Fuzzy J-Means: a new heuristic for fuzzy clustering

A fuzzy clustering problem consists of assigning a set of patterns to a given number of clusters with respect to some criteria such that each of them may belong to more than one cluster with different degrees of membership. In order to solve it, we first propose a new local search heuristic, called Fuzzy J-Means, where the neighbourhood is defined by all possible centroid-to-pattern relocations. The “integer” solution is then moved to a continuous one by an alternate step, i.e., by finding centroids and membership degrees for all patterns and clusters. To alleviate the difficulty of being stuck in local minima of poor value, this local search is then embedded into the Variable Neighbourhood Search metaheuristic. Results on five standard test problems from the literature are reported and compared with those obtained with the well-known Fuzzy C-Means heuristic. It appears that solutions of substantially better quality are obtained with the proposed methods than with this former one.     

Selection and scheduling in a virtual organisation environment with a service broker

This paper addresses the problems of selection and scheduling related to a virtual organisation in which a central service broker communicates with groups of service providers offering services of different types. Our objective is to show that these problems can be formulated in a precise manner and that well-established algorithms can be developed for solving them. Especially, Tabu Search and Simulated Annealing with Variable Neighbourhood search are proposed as options for finding near-optimal solutions to the selection and scheduling problems. In our computational experiments, the algorithms were first validated with the help of benchmark job-shop problems and then applied to a specific cutting stock problem with randomly generated instances.     

双层束搜索算法优化机器人制造单元调度问题

针对混流生产阻塞机器人制造单元调度问题，给出了可行机器人运动插入法，构建可行解。依据可行机器人运动插入法，提出双层过滤变宽度束搜索算法进行求解。搜索过程利用局部评价函数和全局评价函数对节点进行两次择优选取。通过计算随机生成算例，仿真结果表明，相对于以分支定界算法产生的可行解进行变邻域搜索、分支定界算法、局部评价函数束搜索算法、全局评价函数束搜索算法和双层过滤定宽度束搜索算法，双层过滤变宽度束搜索算法不但能显著提高搜索效率，而且解的平均改进度分别为3.07%、6.07%、7.79%、12.62%、14.47%。     

VNS methods for home care routing and scheduling problem with temporal dependencies,and multiple structures and specialties

The need for optimization in the Home Care Service is becoming more and more legitimate in the face of the increase of demand and cost all over the world. Recently, many researchers in the Operation Research community have been attracted by this issue, which presents interesting aspects related to the vehicle routing problems. In this paper, we consider a new variant called the vehicle routing problem with time windows, temporal dependencies (synchronization, precedence, and disjunction), multi‐structures, and multispecialties problem (VRPTW‐TD‐2MS). This new variant is an extension of the vehicle routing problems with time windows and synchronization constraints (VRPTW‐S) that is well‐studied in literature. We present a Mixed Integer Programming method, and propose three Variable Neighborhood Search approaches. Extensive experiments show the effectiveness and efficiency of the General Variable Neighborhood Search with Ejection Chains‐based local search for solving VRPTW‐TD‐2MS and VRPTW‐S.     

COMPARATIVE ANALYSIS OF A HYBRID DE ALGORITHM WITH THE VNS ALGORITHM AND ITS VARIATION SVNS TO SOLVE A REAL-WORLD FREQUENCY ASSIGNMENT PROBLEM

The Frequency Assignment is a very important task in the planning of the GSM networks, and it still continues to be a critical task for current (and future) mobile communication operators. In this work, we compare a hybrid Differential Evolution algorithm with the Variable Neighbourhood Search algorithm and also its variant Skewed Variable Neighbourhood Search to solve a real-world Frequency Assignment problem (FAP) in GSM Networks. The results that are shown use accurate interference information. That information was also adopted by other researchers and it represents a real GSM network, granting, therefore, an really important applicability. Furthermore, we have analyzed and compared our approach with other algorithms proposed so far to this problem. Hence, our approach using the SVNS algorithm has proven to be efficient in solving this problem, and permitted us to obtain good results. In fact, with this work we have contributed to the FAP problem with an additional comparison between approaches using metaheuristics based on trajectory (VNS and SVNS) and others based on population (DE).     

Embedded local search approaches for routing optimization

This paper presents a new class of heuristics which embed an exact algorithm within the framework of a local search heuristic. This approach was inspired by related heuristics which we developed for a practical problem arising in electronics manufacture. The basic idea of this heuristic is to break the original problem into small subproblems having similar properties to the original problem. These subproblems are then solved using time intensive heuristic approaches or exact algorithms and the solution is re-embedded into the original problem. The electronics manufacturing problem where we originally used the embedded local search approach, contains the Travelling Salesman Problem (TSP) as a major subproblem. In this paper we further develop our embedded search heuristic, HyperOpt, and investigate its performance for the TSP in comparison to other local search based approaches. We introduce an interesting hybrid of HyperOpt and 3-opt for asymmetric TSPs which proves more efficient than HyperOpt or 3-opt alone. Since pure local search seldom yields solutions of high quality we also investigate the performance of the approaches in an iterated local search framework. We examine iterated approaches of Large-Step Markov Chain and Variable Neighbourhood Search type and investigate their performance when used in combination with HyperOpt. We report extensive computational results to investigate the performance of our heuristic approaches for asymmetric and Euclidean Travelling Salesman Problems. While for the symmetric TSP our approaches yield solutions of comparable quality to 2-opt heuristic, the hybrid methods proposed for asymmetric problems seem capable of compensating for the time intensive embedded heuristic by finding tours of better average quality than iterated 3-opt in many less iterations and providing the best heuristic solutions known, for some instance classes.     

The classroom assignment problem: Complexity,size reduction and heuristics

In this paper, we investigate a compound of the exam timetabling problems which consists of assigning a set of independent exams to a certain number of classrooms. We can define the exam timetabling problem as the scheduling of exams to time slots in first stage and at a second stage, the assignment of a set of exams extracted from one time slot to some available classrooms.Even though the formulation of this problem looks simple as it contains only two sets of constraints including only binary variables, we show that it belongs to the class of NP hard problems by reduction from the Numerical Matching with Target Sum problems (NMTS).In order to reduce the size of this problem and make it efficiently solvable either by exact method or heuristic approaches, a theorem is rigorously demonstrated and a reduction procedure inspired from the dominance criterion is developed. The two methods contribute in the search for a feasible solution by reducing the size of the original problem without affecting the feasibility. Since the reduction procedures do not usually assign all exams to classrooms, we propose a Variable Neighbourhood Search (VNS) algorithm in order to obtain a good quality complete solution. The objective of VNS algorithm is to reduce the total classroom capacity assigned to exams. A numerical result concerning the exam of the main session of the first semester of the academic year 2009–2010 of the Faculty of Economics and Management Sciences of Sfax shows the good performance of our approach compared with lower bound defined as the sum of the total capacity of all assigned classrooms and the total size of the remaining exams after reduction.     

Bi-objective optimisation model for installation scheduling in offshore wind farms

A bi-objective optimisation using a compromise programming approach is proposed for installation scheduling of an offshore wind farm. As the installation cost and the completion period of the installation are important aspects in the construction of an offshore wind farm, the proposed method is used to deal with those conflicting objectives. We develop a mathematical model using integer linear programming (ILP) to determine the optimal installation schedule considering several constraints such as weather condition and the availability of vessels. We suggest two approaches to deal with the multi-objective installation scheduling problem, namely compromise programming with exact method and with metaheuristic techniques. In the exact method the problem is solved by CPLEX whereas in the metaheuristic approach we propose Variable Neighbourhood Search (VNS) and Simulated Annealing (SA). Moreover, greedy algorithms and a local search for solving the scheduling problem are introduced. Two generated datasets are used for testing our approaches. The computational experiments show that the proposed metaheuristic approaches produce interesting results as the optimal solution for some cases is obtained.     

Combining probabilistic algorithms, Constraint Programming and Lagrangian Relaxation to solve the Vehicle Routing Problem

This paper presents an original hybrid approach to solve the Capacitated Vehicle Routing Problem (CVRP). The approach combines a Probabilistic Algorithm with Constraint Programming (CP) and Lagrangian Relaxation (LR). After introducing the CVRP and reviewing the existing literature on the topic, the paper proposes an approach based on a probabilistic Variable Neighbourhood Search (VNS) algorithm. Given a CVRP instance, this algorithm uses a randomized version of the classical Clarke and Wright Savings constructive heuristic to generate a starting solution. This starting solution is then improved through a local search process which combines: (a) LR to optimise each individual route, and (b) CP to quickly verify the feasibility of new proposed solutions. The efficiency of our approach is analysed after testing some well-known CVRP benchmarks. Benefits of our hybrid approach over already existing approaches are also discussed. In particular, the potential flexibility of our methodology is highlighted.     

A two phase variable neighborhood search for the multi-product inventory routing problem

We address a multi-product inventory routing problem and propose a two-phase Variable Neighborhood Search (VNS) metaheuristic to solve it. In the first phase, VNS is used to solve a capacitated vehicle routing problem at each period to find an initial solution without taking into account the inventory. In the second phase, we iteratively improve the initial solution while minimizing both the transportation and inventory costs. For this, we propose two different algorithms, a Variable Neighborhood Descent and a Variable Neighborhood Search. We present an heuristic and a Linear Programming formulation, which are applied after each local search move, to determine the amount of products to collect from each supplier at each period. During the exploration, we use priority rules for suppliers and vehicles, based on the current delivery schedule over the planning horizon. Computational results show the efficiency of the proposed two-phase approach.     

Heuristic methods for the optimal statistic median problem

We consider the problem of finding the convex combination of vectors for which the median is maximum. The objective function of this problem is non-concave and non-differentiable, with many local optima that can trap any subgradient algorithm. So we analyzed and tested some heuristic procedures to find optimal or near-optimal solutions. First, we introduced a variant of Random Restart, in which starting solutions are the vertices of the simplex, and we proved that small size problems are solved by this procedure. Then, we analyzed two versions of Variable Neighborhood Search that are used to explore the whole space of the feasible solutions, and we show that the continuous version is more powerful than the discrete version.     

Gaussian variable neighborhood search for continuous optimization

Variable Neighborhood Search (VNS) has shown to be a powerful tool for solving both discrete and box-constrained continuous optimization problems. In this note we extend the methodology by allowing also to address unconstrained continuous optimization problems.Instead of perturbing the incumbent solution by randomly generating a trial point in a ball of a given metric, we propose to perturb the incumbent solution by adding some noise, following a Gaussian distribution. This way of generating new trial points allows one to give, in a simple and intuitive way, preference to some directions in the search space, or, contrarily, to treat uniformly all directions. Computational results show some advantages of this new approach.     

An Evolutionary Parallel Tabu Search approach for distribution systems reinforcement planning

In this paper a new meta-heuristic optimisation technique is proposed. The method is based on the Parallel Tabu Search (PTS) algorithm and the application is the optimal electrical distribution systems reinforcement planning through the installation of photovoltaic plants, parallel cables, capacitor banks and transformers. The issue is a combinatorial optimisation problem; the objective function is a non-linear expression of a large number of variables. In these cases, meta-heuristics have proved to work well and one of the most efficient is the Tabu Search algorithm. For large-scale problems, parallelisation improves Tabu Search computational efficiency as well as its exploration ability. In this paper, an enhanced version of PTS, Evolutionary Parallel Tabu Search (EPTS), is proposed. It performs reproduction operators on sub-neighbourhoods directing the search towards more promising areas of the search space. The problem of distribution systems reinforcement planning has been studied in detail and the results of the application show that the EPTS outperforms the PTS and Particle Swarm Optimisation algorithms.The algorithm's performance is also tested on mathematical test functions and other properties of the proposed algorithm are examined.     

A great deluge and tabu search hybrid with two-stage memory support for quadratic assignment problem

A two-stage memory architecture is maintained within the framework of great deluge algorithm for the solution of single-objective quadratic assignment problem. Search operators exploiting the accumulated experience in memory are also implemented to direct the search towards more promising regions of the solution space. The level-based acceptance criterion of the great deluge algorithm is applied for each best solution extracted in a particular iteration. The use of short- and long-term memory-based search supported by effective move operators resulted in a powerful combinatorial optimization algorithm. A successful variant of tabu search is employed as the local search method that is only applied over a few randomly selected memory elements when the second stage memory is updated. The success of the presented approach is illustrated using sets of well-known benchmark problems and evaluated in comparison to well-known combinatorial optimization algorithms. Experimental evaluations clearly demonstrate that the presented approach is a competitive and powerful alternative for solving quadratic assignment problems.     

Routing of barge container ships by mixed-integer programming heuristics

We investigate the optimization of transport routes of barge container ships with the objective to maximize the profit of a shipping company. This problem consists of determining the upstream and downstream calling sequence and the number of loaded and empty containers transported between any two ports. We present a mixed integer linear programming (MILP) formulation for this problem. The problem is tackled by the commercial CPLEX MIP solver and improved variants of the existing MIP heuristics: Local Branching, Variable Neighborhood Branching and Variable Neighborhood Decomposition Search. It appears that our implementation of Variable Neighborhood Branching outperforms CPLEX MIP solver both regarding the solution quality and the computational time. All other studied heuristics provide results competitive with CPLEX MIP solver within a significantly shorter amount of time. Moreover, we present a detailed case study transportation analysis which illustrates how the proposed approach can be used by managers of barge shipping companies to make appropriate decisions and solve real life problems.     

Effective local search algorithms for high school timetabling problems

This paper addresses the high school timetabling problem. The problem consists in building weekly timetables for meetings between classes and teachers with the goal of minimizing violations of specific requirements. In the last decades, several mixed-integer programs have been proposed and tested for this family of problems. However, medium and large size instances are still not effectively solved by these programs using state-of-the-art solvers and the scientific community has given special attention to the devising of alternative soft computing algorithms. In this paper, we propose a soft computing approach based on Iterated Local Search and Variable Neighborhood Search metaheuristic frameworks. Our algorithms incorporate new neighborhood structures and local search routines to perform an effective search. We validated the proposed algorithms on variants of the problem using seven public instances and a new dataset with 34 real-world instances including large cases. The results demonstrate that the proposed algorithms outperform the state-of-the-art approaches in both cases, finding the best solutions in 38 out of the 41 tested instances.     

求解护士排班问题的变邻域搜索算法

变邻域搜索算法是求解护士排班问题的一个有效算法,其扰动方法对算法性能有显著影响。为提高护士排班问题中护士的满意度,提出一个改进的变邻域搜索(IVNS)算法。该算法使用了三种邻域结构,而且当使用任意的邻域都不能进一步改进当前解时,设计了一个对当前最优解进行扰动的方法,即在排班期间内随机地选择两天,在不违反硬性约束的条件下选出一组值班护士并交换他们在这两天中的班次。在2010年举行的第一次全球护士排班大赛提供的一组公共测试集上与一个混合变邻域搜索(HVNS)算法进行了比较,在Sprint-early、Medium-early和Long-early组算例上的结果表明,IVNS算法的最优值至少不劣于HVNS,而平均值均优于HVNS;IVNS算法的最大方差为0.72,波动范围小,求解性能稳定。IVNS的扰动方案对现有方案的扰动较小,能有效跳出当前局部最优,增强变邻域搜索算法的优化能力,与HVNS算法相比,其求解性能更优。     

A hybrid VNS approach for the short-term production planning and scheduling: A case study in the pulp and paper industry

Mathematical formulations for production planning are increasing complexity, in order to improve their realism. In short-term planning, the desirable level of detail is particularly high. Exact solvers fail to generate good quality solutions for those complex models on medium- and large-sized instances within feasible time. Motivated by a real-world case study in the pulp and paper industry, this paper provides an efficient solution method to tackle the short-term production planning and scheduling in an integrated mill. Decisions on the paper machine setup pattern and on the production rate of the pulp digester (which is constrained to a maximum variation) complicate the problem. The approach is built on top of a mixed integer programming (MIP) formulation derived from the multi-stage general lotsizing and scheduling problem. It combines a Variable Neighbourhood Search procedure which manages the setup-related variables, a specific heuristic to determine the digester's production speeds and an exact method to optimize the production and flow movement decisions. Different strategies are explored to speed-up the solution procedure and alternative variants of the algorithm are tested on instances based on real data from the case study. The algorithm is benchmarked against exact procedures.     

Variable neighborhood search for the pharmacy duty scheduling problem

In this paper, we study on the Pharmacy Duty Scheduling (PDS) problem, where a subset of pharmacies should be on duty on national holidays, at weekends and at nights in order to be able to satisfy the emergency drug needs of the society. PDS problem is a multi-period p-median problem with special side constraints and it is an NP-Hard problem. We propose four Variable Neighborhood Search (VNS) heuristics. The first one is the basic version, BVNS. The latter two, Variable Neighborhood Decomposition Search (VNDS) and Variable Neighborhood Restricted Search (VNRS), aim to obtain better results in less computing time by decomposing or restricting the search space. The last one, Reduced VNS (RVNS), is for obtaining good initial solutions rapidly for BVNS, VNDS and VNRS. We test BVNS, VNRS and VNDS heuristics on randomly generated instances and report the computational test results. We also use VNS heuristics on real data for the pharmacies in central İzmir and obtain significant improvements.     

具有自加速与变邻域搜索的差分演化算法

在分析差分演化（DE）进化方式基础上,首先利用自加速性改进差异算子与选择算子,然后结合变邻域搜索改善算法的局部搜索能力,提出了一种具有自加速特性与变邻域搜索能力的差分演化算法(SAVNDE);基于DE的三种进化模式,利用5个Benchmark测试函数进行对比计算,实验结果表明：SAVNDE在保持了DE原有特性基础上,以较快的速度获得更好的结果。