Multifacility Location Problem using Scaled Conjugate Gradient Algorithm under Triangular Area Constraints

Abstract

The problem of finding the multiple locations for new facilities with respect to the multiple existing facilities in a given environment is known as Multifacility Location Problem (MLP).Every location problem is normally bounded by some sort of area constraint. But the fact that much of the work carried out in the literature has almost neglected the area constraint which has motivated us to work on Multifacility Location Problem taking the area constraint into consideration. The mathematical model of the multifacility location problem with area constraint has been developed and the solution has been obtained using Kuhn-Tucker theory. This mathematical analysis and solution procedure is highly complex and time consuming. Hence, an attempt has been made to get the solution of a complex, constrained multifacility location problem using Scaled Conjugate Gradient Algorithm (SCGA) in Artificial Neural Networks (ANN). With the help of Numerical examples, it has been established that the solution obtained through ANN model compares well within the acceptable limits with those obtained through analytical method. Indexterms Multifacility Location Problem, Area Constraint, Kuhn-Tucker theory, Artificial Neural Networks (ANN), Scaled Conjugate Gradient Algorithm (SCGA). INTRODUCTION A number of real life situations involve the problem of locating new facilities with respect to the existing facilities. The multifacility location problems refer to the process of finding the multiple locations for the new facilities with respect to the existing multiple facilities in the prevailing environment. The past investigations into Multifacility Location problems have not taken the availability of area for locating the new facilities into consideration. In this regard we may mention the work of Cabot, Francis and Stray[1],Pritsker and Ghare[2], Kuhn and Kuenne [3], Eyster and White[4] and McHose[5]. Hence in our study of MLP, we have taken the area constraints into consideration. We have considered a deterministic model with the concept of Euclidean distance between facilities with the requirement that the new facilities must be located in a restricted area of triangular shape. This leads to linear constraints. We have solved the problem by using Kuhn-Tucker condition and illustrated the solution procedure of the problem by using some numerical data. 1.1 Applications of ANN The training and testing are two important phases in the development of ANN. The large number of training data sets containing input data and target output are to be fed to the ANN in the training phase. The neural networks create connections and learn patterns based on these data sets as pointed out by Rumelhart [7].Each pattern creates a unique configuration of network structure with a unique set of connection strengths or weights. Similarly, if the neural network does not match the pattern within the given tolerance, it will adjust and try again. A neural network adapts in changing inputs and learns trends from data. Each connection weight builds on previous decision nodes, propagating down to a final decision. After the neural network reaches a final decision, it compares its answer against an answer provided in the training set as target output. If there is a match, within a predefined tolerance, the neural network stores these connection weights as successful. If the decision outcome is outside the tolerance, then the neural network cycles through the training set again. A neural network may cycle thousands of times to reach an acceptable tolerance. The optimal network performance is quite dependent upon making the proper network structure [8]. We have first solved a set of multifacility location problems using the analytical method and then we have again solved the same set of problems through ANN technique using SCGA training algorithm. It is found that the solutions obtained through prediction by using this training algorithm compare well with those obtained using analytical method. The prediction error percentage vary between 0 − 13 % which is well within the acceptable limits. Further, the graphical representation of the results obtained analytically (Anal) and through ANN techniques have also been made to have visual clarity in comparison. 1.2 II. PROBLEM FORMULATION ANDANALYTICAL SOLUTION The MLP with triangular area constraint can be stated as: