International Conference on Mathematical Computer Engineering - ICMCE - 8 MALAB Simulink Modeling and Simulation of Recurrent Neural Network for Solving Linear Programming Problems Raja Das a a School of Advanced Sciences, VI University, Vellore, amil Nadu, India Abstract In this paper, a recurrent neural network for solving linear programming problems is presented that is simpler, intuitive and fast converging. o achieve optimality in accuracy and also in computational effort, an algorithm is presented. We investigate in this paper the MALAB Simulink modeling and simulative verification of such a recurrent neural network. Modeling and simulative results substantiate the theoretical analysis and efficacy of the recurrent neural network for solving the linear programming problem. A detailed eample has been presented to demonstrate the performance of the recurrent neural network. Keywords: Linear Programming Problem; Recurrent Neural Network; MALAB Simulink. Introduction Linear programming techniques are widely used to solve a number of military, economic, industrial, and social problems. In the last 5 years, researchers have proposed various dynamic solvers for solving linear programming problems. he dynamic systems approach to solve constrained optimization problems was first proposed by Pyne []. he recent developments in this field have redefined the application of neural network by dynamic solvers [-]. In recent studies, many researches have been done in relation to the application of the technology to knowledge engineering such as Artificial Neural Network (ANN) to the engineering field. Linear Programming (LP), one of the oldest disciplines of Operations Research, continues to be the most active branch. LP has been widely applied practically in many sectors such as production, financial, human resources, governing and planning. Now, with high-speed computers and multiprocessors it is possible to construct and solve linear programs that were impossible just a few years ago. In 97 Dantzig [5] developed a method for solving linear programming problems which is known today as the Simple method. Brown and Koopmans [6] described the first of a series of interior point method for solving linear programs. Karmakar [7] developed an algorithm which appears to be more efficient than the Simple Method on some complicated real-world problems of scheduling, routing and planning. Conn [7] developed an alternative method for solving linear programs by using unconstrained optimization technique combined with penalty function methods. he first neural approach applied to linear programming problems was proposed by ank and Hopfield []. he simulation models are developed as a stand alone application using MALAB Simulink [8]. Matlab Simulation Models have been widely used [9,]. he primary objective of this paper is to present a recurrent neural network for finding the solution of linear programming problems. Here I describe circuit implementation of proposed neural network using MALAB Simulink. he most important advantages of the neural networks are their massive parallel processing capacity and fast convergence properties.. Linear Programming Linear Programming is the term used for defining a wide range of optimization problems in which the objective function to be minimized or maimized is linear in the unknown variables and the constraints are a combinations of linear equalities and inequalities. LP is one of the most widely applied techniques of operations research in business, industry and numerous other fields. he objective function may be profit, cost, production capacity or any other measure of effectiveness, which is to be obtained in the best possible ISBN 978-9-88-9-8 Bonfring
International Conference on Mathematical Computer Engineering - ICMCE - 9 or optimal manner. he constraints may be imposed by different resources such as market demand, production process and equipment, storege capacity, raw material availibilty, etc. First, the given problem must be presented in linear programming form. his requires defining the variables of the problem, establising inter-relationship between them and formulating the objective function and constraints. A model, which approimates as closely as possible to the given problem, is then to be developed. If some constraints happen to be non-linear, they are approimated to appropiate linear functions to fit the linear programming format.. Problem Statement Consider a standard form of Linear Programming Problems described as Minimize C X () Subject to AX = B () X () n n m where X R is a column vector of decision variables, C R and B R are column vector of m n cost coefficient and right hand side parameters, respectively, A R is a constraint coefficient matri, and the subscript denotes the transpose operator.. Solution to the LP Problem In general, the LP problem can have four possible solution types:. Unique Solution: here is only one solution that satisfies all constraints, and the objective function reaches a minimum within the feasible region.. Nonunique Solution: here are several feasible solution where the objective function reaches a minimum. An unbounded Solution: he objective function is not bounded in the feasible region and it approaches -. No feasible Solution: Constraint provided in () and () are too restrictive, and the set of feasible solution is empty. Although theatrically valid, cases and appear rarely in engineering and scientific applications. Furthermore, they can be easily detected, and in further consideration of the LP problem we will assume that it is formulated in such a way that there eist at least one feasible solution. 5. he Neural Network An artificial Neural Network (ANN) is a dynamic system, consisting of highly interconnected and parallel non-linear processing elements, that is highly efficient in computation. In this paper, a recurrent neural network [7] with equilibrium points representing a solution of the constrained optimization problem has been developed. As introduced in Hopfield [] these network are composed with feed back connection between nodes. In the standard case, the nodes are fully connected i.e., every node is connected to all other nodes, including itself. he node equation for the discrete-time network with n-neurons is given by i i i + = ( k ) ( k ) µ c + α y ( k ) if ( k + ) and y ( k + ) = y ( k ) + η z ( k ) α y ( k ) { } j j j. where k z j ( ) = AX B ij j j if i i ( k + ) > ISBN 978-9-88-9-8 Bonfring
International Conference on Mathematical Computer Engineering - ICMCE - 5 he first step in a neural network implementation for solving the LP problem is to define an energy function that can be optimized in an unconstrained fashion. herefore, the method of Lagrange multiplier is applied in the network. o accomplish this, the linear constraints AX = B and non negativity constraints X > are appended to the objective function in some convenient way. Commonly the constraints are incorporated as penalty terms that, when ever violated, increase the value of the energy function. One energy function that can be derived using the Lagrange multiplier method are defined as E X, Y = C X + Y AX B α Y () with ( ) ( ) Y m α, Y R, and X. Applying the method steepest descent in discrete time, I compute the gradient of the energy function in () with respect to X and obtain E = [ C X + X X Y ( AX B) α Y Y ] and E = C + A Y = C + A ( KZ + Y ) X where Z m R is defined as Z = AX B In a similar manner we have E = Y AX B α Y 6. MALAB Simulink he Simulink toolbo is a useful software package to develop simulation models for recurrent neural network applications in the MALAB Simulink environment. With its graphical user interface and etensive library, it provides researchers with a modern and interactive design tool build simulation models rapidly and easily. Simulink is an input/output device GUI block diagram simulator. It opens with the library browser and library browser is used to build simulation models. he library browser contains continuous system model elements, discontinuous system models elements, and list of math operation elements. In the library browser, Sink elements are used for displaying and Source elements are used for model source functions. Model elements are added by selecting the appropriate elements from the library browser and dragging them into model window to create the required model. 7. Numerical Eample First, we give numerical eample to demonstrate the solution of linear programming problem through the proposed recurrent neural network. Eample. Consider the linear programming problem Ma Z = 5 + 8 Subject to 5, + 5 + 5 ISBN 978-9-88-9-8 Bonfring
International Conference on Mathematical Computer Engineering - ICMCE - 5 From the problem statement it can be seen that the linear programming problem is not in the standard form. By adding surplus variables and, the linear programming problem can be transferred in the standard form as follows: Ma Z = 5 Subject to 5, + 5 + 8 +,, + + + = = + 5 o solve this linear programming problem, the MALAB Simulink model for recurrent neural network in Figure. is simulated. he parameters of the network were chosen as µ =., η =.. Zero initial conditions were assumed both for X andy. he network converges in approimately, iterations. Fig.. (Recurrent Neural Network) X =, he solution to the linear programming problem is given as [ ] X =.,. learning rate parameter accuracy from the eact solution [ ], which is within the ISBN 978-9-88-9-8 Bonfring
International Conference on Mathematical Computer Engineering - ICMCE - 5 able (Comparison of the simulation result with the eact solution of the eample) Variables Neural Network Result Eact Solution.. he results are also given in able and they are very close to the eact solution. he trajectories of the neural network variables are plotted in Fig. rajectories of the independent Variable for Linear Programming Problem Independent Variables 5 5 5 68 6 68 7 5 9 76 78 76 Ite rations 8.Conclusion Fig. rajectories of the independent Variable for linear Programming Problem We investigated in this paper the MALAB Simulink modeling and simulative verification of such a recurrent neural network. Finding solution of linear programming problems through recurrent neural network approach is an interesting area of research. he energy function of the linear programming is defined in this paper. A circuit has been designed for the purpose. By using click-and-drag mouse operations in MALAB Simulink environment, we could quickly model and simulate complicated dynamic systems. It has been concluded that recurrent neural network can be used for determining the solution of Linear programming problem. It converges to the eact solutions of the Problem. Modeling and simulative results substantiate the theoretical analysis and efficacy of the recurrent neural network for solving the linear programming problem. his model is very simple. he model can be etended to solve other optimization problems, including conve optimization and other nonlinear programming problems. ISBN 978-9-88-9-8 Bonfring
International Conference on Mathematical Computer Engineering - ICMCE - 5 References [] I. B. Pyne, 956. Linear programming on an electronic analogue computer. rans. Amer. Inst.Eng., vol. 75, pp. 9- [] D. W. ank and J. J. Hopfield, 986. Simple neural optimization network s: An A/D converter, signal decision circuit and a linear programming circuit, IEEE rans. Circ. Syst., vol. CAS-. pp. 5-5, May. [] M. P. Kennedy and L. O. Chua. 988. Neural networks for nonlinear programming. IEEE rans. Circ. Syst., vol. 5. pp. 55-56, May [] W. E. Lillo, M. H. Loh, S Hui and S. H. Zak, 99. On solving constrained optimization problems with neural networks: a penalty function method approach. ech. rep. R-EE 9-, Purdue Univ., W. Lafayette IN. [5] G. B. Dantzig, 96. Linear Programming and Etensions. Princeton. NJ: Princeton Press,. [6] G. W. Brown and. C. Koopmans, 95. Computation suggestions for maimizing a linear function subject to linear equalities, in Activity Analysis of Production and Allocation,. Koopmans. Ed. New York: J Wiley,, pp. 77-8. [7] A. R. Conn, 976. Linear programming via a non differentialable penalty function, SIAM J. Num. Anal., vol., pp. 5-5. [8] Book,. SIMULINK, Model-Based and System-Based Design, using simulink, Math Works Inc., Natick, MA. [9] C. D. Vournas, E. G. Potamianakis, C. Moors, and. V. Cutsem,. An educational simulation tool for power system control and stability, IEEE rans Power Syst 9, 8-55. [] R. Patel,. S. Bhatti, and D. P. Kothari,. MALAB/ Simulink-based transient stability analysis of a multimachine power system, Int J Electr Eng Educ 9, -6. ISBN 978-9-88-9-8 Bonfring