Fuzzy ANP-based PROMETHEE II approach for evaluation of machine tool alternatives

3.1 Finding importance weights of evaluation criteria with F-ANP

The hierarchy of machine tool selection needs to be established before performing the pairwise comparison of F-ANP. After constructing a hierarchy, the decision maker(s) is asked to compare the elements at a given level on a pairwise basis to estimate their relative importance in relation to the element at the immediate proceeding level. In conventional ANP, the pairwise comparison is made by using a ratio scale. A frequently used scale is the nine-point scale [42] that is shown in Table 1. Even though the discrete scale of 1–9 has the advantages of simplicity and easiness for use, it does not take into account the uncertainty associated with the mapping of one’s perception or judgment to a number.

In this study, due to its simplicity, triangular fuzzy numbers, 1 ˜ to 9 ˜, are used to represent subjective pairwise comparisons of selection process in order to capture the vagueness. A fuzzy number is a special fuzzy set F ={ (x, μ _F  (x)) , x ∈ R }, where x takes it values on the real line, R : -∞ < x < + ∞ and μ _F  (x) is a continuous mapping from R to the closed interval [0, 1]. A triangular fuzzy number denoted as M ∼ = ( l , m , u ), where l ≤ m ≤ u, has the following triangular type membership function; μ F ( x ) = { 0 x - l / m - l u - x / u - m 0 x < l l ≤ x ≤ m m ≤ x ≤ u x > u(1)

Alternatively, by defining the interval of confidence level α, the triangular fuzzy number can be characterized as: M ˜ α = [ l α , u α ] ∀ α ∈ [ 0 , 1 ] = [ ( m - l ) α + l , - ( u - m ) α + u ] .(2)

Some main operations for positive fuzzy numbers are described by the interval of confidence, by Kaufmann and Gupta [25] as given below; ∀ m L , m R , n L , n R ∈ R + , M ˜ α = [ m L α , m R α ] , N ˜ α = [ n L α , n R α ] , α ∈ [ 0 , 1 ](3)M ˜ ⊕ N ˜ = [ m L α + n L α , m R α + n R α ] , M ˜ Θ N ˜ = [ m L α - n L α , m R α - n R α ](4)M ˜ ⊗ N ˜ = [ m L α n L α , m R α n R α ] , M ˜ / N ˜ = [ m L α / n L α , m R α / n R α ](5)

The triangular fuzzy numbers, 1 ˜ to 9 ˜, are utilized to improve the conventional nine-point scaling scheme. In order to take the imprecision of human qualitative assessments into consideration, the five triangular fuzzy numbers are defined with the corresponding membership function as shown in Fig. 2.

Computational Steps of F-ANP:

Step 1: Comparing the performance score: Triangular fuzzy numbers (1 ˜, 3 ˜, 5 ˜, 7 ˜, 9 ˜) are used to indicate the relative strength of each pair of elements in the same hierarchy.

Step 2: Constructing the fuzzy comparison matrix: By using triangular fuzzy numbers, via pair wise comparison, the fuzzy judgment matrix A ˜ ( a ij ) is constructed as given below. Here, A ˜ = [ 1 a 12 ˜ . . . . a 1 n ˜ a 21 ˜ 1 . . . . a 2 n ˜ . . . . . . . . . . . . . . . . . . . . a n 1 ˜ a n 2 ˜ . . . . 1 ] where, a ij α ˜ = 1, if i is equal j, and a ij α ˜ = 1 ˜ , 3 ˜ , 5 ˜ , 7 ˜ , 9 ˜ or 1 ˜ - 1 , 3 ˜ - 1 , 5 ˜ - 1 , 7 ˜ - 1 , 9 ˜ - 1 if i is not equal j.

Step 3: Solving fuzzy eigenvalue: A fuzzy eigenvalue, λ ˜ is a fuzzy number solution to A ˜ x ˜ = λ ˜ x ˜(6) where A ˜ is nxn fuzzy matrix containing fuzzy numbers a ij ˜ and x ˜ is a non-zero nx1 fuzzy vector containing fuzzy number x ˜ i. To perform fuzzy multiplications and additions by using the interval arithmetic and α - cut, the equation A ˜ x ˜ = λ ˜ x ˜ is equivalent to [ a i 1 l α x 1 l α , a i 1 u α x 1 u α ] ⊕ … … ⊕ [ a inl α x nl α , a inu α x nu α ] = [ λ x il α , λ x iu α ] where, for 0 < α ≤ 1 and i = 1, 2 …  n, j = 1, 2 …  n, A ˜ = [ a ˜ ij ] , x t ˜ = ( x 1 ˜ , … … , x n ˜ ) , a ˜ ij α = [ a ijl α , a iju α ] , x ˜ i α = [ x il α , x iu α ] , λ ˜ α = [ λ l α , λ u α ] .(7)α - cut is known to incorporate the experts or decision maker(s) confidence over his/her preference or the judgments. Degree of satisfaction for the judgment matrix A ˜ is estimated by the index of optimism μ. The larger value of index μ indicates the higher degree of optimism. The index of optimism is a linear convex combination [27] defined as: a ij α ˜ = μ a iju α + ( 1 - μ ) a ijl α , ∀ μ ∈ [ 0 , 1 ](8)

While α is fixed, the following matrix can be obtained after setting the index of optimism, μ in order to estimate the degree of satisfaction. A ˜ = [ 1 a 12 ˜ . . . . a 1 n α ˜ a ˜ 21 α 1 . . . . a 2 n α ˜ . . . . . . . . . . . . . . . . . . . . a n 1 α ˜ a n 2 α ˜ . . . . 1 ](9)

The eigenvector is calculated by fixing the μ value and identifying the maximal eigenvalue. Normalization of both the matrix of paired comparisons and calculation of priority weights (approx. attribute weights), and the matrices and priority weights for alternatives are also done before calculating λ_max. In order to control the result of the method, the consistency ratio for each of the matrices and overall inconsistency for the hierarchy are calculated. The deviations from consistency are expressed by the following equation consistency index, and the measure of inconsistency is called the consistency index (CI), CI = λ max - n n - 1(10)

The consistency ratio (CR) is used to estimate directly the consistency of pair wise comparisons. The CR is computed by dividing the CI by a value obtained from a table of Random Consistency Index (RI); CR = CI RI(11)

If the CR is less than 0.10, the comparisons are acceptable, otherwise they are not. RI is the average index for randomly generated weights [39].

Step 4: In this step, a super-matrix is constructed to handle interrelationships between the decision levels and criteria by obtaining the composite weights through the development of a supermatrix, which is a partitioned matrix, where each submatrix is composed of a set of relationships between two components or clusters in a connection network structure. Saaty [42] explains the concept corresponding to the markov chain process. In this work, we utilize the matrix manipulation on the concept of Saaty and Takizawa [40] instead of Saaty’s original supermatrix, because of its ease of understanding. We also utilized the work of Shyur and Shih [43] which was implemented for vendor selection problem. In order to reflect the interdependencies in the network, a set of pair wise comparison matrices are constructed for each criterion and their consistency ratios are calculated. These matrices are used to identify the relative impacts of the criteria interdependent relationships. The normalized principal eigenvectors for the matrices are calculated and shown as column component in the manipulated matrix, S, where zeroes are assigned in the matrix if there is no relationship between the related criteria. Finally, we can obtain the interdependence priorities of the criteria by synthesizing the results of previous calculations as follows; w criteria = S * w T(12) where, S is the manipulated matrix and w is the weight line vector of the criteria.

There are four steps in F-PROMETHEE II as fol-lows [47]:

Step 1: Formulate a fuzzy decision making matrix: In the previous section, w _criteria  (w₁ … w _j  … w _n ) values were obtained through F-ANP. A typical m by n fuzzy decision matrix is shown as below; ( w 1 … w j … w n ) ( c ˆ 1 … c ˆ j … c ˆ n ) A ˆ 1 ⋮ A ˆ i ⋮ A ˆ m [ 1 r 12 ˜ . . . . r 1 n ˜ r 21 ˜ 1 . . . . r 2 n ˜ . . . . . . . . . . . . . . . . . . . . r m 1 ˜ r m 2 ˜ . . . . 1 ](13)

Here, c ˆ j ∈ C ˆ is a fuzzy positive criterion. The criterion is a maximum, if decision maker prefers more value for it. Otherwise, it is a minimum criterion. A ˆ i ∈ A ˆ is fuzzy alternative. A ˆ * is the fuzzy alternative from A ˆ. r ˆ ij ∈ r ˆ is the utility value. w ˆ j ∈ W ˆ is the fuzzy weight of c ˆ j. We used the fuzzy number form (l,  m,  u) for triangular fuzzy numbers, where l is the lower boundary, m is the modal value, and u is the upperboundary.

Step 2: Index fuzzy numbers in the fuzzy decision matrix: the fuzzy number in the fuzzy decision matrix can be “defuzzified” with centroid defuzzification approach [45] to the crisp number by the following function: I ( l , m , u ) = ( l + m + u ) / 3(14)

In other words, the above process converts a fuzzy decision matrix as a crisp decision matrix as follows: ( w 1 … w j … w n ) ( c 1 … c j … c n ) A 1 ⋮ A i ⋮ A m [ 1 r 12 . . . . r 1 n r 21 1 . . . . r 2 n . . . . . . . . . . . . . . . . . . . . r m 1 r m 2 . . . . 1 ](15) where, c _j  ∈ C is the positive criterion, A _i  ∈ A is the alternative, A^* is the ideal alternative from A, r _ij  ∈ r is the utility value, w _j  ∈ W is the weight of c _j . The cap removal from the notations is crisp value.

Step 3: Calculate aggregated preference indices: P _j  (A _i ,  A _k ) = P _j  (d (A _i ,  A _k )) = P _j  (r _ij  - r _kj ) is a preference function showing how much A _i prefers to A _k with respect to c _j . According to Brans et al. [9], out of the six types of generalized functions in Table 2, Gaussian criterion was selected.

Brans et al. [9] pointed out that the Gaussian criterion was preferred by most of the users for practical applications especially in the case of continuing data. Since the major criteria in this study contain continuing data, the Gaussian criterion was selected for evaluation, and the preference function is as follows: P ( d ) = { 0 d ≤ 0 1 - e - d 2 2 s 2 d < 0 } , if the criterion is a maximum(16)P ( d ) = { 0 d ≥ 0 1 - e - d 2 2 s 2 d < 0 } , if the criterion is a minimum(17)

Aggregated preference index π (A _i ,  A _k ) expresses the degree of how much A _i prefers to A _k over all the criteria. The aggregated preference indices are of the form:

π ( A i , A k ) = ∑ j = 1 n P j ( A i , A k ) . w j ∑ j = 1 n w j , ∀ A i , A k ∈ A and i ≠ k(18)

Step 4: Calculate outranking flow. Each alternative A _i is facing (m - 1) other alternatives in A. In order to rank the alternatives, the outranking flows are defined as follows:

The positive outranking flow is of the form: φ + ( A i ) = ∑ k = 1 m π ( A i , A k )(19)

The negative outranking flow is of the form: φ - ( A i ) = ∑ k = 1 m π ( A k , A i )(20)

The net outranking flow is applied and is in the form of: φ ( A i ) = φ + ( A i ) - φ - ( A i ) , ∀ i ∈ { 1 , … , m }(21)

The positive outranking flow expresses how an alternative A _i is outranking all the others. Higher φ⁺ (A _i ) gives a better alternative. On the other hand, the negative outranking flow expresses how an alternative A _i is outranked by all the others. The lower φ^- (A _i ) gives a better alternative. The higher φ (A _i ) specifies the final better alternative.