A quadtree-based allocation method for a class of large discrete Euclidean location problems

Discrete facility location problems are classified as types of facility location problems, wherein decisions on choosing facilities in specific locations are made to serve the demand points of customers, thus minimizing the total cost. The covering- and median-based problems are the common classified types of discrete facility location problems, which both comprise different classes of discrete problems as reviewed in this research. However, the discrete facility location problems shown in deterministic and known information and data under uncertain, vague, and ambiguous environments have usually been solved using intuitionistic fuzzy approaches. Neutrosophic is recently applied to tackle the uncertainty and ambiguity of information and data. This paper considered solving the discrete facility location problems under the neutrosophic environment, wherein the information of the locations, distances, times, and costs is uncertain. The mathematical models for the main types of neutrosophic discrete facility location problems, which remain unclear till now despite previous related works, are formulated in this study. Numerical examples demonstrated testing of the neutrosophic discrete models and comparison with the optimization solutions obtained from the normal situations.

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 multiple facility location problem (MFLP) is to locate certain facilities so as to serve optimally a given set of customers, whose locations and requirements are known. When the facility locations have to be selected from a given set of candidate locations, the corresponding location problem becomes a discrete multiple facility location problem (DMFLP). In this study we consider a special case of DMFLP where multiple facilities that are of different type are to be placed (location decision) and assign customers to these facilities (allocation or assignment). The case will be discussed based on with and without interactions among new facilities. We then propose new heuristic solution methods and branch-and-bound algorithms. Computational results on randomly generated data in comparison with optimal solutions indicate that the new methods are both accurate and efficient.

Researchers have studied discrete location problems for a long time because of their importance in practice. The Discrete Ordered Median Problem (DOMP) generalizes discrete facility location problems. The DOMP generalizes the main facility location problems' objective functions such as the p-median, p-center and p-centdian location problems. As these problems, also known as the problems of location-allocation, have NP-hard structure, it is inevitable to use heuristic methods for solution. In this study, a metaheuristic algorithmic suggestion will be put forward by examining the DOMP to find optimal solutions. For that purpose, we proposed a Simulated Annealing (SA) metaheuristic with K-means Clustering Algorithm in initialization for the DOMP. Novel approaches for initial solution and K-exchange algorithm-based neighborhoods for local search were analysed. In addition, best level of selected parameters were determined by Taguchi method. Forty common p-median instances derived from OR-LIB were used to test the SA performance, and the results were compared with three state-of-art algorithms in the literature. According to the computational results, 21 best solutions were obtained on instances despite gap values and CPU times increasing proportionally to the scale of the instances. In a conclusion, the proposed clustering-based SA algorithm is competitive and can be a robust alternative for the DOMP.

To model flexible objectives for discrete location problems, ordered median functions can be applied. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. In this paper a reformulated and more compact version of a covering model for the discrete ordered median problem (DOMP) is considered. It is shown that by using this reformulation better solution times can be obtained. This is especially true for some objectives that are often employed in location theory. In addition, the covering model is extended so that ordered median functions with negative weights are feasible as well. This type of modeling weights has not been treated in the literature on the DOMP before. We show that several discrete location problems with equity objectives are particular cases of this model. As a result, a mixed-integer linear model for this type of problems is obtained for the first time.

Locating certain facilities in predetermined sites is named the multiple facility location problems (MFLP). The objective of these kinds of problems is locating facilities to serve a given set of customers so that candidate sites and requirements are known. When the new facility sites have to be selected from a given set of candidate sites, the mentioned location problem becomes a discrete multiple facility location problem (DMFLP). In this paper, a special approach of DMFLP is considered where different multiple facilities have to be placed (location decision) and also customers have to be assigned to these facilities (allocation or assignment). The mathematical model of the proposed problem is developed, and with respect to the complexity of solving the mathematical model, especially in large scale, a new hybrid approach is proposed based on tabu search algorithm to solve the problem at each scale. Computational results on several randomly generated problems in comparison with a new proposed lower bound obtained from Lagrangian relaxation indicate that the proposed hybrid approach is both accurate and efficient.

We consider a class of problems where a given number p of facility locations must be selected from a set of s potential locations so as to optimize a pre-determined fitness function. There exist outstanding location problems in this class, such as the p-median, max-covering, MAXCAP, and MAXPROFIT problems. These location problems may contain more than a single global optimal solution. Obtaining multiple global optimal solutions allows us to consider other characteristics in the process of selecting the preferred solution. In [B. Pelegrín, J. Redondo, P. Fernández, I. García, and P. Ortigosa, GASUB: Finding global optima to discrete location problems by a genetic-like algorithm, J. Glob. Optim. 38 (2007), pp. 249–264], a new ‘multimodal’ algorithm (gasub) was presented for solving the previously mentioned location problems. In the paper, the algorithm was fine-tuned to obtain a single global optimum (with 100% success) as fast as possible, and it was compared with two widely used techniques, that is, the standard optimizer Xpress-MP [Xpress-MP, Dashoptimization, 2004.] and the Multi Start Heuristic. In this paper, we propose a new set of parameter values so that gasub can explore the space more deeply and consequently find several alternative global optimal solutions. Moreover, two coarse-grain parallelizations of gasub are presented, cggasub and cggasub_mo. cggasub is able to reduce the computational time of gasub where solutions of the same quality are generated, while cggasub_mo finds more alternative global optima than the sequential version and spending similar computational time in the process. gasub, with the new parameter setting and also the parallel algorithms, will be evaluated using a comprehensive computational set of experiments. Additionally, some criteria can also be used to select one of the known global optimal solutions when several alternatives optimize the corresponding objective function.

On solving the discrete location problems when the facilities are prone to failure

Some new algorithms for location problems on networks

In this paper, we extend the ordinary discrete type facility location problems to continuous type ones. Unlike the discrete type facility location problem in which the objective function isn‘t everywhere differentiable, the objective function in the continuous type facility location problem is strictly convex and continuously differentiable. An algorithm without line search for solving the continuous type facility location problems is proposed and its global convergence, linear convergence rate is proved. Numerical experiments illustrate that the algorithm suggested in this paper have smaller amount of computation, quicker convergence rate than the gradient method and conjugate direction method in some sense.

Integer programming formulations of discrete hub location problems

Multi-supply multi-capacitated p-median location optimization via a hybrid bi-level intelligent algorithm

A computational study of a nonlinear minsum facility location problem

This book presents an up-to-date review of modeling and optimization approaches for location problems along with a new bi-level programming methodology which captures the effect of competition of both producers and customers on facility location decisions. While many optimization approaches simplify location problems by assuming decision making in isolation, this monograph focuses on models which take into account the competitive environment in which such decisions are made. New insights in modeling, algorithmic and theoretical possibilities are opened by this approach and new applications are possible. Competition on equal term plus competition between market leader and followers are considered in this study, consequently bi-level optimization methodology is emphasized and further developed. This book provides insights regarding modeling complexity and algorithmic approaches to discrete competitive location problems. In traditional location modeling, assignment of customer demands to supply sources are made for which the associated costs target the firm and not the customers, though in many real world situations the cost isincurred by the customers. Moreover, there may be customer competition for the provided services. Thus, a new methodological framework is needed in order to encompass such considerations into the modeling and solution process. This book offers initial directions for further research and development along these lines. Aimed toward graduate students and researchers in the field of mathematics, computer science, operational research and game theory, this title provides necessary information on which further research contributions can be based

Location Problems with Multiple Criteria