In this paper, we present a new approach for solving distance measures using neural networks with suitable strategies. These measures may be of different norms, viz., Euclidean, Squared Euclidean, Rectilinear or Geodesic. Euclidean distance is most frequently used in many realistic problems and has received considerable attention in the literature. Some examples are network location problems, location-allocation problems and instanced involving conveyors and air travel. Problems of electrical wiring and pipe line design may also be mentioned in this regard. It has been shown by many researchers that facility location problems are used in the Euclidean, Squared Euclidean and Rectilinear norms for distance measures. The two major distance measurements Euclidean distance, with no restrictions of orientations to travel, the minimax location problems with Euclidean distance and rectilinear distance are already investigated by many authors. When the Euclidean norm is used one comes across various names given to the facility location problems, such as: the generalized format problem', the p-median problem', and the generalized weber problem 'or Steiner problem. When the squared Euclidean norm is used in measuring the distances between the facilities the problems are often called quadratic facility location problems' or the gravity problems'. Though in neural networks, many researches has already made for application in many branches using Euclidean norms, a detailed literature needs that no researchers has established their work to develop a network structure for Euclidean norms. As in analytical approaches measuring distance for huge data becomes incredibly difficult and since neural network does parallel operations, finding solution for distance measures using neural network in batch mode improves in performance and quick processing also. Hence an attempt has been made to construct a common network structure for the different distance norms. The basic idea is to start with a small network then add hidden units and weights incrementally until a satisfactory solution is found where we will mainly concentrate on distance measures.
[...] The network is composed of a large number of highly interconnected processing elements (neurons) working in parallel to solve a specific problem. A detailed literature survey was made in the area of neural network which has motiveted us to apply this technique to solve the problem Neural Network Architecture Neural network can be divided into three architectures, namely 1. Single layer Multilayer network 3. Competitive layer The number of layers in a net is defined based on the number of interconnected weight in the neuron. Single layer network consists only one layer of connection weights. [...]
[...] The network also consists of additional layer called hidden layer Learning The input layer, activation function and output layer in artificial neuron are similar to the function of dendrites, soma and axon in biological neuron. The input layer, activation function and output layer in artificial neuron are similar to the function of dendrites, soma and axon in biological neuron.Learning falls into three types i. Supervised learning The input-output pairs can be provided by an external teacher, or by the system which contains the neural network (self-supervised). [...]
[...] S.No x1 y1 x2 y2 Euclidean Solutions for Squared Euclidean Rectilinear Table showing Sample Input and Output Trained Dataset Figure 4.1 shows the network structure used for training the Euclidean distance. Fig 4.1 Network Structure for Euclidean Figure 4.2 shows the network structure used for training the Squared Euclidean distance. Fig 4.2 Network Structure for Squared Euclidean Figure 4.3 shows the network structure used for training the Rectilinear distance. Fig 4.3 Network Structure for Rectilinear The network training was completed with 781 epochs for Euclidean distance. [...]
[...] The squared Euclidean distance d2 between two n-dimensional vectors x1 and x2 is d = ( x1 y1 ) 2 + ( x 2 y 2 ) Rectilinear The Rectilinear distance between two points in a Euclidean space with fixed Cartesian coordinate system is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. For example, in the plane, the taxicab distance between the point P1 with coordinates y1) and the point P2 at y2) is 3. [...]
[...] The sample data for Euclidean, Squared Euclidean and Rectilinear distance measures are assembled, trained and simulated with the network designed Proposed Work and Analytical Solutions Obtained The standard Backpropagation algorithm is implemented for the problem of solving distance measures. We have solved 325 numerical examples out of which sets have been taken for training the net and 25 sets for testing the net.We have used a network structure consisting of i. One Input layer having 4 neurons. ii. One Hidden Layer having 50 neurons. [...]
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