Improving Reliability for Linguistic Preference Relations Considering New Ordinal and Cardinal Consistency Measures

Linguistic preference relations (LPRs) and its variations served for different decision-making situations are significantly important instruments of qualitative decision-making. The consistency analysis of LPRs is a necessary prerequisite for further operations of these original preference relations so as to ensure that the final decision results are convincing. However, the existing consistency improvement methods for LPRs are difficult to guarantee the reliability of the revised LPRs. Because the suggested LPRs from these methods either not eliminate the ordinal inconsistency or not satisfy the expectation of minimum modification. In this paper, a new definition of ordinal consistency of LPRs is first proposed and then an optimization approach is constructed to eliminate ordinal inconsistency for LPRs. Secondly, a new cardinal consistency index for LPRs is proposed and a corresponding optimization model to increase the cardinal consistency level is presented. After that, an optimization model is proposed to simultaneously manage ordinal and cardinal inconsistency for LPRs. Last, the proposed models are applied to a real linguistic decision-making problem involving evaluation and selection of investment projects. The comparative analysis and discussion illustrate the applicability and effectiveness of the proposed models.


Introduction
Preference relation is one of the most popular decisionmaking techniques for the expression and modeling of decision makers' (DMs) preference information regarding decision-making problems [1][2][3]. The consistency analysis of preference relation, which includes ordinal and cardinal consistency analyses, is of great importance in decisionmaking using preference relations and it is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons [4][5][6][7][8]. The lack of consistency in decision-making using preference relations can lead to unreasonable conclusions. However, inconsistency of original individual preference relation is always hard to avoid due to DMs' limitations on knowledge and global perspective. Therefore, it is necessary to grapple with the inconsistency prior to the further operations of preference information. Furthermore, the key to deal with inconsistency of preference relations is to improve the reliability of the revised preference relations. In various decision making contexts such as in social choice and group decision making, both individual preferences and the aggregated group preferences are required to be transitive [9][10][11]. For preference relations, transitivity implies ordinal consistency. Therefore, for any kind of preference relations, ordinal consistency as a basic requirement ensuring the reliability of preference relations must be satisfied.
In plenty of real decision-making situations, the linguistic decision-making approach [12][13][14] has attracted a lot of attentions, because DMs often feel more comfortable providing their pairwise comparison judgements by means of linguistic variables initiated by Zadeh [15]. The consistency analysis of LPRs has been a research hotspot of many literature in related fields. Dong et al. [6] defined a consistency index to quantify the additive consistency of LPRs and proposed an optimization model and an iterative algorithm to deal with the additive inconsistency of LPRs. Alonso et al. [16] adapted the definition of additive transitivity of LPRs and proposed an interesting method to measure the consistency level of LPRs. Dong et al. [17] proposed a 2-tuple linguistic index that based on the individual consistency evaluation to measure the consistency level of LPRs. Jin et al. [18] introduced a new concept of order consistency and additive consistency for LPRs, and then proposed a consistency index to measure whether a LPR has an acceptable additive consistency level. Jin et al. [19] proposed a multiplicative consistency index of LPRs and developed an automatic iterative algorithm to improve the multiplicative consistency level for LPRs. Xu et al. [20] proposed a graphic method of gower plot to simultaneously ascertain and adjust the ordinal and additive inconsistencies for LPRs. Xu et al. [21] developed a distance-based nonlinear programming method to identify and adjust the ordinal and additive inconsistencies for 2-tuple LPRs. Tian et al. [22] offered an additive consistency index for LPRs and formulated an optimization model to obtain a new LPR that has an acceptable additive consistency level. Wu et al. [23] introduced an additive consistency index on the basis of the information of the original LPRs to check whether a LPR has an acceptable additive consistency level. Zhang et al. [24] proposed a multiplicative consistency index of LPRs on the basis of the LPRs' original information and then developed the consistency checking and repairing models to ensure the LPR has an acceptable multiplicative consistency level.
In the aforementioned literature, although many methods and models have been proposed to deal with the inconsistency of LPRs, there still are some gaps needing to be addressed.
1. Most proposed methods and models used to deal with the inconsistency of LPRs only took into account the cardinal consistency (additive consistency or multiplicative consistency) [6,16,19,22,24]. These models considering multiplicative or additive consistency can increase the cardinal consistency level of LPRs. Nevertheless, higher cardinal consistency level does not always imply ordinal consistency. For instance, a revised LPR obtained by the model in [22] is still ordinally inconsistent according to the definiton of ordinal consistency of LPRs. Even if the revised LPRs in these related models meet an acceptable cardinal consistency level, it is probable that the revised LPRs is also of ordinal inconsistency. Therefore, only the requirement for an acceptable cardinal consistency level is not enough to ensure that the final decision result is logically consistent. 2. Although the proposed models in [20,21] considered the ordinal inconsistency of LPRs, the definition of the ordinal consistency is not directly defined on the basis of LPRs and their model can not obtain the optimal solution directly. For instance, in [21], the distancebased nonlinear programming model used to identify and adjust ordinal inconsistency of LPRs is established based on the adjacency matrices of LPRs. However, it may lead to loss of information when converting original LPRs into its corresponding adjacency matrix. In this paper, we directly define some conditions of ordinal consistency on the basis of original LPRs and use several systems of inequalities to explicitly characterize these conditions. An optimization model will be then proposed to eliminate the ordinal inconsistency of LPRs.
After that, an optimization model is proposed to simultaneously manage ordinal and cardinal inconsistency for LPRs. These proposed models are effective offering DMs the suggestions of modification and guarantee the reliability of the revised LPRs.
The remainder of this paper is organized as follows. Some basic knowledge and laws of linguistic term sets (LTSs) and LPRs are introduced in Sect. 2. Two sets of different conditions for ordinal consistency of LPRs and an additive consistency index of LPRs are given in Sect. 3. Section 4 consists of several optimization models respectively to eliminating ordinal inconsistency of LPRs, to obtaining an acceptable additive consistency level, and to simultaneously managing ordinal and cardinal inconsistency of LPRs. An application case will be given in Sect. 5. Comparative analysis and discussion will be given in Sect. 6. Some concluding remarks will be given in Sect. 7.

Preliminaries
In this section, some basic concepts of LTSs and LPRs will be introduced respectively.

LTSs and Its 2-Tuple Linguistic Representation Model
The values of a linguistic variable are collected by a LTS in which each linguistic term is associated with its syntax and semantics. In this paper, suppose that S = {s t |t = −g, −g + 1, … , −1, 0, 1, … , g − 1, g} i s a finite and totally ordered discrete LTS whose granularity value is odd, such as 7 and 9, where s t represents a possible value of a linguistic variable. Note that these linguistic terms are assumed to be uniformly and symmetrically distributed and the subscripts of linguistic terms are symmetric about zero. It is required that s i and s j ( i, j ∈ {−g, −g + 1, … , −1, 0, 1, … , g − 1, g} ) satisfy the following properties: (1) The set is ordered. i.e., s i > s j if and only if i > j . Therefore, there exist a minimization and a maximization operator. Note that " s i > s j " means that the semantics of the linguitic term, s i , is better than s j . (2) There is a negation operator: neg(s i ) = s (−i) , most notably, neg(s 0 ) = s 0 . An example of an uniformly and symmetrically distributed LTS with a cardinality of 9 can be given as follows [25]: Without losing information during the operation, the discrete LTS S is extended to a continuous LTS S = {s t |t ∈ [−q, q]} , where q (q ≥ g) is a sufficiently larger positive integer. If s t ∈ S , then we call s t the original linguistic term; otherwise, we call it the virtual linguistic term. In general, DMs use the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in operation processes. Consider any two linguistic terms s , s ∈ S , and , 1 , 2 ∈ [0, 1] , there are some operational laws as follows [6].
Information representation models and the computation methods are needed when fusion processes are performed on linguistic terms. The 2-tuple linguistic representation model based on the LTS S = {s t |t = −g, −g + 1, … , −1, 0, 1, … , g − 1, g} , represents the linguistic information by a 2-tuple (s t , ) which belongs to the set U = {(s t , )|s t ∈ S, ∈ [−0.5, 0.5)} . Let ∈ [−g, g] be a value representing the result of a symbolic aggregation operation. The 2-tuple that expresses the equivalent information to is then obtained as follows [26]: s −4 = extremely poor, s −3 = very poor, s −2 = poor, s −1 = slightly poor, s 0 = fair, s 1 = slightly good, s 2 = good, s 3 = very good, s 4 = extremely good Numerical scale (NS), a concept initiated by [27], is defined as a one-to-one mapping used to convert linguistic term into numerical value. It is argued that the key of the computational techniques based on 2-tuple fuzzy linguistic representation models is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values.
Definition 1 [27] Let S = {s t |t = −g, −g + 1, … , −1, 0, 1, … , g − 1, g} be a LTS and R be a real number set. The function NS ∶ S → R is defined as a numerical scale of S and NS(s t ) is called as the numerical index of s t .
Without loss of generality, the numerical scale is defined as NS(s t ) = t when managing the inconsistency of LPRs in this paper. Therefore, the above operation on two linguistic terms can be converted to an operation on the subscripts of the corresponding terms. Let s t ∈ S , the subscript of the linguistic term s t can be obtained by the function: NS(s t ) = t . There exists an inverse function: Let S be a LTS. For convenience, four subsets with regard to S are given as follows:

Pairwise Comparisons and LPRs
Let A = {A 1 , A 2 , … , A n }(n ≥ 2) be a finite set of alternatives. When a DM provides pairwise comparisons using a LTS S, he/ she can construct a LPR. Definition 2 [24] A LPR L on the alternatives set A is a matrix L = (l ij ) n×n ∈ A × A , where l ij ∈ S estimates the preference strength of alternative A i over A j . For all i, j = 1, 2, … , n , l ij satisfy the following conditions: Definition 3 Let L = (l ij ) n×n be a LPR. The different value of l ij implies that A i has a different relationship with A j , which is defined as follows: (2) where A i ∼ A j means that the ith alternative is indifferent with the jth one, A i ≻ A j means that the ith alternative is superior to the jth one, A i ≺ A j means that the ith alternative is inferior to the jth one, A i ⪰ A j means that the ith alternative is not inferior to the jth one, and A i ⪯ A j means that the ith alternative is not superior to the jth one.

Consistency Analysis of LPRs
Two sets of ordinal consistent conditions, the definition of cardianl consistency and an acceptable additive consistency index for LPRs are respectively introduced in this section.

Ordinal Consistency of LPRs
Ordinal consistency of preference relation is on the basis of preference transitivity, which is the cornerstone of normative decision theory [4,28]. Ordinal consistency have been fully discussed for fuzzy preference relations [29][30][31], Inspired by these research, ordinal consistency for LPRs could be defined.
Definition 4 [21] Let L = (l ij ) n×n be a LPR. If for any i, k, j = 1, 2, … , n and i ≠ k ≠ j , the following four conditions are not violated.
then, the LPR L is of weak ordinal consistency.
Definition 5 [21] Let L = (l ij ) n×n be a LPR. If for any i, k, j = 1, 2, … , n and i ≠ k ≠ j , one of the following cases holds: then, the LPR L is ordinally inconsistent.
The conditions of the ordinal consistency of LPRs defined in Definition 4 is not that strict, and it is defined in a way that may be easily achievable in some real linguistic decision-making situations. For example, using the LTS S in Some strict conditions of ordinal consistency for LPRs should be considered. Let S be a LTS as before. If the preference intensity of alternative A is s t (t > 0) over alternative B, and alternative B is s l (l > 0) over C, then the strict conditions of ordinal consistency for LPRs require that the preference degree of alternative A over C should greater than or equal to the lager one between s t and s l . Inspired by the weakconsistency conditions of fuzzy preference relations in [32], several new conditions of the ordinal consistency for LPRs are proposed. Let L = (l ij ) n×n be a LPR. It is obvious that L is of weak ordinal consistency under the conditions of ordinal consistency in Definition 6.

Example 1 An expert use the LTS S in Eq.
(1) to compare alternatives in pairs for a given alternative set Then a corresponding LPR L = (l ij ) n×n can be specified as follows [20,33]: According to Definition 3, l 12 = s 0 and l 23 = s −2 imply that A 1 ∼ A 2 and A 2 ≺ A 3 . Then based on Definition 4, l 13 ∈ S L should hold if L is required to be of weak ordinal consistency. Or based on Definition 6, l 13 = s −2 should hold if L is required to be of ordinal consistency. But l 13 = s 3 , which means A 1 ≻ A 3 . Therefore, there exists a three-way cycle for alternatives A 1 , A 2 and A 3 . Based on Definition 6, other intransitive preferences of L are obtained from sets

Additive Consistency of LPRs
It is well known that the cardinal consistency of preference relations is stricter than the ordinal consistency of preference relations [30,31]. For LPRs, the additive consistency index and the multiplicative consistency index are two general tools to characterize the cardinal consistency level. In this paper, the additive consistency index is adopted to measure the cardinal consistency level of LPRs.
then the LPR L is of additive consistency.
Equation (3) is also called perfect consistency. However, this property is so strong that the LPRs derived from DMs in reality in general do not confirm to such property. A more realistic way is to relax the condition Eq. (3) and consider an acceptable additive consistency level.

Definition 8
Let L = (l ij ) n×n be a LPR. An additive consistency index (ACI) for L is defined as follows: Let ACI be a threshold determined in advance. Then L has an acceptable additive consistency level if ACI(L) ≥ ACI . Furthermore, the higher ACI(L), the better the additive consistency level. Most notably L is said to be consistent when ACI(L) = 1 . If ACI(L) < ACI , L needs to be revised until that ACI(L) ≥ ACI.
Example 2 Consider the LPR L in Example 1. Let ACI = 0.9 . According to Eq. (4), we can get that ACI(L) = 0.817 , which does not satisfy that ACI(L) ≥ ACI . Therefore L needs to be revised.

The Proposed Optimization Methodology to Improve Consistency of LPRs
In this section, two sets of inequalities are constructed to explicitly characterize the conditions of the ordinal consistency for LPRs. After that, an optimization model is proposed to eliminate the ordinal inconsistency of a LPR. Then, a second optimization model is developed to obtain a LPR with an acceptable additive consistency level. At last, a mathematical optimization model is proposed to simultaneously address the ordinal and additive inconsistency problems for LPRs.

Explicit Representation for the Ordinal Consistency of LPRs
Ordinal consistency is a minimum requirement to ensure that there is no mutually contradict judgments for alternatives over any combination {i, k, j}, i, k, j = 1, 2, … , n , for a given LPR. So, one needs to detect whether the original LPR is of ordinal consistency prior to using it. A straightforward way is to use the four conditions given in Definition 4 or 6. If one of the conditions is violated, then the given LPR is not of ordinal consistency and needs to be revised. Let L = (l ij ) n×n , where l ij ∈ S , be a LPR. The value of every element l ik ( i, k = 1, 2, … , n ) must belong to one of the following three situations: (1) l ik ∈ S U , (2) l ik = s 0 , or (3) l ik ∈ S L . In order to characterize these situations by means of inequalities, two binary variables, u ik and v ik ( i, k = 1, 2, … , n ), are introduced.

Definition 9
Let L = (l ij ) n×n be a LPR. The relationships between l ik and the two binary variables, u ik and v ik , are defined as follows:

be a LPR and M be a large positive integer. The three cases in Definition 9
can be constructed into the following system of inequalities: Proof (1) First, we prove that l ik ∈ S U ⇒ u ik = 1, v ik = 0 : When l ik ∈ S U , i.e., NS(l ik ) > 0 , so (5-2) enforces u ik = 1 , then (5-6) enforces v ik = 0 . Meanwhile, other expressions are always true. Then we prove that u ik = 1, v ik = 0 ⇒ l ik ∈ S U : When u ik = 1, v ik = 0 , (5-1) enforces l ik ∈ S U . Meanwhile, other expressions are always true.
This completes the proof. ◻

Optimization Model to Obtain Ordinally Consistent LPRs
Let N L = {(l ij ) n×n |l ij ∈ S, l ii = s 0 , l ij = neg(l ji )} be the set of all LPRs. Suppose that L = (l ij ) n×n is a revised LPR that meets all the conditions of ordinal consistency defined in Definition 4 or Definition 6. In general, in order to guarantee the reliability of revised LPRs, there are two ways to construct the objective function to maintain the original preference information as much as possible. One of them is that the DMs expect to minimize the amount of change (AOC) defined by some distance function d(L,L) . In this way, the optimization model can be obtained: In addition, the cost spent on persuading a DM to revise his/ her judgments depends on the number of changed judgments (NOC). Introducing a set binary variables ij , where ij = 1 if ̄l ij ≠ l ij ; otherwise, ij = 0 . Therefore, the corresponding optimization model is obtained as follows: The difference between model (8) and model (9) is the objective function. The requirement that L meets ordinal consistency can be achieved by considering the system of inequalities in Theorem 2 or Theorem 3 . Moreover, the claim that L is also a LPR can be realized by the constraints in Definition 2. Here, we only specify the optimization model with NOC. Therefore, based on Theorem 2 or Theorem 3, the panorama of model (9) can be respectively specified as follows.

Optimization Model to Obtain Acceptable Additively Consistent LPRs
Cardinal consistency improvement is needed even if a given preference relation meets ordinal consistency. Because ordinal consistency is only the minimum requirement. In this subsection, a mathematical optimization model is proposed to improve the additive consistency level of LPRs. The aim is to obtain a revised LPR with an acceptable additive consistency level. As mentioned in the previous subsection, there are also two ways to describe the objective function.
Here we only specify the optimization model with NOC. Let ij be defined as before and let ACI be a threshold. Then the model to obtain a LPR with an acceptable additive consistency level can be given as follows and it is denoted as P 3 : In model P 3 , the decision variables are ̄l ij , u ik , v ik , ij , d ij , f ikj and h ikj , for all i, k, j = 1, 2, … , n . Constrains (14-1) to  are used to guarantee that L has an acceptable additive consistency level. Constrains  to , which NS(l ij ) + NS(l ji ) = 0,  l ij ∈ S,  ij ∈ {0, 1},  i, k, j = 1, 2, … , n.  are transformed from (11), are used to maintain the original preference information as much as possible. Constraints  to  are used to guarantee that L is a LPR. Most notably, constraints (14-1) to  are transformed from the nonlinear constraint:

Optimization Model to Address Both Ordinal and Cardinal Consistency
For an original LPR, if merging model P 1 and P 3 , a revised LPR L with weak ordinal consistency and an acceptable cardinal consistency level can be obtained. If merging P 2 and P 3 , a revised LPR L with ordinal consistency and an acceptable cardinal consistency level can be obtained. Because the optimal solution of model P 2 must satisfy the four conditions in Definition 4. So by merging model P 2 and P 3 , an optimization model that simultaneously obtains a revised LPR with ordinal consistency and an acceptable additive consistency level can be constructed as model P 4 shown in the following: According to Sects. 4.1 to 4.3, the model P 4 is clearly valid.

Application to Investment Projects Evaluation and Selection
In this section, the proposed optimization model P 4 will be applied to the decision problem of evaluation and selection of investment projects. Investment companies continuously make investments for many reasons such as capacity building and technological innovation. It is necessary to consider many criteria for the evaluation of an investment project. These criteria are generally subjective and extremely difficult to express by numerical scales. Often, there are several alternatives that fit for the purpose of investment. Among these alternatives, the best option must be chosen.

Problem Description and Experts Evaluation
Suppose that an investment company intends to invest some money in one of five possible options: a car company, a food company, a computer company, an arms company and a TV company [22]. To make a better decision, a group with four investment risk assessment experts in related fields is set up to evaluate the five companies through the pairwise comparison method based on a linguistic scale. The alternatives set can be denoted as A = {a 1 , a 2 , a 3 , a 4  These individual LPRs are cardinal inconsistent if the threshold is set to 0.9. In addition, these individual LPRs are of ordinal inconsistency. Because there are at least one three-way circle in each LPR: In L 3 : a 1 ∼ a 2 ≺ a 3 ≺ a 1 .

Modification and Selection Processes
Using the proposed optimization model P 4 to these inconsistent LPRs. Then the revised LPRs L h (h = 1, 2, 3, 4) with ordinal consistency and an accepatable additive consistency level are obtained as follows: According to Eq. (4), the acceptable additive consistency levesl for these revised LPRs can be obtained as: ACI Bases on these revised LPRs, the process of aggregation can be started.
It must be mentioned that, in the next process of consensus and aggregation, we use the same methods proposed in [22] so as to control the comparability of results. Using the consensus measure (CM) computational formula (28) developed in [22], we can obtain the consensus level for these four revised LPRs as CM(e 1 , e 2 , e 3 , e 4 ) = 0.972 . Let the given consensus threshold CM = 0.9 , then these revised LPRs already reach a consensus. On this basis, the determination of the DMs' weights and the aggregation of individual LPRs are conducted as follows. Adopting the formulas (34) and (36) developed in [22], the weights of experts and the collective LPR L C can be obtained, respectively.  [22], the optimal priority weight vector of L C can be obtained as: Then the five investment projects are ranked as: Therefore, the most desired inverstment project is the car company.

Comparative Analysis and Discussion
In this section, two comparative analyses will be carried out. The first one is to compare the results of the proposed optimization model P 4 with the method developed in [22] which only considers cardinal consistency analysis. Then a comparison between the proposed optimizaion model and the existing method considering ordinal and cardinal consistency analysis is given.

Comparison with the Methods Only Considering Cardinal Consistency
For the case discussed in Sect. 5, Tian et al. [22] only considered to ascertain and improve the cardinal consistency level for individual LPRs prior to aggregate these LPRs into a collective LPR. Using models (32) and (33) developed in [22], the new LPRs with acceptable consistency levels and acceptable consensus levels are obtained as follows:  Although these revised individual LPRs have acceptable consistency level and acceptable consensus level, they are all not of ordinal consistency. Because there exist more than one three-way circles in each revised LPR, such as: The decision results based on intransitive perferences are unconvincing. Moreover, the linguistic scale of these revised LPRs is not in line with the original discrete LTS, which is hard to be interpreted and accepted by DMs. Based on these revised LPRs with ordinal inconsistency, Tian et al. [22] computed the weights of the DMs and aggregated them into the collective LPR.
Obviously, the collective LPR L Tian C is also ordinally inconsistent. Based on L Tian C , Tian et al. [22] derived the priority weight vector as: The rank is: This result is partly different from ours. Although the best alternative is the same, their result is based on self-contradictory LPRs. Therefore, the proposed optimizaton method in this paper can improve reliability of LPRs and ensure that the decision result is more convincing.

Comparison with the Methods Considering Ordinal and Cardinal Consistency
Let ACI = 0.9 . The LPR L in Example 1 is of ordinal inconsistency. Using the proposed optimization model P 4 to L , a revised LPR with ordinal consistency and an acceptable additive consistency level can be obtained as follows: ACI(L) = 0.967 according to Eq. (4). The number of changed judgement for L is 3, i.e., NOC(L) = 3 . The optimal solution L not only meets the expectation of minimum modification, but satisfies the requirment of consistency. That is, the proposed optimization model P 4 can improve reliability of LPRs. For L, the optimal revised result from the model proposed by Xu et al. [20] is as follows and denoted as L Xu : ACI(L Xu ) = 0.917 according to Eq. (4). The number of changed judgement for L Xu is 6, i.e., NOC(L Xu ) = 6.
Comparing the revised LPR L suggested by the proposed model P 4 with the revised LPR L Xu suggested by the model in [20]: So the proposed optimization model can provide a better solution under the condition of minimum adjustment. In other words, the proposed optimization method can offer DMs more advisable modified suggestions and improve the reliability of the revised LPRs.

Discussion
As mentioned in the introduction, extensive researches on consistency of LPRs focused on the cardinal consistency, including additive consistency and multiplicative consistency. Few researches studied the ordinal consistency of LPRs. An intuitive comparison of consistency analysis of LPRs is shown in Table 1. More details about these models and the proposed models are analyzed in the following.
1. Comparison with the methods in [6,18,22,23]: Dong et al. [6] defined an additive consistency index and established the consistency thresholds. Jin et al. [18] proposed two automatic iterative algorithms to improve the additive consistency level. Wu et al. [23] proposed an additive consistency index based on hamming distance and developed an integer optimization model to derive the acceptable additively consistent LPRs. Tian et al. [22] offered an additive consistency index to quantify the consistency level for LPRs. These methods all considered the measurment and improvement of additive consistency. But they did not guarantee that there is no self-contradictory judgement in their revised LPRs. 2. Comparison with the methods in [19,24]: Jin et al. [19] developed an automatic iterative algorithm to improve the multiplicative consistency level. A multiplicative consistency index of LPRs is proposed in [24] that depends on the LPR's original information. Then, the consistency checking and repairing optimization models are proposed to ensure the revised LPR has an acceptable consistency level. These methods considered the measurement and improvement of multiplicative consistency. But these models can not really guarantee that there is no contradictory judgement in the revised LPRs. 3. With regard to the research on ordinal consistency of LPRs: Xu et al. [20] used Gower plot to ascertain the cause of the ordinal inconsistency and additive inconsistency of LPRs. A procedure based on Gower Plot was proposed to adjust the elements of the inconsistent nodes. But the revised LPRs may still violate ordinal consistency. Xu et al. [21] developed a definition of ordinal consistency based on the adjacency matrices corresponding to LPRs. This definition is essentially in line with Definition 4. However, their proposed nonlinear programming model depended on a parameter, so it takes many iterations and comparisons to get the best solution.
In this paper, the proposed optimization models can directly eliminate the ordinal inconsistency as the definition of ordinal consistency of LPRs is directly based on the original linguistic preference information.

Concluding Remarks
These proposed optimization models improve the reliability of the revised LPRs. There are several contributions in this paper: (1) Several straightforward conditions of ordinal consistency for LPRs are put forward. The four conditions of ordinal consistency defined in Definition 6 guarantee that there are no conflicting judgments in DMs' revised LPRs. An acceptable additive consistency index for LPRs is given. It is potential to use the proposed methods to grapple with the consistency analysis of other popular liguistic decisionmaking approaches using preference relations, such as hesitant fuzzy linguistic preference relations, distribution linguistic preference relations, probabilistic linguistic preference relations and so forth. In group decision-making under linguistic preference information context, the proposed methods can be used in the consensus reaching process such as the ones in [34,35]. A further direction is to extend the proposed approach to incomplete LPRs. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.