Capacitated Clustering Research Articles

Capacitated Clustering via Majorization-Minimization and Collaborative Neurodynamic Optimization.

This paper addresses capacitated clustering based on majorization-minimization and collaborative neurodynamic optimization (CNO). Capacitated clustering is formulated as a combinatorial optimization problem. Its objective function consists of fractional terms with intra-cluster similarities in their numerators and cluster cardinalities in their denominators as normalized cluster compactness measures. To obviate the difficulty in optimizing the objective function with factional terms, the combinatorial optimization problem is reformulated as an iteratively reweighted quadratic unconstrained binary optimization problem with a surrogate function and a penalty function in a majorization-minimization framework. A clustering algorithm is developed based on CNO for solving the reformulated problem. It employs multiple Boltzmann machines operating concurrently for local searches and a particle swarm optimization rule for repositioning neuronal states upon their local convergence. Experimental results on ten benchmark datasets are elaborated to demonstrate the superior clustering performance of the proposed approaches against seven baseline algorithms in terms of 21 internal cluster validity criteria.

A Metaheuristic to Solve the Capacitated Clustering Problem

A Metaheuristic to Solve the Capacitated Clustering Problem

Parameterized complexity of categorical clustering with size constraints

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a1,…,an} over Σm, where Σ is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable for the binary case Σ={0,1}. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters are lower and upper bounded by integer parameters p and q, respectively. Our main theorem is that the problem is solvable in time 2O(Blog⁡B)|Σ|B⋅(mn)O(1).

Resilient route design for collection of material from suppliers with split deliveries and stochastic demands

The efficient design of routes and schedules for moving materials into manufacturing or assembly plants is a central element of freight logistics systems. However, such route design faces many challenges in practice, including uncertain supplies and operational effectiveness. The work in this paper addresses both the uncertainty in supplier quantities and the need for operational constraints on supplier splitting in the route design process. The model developed here accomplishes this resilient design of routes using a stochastic version of a capacitated clustering formulation. The model is constructed as a two-stage stochastic mixed integer program, and an effective solution method is developed using progressive hedging. An application to logistics operations in the automotive industry is used to validate the model, and the efficiency and stability of the proposed algorithm has been verified by computational experiments.

Hybrid matheuristics for the multi-capacitated clustering problem

The capacitated clustering problem is a well-known and largely studied combinatorial optimization problem with several industrial applications. Although a great attention has been paid to this problem in the literature, the deeming of the problem with clusters centers with multiple types and a unique capacity per type is quite limited. We introduce a novel variant of capacitated clustering problems named multi-capacitated clustering problem (MCCP), a NP-hard optimization problem in which there are clients with different types and units of services to offer that must be grouped into given centers that demand with limited capacity per type the services. It is taken into account the distance between each one of these clients and the potential clusters to which they can be allocated, aiming to minimize the sum of such distances. It is presented an integer programming model for this problem, which it is shown to have limited application solving large-sized instances. As solution procedures, we present the following algorithms. We propose a greedy heuristic to generate a tentative feasible solution within a negligible computational effort. We adapt a size-reduction (SR) matheuristic to solve the problem under study. Furthermore, we introduce an innovative matheuristic that hybridizes the constructive phase of the well-known GRASP metaheuristic with the SR algorithm. Also, we develop a variable fixing (VF) heuristic. Finally, we propose a hybrid matheuristic based on the SR and VF algorithms. Computational results over a set of 100 randomly generated test instances point out the quality of the solutions found by the proposed algorithms. Besides, the results are statistically tested, and thus, our proposals are recommended to solve the problem under study.

MEACCP: A membrane evolutionary algorithm for capacitated clustering problem

The Capacitated Clustering Problem (CCP) is a constrained clustering problem with a wide range of applications in VLSI design, vehicle routing, and mail delivery. In order to improve the performance of solving CCP, this paper proposes a membrane evolutionary algorithm MEACCP based on the Membrane Evolutionary Algorithm Framework (MEAF). First, the membrane structure of MEACCP and its object representation are designed. Second, six evolution operators are designed to evolve objects within the membrane and membrane structures. Third, the MEACCP and the method of membrane population initialization are discussed. Finally, MEACCP experiments were conducted on 100 instances of four different types of benchmark datasets, and the experimental results were compared with the results of the current state-of-the-art CCP algorithms. The experimental results show that the number of instances where MEACCP can find the optimal solution is 4.3 ∼ 9.1 times that of the algorithms in comparison, and the stability is better than all comparison algorithms.

HA-CCP: A Hybrid Algorithm for Solving Capacitated Clustering Problem.

The capacitated clustering problem (CCP) divides the vertices of the undirected graph into several disjoint clusters so that the sum of the node weights in each cluster meets the capacity limit while maximizing the sum of the weight of the edges between nodes in the same cluster. CCP is a typical NP-hard problem with a wide range of engineering applications. In recent years, heuristic algorithms represented by greedy random adaptive search program (GRASP) and variable neighborhood search (VNS) have achieved excellent results in solving CCP. To improve the efficiency and quality of the CCP solution, this study proposes a new hybrid algorithm HA-CCP. In HA-CCP, a feasible solution construction method is designed to adapt to the CCP with stricter upper and lower bound constraints and an adaptive local solution destruction and reconstruction method is designed to increase population diversity and improve convergence speed. Experiments on 90 instances of 4 types show that the best average solution obtained by HA-CCP on 58 instances is better than all comparison algorithms, indicating that HA-CCP has better solution stability. HA-CCP is also superior to all comparison algorithms in average solving efficiency.

Integrated student to school assignment and school bus routing problem for special needs students

In this paper, we address the combined problem of assigning students to schools and generating school bus routes for special needs students. Special needs students typically require additional accommodations, both in terms of the services available at school and during pick-ups and drop-offs. This introduces additional complexities to the problems of assignment and routing for students. We develop a mathematical model that jointly addresses these problems, and incorporate multi-loads and heterogeneous bus fleet features. The model simultaneously assigns special needs students to schools based on their level of need and the availability of resources, and generates efficient routes to transport them to their designated schools. We propose a two-phase heuristic approach using the capacitated clustering problem, which is capable of generating high-quality solutions. We evaluate our proposed methodology using several randomly generated instances, and a real-world case study in Fort Smith, Arkansas. The results from the case study demonstrate that the proposed methodology can reduce the total miles traveled by buses by over 35% on average, when compared to previously developed heuristics in the literature for the same case study.

Neighborhood decomposition-driven variable neighborhood search for capacitated clustering

The capacitated clustering problem (CCP) is a general model relevant for a variety of important applications in areas such as parallel computing and very large scale integration design. However, the problem is known to be NP-hard, and thus computationally challenging. In this work, we present an original and highly effective variable neighborhood search algorithm for the problem, which is characterized by its neighborhood decomposition technique and a probability-based diversification strategy. The proposed algorithm is assessed via extensive experiments on 110 benchmark instances commonly used in the literature. Computational results show that the algorithm significantly outperforms the existing state-of-the-art algorithms in the literature. This work advances the state-of-the-art of solving the capacitated clustering problem and can be useful for the related practical applications. The key feature of the algorithm, i.e., combining the neighborhood decomposition-driven local search with the perturbation, is of general interest and can help to design effective heuristic algorithms for other important clustering problems.

A matheuristic for large-scale capacitated clustering

Clustering addresses the problem of assigning similar objects to groups. Since the size of the clusters is often constrained in practical clustering applications, various capacitated clustering problems have received increasing attention. We consider here the capacitated p-median problem (CPMP) in which p objects are selected as cluster centers (medians) such that the total distance from these medians to their assigned objects is minimized. Each object is associated with a weight, and the total weight in each cluster must not exceed a given capacity. Numerous exact and heuristic solution approaches have been proposed for the CPMP. The state-of-the-art approach performs well for instances with up to 5,000 objects but becomes computationally expensive for instances with a much larger number of objects. We propose a matheuristic with new problem decomposition strategies that can deal with instances comprising up to 500,000 objects. In a computational experiment, the proposed matheuristic consistently outperformed the state-of-the-art approach on medium- and large-scale instances while having similar performance for small-scale instances. As an extension, we show that our matheuristic can be applied to related capacitated clustering problems, such as the capacitated centered clustering problem (CCCP). For several test instances of the CCCP, our matheuristic found new best-known solutions.

Solution strategy based on Gaussian mixture models and dispersion reduction for the capacitated centered clustering problem.

The Capacitated Centered Clustering Problem (CCCP)—a multi-facility location model—is very important within the logistics and supply chain management fields due to its impact on industrial transportation and distribution. However, solving the CCCP is a challenging task due to its computational complexity. In this work, a strategy based on Gaussian mixture models (GMMs) and dispersion reduction is presented to obtain the most likely locations of facilities for sets of client points considering their distribution patterns. Experiments performed on large CCCP instances, and considering updated best-known solutions, led to estimate the performance of the GMMs approach, termed as Dispersion Reduction GMMs, with a mean error gap smaller than 2.6%. This result is more competitive when compared to Variable Neighborhood Search, Simulated Annealing, Genetic Algorithm and CKMeans and faster to achieve when compared to the best-known solutions obtained by Tabu-Search and Clustering Search.

Capacitated clustering problems applied to the layout of IT-teams in software factories

This work studies Heterogeneous Capacitated Clustering Problems (HCCP) which are variations of the Capacitated Clustering Problem (CCP). They are applied to the layout of IT-Teams in software factory. In the real problem, there are workstations placed along the software factory where they are disposed in a pattern the individuals working in the same project form a group that would be geographically close to each other. The groups are heterogeneous in the number of individuals. The general problem discussed here consider a given number of individuals with attributes (weight and coordinates in Euclidean space), and wishes to determine minimum dissimilar clusters constrained to a given maximum capacity for each cluster. The groups are formed to achieve a specific objective of just forming feasible capacitated compact groups and/or also including the position of the group manager. The models here presented and evaluated consider a new generalized version of the heterogeneous capacitated median problem (gHCMP) which extracts the medians from the set of individuals; a new formulation is introduced for the min-max diameter heterogeneous capacitated clustering problem; a new formulation for the minimum group distance heterogeneous capacitated clustering problem; and a new formulation using the HCCCP. We also discuss ways to obtain the best results from the proposed formulations using adequate solvers for each problem; and it is presented a metaheuristic framework for the HCCCP that solves close to optimality all the real instances here tested. We also present the results obtained from the formulations and metaheuristic related to instances that represent real situations about organizing the layout of two Brazilian software factories. The proposed gHCMP model outperform the solutions obtained for the evaluated instances also using the other models.

Machine learning methods to improve the operations of 3PL logistics

Nowadays, the variety in the product mix, unpredictable customer demand and the need for a high level of service are crucial challenges in the management of a supply chain. Flexible processes are needed to gain competitive advantage and economic edges. This paper presents a data-driven application of unsupervised machine learning clustering algorithms to a real-world case study in the automotive industry. The clustering input dataset collects the data available to a third-party logistics (3PL) provider. Clustering algorithms are used to define product families for the assignment of the workload to the processing resources. Several clustering algorithms (k-means, Gaussian mixture models and hierarchical clustering) define different product families scenarios using different tuning parameters. The impact of each clustering scenario on the operations is assessed via a dashboard of logistics KPIs to identify the best performing clustering algorithm. The performance of each clustering is, then, compared to a logistic benchmark given by a capacitated clustering to identify the best compromise between a logistic-constrained algorithm with a long runtime and fast data-driven uncapacitated algorithm.

RCMeans: A Recursive Capacitated Means for Districting Problem

The billing process of an energy distributor in Brazil is connected to reading energy consumption logistics. An efficient, balanced and capacity enabled process has benefits with cost reducing and quality perception of the service provided. In capacitated clustering, the elements are associated with weights for construction of groups with limited capacities. This paper presents an approach to the capacitated clustering problem, applied to the consumer unit measurement groups organization in Brazil's energy distributors companies. The process of creating those measurement groups, in general, is carried out manually by expert analysts. The purpose of this problem modality is to create partitions that minimize the internal dispersion of the associated group. In this work, the RCMeans method, which is based on the K-Means technique applied to data groupings with the inclusion of the capacity constraint for the group definition, is presented. The obtained results show a comparison between the current situation and the result with the proposed method, under the cohesion analysis, separation, number of groups, silhouette index, and consumer units measurement mean time of the groups.

Preface: advances in theoretical and practical combinatorial optimization

Preface: Advances in Theoretical and Practical Combinatorial Optimization Said Salhi1, Richard Eglese2, Eleni Hadjiconstantinou3 This issue consists of research papers dedicated, though not limited, to the International Symposium in Combinatorial Optimisation (CO2016), chaired by Said Salhi and held at the University of Kent, Canterbury, 1-3 September 2016. Of the many good papers it attracted, 35 were considered suitable for the next stage of review, resulting in 20 being accepted mostly after two rounds of reviews. We divided the issue into three parts, each covering a wider though well-defined area within combinatorial optimization (CO). These three parts, which are not completely distinct, are referred to for simplicity as Mathematically-Oriented (MO), Heuristically-Based (HB) and Practically-Focussed (PF). MO contains seven papers. These include a paper on the integration of network flows in a coloring algorithm by Koster, Scheidweiler, and Tieves; one on a polynomial time algorithm for the minimum flow problem by Khodayifar, Raayatpanah, and Pardalos, and one on a formulation in multi-level capacitated facility location by Irawan and Jones. A.Salhi, Alsoufi, and Yang provide a mixed integer program combined with an evolutionary approach for seaside operations in container ports, while Zare, Borrero, Zeng and Prokopyev have contributed a note on new linearized formulations in bilevel linear integer problems. Fuzzy logic is explored for portfolio optimization by Liagkouras and Metaxiotis, and for supplier selection by Chang. HB covers six papers. Azizi, using a genetic algorithm, explores hub and spoke networks under disruption. Knust and Xie produce a simulated annealing algorithm for nurse rostering, while Pei, Cheng, Liu, Pardalos, and Kong investigate Variable Neighbourhood Search (VNS) for single and parallel machine serial batching. A skewed VNS is adopted by Derbel, Jarbaoui, and Bhiri for the heterogeneous fleet vehicle routing problem, and an ascent-escent VNS for the detection of community (cluster) structures in a network is discussed by Džamic, Aloise, and Mladenovic. The last paper in this category is by Brimberg, Mladenovic, Todosijevic, and Urosevic, who investigate the capacitated clustering problem using a variant of VNS. PF has seven papers that treat inspiring implementations of CO for real-life applications. For example, in the area of scheduling, the use of spare parts in vessel maintenance is dealt with by Kian, Bektas, and Ouelhadj, while shift scheduling for power station staff is explored by Shuib and Kamarudin. The avoidance of obstacles in off shore wind farms is optimized by Klein and Haugland, and the costing of a product within a supply chain is performed by de Matta. In the health area, the maximization of the expected number of kidney transplants is performed by Alvelos, Klimentova, and Viana; a preventive health care network is designed by Davari; and the layout of an operating theater is produced by Chraibi, Osman, and Kharraja. Finally, we are grateful to Professor E. Boros for trusting our academic judgment in acting as guest editors of the Annals of Operations Research special issue devoted, but not restricted, to CO2016. We are also thankful to the managing editor, Katie D’Agosta, for her patience and valuable support in preparing this special issue. 1 Centre for Logistics and Heuristic Optimisation, Kent Business School, University of Kent, Canterbury, UK 2 Lancaster University Management School, Lancaster, UK 3 School of Business and Law, Frederick University Cyprus, Cyprus.

Heuristic search to the capacitated clustering problem

Given a weighted graph, the capacitated clustering problem (CCP) is to partition a set of nodes into a given number of distinct clusters (or groups) with restricted capacities, while maximizing the sum of edge weights corresponding to two nodes from the same cluster. CCP is an NP-hard problem with many relevant applications. This paper proposes two effective algorithms for CCP: a Tabu Search (denoted as FITS) that alternates between exploration in feasible and infeasible search space regions, and a Memetic Algorithm (MA) that combines FITS with a dedicated cluster-based crossover. Extensive computational results on five sets of 183 benchmark instances from the literature indicate that the proposed FITS competes favorably with the state-of-the-art algorithms. Additionally, an experimental comparison between FITS and MA under an extended time limit demonstrates that further improvements in terms of the solution quality can be achieved with MA in most cases. We also analyze several essential components of the proposed algorithms to understand their importance to the success of these approaches.

Model-based capacitated clustering with posterior regularization

We propose a heuristic approach to address the general class of optimization problems involving the capacitated clustering of observations consisting of variable values that are realizations from respective probability distributions. Based on the expectation-maximization algorithm, our approach unifies Gaussian mixture modeling for clustering analysis and cluster capacity constraints using a posterior regularization framework. To test our algorithm, we consider the capacitated p-median problem in which the observations consist of geographic locations of customers and the corresponding demand of these customers. Our heuristic has superior performance compared to classic geometrical clustering heuristics, with robust performance over a collection of instance types.

A tabu search algorithm for a capacitated clustering problem

This paper investigates a clustering problem in a production and routing environment where a centralised facility uses a fleet of vehicles to serve a set of customers. A multi-period capacitated clustering problem is solved to partition customer into clusters with constraints that the accumulated customer demand in every period of the planning horizon is satisfied. The resulting mathematical model is hard to solve exactly and efficient heuristics are thus developed in this paper. The heuristics are based on the tabu search but include two novel features in the design of neighbourhood search; the first one is the dynamic combination of customers with close proximity into supernodes to eliminate ineffective moves and the second one is an ejection chain approach to perform composite moves of variable length from a series of simple moves. Computational results showed that they were able to provide superior solutions, especially in certain cases of tight capacity constraints.

A tabu search algorithm for a capacitated clustering problem

This paper investigates a clustering problem in a production and routing environment where a centralised facility uses a fleet of vehicles to serve a set of customers. A multi-period capacitated clustering problem is solved to partition customer into clusters with constraints that the accumulated customer demand in every period of the planning horizon is satisfied. The resulting mathematical model is hard to solve exactly and efficient heuristics are thus developed in this paper. The heuristics are based on the tabu search but include two novel features in the design of neighbourhood search; the first one is the dynamic combination of customers with close proximity into supernodes to eliminate ineffective moves and the second one is an ejection chain approach to perform composite moves of variable length from a series of simple moves. Computational results showed that they were able to provide superior solutions, especially in certain cases of tight capacity constraints.

Faster Algorithms for the Constrained k-means Problem

The classical center based clustering problems such as k-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r -gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O 1, ..., O k are an arbitrary partition of the dataset and the goal is to output k-centers c 1, ..., c k such that the objective function ${\sum }_{i = 1}^{k} {\sum }_{x \in O_{i}} ||x - c_{i}||^{2}$ is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter e > 0, let l denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1 + e) approximation w.r.t. the objective function above. In this paper, we show an upper bound on l by giving a randomized algorithm that outputs a list of $2^{\tilde {O}(k/\varepsilon )}$ k-centers. We also give a closely matching lower bound of $2^{\tilde {\Omega }(k/\sqrt {\varepsilon })}$ . Moreover, our algorithm runs in time $O \left (n d \cdot 2^{\tilde {O}(k/\varepsilon )} \right )$ . This is a significant improvement over the previous result of Ding and Xu (2015) who gave an algorithm with running time O(n d ⋅ (log n) k ⋅ 2 p o l y(k/e)) and output a list of size O((log n) k ⋅ 2 p o l y(k/e)). Our techniques generalize for the k-median problem and for many other settings where non-Euclidean distance measures are involved.