Weber Problem
As it offered before the facility location theory has it is origins in the formulation of the French mathematician, Pierre sobre Fermat who also stated the question about the existence of the three factors in the planes and the locating of a new forth reason for a spot that will minimize the whole sum of distances towards the three previous ones. In a same manner Weber generalized Fermat’s initial formulation and assigned dumbbells to the previously mentioned points.
As Eiselt and Marianov (2011) condition, Weber shown Fermat’s strategy in more realistic cases simply by identifying one particular new reason for the map, that symbolize one grow, in order to be reduced the amount of distances, representing the transportation expense, from suppliers to consumers, that symbolize the known points which will reflect several values of demand, the called designated weights. Because Fermat’s formulation has many applications and had being studied simply by different researchers in the materials it can be known additionally while the Fermat ” Torricelli problem, the Steiner difficulty, the Weber problem, the Steiner ” Weber difficulty, the One median problem [Eiselt and Marianov (2011) add that the necessity points are situated on the nodes of an network], the single center 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 always be represented inside the reality while the situation where it is must be opened a brand new warehouse (with coordinates By, Y) within an area in order to serve different amounts of products (the weights) to existing demand items (with runs ai, bi) in a such a way that the total transportation price will be reduced (represented since the sum of miles in correlation to the number of products). It is mathematical ingredients is depicted in the pursuing format. Min¡ z(X)=‘_(i=1)^n’〖w_i †(X, P_i ) 〗 (2. 1) in which d (X, P_i) may be the distance between your warehouse plus the demand factors i. One of the most commonly used range metric is a Euclidean 1, d(X, P_i)=š((X-ai)^2+(Î¥-bi)^2 ). Weiszfeld (1930s) (Eiselt and Marianov, 2011) was the first who also discovered the practical way to Weber’s problem. Its answer is a great iterative algorithm that requires as a preliminary solution a place that minimizes the amount of the squares of the ranges. On opposite, more recently, Chen (2011) aknowledges the performance of interative methods nevertheless judging their very own solution process as quite long. Resulting from this in his research article he offers an noniterative solution. There are many extensions and different approaches inside the investigation from the initial Weber problem. An exclusive one is the effort of Cooper (1963, 1964) barbati cooper rererence by which there are a lot more than three demand points and even more than one particular new below investigated service while a heuristic solution is recommended. It is known in the materials as the Multi ” Weber or perhaps can be met as the place allocation trouble. In this kind of problem, it is necessary to investigate which usually facility will serve which in turn demand stage.
1 different approach to the aforementioned problem is the anti ” Weber problem presented by Hansel et approach. (1981) inside the work of Melachrinoudis (2011) refering to Undesirable facility location concerns. Specifically, they will investigated the locational patterns of indivisible power vegetation in England and offered a solution by using the brand and bound approach. Another procedure is refering to the capacitated multi ” facilty Weber problem analyzed by Aras et ‘s (2007) Xatzigiannis refernce that took into consideration different range metrics. Especially they applied except from Euclidean distance, the Squared Euclidean distance and the Lp Norm Range.
Inside the same problematic vein, Plastria (2011) noted that different types of metrics are commonly used in the investigation of Weber problem. Additionally , Kara and Taner (2011) concluded that the single ” centre location problem seems to act in a same manner to that particular of the time-honored Weber difficulty. According to Plastria (2011), further exts include the job of unfavorable weights (Drezner and Wesolowsky 1991), or perhaps considering the first problem in to buildings (Arriola et ‘s. 2005) or perhaps taking into account cost decisions (Fernández et ‘s. (2007).
