A fuzzy clustering-based hybrid method for a multi-facility location problem

A fuzzy clustering-based hybrid method for a multi-facility location problem is presented in this study. It is assumed that capacity of each facility is unlimited. The method uses different approaches sequentially. Initially, customers are grouped by spherical and elliptical fuzzy cluster analysis methods in respect to their geographical locations. Different numbers of clusters are experimented. Then facilities are located at the proposed cluster centers. Finally each cluster is solved as a single facility location problem. The center of gravity method, which optimizes transportation costs is employed to fine tune the facility location. In order to compare logistical performance of the method, a real world data is gathered. Results of existing and proposed locations are reported.     

Heuristics for a continuous multi-facility location problem with demand regions

We consider a continuous multi-facility location allocation problem where the demanding entities are regions in the plane instead of points. The problem can be stated as follows: given m (closed, convex) polygonal demand regions in the plane, find the locations of q facilities and allocate each region to exactly one facility so as to minimize a weighted sum of squares of the maximum Euclidean distances between the demand regions and the facilities they are assigned to.We propose mathematical programming formulations of the single and multiple facility versions of the problem considered. The single facility location problem is formulated as a second order cone programming (SOCP) problem, and hence is solvable in polynomial time. The multiple facility location problem is NP-hard in general and can be formulated as a mixed integer SOCP problem. This formulation is weak and does not even solve medium-size instances. To solve larger instances of the problem we propose three heuristics. When all the demand regions are rectangular regions with their sides parallel to the standard coordinate axes, a faster special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.     

Supply facility and input/output point locations in the presence of barriers

This paper studies a facility location model in which two-dimensional Euclidean space represents the layout of a shop floor. The demand is generated by fixed rectangular-shaped user sites and served by a single supply facility. It is assumed that (i) communication between the supply point and a demand facility occurs at an input/output (I/O) point on the demand facility itself, (ii) the facilities themselves pose barriers to travel and (iii) distance measurement is as per the L₁-metric. The objective is to determine optimal locations of the supply facility as well as I/O points on the demand facilities, in order to minimize total transportation costs. Several, increasingly more complex, versions of the model are formulated and polynomial time algorithms are developed to find the optimal locations in each case.Scope and purposeIn a facility layout setting, often a new central supply facility such as a parts supply center or tool crib needs to be located to serve the existing demand facilities (e.g., workstations or maintenance areas). The demand facilities are physical entities that occupy space, that cannot be traveled through, and that receive material from the central facility, through a perimeter I/O (input/output or drop-off/pick-up) point. This paper addresses the joint problem of locating the central facility and determining the I/O point on each demand facility to minimize the total material transportation cost. Different versions of this problem are considered. The solution methods draw from and extend results of location theory for a class of restricted location problems. For practitioners, simple results and polynomial time algorithms are developed for solving these facility (re) design problems.     

Integrated Inventory Control and Facility Location Decisions in a Multi-Echelon Supply Chain Network with Hubs

This paper develops mathematical models to coordinate facility location and inventory control for a four-echelon supply chain network consisting of multiple suppliers, warehouses, hubs and retailers. The hubs help in reducing transportation costs by consolidating products from multiple warehouses and directing the larger shipments to the retailer. The integrated models studied in this paper simultaneously determines three types of decisions: (i) facility location—the number and location of warehouses and hubs, (ii) allocation—assignment of suppliers to located warehouses and retailers to located warehouses via the location hubs, and (iii) inventory control decisions at each located warehouse. The goal is to minimize the facility location, transportation and the inventory costs. A mixed integer nonlinear programming formulation is first presented. The nonlinear integer programming formulation is then transformed into a conic mixed integer program and a novel and compact conic mixed integer programming formulation. Computational runs are conducted using commercial solvers to compare the performance of the different formulations. The compact conic mixed integer programming formulation was found to significantly outperform the other formulations by achieving significant computational savings. The results demonstrate that large scale instances of certain multi-echelon supply chain network design problems can be solved using commercial solvers through intelligent reformulation of the model.     

Efficient approximate solution methods for the multi-commodity capacitated multi-facility Weber problem

The capacitated multi-facility Weber problem is concerned with locating I capacitated facilities in the plane to satisfy the demand of J customers with the minimum total transportation cost of a single commodity. This is a nonconvex optimization problem and difficult to solve. In this work, we focus on a multi-commodity extension and consider the situation where K distinct commodities are shipped subject to capacity constraints between each customer and facility pair. Customer locations, demands and capacities for each commodity, and bundle restrictions are known a priori. The transportation costs, which are proportional to the distance between customers and facilities, depend on the commodity type. We address several location-allocation and discrete approximation heuristics using different strategies. Based on the obtained computational results we can say that the alternate solution of location and allocation problems is a very efficient strategy; but the discrete approximation has excellent accuracy.     

Stochastic facility and transfer point covering problem with a soft capacity constraint

Existing models for transfer point location problems (TPLPs) do not guarantee the desired service time to customers. In this paper, a facility and TPLP is formulated based on a given service time that is targeted by a decision maker. Similar to real‐world situations, transportation times and costs are assumed to be random. In general, facilities are capacitated. However, in emergency services, they are not allowed to reject the customers for out of capacity reasons. Therefore, a soft capacity constraint for the facilities and a second objective to minimize the overtime in the facility with highest assigned demand are proposed. To solve the biobjective model with random variables, a variance minimization technique and chance‐constraint programming are applied. Thereafter, using fuzzy multiple objective linear programming, the proposed biobjective model is converted to a single objective. Computational results on 30 randomly designed experimental problems confirm satisfactory performance of the proposed model in reducing the variance of solutions as well as the overtime in the busiest facility.     

战术层库存/运输优化问题的一种求解算法

讨论了现代物流管理中比较重要、复杂的战术层随机库存/运输联合优化问题的一种求 解思路,并根据求解约束集中器选址问题(CCLP)的方法和禁忌搜索算法设计了一个启发式算法。重 点介绍了编程实现此算法的要点,并用模拟算例对设计的算法进行了验证,计算结果是比较理想的。     

Fuzzy c-means improvement using relaxed constraints support vector machines

Fuzzy clustering is a widely applied method for extracting the underlying models within data. It has been applied successfully in many real-world applications. Fuzzy c-means is one of the most popular fuzzy clustering methods because it produces reasonable results and its implementation is straightforward. One problem with all fuzzy clustering algorithms such as fuzzy c-means is that some data points which are assigned to some clusters have low membership values. It is possible that many samples may be assigned to a cluster with low-confidence. In this paper, an efficient and noise-aware implementation of support vector machines, namely relaxed constraints support vector machines, is used to solve the mentioned problem and improve the performance of fuzzy c-means algorithm. First, fuzzy c-means partitions data into appropriate clusters. Then, the samples with high membership values in each cluster are selected for training a multi-class relaxed constraints support vector machine classifier. Finally, the class labels of the remaining data points are predicted by the latter classifier. The performance of the proposed clustering method is evaluated by quantitative measures such as cluster entropy and Minkowski scores. Experimental results on real-life data sets show the superiority of the proposed method.     

A simple and effective genetic algorithm for the two-stage capacitated facility location problem

This paper presents a simple and effective Genetic Algorithm (GA) for the two-stage capacitated facility location problem (TSCFLP). The TSCFLP is a typical location problem which arises in freight transportation. In this problem, a single product must be transported from a set of plants to meet customers demands, passing out by intermediate depots. The objective is to minimize the operation costs of the underlying two-stage transportation system thereby satisfying demand and capacity constraints of its agents. For this purpose, a GA is proposed and computational results are reported comparing the heuristic results with those obtained by two state-of-the-art Lagrangian heuristics proposed in the literature for the problem.     

Robust and fuzzy goal programming optimization approaches for a novel multi-objective hub location-allocation problem: A supply chain overview

In this paper, a novel multi-objective mathematical model is developed to solve a capacitated single-allocation hub location problem with a supply chain overview. Three mathematical models with various objective functions are developed. The objective functions are to minimize: (a) total transportation and installation costs, (b) weighted sum of service times in the hubs to produce and transfer commodities and the tardiness and earliness times of the flows including raw materials and finished goods, and (c) total greenhouse gas emitted by transportation modes and plants located in the hubs. To come closer to reality, some of the parameters of the proposed mathematical model are regarded as uncertain parameters, and a robust approach is used to solve the given problem. Furthermore, two methods, namely fuzzy multi-objective goal programming (FMOGP) and the Torabi and Hassini's (TH) method are used to solve the multi-objective mathematical model. Finally, the concluding part presents the comparison of the obtained results.     

不确定条件下战时应急物资配送中心选址研究

应急物资配送中心是战时运送应急物资的重要枢纽,面对战场环境的不确定性,设计高效、可靠的战时应急物资配送网络具有重要战略意义。将需求不确定性和设施损毁同时纳入考量,建立了模糊需求和设施损毁场景下带容量约束的可靠性选址模型。设计了一种改进的免疫遗传算法对模型进行求解。算法中针对模型特点加入动态交叉变异算子和客户优先级算法,通过仿真实验验证了模型和算法的有效性,为可靠性选址问题提供模型和求解思路。通过数值实验分析了不确定水平和损毁概率对实验结果的影响,结果表明损毁概率对最终选址方案有更大的影响。     

Optimizing the Location of a Production Firm

A facility needs to be located in the plane to sell goods to a set of demand points. The cost for producing an item and the actual transportation cost per unit distance are given. The planner needs to determine the best location for the facility, the price charged at the source (mill price) and the transportation rate per unit distance to be charged to customers. Demand by customers is elastic and assumed declining linearly with the total charge. For each customer two parameters are given: the demand at charge zero and the decline of demand per unit charge. The objective is to find a location for the facility in the plane, the mill price charged to customers and the unit transportation rate charged to customers such that the company’s profit is maximized. The problem is formulated and an algorithm that finds the optimal solution is designed and tested on randomly generated problems.     

Integrated mathematical optimisation approach for the tower crane hook routing problem to satisfy material demand requests on-site

Optimising decisions around the location and operation of tower cranes can improve the workflow in construction projects. Traditionally, the location and allocation problems involved in tower crane operations in the literature have been solved separately from the assignment of material supply points to demand points and the scheduling of the crane’s activity sequence across supply and demand points on a construction site. To address the gap, this paper proposes a binary integer programming problem, where location of the tower crane, allocation of supply points to material-demanding regions, and routing of hook of the crane based on activity sequencing of the hook across supply and material-demanding regions on site are optimised. The novelty in this work is in the way the crane’s activity scheduling is modelled via mathematical programming, based on routing the hook movement to meet material demand, through minimising tower crane operating costs. A realistic case study is solved to assess the validity of the model. The model is contrasted with results obtained from other solving algorithms commonly adopted in the literature, along with a solution proposed by an experienced practitioner. Results indicate that all instances can be solved when compared to other meta-heuristics that fail to achieve an optimum solution. Compared to the solution proposed by the practitioner, the results of the proposed model achieve a 46% improvement in objective function value. Planners should optimise decisions related to the location of the crane, the crane’s hook movement to meet service requests, and supply points’ locations and assignment to material-demanding regions simultaneously for effective crane operations.     

A cross decomposition procedure for the facility location problem with a choice of facility type

Consider a capacitated facility location problem in which each customer is assumed to have a unit demand, and each facility capacity has to be chosen from the given set of admissible levels. Under the restriction that each customer's unit demand be met by exactly one facility, the objective is to select a set of facilities to open, along with their capacities, and to assign customer's demand to them so as to minimize the total cost which includes fixed costs of opening facilities as well as variable assignment costs. The problem is modelled as a pure 0–1 program which extends the scope of applicability significantly over that by conventional location models. Based on Cross Decomposition recently developed by Van Roy, a solution procedure is proposed, when exploits the special structure of the problem. Computational results with a set of test problems shows the superiority of our solution procedure to other related ones.     

Solving fuzzy capacitated location routing problem using hybrid variable neighborhood search and evolutionary local search

A fuzzy capacitated location routing problem (FCLRP) is solved by using a heuristic method that combines variable neighborhood search (VNS) and evolutionary local search (ELS). Demands of the customer and travel times between customers and depots are considered as fuzzy and deterministic variables, respectively in FCLRP. Heterogeneous and homogeneous fleet sizes are performed together to reach the least multi-objective cost in a case study. The multi-objective cost consists of transportation cost, additional cost, vehicle waiting cost and delay cost. A fuzzy chance constrained programming model is added by using credibility theory. The proposed method reaches the solution by performing four stages. In the first stage, initial solutions are obtained by using a greedy heuristic method, and then VNS heuristic, which consists of seven different neighborhood structures, is performed to improve the solution quality in the second stage. In the third stage, a perturbation procedure is applied to the improved solution using ELS algorithm, and then VNS heuristic is applied again in the last stage. The combination of VNS and ELS is called VNSxELS algorithm and applied to a case study, which has fifty-seven customers and five distributing points, effectively in a reasonable time.     

A solid transportation problem with type-2 fuzzy variables

This paper focuses on generating the optimal solutions of the solid transportation problem under fuzzy environment, in which the supply capacities, demands and transportation capacities are supposed to be type-2 fuzzy variables due to the instinctive imprecision. In order to model the problem within the framework of the credibility optimization, three types of new defuzzification criteria, i.e., optimistic value criterion, pessimistic value criterion and expected value criterion, are proposed for type-2 fuzzy variables. Then, the multi-fold fuzzy solid transportation problem is reformulated as the chance-constrained programming model with the least expected transportation cost. To solve the model, fuzzy simulation based tabu search algorithm is designed to seek approximate optimal solutions. Numerical experiments are implemented to illustrate the application and effectiveness of the proposed approaches.     

Capacitated facility location problem with general setup cost

This paper presents an extension of the capacitated facility location problem (CFLP), in which the general setup cost functions and multiple facilities in one site are considered. The setup costs consist of a fixed term (site setup cost) plus a second term (facility setup costs). The facility setup cost functions are generally non-linear functions of the size of the facility in the same site. Two equivalent mixed integer linear programming (MIP) models are formulated for the problem and solved by general MIP solver. A Lagrangian heuristic algorithm (LHA) is also developed to find approximate solutions for this NP-hard problem. Extensive computational experiments are taken on randomly generated data and also well-known existing data (with some necessary modifications). The detailed results are provided and the heuristic algorithm is shown to be efficient.     

HSI颜色模型下基于空间信息的FCM算法

针对提花毛皮样片的花型识别技术,在HSI颜色模型下提出了一种基于空间信息的FCM图像分割算法。算法在HSI颜色模型下获得FCM算法的初始聚类中心,并采用了基于空间信息的模糊C均值聚类方法对图像进行分割。经C++编程验证,算法能有效去除花型图像中的噪声,获得较理想的花型识别结果。     

Weighted Dantzig–Wolfe Decomposition for Linear Mixed-integer Programming

Dantzig–Wolfe decomposition can be used to solve the Lagrangian dual of a linear mixed-integer programming problem ( MIP ) if the dual structure of the ( MIP ) is exploited via Lagrangian relaxation with respect to the complicating constraints. In the so-called weighted Dantzig–Wolfe decomposition algorithm, instead of the optimal solution of the Dantzig–Wolfe master problem a specially weighted average of the previously constructed Lagrangian multipliers and the optimal solution of the master problem is used as Lagrangian multiplier for the next Lagrangian subproblem to be solved. A convergence proof of the weighted Dantzig–Wolfe decomposition algorithm is given, and some properties of this procedure together with computational results for the capacitated facility location problem are discussed.     

The congested facility location problem

Consider a network facility location problem where congestion arises at facilities, and is represented by delay functions that approximate the queueing process. We strive to minimize the sum of customers' transportation and waiting times, and facilities' fixed and variable costs. The problem is solved using a column generation technique within a branch-and-bound scheme. Numerical results are reported and a bilevel (user-optimized) formulation considered, among other extensions.