Solving Fuzzy Sequential Linear Programming Problem by Fuzzy Frank Wolfe Algorithm

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 3, Number (07), pp Research India Publications Solving Fuzzy Sequential Linear Programming Problem by Fuzzy Frank Wolfe Algorithm A. Nagoor Gani and R. Abdul Saleem PG and Research Department of Mathematics, Jamal Mohamed College (Autonomous), Tiruchirappalli-0, Tamilnadu, India. PG and Research Department of Mathematics, A.V.C. College (Autonomous), Mannampandal , Tamilnadu, India. Abstract In this paper an algorithm is proposed to solve Fuzzy sequential non-linear programming problem. This algorithm is applicable to the problem when the objective function is of non-linear and the constraints are all of linear. Initially, the Fuzzy sequential linear programming problem is converted in to fuzzy linear programming problem using fuzzy Frank Wolfe algorithm and then it is solved by Fourier Motzkin Elimination Method. Key words: Triangular fuzzy numbers, Fuzzy Linear system of equations, Fourier Motzkin Elimination Method, Sequential Linear Programming Problem. AMS subject classification: 03E7, 90C70, 5A39, 90C7. INTRODUCTION Sequential Linear Programming problem is originally proposed by Griffith and Stewart [8]. Bhavikatti [] suggested several progresses to the method to make sure that the method can be used nearly as a black bo for real time problems. A upright discussion of the earlier growths can be found in Himmelblaue [9] who calls the method as approimate programming. Sequential Linear Programming (SLP) [, 6, ] is one of the dominant methods for solving nonlinear optimization problems. The

2 750 A. Nagoor Gani and R. Abdul Saleem SLP [3] consists in linearising the constraints and the objective function in the neighbourhood of a design vector and solving the resulting linear programming problem to get a new design vector. The linearization of non-linear equations and the solution of linear programming problem is sustained in a sequence till optimal solution is reached. The concept of fuzzy numbers was first introduced and investigated by Zadeh [4], etc. One of the major applications of fuzzy arithmetic is treating linear systems and their parameters that are all partially represented by fuzzy number. Friedman etal [7] introduced a general model for solving a fuzzy n n linear system whose coefficient matri is crisp and the right-hand side column is an arbitrary fuzzy number vector. They used the parametric form of fuzzy numbers and replaced the original fuzzy n n linear system by a crisp n n linear system. In this paper, all the variables are considered as fuzzy variables. In solving the fuzzy linear programming problem, to linearize the objective function we applied the Fuzzy Frank Wolfe algorithm [6] and then it is solved by Fourier Motzkin elimination method []. Some basic definitions about the fuzzy concept, the fuzzy variables and fuzzy linear programming problem are given for the clear understanding of the nature of the problem considered here. We also eplained the Fuzzy Frank Wolfe algorithm and numerical eamples are illustrated by the proposed method.. PRELIMINARIES.. Fuzzy set []: A fuzzy set A is defined by A = (,μ A()) : A,μ A() [0,]}. In the pair (,μ A()),the first element belongs to the classical set A and the second element μ A() belongs to the interval [0, ] called membership function... Fuzzy number []: A fuzzy set A on R must possess at least the following three properties to qualify as a fuzzy number: (i) A must be a normal fuzzy set; (ii) must be a closed interval for every α [0, ]; and A (iii) The support of A must be bounded..3. Triangular fuzzy number[]: It is a fuzzy number represented with three points as follows: A ( a, a, a3). This representation is interpreted as membership functions and holds the following conditions: (i) a to a is an increasing function; (ii) a to a 3 is a decreasing function; and (iii) a a a3.

3 Solving Fuzzy Sequential Linear Programming Problem by Fuzzy a A( ) a for a a a a 3 for a a3 a3 a 0 for for.4. Operation of triangular fuzzy number using function principle []: Let A ( a, a, a3) and B ( b, b, b3 ) Then (i) Addition: A B ( a b, a b, a3 b3 ). (ii) Subtraction: A B ( a b3, a b, a3 b ). (iii) Multiplication: AB (min( a b, a b, a b, a b ), a b,ma( a b, a b, a b, a b )) (iv) Division: A/ B (min( ab, ab 3, a3b, a3b3 ), ab,ma( ab, ab 3, a3b, a3b3 )). (v) Scalar Multiplication: K( A) ( Ka, Ka, Ka ), if K is positive and 3 K( A) ( Ka, Ka, Ka ), if K is negative 3 a 3 a.5. Fuzzy linear system of equations Consider the m n fuzzy linear system of equations []: a a... a n b n a a... a b.. n n a a... a b m m m m mn mn m The matri form of the above equations is A b, where the coefficient matri A is (aij) where i = to m and j = to n, is a fuzzy variable and b is also a fuzzy variable..6. Graded mean integration method The graded mean integration method [4] is used to defuzzify the triangular fuzzy number. The representation of triangular fuzzy number is A ( a, a, a3) and its a 4a a3 defuzzified value is obtained by A. 6

4 75 A. Nagoor Gani and R. Abdul Saleem 3. AN ALGORITHM TO SOLVE THE FUZZY SEQUENTIAL LINEAR PROGRAMMING PROBLEM 3. Proposed Algorithm (0) Step : Assume an initial point, two convergence parameters and. Set an iteration Counter t =0. ( t) ( t) Step : Calculate f ( ). If f ( ). Terminate; Else go to step 3. Step 3: Frame the Fuzzy Linear Programming Problem as Maimize or Minimize f ( ) f ( )( ) Subject to constraints g ( ) g ( )( ) 0 ; j,... J j g ( ) g ( )( ) 0 ; j,... J j h ( ) h ( )( ) 0 ; k,... K k ( L) ( U) i k j j Step 4: In the above fuzzy linear programming problem include the objective function in the constraints, for maimization problem, change the equal = in the objective as and (for minimization problem, change ). Step 5: Eliminate the variable one by one in the order as i. Divide each equation by its modulus value of coefficient or all the equations. ii. Now we have three classes of coefficient, i.e or or linear equations. iii. Adding or subtracting any two classes of equations to eliminate Step 6: Repeat the above process until all the fuzzy variables are eliminated. Step 7:.After eliminating all the fuzzy variables, we get the values and substitute the in above,we get the values of fuzzy variables in back to back substitution. Then we get Step 8: Find y be the optimal solution to the above fuzzy LPP. Step 9: Calculate Step 0: If that minimizes ( ) ( t) ( t) t ( y ) ( t) and if ( t) ( t) f (( ) ( y ) in the range α (0, ). ( t) f ( ) f ( ) f ( ) Terminate. Else t = t+ and go to step.

5 Solving Fuzzy Sequential Linear Programming Problem by Fuzzy NUMERICAL EXAMPLE Consider the Fuzzy Sequential Linear Programming Problem with triangular fuzzy number is, Maimize Subject to f ( ) (5,6,7) Non negative restriction (0,0,0) (0,0,0) Applying the above algorithm step by step we get the following solutions: Initial Point t=0 (0) (0) (0) Ma f ( ) f ( ) f ( )( ) f f( ) f (, ) f 5 5 f (0,0) (0) f( ) (0) f( ) =0 (0,0) 5 0 Ma f ( ) Ma f ( ) 0 8 Ma f ( ) 5 8 Linearized Fuzzy problem is given by Ma f ( ) 5 8 Subject to 3 (5,6,7) Non negative restrictions are (0,0,0) (0,0,0)

6 754 A. Nagoor Gani and R. Abdul Saleem Include the objective function in the constraints, for maimization problem, change the equal = in the objective as and (for minimization problem, change ). z (5,6,7) (4..) (0,0,0) (0,0,0) Change all the inequalities in the system as for maimization (and for minimization) 5 8 z (0,0,0) 3 (5,6,7) (4..) (0,0,0) (0,0,0) To eliminate, divide each coefficient of the system (4..) by the coefficient of we have.6 0. z (0, 0, 0) 0.66 (.66,,.33) (4..3) (0,0,0) (0,0,0) Rearranging the equations in (4..3), to eliminate z (.66,,.33).6 0. z (0, 0, 0) 0.66 (.66,,.33) (0,0,0) (4..4) Using the same procedure as the elimination of, we get 0.5 z (0, 0, 0) 0.3 z (.76,.3,.47) (.5,3.03,3.53) (0,0,0) (4..5)

7 Solving Fuzzy Sequential Linear Programming Problem by Fuzzy. 755 Now, the set equations were obtained by eliminating z (.76,.3,.47) 0.5 z (.5,3.03,3.53) 0.5 z (0, 0, 0) (4..6) 0.3 z (4.7,5.6, 6.00) 0.3 z (.76,.3,.47) 0 (.5,3.03,3.53) From the above equation (4..6), we have z (0, 4.0, 8.06) z (0.6, 4.4, 8.4) z (0,0,0) z (0.04, 4., 8.6) z (8.6,0,.59) Now choosing the value for z which satisfies all the constraints. So, the optimal solution is given by z (0,4.0,8.06). Using the obtained z in (4..5), 0.3(0, 4.0, 8.06) (.76,.3,.47) 0.5(0, 4.0, 8.06) (0, 0, 0) (.5,3.03,3.53) (0,0,0) We get, Then (4.6,5.5,5.97) (.76,.3,.47) (.5,3.0,3.50) (0, 0, 0) (.5,3.03,3.53) (0,0,0) (.76,.3,.47) (4.6,5.5,5.97) (0, 0, 0) (.5,3.0,3.50) (.5,3.03,3.53) (0,0,0)

8 756 A. Nagoor Gani and R. Abdul Saleem ( 4., 3.0,.79) (.5,3.0,3.50) (.5,3.03,3.53) (0,0,0) (.79,3.0, 4.) (.5,3.0,3.50) (.5,3.03,3.53) (0,0,0) (.79,3.0, 4.) (.5,3.0,3.50) (.5,3.03,3.53) (0,0,0) (.79,3.0, 4.) (.5,3.03,3.53) The defuzzified value on both sides is nearly 3.0, therefore, select any one (.5,3.03,3.53) Substitute and z in (4..3)..6(.5,3.03,3.53) 0.(0, 4.0, 8.06) (0, 0, 0) 0.66(.5,3.03,3.53) (.66,,.33) We get, (0,0,0) (.5,3.03,3.53) (0, 0, 0) (4.03, 4.84,5.64) (4, 4.84,5.6) (0, 0, 0) (.66,3.03,3.53) (.66,,.33) Then, (0,0,0) (.5,3.03,3.53) (0, 0, 0) (.6, 0,.6) ( 0.67, 0, 0.67) (0,0,0)

9 Solving Fuzzy Sequential Linear Programming Problem by Fuzzy. 757 From the above equations, we get (.6,0,.6) (0,0,0) The defuzzified value of on both the inequalities are same. (0,0,0) Then y Find : (0,0,0), (.5,3.03,3.53) z (0, 4.0, 8.06). => ( t) ( t) f (( ) ( y ) 0,0,0 (0,0,0) 0,0,0 f (0,0,0) (.5,3.03,3.53) (0,0,0) 0,0,0 (0,0,0) f (0,0,0) (.5,3.03,3.53) 0,0,0 (0,0,0) f (0,0,0) (.5,3.03, 3.53) f (0, 0, 0),(.5,3.03,3.53) f( ) 5[(0, 0, 0) ] [(0, 0, 0) ] 8[(.5,3.03,3.53) ] [(.5,3.03,3.53) ] (0.08, 4.4, 8.4) (.5,8.4, 4.5) f( ) (0.08, 4.4, 8.4) (.5,8.4, 4.5) (0.08, 4.4, 8.4) (.5,8.4, 4.5) (0.4, 0.66,.) in the range (0, ). f f f (0, 0, 0),(.5,3.03,3.53) (0, 0, 0)(0.4, 0.66,.),(.5,3.03,3.53)(0.4, 0.66,.) (0, 0, 0),(.8,.00,.45) The Optimal solution is (0,0,0) (.8,.00,.45) CONCLUSION The proposed Fuzzy Sequential Linear Programming algorithm offers a suitable and well-organized method to solve the nonlinear programming problems. It provides not only the optimal solution but also near optimal solutions. So, it paves the way for the performance analysis of the solutions obtained. The optimal solutions obtained from

10 758 A. Nagoor Gani and R. Abdul Saleem the proposed algorithm are more approimate than many other solutions obtained from the eisting methods. REFERENCES [] Arora, J.S., 989, Introduction to optimum design, McGraw-Hill, New York. [] Bhavikatti, S.S., 980, Computational efficiency of Improved Move Limit Method of Sequential Linear Programming for structural optimization, Computers and Structures,, pp [3] Carpentieri, G and Tooren, J.L., 006, Improving the efficiency of Aerodynamic Shape optimization on Unstructured Meshes, 44th AIAA Aerospace Sciences Meeting and Ehibit, Reno, NV. [4] Chen, S.H., and Hsieh, C.H., 999, Graded mean integration representation of generalized fuzzy number, J. Chinese Fuzzy System Assoc., 5(), pp. -7. [5] Chen, S.H., and Hsieh, C.H., 999, Graded mean integration representation of generalized fuzzy number, J. Chinese Fuzzy System Assoc., 5(), pp. -7. [6] Deb, K., 995, Optimization for engineering design: Algorithms and eamples, Prentice Hall of India. [7] Friedman, M., Ming, M., and Kandel, A., 998, Fuzzy linear systems, Fuzzy Sets and Systems, 96, pp [8] Griffith, R.E., and Steward, R.A., 96, A nonlinear programming technique for the Optimization of continuous processing systems, Journal of Management Science, 7, pp [9] Himmelblaue, D.M., 97, Applied Nonlinear Programming, McGraw Hill, New York. [0] Motzkin, T.S., 934, Beitrage zur theorie der linear en ungleichungen, Doctoral Thesis, University of Basel. [] Nagoor Gani, A., and Mohamed Assarudeen, S.N., 0, An Algorithmic Approach of Solving Fuzzy Linear System Using Fourier Motzkin Elimination Method, Advances in Fuzzy Sets and Systems, 0(), pp [] Rao, S.S., 996, Engineering optimization-theory and practice, New Age International (P) Ltd. Publishers, New Delhi. [3] Zadeh, L.A., 965, Fuzzy sets, Information and Control, 8, pp [4] Zadeh, L.A., 976, The concept of a linguistic variable and its applications to approimate reasoning, Information Science, 9, pp