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Applied Mathematical Scie nces, Vol. 7, 2013, no. 54, 2661 - 2673 HIKARI Ltd, www.m - hikari.com An Approach for Solving Fuzzy Transportation Problem Using Octagonal Fuzzy Numbers S. U. Malini Research Scholar, Stella Maris College (Autonomous), Che nnai Felbin C. Kennedy Research Guide, Associate Professor, Stella Maris College, Chennai. Copyright © 2013 S. U. Malini and Felbin C. Kennedy . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, a general fuzzy transportation problem model is discussed. There are several approaches by different authors to solve su ch a problem viz., [1,2 ,3,6,7,8 ] . We introduce octagonal fuzzy numbers using which we develop a new model to solve the problem . By defining a ranking to the octagonal fuzzy numbers, it is possible to compare them and using this we convert the fuzzy valued transportation problem (cost, supply and demand appearing as octagonal fuzzy numbers) to a crisp valued transportation problem, which then can be solved using the MODI Method. W e have proved that the optimal value for a fuzzy transportation problem, when s olved using octagonal fuzzy number gives a much more optimal value than when it is solved using trapezoidal fuzzy number as done by Basirzadeh [ 3 ] which is illustrated through a numerical example. Mathematics Subject Classification: 03B52 , 68T27, 68T37, 9 4D05 Keywords : Octagonal Fuzzy numbers, Fuzzy Transportation Problem. 1 Intro duction The transportation problem is a special case of linear programming problem, which enable us to determine the optimum shipping patterns bet ween origins and 2662 S. U. Malini and Felbin C. Kennedy destinations. Suppose that there are m origins and n destinations. The solution of the problem will enable us to determine the number of units to be transported from a particular origin to a particular destination so that the cost incurred is least or the time taken is least or the profit obtained is maximum. Let a i be the number of units of a prod uct available at origin i , and b j be the number of units of the product required at destinatio n j . Let c ij be the cost of transporting one unit from origin i to destination j and l et x ij be the amount of quantity transported or shipped from origin i to destination j . A fuzzy transportation problem is a transportation problem in which the transport ation costs, supply and demand quantities are fuzzy quantities. Michael [ 11 ] has proposed an algorithm for solving transportation problems with fuzzy constraints and has investigated the relationship between the algebraic structure of the optimum solution of the deterministic problem and its fuzzy equivalent. Chanas et al [ 4 ] deals with the transportation problem w here i n fuzzy supply values of the deliverers and th e fuzzy demand values of the receivers are analysed. For the solution of the problem the tech nique of parametric programming is used. Chanas and Kuchta [ 5 ] have given a definition for the optimal solution of a transportation problem and as also proposed an algorithm to determine the optimal solution . Shiang - Tai Liu and Chiang Kao [1 4 ] have given a pr ocedure to derive the fuzzy objective value of the fuzzy transportation problem based on the extension principle. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constrain ts. Nagoor Gani and Abdul Razack [1 2 ] obtained a fuzzy solution for a two stage cost minimising fuzzy transportation problem in which supplies and demands are trapezoidal fuzzy numbers. Pandian et al [ 1 3 ] proposed a method namely fuzzy zero point method for finding fuzzy optimal solution for a fuzzy transportation problem where all parameters are trapezoidal fuzzy numbers. In a fuzzy transportation problem, all parameters are fuzzy numbers. Fuzzy numbers may be normal or abnormal, triangular or trapezoida l or it can also be octagonal. Thus, they cannot be compared directly. Several methods were introduced for ranking of fuzzy numbers, so that it will be helpful in comparing them. Basirzadeh et al [ 2 ] have also proposed a method for ranking fuzzy numbers u sing α – cuts in which he has given a ranking for triangular and trapezoidal fuzzy numbers . $ rnking using α - cut is introduced on octagonal fuzzy numbers. Using this ranking the fuzzy transportation problem is converted to a crisp valued problem, which can be solved using VAM for initial solution and MODI for optimal solution. The optimal solution can be got either as a fuzzy number or as a crisp number. 2 . Octagonal fuzzy numbers Two relevant classes of fuzzy numbers, which are frequently used in practical purposes so far, re “ triangular a nd trpezoidl fuzzy numbers”. I n this paper we introduce o ctagonal fuzzy numbers which is much useful in solving Solving fuzzy t ransportation p roblem 2663 fuzzy transportation problem ( FTP ) . Definition 2 .1: An o ctagonal fuzzy number denoted by ω is defined to be the ordered quadruple ω ଵ ݎ ݏ ଵ ݐ ݏ ଶ ݐ ଶ ݎ , for ݎ ሾ Ͳ ሿ , and t ሾ ሿ where (i) ଵ ݎ is a bounded left continuous non decreasing function over [ , ω 1 ], [ 0 ω 1 k ] (ii) ݏ ଵ ݐ is a bounded left cont inuous non decreasing function over [ k, 2 ], [ k ω 2 ] (iii) ݏ ଶ ݐ is bounded left continuous non increasing function over [ k, ω 2 ], [ k ω 2 ] (iv) ଶ ݎ is bounded left continuous non increasing function over [ ,ω 1 ]. [ 0 ω 1 k ] Remark 2 .1: If ω = 1 , then th e above - defined number is called a normal o ctagonal fuzzy number. The octagonal number s we consider for our study is a subclass of the octagonal fuzzy numbers (Definition 2.1) defined as follows: Definition 2 .2: A fuzzy number is a normal octagon al fuzzy number denoted by ( a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,a 7 ,a 8 ) where a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 are real numbers and its membership function ( x ) is given below μ à (x) = ە ۖ ۖ ۖ ۖ ۖ ۔ ۖ ۖ ۖ ۖ ۖ ۓ Ͳ ݎ ܽ ଵ ቀ ି à°­ à°® ି à°­ ቁ ݎ ܽ ଵ ܽ ଶ ݎ ܽ ଶ ܽ ଷ ͳ ቀ ି à°¯ à°° ି à°¯ ቁ ݎ ܽ ଷ ܽ ସ ͳ ݎ ܽ ସ ܽ ହ ͳ ቀ à°² ି à°² ି à°± ቁ ݎ ܽ ହ ܽ ଺ ݎ ܽ ଺ ܽ ଻ ቀ à°´ ି à°´ ି à°³ ቁ ݎ ܽ ଻ ܽ ଼ Ͳ ݎ ܽ ଼ where 0 k 1 . Remark 2 . 2 : If k = 0 , the octagonal fuzzy number reduces to the trapezoidal number ( a 3 ,a 4 ,a 5 ,a 6 ) and if k= 1 , it reduces to the trapezoidal number ( a 1 ,a 4 ,a 5 ,a 8 ). Remark 2. 3 : According to the above mentioned defin ition , octagonal fuzzy number ω i s the ordered quadruple ଵ ݎ ݏ ଵ ݐ ݏ ଶ ݐ ଶ ݎ , for ݎ ሾ Ͳ ሿ , and t ሾ ሿ where ଵ ݎ = ቀ ି à°­ à°® ି à°­ ቁ , ݏ ଵ ݐ = ͳ ቀ ି à°¯ à°° ି à°¯ ቁ , ݏ ଶ ݐ ͳ ቀ à°² ି à°² ି à°± ቁ and ଶ ݎ ቀ à°´ ି à°´ ି à°³ ቁ 2664 S. U. Malini and Felbin C. Kennedy Remark 2. 4 : Membership functions are continuous functions. Remark 2. 5 : Here ω represents  fuzzy number in which “ ω ” is the mximum membership value that a fuzzy number takes on. Whenever a normal fuzzy number is meant, the fuzzy number is shown by , for convenience. Definition 2 . 3 : If ω be an oct gonl fuzzy number, then the α - cut of ω is [ ω ] α = ω = ሾ ଵ ଶ ሿ α ሾ Ͳ ሾ ଵ α ଶ α ሿ α ሾ ω Remark 2 . 6 : The octagonal fuzzy number i s convex as th eir α - cuts are convex sets in the classical sense . Remark 2. 7 : T he collection of all octagonal fuzzy real numbers from R to I is denoted as R ω ( I ) and if ω = 1 , then the collection of normal octagonal fuzzy numbers is denoted by R ( I ). G raphical representati on of a n ormal octagonal fuzzy number for k=0.5 is Working Rule I : Using interval arithmetic given by K aufmann A, [ 10 ] we obtin α - cuts, α ∊ (0, 1] , addition, subtraction and multiplication of two octagonal fuzzy numbers as follows: a ) α - cut of an oct agonal fuzzy number: T he α - cut of a normal octagonal fuzzy number = n ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) given by D efinition 2 . 3 ( i.e. ͳ ) , for α ∊ ( 0, 1 ] is: ሾ ሿ = ቂ ܽ ଵ ቀ ௞ ቁ ܽ ଶ ܽ ଵ ܽ ଼ ቀ ௞ ቁ ܽ ଼ ܽ ଻ ቃ ∊ ሾ Ͳ ሿ ሾ ܽ ଷ ቀ ି ௞ ଵ ି ௞ ቁ ܽ ସ ܽ ଷ ܽ ଺ – ቀ ି ௞ ଵ ି ௞ ቁ ܽ ଺ ܽ ହ ሿ ∊ ͳ ሿ 0 0.2 0.4 0.6 0.8 1 1.2 a ₁ a ₂ a ₃ a ₄ a ₅ a ₆ a ₇ a ₈ s 1 ( t ) s 2 ( t ) l 1 ( r ) l 2 ( r ) Solving fuzzy t ransportation p roblem 2665 b ) Addition of octagonal fuzzy Numbers : Le t = ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) and = ( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 , b 8 ) be two o ctagonal fuzzy numbers. To calculate addition of fuzzy numbers and we first add the α – cuts of and using interval arithmetic. c ) Subtraction of two octagonal fuzzy numbers: Let = ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) and = ( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 , b 8 ) be two octagonal fuzzy numbers. To calculate subtraction of fuzzy numbers and we fir st subtract the α – cuts of and using interval arithmetic. ሾ ሿ ሾ ሿ = [ ݍ , ݍ ሿ w here and d ) Multiplication of two o ctagonal fuzzy numbers: Let = ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , a 8 ) and = ( b 1 , b 2 , b 3 , b 4 , b 5 , b 6 , b 7 , b 8 ) be two octagonal fuzzy numbers . To calculate multiplication of fuzzy numbers and we first multiply the α – cuts of and using interval arithmetic. ሾ ሿ ሾ ሿ = [ ݍ , ݍ ሿ where 2666 S. U. Malini and Felbin C. Kennedy ሾ ሿ ሾ ሿ = [ ݍ , ݍ ሿ where 3 Ranking of o ctagonal fuzzy numbers [3] The parametric methods of comparin g fuzzy numbers, especially in fuzzy decision making theory are more efficient than non - prmetric methods. Cheng’s centroid point method [6], Chu nd Tso’s method [7] , Abbasbandy and $ssdy’s [1] sign - distance method was all non - parametric and was applic able only for normal fuzzy numbers. The non - parametric methods for comparing fuzzy numbers have some drawbacks in practice. Definition 3.1: A measure of fuzzy number is a function M α : R ω ( I ) → R + which assigns a non - negative real number M α ( ) that expresses the measure of . ( à ω ) = ଵ ଶ ଵ ݎ ଶ ݎ ௞ ݀ݎ + ଵ ଶ ݏ ଵ ݐ ݏ ଶ ݐ ݀ݐ ௞ where 0 ͳ Def inition 3.2: The measure of a n octagonal fuzzy number is obtained by the average of the two fuzzy side areas, left side area and right side area, from membership function to α xis. Definition 3 . 3 : Let be a normal octagonal fuzzy number . The value ଴ , called the measure of is calculated as follows: M 0 Oct ( ) = ଵ ଶ ଵ ௞ ଴ ݎ ଶ ݎ ݀ݎ ଵ ଶ ݏ ଵ ଵ ௞ ݐ ݏ ଶ ݐ ݀ݐ where 0 k 1 = ଵ ସ [( a 1 + a 2 +a 7 + a 8 ) k + ( a 3 +a 4 +a 5 +a 6 )( 1 - k )] - ------- ------- ( 3 .1 ) Solving fuzzy t ransportation p roblem 2667 Remark 3.1 : Consider the t rapezoidal number ܽ ଵ ܾ ܽ ସ ܽ ହ Ü¿ ܽ ଼ which is got from the above octagonal number by equating ܽ ଶ ܽ ଷ ܾ ܽ ଺ ܽ ଻ Ü¿ , for k= 0.5 and ω 1 T he measure of the normal fuzzy trapezoidal number is given by M 0 tra ( ) = ଵ ଼ ܽ ଵ Í´ ܾ ܽ ସ ܽ ହ Í´ Ü¿ ܽ ଼ ------- ( 3 . 2 ) Remark 3.2 : If k= 0.5 , M 0 Oct ( ) =  (a 1 + a 2 + a 3 +a 4 +a 5 +a 6 +a 7 + a 8 ) When a 2 coincides with a 3 and a 6 coincides with a 7 it reduces to trapezoidal fuzzy number, which is given by E quation ( 3 .2). Remark 3.3 : If a 1 + a 2 +a 7 + a 8 = a 3 +a 4 +a 5 +a 6 --------------- ( 3 .3) then the measure of a n octagonal number is the same for any value of k ( 0 k 1 ) . Remark 3. 4 : I f ω =( a 1 + ௞ ( a 2 - a 1 ), a 3 + ି ௞ ଵ ି ௞ ( a 4 - a 3 ), a 6 – ି ௞ ଵ ି ௞ ( a 6 - a 5 ), a 8 - ௞ ( a 8 - a 7 )) for ݎ ሾ Ͳ ሿ , and t ሾ ሿ be an arbitrary octagonal fuzzy number at decision level higher than “ α ” and α , ω ∊ [ 0,1 ], t he value α 2 ct ( $ ω ), assigned to $ ω may be calculated as follows: Working Rule I I : If ω>α , then ( à ω ) = ଵ ଶ ଵ ݎ ଶ ݎ ௞ ݀ݎ + ଵ ଶ ݏ ଵ ݐ ݏ ଶ ݐ ݀ݐ ௞ ; = ଵ ଶ { [ ܽ ଵ ܽ ଼ + ቀ à°® ା à°³ ି à°­ ି à°´ ଶ ௞ ቁ ] + ܽ ଷ ܽ ଺ + ቀ à°° ା à°± ି à°¯ ି à°² ଶ ଵ ି ௞ ቁ ( ω + k - 2 ω )] ( ω - k )} Obviously if ω α then the bove quntity will be zer o. ,f ω   it becomes  norml octagonal number , then α 2ct ( $ ) = ଵ ଶ ଵ ݎ ଶ ݎ ௞ ݀ݎ + ଵ ଶ ݏ ଵ ݐ ݏ ଶ ݐ ݀ݐ ଵ ௞ ; α ∊ ሾͲ ͳ = ଵ ଶ { [ ܽ ଵ ܽ ଼ + ቀ à°® ା à°³ ି à°­ ି à°´ ଶ ௞ ቁ ] + ܽ ଷ ܽ ଺ + ቀ à°° ା à°± ି à°¯ ି à°² ଶ ଵ ି ௞ ቁ ( 1 - k )] ( 1 - k )} α ∊ [ 0,1 ) 0 0.5 1 1.5 a ₁ b a ₄ a ₅ c a ₈ 2668 S. U. Malini and Felbin C. Kennedy Remark 3. 5 : If ω and ω’ are two o ctagonal fu zzy numbers and ω, ω’ [ 0, 1 ], then we have: 1. ω ω’ ∀ α [ 0 , 1 ] ( ω ) ( ω ’ ) 2 . ω = ω’ ∀ α [ 0 , 1 ] ( ω ) = ( ω ’ ) 3 . ω ω’ ∀ α [ 0 , 1 ] ( ω ) ( ω ’ ) Remark 3. 6 : If α is close to one, the p ertining decision is clled  “ high level decision”, in which cse only prts of the two fuzzy numbers, with membership values between “ α ” nd “ 1 ”, will be compred. Likewise, if “ α ” is close to zero, the pertining decision is referred to s  “low level decision”, since members with membership values lower than both the fuzzy numbers are involved in the comparison. 4 Mathematical formulation of a F uzzy T ransportation Problem Consider the following fuzzy transportation problem (FTP) having fuzzy costs, fuzzy sources and fuzzy demands, (FTP) Minimize z = Ü¿ ௜௝ ௝ ୀ ଵ ௜ ୀ ଵ Subject to ௜௝ ௝ ୀ ଵ  ã i , for i= ,,… m ( 4. 1) ௜௝ ௜ ୀ ଵ ≈ ܾ ௝ , for j= , ,… n ( 4. 2) ≽ Ͳ for i= , , … m and j= , ,… n ( 4. 3) where m = the number of supply points; n = the number of demand points; ௜௝ ≈ ௜௝ ଵ ௜௝ ଶ ௜௝ ଷ ௜௝ ସ ௜௝ ହ ௜௝ ଺ ௜௝ ଻ ௜௝ ଼ is the uncertain number of units shipped from supply point i to demand point j ;  Ü¿ ௜௝ ଵ Ü¿ ௜௝ ଶ Ü¿ ௜௝ ଷ Ü¿ ௜௝ ସ Ü¿ ௜௝ ହ Ü¿ ௜௝ ଺ Ü¿ ௜௝ ଻ Ü¿ ௜௝ ଼ is the un certain cost of shipping one unit from supply point i to the demand point j ; ã i  ܽ ௜ ଵ ܽ ௜ ଶ ܽ ௜ ଷ ܽ ௜ ସ ܽ ௜ ହ ܽ ௜ ଺ ܽ ௜ ଻ ܽ ௜ ଼ is the uncertain supply at supply point i and ܾ ௝ ≈ ܾ ௝ ଵ ܾ ௝ ଶ ܾ ௝ ଷ ܾ ௝ ସ ܾ ௝ ହ ܾ ௝ ଺ ܾ ௝ ଻ ܾ ௝ ଼ is the uncertain demand at demand point j . The necessary and sufficient condition for the linear programming problem given above to have a solution is that ܽ ௜ ௜ ୀ ଵ ≈ ܾ ௝ ௝ ୀ ଵ The above problem can be put in table namely fuzzy transportation table given b elow: Supply ܽ ଵ ܽ Demand ܾ ଵ ……………. ܾ Ü¿ ଵଵ ……………… Ü¿ ଵ . . . . . . . . . Ü¿ ଵ ……………… Ü¿ Solving fuzzy t ransportation p roblem 2669 5 . Procedure for S ol ving Fuzzy Transportation Problem We shall present a solution to fuzzy transportation prob lem involving shipping cost, customer demand and availability of products from the producers using octagonal fuzzy numbers . Step 1: First convert the cost, demand and supply values which are all octagonal fuzzy numbers into crisp values by using the measu re defined by ( D efinition 3 . 3 ) in S ection 3 . Step 2: [ 9 ] we solve the transportation problem with crisp values by using the VAM procedure to get the initial solution and then the MODI Method to g et the optimal solution and obtain the allotment table. Rema rk 5 .1 : A solution to any transportation problem will contain exactly ( m+n - 1 ) basic feasible solutions. The allotted value should be some positive integer or zero, but the solution obtained may be an integer or non - integer, because the original problem in volves fuzzy numbers whose values are real numbers. If crisp solution is enough the solution is complete but if fuzzy solution is required go to next step. Step 3: Determine the locations of nonzero basic feasible solutions in transportation table. There must be atleast one basic cell in each row and one in each column of the transportation table. Also the m+n - 1 basic cells should not contain a cycle. Therefore, there exist some rows and columns which have only one basic cell. By starting from these ce lls , we calculate the fuzzy basic solutions, and continue until ( m+n - 1 ) basic solution s are obtained. 6 . Numerical Example Consider the following fuzzy transportation problem . Acc ording to the definition of an o ctag onal fuzzy number Ã, the measure of à is calculated as M 0 Oct ( $ ) = ଵ ଶ ଵ ௞ ଴ ݎ ଶ ݎ ݀ݎ ଵ ଶ ݏ ଵ ଵ ௞ ݐ ݏ ଶ ݐ ݀ݐ where 0 k 1 2670 S. U. Malini and Felbin C. Kennedy = ଵ ସ [( a 1 + a 2 +a 7 + a 8 ) k + ( a 3 +a 4 +a 5 +a 6 )( 1 - k )] where 0 k 1 Step 1: Convert the given fuzzy problem into a crisp value problem by using the measure given by D efinition 3.3 in S ection 3 . This problem is done by taking the value of k as 0.4, we obtain the val ues of M 0 Oct ( Ü¿ ௜௝ ), M 0 Oct ( ܽ ௜ ) and M 0 Oct ( ܾ ௝ ) as c  ( - 1,0,1,2,3,4,5,6) M 0 Oct ( c  )= ଵ ସ ሾ  ( - 1+0+5+6)+  (1+2+3+4) ] = 2.5 c  (0,1,2,3,4,5,6,7) M 0 Oct ( c  )= ଵ ସ ሾ  (0+1+6+7)+  (2+3+4+5) ] = 3.5 c  ( 8,9,10, 11,12,13,14,15) M 0 Oct ( c  ) = ଵ ସ ሾ  (8+9+14+15)+  (10+11+12+13) ] = 11.5 c  (4, 5,6,7,8,9,10,11) M 0 Oct ( c  )= ଵ ସ [  (4+5+10+11)+  (6+7+8+9) ] = 7.5 c  ( - 2, - 1,0,1,2,3,4,5) M 0 Oct ( c  )= ଵ ସ ሾ  ( - 2 - 1+4+5)+  (0+1+2+3) ] = 1.5 c  = ( - 3, - 2, - 1,0,1,2,3,4) M 0 Oct ( c  )= ଵ ସ ሾ  ( - 3 - 2+3+4)+  ( - 1+0+1+2) ] = 0.5 c  = (2,4,5,6,7,8,9,11) M 0 Oct ( c  )= ଵ ସ ሾ  (2+4+9+11)+  (5+6+7+8) ] = 6.5 c  = ( - 3, - 1,0,1,2,4,5,6) M 0 Oct ( c  )= ଵ ସ ሾ  ( - 3 - 1+5+6)+  (0+1+2+4) ] = 1.75 c  (2,3,4,5,6,7,8,9) M 0 Oct ( c  )= ଵ ସ ሾ  (2+3+8+9)+  (4+5+6+7) ] = 5.5 c  (3,6,7,8,9,10,12,13) M 0 Oct ( c  )= ଵ ସ ሾ  (3+6+12+13)+  (7+8+9+10) ] = 8.5 c  = (11,12,14,15,16,17,18 ,21) M 0 Oct ( c  )= ଵ ସ ሾ  (11+12+18+21)+  (14+15+16+17) ] =15.5 c  (5,6,8,9,10,11,12,15) M 0 Oct ( c  )= ଵ ସ ሾ  (5+6+12+15)+  (8+9+10+11) ] = 9.5 And the fuzzy supplies are ଵ = (1,3,5,6,7,8,10,12) M 0 Oct (  ) = ଵ ସ ሾ  (1+3 +10+12)+  (5+6+7+8) ] = 6.5 ଶ = ( - 2, - 1,0,1,2,3,4,5) M 0 Oct (  ) = ଵ ସ ሾ  ( - 2 - 1+4+5)+  (0+1+2+3) ] = 1.5 ଷ = (5,6,8,10,12,13,15,17) M 0 Oct (  ) = ଵ ସ ሾ  (5+6+15+17)+  (8+10+12+13) ] = 10.75 And the fuzzy demands are (4,5,6,7, 8,9 ,10,11) M 0 Oct ( ܾ ଵ ) = ଵ ସ [  ( 4+5 +10+11 )+  ( 6 + 7 + 8 + 9 ) ] = 7 .5 (1,2,3,5,6,7,8,10) M 0 Oct ( ܾ ଶ ) = ଵ ସ ሾ  ( 1+ 2+8 + 10 )+  ( 3+5+6+7 ) ] = 5 . 2 5 (0,1,2,3, 4,5,6,7) M 0 Oct ( ܾ ଷ ) = ଵ ସ ሾ ଶ ହ ( 0+ 1+ 6 + 7 )+ ଷ ହ (2+3 +4+5 ) ] = 3 .5 ( - 1,0,1,2,3,4,5,6) M 0 Oct ( ܾ ସ ) = ଵ ସ ሾ ଶ ହ ( - 1 +0 +5 +6 )+ ଷ ହ (1+2+3 +4 ) ] = 2 .5 Solving fuzzy t ransportation p roblem 2671 Remark 6 .1 : In the above problem since condition 3 . 3 (E quation 3.3) is satisfied by all the octagonal numbers (cost, supply and demand) , for any value of k we w ill get the same table as below . 6.5 1.5 10.75 7.5 5.25 3.5 2.5 Step 2: Using VAM procedure we obtain the initial soluti on as 1.25 5.25 1.5 6.25 3.5 1 w hich is not an optimal solution. Hence by using the MODI method we shall improve the solution and get the optimal solution as 5.25 1.25 1.5 7.5 0.75 2.5 Step 3: Now using the allotment rules, the solu tion of the problem can be obtained in the form of octagonal fuzzy numbers Therefore the fuzzy optimal solution for the given transportation problem is ଵଶ (1,2,3,5,6,7,8,10), ଵଷ = ( - 9, - 5, - 2,0,2,5,8,11) ଶଷ = ( - 2, - 1,0,1,2,3,4,5), ଷଵ =(4,5,6,7,8,9,10,11), ଷଷ = ( - 12, - 9, - 5, - 1,3,6,10,14), ଷସ = ( - 1,0,1,2,3,4,5,6) and the fuzzy optimal value of z = ( - 416, - 224, - 73,58,188,333,516,773). And the crisp solution to the problem i s Minimum cost = 119.125 . Also for diffe rent values of k ( 0 k 1 ) we obtain the same solution. Hence the solution is 2.5 3.5 11.5 7.5 1.5 0.5 6.5 1.75 5.5 8.5 15.5 9.5 2672 S. U. Malini and Felbin C. Kennedy independent of k . Remark 6 .2 : When we convert the octagonal fuzzy transportation problem into trapezoida l fuzzy transportation problem we get Supply (1,6,7,12) ( - 2, 1,2,5) (5, 10,12,17) Demand (4, 7,8,11) (1,5, 6,10) (0,3, 4,7) ( - 1,2,3,6) When this problem is solved as in [ 13 ] we would get the fuzzy optimal solution for the given transportation problem as ଵଶ (1,5,6,10), ଵଷ = ( - 9,0,2,11) ଶଷ = ( - 2, - 1,2,5), ଷଵ =(4,7,8,11), ଷଷ = ( - 12, - 1,3,14), ଷସ = ( - 1,2,3,6) , the fuzzy optimal value of z as z = ( - 416,5 9,185 ,773) and the crisp value of the optimum fuzzy transportation cost for the problem is z = 140.83 . I f it is solved as in [ 3 ] , we will get the same value for the variables but the optimal cost will be 121.375. O n the other hand if the problem is solved using octagonal fuzzy numbers we get the optimum cost as 119.125 Remark 6 .3 : If the octagona l numbers are slightly modified so that the condition ( E quation (3 .3 ) ) is not satisfied, i.e. a 1 + a 2 +a 7 + a 8  a 3 +a 4 +a 5 +a 6 , then for such a problem the optimal solution for different values of k ( 0 k 1 ) can be easily checked to lie in a finite interval. 7 . Conclusion In this paper a simple method of solving fuzzy transportation problem ( supply, demand, and cost are all octagonal fuzzy numbers ) were introduced by using ranking of fuzzy numbers. The shipping cost , availability at the origins and requ irements at the destinations are all octagonal fuzzy numbers and the solution to the problem is given both as a fuzzy number and also as a ranked fuzzy number . It also gives us the optimum cost which is much lower than , when it is done using trapezoidal f uzzy numbers. Acknowledgement. The authors wish to thank Professor M.S. Rangachari, Former Director and Head, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai and Professor P.V. Subramanyam, Department of Mathema tics, IIT Madras, Chennai for their valuable suggestions in the preparation of this paper. ( - 1,2,3,6) (0,3,4,7) (8,11,12,15) (4,7,8,11) ( - 2,1,2,5) ( - 3,0,1,4) (2,6,7,11) ( - 3,1 , 2,6) (2,5,6,9) (3,8,9,13) (11,15,16,21) (5,9,10,15) Solving fuzzy t ransportation p roblem 2673 References [1]. S. Abbasbandy, B.Asady, Ranking of fuzzy numbers by sign distance, Information Science, 176 (2006) 2405 - 2416. [2]. H. Basirzadeh, R. Abbasi, A new approach for ranking fuzzy numbers based on α cuts, JAMI, Journal of Applied Mathematics & Informatic, 26(2008) 767 - 778. [3]. H. Bazirzadeh, An approach for solving fuzzy transportation problem, Ap plied Mathematical Sciences, 5(32)1549 - 1566. [4]. S.Chanas and W.Kolodziejczyk and A. 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