A personalized individual semantics model for computing with linguistic intuitionistic fuzzy information and application in MCDM

This paper develops a personalized individual semantics (PISs) model for computing with linguistic intuitionistic fuzzy information and applies to evaluate different brands of mobile phones. First, considering that a linguistic term means different things to different decision-makers, a consistency-driven optimization model for checking the additive consistent linguistic intuitionistic fuzzy preference relations (LIFPRs) is constructed by considering the PISs model. Besides, several optimization models are built to determine the PISs of linguistic terms with LIFPRs and obtain the acceptable additive consistent LIFPRs. Second, a new definition of Hamming distance for measuring linguistic intuitionistic fuzzy numbers (LIFNs) is developed by considering the PISs model, and several desirable properties are discussed. Then, the method of deriving the weight vectors of criteria is calculated based on the proposed distance measure. Subsequently, a framework of group decision-making (GDM) process with LIFPRs is offered, and the application of the proposed method is illustrated by using a multi-criteria decision-making (MCDM) problem about evaluating different brands of mobile phones. Finally, the comparative analysis is conducted to show the feasibility of proposed method.


Introduction
Group decision-making (GDM) is widespread in real-world decision-making problems. Many studies focused on all kinds of GDM problems and obtained fruitful results (Zhang et al. 2022a, Rabiee et al. 2021, Ding et al. 2020. For GDM under fuzzy and uncertain environment, linguistic information is common way to express evaluation information. These GDM problems are named linguistic GDM problems. Traditionally, linguistic GDM utilizes single linguistic terms to express evaluation information. In order to enrich linguistic expressions for decision-makers, a variety of linguistic formats are developed, such as interval linguistic (Liao et al. 2018), hesitant fuzzy linguistic term set (Rodríguez et al. 2021), probabilistic linguistic term set , flexible linguistic expression (Fan et al. 2022), linguistic distribution (Wu et al. 2021), multi-granular linguistic distribution , and hesitant intuitionistic linguistic distribution (Mahboob et al. 2022). Wu et al. (2021) provided a comprehensive review of the distributed linguistic representations in decision-making, including: taxonomy, key elements, applications, challenges in data science and explainable artificial intelligence. Herrera-Viedma et al. (2021) provided a brief tour through the linguistic decision-making trends, studies, methodologies, and models developed in the last 50 years.
Motivated by intuitionistic fuzzy set and linguistic term set, Chen et al. (2015) introduced the concept of linguistic intuitionistic fuzzy numbers (LIFNs). A LIFN expresses the decision-makers preferences with linguistic membership degree and linguistic non-membership, which the membership and non-membership are presented by linguistic terms. Since LIFNs can represent preferred and non-preferred qualitative judgments of decision-makers, the LIFNs are considered a flexible linguistic model for decision-makers to express their evaluation information. At present, LIFNs have been applied in many real-world decision-making problems, such as individual financial investment decision-making (Zou et al. 2021), three-way decision-making , scheme selection of design for disassembly (Wang et al. 2021a), analysis of socioeconomic development (Imanov et al. 2017), and system failure probability evaluation (Kumar and Kaushik 2020). And several GDM methods with LIFNs have been developed, such as TOPSIS method (Kumar and Chen 2022), MULTIMOORA method , and PROMETHEE method (Zhu and Zhao 2022). One can observe that several new multi-criteria decision-making (MCDM) methods such as MARCOS, CoCoSo, and MACONT have not been developed in linguistic intuitionistic fuzzy environments.
In linguistic decision-making processes, computing with words is an important point to note about. Many studies focused on the method of computing with words and obtained fruitful results. The common used method is by means of mathematical functions between numerical values and linguistic terms. Among them, the most notable ones are the following: (1) The method is based on 2-tuples linguistic representation model. This method is based on the canonical characteristic values of the linguistic terms to deal with linguistic term sets that are symmetrically distributed and not uniformly. Based on 2-tuples linguistic representation model, the numerical scales model is developed. (2) The method is based on numerical scales model. This method establishes a one-toone mapping between a numerical scale and the linguistic terms. Personalized individual semantics (PISs) model (Muhuri and Gupta 2020) is a common used method. (3) The method is based on specific mathematical functions. This method assigns numerical values to linguistic terms to represent its corresponding semantic. Utilizing subscript values of the linguistic terms for computing with words is a common method (Fu and Liao 2019). (4) The method is based on cloud model. Cloud model can characterize the uncertainties of linguistic information and identify the certainty degree of random variables using probability distributions. In order to make a comprehensive understanding to this computing with words methods, summary of these methods is listed in Table 1.
Although there are several methods for computing with words, shortcomings still exist in some methods. Such as, the method based on cloud model cannot guarantee the loss of information in the process of linguistic conversion, and the method based on 2-tuples linguistic representation model the specific mathematical functions are difficult to reflect words mean different things to different people, etc. In GDM, different decision-makers may have different understandings of words. There are mainly two methods to address this issue. The first one is utilizing multi-granularities linguistic term sets for decision-makers, and the second one is using a same linguistic term set for all decision-makers, while the decisionmakers' PISs of the linguistic information are considered. For the latter one, constructing some optimization models to maximize the consistency of linguistic preference relations is a common used method (Li et al. 2021a, Zhang andLiang et al. 2021), and it has been studied in several linguistic environments, such as incomplete linguistic , flexible linguistic expressions (Jiang et al. 2022), comparative linguistic expressions (Li et al. 2018;Zhang et al. 2021), probabilistic linguistic (Wan et al. 2022a), and linguistic distribution (Xiao et al. 2020;Tang et al. 2020).
Although the existing studies are effective for solving MCDM problems with LIFNs, there are still some limitations as follows. (1) Although several studies focused on solving the MCDM problems with LIFNs. Unfortunately, these studies did not take into account PISs. At present, some existing PISs model-based MCDM methods have been proposed for managing the PISs of linguistic term with comparative linguistic expressions, probabilistic linguistic, etc. But these methods are not suitable for managing the PISs in MCDM problems with LIFNs. (2) The existing studies assumed that the criteria weight vectors are the same for different decision-makers. However, the individuals may have their own criteria weight vectors due to the difference of preference, interest and background. PISs and criteria weight vectors have an important impact on GDM results.
The above analysis highlights some research achievements based on MCDM problems with LIFNs. However, there are still some shortcomings that remain to be addressed. The research motivations of this study are summarized as follows: (1) the advantage of LIFNs indicates that it will play an important role in MCDM problems, developing some measuring methods such as distance and consistency becomes necessary.
(2) Considering that PISs has an important impact on GDM problems with computing with words, it is necessary to develop a new method for managing PISs in MCDM problems with LIFNs.
In order to achieve these goals, the consistency measure from the perspective of additive consistent linguistic intuitionistic fuzzy preference relations (LIFPRs) is defined. And several optimization models are built to determine the PISs of linguistic terms with LIFPRs and obtain the acceptable additive consistent LIFPRs. The primary contributions of this study are summarized as follows.
(1) Considering that a linguistic term means different things to different decision-makers, a consistencydriven optimization model for checking the additive consistent LIFPRs is constructed by considering the PISs model. Moreover, several optimization models are built to determine the PISs of linguistic terms with LIFPRs and obtain the acceptable additive consistent LIFPRs.
(2) A new definition of Hamming distance for measuring LIFNs is developed by considering the PISs model, and several desirable properties are discussed. Besides, based on the proposed distance measure, the method of deriving the weight vectors of criteria is introduced.
The remainder of the paper is organized as follows. In Sect. 2, basic concepts related to LIFNs, numerical scale model, and PISs model are reviewed. In Sect. 3, the concept of acceptable additive consistent LIFPRs based on PISs model is introduced, and several programming models are developed for deriving the acceptable additive consistent LIFPRs. Besides, an algorithm is introduced to obtain complete or acceptable additive consistent LIFPRs. In Sect. 4, the MCDM problems with LIFPRs are introduced, and an optimization model is constructed for determining the weight vectors of criteria. Moreover, a framework of MCDM procedure with LIFPRs is developed. In Sect. 5, the proposed method is illustrated by an example, and a comparative analysis is provided. Finally, conclusions are presented in Sect. 6.

Preliminaries
To carry out the following research, this part briefly reviews some basic concepts, including the concepts of LIFNs, numerical scale model, and PISs model.

LIFNs
Considering the fact that linguistic variables can denote the qualitative preferred recognitions of decision-makers, Chen et al. (2015) introduced the concept of LIFNs, which denoted the qualitative preferred and non-preferred judgments of decision-makers simultaneously.
Definition 1 (Chen et al. 2015). A LIFN s $ on the continuous linguistic term set S c ¼ s a ja 2 0; 2t ½ f gis expres- where s a denoted the preferred qualitative degree, and s b denoted the non-preferred qualitative degree, such that s a È s b s 2t .
Obviously, LIFNs follow the principle of intuitionistic fuzzy numbers (IFNs). The difference between them is LIFNs used qualitative degree to denote the preferred and non-preferred degrees, while IFNs used to quantify numbers. Afterward, Meng et al. (2019a) proposed the concept of LIFPRs.
Definition 2 (Meng et al. 2019a). Let x ¼ x 1 ; x 2 ; . . .; x n f g be a finite object set, and S c ¼ s a ja 2 0; 2t ½ f gbe a continuous linguistic term set. The concept of LIFPRs R $ is defined as R i; j ¼ 1; 2; . . .; n, and s l ij denoted the preferred qualitative degree and s v ij denoted the non-preferred qualitative degree of the object x i over x j under the following conditions: To develop the concept of additive consistent LIFPRs, Meng et al. (2019a) first introduced the concept of reverse complementary LIFNs (RCLIFNs) s c ¼ s 2t s a ; s 2t s b À Á , where s 2t s a ¼ s 2tÀa and s 2t s b ¼ s 2tÀb , and then developed the concept of additive consistent LIFPRs.
Definition 3 (Meng et al. 2019a). Let x ¼ x 1 ; x 2 ; . . .; x n f g be a finite object set, S c ¼ s a ja 2 0; 2t ½ f gbe a continuous The method based on 2-tuples linguistic representation model Uses the ordinary for representing linguistic information Muhuri and Gupta (2020), Wang et al. (2022) and Giráldez-Cru et al. (2021) The method based on numerical scales model Establishes a one-to-one mapping between the linguistic terms and a numerical scale Dong and Herrera-Viedma (2015), Fan et al. (2022) and Jiang et al. (2022) The method based on specific mathematical functions Assigns numerical values to linguistic terms to represent its corresponding semantic Fu and Liao (2019), Zhang et al. (2022b) and Mi et al. (2021) The method based on cloud model Identifies the certainty degree of random variables using probability distributions Wang et al. (2021b), Xiao and Wang (2019) and Peng et al. (2019) A personalized individual semantics model for computing with linguistic intuitionistic... 4503 linguistic term set, and R $ ¼ r $ ij nÂn be a LIFPR. Then, R $ is additive consistency if and only if there is a set of the 0-1 indicator variables a ij , i; j ¼ 1; 2; . . .; n, and i\j such that: for each triple of i; j; k ð Þ, i\k\j, andr c ji is a RCLIFN. According to the relationship of interval linguistic fuzzy preference relations and LIFPRs, Meng et al. (2019a) provided the proof of Eq. (2) is equivalent to the following formulas: 2.2 Numerical scale model Definition 5 (Dong et al. 2009). The numerical scale model NS for s i ; a ð Þ is defined as follows: Obviously, the numerical scale defines a one-to-one mapping between a linguistic term set and a numerical scale. Afterward, Dong and Herrera-Viedma (2016) proposed the inverse operator of numerical scale model NS, which defines a one-to-one mapping between a numerical scale and a linguistic term set.
Definition 6 (Dong and Herrera-Viedma 2016). The inverse operator of numerical scale model NS is defined as follows: To demonstrate the validity of proposed numerical scale model,  provided a unified framework work of numerical scale connected to the 2-tuple linguistic model, the unbalanced linguistic model, and the proportional 2-tuple linguistic model.

Personalized individual semantics model
Since different decision-makers might have different understandings of the linguistic terms. In other words, words mean different things for different people. To address this issue, Li et al. (2017) developed the concept of PISs. Owing to its practicability and effectiveness, the use of PISs has been developed in various linguistic forms, such as, comparative linguistic expression (Fan et al. 2021), flexible linguistic expression (Jiang et al. 2022), probabilistic linguistic term sets (Wan et al. 2022a), and distribution linguistic term sets (Tang et al. 2020).
The PISs of linguistic terms have different formats, such as exact values (Li et al. 2017) and unit intervals (Li et al. 2018). In the following section, the method of computing PISs of the linguistic terms in linguistic preference relations is reviewed.
Let A ¼ a 1 ; a 2 ; . . .; a n f g and E ¼ e 1 ; e 2 ; . . .; e m f g , respectively, be the set of alternative and decision-makers. The decision-makers e 1 ; e 2 ; . . .; e m provide the pairwise comparisons in the form of linguistic preference relations . . .; n, k ¼ 1; 2; . . .; m. To derive the PISs of linguistic terms, Li et al. (2020) developed the following model: In Eq. (5), the objective function indicates the larger the value CI P k À Á , the more consistent P k is. The first to fourth constraints determined the range of numerical scales, and the fifth constraint assured the ordered of numerical scales. The constraint value k 2 0; 1 ð Þ is a small positive number, it can be set priori, such as k ¼ 0:01.

Acceptable additive consistent LIFPRs
In this section, the concept of acceptable additive consistent LIFPRs based on PISs model is introduced, and then, several programming models are developed for deriving the acceptable additive consistent LIFPRs. Finally, an algorithm is introduced to obtain complete or acceptable additive consistent LIFPRs.

The consistency of LIFPRs
Let s a and s b be any two linguistic numbers, k be a nonnegative real number within interval 0; 1 To further consider Eq. (3), according to the operations list above, it can be further simplified into: To facilitate following discussion, some notations are used to denote the subscript values that are listed in Eq. (7), In this way, Eq. (7) is equivalent to the following formulas: Let nÂn be a LIFPR on the finite object set If there is a set of the 0-1 indicator variables a ij , i; j ¼ 1; 2; . . .; n, and i\j makes Eq. (8) hold, then R $ is additive consistent LIFPR. Unfortunately, we cannot guarantee that Eq. (8) always holds. In other words, when the LIFPR R $ is inconsistent, Eq. (8) will not hold. Considering that the above-mentioned equations do not constantly hold in general given a deviation between s p ij;1 and s p k ij;2 , s q ij;1 and s q k ij;2 for a set of the 0-1 indicator variables a ij , i; j ¼ 1; 2; . . .; n, and i\j. Moreover, the more deviation s p ij;1 Ès p k ij;2 and s q ij;1 Ès q k ij;2 approach to 0, the more the consistency is. Based on this fact, the following programming model is constructed based on PISs model: In Eq. (9), the symbol d is the objective function. The larger the value d, indicates the more deviation s p ij;1 És p k ij;2 and s q ij;1 És q k ij;2 approach to 0. The constraints (9-1) and (9-2), respectively, denote the distance between s p ij;1 and s p k ij;2 , s q ij;1 and s q k ij;2 based on PISs. The constraints from (9-3) to (9-6), respectively, denote the subscript values of linguistic variables s p ij;1 , s p k ij;2 , s q ij;1 and s q k ij;2 . The constraints from (9-7) to (9-10) denote the range of numerical scales, and the constraint (9-11) assures the ordered of numerical scales. And constraint (9-12) is a set of the 0-1 indicator variables. Solving Eq. (9) by the software packages such as Lingo, MATLAB. After solving Eq. (9), we obtain the objective function value d and the personalized individual numerical scales for each linguistic term set words, there is a set of the 0-1 indicator variables a ij , i; j ¼ 1; 2; . . .; n, and i\k\j makes s p ij;1 ¼ s p k ij;2 and s q ij;1 ¼ s q k ij;2 hold simultaneously. In this case,R is a complete additive consistent LIFPR. In contrary, if the objective function value d 6 ¼ 1, thenR is not a complete additive consistent LIFPR. Obviously, the objective function value d indicates the optimal consistency of LIFPR. For that, the consistency index of LIFPR can be defined as follows.
Definition 7 LetR ¼r ij À Á nÂn be a LIFPR on the finite object set x ¼ x 1 ; x 2 ; . . .; x n f g , d be the objective function value derived from Eq. (9). Then, the consistency index of In the actual decision-making process, it is difficult to obtain complete consistent LIFPR and it is not necessary to obtain it sometimes. On the basis of above analysis, acceptable additive consistent LIFPRs are available. The concept of it is developed as follows.
nÂn be a LIFPR on the finite object set x ¼ x 1 ; x 2 ; . . .; x n f g , d 0 be the threshold value, and d be the objective function value derived from Eq. (9).
If the relationship d ! d 0 holds, then R $ is considered as an acceptable additive consistent LIFPR.
From definition 8 can be seen that, if d\d 0 , indicatives that R $ is not an acceptable additive consistent LIFPR, its consistency needs to be further improved, this will be conducted in the following section.

Consistency improving process from unacceptable ones
Consistency of preference relations is related to rationality. By comparison, inconsistent preference relations often lead to misleading solutions. Therefore, developing some approaches to obtain the expected consistency level is necessary. However, only few scholars focus on optimization-based method to obtain the expected consistent LIFPR at present. Therefore, in this section, several mathematical programming models are proposed to obtain acceptable additive consistent LIFPR which considering the PISs. There are two stages including, namely the first stage is derived the LIFPR with largest number of LIFNs in the upper triangular part, and the second stage is obtained the adjust LIFPR, which considering the minimum adjustment and PISs.

Stage 1. Derive the LIFPR with largest number of LIFNs
By solving Eq. (9), if we have d ! d 0 , then R $ is an acceptable additive consistent LIFPR. In contrary, when d\d 0 , R $ is unacceptable consistency. Equation (9) shows that there are maybe more than two LIFPRs have the same objective function value d Ã . To determine the unique LIFPR with the highest additive consistency level, the following programming model is developed: In Eq. (10), the sum of the 0-1 indicator variables a ij is the objective function. The larger the value z, indicates the more indicator variables set value 1, and the LIFPR with largest number of LIFNs is derived. The constraints (10-1) and (10-2) are based on the objective function value d Ã derived from Eq. (9), they indicate that the largest number of LIFNs is derived based on d Ã . In other words, constraints (10-1) and (10-2) make sure the derived LIFNs not only have the largest number of elements, but also have the optimal consistency level d Ã . Others constraints are the same as those given in Eq. (9). After solving Eq. (10), a unique set of the 0-1 indicator variables a Ã ij , i; j ¼ 1; 2; . . .; n, and i\j are obtained. If we return the indicator variables a Ã ij to Eq. (3), the associated LIFPR is derived, which has the largest number of LIFNs in the upper triangular part.

Stage 2. Obtain the adjust LIFPR
IN this regard, when R $ is confirmed an unacceptable consistency, we need to adjust the original evaluation values provided by the decision-makers to ensure the ranking of objects reasonably. Meanwhile, in order to avoid the loss and distortion of evaluation information, the adjustment should be as small as possible to retain the decision-makers' original evaluation.
To do this, some notations are developed, let / þ ij and / À ij , i; j ¼ 1; 2; . . .; n, and i\j, respectively, denote the adjustment positive deviation and adjustment negative deviation relate to the preferred qualitative degree . . .; n, and i\j, respectively, denote the adjustment positive deviation and adjustment negative deviation relate to the non-preferred qualitative degree s a Ã be the subscript values of adjustment LIFPR. Then, the following programming model is developed: A personalized individual semantics model for computing with linguistic intuitionistic... 4507 In Eq. (11), the objective function is constructed by the sum of deviation variables, which ensures the adjustment as small as possible. The constraints (11-1) and (11-2), respectively, denote the distance between s p Ã ij;1 and s p kÃ ij;2 , s q Ã i;1 and s q kÃ ij;2 , which are based on the objective function value a Ã ij derived from Eq. (10). In other words, the distances developed in constraints (11-1) and (11-2) in view of the largest number of LIFNs. Furthermore, the threshold value d 0 ensures the adjustment LIFPR meets the acceptable additive consistency level. The constraints from (11-3) to (11-6), respectively, denote the subscript values of linguistic variables s p Ã ij;1 , s p kÃ ij;2 , s q Ã i;1 and s q kÃ ij;2 , a Ã ij are the objective function value derived from Eq. (10). The constraints from (11-7) to (11-10) denote the range of numerical scales, and the constraint (11-11) assures the ordered of numerical scales. And constraints (11-12) and (11-13) ensure the adjustment subscript values within 0; 2t ½ . The constraints from (11-14) to (11-16) denote adjustment variables constraints. After solving Eq. (11), we obtain the adjustment variables / þ ij , / À ij , u þ ij , u À ij and the personalized numerical scales for each linguistic term set S c ¼ s a ja 2 0; 2t ½ f g , that is, NS s 0 ð Þ, NS s 1 ð Þ, . . ., NS s 2t ð Þ. According to the adjustment variables, the adjustment LIFPRR Ã ¼r Ã ij nÂn that meets the acceptable additive consistency is determined, where: 3.3 An algorithm for obtaining complete or acceptable additive consistent LIFPRs On the basis of above discussion, this subsection develops an algorithm for obtaining complete or acceptable additive consistent LIFPRs. The main steps are described in Fig. 1 and listed as follows.
To show the concrete application of the above algorithm for obtaining complete or acceptable additive consistent LIFPRs, an example from Meng et al. (2019a) is conducted as follows.
Example 1 Let x ¼ x 1 ; x 2 ; x 3 ; x 4 f gbe the given object set. The LIFPRR ¼r ij À Á 4Â4 on x for the continuous linguistic term set S c ¼ s a ja 2 0; 10 ½ f gis defined as follows: To judge the additive consistency of the LIFPRR, the following steps are conducted.
Step 1 Derive the consistency index ofR. Set k ¼ 0:01, according to Eq. (9), following model is constructed By solving the above model, we have d ¼ 0:93. If we set the threshold value d 0 ¼ 0:9, since the relationship d [ d 0 hold,R is the acceptable additive consistent LIFPR.

A framework of MCDM procedure with LIFPRs
In this section, the MCDM problems with LIFPRs are first introduced, and then, an optimization model is constructed for determining the weight vectors of criteria, which is taken into accounted PISs. Finally, a framework of MCDM procedure with LIFPRs is introduced.

The MCDM problems with LIFPRs
The MCDM problems involve n alternatives denoted as A ¼ a 1 ; a 2 ; . . .; a n f g . Each alternative is assessed based on several feature criteria, denoted as C ¼ c 1 ; c 2 ; . . .; c m f gand x ¼ x 1 ; x 2 ; . . .; x m ð Þis the criteria' weight vector. We assume that the weights of criteria are completely unknown. The moderator provides the evaluation of alternative a i ,i ¼ 1; 2; . . .; n under criterion c j , j ¼ 1; 2; . . .; m and denotes as R , which are LIFNs for the continuous linguistic term set S c ¼ S a ja 2 0; 2t ½ f g . Besides, an expert term is invited to comment these n alternatives for each criterion using LIFPRs. Let R c l ¼r c l ij nÂn , l ¼ 1; 2; . . .; m denote the LIFPRs on A for the continuous linguistic term set. The objective of the decision-making process is to find out the best choice, which utilizes the evaluation matrixR and pairwise judgment matricesR c l , l ¼ 1; 2; . . .; m. max d s:t: . . .; 9; i 6 ¼ 5 NS s iþ1 ð ÞÀNS s i ð Þ ! 0:01; i ¼ 0; 1; . . .; 9 a ij ¼ 0 _ 1; i; j ¼ 1; 2; . . .; 4; i\j

Calculate the weight vectors of the criteria
In the above-mentioned MCDM problems, the integrated pairwise judgment information is needed to derive best choice. For that, the development of the method to determine the weight vectors of the criteria is necessary. There are two stages including, namely the first stage is calculated the distance of any two LIFNs, and the second stage is obtained the weight vectors of the criteria, which based on the proposed distance measure.

Stage 1. Calculate the distance between any two LIFNs
The distance measure is a classical topic in fuzzy set theory. As for LIFNs, Peng et al. (2018) developed the Hamming distance utilized 2-tuples, and Zhang et al. (2017) in view of linguistic-scale functions introduced the generalized distance. Although some of distance measures have been proposed and successfully applied to the decision-making problems, it is found that there still exists some shortcoming where the PISs are seldom considered in the existing distance measures. Based on this fact, the new distance measure is developed as follows.
Definition 9 Lets 1 ¼ s a 1 ; s b 1 À Á ands 2 ¼ s a 2 ; s b 2 À Á be any two LIFNs. The Hamming distance betweens 1 ands 2 is defined as follows: where NS is the number scale function on S c ¼ S a ja 2 0; 2t ½ f g , and NS s 0 ð Þ, NS s 1 ð Þ, . . ., NS s 2t ð Þ is the PISs of the number scale function.
To derive the PISs of linguistic terms, the following model is developed: To show the concrete application of the proposed distance measure, the following example is conducted.  Property 1 Let s be any three LIFNs. The Hamming distance defined above has the following properties: (1) 0 Ds 1 ;s 2 ð Þ 1; (2) D s Proof Obviously, the Hamming distance D s (1) and (2) of Property 1, the proof of them is omitted, and the proof of (3) of is provided as follows.
If s $ 1 s $ 2 s $ 3 , then s a 1 s a 2 s a 3 and s b 1 ! s b 2 ! s b 3 . Meanwhile NS is a strictly monotonically increasing and continuous function. Therefore, the following inequalities can be obtained: Thus.
can be proven in a similar manner.

Stage 2. Obtain the weight vectors of the criteria
In the above-mentioned MCDM problems, the moderator provides the evaluation matrix R denotes the evaluation of alternative a i , i ¼ 1; 2; . . .; n under criterion c j , j ¼ 1; 2; . . .; m. In following section, motivated by the maximizing deviation method (Xu and Cai 2010), a maximizing Hamming distance deviation is developed to obtain the criteria weight vectors under linguistic intuitionistic fuzzy environments. First, the Hamming distance between the criteria c j and other criteria c g , g ¼ 1; 2; . . .; m, g 6 ¼ j with respective to the alternative a i is calculated as follows: where i ¼ 1; 2; . . .; n and j ¼ 1; 2; . . .; m. Second, the distance between alternative a i and other alternatives a j , j ¼ 1; 2; . . .; n, with respective to the criteria c j is derived as follows: where j ¼ 1; 2; . . .; m. Third, the weighted distance function is then constructed: where x j is the weight vectors of criteria. Next, the following optimization model for computing the optimal weight vectors of criteria is constructed as follows: The Lagrange function is constructed to obtain the solution of Eq. (18): The criteria weight vector is derived by solving the Lagrange function: where j ¼ 1; 2; . . .; m.

A framework of MCDM procedure with LIFPRs
The proposed decision-making procedure is summarized in the following steps.
Step 1 Form pairwise judgment matrices. According to the determine criteria and alternatives, the expert team provides their comment on these n alternatives for each criterion c l using LIFPRs and denotes as R c l ¼r c l ij nÂn , l ¼ 1; 2; . . .; m.
Step 2 Check and improve the consistency of LIFPRs. With respective to algorithm 1, the complete or acceptable additive consistent LIFPRs are obtained, which is denoted byR Ã;c l ¼r Ã;c l ij nÂn , l ¼ 1; 2; . . .; m, and the corresponding PISs of number scale functions NS sr Ã;c l ij , l ¼ 1; 2; . . .; m are derived.
Step 3 Determine the weight vectors of criteria. The weight vectors of criteria are determined according to Eq. (20).
Step 4 Compute the collective number scale function. The collective number scale function is determined by the following formula: where x c l is the weight vector of criteria, which is determined in Step 3.
Step 5 Calculate the score and accuracy functions of collective number scale function.
The score function of collective number scale function is determined by the following formula: The accuracy function of collective number scale function is determined by the following formula: Step 6 Rank the alternatives.
The ranking order of all alternatives is obtained by the value of score and accuracy functions of collective number scale function. The order relationship is defined as follows: The proposed decision-making procedure is depicted in Fig. 2. Fig. 2 The proposed framework of decision-making process with LIFPRs 5 Illustrative example In this section, evaluation of different brands of mobile phones (revised from Meng et al. 2019a) is provided to illustrate the application of the proposed method, and conjunction with comparative analysis and discussion is conducted.
Since the Motorola Corporation invented the first mobile phone in 1983. After decades of development, the function of mobile phones has undergone tremendous changes and the mobile phones have become one of the most important daily necessities. In the first half of 2016, the sales of mobile phones exceed 250 million in China. According to the sales, there are four major brands of mobile phones, including: (1) a 1 HUAWEI; (2) a 2 OPPO; (3) a 3 APPLE; and (4) a 4 VIVO. In the above list brands of mobile phones, HUAWEI, OPPO and VIVO are made in China, while APPLE is made in America. There are different characteristics of each brand of mobile phones. However, several factors should be considered when evaluate these brands of mobile phones, such as (1) c 1 appearance; (2) c 2 price; (3) c 3 performance; (4) c 4 quality, and (5) c 5 reputation. To purchase a number of cost-effective mobile phones, a business company invited an expert team to evaluate these brands of mobile phones. To fully express the recognitions of the experts, they are permitted to apply linguistic variables in the predefined linguistic term set {s 0 : extremely bad; s 1 : very bad; s 2 : bad; s 3 : relatively bad; s 4 : a little bad; s 5 : fair; s 6 : a little good; s 7 : relatively good; s 8 : good; s 9 : very good; s 10 : extremely good}. Furthermore, the experts are allowed to express the preferred and nonpreferred opinions for each pair of brands mobile phones. With respective to these brands of mobile phones for each criterion, LIFPRs are listed in matrices R $ c l , l ¼ 1; 2; . . .; 5, and the moderator provides the evaluation of alternatives with LIFNs, which are listed in matrix R $ . Take the evaluation values s 3 ; s 6 ð Þ from matrix R $ c 1 , for example. The expert term provides the preferred value s 3 and non-preferred value s 6 when assesses the brands of mobile phones HUAWEI to OPPO and cannot determine exact numerals. In such case, the evaluation value can be modeled by a LIFPR s 3 ; s 6 ð Þ. Other entries, that is, LIFPRs, in matrices R $ c l , l ¼ 1; 2; . . .; 5 are similarly explained.R c 1 ¼

Illustration of the proposed method
The procedure for evaluating different brands of mobile phones using the proposed method is showed below.
Step 1 Form pairwise judgment matrices. All pairwise judgment matrices for each criterion in C have been provided, as demonstrated in matrices 1-5.
Step 2 Check and improve the consistency of LIFPRs.
To check and improve the consistency of LIFPRs, algorithm 1 is developed in this subsection.   In a same way, for LIFPRR c 5 , by solving Eq. (9), we have: Step 3 Determine the weight vectors of criteria. The weight vectors of criteria are determined according to Eq. (20): x c 1 ¼ 0:15, x c 2 ¼ 0:16, x c 3 ¼ 0:23, x c 4 ¼ 0:15 and x c 5 ¼ 0:30.
Step 4  Step Step 6 Rank the alternatives.
based on Eq. (24), by which we have APPLE1OPPO1HUAWEI1VIVO.

Comparative analysis
To validate the feasibility of the proposed method, we conducted a comparative study with other methods based on the same illustrative example. Meng et al. (2019a) first proposed the concept of LIFPR and then introduced the concept of additive consistent LIFPR. Moreover, to obtain the complete consistent LIFPR, several goal programming models are developed based on additive consistency. Finally, these goal programming models have been extended to incomplete LIFPR. To better comparison, the results obtained by Meng et al. (2019a)'s method and the proposed method are summarized in Table 2. The detailed calculation process of Meng et al. (2019a)'s method can be found in (Meng et al. 2019a).
As shown in Table 1, it can be easily found that the best alternative obtained from Meng et al. (2019a)'s method is the same as the proposed method, and the ranking results are also the same. This also confirms the effectiveness of the proposed method. Although these two methods both consider the additive consistency checking and improving processes and take into accounted the weight vectors of criteria. They are some difference. First, in linguistic decision-making, computing with words is an important point to note about. Meng et al. (2019a)'s method conducted it with subscript values, while the proposed method utilized PISs model. Based on the fact that words mean different things for different people, the proposed method utilized PISs model seems more reasonable. Second, consistency of preference relations is related to rationality. By comparison, inconsistent preference relations often lead to misleading solutions. Meng et al. (2019a)'s method only developed the complete consistency checking and improving process, while the proposed method not only considers the complete consistency, but also the acceptable consistency. In the actual decision-making process, it is difficult to obtain complete consistent LIFPR and it is not necessary to obtain it sometimes. On the basis of above analysis, acceptable additive consistent LIFPRs are available. For that, the proposed method has a wider background application. Moreover, the objective functions are different when considering the consistency checking process. In Meng et al. (2019a)'s method, the objective functions are constructed based on minimizing the deviation from the target of the goal. However, the proposed method focuses on maximizing the parameter of satisfaction degree. The different perspectives for solving the problems lead to different decision-making results, but the proposed method takes the decision-makers' satisfaction

Discussion
To verify the advantages of proposed method, we discuss it with several representative models under the MCDM environment with LIFPRs. Table 3 presents the performances of these approaches regarding several indexes.
(1) Meng et al. (2019b)'s method: This method derived priority weights of alternatives based on two stages strategy. The first stage is estimating the missing elements in LIFPRs based on the properties of multiplicative consistent LIFPRs, and the second stage is deriving the priority weights based on complete LIFPRs. The method developed these processes in view of the subscript values of linguistic variables and does not take into account the psychological characteristics of decision-makers. Compared with Meng et al. (2019b)'s method, the proposed method utilized PISs model to compute with words and take into account the psychological characteristics of decision-makers. The proposed method has advantage in representing the specific semantics of each individual.
(2) Wan et al. (2022b)'s method. This method studied the consensus-reaching process with LIFNs. The process included two-stage consensus-reaching method. LIFNs evaluation values with high linguistic indeterminacy degrees are modified in the first stage, and evaluation elements with high deviation elements are modified in the second stage. The method developed these stages in view of the 2-tuple of linguistic variables and does not take into account the psychological characteristics of decision-makers. Compared with Wan et al. (2022b)'s method, the proposed method utilized PISs model to compute with words and obtain the priority weight vectors by taking into account decision-makers' satisfaction degree. On account of these, the proposed method has advantage in avoiding the loss of individual information and considering the psychological characteristics of decision-makers.
(3) Jin et al. (2019)'s method. This method developed a decision support model that simultaneously considered the individual consistency and group consensus for GDM with LIFPRs. The method defined the concept of multiplicative consistency of LIFPRs directly applies the concept of multiplicative consistency of linguistic fuzzy preference relations. As Meng et al. (2017) noted, issues may exist as that for intuitionistic fuzzy preference relations. Compared with Jin et al. (2019)'s method, the proposed method considered all the cases correspond to the consistency LIFPRs. In view of these, the proposed method has advantage in avoiding the loss of information and the calculation process seems more reasonable.
According to the comparison analysis, the method proposed in this study has the following advantages over other existing approaches.
(1) In linguistic decision-making, computing with words is an important point to note about. The proposed method utilized PISs model to compute with words; this ensures the proposed method has advantage in representing the specific semantics of each individual.
(2) The proposed method focuses on maximizing the parameter of satisfaction degree to construct objective functions when considering the consistency checking process. The decision-making process considers the psychological characteristics of decision-makers, and more suitable for decision-making problems in some backgrounds. (3) The method of determining the weight vectors of criteria is developed. This method uses PISs model to compute with words.

Conclusion
This paper develops a PISs model for computing with linguistic intuitionistic fuzzy information and applies to evaluate different brands of mobile phones. First, a consistency-driven optimization model for checking the additive consistent LIFPRs is constructed by considering the PISs model. Besides, several optimization models are built to determine the PISs of linguistic terms with LIFPRs and obtain the acceptable additive consistent LIFPRs. Second, a new definition of Hamming distance between LIFNs is introduced. Then, the method of deriving the weight vectors of criteria is developed based on the proposed distance measure. Subsequently, a framework of GDM process with LIFPRs is offered, and the application of the proposed method is illustrated by evaluating different brands of mobile phones. Finally, the comparative analysis is presented to show the feasibility of the GDM method.
The present study provides several significant contributions for MCDM problems with LIFPRs. They are summarized as follows: (1) the proposed method utilized PISs model to compute with words; this ensures the proposed method has advantage in representing the specific semantics of each individual. Take linguistic value s 3 in illustrative example, for example, it set 0.4 in R (2) The proposed method focuses on maximizing the parameter of satisfaction degree to construct objective function, and more suitable for decisionmaking problems in some backgrounds. (3) A new definition of Hamming distance for measuring LIFNs is introduced considering the PISs model. In our future research, the framework of GDM process with LIFPRs is designed by considering the consistency and consensus and applied the proposed method to solve other practical MCDM problems.