软容量限制设施选址问题的竞争决策算法

软容量设施选址问题是NP-Hard问题之一，具有广泛的应用价值。为了求解软容量设施选址问题，提出一种基于数学性质的竞争决策算法。首先研究该问题的数学性质，运用这些数学性质不仅可以确定某些设施必定开设或关闭，还可以确定部分顾客由哪个设施提供服务，从而缩小问题的规模，加快求解速度。在此基础上设计了求解该问题的竞争决策算法，最后经过一个小规模的算例测试并与精确算法的结果比较，得出了最优解；针对大规模的问题快速地求出了可行解，得到了令人满意的结果。     

Exact optimal solutions of the minisum facility and transfer points location problems on a network

We consider hierarchical facility location problems on a network called Multiple Location of Transfer Points (MLTP) and Facility and Transfer Points Location Problem (FTPLP), where q facilities and p transfer points are located and each customer goes to one of the facilities directly or via one of the transfer points. In FTPLP, we need to find an optimal location of both the facilities and the transfer points while the location of facilities is given in MLTP. Although good heuristics have been proposed for the minisum MLTP and FTPLP, no exact optimal solution has been tested due to the size of the problems. We show that the minisum MLTP can be formulated as a p‐median problem, which leads to obtaining an optimal solution. We also present a new formulation of FTPLP and an enumeration‐based approach to solve the problems with a single facility.     

A Cross Entropy-based heuristic for the capacitated multi-source Weber problem with facility fixed cost

This paper investigates a capacitated planar location–allocation problem with facility fixed cost. A zone-based fixed cost which consists of production and installation costs is considered. A nonlinear and mixed integer formulation is first presented. A powerful three stage Cross Entropy meta-heuristic with novel density functions is proposed. In the first stage a covering location problem providing a multivariate normal density function for the associated stochastic problem is solved. The allocation values considering a multinomial density function are obtained in the second stage. In the third stage, single facility continuous location problems are solved. Several instances of various sizes are used to assess the performance of the proposed meta-heuristic. Our approach performs well when compared with the optimizer GAMS which is used to provide the optimal solution for small size instances and lower/upper bounds for some of the larger ones.     

Optimal facility location with random throughput costs

In this paper we consider the optimal location and size of facilities where the throughput costs for each facility are random. Given a set of origins and a set of destinations, we want to determine the optimal location and size of a set of intermediate facilities in order to minimize the expected total generalized transportation cost. The generalized transportation cost of a freight unit from an origin to a destination passing through a facility is the sum of two terms: the transportation cost from the origin to the destination through the facility and the throughput cost of the facility. While the first term is deterministic, the second one is stochastic with a Gumbel probability distribution. Looking for the expected value of the optimal solution, a mixed deterministic nonlinear problem for the optimal location of the facilities is derived. Two heuristics, which give very good approximations to the optimum, are proposed.     

On the competitive facility location problem with a free choice of suppliers

We consider a mathematical model from the class of competitive sequential facility location problems. In these problems, the competitors sequentially open their facilities, and each side aims to “capture” the consumers and maximize its profits. In the proposed model, we consider a situation of a “free” choice by each side of an open facility to service a customer. The model is formulated as a bilevel integer programming problem. We show that the problem of finding an optimal noncooperative solution can be represented as a maximization problem for a pseudo-Boolean function. We propose an algorithm for constructing an admissible noncooperative solution for fixed values of the variables in this pseudo-Boolean function. We also propose a method for constructing an upper bound on the maximal value of the pseudo-Boolean function on subsets of solutions defined by partial (0, 1)-vectors.     

On solving unreliable planar location problems

In this paper, we generalize conventional P-median location problems by considering the unreliability of facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. We proposed efficient solution methods to determine locations of these facilities in the unreliable location model. Space-filling curve-based algorithms are developed to determine initial locations of these facilities. The unreliable P-median location problem is then decomposed to P 1-median location problems; each problem is solved to the optimum. A bounding procedure is used to monitor the iterative search, and to provide a consistent basis for termination. Extensive computational tests have indicated that the heuristics are efficient and effective for solving unreliable location problems.Scope and purposeThis paper addresses an important class of location problems, where p unreliable facilities are to be located on the plane, so as to minimize the expected travel distance or related transportation cost between the customers and their nearest available facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. Potential application of the unreliable location problem is found in numerous areas. The facilities to be located can be fire station or emergency shelter, where it fails to provide service during some time window, due to the capacity or resource constraints. Alternatively, the facilities can be telecommunication posts or logistic/distribution centers, where the service is unavailable due to breakdown, repair, shutdown of unknown causes. In this paper, we prescribed heuristic procedures to determine the location of new facilities in the unreliable location problems. The numerical study of 2800 randomly generated instances has shown that these solution procedures are both efficient and effective, in terms of computational time and solution quality.     

Optimal location of a single facility with circular demand areas

The single facility minimum location problem in Euclidean space has usually been studied with a certain number of discrete demand points. Some authors have also described the possibility of demand areas. In the present work, a new approach is offered to the optimal location of a single facility, which should serve a number of circular demand areas, each with uniform demand density, along with some discrete demand points. The effect of a circular demand area on the service facility at each stage of the Weiszfeld-like iterative procedure is evaluated for the three possible cases of the incumbent service point being outside, inside, or on the circumference of such a circle. Some limiting cases are considered, such as that of the demand area being very far from the service point to be optimally located. The amended Weiszfeld iterative procedure is described, and some numerical experience of solving such problems is reported.     

Sequential versus simultaneous approach in the location and design of two new facilities using planar Huff-like models

Companies frequently decide on the location and design for new facilities in a sequential way. However, for a fixed number of new facilities, the company might be able to improve its profit by taking its decisions for all the facilities simultaneously. In this paper we compare three different strategies: simultaneous location and independent design of two facilities in the plane, the same with equal designs, and the sequential approach of determining each facility in turn. The basic model is profit maximization for the chain, taking market share, location costs and design costs into account. The market share captured by each facility depends on the distance to the customers (location) and its quality (design), through a probabilistic Huff-like model. Recent research on this type of models was aimed at finding global optima for a single new facility, holding quality fixed or variable, but no exact algorithm has been proposed to find optimal solutions for more than one facility. We develop such an exact interval branch-and-bound algorithm to solve both simultaneous location and design two-facility problems. Then, we present computational results and exhibit the differences in locations and qualities of the optimal solutions one may obtain by the sequential and simultaneous approaches.     

韦伯型多点设施优化选址的组合算法研究

韦伯型设施选址问题是组合优化领域中的一类重要问题,其核心内容是如何在离散的需求空间域内,寻找到最优决策关注点,即设施点。对于单点设施最优规划问题,由于不存在设置点之间的作用,仅考虑设施点与需求点之间的引力作用问题即可。对于多点设施的最优规划问题,不仅存在着设施点与需求点之间的引力作用问题,而且从资源优化配置的角度,还存在着设施点之间的斥力问题。因此,需要从系统整体优化的角度进行选择规划。目前解决韦伯型设施多点的优化选址问题,一般是通过寻找局部最优解的逐次递阶法来确定最优设施点。但由于该方法没有考虑到设施点间的斥力问题,容易导致设施点间的粘连。针对此问题,提出了一种PGSA-GA组合算法,通过建立模拟植物生长算法得到全局最优解的单点坐标,将其与需求点结合构建遗传算法优化的多目标规划多点设施选址模型求出Pareto最优解,并依此推广到多次选址方案。     

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.     

Constrained location of competitive facilities in the plane

This paper examines a competitive facility location problem occurring in the plane. A new gravity-based utility model is developed, in which the capacity of a facility serves as its measure of attractiveness. A new problem formulation is given, having elastic gravity-based demand, along with capacity, forbidden region, and budget constraints. Two solution algorithms are presented, one based on the big square small square method, and the second based on a penalty function formulation using fixed-point iteration. Computational testing is presented, comparing these two algorithms along with a general-purpose nonlinear solver.Scope and purposeIn a competitive business environment where products are not distinguishable, facility location plays an important role in an organization's success. This paper examines a firm's problem of selecting the locations in the plane for a set of new facilities such that market capture is maximized across all of the firm's facilities (both new and pre-existing). Customers are assumed to divide their demand among all competing facilities according to a utility function that considers facility attractiveness (measured by facility capacity for satisfying demand) and customer-facility distance. The level of customer demand is assumed to be a function of the facility configuration. Three types of constraints are introduced, involving facility capacity, forbidden regions for new facility location, and a budget function. Two solution algorithms are devised, one based on branch-and-bound methods and the other based on penalty functions. Computational testing is presented, comparing these two algorithms along with a general-purpose nonlinear solver.     

Planar expropriation problem with non-rigid rectangular facilities

In this paper, we introduce the planar expropriation problem with non-rigid rectangular facilities. The facilities considered in this study are two-dimensional facilities of rectangular shape. Moreover, we allow the facility dimensions to be decision variables and introduce the concept of non-rigid facilities. Based on the geometric properties of such facilities, we developed a new formulation for this continuous covering location model which does not require employing distance measures. This model is intended to determine the location and formation of facilities simultaneously. For solving this new model, we proposed a continuous branch-and-bound framework utilizing linear approximations for the tradeoff curve associated with the facility formation alternatives. Further, we developed new problem generation and bounding strategies suitable for our particular problem structure. Computational experience shows that the branch-and-bound procedure we developed performs better than conventional mixed-integer nonlinear programming solvers BARON and SBB for solving this particular location model.     

A leader-follower game in competitive facility location

We address the problem of locating new facilities of a firm or franchise that enters a market where a competitor operates existing facilities. The goal of the new entrant firm is to decide the location and attractiveness of its new facilities that maximize its profit. The competitor can react by opening new facilities, closing existing ones, and adjusting the attractiveness levels of its existing facilities, with the aim of maximizing its own profit. The demand is assumed to be aggregated at certain points in the plane and the new facilities of both the firm and the competitor can be located at predetermined candidate sites. We employ the gravity-based rule in modeling the behavior of the customers where the probability that a customer visits a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. We formulate a bilevel mixed-integer nonlinear programming model where the firm entering the market is the leader and the competitor is the follower. We propose heuristics that combine tabu search with exact solution methods.     

The center location-dependent relocation problem with a probabilistic line barrier

This paper presents a new mixed-integer nonlinear programming (MINLP) for a multi-period rectilinear distance center location-dependent relocation problem in the presence of a probabilistic line-shaped barrier that uniformly occurs on a given horizontal route. In this problem, the demand and location of the existing facilities have a dynamic nature and the relocation is dependent to the location of new facilities in previous period. The objective function of the presented model is to minimize the maximum expected weighted barrier distance between the new facility and the existing facilities during the planning horizon. The optimum solution of small-sized test problems is obtained by the optimization software. For large-size test problems which the optimization software is unable to find the optimum solution in the runtime limitation, two meta-heuristics based on the genetic algorithm (GA) and imperialist competitive algorithm (ICA) are applied. To validate the meta-heuristics, a lower bound problem based on the forbidden region instead of the line barrier is generated. Related results of numerical experiments are illustrated and are then compared.     

Risk management in uncapacitated facility location models with random demands

In this paper we consider a location-optimization problem where the classical uncapacitated facility location model is recast in a stochastic environment with several risk factors that make demand at each customer site probabilistic and correlated with demands at the other customer sites. Our primary contribution is to introduce a new solution methodology that adopts the mean–variance approach, borrowed from the finance literature, to optimize the “Value-at-Risk” (VaR) measure in a location problem. Specifically, the objective of locating the facilities is to maximize the lower limit of future earnings based on a stated confidence level. We derive a nonlinear integer program whose solution gives the optimal locations for the p facilities under the new objective. We design a branch-and-bound algorithm that utilizes a second-order cone program (SOCP) solver as a subroutine. We also provide computational results that show excellent solution times on small to medium sized problems.     

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.     

Solving dynamic double row layout problem via combining simulated annealing and mathematical programming

Double row layout problem (DRLP) is to allocate facilities on two rows separated by a straight aisle. Aiming at the dynamic environment of product processing in practice, we propose a dynamic double-row layout problem (DDRLP) where material flows change over time in different processing periods. A mixed-integer programming model is established for this problem. A methodology combining an improved simulated annealing (ISA) with mathematical programming (MP) is proposed to resolve it. Firstly, a mixed coding scheme is designed to represent both of sequence of facilities and their exact locations. Secondly, an improved simulated annealing algorithm is suggested to produce a solution to DDRLP. Finally, MP is used to improve this solution by determining the optimal exact location for each facility. Experiments show that this methodology is able to obtain the optimal solutions for small size problems and outperforms an exact approach (CPLEX) for problems of realistic size.     

A mixed integer linear program and tabu search approach for the complementary edge covering problem

In this paper we study a complementary edge covering problem (CECP) as a variant of the total edge covering problem (TECP) which has application in the area of facility location. Unlike TECP, the partial cover of an edge through vertices is allowed in CECP such that in a feasible solution the entire edge will be covered. We propose a new mixed integer linear formulation for the problem. A number of size reduction rules are proposed which speed up getting optimal solution(s). Since the CECP is NP-Hard, a solution method based on tabu search is designed to search for optimal or near-optimal solutions. Computational experiments are carried out to evaluate effectiveness of the proposed mathematical formulation and the modified tabu search algorithm. Results justify that the proposed mathematical model can solves problems with up to 40 vertices and 456 edges optimally. Our computational efforts signify that the proposed tabu search is very effective and can find high quality solutions for larger problems in reasonable amount of time.     

Connected facility location via random facility sampling and core detouring

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. The mentioned results can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems.     

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.