Solving the bi‐objective capacitated p‐median problem with multilevel capacities using compromise programming and VNS

A bi‐objective optimisation using a compromise programming (CP) approach is proposed for the capacitated p‐median problem (CPMP) in the presence of the fixed cost of opening facility and several possible capacities that can be used by potential facilities. As the sum of distances between customers and their facilities and the total fixed cost for opening facilities are important aspects, the model is proposed to deal with those conflicting objectives. We develop a mathematical model using integer linear programming (ILP) to determine the optimal location of open facilities with their optimal capacity. Two approaches are designed to deal with the bi‐objective CPMP, namely CP with an exact method and with a variable neighbourhood search (VNS) based matheuristic. New sets of generated instances are used to evaluate the performance of the proposed approaches. The computational experiments show that the proposed approaches produce interesting results.     

The gravity multiple server location problem

Models for locating facilities and service providers to serve a set of demand points are proposed. The number of facilities is unknown, however, there is a given number of servers to be distributed among the facilities. Each facility acts as an M/M/k queuing system. The objective function is the minimization of the combined travel time and the waiting time at the facility for all customers. The distribution of demand among the facilities is governed by the gravity rule. Two models are proposed: a stationary one and an interactive one. In the stationary model it is assumed that customers do not consider the waiting time at the facility in their facility selection decision. In the interactive model we assume that customers know the expected waiting time at the facility and consider it in their facility selection decision. The interactive model is more complicated because the allocation of the demand among the facilities depends on the demand itself. The models are analyzed and three heuristic solution algorithms are proposed. The algorithms were tested on a set of problems with up to 1000 demand points and 20 servers.     

Bi-objective vibration damping optimization for congested location–pricing problem

This paper presents a bi-objective mathematical programming model for the restricted facility location problem, under a congestion and pricing policy. Motivated by various applications such as locating server on internet mirror sites and communication networks, this research investigates congested systems with immobile servers and stochastic demand as M/M/m/k queues. For this problem, we consider two simultaneous perspectives; (1) customers who desire to limit waiting time for service and (2) service providers who intend to increase profits. We formulate a bi-objective facility location problem with two objective functions: (i) maximizing total profit of the whole system and (ii) minimizing the sum of waiting time in queues; the model type is mixed-integer nonlinear. Then, a multi-objective optimization algorithm based on vibration theory (so-called multi-objective vibration damping optimization (MOVDO)), is developed to solve the model. Moreover, the Taguchi method is also implemented, using a response metric to tune the parameters. The results are analyzed and compared with a non-dominated sorting genetic algorithm (NSGA-II) as a well-developed multi-objective evolutionary optimization algorithm. Computational results demonstrate the efficiency of the proposed MOVDO to solve large-scale problems.     

A biobjective model for production planning in a cement factory

This paper is an illustration of the application of biobjective linear programming to determining the optimal allocation of production materials to the major production facilities. The system under consideration is a cement factory. However, the approach and methodology are sufficiently general that the method can be applied to similar problems in other firms.Two objectives of the factory are distinguished: minimization of manufacturing cost and maximization of capacity utilization. Addressed to the needs of the biobjective case, a technique, which is referred to as ‘compromise constraint’ technique, is introduced.     

Hierarchical facility location problem: Models,classifications, techniques,and applications

The primary objective in a typical hierarchical facility location problem is to determine the location of facilities in a multi-level network in a way to serve the customers at the lowest level of hierarchy both efficiently (cost minimization objective) and effectively (service availability maximization objective). This paper presents a comprehensive review of over 40 years of hierarchical facility location modeling efforts. Published models are classified based on multiple characteristics including the type of flow pattern, service availability, spatial configuration, objective function, coverage, network levels, time element, parameters, facilities, capacity, and real world application. A second classification is also presented on the basis of solution methods adopted to solve various hierarchical facility location problems. The paper finally identifies the gaps in the current literature and suggests directions for future modeling efforts.     

基于混合需求的设施选址问题研究

研究网络中设施的需求一部分来自于网络节点, 一部分来自于过往流量的基于混合需求的设施选址问题。引入引力模型, 以新建设施获得总利润最大为目标建立非线性整数规划模型, 并构造启发式算法, 通过MATLAB进行仿真实验, 将求解结果与GPAH算法及精确算法的结果进行比较。比较结果表明, 提出的算法求解质量高、运行速度快, 可用于大中型网络设施的选址问题。     

基于改进遗传算法的省级医疗中心选址研究

针对公共设施选址问题中因多目标约束条件造成的复杂空间搜索问题,提出了一种基于遗传算法的P-中值模型,以设施点与供应点间的分配关系作为基因序列,将出行时间消耗、建设投入成本、容量限制条件等因素构成目标函数,用于设施供需分配过程中的优化求解。同时从初始种群构成方式和变异率两方面对遗传算法进行改进以提高求解准确性。实验将该模型运用于河南省省级医疗中心选址,并结合多种评价指标得出多样化的医疗中心布局方案,验证了模型的有效性和可行性。     

A bilevel fixed charge location model for facilities under imminent attack

We investigate a bilevel fixed charge facility location problem for a system planner (the defender) who has to provide public service to customers. The defender cannot dictate customer-facility assignments since the customers pick their facility of choice according to its proximity. Thus, each facility must have sufficient capacity installed to accommodate all customers for whom it is the closest one. Facilities can be opened either in the protected or unprotected mode. Protection immunizes against an attacker who is capable of destroying at most r unprotected facilities in the worst-case scenario. Partial protection or interdiction is not possible. The defender selects facility sites from m candidate locations which have different costs. The attacker is assumed to know the unprotected facilities with certainty. He makes his interdiction plan so as to maximize the total post-attack cost incurred by the defender. If a facility has been interdicted, its customers are reallocated to the closest available facilities making capacity expansion necessary. The problem is formulated as a static Stackelberg game between the defender (leader) and the attacker (follower). Two solution methods are proposed. The first is a tabu search heuristic where a hash function calculates and records the hash values of all visited solutions for the purpose of avoiding cycling. The second is a sequential method in which the location and protection decisions are separated. Both methods are tested on 60 randomly generated instances in which m ranges from 10 to 30, and r varies between 1 and 3. The solutions are further validated by means of an exhaustive search algorithm. Test results show that the defender's facility opening plan is sensitive to the protection and distance costs.     

New MAXCAP related problems: Formulation and model solutions

In this paper three related problems of the maximum capture (MAXCAP) model are proposed. These include the case where facilities provide a certain amount of service level for the customers, the possibility where customers do not allocate their demand completely to one facility but prorate their demand based on the service level, and finally we explore the situation where customers will not opt for sharing their demand irrespective of the service level if the next attractive facility is too far way which we express by a distance threshold. These models are put forward to mimic realistic situations related to customer behavior when it comes to selecting a facility. Their respective mathematical formulations are put forward and tested on a case study and also over a range of larger data sets.     

Locating a semi-obnoxious facility with expropriation

This paper considers the problem of locating semi-obnoxious facilities assuming that “too close” demand nodes can be expropriated by the developer at a given price. The objective is to maximize the minimum weighted distance from the facility to the non-expropriated demand nodes given a limited budget while taking into account the fact that customers do not want to be too far away from the facility. Two models of this problem on a network are presented. One is to minimize the difference between the maximum and the minimum weighted distances. The other one is to maximize the minimum weighted distance subject to an upper bound constraint on the maximum weighted distance. The dominating sets are determined and efficient algorithms are presented.     

Solving an extended multi-row facility layout problem with fuzzy clearances using GA

Multi-row facility layout problem (MRFLP) is a class of facility layout problems, which decides upon the arrangement of facilities in some fixed numbers of rows in order to minimize material handling cost. Nowadays, according to the new layout requirements, the facility layout problems (FLPs) have many applications such as hospital layout, construction site layout planning and layout of logistics facilities. Therefore, we study an extended MRFLP, as a novel layout problem, with the following main assumptions: 1) the facilities are arranged in a two-dimensional area and without splitter rows, 2) multiple products are available, 3) distance between each pair of facilities, due to inaccurate and flexible manufacturing processes and other limitations (such as WIPs, industrial instruments, transportation lines and etc.), is considered as fuzzy number, and 4) the objective function is considered as minimizing the material handling and lost opportunity costs. To model these assumptions, a nonlinear mixed-integer programming model with fuzzy constraints is presented and then converted to a linear mixed-integer programming model. Since the developed model is an NP-hard problem, a genetic algorithm approach is suggested to find the best solutions with a minimum cost function. Additionally, three different crossover methods are compared in the proposed genetic algorithm and finally, a sensitivity analysis is performed to discuss important parameters.     

The Capacitated p-facility Location Problem on the Real Line

The problem we address involves locating p new facilities to service a set of customers or fixed points on the real line such that a measure of total cost will be minimized. A basic form of this problem was investigated by Love (1976), who observed that the fixed points must be allocated in sequence to the new facilities in an optimal solution, and thus, the problem can be solved by a dynamic programming algorithm. Since then, other forms of the model have been investigated; however, in all cases it is assumed that the new facilities have unlimited capacity so that customer flows are always allocated to the nearest facility. The objective of this paper is to analyze the effect of capacity constraints on the optimal locations of the new facilities. A general fixed-cost function is also included to account for practical considerations such as zoning regulations, and to permit the facilities to be located anywhere on the line instead of only at the fixed vertices. A dynamic programming method is formulated to solve the problem when the variable cost components are increasing convex functions of travel distance. The problem is shown to be NP-hard under more general cost structures.     

A bi-objective reverse logistics network analysis for post-sale service

Reverse logistics, induced by various forms of return, has received growing attention throughout this decade. Reverse logistics network design is a major strategic issue. This paper addresses the analysis of reverse logistic networks that deal with the returns requiring repair service. A problem involving a manufacturer outsourcing to a third-party logistics (3PLs) provider for its post-sale services is proposed. First, a bi-objective optimization model is developed. Two objectives, minimization of the overall costs and minimization of the total tardiness of cycle time, are addressed. The facility capacity option at each potential location is treated as a discrete parameter. The purpose is to find a set of non-dominated solutions to the facility capacity arrangement among the potential facility locations, as well as the associated transportation flows between customer areas and service facilities. Then, a solution approach is designed for solving this bi-objective optimization model. The solution approach consists of a combination of three algorithms: scatter search, the dual simplex method and the constraint method. Finally, computational analyses are performed on trial examples. Numerical results present the trade-off relationship between the two objectives. The numerical results also show that the optimization for the first objective function leads to a centralized network structure; the optimization for the second objective function results in a decentralized network structure.     

A study on facility location–allocation problem in mixed environment of randomness and fuzziness

In logistics system, facility location–allocation problem, which can be used to determine the mode, the structure and the form of the whole logistics system, is a very important decision problem in the logistics network. It involves locating plants and distribution centers, and determining the best strategy for allocation the product from the plants to the distribution centers and from the distribution centers to the customers. Often uncertainty may be associated with demand, supply or various relevant costs. In many cases, randomness and fuzziness simultaneously appear in a system, in order to describe this phenomenon; we introduce the concept of hybrid variable and propose a mixed-integer programming model for random fuzzy facility location–allocation problem. By expected value and chance constraint programming technique, this model is reduced to a deterministic model. Furthermore, a priority-based genetic algorithm is designed for solving the proposed programming model and the efficacy and the efficiency of this method and algorithm are demonstrated by a numerical example. Till now, few has formulated or attacked the FLA problems in the above manner. Furthermore, the techniques illustrated in this paper can easily be applied to other SCN problems. Therefore, these techniques are the appropriate tools to tackle other supply chain network problems in realistic environments.     

考虑客户分类的随机时间车辆路径优化模型与算法

针对一类考虑客户分类、随机旅行时间、随机服务时间及时间窗约束的车辆路径问题构建了机会约束规划模型,该模型考虑两类客户（普通客户与优质客户）并通过添加机会约束条件确保优质客户获得准时服务的概率。同时,设计了变邻域迭代局部搜索算法,并给出了一种基于最小等待时间的初始解生成启发式规则。基于Solomon算例进行了多组仿真实验。仿真实验结果表明,所设计生成初始解的启发式规则是有效的;所给算法能够在短时间内找到确定问题和随机问题的近似最优解;客户比与车辆使用数目呈正相关关系。研究结果对解决资源有限条件下克服随机不确定性因素带来的不利影响、保证客户服务水平等问题有一定的参考意义。     

基于竞争环境的截流设施选址与车辆路径问题

研究竞争环境下截流设施选址与带时间窗的多中心车辆路径问题。首先,在考虑设施覆盖范围衰退的情况下,利用阶梯型效用函数和偏离距离描述消费者的选择行为,并确定截流设施的需求量;然后,采用基于聚集度的启发式算法对门店进行分类,借助双层规划法,建立门店选址与车辆路径安排的多目标整数规划模型;最后,采用改进的蚁群算法进行求解。通过分析对比实验结果,验证了模型的有效性和可行性。     

Optimal control of service for facilities holding inventory

This paper addresses the problem of optimally controlling service rates for an inventory system of service facilities. We consider a finite capacity system with Poisson arrivals and exponentially distributed leadtimes and service times. For given values of maximum inventory and reorder levels, we determine the service rates to be employed at each instant of time so that the long-run expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. We establish the existence of a stationary optimal policy and we solve it by employing linear programming. Several instances of a numerical example, which provide insight into the behaviour of the system, are presented.Scope and purposeIn this article we discuss the problem of inventory control of service parts at a service facility where there is only a limited waiting space for customers. If a customer enters the service facility and sees all the waiting spaces occupied he/she will leave the facility, which results in both intangible losses (loss of goodwill) and tangible losses (loss in profit). Hence, the service provider aims at obtaining an optimal rate at which service is to be provided by balancing costs due to waiting time and limited waiting spaces against costs due to ordering and overheads due to storing items. We develop an algorithm that controls the service rate as a function of the number of customers waiting for service.     

网络中设备位置问题分析

通过引入虚拟的质量保证协议（SLA）服务条款，提出网络中形式化的设备位置问题，并根据网络特点改进了现有的静态局部优化算法，证明了在常见的分区低服务质量敏感类型的k median问题和低服务质量敏感类型的UFL问题中的近似度上界。测试结果表明改进后的算法在较小增加现有静态算法近似度的情况下运算速度有较大的提高。     

Managing service facilities with endogenous arrival and service rates

We conduct a simulation‐based experiment to analyze how past experiences in a service facility system affect customers’ and service providers’ behavior. We study the problem faced by service providers in deciding by how much to adjust the capacity of their facility when captive repeat customers choose which facility to join based on their expected sojourn times. The customers’ decision‐making process differs from that of service providers in that the former is based on the customers’ experience, whereas the latter is based on the most recent information service providers have regarding demand. Customers use their previous experience and that of their neighbors to update their perceptions of the future average sojourn time, whereas service providers form perceptions of the future arrival rate. We use cellular automata to model the interaction between customers and service providers. We perform simulations to assess the way the customers’ and service providers’ decisions evolve and affect system behavior. Our findings demonstrate that the system we study possesses a certain degree of path dependence. The primary conclusion is that the more conservative service providers are regarding new information, the larger the market share they achieve, and the lower the probability that their facility will shut down.     

A class of multi-objective supply chain networks optimal model under random fuzzy environment and its application to the industry of Chinese liquor

An important issue, when shipping cost and customers demand are random fuzzy variables in supply chain network (SCN) design problem, is to find the network strategy that can simultaneously achieve the objectives of minimization total cost comprised of fixed costs of plants and distribution centers (DCs), inbound and outbound distribution costs, and maximization customer services that can be rendered to customers in terms of acceptable delivery time. In this paper, we propose a random fuzzy multi-objective mixed-integer non-linear programming model for the SCN design problem of Luzhou Co., Ltd. which is representative in the industry of Chinese liquor. By the expected value operator and chance constraint operator, the model has been transformed into a deterministic multi-objective mixed-integer non-linear programming model. Then, we use spanning tree-based genetic algorithms (st-GA) by the Prüfer number representation to find the SCN to satisfy the demand imposed by customers with minimum total cost and maximum customer services for multi-objective SCN design problem of this company under condition of random fuzzy customers demand and transportation cost between facilities. Furthermore, the efficacy and the efficiency of this method are demonstrated by the comparison between its numerical experiment results and those of tradition matrix-based genetic algorithm.