A New Approach to Derive Priority Weights from Additive Interval Fuzzy Preference Relations Based on Logarithms

This paper investigates the consistency definition and the weight-deriving method for additive interval fuzzy preference relations (IFPRs) using a particular characterization based on logarithms. In a recently published paper, a new approach with a parameter is developed to obtain priority weights from fuzzy preference relations (FPRs), then a new consistency definition for the additive IFPRs is defined, and finally linear programming models for deriving interval weights from consistent and inconsistent IFPRs are proposed. However, the discussion of the parameter value is not adequate and the weights obtained by the linear models for inconsistent IFPRs are dependent on alternative labels and not robust to permutations of the decision makers’ judgments. In this paper, we first investigate the value of the parametermore thoroughly and give the closed form solution for the parameter.Then, we design a numerical example to illustrate the drawback of the linear models. Finally, we construct a linear model to derive interval weights from IFPRs based on the additive transitivity based consistency definition. To demonstrate the effectiveness of our proposed method, we compare our method to the existing method on three numerical examples.The results show that our method performs better on both consistent and inconsistent IFPRs.


Introduction
Since its introduction, the analytic hierarchy process (AHP) [1] method has been widely used in many applications and intensively studied by lots of researchers [2].AHP derives priority weights from pairwise reciprocal matrix.With the increase of the complexity of decision-making problems, the classical AHP has been extended in many aspects (scales used to measure the results of pairwise comparisons [3][4][5], the styles in which the pairwise comparisons carried out [6][7][8], combinations with other methods [9,10], uncertainty concerns [11][12][13][14][15][16], etc.).FPR which introduces fuzzy thoughts and methods into pairwise comparison is an important extension of AHP.The studies of FPR [17][18][19][20] mainly focus on the consistency issues and the derivation of priority weights.In FPR, the decision-makers (DMs) assign a real number between 0 and 1 to represent the degree of a preference relation.However, in some circumstances, the DMs would choose an interval number rather than a crisp value to represent preferences due to the uncertainty or lack of information.
IFPRs are FPRs with interval judgments.Many researchers have paid attention to the definition of consistency of IFPRs and the derivation of interval priority weights from IFPRs [21][22][23].The consistency issue is the foundation of the study on IFPR.There are two kinds of consistency: additive consistency [24] and multiplicative consistency [14].In this paper, we focus on the additive consistency, and for multiplicative consistency we refer to [14].
According to the additive consistency of FPR defined by Tanino [18], Xu et al. [25] proposed an additive consistency definition based on the feasible region.An IFPR is consistent if the feasible region is not empty.The interval priority weights are derived by minimizing and maximizing each weight in the feasible region for the consistent IFPRs and by minimizing the sum of deviations for the inconsistent IFPRs.Xu et al. [26] defined an additive transitivity based consistency, but Wang [27] pointed out that this definition was highly dependent on alternative labels and not robust to the permutations of the DMs' judgments.Wang et al. [28] defined another additive transitivity based consistency and proposed a goal programming model to obtain the interval priority weights.Liu et al. [29] transformed the interval fuzzy preference relation into an interval multiplicative preference relation and used the method in [30] to check the consistency.But Li et al. [13] pointed out that the definition in [30] was technical deficiency and yielded contradictory results for the same judgment matrix after the alternatives were relabeled.Dong et al. [12] defined the average-case consistency index as the average consistency degree of all FPRs associated with the IFPR.Krejčí [24] reviewed the definitions of additive consistency and proposed two new definitions.
In a recent paper, Wang et al. [19] put forward a new method with a parameter using a particular characterization based on logarithms to obtain priority weights from FPR and defined a new consistency definition based on the feasible region restricted by the characterization for additive IFPR.Based on the new definition, linear programming models for deriving interval priority weights from consistent IFPRs were proposed.For inconsistent IFPRs, they proposed to revise the inconsistent IFPR to a new consistent IFPR and then use the linear models designed for consistent IFPRs to derive interval weights.However, the value of the parameter is not fully investigated and the weights obtained by the proposed method for inconsistent IFPRs are dependent on alternative labels and not robust to permutations of the DMs' judgments.Although there are some drawbacks in [19], the basic idea is very interesting.The purpose of this paper is to illustrate the drawbacks in [19] and to improve these drawbacks.Firstly, we investigate the value of the parameter more thoroughly and we give the closed form solution for the parameter.Secondly, we show that the results obtained by the linear models are not robust to permutations of the DMs' judgments by a numerical example.Then, we propose a new method based on logarithms and the consistency definition proposed by Wang et al. [28] to derive interval weights from IFPRs.Our proposed method is robust to permutations of the DMs' judgments.Finally, we compare our method to the method in [19] on three numerical examples with respect to the fitted error.The results demonstrate the effectiveness of our method.
The rest of the paper is organized as follows.Some basic concepts and the main idea of Wang et al. 's method [19] are briefly reviewed in Section 2. The value of the parameter is discussed in Section 3. Section 4 illustrates the rankings derived by the linear models in [19] from inconsistent IFPRs are not robust to permutations of DMs' judgments.A linear model is proposed to derive interval weights from IFPRs in Section 5. Section 6 shows the effectiveness of the proposed model by numerical examples.The innovations of this paper and the future research directions are concluded in Section 7.
The value of   represents the preference degree of   over   .  > 0.5 means   is preferred to   and   < 0.5 indicates   is preferred to   and   = 0.5 shows that   and   are indifferent.Definition 2. A FPR  = (  ) × is additive consistent if it satisfies the additive transitivity [20]: =   −   + 0.5, for all , ,  = 1, 2, . . ., . ( Xu [20] points out that a FPR is additively consistent if there exists a normalized weight vector  = ( Wang et al. [19] proved that, given the parameter , the weight derived from a FPR can be represented as In order to determine the value of , Wang et al. [19] proposed the following constrained programming model in which  = ln .(5) Based on Theorem 5, Wang et al. [19] defined the additive consistency of an IFPR R = (r  ) × , r = [ −  ,  +  ] using the feasible region   .
For an inconsistent IFPR R, Wang et al. [19] proposed to revise R to acquire a consistent IFPR R * and derive interval weights from R * .

The Value of the Parameter Used in [19]
Wang et al. [19] ended the discussion of the value of  with the programming model (5).In this section, we continue the investigation and discuss the value of  more thoroughly.
As the objective function of problem ( 5) is a quadratic function whose quadratic coefficient is nonnegative, the minimum will be obtained at without considering the constraints.
So, we can get This completes the proof.
With this conclusion, it will be more convenient and efficient to derive weights or to revise the inconsistent FPRs by the methods in [19].
It is worth noting that the situation in which 0.5/max ,=1,2,..., (  −   ) ≤ 1 is satisfied is very extreme and unusual.If this situation do appear, the DMs will be able to determine that the ith object is the best or the jth object is the worst intuitively and they could remove the ith or the jth object and then reconstruct the matrix to get a more usual situation in which 0.5/max ,=1,2,..., (  −   ) ≥ 1.
In the following discussion, we will take  = 1 and  =  acquiescently.

The Invalidity of the Weight-Deriving
Method in [19] for Inconsistent IFPRs This section develops a numerical example to illustrate the technical deficiency of the weight-deriving method in [19] for inconsistent IFPRs.It is notable that Wang et al. [19] took  =  when they defined the feasible region   in ( 6), but they did not give the reason.From the discussion in Section 3 we can see that it is reasonable to set  = .) .(14) It can be verified that R is an inconsistent IFPR according to the definition 3 in [19] on page 73, so R has to be revised first.
By dividing R according to (9), we can get ) . ( From Theorem 6, we can get  1 = 1 and  2 = 1.Then, we can get the following two additive consistent FPRs according to (10)  ) . ( R * is an additive consistent IFPR, and the interval priority weights W = (w 1 , w2 , . . ., w ) derived from R * according to the linear models (7) and ( 8 We adopt the method proposed by Xu et al. [32] to compare the interval weights.Let ã = [ − ,  + ] and b = [ − ,  + ] be any interval numbers; then the degree of possibility of ã ≥ b is defined as As per (18), the preference relation matrix  = (  ) × ,   = (w  ≥ w ) can be constructed and the preference degree   = ∑  =1   can be used to rank the weights.
In this example,  = ( Comparing the two rankings, we can find that though the judgment information is the same, the ranks of  2 and  4 are contradictory in the two rankings.
Example 1 demonstrates that the rankings derived by the linear models in [19] from inconsistent IFPRs are not robust to permutations of DMs' judgments and rank reversal problem may arise when the alternatives are relabeled.

New Models to Derive Interval Weights from IFPR
In this section, we devise a formula to transform normalized interval weights into an additive consistent IFPR according to an additive transitivity based consistency definition proposed in Wang et al. [28] and develop a linear models to derive interval weights.

Consistency
As per Definition 7, P is an additive consistent IFPR.This completes the proof.
It is obvious that when the IFPR R = (r  ) × is reduced to a FPR  = (  ) × , Corollary 9 will be equivalent to Theorem 1 in [19] with  = 1.
Proof.As W = (w 1 , w2 , . . ., w ), w = [   ,    ] is a normalized interval weight vector; it is obvious that there exists a weight vector  = ( 1 ,  2 , . . .,   ), such that for all  = 1, 2, . . ., ,    ≤   ≤    , and So, we can say the feasible region   of P contains  = ( 1 ,  2 , . . .,   ), which means   is not empty and P is additive consistent under the definition provided by (6).25) and ( 26) hold for additive consistent IFPRs.However, it is difficult for DMs to provide consistent IFPRs due to the subjectivity of DMs' judgment and complexity of decision problems in many decision situations [33].As per Theorem 8, a normalized interval weight vector can reproduce an additive consistent IFPR.In order to derive suitable decision result from an inconsistent IFPR R, we seek for a normalized interval weight vector such that the reproduced IFPR P is close to R as much as possible under the sense that

Deriving Interval Weights from IFPR. Equations (
for all ,  = 1, 2, . . ., ,  ̸ = . It is obvious that the smaller the deviations between two sides of ( 27) and ( 28) are, the closer the P is to R. Therefore, the following nonlinear programming model is established to derive normalized interval weights from R.
It is obvious that the solution of model ( 29) will not depend on the permutations of the DMs' judgments.So, our nonlinear model will not suffer from the problem suffered by Wang et al. 's method.
The first two inequalities in model ( 29) come from the constraints in the definition of P in (23), and the last three lines are constraints to make sure the interval weights are normalized.
Solving model (34) we can obtain the optimal value  * and the corresponding interval weights.If  * = 0, we can draw the conclusion that R is additive consistent.
It is obvious that model (34) is a nonlinear optimization problem, and we can solve it using the MATLAB toolbox.Now we illustrate how to construct a linear model to derive the interval weights.
It is worth noting that ( 25) and ( 26) can also be written as The objective function of model ( 39) is to minimize the deviations between the two sides of ( 35) and (36).The objective function of model ( 29) is to minimize the deviations between the two sides of ( 25) and (26).As (35) and (36) are transformations of ( 25) and ( 26), the basic idea of model (39) and model (29) It is obvious the solution of model (44) will not change when the DMs' judgments are relabeled; i.e., our linear model will not suffer from the problem suffered by Wang et al. 's method.
Solving model (44) we can obtain the optimal value  * and the corresponding interval weights.If  * = 0, we can draw the conclusion that R is an additive consistent IFPR.

Numerical Examples
In this section, we take the three IFPRs used in [19] to demonstrate the effectiveness of the proposed linear models in Section 5.For convenience, NM, LM are used to denote Wang et al. 's method [19] and our linear model, respectively.For the sake of brevity, we only give the IFPRs used for computation.For detailed information about the three IFPRs, we refer to [19].
We adopt the fitted error [19] to measure the quality of interval weights derived by different models.The fitted error is defined as follows: where R is the original IFPR and P is the fitted IFPR transformed from the interval weights.The transformation can be done as per (22).
Example 2. The IFPR in this example is ) . ( For both models, R3 is additive consistent.Table 1 gives the interval weights, the fitted error, and the ranking of the weights derived by the two models.It is worth noting that the results obtained by NM in the original paper [19] are incorrect due to computational error.Table 1 shows clearly that our model reproduces the original IFPR perfectly and performs better than NM. ) .(47) NM identifies R4 as additive consistent while LM identifies R4 as inconsistent.The reason of this phenomenon is that the additive transitivity based consistency definition is stricter than the feasible region based consistency definition.