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JOURNAL OF COMPUTERS, VOL. 3, NO. 10, OCTOBER 2008 31 If x ij = S , x~ = S , j ∈ I' , (1) i For cost criteria, the normalized formula is: If x = = S n i j ∈ I ′ ij Si , x~ ij + 1− , , (2) where, n is the number of elements in S, here n=7, x~ ij is the normalized form of x ij , I′ is associated with benefit criteria and I ′ is associated with cost criteria. B. Convert the Linguistic Term into Triangular Fuzzy Number Since the subjective judgment or prediction of an expert may be ill expressed, it is reasonable to represent x ij as linguistic variables and then converted it into triangular fuzzy number. Table II shows the linguistic values and their corresponding triangular fuzzy numbers. So the linguistic term x ij can be converted to triangular fuzzy number as L M U v = ( v , v , v ) . ij ij C. Determine the Ideal and Negative Ideal Solutions The fuzzy positive-ideal solution (FPIS,A * ) and the fuzzy negative-ideal solution (FNIS,A - ) are shown in the following equations: A * * * = { v ,... v } = {max v | i = 1,2, K m, j = 1,2, K, n} , (3) 1 j i ij − − − A = { v 1 ,... v } = {min v | i = 1,2 K m, j = 1,2, K, n} . (4) j i ij D. Calculate the Grey Relation Coefficient The relation coefficient between alternative i and FPIS with respect to criterion C j (namely V ) is as follows: d = ij V ij = d( v 1 [( v 3 ij ij ij min min dij + ρ max max dij i j i j = . (5) d + ρ max max d * j L* j ,v ) ij − v L ij 2 ) + ( v M * j i − v i M ij ij ij j 2 ) + ( v ij U * j − v U ij 2 ) ] . (6) The relation coefficient between alternative A i and FNIS with respect to criterion C j (namely W ij ) is as follows: ij W ij 1 [( v 3 min min s i s ij j ij + ρ max max s = . (7) − j s = d( v = L− j , v ) ij − v + ρ max max s i L 2 M − M 2 U − U ij ) + ( v j − vij ) + ( v j − vij i j j ij ij 2 ) ] . (8) TABLE II. LINGUISTIC TERMS AND THEIR CORRESPONDING TRIANGULAR FUZZY NUMBERS Linguistic values Extremely high(EH) (0.95,1,1) Very high(VH) (0.7,0.85,1) High(H) (0.55,0.7,0.85) Medium(M) (0.35,0.5,0.65) Low(L) (0.15,0.3,0.45) Very low(VL) (0,0.15,0.3) Extremely low(EL) (0,0,0.05) Where, i=1,2,…m, j=1,2,…n, d ij is the distance between triangular fuzzy numbers v * j and v ij , s ij is the - distance between triangular fuzzy numbers v j and v ij . ρ is the discrimination coefficient, ρ ∈[0,1] and generally ρ =0.5. The relation coefficient vector between alternative A i and FPIS(A * ) is ( i , Vi2 V , V ) 1 L in , and the relation coefficient vector of alternative A i and FNIS(A - ) is ( i , Wi2 in T W , W ) T 1 L . E. Construct the Single Objective Programming Problem to Determine the Weight Vector Suppose V i = n ∑ j= 1 V ij w j ∑ , W = W w , i n j= 1 Where, i=1,2…,m. For get V i and W i , the vector T ϖ = ( w 1, w2, Lw n ) is need to be determined first. So the multi-objective optimization model is developed to T get ϖ . = ( 1, w2, Lw n w ) maxV minW i i s. t. w w = = j j n ∑ j= 1 n ∑ j= 1 W w , i = 1,2,..., m, ∈ Q, j = 1,2,... n, n ∑ j= 1 V w , i = 1,2,..., m, ij ij w j Triangular fuzzy numbers j = 1, ij ≥ 0, j = 1,2,... n. Since every alternative is fair competition and there is no preference relation, the above multi-objective optimization model can converted into single objective programming problem. j j (9) © 2008 ACADEMY PUBLISHER
32 JOURNAL OF COMPUTERS, VOL. 3, NO. 10, OCTOBER 2008 TABLE III. TRIANGULAR FUZZY NUMBER DECISION MATRIX C 1 C 2 C 3 C 4 C 5 C 6 A 1 (0,0,0.05) (0.95,1,1) (0.15,0.3,0.45) (0.35,0.5,0.65) (0.35,0.5,0.65) (0.35,0.5,0.65) A 2 (0,0.15,0.3) (0.95,1,1) (0.15,0.3,0.45) (0,0,0.05) (0.7,0.85,1) (0.35,0.5,0.65) A 3 (0,0.15,0.3) (0.35,0.5,0.65) (0.35,0.5,0.65) (0.55,0.7,0.85) (0.7,0.85,1) (0,0,0.05) A 4 (0,0.15,0.3) (0.35,0.5,0.65) (0.35,0.5,0.65) (0.55,0.7,0.85) (0,0.15,0.3) (0.15,0.3,0.45) C 7 C 8 C 9 C 10 C 11 A 1 (0.55,0.7,0.85) (0.7,0.85,1) (0.95,1,1) (0.35,0.5,0.65) (0.35,0.5,0.65) A 2 (0.55,0.7,0.85) (0.35,0.5,0.65) (0.35,0.5,0.65) (0.7,0.85,1) (0.95,1,1) A 3 (0.7,0.85,1) (0.35,0.5,0.65) (0.35,0.5,0.65) (0.7,0.85,1) (0.95,1,1) A 4 (0.35,0.5,0.65) (0.95,1,1) (0.55,0.7,0.85) (0.7,0.85,1) (0.35,0.5,0.65) minD = s. t. w w j j m i= 1 j= 1 ( W ∈ Q, j = 1,2,... n, n ∑ j= 1 n ∑∑ w j ij = 1, −V ) w , ij ≥ 0, j = 1,2,... n. And with the single objective programming problem, the weight vector ϖ can be determined. j (10) F. Calculate the Relative Relation Degree of Alternatives C V i i = . (11) Vi + Wi Choose an alternative with maximum C i or rank alternatives according to C i in ascending order. IV. EMPIRICAL STUDY OF SUPPLY CHAIN RISK ASSESSMENT Supply chain risk evaluation is generally complex in operation and it involves a variety of factors arising from environment, social, and even political concerns. Owing to the uncertainty in human judgment, decision making in supply chain risk evaluation is actually a multi-criteria decision making problem under fuzzy environment. Here we use the proposed method to evaluate the supply chain risk of 4 enterprises in Shandong province of China. The partial weight information is given as follows: w 1 / w2 = 0.07 ; w 8 >= 0.03 . w 3 >= 0.09 And the linguistic decision matrix is as follows: ⎡EH EL H M M M L VL EL M M⎤ ⎢ ⎥ = ⎢ VH EL H EH VL M L M M VL EL . D ⎥ ⎢VH M M L VL EH VL M M VL EL⎥ ⎢ ⎥ ⎣VH M M L VH H M EL L VL M ⎦ Step1 normalized the decision matrix Since these criteria are all cost criteria, with (2), the normalized decision matrix is: ⎡EL EH L M M M H VH EH M M ⎤ ⎢ ⎥ = ⎢ VL EH L EL VH M H M M VH EH D ~ . ⎥ ⎢VL M M H VH EL VH M M VH EH⎥ ⎢ ⎥ ⎣VL M M H VL L M EH H VH M ⎦ Step 2 convert the linguistic term into triangular fuzzy number According to Table II, convert the linguistic decision matrix into triangular fuzzy number decision matrix, and get the matrix shown in Table III. Step 3 Determine the ideal and negative ideal solutions With (3) and (4), we get * A = ((0.00,0.15,0.30), (0.95,1.00,1.00), (0.35,0.50,0.65), (0.55,0.70,0.85), (0.70,0.85,1.00), (0.35,0.50,0.65), . (0.70,0.85,1.00), (0.95,1.00,1.00), (0.95,1.00,1.00), (0.70,0.85,1.00), (0.95,1.00,1.00)) A − = ((0.00,0.00,0.05), (0.35,0.50,0.65), (0.15,0.30,0.45), (0.00,0.00,0.05), (0.00,0.15,0.30), (0.00,0.00,0.05), . (0.35,0.50,0.65), (0.35,0.50,0.65), (0.35,0.50,0.65), (0.35,0.50,0.65), (0.35,0.50,0.65)) Step 4 calculate the grey relation coefficient Use (6) and (8) to get V ij and W ij as follows: © 2008 ACADEMY PUBLISHER
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