On Consistency in AHP and Fuzzy AHP

Definition 1

(see [2]) A = (a_ij)_n×n is called a multiplicative reciprocal preference relation, if the entries of A satisfy the following condition

where a_ij stands for the relative importance of alternatives x_i over x_j.

Definition 2

(see [2]) Let A = (a_ij)_n×n be a multiplicative reciprocal preference relation. If the following condition is satisfied

then A is called a consistent multiplicative reciprocal preference relation.

Definition 2 reflects the basic idea that the direct pairwise comparison a_ij should be determined exactly by any pair of indirect pairwise comparisons a_ik and a_kj (k = 1, 2, ⋯ , n) for a consistent comparison matrix. It is based on the ideal assumption that a decision maker is fully rational and exhibits judgements. Obviously, under the consideration of the limitations of human-being to provide this type of numeric quantification and the complexity of a practical decision problem, Definition 2 is idealized and one cannot always generate a consistent preference relation, especially when the number of alternatives is large. Furthermore, Saaty^[2] considered that the departure from consistency to some extent is permitted in practical situations. He introduced the so-called consistency index (C.I.) and the consistency ratio (C.R.) to measure the derivation level from consistency:

where λ_max and n are the largest eigenvalue and the order of pairwise comparison matrices, respectively. R.I. is a random index dependent on the size of the matrices, and it is the average C.I. of a large number of randomly generated multiplicative preference relations. When C.R.(A) ≤ 0.10, A is considered to be acceptably consistent, and for C.I.(A) = 0, A is fully consistent. It is seen that there is a threshold of C.R.(A) = 0.10, which means that the consistency index is ten percent of the mean consistency index of randomly generated matrices. Of much interest is to calculate the thresholds of the largest eigenvalue λmaxT=n+0.1⋅(n−1)⋅R.I. and the relative errors εr=(λmaxT−n)/n, which are given in Table 1. It is seen from Table 1 that with increasing the order n, the acceptable relative errors are increasing. Moreover, when C.R.(A) > 0.10, implying that the comparison matrix is unacceptably consistent, a feasible technique should be used to make it an acceptable one^[3]. Here according to Table 1, one can propose a method to adjust the comparison matrix with unacceptable consistency to that with acceptable consistency such that the greatest eigenvalue is equal to λmaxT or less.

Definition 3

(see [23]) B = (b_ij)_n×n ⊂ X ×X is called as an additive reciprocal matrix, if bij is the preference intensity of the alternatives x_i to x_j with b_ij + b_ji = 1, 0 ≤ b_ij ≤ 1, for ∀_i,j = 1, 2, ⋯ , n.

In what follows, let us focus on the transitivity and the consistency of additive reciprocal matrices. It is considered that if x_i ≻ x_j and x_j ≻ x_k, one should have x_i ≻ x_k with at least the same intensity. The max-min transitivity is defined as^[23]

The restrict max-min transitivity is a stronger condition and it is further defined as^[23]

In addition, the additive transitivity is given as

which is stronger than the restrict max-min transitivity and means that the preference degree b_ik can be derived indirectly by the preference degrees b_ij and b_jk. Specially, it is assumed that b_ij/b_ji stands for the comparison ratio of x_i over x_j. From Definition 2, one has the multiplicative transitivity

or

by considering the case of b_ij = 0, which can be used to give the restrict max-min transitivity^[23]. For convenience, two definitions of additive reciprocal preference relations with consistency are given as follows^{[25, 26]}:

Definition 4

(see [25]) An additive reciprocal matrix B = (b_ij)_n×n is additively consistent, if one has

where 0 < b_ij < 1 for ∀i, j = 1, 2, ··· ,n.

Definition 5

(see [25]) An additive reciprocal matrix B = (b_ij)_n×n has the multiplicative consistency, if it satisfies the multiplicative transitivity

where 0 < b_ij < 1 for Vi, j = 1, 2, ··· ,n.

For the additive consistency in Definition 4, it is interpreted that if the value of (bij−12) is considered as the preference intensity of x_i over x_j, the preference intensity of x_i over x_k should be equal to the sum of the indirect preference intensities of (bij−12)and(bjk−12), and some equivalent propositions have been given in [25]. Definition 5 is completely based on the typical idea of consistent comparison matrices together with the consistent judgements in Definition 2. It seems that Definition 5 has nothing with the fuzzy set theory except b_ij ∈ [0, 1].

Definition 6

(see [8]) An interval multiplicative reciprocal matrix à is defined as

where aij−andaij+ are non-negative real numbers, aij−≤aij+,aij−=1/aji+andaij+=1/aji−.a~ij indicates that the alternative x_i is between aij−andaij+ times as important as the alternative x_j.

For the consistency analysis of Ã, Wang, et al.^[28] have considered that an interval multiplicative reciprocal matrix is consistent, if there exists a consistent matrix with elements lying in the entries of Ã. The corresponding definition is given as follows:

Definition 7

(see [28]) An interval multiplicative reciprocal matrix à is consistent, if the convex feasible region Sw={w=(w1,w2,⋯,wn)|aij−≤wi/wj≤aij+,∑i=1nwi=1,wi>0,∀i} is nonempty.

It is seen that Definition 7 is directly used the consistency definition of multiplicative reciprocal matrices in Definition 2. The reciprocity and the transitivity of fuzzy intervals have been completely ignored. However, as pointed out in [29], the reciprocal condition ã_ij = 1/ã_ij does not mean ã_ij · ã_ji = 1 and the transitivity property of all entries does not hold in the form ã_ji = ã_ik · ã_kj. It implies that the consistency of interval multiplicative reciprocal matrices cannot be defined as that in Definition 2. In other words, the consistency condition in Definition 2 is incompatible with the operation law of fuzzy interval in fuzzy set theory^[30].

Furthermore, the reciprocal property of preference relation is one of the axioms as the foundation of the AHP in [10]. One can observe from Definition 6 that the decision maker must consider the reciprocal property aij−=1/aji+andaij+=1/aji−. Then Liu^[31] has investigated the consistency and acceptable consistency of A. That is, letting C = (cij)_n×n and D = (dij)_n×n where

one has the two definitions:

Definition 8

(see [31]) An interval multiplicative reciprocal preference relation à is said to be consistent, if the multiplicative preference relations C and D determined by using (19) are all consistent.

Definition 9

(see [31]) If the multiplicative reciprocal preference relations C and D determined by utilizing (19) are all of acceptable consistency, then à is acceptably consistent. Otherwise, U is said to be unacceptably consistent.

From Definitions 8 and 9, the conditions of consistency and acceptable consistency can be easily checked by using the idea given in Definition 2. But Definitions 8 and 9 are based on two multiplicative reciprocal matrices and the transitivity of fuzzy intervals is still not considered. Recently, Wang^[32] has pointed out that Definition 8 is technically incorrect by using a numerical example. The reason is based on the fact that interval multiplicative reciprocal matrices may be consistent or inconsistent by only changing the comparison orders of two alternatives. It is considered that the consistency of preference relations should keep the invariance with respect to permutations of decision alternatives. We should confirm that for the typical consistency in the AHP, the invariance of consistency with respect to permutations of the alternatives should be holden due to the reciprocal property a_ij = 1/a_ji. However, when the judgments of the decision maker are expressed by using fuzzy numbers, one should ask if the invariance of consistency with respect to permutations of the alternatives is still holden. This important problem should be answered and studied in the future.

Furthermore, Wang^[32] gave a new definition of consistent interval multiplicative reciprocal matrices to correct Definition 8 as

Definition 10

(see [32]) An interval multiplicative reciprocal matrix à is geometrically consistent, if the following transitive condition is satisfied

Clearly, if one has the following transitive conditions

the interval multiplicative reciprocal matrix à is consistent in terms of Definition 10. Consequently, Definition 10 is an extension of Definition 2 by considering the transitivity of left bounds and right bounds of interval entries in Ã. Moreover, from the process of giving comparison matrices, one should consider the reciprocity of aij−=1/aji+andaij+=1/aji−. Under the consideration of the reciprocity, Eq. (20) can be rewritten in the form

or

It is noted that the transitive conditions in (22) and (23) are identical with that given in Definition 5 for an additive reciprocal matrix with multiplicative consistency. That is to say, the consistency condition in Definition (10) is only a rewritten version of the known ones. In addition, Eqs. (22) and (23) require the left bounds or the right bounds of interval entries in à to satisfy the transitive conditions. It is hard to understand from the viewpoint of giving comparison ratios in a decision making process. Definition 10 does not reflect the reciprocal property and the idea of fuzzy set theory. The requirement of the transitive conditions in (20) seems mainly a mathematical technique to some extent in order to make the definition have the invariance with respect to permutations of alternatives. The basic idea of Definition 10 is to characterize the consistency of interval multiplicative reciprocal matrices by using a special multiplicative reciprocal preference relation in (20). The effectiveness of Definition 10 is in doubt and it should be further investigated in the future.

It is noted that Xia and Chen^[33] have defined the weak transitivity of interval multiplicative reciprocal preference relations by using a possibility degree formula to compare two interval multiplicative reciprocal preference values, namely

Definition 11

(see [33]) An interval multiplicative reciprocal matrix à is weakly transitive if application of P_ãij≻[1, _1]≥ 0.5 and P_ãjk≻[1, _1]≥ 0.5 leads to P_ãik≻[1, _1]≥ 0.5 for i, j, k = 1, 2, ⋯ , n, where P. denotes the possibility degree.

Furthermore, Xia and Chen^[33] have given the definition of consistent interval multiplicative reciprocal matrices, namely

Definition 12

(see [33]) An interval multiplicative reciprocal matrix à is consistent if one has

In virtue of the laws of fuzzy numbers, application of (24) yields

Then according to Lemma 1 in [33], we can further obtain the transitive condition (20). As compared to the studies of consistency in the literature, one can see that the studies of the weak transitivity of interval multiplicative reciprocal matrices in [33] is a feasible attempt and it is worthy of investigating and extending widely.

Definition 13

(see [34]) An interval additive reciprocal matrix B̃ is expressed as

where bij−andbij+ are non-negative real numbers, bij−≤bij+,bij−+bji+=1 for all 1 ≤ i, j ≤ n. b̃_ij indicates the importance degree of alternative x_i over x_j. Especially, under the consideration of the multiplicative consistency of additive reciprocal matrix, we have 0<bij−≤bij+<1.

Two definitions of consistent interval additive reciprocal matrices are further presented as follows^{[26, 34]}.

Definition 14

(see [26]) An interval additive reciprocal matrix B̃ is additively consistent, if there is a vector w = (w₁, w₂, ⋯ , w_n) such that

Definition 15

(see [34]) An interval additive reciprocal matrix B̃ is multiplicative consistent, if there exists a vector w = (w₁, w₂, ⋯ , w_n) such that

It is obvious that Definitions 14 and 15 are only given through a direct application of Definitions 4 and 5, respectively.

Liu, et al.^{[24, 35]} have considered the additive reciprocity of bij−+bji+=1 and defined consistent interval additive reciprocal matrices. Letting P = (p_ij)_n×n and Q = (q_ij)_n×n be expressed as

The following definitions are given:

Definition 16

(see [35]) An interval additive reciprocal matrix B̃ is additively consistent, if P = (p_ij)_n×n and Q = (q_ij)_n×n are additively consistent.

Definition 17

(see [24]) An interval additive reciprocal matrix B̃ is multiplicative consistent, if P = (p_ij)_n×n and Q = (q_ij)_n×n are multiplicative consistent.

An interval additive reciprocal matrix B̃ with the conditions in Definitions 16 and 17 is also consistent by using Definitions 14 and 15, respectively. These definitions are not related to the comparison of interval additive reciprocal preference values. Basically, one should consider the fuzzy relation of interval numbers and study the transitivity of interval additive reciprocal matrices in the future.

On the other hand, based on the operation laws of interval fuzzy number, Wang and Li^[36] extended the definition of consistent additive reciprocal matrices to that of consistent interval additive reciprocal matrices. They obtained the following two definitions:

Definition 18

(see [36]) An interval additive reciprocal matrix B̃ is called to be additively consistent, if the following additive transitivity holds:

Definition 19

(see [36]) An interval additive reciprocal matrix B̃ is called multiplicative consistent, if one has the following multiplicative transitivity:

If one considers that the equal sign “=“ in (28) and (29) implies two fully identical interval number, it yields^[37]

That is, the conditions in Definitions 18 and 19 require that two numeric matrices (bij−)n×n and (bij+)n×n are additive consistent and multiplicative consistent respectively. The requirements are much stricter than those given in Definitions 14 and 15 and they have nothing to do with the transitivity of interval numbers.

In addition, the consistency and transitivity of triangular and trapezoidal fuzzy comparison matrices have been also investigated in the literature. For example, Buckley^[9] has directly extended the consistency definition of multiplicative reciprocal matrices in [2] and introduced the following definition:

Definition 20

(see [9]) A fuzzy positive reciprocal matrix à = (ã_ij)_n×n is consistent if and only if ã_ij ≈ ã_ikã_kj, where ã_ij denote trapezoidal fuzzy numbers.

Based on Definition 20 and the operation laws of fuzzy numbers^[30], it is the key of how to understand and define the fuzzy relation of “ ≈ “. In [9], for two fuzzy numbers M̃ and Ñ with the membership functions μ_m(x) and μ_n(x) respectively, one defines a comparison function as

When v(M̃ ≤ Ñ) = 1 and v(Ñ ≤)M̃ < θ, where θ is some fixed positive fraction less or equal to one, it means that M̃ is greater than Ñ. When min(v(M̃ ≥ Ñ),v(M ≤ Ñ))≥ θ, one says that M̃ is approximately equal to Ñ, namely M̃ ≈Ñ. In form, Definition 20 is a direct extension of consistent multiplicative reciprocal comparison matrices^[9]. On the other hand, Wang and Chen^[38] considered that Definition 20 is suitable for characterizing the consistency of triangular fuzzy reciprocal preference relations. However, the method of extending directly the typical definition for consistent multiplicative reciprocal matrices^[2] is not suitable for consistent comparison matrices with fuzzy numbers^[29]. The reciprocal property of preference relations in AHP^[10] and fuzzy AHP is a basic requirement and it should be considered^[39]. Under the consideration of the reciprocal property in triangular fuzzy reciprocal matrices, Liu, et al.^[40] gave the corresponding consistency definition and pointed out the existing shortcomings of the propositions given in [38]. The triangular fuzzy reciprocal preference relation is given as:

Definition 21

(see [7]) A triangular fuzzy reciprocal preference relation is represented as

where t̃_ij stands for the triangular fuzzy degree of the alternative (criterion) x_i over x_j. l_ij and u_ij represent the lower and upper bounds of the triangular fuzzy number t̃_ij, respectively and m_ij is the median value. l_ij , m_ij and u_ij are non-negative real numbers with l_ij ≤ m_ij ≤ u_ij and l_ij u_ji = m_ij m_ji = u_ij l_ji = 1.

Then three preference relations T^L, T ^M and T^R are constructed from T = ((l_ij, m_ij, u_ij))_n×n and let TL=(tijL)n×n,TM=(tijM)n×nandTR=(tijR)n×n, where

and tijM=mij, ∀i, j = 1, 2, ⋯ , n. One further has the following definition:

Definition 22

(see [40]) If three reciprocal preference relations T^L, T^M and T^R are consistent, T̃ is said to be a consistent triangular fuzzy reciprocal preference relation. Otherwise, T̃ is said to be inconsistent.

Unfortunately, similar to Definition 8, Wang^[41] also stated that Definition 22 is not ideal and there exists the technical shortcoming. That is, Definition 22 is dependent on the permutations of alternatives. A definition similar to Definition 10 is given as:

Definition 23

(see [41]) T̃ is said to be a consistent triangular fuzzy reciprocal preference relation, if one has

for i, j, k = 1, 2, ⋯ , n.

Clearly, Definition 23 can be analyzed similar to Definition 10 and it is still unsatisfactory. According to the above analysis, it is difficult to define and test the consistency of preference relations with fuzzy numbers. It is found that there is a substantial controversy in analyzing the consistency of interval and triangular fuzzy multiplicative reciprocal matrices and this may be clarified in the coming works.

Theorem 1

(see [44]) Letλmaxkbe the greatest eigenvalue of the multiplicative reciprocal preference relationAk=(aijk)n×n,k=1,2,⋯,m.It is supposed thatAc=(aijc)n×n=(∏k=1m(aijk)vk)n×nwith∑k=1mvk=1,vk≥0,andλmaxAcis the greatest eigenvalue of A_c. One has

It is seen from Theorem 1 that the collective multiplicative reciprocal matrix A_c can be confirmed to be acceptably consistent when all the individual ones are acceptably consistent by using the consistency ratio of Saaty^[2]. Although the above result has been given in [45], the proof was shown to be wrong by using a counter example in [46]. In particular, when the individual multiplicative reciprocal matrices are consistent, the collective one obtained by using the geometric mean method is consistent. Moreover, based on Definition 9, when individual interval multiplicative reciprocal matrices are all acceptably consistent, the collective one obtained by using the induced weighted geometric average (IWGA) operator is acceptably consistent ^[44].

Definition 24

(see [49]) A mapping F : [0, 1]ⁿ → [0,1] is called an OWA operator of dimension n with the associated weight vector w=(w1,w2,⋯,wn)(wi∈(0,1),∑i=1nwi=1) such that

where b_j is the jth largest element in the collection a₁, a₂, ⋯ , a_n.

In the OWA operators, a lot of importance is paid to the determination of the associated weight vector^{[48, 50]}. Yager^[49] stated that there are at least two approaches: Applying some kind of learning mechanism and giving the meaning of the weights w_i. One of the methods for obtaining the weighting vector is to associate the OWA aggregation with a linguistic quantifier Q, which can be represented as a fuzzy subset on the unit interval.

Furthermore, Yager and Filev^[51] proposed the induced ordered weighted averaging (IOWA) Operator, which is a more general type of OWA operator and given as follows:

Definition 25

(see [51]) An IOWA operator of dimension m is a function Φ_w : (ℜ × ℜ)ⁿ → ℜ such that

Here the associated weight vector is w = (w₁,··· ,w_n) with w_i ∈ [0,1] and ∑i=1nwi=1. 〈μ_σ(i), p_σ(i)〉 is the two-tuple with the ith highest value μ_σ(i) in the set {μ₁, μ₂, ⋯ , μ_n}. σ is a permutation of {1, 2, ⋯ , n} such that μ_σ(i) ≥ μ_σ(i+1),∀i = 1, 2, ⋯ , n – 1.

It is obvious that the OWA and the IOWA operators are feasible to aggregate the information expressed by additive reciprocal preference relations in group decision making^{[52, 53]}. Then the consistency and the transitivity of the collective additive reciprocal matrix should be investigated. When individual additive reciprocal matrices are all additive consistent according to Definition 4, we can affirm that the collective additive reciprocal matrix by using the OWA and the IOWA operators is additively consistent. But when individual additive reciprocal matrices are all multiplicative consistent according to Definition 5, one cannot assure that the collective additive reciprocal matrix by using the OWA and the IOWA operators is multiplicative consistent. The main reason is that the additive consistency is archived by using the linear addition method and the multiplicative consistency is reached by using the multiplication method. For the weak transitivity of the collective additive reciprocal matrix, we cannot find any result in the literature, and this topic should be studied in the future. In addition, the IOWA operators can be used to aggregate interval additive reciprocal preference relations and the corresponding group decision making problems have been considered in [54].

Definition 26

(see [55, 56]) An OWG operator of dimension n is a mapping φ^G : Rⁿ → R such that

where w=(w1,w2,⋯,wn)(wi∈(0,1),∑i=1nwi=1) is the associated weight vector and b_j is the jth largest element in the collection a₁, a₂, ⋯ , a_n.

The methods of determining the associated weight vector w are similar to those given in [49]. The induced ordered weighted geometric (IOWG) operator is further defined as follows:

Definition 27

(see [57]) An IOWG operator of dimension n is a mapping ϕwG:Rn→R such that

where w=(w1,w2,⋯,wn)(wi∈(0,1),∑i=1nwi=1) is the associated weight vector and b_j is the jth largest element in the collection a₁, a₂, ⋯ , a_n. 〈u_σ(i), a_σ(i)〉is the two-tuple with the ith highest value u_σ(i) in the set {u₁, u₂, ⋯ , u_n}. σ is a permutation of {1, 2, ⋯ , n} such that

When the associated weight vector w = (1/n, 1/n, ⋯ , 1/n), the OWG and the IOWG operators are reduced to the geometric mean (GM) operator^[57]. Similar to those in applying the OWA and IOWA operators, once a method of determining the associated weight vector w is given, one has the corresponding OWG and IOWG operators. In virtue of Theorem 1, it is found that the collective multiplicative reciprocal matrix by using the OWG and IOWG operators is consistent or acceptably consistent when all individual ones are consistent or acceptably consistent. On the other hand, applying the OWG operator or the IOWG operator to aggregate interval multiplicative reciprocal matrices, the similar results can be obtained if Definition 9 is used to characterize the acceptable consistency of interval multiplicative reciprocal preference relations^[24]. Specially, it should be pointed out that the formula of obtaining the associated exponential weight vector in [24] is only suitable in the case that the consistency indexes are lower than 1. For the case that the applied consistency-index values are higher than 1, the normalized method as the one in [58] can be used to modify the proposed formula in [24]. In the end, it is concluded that the consistency of the group is dependent on the consistency of each individual decision maker, and the inconsistency indexes of the collective preference relations may be dependent on the permutations of alternatives^{[24, 58]}. When an individual decision maker is in extremely contrary to the most decision makers in the group, it should be considered carefully how to get the consensus and achieve a reasonable decision.