An Improved Lower Bound for the Multimedian Location Problem

We consider the problem of locating, on a network, n new facilities that interact with m existing facilities. In addition, pairs of new facilities interact. This problem, the multimedian location problem on a network, is known to be NP-hard. We give a new integer programming formulation of this problem, and show that its linear programming relaxation provides a lower bound that is superior to the bound provided by a previously published formulation. We also report results of computational testing with both formulations.     

Locational analysis

For our purposes, locational analysis is the formulation and solution of location problems. We consider locational analysis which is carried out by the construction and solution of locational models. Such models typically involve locating one or more new facilities, and may include transport costs, fixed costs, constraints on the number of new facilities, upper bounds on distances between new and existing facilities, as well as determining amounts to be shipped between new and existing facilities.We give a selective review of the locational analysis literature, concentrating on models which have been thoroughly tested, and which can be solved by ‘reliable algorithms’. For convenience, we consider four classes of locational models: planar models, warehousing models, network models, and discrete, or mixed-integer programming, models.     

Infrastructure development for conversion to environmentally friendly fuel

An important concern for any nation wishing to convert to alternate, environmentally friendly energy sources is the development of appropriate fuel distribution infrastructure. We address the problem of optimally locating gas station facilities for developing nations, like India, which are in the process of converting from leaded to unleaded fuel. Importantly, a similar approach may be used in developed countries, which are in the process of converting to automobiles using hydrogen or electrical energy. An integer-programming model with the objective of balancing the perspectives of coverage and cost is presented for this facility location problem. Given the existing network of roads, the model considers the traveling population, the location of existing facilities and the cost of, either converting these facilities to carry unleaded fuel, or of installing new facilities in an attempt to minimize cost and simultaneously maximize coverage of population. We develop a heuristic solution procedure for this problem. The methodology is applied to data sets obtained from Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, Decision Sciences 19 (1988) 490] representing the Ohio state limited access highway network, and to the Indian national highway network. Additionally, extensive simulations are carried out in order to examine where our approach compares with the maximum weighted spanning tree approach. This work extends the Maximum Covering/Shortest Path problem (MCSPP) formulated by Current et al. [J.R. Current, C.S. ReVelle, J.L. Cohon, European Journal of Operational Research 21 (1985) 189] to accommodate multiple sources and destinations.     

Incorporating the threat of terrorist attacks in the design of public service facility networks

In this paper, we consider the design problem of a public service facility network with existing facilities when there is a threat of possible terrorist attacks. The aim of the system planner, who is responsible for the operation of the network, is to open new facilities, relocate existing ones if necessary, and protect some of the facilities to ensure a maximum coverage of the demand that is assumed to be aggregated at customer zones. By doing so, the system planner anticipates that a number of unprotected facilities will be rendered out-of-service by terrorist attacks. It is assumed that the sum of the fixed cost of opening new facilities, the relocation costs, and the protection costs cannot exceed a predetermined budget level. Adopting the approach of gradual (or partial) coverage, we formulate a bilevel programming model where the system planner is the leader and the attacker is the follower. The objective of the former is the maximization of the total service coverage, whereas the latter wants to minimize it. We propose a heuristic solution procedure based on tabu search where the search space consists of the decisions of the system planner, and the corresponding objective value is computed by optimally solving the attacker??s problem using CPLEX. To assess the quality of the solutions produced by the tabu search (TS) heuristic, we also develop an exhaustive enumeration method, which explores all the possible combinations of opening new facilities, relocating existing ones, and protecting them. Since its time complexity is exponential, it can only be used for relatively small instances. Therefore, to be used as a benchmark method, we also implement a hill climbing procedure employed with the same type of moves as the TS heuristic. Besides, we carry out a sensitivity analysis on some of the problem parameters to investigate their effect on the solution characteristics.     

Sensitivity analysis of a continuous multifacility competitive location and design problem

A continuous location problem in which a firm wants to set up two or more new facilities in a competitive environment is considered. Other facilities offering the same product or service already exist in the area. Both the locations and the qualities of the new facilities are to be found so as to maximize the profit obtained by the firm. This is a global optimization problem, with many parameters to be estimated, and whose behavior is not really well understood. Using random problems and a robust evolutionary algorithm recently proposed for solving this problem, the behavior of optimal solutions in various environments and changes in the basic model parameters are researched. These comprise the quality of existing and new facilities, cost function and presence of the chain. Some economic implications are derived.     

The solution of the EMFL problem with two new facilities in a quadrangle

This paper is concerned with the analytical solution of the EMFL (Euclidean multifacility location) problem with two new facilities and four existing facilities. In Section 1, the optimality conditions for a general EMFL problem are summarized in the form presented in [1]. In Section 2, they are applied to the considered problem, in order to locate the new facilities and to partition the space of the weights (for a given set of existing facilities) into regions with the same type of solution. However, it is pointed out that a complete solution can be obtained only in particular cases.     

A fuzzy programming approach for dynamic virtual hub location problem

Hub location problem has been used in transportation network to exploit economies of scale. For example, a controversial issue in the planning of air transportation networks is inclement weather or emergency conditions. In this situation, hub facilities would not be able to provide a good service to their spoke nodes temporarily. Thus, some other kinds of predetermined underutilized facilities in the network are used as virtual hubs to host some or all connections of original hubs to recover the incurred incapacitation and increase network flexibility and demand flow. In such an unexpected situation, it is not unreasonable to expect that some information be imprecise or vague. To deal with this issue, fuzzy concept is used to pose a more realistic problem. Here, we present a fuzzy integer liner programming approach to propose a dynamic virtual hub location problem with the aim of minimizing transportation cost in the network. We examine the effectiveness of our model using the well-known CAB data set.     

Planar weber location problems with line barriers

The Weber problem for a given finite set of existing facilities in the plane is to find the location of a new facility such that the weithted sum of distances to the existing facilities is minimized.

A variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier. Such barriers like rivers, highways, borders or mountain ranges are frequently encountered in practice.

Structural results as well as algorithms for this non-convex optimization problem depending on the distance function and on the number and location of passages through the barrier are presented.     

两阶段选址优化问题研究

本文主要考虑如下实际问题:假设选址决策者需要建设p个设施,但是由于资金等等的影响,实际建设时会被要求先建设q个设施,其次再建设p-q个设施(设p>q),同时要求,在建设p-q个设施的时候,已经建设好的q个设施不被删除。本文建立了一个两阶段优化问题,问题的输出是两个待修建的设施的集合Fq,Fp,|Fp|=p,|Fq|=q,且Fq是Fp的子集,问题的目标是最小化这两个设施集合的费用同对应的最优费用的比值的最大值。本文给出一个近似比为9的近似算法,并对一些特殊的情况进行了讨论。所得结论对实际的选址决策具有理论意义,同时也完善已有相关研究结果。     

Developments in network location with mobile and congested facilities

We review four facility location problems which are motivated by urban service applications and which can be thought of as extensions of the classic Q-median problem on networks. In problems P1 and P2 it is assumed that travel times on network links change over time in a probabilistic way. In P2 it is further assumed that the facilities (servers) are movable so that they can be relocated in response to new network travel times. Problems P3 and P4 examine the Q-median problem for the case when the service capacity of the facilities is finite and, consequently, some or all of the facilities can be unavailable part of the time. In P3 the facilities have stationary home locations but in P4 they have movable locations and thus can be relocated to compensate for the unavailability of the busy facilities. We summarize our main results to date on these problems.     

Fixed Point Optimality Criteria for the Location Problem with Arbitrary Norms

The single-facility location problem in continuous space is considered, with distances given by arbitrary norms. When distances are Euclidean, for many practical problems the optimal location of the new facility coincides with one of the existing facilities. This property carries over to problems with generalized distances. In this paper a necessary and sufficient condition for the location of an existing facility to be the optimal location of the new facility is developed. Some computational examples using the condition are given.     

A review of extensive facility location in networks

A facility is called ‘extensive’ if, for purposes of location, it is too large in relation to its environment regarding the activities of interest for it to be considered as a point. The literature on location on a network of ordinary and obnoxious extensive facilities is reviewed. Suggestions are made for possible directions of future research.     

A Linear Programming Approach to the Solution of Constrained Multi-Facility Minimax Location Problems where Distances are Rectangular

The problem of locating new facilities with respect to existing facilities is stated as a linear programming problem where inter-facility distances are assumed to be rectangular. The criterion of location is the minimization of the maximum weighted rectangular distance in the system. Linear constraints which (a) limit the new facility locations and (b) enforce upper bounds on the distances between new and existing facilities and between new facilities can be included. The dual programming problem is formulated in order to provide for an efficient solution procedure. It is shown that the duLal variables provide information abouLt the complete range of new facility locations which satisfy the minimax criterion.     

Optimal location of interconnected facilities on tree networks subject to distance constraints

Optimal location of interconnected facilities on tree networks is considered in the case when some of the nodes of the network contain existing facilities. The distances between the facilities must satisfy maximum constraints. Polynomial algorithms for the solution of this problem are proposed.     

Robust supply chain network design with service level against disruptions and demand uncertainties: A real-life case

We have developed a stochastic mathematical formulation for designing a network of multi-product supply chains comprising several capacitated production facilities, distribution centres and retailers in markets under uncertainty. This model considers demand-side and supply-side uncertainties simultaneously, which makes it more realistic in comparison to models in the existing literature. In this model, we consider a discrete set as potential locations of distribution centres and retailing outlets and investigate the impact of strategic facility location decisions on the operational inventory and shipment decisions of the supply chain. We use a path-based formulation that helps us to consider supply-side uncertainties that are possible disruptions in manufacturers, distribution centres and their connecting links. The resultant model, which incorporates the cut-set concept in reliability theory and also the robust optimisation concept, is a mixed integer nonlinear problem. To solve the model to attain global optimality, we have created a transformation based on the piecewise linearisation method. Finally, we illustrate the model outputs and discuss the results through several numerical examples, including a real-life case study from the agri-food industry.     

Determining an optimum location for an undesirable facility in a workroom environment

This paper deals with the problem of placing an undesirable but necessary piece of equipment, process or facility into a working environment. Locating a piece of equipment that produces contaminants or creates stresses for nearby workers, placing a storage facility for flammable materials or locating hazardous waste in the workroom environment, are all typical examples of the undesirable facility location problem. The degree of undesirability between an existing facility or worker and the new undesirable entity is reflected through a weighting factor. The problem is formally defined to be the selection of a location within the convex region that maximizes the minimum weighted Euclidean distance with respect to all existing facilities. A ‘Maximin’ model is formulated and two solution procedures introduced. A geometrical approach and an algorithmic approach are described in detail. An example is provided for each solution procedure and the computational efficiency of the algorithm is discussed and illustrated.     

Stop location design in public transportation networks: covering and accessibility objectives

We consider the location of new stops along the edges of an existing public transportation network. Examples of StopLoc include the location of bus stops along some given bus routes or of railway stations along the tracks in a railway system. In order to evaluate the decision assume that potential customers in given demand facilities are known. Two objectives are proposed. In the first one, we minimize the number of stations such that any of the demand facilities can reach a closest station within a given distance of r. In the second objective, we fix the number of new stations and minimize the sum of the distances between demand facilities and stations. The resulting two problems CovStopLoc and AccessStopLoc are solved by a reduction to a classical set covering and a restricted location problem, respectively. We implement the general ideas in two different environments, the plane, where demand facilities are represented by coordinates, and in networks, where they are nodes of a graph.     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

Weber problems with mixed distances and regional demand

We consider a location problem where the distribution of the existing facilities is described by a probability distribution and the transportation cost is given by a combination of transportation cost in a network and continuous distance. The motivation is that in many cases transportation cost is partly given by the cost of travel in a transportation network whereas the access to the network and the travel from the exit of the network to the new facility is given by a continuous distance.