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The new definition of a fuzzy vertex cover of a fuzzy graph depending on the fuzzy covering radius is introduced in this paper. A fuzzy covering problem for a fuzzy network in modification of the fuzzy covering radius as per requirement is considered and solved in an optimization sense. The covering problem is customized with the help of a set of programming problems, and they are solved by using mathematical software “LINGO”. Two relevant algorithms are designed in this paper. Also, one of the burning issues of today’s world, “Climate Action” (one of the most important sustainable developmental goals), together with various weather parameters, is taken in the application part. The fuzzy environment for this issue of India to save our nature is modeled for making some decisions to improve the situations for this purpose, and the whole methodology reflects the importance of the applicability of our proposed model.

This paper addresses the multi-period two-echelon integrated competitive/uncompetitive facility location problem in a distribution system design that involves locating regional distribution centers (RDCs) and stores, and determining the best strategy for distributing the commodities from a central distribution center (CDC) to RDCs and from RDCs to stores. The goal is to determine the optimal numbers, locations and capacities of RDCs and stores so as to maximize the total profit of the distribution system. Unlike most of past research, our study allows for dynamic planning horizon, distribution of commodities, configuration of two-echelon facilities, availability of capital for investment, external market competition, customer choice behavior and storage limitation. This problem is formulated as a bi-level programming model and a mutually consistent programming mode, respectively. Since such a distribution system design problem belongs to a class of NP-hard problem, a genetic algorithm-based heuristic (GA) is presented and compared with random search solution and mutually consistent solution (MC) using numerical example. The computational results show that the GA approach is efficient and the values of the performance index were significantly improved relative to the MC.

Multi-source Weber problem (MWP) is an important model in facility location, which has wide applications in various areas such as health service management, transportation system management, urban planning, etc. The location-allocation algorithm is a well-known method for solving MWP, which consists of a location phase and an allocation phase at each iteration. In this paper, we consider more general and practical case of MWP–the constrained multi-source location problem (CMSLP), i.e., the location of multiple facilities with considering interactive transportation between facilities, locational constraints on facilities and the gauge for measuring distances. A variational inequality approach is contributed to solving the location subproblem called the constrained multi-facility location problem (CMFLP) in location phase, which leads to an efficient projection-type method. Then a new location-allocation algorithm is developed for CMSLP. Global convergence of the projection-type method as well as local convergence of new location-allocation algorithm are proved. The efficiency of proposed methods is verified by some preliminary numerical results.

A self-organizing neural network model that computes the smallest circle (also called minimum spanning circle) enclosing a finite set of given points was proposed by Datta.³ In the article,³ the algorithm is stated and it is demonstrated by simulation that the center of the smallest circle can be achieved with a given level of accuracy. No rigorous proof was given in support of the simulation results. In this paper, we make a rigorous analysis of the model and mathematically prove that the model converges to the desired center of the minimum spanning circle. A suitable neural network architecture is also designed for parallel implementation of the proposed model. Time complexity of the algorithm is worked out under the proposed architecture. Extension of the proposed model to higher dimensions is discussed and demonstrated with some applications.

We present efficient algorithms for two problems of facility location. In both problems we want to determine the location of a single facility with respect to n given sites. In the first we seek a location that maximizes a weighted distance function between the facility and the sites, and in the second we find a location that minimizes the sum (or sum of the squares) of the distances of k of the sites from the facility.

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p∈S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n³logn) time and O(n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k∈O(n) our algorithm runs in O(n³logn) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n logn) time and O(n) space.

For a given set S of n points in the plane and a fixed point o, we introduce the Voronoi diagram of S anchored at o. It will be defined as an abstract Voronoi diagram that uses as bisectors the following curves. For each pair of points p, q in S, the bisecting curve between p and q is the locus of points x in the plane such that the line segment is equidistant to both p and q. We show that those bisectors have nice properties and, therefore, this new structure can be computed in O(n log n) time and O(n) space both for nearest-site and furthest-site versions. Also, we prove that the dynamic version of this diagram can be built in O(n²λ_6s+2(n) log n) time complexity, where s is a constant depending on the function that describes the motion of the points. Finally, we show how to use these structures for solving several locational optimization problems.

The Euclidean centre (centre of the smallest enclosing sphere) of a set of points P in two or more dimensions is unstable; small perturbations at only a few points of P can result in an arbitrarily large relative change in the position of the Euclidean centre. Any centre function more stable than the Euclidean centre is eccentric; that is, its associated radius exceeds the radius of the smallest enclosing circle for some point sets P. Motivated by applications in mobile facility location (in which clients move continuously with some maximum velocity) we seek alternative notions of centrality that are stable while maintaining low eccentricity. In general there is a trade-off; centre functions with lower eccentricity are less stable. In an attempt to balance the conflicting goals of closeness of approximation and stability, we apply the Steiner centre, traditionally defined for convex polytopes, as a centre function of a set of points in the plane. Although previously defined, the notion of a Steiner centre had not been analyzed in terms of its approximation of the Euclidean centre. Exploiting the equivalence of the two definitions of the Steiner centre established by Shephard,²⁷ we prove the stability of the Steiner centre is π/4 and show that the associated radius is at most 1.1153 times the Euclidean radius of any point set P. It follows that a mobile facility located at the Steiner centre of the positions of a set of mobile clients remains close to the Euclidean centre of the clients yet never moves with relative velocity that exceeds 4/π.

Given a set P of n points in the plane such that each point has a positive weight, we study the problem of finding an obnoxious line that intersects the convex hull of P and maximizes the minimum weighted Euclidean distance to all points of P. We present an O(n²log n) time algorithm for the problem, improving the previously best-known O(n²log³ n) time solution. We also consider a variant of this problem whose input is a set of m polygons with a total of n vertices in the plane such that each polygon has a positive weight and whose goal is to locate an obnoxious line with respect to the weighted polygons. An O(mn + n log² n log m + m²log n log² m) time algorithm for this variant was known previously. We give an improved algorithm of O(mn + n log² n + m²log n) time. Further, we reduce the time bound of a previous algorithm for the case of the problem with unweighted polygons from O((m² + n log m) log n) to O(m² + n log m).

Let S be a set of n points in ℝ^d, and let each point p of S have a positive weight w(p). We consider the problem of computing a ray R emanating from the origin (resp. a line l through the origin) such that min_{p ∈ s} w(p) · d(p,R) (resp. min_{p ∈ S} w(p) · d(p,l)) is maximal. If all weights are one, this corresponds to computing a silo emanating from the origin (resp. a cylinder whose axis contains the origin) that does not contain any point of S and whose radius is maximal. For d = 2, we show how to solve these problems in O(n log n) time, which is optimal in the algebraic computation tree model. For d = 3, we give algorithms that are based on the parametric search technique and run in O(n log⁴ n) time. The previous best known algorithms for these three-dimensional problems had almost quadratic running time. In the final part of the paper, we consider some related problems.

This work presents a fuzzy TOPSIS model under group decisions for solving the facility location selection problem, where the ratings of various alternative locations under different subjective attributes and the importance weights of all attributes are assessed in linguistic values represented by fuzzy numbers. The objective attributed are transformed into dimensionless indices to ensure compatibility with the linguistic ratings of the subjective attributes. Furthermore, the membership function of the aggregation of the ratings and weights for each alternative location versus each attribute can be developed by interval arithmetic and α-cuts of fuzzy numbers. The ranking method of the mean of the integral values is applied to help derive the ideal and negative-ideal fuzzy solutions to complete the proposed fuzzy TOPSIS model. Finally, a numerical example demonstrates the computational process of the proposed model.

This paper deals with a mathematical model for reduction of the lack of coverage (LC) involving multiple coverage in presence of partial covering. The model proposes a new structure of assignment of facilities in a facility location system to cover in greater proportion of the demand territory, avoiding assignment of several facilities in the same space of the territory. A comparison between the engendered solution and its representation is carried out through known indicators to measure the improvement of the solution. The results of our proposed model are contrast and better compared to defined referred models in order to evaluate the reduction of LC.

This paper focuses on a new Stackelberg location problem on a network with demands whose weights are given uncertainly and vaguely. By representing them as fuzzy random variables, the optimal location problem can be formulated as a fuzzy random programming problem for finding Stackelberg equilibrium. By using both their α-level sets for fuzziness and their satisfaction level for a given probability for randomness, it can be reformulated as a version of conventional Stackelberg location problem on the network. Theorems for its complexity are shown based upon the characteristics of the facility location.