IP modeling of the survivable hop constrained connected facility location problem

We consider a generalized version of the rooted connected facility location problem which occurs in planning of telecommunication networks with both survivability and hop-length constraints. Given a set of client nodes, a set of potential facility nodes including one predetermined root facility, a set of optional Steiner nodes, and the set of the potential connections among these nodes, that task is to decide which facilities to open, how to assign the clients to the open facilities, and how to interconnect the open facilities in such a way, that the resulting network contains at least λ edge-disjoint paths, each containing at most H edges, between the root and each open facility and that the total cost for opening facilities and installing connections is minimal. We study two IP models for this problem and present a branch-and-cut algorithm based on Benders decomposition for finding its solution. Finally, we report computational results.     

Designing robust coverage networks to hedge against worst-case facility losses

In order to design a coverage-type service network that is robust to the worst instances of long-term facility loss, we develop a facility location–interdiction model that maximizes a combination of initial coverage by p facilities and the minimum coverage level following the loss of the most critical r facilities. The problem is formulated both as a mixed-integer program and as a bilevel mixed-integer program. To solve the bilevel program optimally, a decomposition algorithm is presented, whereby the original bilevel program is decoupled into an upper level master problem and a lower level subproblem. After sequentially solving these problems, supervalid inequalities can be generated and appended to the upper level master in an attempt to force it away from clearly dominated solutions. Computational results show that when solved to optimality, the bilevel decomposition algorithm is up to several orders of magnitude faster than performing branch and bound on the mixed-integer program.     

A polyhedral approach to the single row facility layout problem

The single row facility layout problem (SRFLP) is the NP-hard problem of arranging facilities on a line, while minimizing a weighted sum of the distances between facility pairs. In this paper, a detailed polyhedral study of the SRFLP is performed, and several huge classes of valid and facet-inducing inequalities are derived. Some separation heuristics are presented, along with a primal heuristic based on multi-dimensional scaling. Finally, a branch-and-cut algorithm is described and some encouraging computational results are given.     

A neural network approach to facility layout problems

A near-optimum parallel algorithm for solving facility layout problems is presented in this paper where the problem is NP-complete. The facility layout problem is one of the most fundamental quadratic assignment problems in Operations Research. The goal of the problem is to locate N facilities on an N-square (location) array so as to minimize the total cost. The proposed system is composed of N × N neurons based on an artificial two-dimensional maximum neural network for an N-facility layout problem. Our algorithm has given improved solutions for several benchmark problems over the best existing algorithms.     

Isodistant points in competitive network facility location

An isodistant point is any point on a network which is located at a predetermined distance from some node. For some competitive facility location problems on a network, it is verified that optimal (or near-optimal) locations are found in the set of nodes and isodistant points (or points in the vicinity of isodistant points). While the nodes are known, the isodistant points have to be determined for each problem. Surprisingly, no algorithm has been proposed to generate the isodistant points on a network. In this paper, we present a variety of such problems and propose an algorithm to find all isodistant points for given threshold distances associated with the nodes. The number of isodistant points is upper bounded by nm, where n and m are the number of nodes and the number of edges, respectively. Computational experiments are presented which show that isodistant points can be generated in short run time and the number of such points is much smaller than nm. Thus, for networks of moderate size, it is possible to find optimal (or near-optimal) solutions through the Integer Linear Programming formulations corresponding to the discrete version of such problems, in which a finite set of points are taken as location candidates.     

The p/q-active uncapacitated facility location problem: Investigation of the solution space and an LP-fitting heuristic

The p/q-active uncapacitated facility location problem is the problem of locating p facilities on n possible sites each serving at least q of the m clients at the minimum cost. The problem is an extension of the uncapacitated facility location problem (UFL) where constraints on the number of facilities and their minimum activity have been added. A use of this formulation could be the opening of p new schools where each must have at least q pupils. p/q-active is NP-hard like the UFL.     

Solving the Passive Optical Network with Fiber Duct Sharing Planning Problem Using Discrete Techniques

Similar to the constrained facility location problem, the passive optical network (PON) planning problem necessitates the search for a subset of deployed facilities (splitters) and their allocated demand points (optical network units) to minimize the overall deployment cost. In this paper we use a mixed integer linear programming formulation stemming from network flow optimization to construct a heuristic based on limiting the total number of interconnecting paths when implementing fiber duct sharing. Then, a disintegration heuristic involving the construction of valid clusters from the output of a k means algorithm, reduce the time complexity while ensuring close to optimal results. The proposed heuristics are then evaluated using a real-world dataset, showing favourable performance.     

The <Emphasis Type="Italic">p</Emphasis>-Centre Problem—Heuristic and Optimal Algorithms

The p-centre problem, or minimax location-allocation problem in location theory terminology, is the following: given n demand points on the plane and a weight associated with each demand point, find p new facilities on the plane that minimize the maximum weighted Euclidean distance between each demand point and its closest new facility. We present two heuristics and an optimal algorithm that solves the problem for a given p in time polynomial in n. Computational results are presented.     

Location problems

Recent developments for several location problems are surveyed. These include: graph theoretic and combinatorial formulations of the simple plant location problem, the NP-hardness of some p-center problems, worst-case bounds for several polynomial-time heuristics for some p-center problems, and a general solution to a class of one facility network problems with convex cost functions.     

Capacitated facility location/network design problems

We introduce a combined facility location/network design problem in which facilities have constraining capacities on the amount of demand they can serve. This model has a number of applications in regional planning, distribution, telecommunications, energy management, and other areas. Our model includes the classical capacitated facility location problem (CFLP) on a network as a special case. We present a mixed integer programming formulation of the problem, and several classes of valid inequalities are derived to strengthen its LP relaxation. Computational experience with problems with up to 40 nodes and 160 candidate links is reported, and a sensitivity analysis provides insight into the behavior of the model in response to changes in key problem parameters.     

A k-product uncapacitated facility location problem

A k-product uncapacitated facility location problem can be described as follows. There is a set of demand points where clients are located and a set of potential sites where facilities of unlimited capacities can be set up. There are k different kinds of products. Each client needs to be supplied with k kinds of products by a set of k different facilities and each facility can be set up to supply only a distinct product with a non-negative fixed cost determined by the product it intends to supply. There is a non-negative cost of shipping goods between each pair of locations. These costs are assumed to be symmetric and satisfy the triangle inequality. The problem is to select a set of facilities to be set up and their designated products and to find an assignment for each client to a set of k   facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, an approximation algorithm within a factor of 2k+1$2k+1$ of the optimum cost is presented. Assuming that fixed setup costs are zero, we give a 2k-1$2k-1$ approximation algorithm for the problem. In addition we show that for the case k=2$k=2$, the problem is NP-complete when the cost structure is general and there is a 2-approximation algorithm when the costs are symmetric and satisfy the triangle inequality. The algorithm is shown to produce an optimal solution if the 2-product uncapacitated facility location problem with no fixed costs happens to fall on a tree graph.     

Incorporating congestion in preventive healthcare facility network design

Preventive healthcare aims at reducing the likelihood and severity of potentially life-threatening illnesses by protection and early detection. The level of participation to preventive healthcare programs is a crucial factor in terms of their effectiveness and efficiency. This paper provides a methodology for designing a network of preventive healthcare facilities so as to maximize participation. The number of facilities to be established and the location of each facility are the main determinants of the configuration of a healthcare facility network. We use the total (travel, waiting and service) time required for receiving the preventive service as a proxy for accessibility of a healthcare facility, and assume that each client would seek the services of the facility with minimum expected total time. At each facility, which we model as an M/M/1 queue so as to capture the level of congestion, the expected number of participants from each population zone decreases with the expected total time. In order to ensure service quality, the facilities cannot be operated unless their level of activity exceeds a minimum workload requirement. The arising mathematical formulation is highly nonlinear, and hence we provide a heuristic solution framework for this problem. Four heuristics are compared in terms of accuracy and computational requirements. The most efficient heuristic is utilized in solving a real life problem that involves the breast cancer screening center network in Montreal. In the context of this case, we found out that centralizing the total system capacity at the locations preferred by clients is a more effective strategy than decentralization by the use of a larger number of smaller facilities. We also show that the proposed methodology can be used in making the investment trade-off between expanding the total system capacity and changing the behavior of potential clients toward preventive healthcare programs by advertisement and education.     

The uncapacitated facility location problem with demand-dependent setup and service costs and customer-choice allocation

We consider a generalization of the uncapacitated facility location problem, where the setup cost for a facility and the price charged for service may depend on the number of customers patronizing the facility. Customers are represented by the nodes of the transportation network, and facilities can be located only at nodes; a customer selects a facility to patronize so as to minimize his (her) expenses (price for service + the part of transportation costs paid by the customer). We assume that transportation costs are paid partially by the service company and partially by customers. The objective is to choose locations for facilities and balanced prices so as to either minimize the expenses of the service company (the sum of the total setup cost and the total part of transportation costs paid by the company), or to maximize the total profit. A polynomial-time dynamic programming algorithm for the problem on a tree network is developed.     

A hybrid multistart heuristic for the uncapacitated facility location problem

We present a multistart heuristic for the uncapacitated facility location problem, based on a very successful method we originally developed for the p-median problem. We show extensive empirical evidence to the effectiveness of our algorithm in practice. For most benchmarks instances in the literature, we obtain solutions that are either optimal or a fraction of a percentage point away from it. Even for pathological instances (created with the sole purpose of being hard to tackle), our algorithm can get very close to optimality if given enough time. It consistently outperforms other heuristics in the literature.     

The multifacility maximin planar location problem with facility interaction

^** Email: s.salhi{at}kent.ac.uk Two branch-and-bound algorithms are proposed to optimally solvethe maximin formulation for locating p facilities in the plane.Tight upper and lower bounds are constructed and suitable methodsof guiding the search developed. To enhance the method, efficientmeasures for identifying specific squares for subdivision aresuggested. The proposed algorithms are evaluated on a set ofrandomly generated problems of up to five facilities and 120nodes.     

Heuristics for multiple facility location in a network using the variance criterion

In cyclic networks the p-variance location problem is NP-hard, and therefore it is suitable to use heuristic methods to find approximate solutions to the problem. To this end, two strategies are explored, the first using combinatorial search procedures over the vertex set, whereas the second searches for the solution over the entire network. The initial vertex set solutions are generated by using tabu search, variable neighbourhood search and interchange procedures. The heuristics have been tested on instances of up to 30 vertices and 70 edges, and their performances compared.     

Heuristics for the facility location and design (1|1)-centroid problem on the plane

A chain (the leader) wants to set up a single new facility in a planar market where similar facilities of a competitor (the follower), and possibly of its own chain, are already present. The follower will react by locating another single facility after the leader locates its own facility. Fixed demand points split their demand probabilistically over all facilities in the market in proportion to their attraction to each facility, determined by the different perceived qualities of the facilities and the distances to them, through a gravitational model. Both the location and the quality (design) of the new leader’s facility are to be found. The aim is to maximize the profit obtained by the leader following the follower’s entry. Four heuristics are proposed for this hard-to-solve global optimization problem, namely, a grid search procedure, an alternating method and two evolutionary algorithms. Computational experiments show that the evolutionary algorithm called UEGO_cent.SASS provides the best results.     

LP-based approximation algorithms for capacitated facility location

In the capacitated facility location problem with hard capacities, we are given a set of facilities, ${\mathcal{F}}$ , and a set of clients ${\mathcal{D}}$ in a common metric space. Each facility i has a facility opening cost f _i and capacity u _i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set ${\mathcal{F}}$ and assign each client to an open facility so that at most u _i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c _ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.     

Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.