Solving Fuzzy Linear Programming by Using Revised Tsao s Method

Transcription

1 ustralian Journal of asic and pplied Sciences, 4(1): , 21 ISS Solving Fuzz inear Programming b Using evised Tsao s Method S.H. asseri, M. Sohrabi Department of Mathematical Sciences, Mazandaran Universit, P.O.ox , abolsar, Iran. bstract: aning fuzz numbers plas an important role in linguistic decision maing and some other fuzz application sstems such as management, operations research and etc. Man methods have been proposed to deal ith raning fuzz numbers. In 26, Yong Deng, Zhu Zhenfu and iu Qi presented a method to ran fuzz numbers. The emploed radius of gration ran fuzz numbers; hoever there ere some problems ith the raning method. ecentl the revised Tsao method is proposed for raning fuzz number hich as over-came the short coming of the privous method. On the other hand, usuall the fuzz primal simplex algorithm is designed based on raning fuzz numbers. Hence, in this paper, using the revised Tsao method for solving fuzz linear programming is proposed. Ke ords: triangular fuzz number, raning fuzz number, fuzz linear programming, revised Tsao method, fuzz primal Simplex algorithm. MS subect classification ITODUCTIO In fuzz decision maing problems, the concept of maximizing decision as proposed b ellman and Zadeh (197). This concept as adopted to problems of mathematical programming b Tanaa et al. (1984). Zimmermann (1983) presented a fuzz approach to multi obective linear programming. He also studied the dualit relations in fuzz linear programming. Dubois and Prade (1978) investigated linear fuzz constraints. Tanaa and sai (1984) also proposed a formulation of fuzz linear programming ith fuzz constraints and gave a method for its solution hich based on inequalit relations beteen fuzz numbers. Shaocheng (1994) considered the fuzz linear programming problem ith fuzz constraints and defuzzificated it b first determining an upper bound for the obective function. Further he solved the so-obtained crisp problem b the fuzz decisive set method introduced b Saaa and Yana (1994). Malei et al. (2) considered linear programming problem ith fuzz variables and defined an auxiliar problem (a linear programming including the trapezoidal fuzz numbers in the cost coefficients) for solving them. The solved the auxiliar problem b using Yager s raning method. fter that, Mahdavi-miri and asseri extended their ors to linear raning functions and established the dualit on fuzz number linear programing and linear programming ith fuzz variables. o e are going to propose the revised Tsao s method as a tool for using the fuzz primal simplex algorithm in solving the auxiliar problems or generall the linear programming ith fuzz cost coefficients. This paper is organized in 6 sections. In Section 2, e introduce some preliminaries of fuzz sets. In Section 3, e explain the revised Tsao s method for raning fuzz numbers. Fuzz linear programming is defined in Section 4. n illustrative example is given in Section 5. We conclude in Section 6. 2 Preliminaries: Fuzz numbers are a special ind of fuzz set, hich are normal and convex. lthough these numbers can be described b using man special methods and shapes, triangular and trapezoidal shapes are idel best used for solving practical applications. In this paper, e consider these to fuzz numbers. In man fuzz multiple criteria decision-maing problems, the final scores of alternatives are represented in terms of fuzz numbers. In order to choose best alternatives, e need a method for building a crisp total ordering from fuzz numbers. Man methods for raning of fuzz numbers have been suggested. Each method appears to have some advantages as ell as disadvantages. In fuzz multiple criteria decision maing problems, man triangular Corresponding uthor: S.H. asseri, Department of Mathematical Sciences, Mazandaran Universit, P.O.ox , abolsar, Iran. nasseri@umz.ac.ir (Hadi asseri) 4592

2 ust. J. asic & ppl. Sci., 4(1): , 21 fuzz numbers can intuitivel ran its ordering b draing its curves. If its ordering cannot be raned b figures, e can use man other methods of raning fuzz numbers. Definition 2.1: et U be a universe set. fuzz set of U is defined b a membership function μ (x),x U, indicates the degree of x in. [,1], here Definition 2.2: fuzz subset of universe set U is normal if and onl if sup xu μ (x) = 1, here U is the universe set. Definition 2.3: fuzz subset of universe set U is convex if and onl if μ (x + (1 - )) (μ (x), μ ()), x, U, [, 1], here ^ denotes the minimum operator. Definition 2.4: fuzz set is a fuzz number if and onl if is normal and convex on U. Definition 2.5: triangular fuzz number is a fuzz number ith a pieceise linear membership function μ defined b: x a1, a1 xa2 a2 a1 a3 x ( x), a2 xa3 a3 a2, otherise, hich can be denoted as a triplet ( 1, 2, 3 ). emar 2.1: We consider (,,) as the zero triangular fuzz number. Definition 2.6: The membership function f of can be expressed as f,a xb, b xc f( x) f, c xd, otherise, (2.1) here f : [a, b] [, ] and f : [c, d] [, ]. Since f : [a, b] [, ] is continuous and strictl increasing, the inverse function of f exists. Similarl, since f : [a, b] [, ] is continuous and strictl decreasing, the inverse function of f also exists. The inverse functions of f and f can be denoted b g and g, respectivel. Since f : [c, d] [, ] is continuous and strictl increasing, g : [, ] [a, b] is also continuous and strictl increasing. Similarl, since f : [a, b] [, ] is continuous and strictl 4593

3 ust. J. asic & ppl. Sci., 4(1): , 21 decreasing, g : [, ] [c, d] is also continuous and strictl increasing, g and g are continuous on [, ]; the are integrable on [, ]. That is, both gdand gd 3 The evised Method of aning Fuzz umbers ith an rea eteen the Centroid and Original Points: aning fuzz numbers is important in decision-maing, data analsis operations re-search, artificial intelligence and socioeconomic sstems. Jain, Dubois and Prade (Dubois, D., H. Prade, 1978; Jain,., 1976; Jain,., 1978) introduced the relevant concepts of fuzz numbers. ortolan and Degani (1985) revieed some methods to ran fuzz numbers in 1985, Chen and Hang (1992) proposed fuzz multiple attribute decision maing in 1992, Choobineh and i (1993) proposed an index for ordering fuzz numbers in 1993, Dias (1993) raned alternatives b ordering fuzz numbers in 1993, ee et al. (1994) raned fuzz numbers ith a satisfaction function in 1994, equena et al. (1994) utilized artificial neural netors for the automatic raning of fuzz numbers in 1994, Fortemps and oubens (1996) presented raning and defuzzication methods based on area compensation in 1996, and a et al. (1999) investigated maximizing and minimizing sets to ran fuzz alternatives ith fuzz eights in Hoever, Chu and Tsao (22) proposed a method of raning fuzz numbers ith an area beteen the centroid and original points. Chu and Tsaos method originated from the concepts of ee, i (1988) and Cheng (1998). In 1988, ee and i proposed the comparison of fuzz numbers, for hich the considered mean and standard deviation values for fuzz numbers based on the uniform and proportional probabilit distributions. Then Cheng proposed the coefficient of variance in 1998 to improve ee and i s method based Cheng also proposed a ne distance index to improve the method proposed b Muraami et al (1983). The centroid point of a fuzz number corresponds to an value on the horizontal axis and a value on the vertical axis. The centroid point ( x, ) for a fuzz number : x( ) ( ) b c d a b c b c d a b c ( xf ) dx xdx ( xf ) dx ( f ) dx xdx ( f ) dx ( g ) d ( g ) d ( g ) d ( g ) d here f and f are the left and right membership functions of fuzz number, respectivel. g and g are the inverse functions of f and f, respectivel. The area beteen the centroid point ( x, ) S( ) x., and original point (,) of the fuzz number is then defined as here x and are the centroid points of fuzz number. To ran fuzz numbers, e no that the importance of the degree of representative location is higher than average height. ased on this concept, a revision of Chu and Tsao s method (Wang, Y.J., H.S. ee, 28) is presented as follos. For an to fuzz numbers and, e have folloing situations. if x( ) x( ) if x( ) x( ) 4594 x exist.

4 ust. J. asic & ppl. Sci., 4(1): , 21 If ( ) ( ) If x( ) x( ) If ( ) ( ) If ( ) ( ) Example 3.1: We ant to compare to fuzz numbers (5,2,2) and (7,2,3) ith the revised Tsao s method. Since x( ) = 5. and x( ) = 7.333, thus the fuzz number is bigger than the fuzz number. The folloing figure verifies this result. Fig. 1: 4 Fuzz inear Programming: linear programming problem is defined as: Max z cx s.t. x b x (4.1) T T n T m mn here c ( c,..., c ), b( b,..., b ) and and the vector x 1 n 1 m [ a i ] m is an unnon vector. In this problem, all parameters are crisp. If some parameters be fuzz numbers, e have a ind of fuzz linear programming. In man real situations the cost coefficients in linear programming problems are fuzz numbers. Here, e consider these ind problems including to the triangular fuzz numbers and is defined as folloing: Max z cx s.t. x b x (4.2) 4595

5 ust. J. asic & ppl. Sci., 4(1): , 21 here b, x,, c F( ) m n m n T n. We denote these problems ith FP and an x satisfies the sstem x b, x is said a feasible solution to (4.2). o for defining a basic feasible solution for FP problems, consider the sstem x = b ith x, here is an m n matrix and b is an m vector. Suppose that ran() = m. Partition after possibl rearranging the column of as [, ] that T T T x ( x, x ) x 1 b, x is an m m matrix. The vector, here is called a basic feasible solution (FS) of sstem. Here is called the basic matrix and is called the nonbasic matrix. The x x components of are called the basic variables. If, then x is called a nondegenerate basic feasible solution, and the corresponding fuzz obective value is, here c ( c,..., c ). o, corresponding to ever index, 1 n, define. z c c a 1 z cx 1 m Observe that for an basic index = i, 1 i m, e have -1 a = e i, here e i = (,...,, 1,,...,) T is ith unitvector, since e [ a,..., a,..., a ] e e a i 1 i m i i and so e have: z c c a c c c c c 1 (4.3) Max z c x c x s.t. x x b x, x Definition 4.1: feasible solution x * is an optimal solution for (4.2), if x x b 1 1 cx * cx rite. Therefore,. 1 z( cc) x c b, for all feasible solution x. We can Theorem 4.1: et the FP problem be nondegenerate. basic feasible solution x = -1 b, x = is optimal to (4.2) if and onl if z c, for all, 1 n. Proof: Suppose that x* ( x, x) T T T is a basic feasible solution to (4.2), here x = -1 b, x =. Then z c x c b 1 ` (4.4) On the other hand, for ever feasible solution x to (4.2), e have b xx x (4.5) 4596

6 Hence, e can rerite (4.5) as follos: ust. J. asic & ppl. Sci., 4(1): , xb b x (4.6) Then, for an fuzz basic feasible solution to (4.2), e have z cx c x c x c b ( c a c ) x 1 1 Hence, using (4.3) and (4.4) e have z z* ( z c ) x o, if for all, 1 n e have, then from feasibilit of x e have, (4.7) z c ( z c ). x and then e obtain ( z c ) x. Therefore, it follos from (4.7) that, and so x * i z z * is optimal. For onl if part, let x * be a fuzz optimal basic feasible solution to (4.2). For = i, 1 i z c m, from (4.3) e no that. From (4.7) it is obvious that if for an nonbasic variable x e have z c, then e can enter x into the basic and obtain * (because the problem is z z nondegenerate and x > in the ne basis). This is a contradiction to ( z c ),1 n e must have. z * begin optimal. Hence emar 4.1: ccording the optimalit conditions (Theorem 4.1) e are at the optimal solution if nonbasic variables. On the other hand, if z i c i z c, for, for some nonbasic variables, then the problem is either unbounded or exchange of a basic variable x r and nonbasic variable x can be made to increase the ran of the obective value. Theorem 4.2: If in an FP simplex tableau, there is a column (not is basis) so that then the FP problem is unbounded. z c and i. i = 1,...,m, Proof. See in (Mahdavi-miri,., S.H. asseri, 27; Malei, H.., M. Tata, M. Mashinchi, 2). Theorem 4.3: If in an FP simplex tableau, a exists such that z c and there exists a basic index i such that i >, then a pivoting ro r can be found so that pivoting on Y r ields a feasible tableau ith a corresponding nondecreasing (increasing under nondegenerac assumption) fuzz obective value. Proof. See in (Mahdavi-miri,., S.H. asseri, 27; Malei, H.., M. Tata, M. Mashinchi, 2). emar 4.2: If exists such that z c and the problem is not unbounded then r can be chosen so that 4597

7 ust. J. asic & ppl. Sci., 4(1): , 21 ro r io min{ i,1 im} i x br In order to replace in the basis b x, resulting in a ne basis ( a,..., a, a, a,..., a ) 1 r1 r1 m The ne basis is primal feasible and the corresponding fuzz obective value is nondecreasing (increasing under nondegenerac). It can be shon that the ne simplex tableau is obtained b pivoting on r, that is, doing Gaussian elimination on the th column using the pivot ro r, ith the pivot r, to transform the th column to the unit e r. It is easil seen that the ne fuzz obective value is: ro oo oo o oo r emar 4.3: If x enters the basis and x r leaves the basis, then pivoting on r in the simplex tableau is stated as follos: First divide ro r b r (Since r > ). So for i =, 1,..., m and i r, update the ith ro b adding to it - r times the ne rth ro. 5 umerical Example: Example 5.1: Consider the folloing FP problem Max z (3,2,2) x (5,2,4) x s.t. x13x2 8 x1x2 8 x x First exchange into normal form: x 3x x x 2x x The FP simplex tableau corresponding to = I, 1 = 3, 2 = 4. 1 z 1 c1 ( 3,2,2) 2 z 2 c2 ( 5,4,2) Since and, then, usingevised Tsao method and 1 2 for ordering, e have and hence x 2 enters the basis and the leaving variable is 1 2 x 3. Pivoting on 23 = 3, results in the next table: 4598

8 ust. J. asic & ppl. Sci., 4(1): , 21 gain using evised Tsao method conclude that and, for all i {1} and hence 1 x 1 enters the basis and the leaving variable is x 4. Pivoting on ote that, this table is optimal, because the obective function is z (,, ) ( z c, z c ) 14 3, results in the next table:. Therefore, the fuzz optimal value of 6 Conclusion: evised Tsao method is one of the most efficient methods for raning fuzz numbers. In this paper, e suggested using this method in the fuzz primal simplex here e need to evaluate the optimalit condition in each iteration. The mentioned approach ill be useful hen one have to use the fuzz dual simplex too. CKOWEDGMET The first author thans to the esearch Center of lgebraic Hperstructures and Fuzz Mathematics, abolsar, Iran and ational Elite Foundation, Tehran, Iran for their supports. EFEECES ortolan, G.,. Degani, revie of some methods for raning fuzz numbers, Fuzz Sets and Sstems, 15: ellman,.e.,.. Zadeh, 197. Decision- maing in a fuzz environment, Management Science, 17: Chen, S.J., C.. Hang, Fuzz Multiple ttribute Decision Maing, Springer, e Yor. Cheng, C.H., ne approach for raning fuzz numbers b distance method, Fuzz Sets and Sstems, 95: Choobineh, F., H. i, n index for ordering fuzz numbers, Fuzz Sets and Sstems, 54: Chu, T.C., C.T. Tsao, 22. aning fuzz numbers ith an area beteen the centroid point and the original point, Computers and Mathematics ith pplications, 43: Dias, G., aning alternatives using fuzz numbers: computational approach, Fuzz Sets and Sstems, 56: Dubois, D., H. Prade, Operations on fuzz numbers, The International Journal of Sstems Sciences, 9: Fortemps, P., M. oubens, aning and defuzzification methods based on area compensation, Fuzz Sets and Sstems, 82(3): Hibbeler,.C., 2. Mechanics of Materials, Prentice Hall, e Jerse. 4599

9 ust. J. asic & ppl. Sci., 4(1): , 21 Jain,., Decision-maing in the presence of fuzz variables, IEEE Transactions on Sstems, Man and Cbernetics, 6: Jain,., procedure for multi-aspect decision maing using fuzz sets, The International Journal of Sstems Sciences, 8: 1-7. ee, K.M., C.H. Cho, H. ee-kang, aning fuzz values ith satisfaction function, Fuzz Sets and Sstems, 64: ee, E.S.,.J. i, Comparison of fuzz numbers based on the probabilit measure of fuzz events, Computers and Mathematics ith pplications, 15: Mahdavi-miri,., S.H. asseri, 27. Dualit results and a dual simplex method for linear programming problems ith trapezoidal fuzz variables, Fuzz Sets and Sstems. 158: Mahdavi-miri,., S.H. asseri, 26. Dualit in fuzz number linear programming b use of a certain linear raning function, pplied Mathematics and Computation, 18: Malei, H.., M. Tata, M. Mashinchi, 2. inear programming ith fuzz variables, Fuzz Sets and Sstems, 19: Muraami, S., S. Maeda, S. Imamura, Fuzz decision analsis on the development of centralized regional energ control sstem, in: IFC Smp. on Fuzz Inform. Knoledge epresentation and Decision nal, a,., D.. Kumar, aning alternatives ith fuzz eights using maximizing set and minimizing set, Fuzz Sets and Sstems, 15: equena, I., M. Delgado, J.I. Verdaga, utomatic raning of fuzz numbers ith the criterion of decision-maer learntb an artificial neural netor, Fuzz Sets and Sstems, 64: Saaa, M., H. Yana, Interactive decision maing for multi- obective linear fractional programming problems, Cbernetics Sstem, 16: Shoacheng, T., Interval number and Fuzz number linear programming, Fuzz Sets and Sstems, 66: Tanaa, H., K. sai, Fuzz linear programming problems ith fuzz numbers, Fuzz Sets and Sstems, 13: 1-1. Tanaa, H., T. Ouda, K. sai, On fuzz mathematical programming, J. Cbernetics, 3: Wang, Y.J., H.S. ee, 28. The revised method of raning fuzz numbers ith an area beteen the centroid and original points, Computers and Mathematics ith pplications, 55: Zimmermann, H.J., Fuzz mathematical programming, Comput. Ops. es., 1(4):