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As it presented before the facility location theory has its origins in the formulation of the French mathematician, Pierre de Fermat who stated the question about the existence of the three points in the plane and the locating of a new forth point in a spot that will minimize the total sum of distances to the three previous ones. In a same manner Weber generalized Fermat’s initial formulation and assigned weights to the aforementioned points.
As Eiselt and Marianov (2011) state, Weber presented Fermat’s approach in more realistic cases by identifying one new point in the map, that represent one plant, in order to be minimized the sum of distances, representing the transportation cost, from vendors to consumers, that represent the known points which reflect different values of demand, the called assigned weights. Due to the fact that Fermat’s formulation has many applications and had being studied by different researchers in the literature it can be referred additionally as the Fermat – Torricelli problem, the Steiner problem, the Weber problem, the Steiner – Weber problem, the One median problem [Eiselt and Marianov (2011) add that the demand points are located on the nodes of an network], the single facility Euclidean Minisum problem, the Minimum aggregate travel point [from the perspective of geographers and economists (Plastria, 2011)], the bivariate median, the spatial median (Xatzigiannis, 2013).
The Weber problem can be represented in the reality as the situation where it is needed to be opened a new warehouse (with coordinates X, Y) in an area in order to serve different amounts of products (the weights) to existing demand points (with coordinates ai, bi) in a such a manner that the total transportation cost will be minimized (represented as the sum of distances in correlation to the number of products). Its mathematical formulation is depicted in the following format. Min z(X)=∑_(i=1)^n▒〖w_i ⅆ(X,P_i ) 〗 (2.1) where d (X,P_i) is the distance between the warehouse and the demand points i. The most commonly used distance metric is the Euclidean one, d(X,P_i)=√((X-ai)^2+(Υ-bi)^2 ). Weiszfeld (1930s) (Eiselt and Marianov, 2011) was the first who discovered the practical solution to Weber’s problem. Its solution is an iterative algorithm that takes as an initial solution a point that minimizes the sum of the squares of the distances. On contrary, more recently, Chen (2011) aknowledges the efficiency of interative methods but judging their solution procedure as quite long. As a result of this in his research article he proposes an noniterative solution. There are many extensions and different approaches in the investigation of the initial Weber problem. A distinctive one is the work of Cooper (1963, 1964) barbati cooper rererence in which there are more than three demand points and more than one new under investigated facility while a heuristic solution is proposed. It is referred in the literature as the Multi – Weber or can be met as the location allocation problem. In this kind of problem, it is necessary to investigate which facility will serve which demand point.
One different approach to the aforementioned problem is the anti – Weber problem presented by Hansel et al. (1981) in the work of Melachrinoudis (2011) refering to Undesirable facility location problems. Specifically, they investigated the locational patterns of nuclear power plants in France and provided a solution by using the brand and bound technique. Another approach is refering to the capacitated multi – facilty Weber problem examined by Aras et al (2007) Xatzigiannis refernce that took into account different distance metrics. Specifically they used except from Euclidean distance, the Squared Euclidean distance and the Lp Norm Distance.
In the same vein, Plastria (2011) noted that different types of metrics are commonly used in the investigation of Weber problem. In addition, Kara and Taner (2011) concluded that the single – hub location problem seems to behave in a same manner to that of the classical Weber problem. According to Plastria (2011), further extensions include the assignment of negative weights (Drezner and Wesolowsky 1991), or considering the initial problem into buildings (Arriola et al. 2005) or taking into account price decisions (Fernández et al. (2007).
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