Modified Method for Solving Fully Fuzzy Linear Programming Problem with Triangular Fuzzy Numbers

Transcription

1 Int. J. Res. Ind. Eng. Vol. 6, No. 4 (7) 93 3 International Journal of Research in Industrial Engineering Modified Method for Solving Full Fuzz Linear Programming Problem with Triangular Fuzz Numbers S. K. Das Department of Mathematics, National Institute of Technolog Jamshedpur, Jharkhand 834, India. A B S T R A C T The fuzz linear programming problem has been used as an important planning tool for the different disciplines such as engineering, business, finance, economics, etc. In this paper, we proposed a modified algorithm to find the fuzz optimal solution of full fuzz linear programming problems with equalit constraints. Recentl, Ezzati et al. (Applied Mathematical Modelling, 39 (5) ) suggested a new algorithm to solve full fuzz linear programming problems. In this paper, we modified this algorithm and compare it with other eisting methods. Furthermore, for illustration, some numerical eamples and one real problem are used to demonstrate the correctness and usefulness of the proposed method. Kewords: Linear programming problem, full fuzz linear programming, multi-obective linear programming, triangular fuzz numbers. Article histor: Received: 9 October 7 Accepted: 9 December. Introduction Modeling and optimization under a fuzz environment is called fuzz modeling and fuzz optimization. Fuzz linear programming is one of the most frequentl applied in fuzz decision making techniques. Although, it has been investigated and epanded for more than decades b man researchers and from the varies point of views, it is still useful to develop new approaches in order to better fihe real world problems within the framework of fuzz linear programming. Traditional optimization techniques and methods have been successfull applied for ears to solve problems with a well-defined structure. Such optimization problems are usuall well formulated b crispl specific obective functions and specific sstem of constraints, and solved b precise mathematics. Unfortunatel, real world situations are often not deterministic. There eist various tpes of uncertainties in social, industrial and economic sstems, such as Corresponding author address: cool.sapankumar@gmail.com DOI:.5/rie

2 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) randomness of occurrence of events, imprecision and ambiguit of sstem data and linguistic vagueness, etc., which come from man was, including errors of measurement, deficienc in histor and statistical data, insufficienheor, incomplete knowledge epression, and the subectivit and preference of human udgment, etc. As talked about b Zimmermann [], several tpes of uncertainties can be categorized as stochastic uncertaint and fuzziness. Stochastic uncertaint pertains to the uncertaint of events of phenomena or events. Its features lie in the fachat descriptions of information are crisp and well defined, however, the var in their frequenc of occurrence. Therefore, sstems with this tpe of uncertaint are called stochastic sstems, which can be solved b stochastic optimization techniques using probabilit theor. In some other circumstances, the decision-maker does nohink the frequentl used probabilit distribution is alwas appropriate, particularl when the information is vague, relating to human language and behavior, imprecise/ambiguous sstem data, or when the information could not be described and defined well due to limited knowledge and deficienc in its understanding. Such tpes of uncertaint are called fuzziness. It cannot be formulated and solved effectivel b traditional mathematics-based optimization techniques nor did probabilit base stochastic optimization approaches. The Fuzz Linear Programming (FLP) problem can be introduced in several was such as (i) the constraints are inequalities with fuzz technological coefficients/fuzz decision variables (ii) the constraints are equalities with fuzz technological coefficients/fuzz decision variables equalities (iii) the goals ma be fuzz (iv) all parameters of LP ma be in terms of fuzz numbers. Man researchers proposed various methods for solving fuzz linear programming problem [- ]. Lotfi et al. [7] proposed full fuzz linear programming problems where all parameters and variables were triangular fuzz numbers. The proposed method is to find the fuzz optimal solution with equalit constraints. Ebrahimnead et al. [4] proposed a method for solving fuzz linear programming problems where all coefficients of obective functions and right hand side are represented b smmetric trapezoidal fuzz numbers. The converhe fuzz linear programming problem into an equivalent crisp linear programming and then solved the LP b standard primal simple algorithm. Kumar et al.[6] proposed a method for solving full fuzz linear programming problem with equalit constraints. The transformed FLP in to equivalent crisp linear programming problem and used regular simple method. Iskander [] introduced a stochastic fuzz linear multi obective programming problem and transformed io a stochastic fuzz linear programming problem using a fuzz weighted obective function. The used chanceconstrained approach as well as the α-level methodolog to transform the stochastic fuzz linear programming problem to its equivalent deterministic-crisp linear programming problem. Veeramani and Duraisam [] discussed the full fuzz linear programming (FFLP) problems in which all the parameters and variables are triangular fuzz numbers. The proposed a new approach of solving FFLP problem using the concept of nearest smmetric triangular fuzz

3 95 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers number approimation with preserve epected interval. The dual problem of the LP with trapezoidal fuzz number and some dualit results to solve fuzz linear programming problem was introduced in [3]. Ezzati et al. [4] introduced a new algorithm to solve full fuzz linear programming problem. The have converted the problem to a multi-obective linear programming (MOLP) problem and then it was solved b a new leicographic ordering method. In this paper, we modified the mentioned algorithm and proposed a new method for finding the fuzz optimal solution of FFLP problem. This paper is organized as follows: Some basic definitions and notations are present in Section. In Section 3, we present a modified method to solve FFLP problem. Some eamples are provided for testing the new proposed method in net section and Kumar et al. method [6], Ezzati et al. method [4] and the proposed method will be compared each other. Finall, in Section 5, concluding remarks are present.. Preliminaries In this section, we have presented some basics concepts of fuzz sets and triangular fuzz number, which was ver useful in this paper. Definition. [5, 6] Let X denotes a universal set. Then a fuzz subset A of X is defined b its membership function : X [, ]; which assigned a real number ( ) in the interval [, A ], to each element X, where the values of ( ) at shows the grade of membership of in A. A fuzz subset A can be characterized as a set of ordered pairs of element and grade ( ) A and is often written A (, ( )) : X is called a fuzz set. A Definition. [5, 6]A fuzz number A ( b, c, a) (where b c a) is said to be a triangular fuzz number if its membership function is given b A ( A ( b), b) ) ( a), a) A b c, c a, Definition.3 [5, 6]A triangular fuzz number ( b, c, a) is said to be non-negative fuzz number if and onl if b. Definition.4 [5, 6]Two triangular fuzz number A ( b, c, a) and B ( e, f, d) are said to be equal if and onl if b e, c f, a d.

4 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Definition.5[7] A ranking is a function R : F( R) R where F(R)is a set of fuzz number defined on set of real numbers, which maps each fuzz number into the real line, where a natural order eists. Let A ( b, c, a) is a triangular fuzz number then b c a ( A). 4 Definition.6 [5, 6] Let A ( b, c, a), B ( e, f, d) be two triangular fuzz numbers then: A B ( b, c, a) ( e, f, d) ( b e, c f, a d), A B ( b, c, a) ( e, f, d) ( b d, c f, a e), Let A ( b, c, a) be an triangular fuzz number and B ( e, f, d) be a non-negative triangular fuzz number then, ( be, cf, ad) A B AB ( bd, cf, ad) ( bd, cf, cd ) if b, if b, a, if c, Definition.7 Let A ( b, c, a), B ( e, f, d) be two triangular fuzz numbers. We sa that A is relativel less than B, if and onl if b e or b e and ( b c) ( e f ) or b e, ( b c) ( e f ) and ( a b) ( d e). Note: It is clear from the Definition.7 that b e, ( b c) ( e f ) and ( a b) ( d e) if and onl if A B. 3. Full Fuzz Linear Programming Problem We consider a standard form of FFLP problem where all parameters and variables are fuzz triangular numbers as follows: Ma c t s.t A b, (). Where t c [ c ] is b n matri; [ ] is n b matri; [ A a i ] is m b n matri; b [ b i ] is m b matri; Here all the parameters c, d,, a are set of fuzz numbers. * * * * Let (,, z ) be an eact optimal solution of the problem (). Now we are going to introduce a new algorithm to find eact optimal solution. The steps of the proposed algorithm are given as follows: i

5 97 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers Step. The problem () can be written as t Ma ( ), ), z) ) s.t. ( ( A),( A), ( Az) ( b ), ( b ), ( b3 ) ), (),, z. Step. In problem () ma be written as t Ma ( ), ), z) ) s.t. ( A) ( b ), (( A) ( A) ( b ) ( b )), ( ( A) ( Az) ( b ) ( b3 ) ), (3), z,. Step 3. Based on Definition.6 the problem (3) can be transformed into MOLP problem with three crisp linear programming problem as follows: Ma c t Ma ) ) Ma ) z) (4) s.t. ( A) ( b ), (( A) ( A) ( b ) ( b )), ( ( A) ( Az) ( b ) ( b3 ) ),, z,. Step 4. The leicographic method will be used to obtain optimal solution of problem (4), we get Ma c t s.t. ( A) ( b ), (( A) ( A) ( b ) ( b )), ( ( A) ( Az) ( b ) ( b3 ) ), (5), z,. Step 5. Using Steps 3-4 we have, Ma ) ) s.t. c t =t, ( A) ( b ), (( A) ( A) ( b ) ( b )), ( ( A) ( Az) ( b ) ( b3 ) ), (6), z,. Where t is the optimal solution of problem (5). Step 6. Solve this problem used in Step 5 as follows: Ma ) z) s.t. ) ) =u, c t ( =t, A) ( b ),

6 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) (( A) ( A) ( b ) ( b )), (7) ( ( A) ( Az) ( b ) ( b3 ) ),, z,. Where u is the optimal solution of problem (6). In Step 6, we get an eact optimal solution, which is equivaleno the problem (). * * * * Theorem 3. If ( ),( ),( z ) be an optimal solution of problems (5-7), then it is also an eact optimal solution of problem (). * * * * Proof. B the method of contradiction, let ( ),( ),( z ) be an optimal solution of (5)-(7), but it is nohe eact optimal solution of problem (). Let us we consider ( ),( ),( z ), such that in the case of maimization t * t ), * t * t t ), z ) ), t ), z ). Based on Definition.7, we have three conditions as follows: t * t Case (i) In case of maimization, we consider let ) ). Also, with respeco the assumption we have: ( A ) ( b ), ( A ) ( A ) ( b ) ( b ), A ) ( Az ) ( b ) ( b ), ( 3, z,. Therefore, ( ),( ),( z ) is a feasible solution of problem (5) in which the obective value in * * * ( ), ( ), ( z ) is greater than the obective value in ( ), ( ), ( z ). However, it is contradiction. t * t Case(ii) In case of maimization, let us we consider ) ( c ), and t * t * t t ) ( c ) ) ( c ) Also, with respeco the assumption we have: ( A ) ( b ), ( A ) ( A ) ( b ) ( b ), A ) ( Az ) ( b ) ( b ), ( 3, z,. Therefore, ( ),( ),( z ) is a feasible solution of problem (6) in which the obective value in * * * ( ), ( ), ( z ) is less than the obective value in ( ), ( ), ( z ). However, it is contradiction.

7 99 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers t * t Case (iii) In case of maimization, let us we consider ) ( c ), t * t * t t t * t * t t ) ) ( c ) ) and ) z ) ) z ). In addition, with respeco the assumption we have: A ) ( b ), A ) ( A ) ( b ) ( b ), A ) ( Az ) ( b ) ( b ), ( ( ( 3, z,. Therefore, ( ),( ),( z ) is a feasible solution of problem (7) in which the obective value in * * * ( ), ( ), ( z ) is greater than the obective value in ( ), ( ), ( z ). However, it is contradiction. * * * * Therefore ( ),( ),( z ) is an eact optimal solution of problem (). The flow chart describes the procedure of the proposed method as shown in Figure. Figure. Flowchart for solving FFLP problem.

8 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Numerical Eample In this section we consider some eamples to illustrate our proposed method and compare it. Eample 4.. [6]: Let us consider the following FFLP problem and solve it. Maimize (,6,9) (,3,8), subeco (,3,4) (,,3) (6,6,3), (8) (,,) (,3,4) (,7,3),,. Solution: Let us consider (,, ) and (,, ) then the given FFLP problem (8) z can be written as follows: Maimize (,6,9) (,, z) (,3,8) (,, ), Subeco (,3, 4) (,, z) (,,3) (,, ) (6,6,3), (9) (,,) (,, z) (,3,4) (,, ) (,7,3), (,, z),(,, ). Using Step, the problem (9) can be converted in to MOLP problem as follows: Ma Ma 6 3 Ma 9z 8 Subeco 6,, 3, () 3 6, 4z 3 36, z 4 3,, z,,,,. Using Step 3, the problem () can be converted in to MOLP problem as follows: Ma, Ma 6 3, Ma 9z 8, Subeco 6,, 3, () 6, 3

9 3 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers 4z 3 z 4z, z 36, 3,,,,,. Using Steps 4-6, the optimal solution of problem () is (,, z) = (.6,, 3), (,, ) =(.6, 5, 6), Now the optimal value of the problem () ma be written as ) proposed method= ( t ), )= (6.8, 7, 75). In Ezzati s method [4] the optimal solution of the obective functions are (,, ) = (.6,, 3), (,, ) =(.6, 5, 6). z The obective value is ) = ( t ), )= (6.8, 7, 75). In Kumar s method [6] the optimal solution of the obective functions are (,, ) = (,, 3), (,, ) =(4, 5, 6). z The obective value is: ) = ( t ), ) = (9, 7, 75). B comparing the results of proposed method with Ezzati s and Kumar s method, based on Definition.7, we can conclude that our result is equal to Ezzati result, because: 9= t ) Kumar s method> t ) proposed method= t ) Ezzati method=6.8, 84= t ) Kumar s method> ) z) proposed method= ) z) Ezzati method=8.8, )Ezzati method = )proposed method=(6.8,7,75)< t ) Kumar s method=(9, 7, 75). Eample 4. [8]: Consider the following FFLP problem and find the obective value functions. Min (,3,9) (,,8), Subeco (,3,5) (,3,4) (,9,), () (,,3) (,3, 4) (,8,8),,. Solution: Let us consider (,, ) and (,, ) then the given FFLP problem () can be written as follows: z Maimize,3,9) (,, z ) (,,8) (,, ), (

10 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Subeco,3,5) (,, z ) (,3,4) (,, z ) (,9,), ( (,,3) (,, z) (,3, 4) (,, ) (,8,8), (,, z),(,, ). Using Step, the problem (3) can be converted in to MOLP problem as follows: (3) Ma, Ma 3, Ma 9z 8, Subeco,, 3 3 8, (4) 3 7, 5z 4 3, 3z 4 9,, z,,,,. Using Steps 4-6, the optimal solution of problem (4) is (,, z) = (,, ), (,, ) = (.5,, 3). Now the optimal value of the problem () ma be written as ) proposed method= ( t ), ) = (, 7, 4). In Ezzati s method the optimal solution of the obective functions are: (,, z) = (,, ), (,, ) = (.5,, 3). Now the optimal value of the problem () ma be written as ) Ezzati s method= ( t ), ) = (, 7, 4). In Kumar s method the optimal solution of the obective functions are: (,, z) = (,, ), (,, ) = (.5,, 3). Now the optimal value of the problem () ma be written as ) Kumar s method= ( t ), ) = (, 7, 4). B comparing the results of proposed method with Ezzati s and Kumar s method, based on Definition.7, we can conclude that our result is more effective, because: t ) proposed method= t ) Ezzati method= t ) Kumar s method= ) z) proposed method= ) z) Ezzati method= t ) )Ezzati method = )proposed method= t ) Kumar s method=(, 7, 4). Kumar s method=43

11 33 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers Eample 4.3 [4]: Consider the FFLP problem Ma c t s.t. A b,. where c, A and b are given as follows: (,5,7) (,6,) (8,,3) (,,3) (9,,3) (,5,7) c, A (,4,7), (,4,6) (4,8,9) (4,7,) (3,4,8) (,,4) (7.75, 4.75,573.75) b, (385.5,539.5, 759.5) Using Step, the problem ma be written as follows: Ma 3 4, ,7z 7z3 4z4 s.t z 3 3z3 7z4 6z 9 z3 8z , 4.75, 539.5, , 759.5,,,, z,,,3,4. Using Step, problem ma be written as Ma 3 4, Ma , Ma 7z 7z3 4z4, s.t , z 3 3z3 7z z3 8z4 4, 845.5, 54, 45,,,,, z,,,3,4. Using Step 3, problem ma be written as

12 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Ma 3 4, Ma , Ma 3 4 7z 7z3 4z4, s.t , z 3 3z3 7z z 9 z3 8z4 4, 845.5, 54, 45,,,,, z,,,3,4. After solving this problem b using Steps 4-6, we have (7.7,7.7, 7.7) (.6,.6,.6), (4.,9.5,5.85) (6.49, 6.49,6.49) Now the optimal value of the problem () ma be written as ) proposed method= ( t ), )= (34.8, 5.8, 75.9). In Ezzati s method the optimal solution of the obective functions are (7.7,7.7, 7.7) (.6,.6,.6), (4.64,9.97,6.36) (6.36, 6.36,6.36) Now the optimal value of the problem () ma be written as ) Ezzati s method= ( t ), ) = (34.58, 59.79, 74.37). In Kumar s method the optimal solution of the obective functions are (5.8,5.8, 5.8) (.4,.4,9.), (6,.5,.5) (6.49, 6.49,6.49) ) Kumar s method= ( t ), ) = (3.83, 53.3, 74.5). B comparing the results of proposed method with Ezzati s and Kumar s method, based on Definition.7, we can conclude that our result is effective, because

13 35 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers 34.8 = t ) proposed method> t ) Ezzati method=34.58> t ) Kumar s method=3.83 ) z) Kumar s method>.7= ) z) proposed method> ) z) Ezzati method=8.95, (3.83, 53.3, 74.5) = ) Kumar s method>(34.58, 59.79, 74.37)= )Ezzati method> )proposed method=(34.8, 5.8, 75.9). In Figure, we compare the membership function for the proposed method and eisting methods. It shows thahe proposed method is better than the eisting methods. Eample 4.4 [4]. Dali Compan is the leading producer of soft drinks and low-temperature foods in Taiwan. Currentl, Dali plans to develop the south-east Asian market and broaden the visibilit of Dali products in the Chinese market. Notabl, following the entr of Taiwan to the world trade Organization, Dali plans to seek strategic alliance with prominent international companies, and introduced international bread to lighten the embedded future impact. In the domestic soft drinks market, Dali produces tea beverages to meet demand from four distribution centers in Taichung, Chiai, Kaohsiung, and Taipei, with production being based ahree plants in Changhua, Touliu, and Hsinchu. According to the preliminar environmental information, Table summarizes the potential suppl available from these three plants, the forecast demand from the four distribution centers, and the uniransportation costs for each route used b Dali for the upcoming season. The environmental coefficients and related parameters generall are imprecise numbers with triangular possibilit distributions over the planning horizon due to incomplete or unobtainable information. For eample, the available suppl of the Changhua plant is (7., 8, 8.8) thousand dozen bottles, the forecast demand of the Taichung distribution center is (6., 7, 7.8) thousand dozen bottles, and the transportation cost per dozen bottles from Changhua to Taichung is ($8, $, $.8). Due to transportation costs being a maor epense, the management of Dali is initiating a stud to reduce these costs as much as possible.

14 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Figure. Membership function: Proposed method vs eisting methods. Table. Data of Eample 4.4 (in U. S. dollar) Source Destination Suppl (dozenbottles) Taichung Chiai Kaohsiung Taipei Changhua ($8, $, $.8) ($.4, $, $4) ($8, $, $.6) ($8.8, $, $) (7.,8, 8.8) Touliu ($4, $5, $6) ($8., $, $) ($, $, $3) ($6, $8, $8.8) (, 4, 6) Hsinchu ($8.4,$,$) ($9.6,$,$3) ($7.8,$,$.8) ($4,$5,$6) (.,,3.8) Demand (6.,7,7.8) (8.9,,.) (6.5,8,9.5) (7.8,9,.) This real world problem can be formulated to the following FFLP problem: Min((8,,.8) +(.4,,4) +(8,,.6) 3 +(8.8,,) 4 +(4,5,6) +(8.,,) +(,,3) 3 +(6,8,8.8) 4 +(8.4,,) 3 +(9.6,,3) 3 +(7.8,,.8) 33 +(4,5,6) 34 ) s.t. 3 4 (7.,8,8.8 ), 3 4 (,4,6), (.,,3.8), 3 (6.,7,7.8), 3 (8.9,,.), (6.5,8,9.5), (7.8,9,.),

15 37 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers Using Step, problem ma be written as Min , ,.8z 4.6z 3 z z.4 z 3 3z3.8z 33 6z34 s.t. 7., , z3 z z3 z33 z , 3 7, z3 7.8, 3 8.9, 3, z3., , , 3 3 z33 9.5, , , 4 4 z34., z z z z z z z z i i i i i 8.8,, 4, 6,.,, 3.8,, i,,3,,,3,4,, i,,3,,,3,4,, i,,3,,,3,4.

16 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) After solving this problem b using Steps -6, we have: (6., 7, 7.8), (,, ), (,,), (,, ), (,, ), (,,.3),. (4.,5,5.5), (7.8, 9,.), (,,), (8.9,,), (.3,,.8), (,, ), Now the optimal value of the problem () ma be written as ) proposed method= ( t ), ) = (4.98, 34, ). In Ezzati s method the optimal solution of the obective functions are (6.,7,7), (,,), (,,), (,,.8), (,,.), (4.,5,5.9),, (7.8,9,9), (,,.8), (8.9,,), (.3,,.6), (,,.4). Now the optimal value of the problem () ma be written as ) Ezzati s method= ( t ), ) = (4.98, 35, ).

17 39 Modified method for solving full fuzz linear programming problem with triangular fuzz numbers In Kumar s method the optimal solution and optimal valu of the problem is (6.,7,7.8), (,,), (,,), (,,), (,,), (,,), (4., 5,5.8), (7.8, 9,.), (,,), (8.9,,.), (.3,,.7), (,,), ) Kumar s method= ( t ), )= (4.98, 35, ). B comparing the results of proposed method with Ezzati s and Kumar s method, based on Definition.7, we can conclude that our result is effective, because: t ) Proposed method= t ) Ezzati method= t ) Kumar s method= = ) z) proposedmethod> ) z) Ezzati method= ) z) Ezzati method= (4.98, 35, )= ) Kumar s method=(4.98, 35, )= )Ezzati method > )proposed method=(4.98, 34, ). Figure 3. Membership function: Proposed method vs eisting methods. In Figure 3, we compare the membership function for the proposed method and eisting methods. It shows thahe proposed method is better than the eisting methods.

18 S. K. Das / Int. J. Res. Ind. Eng 6(4) (7) Advantages of the Proposed Method The proposed method has been studied in FFLP problems with fuzz coefficients. It is pointed ouhahe proposed method is no restriction of all variables and parameters and the obtained results are satisfied all the constraints. We used in fuzziness of the obective function is neglected b linear ranking function in decision makers. Hence, it is ver eas and comfortable for applied in real life application as compared as to eisting methods. 6. Conclusions In this paper, we proposed a modified technique to solve the FFLP problem in order to obtain the fuzz optimal solution. The main idea behind this paper is LP problem with triangular fuzz numbers is converted into MOLP problem and solved as a crisp linear programming problem. We compare the proposed technique with eisting of two method and also draw graph we shown the eas, applicabilit of our proposed method which we studied in the above result of two eamples and real life problems. The proposed method can be etended to multi-obective linear programming problem, fractional programming problem and multi-obective linear fractional programming problem etc. Moreover, a stochastic approach of the above problem can be studied and the comparison between the approached can be carried out. References [] Zimmermann, H. J. (). Fuzz seheor and its applications. Springer Science & Business Media. [] Bakasoğlu, A., & Subulan, K. (5). An analsis of full fuzz linear programming with fuzz decision variables through logistics network design problem. Knowledge-Based sstems, 9, [3] Dehghan, M., Hashemi, B., & Ghatee, M. (6). Computational methods for solving full fuzz linear sstems. Applied mathematics and computation, 79(), [4] Ebrahimnead, A., & Tavana, M. (4). A novel method for solving linear programming problems with smmetric trapezoidal fuzz numbers. Applied mathematical modelling, 38(7), [5] Khan, I. U., Ahmad, T., & Maan, N. (3). A simplified novel technique for solving full fuzz linear programming problems. Journal of optimization theor and applications, 59(), [6] Khan, I. U., Ahmad, T., & Maan, N. (3). A simplified novel technique for solving full fuzz linear programming problems. Journal of optimization theor and applications, 59(), [7] Lotfi, F. H., Allahviranloo, T., Jondabeh, M. A., & Alizadeh, L. (9). Solving a full fuzz linear programming using leicograph method and fuzz approimate solution. Applied mathematical modelling, 33(7), [8] Kumar, A., Kaur, J., & Singh, P. (). A new method for solving full fuzz linear programming problems. Applied mathematical modelling, 35(), [9] Shamooshaki, M. M., Hosseinzadeh, A., & Edalatpanah, S. A. (4). A new method for solving full fuzz linear programming with lr-tpe fuzz numbers. International ournal of data envelopment analsis and* Operations Research*, (3),

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