Buy Métodos Numéricos 1st by Heitor Pina (ISBN: ) from Amazon’s Book Store. Everyday low prices and free delivery on eligible orders. Buy Métodos Numéricos Complementos e guia prático (Portuguese Editin) by Carlos Lemos e Heitor Pina (ISBN: ) from Amazon’s Book Store. Frequency with two tests and/or examination. Bibliography. Pina, Heitor; Métodos Numéricos, McGraw-Hill. Atkinson, K. E., An Introduction to Numerical Analysis.

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Skip to main content. Log In Sign Up. International Journal Of Industrial Engineering, v. Oliveira 1 1 Dep. Biondi Neto, lbiondi uerj. In addition, the heior programming problem LPP is transformed into an optimization problem without constraints by using a pseudo-cost function, where’s added a term of penalty, causing high cost all time that one of the constraints goes violated.

The problem is converted into a differential equations system.

An unconventional ANN implements a numerical solution based gradient method. Received ; Accepted 1.

FEUP – Numerical Analysis

It also allows the evaluation of the relative operational efficiency of organizations DMUscontemplating each DMU relatively to all the others that compose the investigated DMUs group, Charnes, Cooper and Seiford The DEA technique compares the DMU efficiencies by their abilities in transforming inputs in outputs, measuring the reached output relation in terms of the provision supplied by the input. In the end of the analysis, the DEA technique is able to tell which units are relatively efficient and which units are relatively inefficient, Angulo Optimization modules called Neuro-LP will be used in the neural model proposed Neuro- DEAinspired by the artificial neural network philosophy, Biondi The DEA models can be oriented to inputs or outputs and this orientation ;ina be previously chosen by the analyst as starting point in the DEA analysis.

The orientation to inputs indicates that we want to reduce the inputs, keeping the outputs unaffected. In the other hand, the orientation to outputs indicates that we want to increase the outputs mtodso affecting the inputs, Coelli The most important models are the following: CCR — Model presented by Charnes, Cooper and Rhode that builds a non parametrical surface, linear by parts, over the data and determines the investigated DMUs technical efficiency over this surface.

It was conceived as an input oriented model and it works with constant return of scale CRSwhich means that each variation in the inputs produces a proportional variation in the outputs. The problem consists in determining the uj and vi weight values to maximize the linear combination of the outputs divided by the linear combination of the inputs, Estellita The process must be repeated to each of the n DMUs and, by these processes, determine numricso relative value of each DMU efficiency.

To solve this problem, Charnes and Cooper introduced a linear transformation that allows transforming linear fractional problems into LPPs, creating the model called Multipliers, equation 2. By the exposed reasons, the dual model, called Envelope, being easily solved, is preferred compared to the Multipliers model.

Nhmricos this case, the VRS frontier considers increasing or decreasing returns in the efficient frontier.

Biblioteca do ISEL

So, using the orientation to inputs, we verify that the optimum projection of the DMU 4 happens in a point that reflects the convex linear combination of DMUs 1 and 2.

Using the orientation to inputs, we verify that the optimum projection of the same DMU 4 happens in a point that reflects the convex linear combination of DMUs 2 and 3. The Envelope model, oriented to input and the primal derived model multipliers are given by 4 and 5.

The ANNs are massively paralleled structures, based on simple processing elements PEinspired in the biological neuron and densely interconnected. The main ANN characteristics are: The knowledge is distributed all over the network, available and ready to be used execution step in many different application areas, as in the neural linear programming, Zurada In the case of the Neuro-LP optimization modules, part of the Neuro-DEA model, a structure similar to the ANN is used, where the synaptic weights obtained in the training step are basically formed by the coefficient of the problem constraint groups, Rosenblatt and Wasserman It tries to reproduce, in a simple way, the biological neuron operation.


The PE inputs x1, x So, a good procedure is to associate to each synaptic connection, a positive or a negative weight value Wj1, Wj Wjndetermining the effect that a source PE has over the destination PE.

Consequently, the PE would be able to trigger when the pondered sum of the inputs Xi and the weights Wji, exceed the threshold value Wj0 biasduring the latency period. The PE is defined by a propagation rule NETj, that determines the way the inputs will be computed and by an activation function F that determines the new value of the activation state for the destination PE.

The mostly used formula as propagation rule, equation 6defines NETj as the pondered sum of the values provided as inputs and the weight values of the PE input connections, Harvey There are, basically, two types of architectures used in ANNs: The mostly used network architecture is the feedforward.

This arrangement is composed of a group of PEs, arranged in layers one or more that interconnect themselves in sequence. The most complete configuration presents one or more intermediate or hidden layers between the input and the output layer, and it is known as multi layer network. The fact of having hidden layers allow better results for certain problems, as well as allowing the solution for impossible problems to be solved with single layer networks, Wasserman and Zurada The neural processing is accomplished in two main phases: The training phase or learning is the updating process for the connection weights.

Its goal is to acquire information and store it as a weight matrix WMinsk In the case of the Neuro LP, main cell of the Neuro DEA model, the problem acknowledgement is previously known by the LPP constraint coefficients and eliminates the need for this phase. The execution phase recall calculates the ANNs output Y in terms of the injected stimulus in the input X and the weights obtained in the training phase Y or imposed by the problem itself.

In the Neuro LP case, the goal for this phase is to recover the information, which means, determine the optimum value for the LPP decision variables and that, in the case of the Data Wrapping Analysis, may represent the efficiency value of a DMU, Biondi This process will be done by the resolution of a differential equation system, obtained by the transformation of the original LPP in an optimization problem without constraints.

The numerical method used to solve the differential equation system will be the dynamic gradient method, derived from the Newton method and that is very similar to the ANNs training method.

Initially, the ANN architecture used in the Neuro LP model will be presented, as well as the development of the training algorithm based on the minimization of the sum squared geitor in the network output, by the decreasing gradient method and its variations. In the Neuro LP case, the ANN is used in the execution phase, and it already has the knowledge referred to the LPP, represented here by the problem constraint coefficients.

Then, we mathematically prove the LPP transformation, composed of an objective function and a constraint collection in an optimization problem without constraints, Bazaraa A function called pseudo-cost was adopted where a penalty term was added, causing a high cost every time a constraint is violated. The new problem can be solved by the gradient method, turning it into a njmricos equation system, which can be numerically solved.

Finally, a case study is presented. As shown in Figure 3, the network current output Oj is compared to each new iteration with a desired value Tj associated to the training patterns, generating an error signal ej. Finally, this signal suffers a learning adaptive process updating the weights in each iteration Rosenblatt,Rumelhart, Its operation is based on the following: The signal NETj is then processed by a limiter called activation function F, shown heltor figure 1.

The interactive process of pattern presentation causes the error decrease and when this error reaches a established value, its said that the ANN absorbed the desired knowledge and that the process converged, Skapura In the first experiments done by Rosenblatt, the Perceptron training algorithm was entirely based in the technique developed by Widrow Hoff, where the error signal was obtained before the activation function and therefore, linear.

Even in the mapping of lineally separable functions the technique failed. So, Rosenblatt developed an algorithm adjusting the weights by the minimization of the sum-squared error heitro the decreasing gradient method, Rosenblatt In this case, due to the derived imposed by the method; the activation function must be derivable through the whole domain, as it happens with the tansigmoid. In this phase execution the ANN receives signals in the input, which did not take part in the training phase, and presents the result in the output, according to the knowledge acquired during the training phase and stored in the weight matrix.


In the Neuro LP case, the ANN, where the weights are already known and that represent the problem constraint coefficients, the numricoss of the output is the next step, which indicates the value of the LPP decision variables, Biondi Numricks the multi layer Perceptron, an algorithm similar to the one developed and called back-propagation is used in the training phase. The only difference is in the calculation of the error signal in the intermediate layers of the PEs.

This signal must allow the error propagation to the previous layers back-propagation until it reaches the first one. Although, due to the fact that this subject is not directly related to the development of the Neuro-LP, it will not be investigated, Wasserman Improvement of the ;ina Training.

The necessary and sufficient conditions for the existence of a local minimum are: Its based in the transformation of the optimization problem without constraints in a first order ordinary differential equation system, represented by 8.

The numerical solution can be obtained considering the following: To build the new function E xits incorporated a function or penalty term Pi[Ri x ] to the original objective function Kennedy,Chen,Bargiela,Zhu, The penalty term must penalize big p for the cases of no feasible solutions and inhibit for viable solutions of the LPP, DennisPina and Werner The optimization problem without constraint with penalty term can be solved similarly to the ANN training phase, applying the decreasing gradient method.

So it must be written as an ordinary differential equation system and solved numerically.

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To ensure accuracy of the method, the penalty parameter p must be very high. The practice shows that p values extremely high are not convenient from the computing point of view. In this case, higher values of p are not necessary for the correct convergence of the process. So, with reasonable p values, the minimum of the pseudo-cost function E x, p is equivalent to the optimum solution of the original LPP. According to Cichocki a great choice is to consider the pseudo-cost function Implementation until 5 variables.

The relationship between all inputs and outputs is obtained to each DMU: The implementation was done using the CRS Envelope model, input heitoe.

Numerical and Computational Methods

Some data of a case involving 5DMUs with two inputs and one output are shown in Figure 7 a. Figure 7 c mtovos the 3D frontier and finally, in Figure 7 d there is a table that compares the obtained results, using two commercial and consecrated softwares Lindo and Frontier Analystwith the results of the model proposed in this paper and calculating the error percentage. Block Diagram and Adopted Model. In this case, the observed error did not surpass 0.

Actually, a study using Lagrange Multipliers is being developed, which basically will optimize the step function, shown in figure 5. The solution method for the ordinary differential equation system used in the Neuro-LP model is similar to the technique used in ANN training phase because they use the decreasing gradient method.

The evolution of the solution method for the differential equation system, represented by the solution path curve, indicates in the convergence, the value of the decision variables of the problem. Finally, its important to highlight that the convergence speed can get extremely high, if the proposed modules are integrated in a chip and connected to a free slot in a computer.

Abraham Charnes, Willian W. Lewin and Lawrence M. Moscinski, Zbigniew Ogonowski