Fuzzy Optimal Solution for Fully Fuzzy Linear Programming Problems Using Hexagonal Fuzzy Numbers

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1 Intern. J. uzzy Mathematical Archive Vol. No ISSN: (P (online Published on 29 April 26 International Journal of uzzy Optimal Solution for ully uzzy Linear Programming Problems Using Hexagonal uzzy Numbers K.hurai and A Karpagam 2 epartment of Mathematics Valliammai ngineering ollege Kattangulathur hennai dhurai_msc_mphil@yahoo.com 2 karpagamohan@rediff.com Received 3 April 26; accepted 26 April 26 Abstract. The fuzzy set theory has been applied in many fields such as management engineering and almost in every business enterprise as well as day to day activities. In this paper fully fuzzy linear programming problems with hexagonal fuzzy numbers were discussed. A new approach for solving fully fuzzy linear programming problems (LPP is proposed based upon the new Ranking function which is divided as two Trapezoidal and average values of the same were taken. This paper compares the three different ranking functions by solving some LLP problems which are tabulated. Keywords: uzzy number Trapezoidal fuzzy number Hexagonal fuzzy number Ranking function uzzy variable linear programming uzzy optimal solution. AMS Mathematics Subject lassification (2: 95. Introduction uzzy Logic was initiated in 965 by Lotfi A. Zadeh Professor at the University of alifornia in erkeley. asically uzzy Logic (L is a multivalued logic that allows intermediate values to be defined between conventional evaluations like true/false yes/no high/low etc. Ranking fuzzy number is used mainly in decision-making data analysis artificial intelligence and various other fields of operation research. In fuzzy environment ranking fuzzy numbers is a very important decision making procedure. The idea of fuzzy set was first proposed by ellman and Zadeh [5] as a mean of handling uncertainty that is due to imprecision rather than randomness. The concept of uzzy Linear Programming (LP was first introduced by Tanaka et al. [6 7]. Zimmerman [9] introduced fuzzy linear programming in fuzzy environment. hanas [6] proposed a fuzzy programming in multiobjective linear programming. Allahviranloo et al. [2] proposed a new method for solving fully fuzzy linear programming problems by the use of ranking function. Kumar et al. [] proposed a new method for solving fully fuzzy linear programming problems with inequality constraints. Abbasbandy and Asady [] suggested a sign distance method for ranking fuzzy numbers in 26. Rajarajeswari et al. [5] presented a new operation on hexagonal fuzzy numbers. Liou and Wang [3] presented ranking fuzzy numbers with interval values. Verdegay [8] have developed 7

2 K.hurai and A Karpagam three methods for solving three models of fuzzy integer linear programming based on the representation theorem and on fuzzy number ranking method. Nasseri et.al [4] proposed a new method for solving fuzzy linear programming problems in which he has used the fuzzy ranking method for converting the fuzzy objective function into crisp objective function. Lee and Li [2] discussed the comparison of fuzzy numbers. Amit Kumar et al. [3 4] presented a new method for solving fuzzy linear programs with Trapezoidal fuzzy numbers. heng [8] used a centroid based distance method to rank fuzzy numbers in 998.Kauffmann and Gupta [9] introduced to uzzy Arithmetic. Kolman and Hill [] was introduced a LP problem. hen [7] proposed the ranking trapezoidal fuzzy number using maximizing and minimizing set decomposition principle and sign distance. In this paper some preliminaries are presented in section 2. Section 3 describes the proposed method with one numerical example and obtained results were discussed. Section 4 concludes the paper. 2. Preliminaries efinition 2..The characteristic function µ A of a crisp set A X assigns a value either or to each member in X. This function can be generalized to a function μ such that the value assigned to the element of the universal set X fall within a specified range i.e.μ : X []. The assigned value indicate the membership function and the set = {(x μ (x; x X} defined by μ (x for x X is called fuzzy set. efinition 2.2. An effective approach for ordering the elements of (R is also to define a ranking function R :(R R which maps each fuzzy number into the real line where a natural order exists. We define orders on (R by: if and only if R ( R ( if and only if R ( R ( = if and only if R ( = R ( efinition 2.3. A fuzzy number is a hexagonal fuzzy number denoted by = ( where ( are real numbers and its membership function μ (x is given below. x< a ( + ( x x μ (x = x - ( ( x x x > 8

3 uzzy Optimal Solution or ully uzzy Linear Programming Problems Using Hexagonal uzzy Numbers.5 igure : Graphical representation of a hexagonal fuzzy number for x [ ] 3. Proposed method a a 2 a 3 a 4 a 5 a A A a a2 a3 a4 a5 a6 a a2 a3 a4 a5 a6 igure 2 igure 3:.5 A a a2 a3 a4 a5 a6 In this paper the hexagonal has been divided into two plane figures. These two plane figures are trapezoidal A and (ig.2.then the ranking function were taken for A and and the four lines were joined from x-axis to hexagonal (ig.3. Average has been taken by using all six points (ig.4. Let = ( be a hexagonal fuzzy number. The ranking functions are obtained as below: R( = (3. igure 4 9

4 K.hurai and A Karpagam R( = R( = (3.2 (3.3 Algorithm Step : ormulate the chosen problem into the following fully fuzzy linear programming Problem: Max (or min = Step 2: Using the Ranking functions ( the LPP transformed into VLPP. Step 3: Solve the VLPP by using simplex method / ig-m method. Let the solution be j. Hence the solution of LPP is. Step 4: xpress the problem in standard form by introducing slack / surplus variables to convert the inequality constraints into equations. Step 5: ompute the value of == Y j - j j j=...n. (i If all for maximization problem (ii If all < for minimization problem. Then the current solution is optimal otherwise go to step 6. Step 6: etermine the basic variable k which will be replaced by the non-basic variable Where k = arg min {R( } i= 2.m in maximization problem and k = arg max {R( } i= 2.m in minimization problem. Step 7: Perform the pivot operation and return to step 5. Then repeat the procedure until a fuzzy optimal solution is obtained. xample 3.. Maximize ( ( ( ( ( ( ( ( Solution: Maximize ( ( ( ( ( 3 = ( ( ( ( 4 = ( Ranking function (i Maximize = = ( = (

5 uzzy Optimal Solution or ully uzzy Linear Programming Problems Using Hexagonal uzzy Numbers Initial table asis RHS R( ( ( ( irst iteration asis RHS ( ( ( Since the fuzzy optimal solution of the VLPP and LPP is = ( = ( and = ( 5.54 Ranking function (ii Maximize = = ( = ( The fuzzy optimal solution of the VLPP and LPP is = ( = ( and = ( 5.54 Ranking function (iii Maximize = = ( = ( The fuzzy optimal solution of the VLPP and LPP is = ( = ( and = ( 5.54 S l. N o No. of const raints No. of varia bles Ranking function I Ranking function II Ranking function III uzzy optim al value 2

6 K.hurai and A Karpagam 2 2 = ( = ( = ( = ( = ( = ( = (. = ( = (..... = ( = ( = ( = ( = ( = ( = ( = ( = ( = ( = (..... = (.... = (..... =( = ( = ( = ( = ( = (= ( = ( 29.8 = ( = ( = ( = ( Table: omparison of fuzzy optimal solution and fuzzy optimal values using three ranking functions

7 uzzy Optimal Solution or ully uzzy Linear Programming Problems Using Hexagonal uzzy Numbers 4. onclusion In this paper a new method is proposed for solving the fuzzy optimal solution of LP problem transform into VLP problems. The LP problem is converted into VLP problem using new Ranking function. We have obtained the same results by using the above three ranking functions (3. (3.2 (3.3. Ranking function is reasonable and effective for calculating the hexagonal weights of criteria. Therefore it is easier to solve fully fuzzy linear programming problem. RRNS. S. Abbasbandy and T. Hajjari Ranking of fuzzy numbers by sign distance Information Sciences 76( T.Allahviranloo.H.Lotfi M.K.Kiasary N.A.Kiani and L. Alizadeh Solving fully fuzzy linear Programming problems by the ranking function Applied Mathematical Sciences2 ( A.Kumar and J.Kaur A New Method for Solving uzzy Linear Programs with Trapezoidal uzzy Numbers International Journal of fuzzy set valued Analysis 2(2. 4. Amitkumar Jagdeepkaur and Singh Solving fully fuzzy linear programming problems with inequality constraints International Journal of Physical and Mathematical Sciences( R..ellman and L.A.Zadeh ecision making in a fuzzy environment Management Science7( hanas uzzy programming in multiobjective linear programming a parametric approach uzzy Sets and Systems 29 ( S.H. hen Rank in uzzy numbers with maximizing set and minimizing set uzzy Sets and Systems 7 ( H.heng A New approach for ranking fuzzy numbers by distance method uzzy Sets and Systems 95 ( A.Kauffmann and M. Gupta Introduction to uzzy Arithmetic Theory and Applications Van Nostrand Reinhold New York 98.. Kolman and Hill Applied Mathematics A.Kumar J.Kaur and Singh uzzy optimal solution of fully fuzzy linear programming problems with Inequality constraints International Journal of Applied Mathematics and omputer sciences 6 ( S.Lee and R.J. Li omparison of fuzzy numbers based on the probability number of fuzzy events omputers and Mathematics with Applications 5( T.S.Liou and M.J. Wang Ranking fuzzy numbers with integral value uzzy Sets and Systems 5 ( S.H.Nasseri A new method for solving fuzzy linear programming by solving linear programming Applied Mathematical Sciences 2 ( P.Rajarajeswari A.Sahayasudha and R.Karthika A new operation on hexagonal fuzzy number International Journal of uzzy Logic Systems3 ( H.Tanaka and K.Asai uzzy linear programming problems with fuzzy numbers uzzy Sets and Systems 3 ( H.Tanaka T.Okuda and K.Asai uzzy mathematical programming Journal of ybernetics and Systems 3 ( J.L.Verdegay A dual approach to solve the fuzzy linear programming problem uzzy Sets and Systems 4( H.J. Zimmermann uzzy programming and linear programming with several objective functions uzzy Sets and Systems (