Simulated Annealing for Competitive p-Median Facility Location Problem

Optimizing facility location and design

We consider a variant of the Competitive Location Problem, in which it is necessary to choose not only location, but also the design of facilities in order to maximize the share of customer demand served. It is described as an integer programming model with a nonlinear objective function. Commercial software is not suitable for finding the optimal solution to this problem in a reasonable time. Previously, Aboolian R. et al. proposed an adapted weighted greedy heuristic for this problem. In our earlier study, we developed several variants of local search algorithm. This article is devoted to development of the branch-and-bound algorithm. The scheme of branching is described, the results of a computational experiment are presented, future development possibility is discussed.

An algorithm with different exploration mechanisms: Experimental results to capacitated facility location/network design problem

In this paper, we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median, the most p-median and the expected p-median are introduced. Then, it is shown that the uncertain p-median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.

Network location problems occur when new facilities are to be located on a network. The network of interest may be a road network, an air transport network, a river network, or a network of shipping lanes. For a given network location problem, the new facilities are often idealized as points, and may be located anywhere on the network; constraints may be imposed upon the problem so that new facilities are not too far from existing facilities. Usually some objective function is to be minimized. For single objective function problems, typically the objective is to minimize either a sum of transport costs proportional to network travel distances between existing facilities and closest new facilities, or a maximum of “losses” proportional to such travel distances, or the total number of new facilities to be located. There is also a growing interest in multiobjective network location problems. Of the approximately 100 references we list, roughly 60 date from 1978 or later; we focus upon work which deals directly with the network of interest, and which exploits the network structure. The principal structure exploited to date is that of a tree, i.e., a connected network without cycles. Tree-like networks may be encountered when having cycles is very expensive, as with portions of interstate highway systems. Further, simple distribution systems with a single distributor at the “hub” can often be modeled as star-like trees. With trees, “reasonable” functions of distance are often convex, whereas for a cyclic network such functions of distance are usually nonconvex. Convexity explains, to some extent, the tractability of tree network location problems.

A globally convergent algorithm for facility location on a sphere

Approximation algorithm for squared metric facility location problem with nonuniform capacities

Despite a growing number of studies in stochastic dynamic network optimization, the field remains less well defined and unified than other areas of network optimization. Due to the need for approximation methods like approximate dynamic programming, one of the most significant problems yet to be solved is the lack of adequate benchmarks. The values of the perfect information policy and static policy are not sensitive to information propagation while the myopic policy does not distinguish network effects in the value of flexibility. We propose a scalable reference policy value defined from theoretically consistent real option values based on sampled sequences, and estimate it using extreme value distributions. The reference policy is evaluated on an existing network instance with known sequences (Sioux Falls network from Chow and Regan 2011a): the Weibull distribution demonstrates good fit and sampling consistency with more than 200 samples. The reference policy is further applied in computational experiments with two other types of adaptive network design: a facility location and timing problem on the Simchi-Levi and Berman (1988) network, and Hyytia et al.’s (2012) dynamic dial-a-ride problem. The former experiment represents an application of a new problem class and use of the reference policy as an upper bound for evaluating sampled policies, which can reach 3 % gap with 350 samples. The latter experiment demonstrates that sensitivity to parameters may be greater than expected, particularly when benchmarked against the proposed reference policy.

Competitive facility location (CFL) problem has been proved to be NP-hard. Most location-allocation solution methods rely on heuristic approaches, however, they do not guarantee a global optimal solution. This paper proposes an improved clonal selection algorithm, which is an important theory in artificial immune systems (AIS), and we apply it to the CFL task. A special string is used as the shape-space in immune cell pattern. In addition, we propose an n-hold method to improve the performance of AIS, which can increase the stability and robustness of the clonal selection algorithm. Meanwhile, a special version of the roulette wheel method is proposed to improve the algorithm performance. And we give the evolutionary explain of AIS. Experiments show that the proposed algorithm is effective and efficient to the CFL task

In the universal facility location problem, we are given a set of clients and facilities. Our goal is to find an assignment such that the total connection and facility cost is minimized. The connection cost is proportional to the distance between each client and its assigned facility, whereas the facility cost is a nondecreasing function with respect to the total number of clients assigned to the facility. The universal facility location problem generalizes several classical facility location problems, including the uncapacitated facility location problem and the capacitated facility location problem (both hard and soft capacities). This work considers the universal facility location problem with linear penalties, where each client can be rejected for service with a penalty. The objective is to minimize the total connection, facility and penalty cost. We present a $$(5.83+\epsilon )$$ -approximation local search algorithm for this problem.

The Discrete Ordered Median Problem (DOMP) unifies discrete facility location problems in the sense that classical location problems, such as the N-median, N-centre and Uncapacitated Facility Location problems, are particular cases of the DOMP. In addition, new location problems, which are interesting from a practical point of view (e.g. the k 1 + k 2 trimmed mean problem, see Section 2.1.1), can be formulated as special cases of the DOMP. Therefore, the study of the DOMP is interesting not only from a theoretical point of view (unification of discrete facility location problems) but also from a practical one (possibility to formulate new interesting problems).KeywordsLocation ProblemVariable Neighbourhood SearchFacility Location ProblemSupply Chain PlanningUncapacitated Facility LocationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The sample average approximation (SAA) approach is a widely used technique, based on Monte‐Carlo simulation, often applied to large‐scale stochastic optimization problems. In this approach, a set of sample average problems with multiple copies of sampled scenarios are generated and solved exactly. In other words, there is an implicit assumption that the sample average problems are solvable to optimality. In some instances, however, the sample average problems might be NP‐hard problems, often difficult or impractical to solve to optimality. In this article, we broaden the scope of the SAA approach and show that even without solving the sample problems to optimality, by combining a heuristic and a lower bounding approach, high‐quality solutions with tight confidence bounds on the optimal solution value can be obtained. We demonstrate this “inexact SAA approach” on two problems. First, we apply it to the Stochastic Connected Facility Location (SConFL) problem, the motivating application for this article, that arises in the design of telecommunications networks. As an additional application, we also use it for the Stochastic Uncapacitated Facility Location (SUFL) problem. Our computational results demonstrate the effectiveness of the inexact SAA approach. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 19–33 2017

Some new algorithms for location problems on networks

The multi-period incremental service facility location problem

We consider a mathematical model of market competition between two parties. The parties sequentially bring their products to the market while aiming to maximize profit. The model is based on the Stackelberg game and formulated as a bilevel integer mathematical program. The problem can be reduced to the competitive facility location problem (CompFLP) with a prescribed choice of suppliers which belongs to a family of bilevel models generalizing the classical facility location problem. For the CompFLP with a prescribed choice of suppliers, we suggest an algorithm of finding a pessimistic optimal solution. The algorithm is an iterative procedure that successively strengthens an estimating problem with additional constraints. The estimating problem provides an upper bound for the objective function of the CompFLP and is resulted from the bilevel model by excluding the lower-level objective function. To strengthen the estimating problem, we suggest a new family of constraints. Numerical experiments with randomly generated instances of the CompFLP with prescribed choice of suppliers demonstrate the effectiveness of the algorithm.