In this work, we investigate the adaptation of the Greedy Randomized Adaptive Search Procedure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the performance of these multi-start heuristic search methods when solving this NP-hard problem. The former is a memory-less approach that constructs independent solutions, while the latter is a memory-based method that constructs linked solutions, obtained by partially rebuilding previous ones. Both are based on the combination of greediness and randomization in the constructive process, and coupled with a subsequent local search phase

Campos, Vicente

Landa-Silva, Dario

Marti, Rafael

Martinez-Gavara, Anna

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Multi-start methods for the capacitated clustering problem

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Martinez-Gavara, Anna and Campos, Vicente and Landa-Silva, Dario and Marti, Rafael (2017) Multi-start methods for the capacitated clustering problem. In: 12th Metaheuristics International Conference (MIC 2017), 4-7 July 2017, Barcelona, Spain. Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/44823/1/dls_mic2017.pdfCopyright and reuse: The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions.This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence.  For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdfA note on versions: The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.ukMIC/MAEB 2017  id-1  Barcelona, July 4-7, 2017 Multi-Start Methods for the  Capacitated Clustering Problem 	Anna Martínez-Gavara 1,Vicente Campos1, Dario Landa-Silva 2, Rafael Martí1,   		1 Universitat de València C/Doctor Moliner, 50, 46100, Burjassot (València), Spain gavara@uv.es, Vicente.Campos@uv.es, Rafael.Marti@uv.es  2  University of Nottingham Wollaton Road, Nottingham, NG8 1BB, UK dario.landasilva@nottingham.ac.uk  Abstract In this work, we investigate the adaptation of the Greedy Randomized Adaptive Search Proce-dure (GRASP) and Iterated Greedy methodologies to the Capacitated Clustering Problem (CCP). In particular, we focus on the effect of the balance between randomization and greediness on the per-formance of these multi-start heuristic search methods when solving this NP-hard problem. The former is a memory-less approach that constructs independent solutions, while the latter is a memory-based method that constructs linked solutions, obtained by partially rebuilding previous ones. Both are based on the combination of greediness and randomization in the constructive pro-cess, and coupled with a subsequent local search phase.  1 Introduction Multi-start heuristic procedures were originally conceived as a way to exploit local or neighborhood search, by simply apply the search multiple times starting from different random initial solutions. Mod-ern multi-start heuristic methods for combinatorial optimization problems usually incorporate a power-ful form of diversification in the generation of solutions to help overcome local optimality. Without this diversification, such methods can become confined to a small region of the solution space, making it difficult, if not impossible, to find a global optimum. Most of such methods perform these steps itera-tively: apply a randomized constructive method followed by a local search procedure.  In these methods, diversification comes from the iterative randomized construction of solutions.  Multi-start heuristic methods for combinatorial optimization can be classified as suggested by [5] in memory-based and memory-less procedures. GRASP (Greedy Randomized Adaptive Search procedure) is probably the best well-known memory-less multi-start heuristic method [8], while Tabu Search [3] is nowadays a reference for memory based approaches.  In this paper, we focus on both memory-based and memory-less multi-start heuristic methods, and investigate the effect of randomization in these de-signs.  We use the Capacitated Clustering Problem (CCP), an NP-hard combinatorial optimization prob-lem, as a test case for our proposals and findings. Another memory-based multi-start method is the Iterated Greedy (IG). This method generates a sequence of solutions by iterating over a greedy construc-tive heuristic using two main phases: destruction and construction. IG is easy to implement that has exhibited state-of-the-art performance in some settings [9]. In this work, we investigate these two successful methodologies in multi-start methods: GRASP and Iterated Greedy, and their hybridization. The former constructs independent solutions, while the latter can be viewed as a constructive method of linked solutions. These are two very different approaches to construct a solution. Both methods combine greediness and randomization in different ways.  The aim of this investigation is to identify ways to exploit better greediness and randomization. For our experi-ments, we consider the CCP, which is a difficult optimization problem. However, our objective is to id-2 MIC/MAEB 2017  Barcelona, July 4-7, 2017 identify effective strategies and patterns that could succeed in other settings. Hence, the intended con-tribution of this paper is to exploit greediness and randomization within the context of multi-start heu-ristic search effectively. In a broader sense, we can say that we are comparing memory-less and memory-based designs within constructive methods. 2 Capacitated Clustering Problem The aim of the Capacitated Clustering Problem (CCP) is to obtain a partition of the set of points into different groups in order to optimize some weighted measure of distance among the points in the same group. The most recent applications of this problem are in the context of facility planners at mail pro-cessing and distribution centers within the US Postal Service. In particular, the design of the zones to help rationalize the bulk movement of mail, see [1]. Furthermore, Morán-Mirabal et al. in [7] applied an equivalent in the context of mobility networks.  Given a graph 𝐺 = (𝑉, 𝐸) where 𝑉 is a set of 𝑛 nodes and 𝐸 is a set of edges, let 𝑤* ≥ 0 be the weight of node 𝑖 ∈ 𝑉 and let 𝑐*0 be the benefit of edge 𝑖, 𝑗 ∈ 𝐸. The Capacitated Clustering Problem (CCP) consists of partition 𝑉 into 𝑝 clusters in such a way that the sum of the weights of the elements in each cluster is within some integer capacity limits, 𝐿 and 𝑈, and the sum of the benefits between the pairs of elements in the same cluster is maximized. The CCP can be formulated as a quadratic integer program with binary variables 𝑥*6 that take the value of 1 if element 𝑖 is in cluster 𝑘 and 0 otherwise. The objective function adds the total benefit of all pairs of elements that belong to the same cluster.  																								 𝐶𝐶𝑃 	𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒									 𝑐*0𝑥*6𝑥06@0A*@BC*DCE6DC  																									𝑠𝑢𝑏𝑗𝑒𝑐𝑡	𝑡𝑜																				 𝑥*6E6DC = 1																			𝑖 = 1,2, … , 𝑛 																																																								𝐿 ≤ 𝑤*𝑥*6@*DC ≤ 𝑈														𝑘 = 1,2, … , 𝑝 																																																							𝑥*6 ∈ 0,1 																														𝑖 = 1, … , 𝑛		𝑘 = 1, … , 𝑝  3 Iterated Greedy and GRASP The GRASP that we consider in this work, called GRASP2-1, has the constructive method in [6] but the improvement method performs 2-1 exchanges (IM2-1). This neighborhood explores the exchange of two nodes, say i and j, in the same cluster k with a node l in another cluster s.  This move [6] can be simply called a 2-1 exchange, and it makes possible to swap nodes that individually are not allowed for reasons of capacity. The Iterated Greedy method (IG) alternates between destructive and constructive phases. During the destructive phase, some elements are removed from the solution. Next, it applies a greedy constructive method to reconstruct the partial solution and obtain a new solution. Then, an acceptance criterion is applied to decide whether the new solution replaces the current solution or not. The method iterates following this pattern until a stopping criterion is met. We investigate two different IG algorithms, IG and a hybridization between IG and GRASP called IG-GRASP. Our first implementation of the Iterated Greedy methodology, called simply IG, starts from an initial solution 𝑥, built with the CM algorithm and improved with IM2-1. Then, IG iteratively alternates between destructive and constructive phases. MIC/MAEB 2017  id-3  Barcelona, July 4-7, 2017 In the destructive phase, a percentage of the nodes are removed randomly from each cluster. Then, the constructive phase applies the greedy heuristic CM to reconstruct the solution. Additionally, the local search phase IM2-1 is applied to improve the new solution. Our hybridization initially, as in IG, it builds a complete solution with CM and then improves it with IM2-1. Then, the algorithm iteratively applies a destructive algorithm based on a greedy, then the constructive method CM, and finally the improvement procedure IM2-1. However, after a number of pre-established iterations applying these three methods consecutively with no improvement, instead of ending the procedure (as it is the case of IG), the hybrid algorithm resorts to GRASP2-1 to generate a new solution (built from scratch) to start again.  An interesting distinction between different IG methods is in the acceptance criterion to select the solu-tion for applying the destructive method.  As described in [4], in the ‘Replace if better’ acceptance criterion, the new solution is accepted only if it provides a better objective function value. In other words, the IG iterates over the best solution found. However, this can lead to stagnation situations of the search due to insufficient diversification. On the other hand, the “Always replace” acceptance criterion applies the destruction phase to the most recently visited solution, independently to its objective function value. This criterion clearly favors diversification over intensification, because it promotes a stochastic search in the space of local optima. We applied the latter one to our IG variants. We compare our three new methods: GRASP2-1, IG and IG-GRASP on a previously reported bench-mark (CCPLIB) with 60 instances, available at http://www.optsicom.es/ccp. The results of our experi-mentation provide an important lesson about the way in which greediness and randomization are com-bined in the different methods.  It turns out that the IG approach seems more effective than the GRASP methodology to solve the CCP instances.  In particular, IG is able to match 38% of best-known solutions with a percent deviation from the best value of 0.54% while GRASP2-1 only matches 2% of them with a 2.39% of deviation.  As expected, improved outcomes are obtained when these methodologies are hybridized in IG-GRASP, which obtains a 70% of the best-known solutions, and exhibits a percent de-viation of 0.24%. References [1] Y. Deng and J.F. Bard. A reactive GRASP with path relinking for the capacitated clustering.  Journal of Heuristics. 17:119-152, 2011. [2] T. A. Feo and M.G.C. Resende. A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8(2):67-71, 1989.  [3] F. Glover and M. Laguna. Tabu Search. Kluwer Academic Publisher: Boston, 1997. [4] M. Lozano, F. Glover, C. García-Martínez, F.J. Rodríguez and R. Martí. Tabu search with Strategic Oscilla-tion for the Quadratic Minimum Spanning Tree. IIE Transactions, 46:414-428, 2014. [5] R. Martí, M. Resende and C. Ribeiro. Multi-Start Methods for Combinatorial Optimization. European Jour-nal of Operational Research, 226:1-8, 2013. [6] A. Martínez-Gavara, V. Campos, M. Gallego, M. Laguna and R. Martí. Tabu Search and GRASP for the capacitated clustering problem. Computational Optimization and Applications, 62:589-607, 2015. [7] L. F. Morán-Mirabal, J. L. González-Velarde, M. G. C. Resende and R. M. A. Silva. Randomized heuristics for Handover minimization in mobility networks. Journal of Heuristics, 19:845-880, 2013. [8] M.G.C Resende and C.C. Ribeiro. Greedy randomized adaptive search procedures: Advances and applica-tions. In M. Gendreau and J.-Y. Potvin, editors, Handbook of Metaheuristics, pages 293–319. Springer, 2nd edition, 2010. [9] R. Ruiz and T. Stützle. An Iterated Greedy heuristic for the sequence dependent setup times flowshop prob-lem with makespan and weighted tardiness objectives. European Journal of Operational Research, 187(3):1143-1159, 2008. 

Multi-start methods for the capacitated clustering problem

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