Developing a transfer point location problem considering normal demands distribution

The significance of emergency transfer point and facility location lies in its potential to save lives, optimize resource allocation, and enhance the efficiency of healthcare services during emergencies. It ensures timely patient access and cost-effective care. So, this study introduces a novel Mixed Integer Programming Model (MIP) for addressing the Multi-Facility and Multi-Transfer Point Location Problem (MFMTPLP). The problem involves determining the optimal locations for q facilities and p transfer points to facilitate direct customer access to facilities or through transfer points. The objective is to minimize the maximum expected weight distance for all the required points via transfer points. The study focuses on identifying the required points that carry weight and proposes a viable solution for solving the associated mathematical problems. An important contribution of this study, which has been overlooked in previous research, is the exploration of relevant costs associated with not transferring patients from demand points and hospitalizing them at transfer points. Additionally, the study emphasizes the significance of prioritizing different points and considering the costs associated with not transferring injured individuals from demand points and admitting them to transfer points. Previous models for locating facility and transfer points did not account for these factors. The study concludes by presenting the computed results, which validate the feasibility and effectiveness of the proposed model.

The transfer point location problem has been introduced recently and for the case of minimax objective and planar topology, has only been studied for situations in which demand points are not weighted and have known coordinates. In this paper, we consider the case in which demand points are weighted and their coordinates have bivariate uniform distribution. Also, the problem is developed from a conceptual view and different distance measures are used to make models more applicable in real world situations. The problem is to find the best location for the transfer point such that the maximum expected weighted distance to all demand points through the transfer point is minimized. Depending on assumptions for uniform distributions, two models are considered, convexity conditions are discussed, properties of the optimal solution are obtained and methods to solve the problems are proposed. Finally, numerical examples are given.

The transfer point location problem has been introduced recently and for the case of minimax objective and planar topology, has only been studied for situations in which demand points are not weighted and have known coordinates. In this paper we consider the case in which demand points are weighted and their coordinates have a bivariate uniform distribution. Also the problem is developed from a conceptual view and different distance measures are used to make models more applicable in real world situations. The problem is to find the best location for the transfer point such that the maximum expected weighted distance to all demand points through the transfer point is minimized. Depending on assumptions for uniform distributions, two models are considered, convexity conditions are discussed, properties of the optimal solution are obtained and methods to solve the problems are proposed. Finally numerical examples are given.

In this paper, we investigate the location of a facility and several transfer points to serve as collector points for customers who need the services of the facility. For example, demand for emergency services is generated at a set of demand points that need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The model involves the location of multiple transfer points and one facility. Locating one transfer point when the set of demand points and the location of the facility are known was investigated in Berman et al. (2004a). Location of several transfer points when the location of the facility is given is investigated in Berman et al. (2004b). In this paper, we propose heuristic approaches for the solution of this problem and report computational experiments on a test set of 40 problems.

We consider hierarchical facility location problems on a network called Multiple Location of Transfer Points (MLTP) and Facility and Transfer Points Location Problem (FTPLP), where q facilities and p transfer points are located and each customer goes to one of the facilities directly or via one of the transfer points. In FTPLP, we need to find an optimal location of both the facilities and the transfer points while the location of facilities is given in MLTP. Although good heuristics have been proposed for the minisum MLTP and FTPLP, no exact optimal solution has been tested due to the size of the problems. We show that the minisum MLTP can be formulated as a p‐median problem, which leads to obtaining an optimal solution. We also present a new formulation of FTPLP and an enumeration‐based approach to solve the problems with a single facility.

In Sections 4.2.4 and 4.3.1 we defined the normal (or Gaussian) distributions for both single and multiple variables and discussed their properties. The normal distribution plays a central role in the mathematical theory of statistics for at least two reasons. First, the normal distribution often describes a variety of physical quantities observed in the real world. In a communication system, for example, a received waveform is often a superposition of a desired signal waveform and (unwanted) noise process, and the amplitude of the noise is often normally distributed, because the source of such noise is usually what is known as thermal noise at the receiver front. The normality of thermal noise is a good example of manifestation in the real world of the CLT, which says that the sum of a large number of independent RVs, properly scaled, tends to be normally distributed. In Chapter 3 we saw that the binomial distribution and the Poisson distribution also tend to a normal distribution in the limit. We also discussed the CLT and asymptotic normality. The second reason for the frequent use of the normal distribution is its mathematical tractability. For instance, sums of independent normal RVs are themselves normally distributed. Such reproductivity of the distribution is enjoyed only by a limited class of distributions (that is, binomial, gamma, Poisson). Many important results in the theory of statistics are founded on the assumption of a normal distribution.

In this paper, we investigate the location of several transfer points to serve as collector points for customers who need the services of a facility. For example, demand for emergency services by patients is generated at a set of demand points that need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The general model involves the location of multiple transfer points and one facility. Locating one transfer point when the set of demand points and the location of the facility are known was investigated in a previous paper by the authors. In this paper, we apply the results of that paper to solve the problem when the location of the facility is known. Both minisum and minimax versions of the models are investigated both in the plane and on the network.

In this paper we propose the Weber obnoxious facility location problem. As in the classic Weber location problem, the objective is to minimize the weighted sum of distances between the facility and demand points. However, the facility location is required to be at least a given distance from demand points because it is “obnoxious” to them. A practical example is locating an airport. Since in most applications the nuisance generated by the facility “travels by air”, we concentrate on the case where the required minimum distance between the facility and demand points is Euclidean. The Weber objective distance can be measured by a different norm. We develop very efficient algorithms to optimally solve the single facility problem based on geometric branch and bound and on a finite candidate set. We tested it on problems with up to 10,000 demand points using Euclidean, Manhattan, and ℓp for p=1.78 norms for the Weber objective. The largest problems were optimally solved in a few seconds of computer time. Many extensions to the basic Weber obnoxious facility location problem are proposed for future research.

In this paper, we consider the multiple facility location problem with gradual cover. Gradual cover means that up to a certain distance from the facility a demand point is fully covered. Beyond another distance the demand point is not covered at all. Between these two distances the demand point is partially covered. When there are $$p$$ p facilities, the cover of each demand point can be calculated by a given formula. One objective in this setting is to find locations for $$p$$ p facilities that maximize the total cover. In this paper we consider another objective of maximizing the minimum cover of every demand point. This guarantees that every demand point is covered as much as possible and there are no demand points with low cover. The model is formulated and heuristic algorithms are proposed for its solution. We solved a real-life problem of locating cell phone towers in northern Orange County, California and demonstrated the solution approach on a set of 40 test problems.

Transfer Points Location Problem and Optimal Allocation of Injuries in the Crisis Relief Process Transfer Points Location in Crisis

Demand point aggregation method for covering problems with gradual coverage

Given a set ofn demand points with weightWi,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An Ω(n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.

The problems of facility location and the allo- cation of demand points to facilities are crucial research issues in spatial data analysis and urban planning. It is very important for an organization or governments to best locate its resources and facilities and efficiently manage resources to ensure that all demand points are covered and all the needs are met. Most of the recent studies, which focused on solving facility location problems by performing spatial clustering, have used the Euclidean distance between two points as the dissimilarity function. Natural obstacles, such as mountains and rivers, can have drastic impacts on the distance that needs to be traveled between two geographical locations. While calculating the distance between various supply chain entities (including facilities and demand points), it is necessary to take such obstacles into account to obtain better and more realistic results regarding location- allocation. In this article, new models were presented for location of urban facilities while considering geographical obstacles at the same time. In these models, three new distance functions were proposed. The first function was based on the analysis of shortest path in linear network, which was called SPD function. The other two functions, namely PD and P2D, were based on the algorithms that deal with robot geometry and route-based robot navigation in the presence of obstacles. The models were implemented in ArcGIS Desktop 9.2 software using the visual basic programming language. These models were evaluated using synthetic and real data sets. The overall performance was evaluated based on the sum of distance from demand points to their corresponding facilities. Because of the distance between the demand points and facilities becoming more realistic in the proposed functions, results indicated desired quality of the proposed models in terms of quality of allo- cating points to centers and logistic cost. Obtained results show promising improvements of the allocation, the logis- tics costs and the response time. It can also be inferred from this study that the P2D-based model and the SPD-based model yield similar results in terms of the facility location and the demand allocation. It is noted that the P2D-based model showed better execution time than the SPD-based model. Considering logistic costs, facility location and response time, the P2D-based model was appropriate choice for urban facility location problem considering the geo- graphical obstacles.

Location problems on a sphere can be solved easily by mathematical programming or geometrical methods if it is known that all the demand points are located on a hemisphere. This paper presents an algorithm for determining whether m given demand points are on a hemisphere or not. In this algorithm, the great circle is rotated successively until the hemisphere which contains the m given demand points is found or it becomes known using our criteria that the m given demand points are not on a hemisphere. The convergence of this algorithm is proved, and two illustrative examples are presented.

A fuzzy max-T location problem is considered. The fuzzy max-T location problem is a generalization of a fuzzy maximin location problem by allowing in the objective function arbitrary triangular norms instead of the triangular norm defined by the minimum operation. Then we give conditions for the existence of its optimal solutions, and derive a relationship between its optimal solutions and efficient solutions of a fuzzy multicriteria location problem. Furthermore, we give some properties of triangular norms, and for triangular norms, we investigate the stability of optimal solutions of the fuzzy max-T location problem. 1 Introduction and preliminaries In a general continuous location model, finitely many points called demand points in R n , modeling existing facilities or customers, are given. Let di ∈ R n , i =1 , 2, ··· , � (≥ 2) be demand points. We put I ≡{ 1, 2, ··· , � }. Then a problem to locate a new facility in R n is called a single facility location problem. If one prefers the location of the facility near demand points, then the problem is formulated as follows: (1) min x∈Rn f (γ1(x − d1) ,γ 2(x − d2), ··· ,γ � (x − d� ))