solving a minimum-power covering problem with overlap constraint for cellular network design

We consider a type of covering problem in cellular networks. Given the locations of base stations, the problem amounts to determining cell coverage at minimum cost in terms of the power usage. Overlap between adjacent cells is required in order to support handover. The problem we consider is NP-hard. We present integer linear models and study the strengths of their continuous relaxations. Preprocessing is used to reduce problem size and tighten the models. Moreover, we design a tabu search algorithm for finding near-optimal solutions effectively and time-efficiently. We report computational results for both synthesized instances and networks originating from real planning scenarios. The results show that one of the integer models leads to tight bounds, and the tabu search algorithm generates high-quality solutions for large instances in short computing time.     

A tabu search heuristic for ship routing and scheduling

The purpose of this paper is to solve a planning problem faced by many shipping companies dealing with the transport of bulk products. These shipping companies are committed to carrying some contract cargoes and will try to derive additional revenue from optional spot cargoes. An efficient tabu search algorithm has been developed to ensure quick decision support for the planners. The solutions generated by the tabu search heuristic are compared with those produced by a previously published multi-start local search heuristic. Computational results show that the tabu search heuristic yields optimal or near-optimal solutions to real-life instances within reasonable time. For large and tigthly constrained cases, the tabu search heuristic provides much better solutions than the multi-start local search heuristic. A version of the tabu search heuristic will be integrated as an improved solver in a prototype decision support system used by several shipping companies.     

Multistart Tabu Search Strategies for the Unconstrained Binary Quadratic Optimization Problem

This paper describes and experimentally compares five rather different multistart tabu search strategies for the unconstrained binary quadratic optimization problem: a random restart procedure, an application of a deterministic heuristic to specially constructed subproblems, an application of a randomized procedure to the full problem, a constructive procedure using tabu search adaptive memory, and an approach based on solving perturbed problems. In the solution improvement phase a modification of a standard tabu search implementation is used. A computational trick applied to this modification – mapping of the current solution to the zero vector – allowed to significantly reduce the time complexity of the search. Computational results are provided for the 25 largest problem instances from the OR-Library and, in addition, for the 18 randomly generated larger and more dense problems. For 9 instances from the OR-Library new best solutions were found.     

An Application of Tabu Search Heuristic for the Maximum Edge-Weighted Subgraph Problem

The purpose of this article is to describe an efficient search heuristic for the Maximum Edge-weighted Subgraph (MEwS) problem. This problem requires to find a subgraph such that the sum of the weights associated with the edges of the subgraph is maximized subject to a cardinality constraint. In this study a tabu search heuristic for the MEwS problem is proposed. Different algorithms to obtain an initial solution are presented. One neighborhood search strategy is also proposed. Preliminary computational results are reported for randomly generated test problems of MEwS problem with different densities and sizes. For most of test problems, the tabu search heuristic found good solutions. In addition, for large size test problems, the tabu search outperformed the local search heuristic appearing in the literature.     

带有资源转移时间的RCPSP资源流模型及算法

本文在传统资源受限项目调度问题(resource-constrained project scheduling problem, RCPSP)中引入资源转移时间,为有效获得问题的最优解,采用资源流编码方式表示可行解,建立了带有资源转移时间的RCPSP资源流优化模型,目标为最小化项目工期。根据问题特征设计了改进的资源流重构邻域算子,分别设计了改进的禁忌搜索算法和贪心随机自适应禁忌搜索算法求解模型。数据实验结果表明,相较于现有文献中的方法,所提两种算法均可针对更多的项目实例求得最优解,并且得到最优解的时间更短,求解效率更高。此外,分析了算法在求解具有不同特征的项目实例时的性能,所得结果为项目经理结合项目特征评价算法适用性提供了指导。     

An Application of a Multi-Objective Tabu Search Algorithm to a Bicriteria Flowshop Problem

This paper proposes a new tabu search algorithm for multi-objective combinatorial problems with the goal of obtaining a good approximation of the Pareto-optimal or efficient solutions. The algorithm works with several paths of solutions in parallel, each with its own tabu list, and the Pareto dominance concept is used to select solutions from the neighborhoods. In this way we obtain at each step a set of local nondominated points. The dispersion of points is achieved by a clustering procedure that groups together close points of this set and then selects the centroids of the clusters as search directions. A nice feature of this multi-objective algorithm is that it introduces only one additional parameter, namely, the number of paths. The algorithm is applied to the permutation flowshop scheduling problem in order to minimize the criteria of makespan and maximum tardiness. For instances involving two machines, the performance of the algorithm is tested against a Branch-and-Bound algorithm proposed in the literature, and for more than two machines it is compared with that of a tabu search algorithm and a genetic local search algorithm, both from the literature. Computational results show that the heuristic yields a better approximation than these algorithms.     

Solving the frequency assignment problem with polarization by local search and tabu

In the Frequency Assignment Problem with Polarization (FAPP), a given set of links must each be assigned a frequency and a polarization, while respecting given radio-electric compatibility constraints defined on pairs of links. In this paper, we propose a tabu search algorithm for the FAPP. A specialized neighborhood is proposed for the problem. Other key features of the algorithm are an adaptive technique to adjust the tabu tenure, an original diversification technique, and a pre-processing procedure based on arc-consistency techniques. The algorithm is tested on the 40 instances of the ROADEF Challenge 2001. It reaches the best known feasibility level for all instances and finds or improves on the best known solutions of the Challenge for a majority of the instances.Received: July 2003 / Revised version: September 2004MSC classification: 90B18, 68T20All correspondence to: Philippe Galinier     

A hybrid algorithm for the manufacturing cell formation problem

In this paper we compare different heuristic methods for the manufacturing cell formation problem considering part process sequence: a GRASP algorithm, a reactive GRASP algorithm and a hybrid algorithm which combines reactive GRASP and tabu search. All algorithms are tested with a set of instances from the literature. The results from the GRASP algorithm are compared to those of the reactive GRASP in order to evaluate the advantages of automatically adjusting the parameter value within the randomized greedy procedure. Also the reactive GRASP results are compared to those of the hybrid algorithm to evaluate the contribution to solution quality of replacing the local search phase of the GRASP algorithm with tabu search.