Geometric consistency index for interval pairwise comparison matrices

Abstract Interval pairwise comparison matrices are widely accepted for practical decision making problems when the decision maker is unable to provide an exact judgment on the alternatives. However, as measuring the preference consistencies in pairwise comparison decision making problems is important, this paper proposes a new interval pairwise comparison matrix consistency measure, the average geometric consistency index, that assumes that the preference in a given interval follows the lognormal distribution. This geometric consistency measure accounts for the interval boundaries and uncertainties. As it is often difficult to rationally rank alternatives when interval pairwise comparison matrices are highly uncertain and/or inconsistent, we propose an optimization model to reduce the inconsistencies of these matrices while minimizing information loss and controlling uncertainties. An interval priority vector is derived to rank the alternatives. The feasibility and efficiency of the models are demonstrated using numerical examples.


Introduction
In multi-criteria decision making, when the decision maker (DM) cannot directly assign criteria weight or score to alternatives, pairwise comparison (Thurstone, 1927) has often been employed as an intermediate step. Pairwise comparison matrices (PCMs, also called multiplicative preference relations) in the analytic hierarchy process (AHP) (Saaty, 1980) and fuzzy preference relations (Tanino, 1984) are the most widely used preference relations.
When using PCMs for decision making, it is important that the DM's preferences are consistent. There are two main consistency measures used to assess pairwise comparison matrix inconsistencies: consistency ratio (CR) (Saaty, 1980) and geometric consistency index (GCI) (Crawford & Williams, 1985). Saaty (1980) suggested that a PCM was acceptable when CR 0:1: Aguar on and Moreno-Jim enez (2003) provided the threshold values of geometric consistency index for CR 0:1: For other consistency measures, please refer to Brunelli (2018). Recently, many studies have sought to reduce the inconsistency of PCMs (Aguar on et al., 2021;Boz oki et al., 2015;Wang & Wu, 2021;Wu & Tu, 2021). Boz oki et al. (2015) designed a mixed 0-1 convex programming to modify minimal number of elements to achieve the threshold of acceptable inconsistency (CR 0:1). Aguar on et al. (2021) proposed a procedure to reduce the inconsistency level measured by GCI. Wang and Wu (2021) developed a goal programming model to reduce the inconsistency level measured by GCI. According to Wu and Tu (2021), since transitivity is the basis of normative decision-making, a model is constructed to simultaneously control both PCMs' inconsistency level and intransitivity.
However, due to the complex and uncertain socio-economic environments and the intrinsically subjective nature of human judgments, it is sometimes unrealistic or impossible to obtain exact judgments. A natural way to deal with these kinds of uncertain cases is to depict uncertain judgments using probability distributions (Arbel & Vargas, 1993;Dong et al., 2016;Escobar & Moreno-Jim enez, 2000), fuzzy sets (Buckley, 1985;Ishizaka et al., 2020;Leung & Cao, 2000) or interval ratios (Conde & de la Paz Rivera P erez, 2010;Saaty & Vargas, 1987). Similar to PCMs, an interval pairwise comparison matrices (IPCM) consistency measure ensures that the DMs are neither illogical nor arbitrary in their preferences, with a highly consistent IPCM generally able to significantly improve the judgment quality of the subjects and ensure a more credible ranking order for the comparison objects. However, no common consistency index can be computed directly from the IPCM. Wang et al. (2005a) provided a simple but practical method to check whether IPCMs were consistent. In Wang et al. (2005a), the consistency definition is based on a convex feasible region concept that ignores the transitivity between three or more comparisons in an IPCM. It therefore leads to two problems: IPCMs are often deemed consistent when interval judgements are highly uncertain, and optimal solutions are not unique and different solutions correspond to different ranking vectors. Krej c ı (2017) defined an IPCM as consistent by adding some special constraints to the IPCM consistency definition proposed in Wang et al. (2005a). Although the consistency definition given by Krej c ı (2017) is stricter than the consistency definition in Wang et al. (2005a), they have the same shortcomings. Liu (2009) defined an acceptable IPCM based on two PCMs obtained from IPCM's boundary information. However, the consistency definition in Liu (2009) highly depended on alternative labels and was not robust to decision alternative permutations. To cope with this problem, Liu et al. (2017) proposed a new concept of approximative consistency to measure the inconsistency level in interval judgements with permutations. Conde & de la Paz Rivera P erez (2010) proposed a linear optimization model to define a consistency index for IPCMs. Based on quasi-positive interval and quasi-consistent IPCM ideas, Meng and Tan (2017) proposed an IPCM consistency definition. It might require 2 0:5nðnÀ1Þ À1 QIPCMs to judge the consistency of an n-order IPCM using this definition, which means that to test an IPCM based on the definition of Meng and Tan (2017) is a complex work (Cheng et al., 2021). Cheng et al. (2021) presented a new consistency concept of IPCM having robustness and invariance regarding permutation of the compared objects' labels, which is stricter than Meng and Tan (2017) and Li et al. (2016). However, Cheng et al. (2021) also did not take into account the interval uncertainty.
In summary, the existing consistency measures for IPCM have the following shortcomings: 1. Consider Example 2.6. Suppose we only consider integers or reciprocal of integers between the upper and lower boundaries. For example, there are 2bÀ1 judgements in the interval ½ 1 b , b with b being an integer. Clearly,Ã 1 contains ð2bÀ1Þ 10 PCMs. Hence, the consistency index ofÃ 1 should be determined by ð2bÀ1Þ 10 PCMs. If the judgment can take any real number in the interval, the number of associated PCMs will be infinite. However, existing IPCM inconsistency indices (Cheng et al., 2021;Krej c ı, 2017;Liu, 2009;Meng & Tan, 2017;Wang et al., 2005a) are not determined by accounting for all associated PCMs. 2. Existing consistency definitions (Cheng et al., 2021;Krej c ı, 2017;Li et al., 2016;Liu, 2009;Meng & Tan, 2017) do not fully consider IPCM uncertainties. IPCMs with a higher uncertainty are more probable to be acceptable by these consistency measures (see Example 2.6). Saaty and Vargas (1987) believed that "large ambiguity in the judgments can render ranking a useless pursuit", that is, a highly uncertain IPCM with little or no useful decision information is not acceptable for actual decision-making (Entani & Sugihara, 2012;Kuo, 2020;Li et al., 2016). Hence, the IPCM consistency index, an index to determine whether an IPCM is acceptable needs to account for uncertainty.
It is difficult to reasonably rank alternatives when the IPCMs are highly inconsistent. There have been a few studies on improving inconsistent IPCMs. For example, Liu et al. (2017) improved the consistency of the boundary PCMs by the consistency improving method in Xu and Wei (1999); Cheng et al. (2021) proposed an iterative algorithm of improving the consistency level of an IPCM; Dong et al. (2015) proposed an LP-based model to reduce IPCM inconsistency and uncertainty by minimizing the amount of changed judgements; Zhang and Pedrycz (2019) proposed a consistency optimization approach to jointly improve the consistency degrees of several IPCMs that do not satisfy the predefined consistency threshold; Gong et al. (2020) used the linear uncertain distribution to represent DM's interval judgments and proposed an optimization method to reduce the inconsistency of the judgments.
Deriving a reliable and reasonable priority vector of an IPCM is a vital process when applying an AHP with IPCMs. Sugihara et al. (2004) suggested that priority vectors be estimated as intervals because of the uncertainty of the DMs' subjective judgments. So far, there are mainly two approaches to derive priority vector from an IPCM: a simulation-based approach (Gong et al., 2019;Moreno-Jimenez & Vargas, 1993;Saaty & Vargas, 1987), and a deterministic approach (Arbel, 1989;Dong et al., 2015;Gong et al., 2019). Saaty and Vargas (1987) used a sampling experiment to estimate the interval priority vectors and studied the impact of imprecise pairwise judgments on rank reversal. The deterministic approaches to deriving priority vectors (Arbel, 1989;Dong et al., 2015) only consider the boundary PCMs of the IPCMs and ignore DMs' interval information. A simulation-based prioritisation method (Basak, 1991;Saaty & Vargas, 1987;Zhu et al., 2016) models the uncertainty in an IPCM using different probability distributions, which considers more interval information than the deterministic approaches. For example, Zhu et al. (2016) sampled 10,000 PCMs from an IPCM to estimate the priority vector and consistency index. However, whether enough interval information is considered depends on the number of samples. Small samples mean that little interval information is considered.
To deal with the mentioned shortcomings in consistency measures, consistency improving and prioritisation, this paper focused on IPCM consistency measures and prioritisation from a probability distribution perspective. The contributions of this paper are threefold: first, we propose a new consistency measure for IPCM from a probability distribution perspective, called the average geometric consistency index, which is determined by the length of the interval, the lower and upper judgements of the interval. Consequently, more uncertainty implies a higher inconsistency. Second, we are able to directly derive the priority vector from the IPCM as the preferences are assumed to be random variables. Third, when the IPCMs are highly inconsistent, an optimization model is constructed to improve the consistency by minimizing information loss and uncertainty.
The remaining sections are organized as follows. Section 2 gives some basic concepts. In Section 3, we propose a new consistency index and the IPCM prioritisation method. Section 4 constructs an optimization framework to improve the IPCM inconsistency. In Section 5, we provide some examples to validate the feasibility and efficiency of the proposed models. Finally, conclusions are given in Section 6.
Definition 2.1. (Saaty, 1980) Specifically, a ij > 1 denotes that x i is preferred over x j , a ij ¼ 1 denotes that alternatives x i and x j are indifferent, and a ij < 1 denotes that x j is preferred over x i . Definition 2.2. (Saaty, 1980) In practice, a PCM is rarely consistent due to DM's limited global perception. For example, x i is a times better than x k , x k is b times better than x j , but x i is not a Â b better than x j . Objective data also suffers from inconsistency, such as sports results (Boz oki et al., 2016), international remittances (Petr oczy, 2021), student preferences (Csat o & T oth, 2020). Thus, Saaty (1980) proposed the following consistency ratio (CR) to measure the inconsistency of A: where k max ðAÞ is the maximal eigenvalue of A, RI(n) is the random index. If CR 0:1, then A is said to be acceptable.
Definition 2.3. (Crawford & Williams, 1985) Let Q n ¼ fw ¼ ðw 1 , w 2 , :::, w n Þ T jw i >0, P n i¼1 w i ¼ 1g: The geometric consistency index (GCI) is defined as: where GCIðAÞ GCI, then A is acceptable; otherwise, the PCM needs to be adjusted. In many decision-making situations, pairwise comparisons are often made when there is uncertainty and vagueness. Therefore, Saaty and Vargas (1987) presented the IPCM for modeling DM's uncertain judgments.
Definition 2.4. (Saaty & Vargas, 1987 In an IPCM, the uncertainty of DM's judgment is measured by the length of interval. nÂn be an IPCM. Then, its uncertainty index (UI) is defined by where D ij reflects the uncertainty of intervalã ij ¼ ½a À ij , a þ ij and UIðÃÞ measures the average uncertainty of the IPCMÃ: When UIðÃÞ ¼ 0, IPCMÃ degenerates to a crisp PCM. The greater the UIðÃÞ, the more the uncertainty ofÃ: Li et al. (2016) thought that a highly uncertain IPCM with little or no useful decision information should be deemed unacceptable, and they provided the indeterminacy ratio ij À ln a À ij , therefore, D ij and IRðã ij Þ have the same effect in measuring the uncertainty of the intervalã ij : To illustrate the uncertainty of IPCM, an example is given below.
where 1 b 9: If we only consider the judgements in the interval ½ 1 b , b belonging to the Saaty's scale, there are 2bÀ1 judgements in the interval ½ 1 b , b with b being an integer. For example, when b ¼ 3, every element in the upper triangle ofÃ 1 contains 5 judgements, that is, f1=3, 1=2, 1, 2, 3g, andÃ 1 contains 5 10 PCMs. For an n-order IPCM, whose elements in the upper triangle are ½ 1 b , b with b being an integer, this IPCM contains ð2bÀ1Þ 0:5nðnÀ1Þ PCMs.
Definition 2.7. (Escobar & Moreno-Jim enez, 2000) A random variable X is reciprocal with respect to a point c (c > 0), when X/c and c/X are identically distributed, that is, Escobar and Moreno-Jim enez (2000) have proved that the lognormal distribution ln Nðl, r 2 Þ is a reciprocal distribution with respect to e l :

Geometric consistency measure of IPCMs
This section first gives some basic assumptions for modelling interval judgements using lognormal distribution. Then, the average and variance for the IPCM geometric consistency indexes are given. Finally, prioritisation by IPCMs is proposed.

Geometric consistency measure
For the sake of simplicity, let a ij 2ã ij ¼ ½a À ij , a þ ij : The interval judgements are modelled using a lognormal distribution, with the main assumption being as follows: Assumption 1. Let a ij be a random variable, a ij be distributed lognormally with mean l ij and variance r ij . It is denoted as ln a ij $Nðl ij , r ij 2 Þ, where and Consider Example 2.6. As per Equations (10) and (11), it is known that l ij ¼ 0 and r ij ¼ 2 ln b 6 , therefore, ln a ij $N 0, ð 2 ln b 6 Þ 2 for all i, j 2 N: Assumption 1 is based on the following reasons: i. Crawford and Williams (1985) proved the geometric mean method by two different approaches: the logarithmic least squares method and the maximum likelihood estimation by supposing the errors to be lognormal distributions. Other papers also suggested that the subjective errors associated with each comparison ratio in crisp PCMs could be sufficiently modelled using a lognormal distribution with zero mean and constant variance (Altuzarra et al., 2010;Basak, 1991;Shrestha & Rahman, 1991). To ensure the comparison errors follow lognormal distributions in the context of IPCMs, judegments in IPCMs should be modelled by lognormal distributions. ii. Reciprocity is the basis of decision-making with PCMs (Saaty, 1986). To be consistent with the reciprocity of the AHP, the random variable a ij must satisfy Equation (9). That is, Pða ij ! xÞ¼Pða ji 1=xÞ, 8x 2 R þ : Clearly, modelling the preferences in the interval through the lognormal distribution and uniform distribution can satisfy the reciprocity. To better understand the reciprocity, consider an interval a ij ¼ ½ 1 b , b, which means Pðx i ! x j Þ ¼ 0:5: If we first take the logarithm of the interval and assume ln a ij follows the uniform distribution, then Pðx i ! x j Þ ¼ ln bÀ ln 1 ln bÀ ln 1 b ¼ 0:5 which is consistent with the actual ranking. If we assume a ij follows the uniform distribution, indicates that x i is better than x j which is not consistent with the actual ranking. Since the reciprocity of IPCMs, intervalsã ji and a ij contain the same information to rank alternatives x i and x j , that is, the ranking of x i and x j fromã ij should be consistent with the ranking of x i and x j fromã ji : By Equation (6) indicates that x j is better than x i and the ranking of x i and x j is not consistent with the ranking derived from a ij . iii. Equation (8) shows that the uncertainty (also represented the interval information) of interval a ij ¼ ½a À ij , a þ ij is modelled using D ij ¼ ln a þ ij À ln a À ij : Thus, the probability of ln a ij being distributed in the interval ½ln a À ij , ln a þ ij should be close to 1. According to the 3r principle for normally distributed variables, PðjxÀlj 3rÞ ¼ 0:997: By (10) and (11), we know that the interval lnã ij ¼ ½ln a À ij , ln a þ ij is able to retain 99.7% information of the random variable ln a ij : Lemma 3.1. Letã ij ¼ ½a À ij , a þ ij be an interval ofÃ, then a ij in the intervalã ij ¼ ½a À ij , a þ ij can be written as follows: where a ij $N 1 2 , 1 36 À Á : Lemma 3.1 is introduced to show the relationship between the uncertainty (D ij ) and consistency and simplify the calculation of EðGCIðÃÞÞ: The proofs in this paper can be obtained from the supplemental material online.
Definition 3.2. LetÃ be an IPCM. The geometric consistency index ofÃ is where where GCIðÃÞ, a ij , w i and w j are random variables. If PðGCIðÃÞ>0Þ ¼ 0, thenÃ degenerates to a crisp PCM and is perfectly consistent.
Definition 3.3. For an IPCMÃ, if its average geometric consistency index is below the threshold, that is then the inconsistency level ofÃ is acceptable.
Remark 1. The most popular consistency measure is CR proposed by Saaty (1980). Intuitively, the average consistency ratio EðCRðÃÞÞ should be applied to the interval pairwise comparison matrixes rather than EðGCIðÃÞÞ: However, it is difficult to express CRðÃÞ by upper and lower bounds and interval length. Kułakowski et al. (2021) showed that the eigenvalue method (Saaty, 1980) and geometric mean method (Crawford & Williams, 1985) would lead to more and more similar priority vector under conditions of decreasing inconsistency. Besides, the geometric consistency index has been approximated on the basis of the famous 0.1 rule for CR in Aguar on and Moreno-Jim enez (2003). Thus, we use EðGCIðÃÞÞ to measure the inconsistency of IPCM for the ease of calculation.
Lemma 3.4. Let CI ij ¼ ln ða ij ÞÀð ln ðw i ÞÀ ln ðw j ÞÞ, then CI ij $NðEðCI ij Þ, r 2 ðCI ij ÞÞ, 1 i<j n, where and Remark 2. EðCI ij Þ depends on the lower bound and the uncertainty of intervalã ij : r 2 ðCI ij Þ only depends on the uncertainty of intervalã ij , the greater the uncertainty, the larger the r 2 ðCI ij Þ: When IPCMÃ degenerates to a crisp PCM, r 2 ðCI ij Þ ¼ 0 Property 1. Let CI ij ¼ ln ða ij ÞÀð ln ðw i ÞÀ ln ðw j ÞÞ, then the following equations hold.
As Property 1 shows, the computation process of EðGCIðÃÞÞ needs the same operation ðnÀ1ÞðnÀ2Þ=2 times, thus, the time complexity of judging whether an IPCM is acceptable is Oðn 2 Þ: It requires 2 0:5nðnÀ1Þ À1 QIPCMs to judge the consistency of an n-order IPCM by the definition of Meng and Tan (2017), and the time complexity is nðnÀ1ÞðnÀ2Þ=6 to judge whether a QIPCM is acceptable. Thus, the total time complexity is Oðn 3 2 n 2 Þ: The time complexity of Zhu et al. (2016) is OðPn 2 Þ, where P is the number of samples. Compared with the methods proposed by Zhu et al. (2016) and Meng and Tan (2017), the average geometric consistency measure needs less time complexity.
The average geometric consistency measure is extended by the geometric consistency measure. Hence, the average geometric consistency measure can satisfy the following axiomatic properties of inconsistency measures: the invariance under permutation of alternatives (Brunelli & Fedrizzi, 2015

Prioritisation by interval pairwise comparisons
Definition 3.5. LetÃ be an IPCM. The priority vector U ¼ ðu 1 , u 2 , :::, u n Þ T is defined by where a ij and u i are random variables.
Theorem 3.6. Let U ¼ ðu 1 , u 2 , :::, u n Þ T be as before. u i follows the lognormal distribution denoted by ln u i $N P n j¼1 b À ij þ 1 2 P n j¼1 D ij , 1 36n 2 P n j¼1 D ij : Definition 3.7. LetÃ be an IPCM, the normalized priority vector W ¼ ðw 1 , w 2 , :::, w n Þ T is defined by where Property 2. The normalized priority vector W has the following properties: and iii. In practical decision making, alternatives are ranked using a priority vector; using ln w i $Nðlw i , r w i Þ, the probability of x i being better than x j can be written as follows: where ln ðw i Àw j Þ$Nðl w i Àl w j , r 2 w i þ r 2 w i Þ: iv. By the 3r principle of normally distributed variables, PðjxÀlj 3rÞ ¼ 0:997: To transfer the random variablew i into the interval, thew i interval is estimated as follows: ForÃ 1 with b ¼ 2 in Example 2.6., as per Equation (22) and (23), for all i 2 N : ln w i $Nð0, 0:0085Þ: From Equation (26), the priority vector ofÃ 1 can be determined, 4. Optimization models to improve the consistency measure In this section, a two-stage programming model is suggested to improve the inconsistency and minimize the number of changed judgements (NOC) and the amount of changed judgements (AOC). When the PCM suffers from significant inconsistency, it is difficult to get reliable and rational rankings. For an unacceptable IPCM, the DM needs to reconsider and modify their preferences under the guidance of the moderator to reduce the inconsistency. Saaty (2003) suggested that the inconsistency should be improved by making slight changes to judgements that further improve the inconsistency measure. Therefore, this paper provides a framework for improving consistency level based on an optimization method that considers information loss and uncertainty.

Stage 1
The first stage seeks to minimize AOC, that is The optimization model, therefore, is constructed as follows for i 2 N, i<j : UIðÃÞ UI, Constraint (28-1) ensures that the modified IPCM satisfies the predetermined consistency level, however, GCIð AÞ is a quasi v 2 distribution that can not be directly solved. There are two classic approaches to deal with such problems,

PðGCIð
AÞ GCIÞ ! 1Àa, where a is the confidence level, which is usually set as 0.05; and

EðGCIð
AÞÞ GCI: This paper transforms the stochastic optimization model to an optimization model by replacing Constraint (28-1) with Equation (30). Therefore, the optimization model is constructed as follows for i 2 N, i<j : UIðÃÞ UI, where M is a big value and it takes the value of 1000 in this paper, d À ij , d þ ij , a À ij and a þ ij are decision variables for i 2 N, i<j: Constraint (31-1) says that the average of GCIð AÞ must be less than the preset consistency threshold GCI: Constraints (31-2) and (31-3) judge whether the intervalã ij ¼ ½a À ij , a þ ij has been modified, with d À ij being equal to 1 if a À ij has been modified and 0 otherwise, and similarly, d þ ij is equal to 1 if a þ ij has been modified and 0 otherwise. Constraint (31-4) ensures that the uncertainty of the modified IPCM is no more than the predetermined UI: Constraint (31-5) enforces the modified IPCM satisfies the multiplicative reciprocity. Constraint (31-6) limits the range values for the adjusted intervalã ij ¼ ½a À ij , a þ ij :

Stage 2
The second stage minimizes NOC, which is expressed as where d À ij and d þ ij are two binary variables, ij Àa þ ij 6 ¼ 0 and d þ ij ¼ 0 otherwise. The objective of Stage 1 and Stage 2 is to adjust the IPCM to keep as much of the original judgement information as possible, that is, to minimize the information loss. Obviously, the less the information loss, the more the DM is willing to accept the adjusted IPCM.
For model (34), constraint (34-2) is a convex constraint, the other constraints are linear or mixed integer linear. Thus, model (34) is a mixed integer convex programming model. Because we only consider the nðnÀ1Þ 2 elements of the upper triangle of the IPCM, the total number of variables is 6 Â nðnÀ1Þ 2 ¼ 3nðnÀ1Þ where n means the dimension of IPCM. Psychological experiments have shown that individuals cannot simultaneously compare more than seven objects (plus or minus two) (Miller, 1956). Later, Saaty (1977) applied this conclusion to the AHP and suggested that the dimension of the PCM should not exceed 9. Hence, the optimal solution can be obtained quickly using standard software such as Gurobi and Cplex. In this paper, models (31) and (34) are solved using cutting plane method implemented by Gurobi (version 9.1.1).
The above models are called the continuous models since À ln 9 x À ij x þ ij ln 9: In some practical decision-making, the adjusted judgements are asked to belong to Saaty's original 1/9-9 scale, thus, the discrete model to improve the inconsistency is designed and presented in the supplemental material online.

Examples and comparisons
In this section, two numerical examples and comparisons are presented to test the feasibility and effectiveness of the proposed geometric consistency measure, prioritisation method and inconsistency improvement model.
As in Equation (iv) in Property 1, EðGCIðÃ 2 ÞÞ ¼ 0:18<0:35 is obtained, therefore,Ã 2 is acceptable. Using Equation (26), the priority vector is determined as To rank the alternatives, Equation (24) is used to obtain the ranking order x 1 1 1 x 2 1 0:7921 x 4 1 1 x 3 : The results for the different consistency definitions and prioritization methods are listed in Table 1. The ranking orders for the different methods are the same, x 1 1 x 2 1 x 4 1 x 3 , however, the probability of rank reversal,w i \w j 6 ¼ ;, is higher than the proposed model. A higher probability of rank reversal indicates more contradiction in the priority vector. Ahn (2017)'s priority vector is taken as an example,w 2 \w 3 \w 4 6 ¼ ;, it is challenging to correctly rank the alternatives fx 2 , x 3 , x 4 g: Compared to the prioritisation methods in Ahn (2017); Li et al. (2016); Liu (2009);Meng et al. (2015); Wang et al. (2005b); Wang and Lin (2019), this proposed prioritisation method in Equation (26) results in a minimal rank reversal.
EðGCIðÃ 3 ÞÞ ¼ 0:6765>0:350, UIðÃ 3 Þ ¼ 0:3517: Using Equation (26), the priority vector is According toWÃ 2 , the ranking order is x 2 1 x 1 1 x 3 1 x 4 : The priority vectors and ranking orders using methods in Islam et al. (1997); Li et al. (2016); Meng and Tan (2017); Wang et al. (2005b) and the proposed method for Example 5.2 are shown in Table 2. Kress (1991) found thatÃ 3 was inconsistent and therefore could not be solved using linear programming or the proposed modified method in Arbel (1989). Islam et al. (1997) then applied lexicographic goal programming to determine a point estimate for the priority vector from the upper triangular judgments forÃ 2 , i.e.WÃ 3 ¼ ð0:3030, 0:4545, 0:1515, 0:0900Þ T , which found x 2 1 x 1 1 x 3 1 x 4 : However, Wang et al. (2005b) demonstrated thatWÃ 3 ¼ ð0:3636, 0:3636, 0:1818, 0:0909Þ T and x 2 $x 1 1 x 3 1 x 4 when the point estimate for the priority vector determined from the lower triangular judgments. Because judgments in upper and lower triangle had the same information, Islam et al. (1997)'s method was defective in theory. Because of the uncertainties, it is more logical and acceptable to determine satisfactory interval priority vectors from an IPCM. As Table 2 shows, the ranking order obtained using the proposed method was the same as that obtained by the methods in Islam et al. (1997); Li et al. (2016); Meng and Tan (2017); Wang et al. (2005b), that is, x 2 1 x 1 1 x 3 1 x 4 : All proposed methods judgedÃ 3 to be unacceptable, therefore, the inconsistency needs to be further improved.
Using the proposed model (34), set UI ¼ 0:2845, in the first stage, the modified IPCM where the modified judgements are underlined is given as: A Stage 1 ¼   Table 3.
The ranking order for the proposed method was consistent with (Cheng et al., 2021;Liu, 2009;Liu et al., 2017). In Table 4, the EGCI, UIð A 3 Þ, NOC and AOC for the modified IPCMs in (Cheng et al., 2021;Liu, 2009;Liu et al., 2017) are presented. The modified IPCMs in Liu (2009);Liu et al. (2017) were found to be acceptable using Definition 3.3; however, all the judgments in the upper triangle of the IPCM were changed, that is, the modified IPCMs in (Liu, 2009;Liu et al., 2017) had greater information loss than the proposed method. The modified IPCM using the method proposed by Cheng et al. (2021) was not acceptable using Definition 3.3. And UIð A 3 Þ ¼ 0:4071>UIðÃ 3 Þ ¼ 0:3517 indicates that the method in Cheng et al. (2021) increased the IPCM uncertainty, that is, the uncertainty was not considered in the consistency improvement process. Compared with other methods, IPCM modified by the proposed method had less information loss, less uncertainty and greater consistency.

Conclusion
This paper proposed a new consistency index for IPCMs that considered boundary and uncertainty of interval judgements, and a two-stage programming model was designed to improve the consistency. The contributions of this paper are threefold: 1. We propose a new consistency measure for IPCM from a probability distribution perspective. Several useful properties for the new consistency measure, the average geometric consistency index, were given. The average geometric consistency index is independent of permutations of the alternatives, and considers all the PCMs in the IPCM and the interval uncertainty. 2. To determine the ranking order for the compared objects, a normalized prioritization method for the IPCM was proposed. Based on the assumption that the interval followed a lognormal distribution, an interval priority vector was derived. 3. A two-stage programming model was constructed to ensure the IPCM is acceptable, minimize information loss, and control uncertainty. The feasibility and efficiency of the proposed models were then proven using two examples and specific comparisons. Compared to existing consistency improvement methods, the proposed method was shown to provide an adjusted IPCM that had less information loss and lower uncertainty.
This paper was focused on an individual decision making method with an IPCM based on the proposed IPCM consistency measure. The famous inconsistency threshold of 0.1 for Saaty's CR has been recently extended for incomplete pairwise x 2 1 1 x 1 1 1 x 3 1 1 x 4   comparison matrices ( Agoston & Csat o, 2022). One of the future studies may try to extend CR for IPCMs. As group decision-making is becoming more common (Ding et al., 2020), group decisionmaking or large-scale group decision-making based on this consistency measure will be examined in future studies.