Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation

To address the situation where incomplete hesitant fuzzy preference relation (IHFPR) is necessary, this paper develops decision-making models taking into account decision makers’ satisfaction degree. First, the consistency measures, respectively, from the perspectives of additive and multiplicative consistent IHFPR are defined, which is based on the relationships of the IHPFRs and their corresponding priority weight vector. Second, two decision-making models are developed, respectively, in view of the proposed additive and multiplicative consistency measures. The main characteristic of the constructed models is they taking into account the decision makers’ satisfaction degree. The objective functions of the models are developed by maximizing the parameter of the satisfaction degree. Third, a square programming model is developed to obtain the decision makers’ weights by utilizing the optimal priority weight vectors information, the solution of the model is obtained by solving the partial derivatives of Lagrange function. Finally, a procedure for multi-criteria decision-making (MCDM) problems with IHFPRs is given, and an illustrative example in conjunction with comparative analysis is used to demonstrate the proposed models are feasible and efficiency for practical MCDM problems.


Introduction
Group decision-making (GDM) is a specific type of decision problem where several/many decision makers cooperate with each other and choose the best solution from a set of possible alternatives (Rabiee et al. 2021). In the GDM process, decision makers are invited to provide evaluation information by pairwise comparisons of alternatives, and at the end, a collective decision is reached by utilizing the predetermined criteria (Rodríguez et al. 2021). Due to the complexity of decision-making environments, and the limitations of decision makers' knowledge, experience, and ability, decision makers are hesitant about certain evaluation values during assessments (Yazdani et al. 2021). To deal with this, Torra and Narukawa (2009) introduced the concept of the hesitant fuzzy set (HFS). Since that time, an increasing amount of research on the study of HFS has been published (Liu et al. 2021b;Mishra et al. 2021;Gong et al. 2021). Later, Xia and Xu (2013) found the advantages of hesitant fuzzy element (HFE) and introduced the concept of hesitant fuzzy preference relation (HFPR). Following the original work of Xia and Xu (2013), many multi-criteria decision-making (MCDM) approaches based on HFPR have been developed. A concise literature review of these approaches is presented as follows and summarized in Table 1 According to number of fuzzy preference relation (FPR) was used to derive priority weight vector, all these MCDM approaches can be classified into four categories (Meng et al. 2020a): (1) Only considers one FPR derived from HFPR (Song and Li 2019;Zhu and Xu 2014a;Zhu et al. 2014). This method also named optimistic consistency, that is, a reduced FPR with the highest consistency degree is derived from HFPR. The optimistic consistency method can reflect the highest consistency degree of HFPR, but it cannot reflect the hesitancy of decision makers. It leads to substantial information loss. (2) Based on ordered FPRs derived from normalized HFPR (Zhang et al. 2015a(Zhang et al. , 2015bLiu et al. 2016). This method also named normalized consistency. The normalized consistency requires that any two HFEs have an equal number of elements, if two HFEs have an unequal number of elements, a normalized process is needed. Therefore, the normalized consistency method may distort the original information provided by decision makers.
(3) Based on all possible FPRs including in HFPR (Zhang et al. 2018c(Zhang et al. , 2018b. This method also named average consistency. The method defines the concept of consistent HFPR used all possible FPRs, this seems too restricted. It is difficult for decision makers to provide such pairwise judgement in the actual decision-making process. (4) Based on the derived FPRs for each value in HFEs Tang et al. 2017). This method also named partial average consistency. The main feature of this method is that it considers all the evaluation information, and neither adds values into HFEs nor removes values from HFEs. Compared with (3), this method only used some possible FPRs including in HFPR.
Due to the lack of knowledge and decision makers' limited expertise, it may be difficult for decision makers to provide complete preference relations over alternatives (Zhang and Chen 2021a;Wan et al. 2021;Liu et al. 2021a). At the end, lots of MCDM approaches have been developed to manage incomplete information (Xie et al. 2020;Meng and Chen 2021;Dong et al. 2019). According to their principles of derived priority weight vector, all these MCDM approaches can be classified into two categories: (1) deriving the priority weight vector based on complete FPR (Zhang 2016;Zhang et al. 2015b). This method firstly obtained the missing values based on certain rules, and then derived priority weight vector from complete FPR.
However, this method only apply to the situations where each alternative is compared at least once (Ding et al. 2020).
(2) Deriving the priority weight vector based on incomplete FPR (Zhang et al. 2018c;Xu et al. 2016). When implementing this method, the priority weight vector can be derived by using some programming models. It does not need to derive the missing values, and have ability to handle the case where ignored alternatives exist (Tang et al. 2019).
Since decision makers may provide incomplete preference relations over alternatives. It is necessary to develop some approaches to manage incomplete HFPR (IHFPR). For that, several MCDM approaches based on IHFPR have been proposed (Zhang 2016;Zhang et al. 2018cZhang et al. , 2015bXu et al. 2016;Khalid and Beg 2017). Similar to the contents in the previous paragraph, these MCDM approaches also can be classified into two categories: (1) deriving the priority weight vector based on complete HFPR. For example, Zhang et al. (2015b) developed two methods to estimate the missing elements utilizing the properties of additive consistency, while Zhang (2016) in a similarly way based on multiplicative consistency, and then both of them deriving the priority weight vector based on complete HFPR. (2) Deriving the priority weight vector based on IHFPR. Xu et al. (2016) developed two goal programming models to derive the priority weight vector based on additive and multiplicative consistent IHFPR, respectively. Zhang et al. (2018c) proposed an approach to derive priority weight vector using the logarithmic least squares method.
The concepts of additive and multiplicative consistent IHFPR have been introduced, and some MCDM methods with IHFPR have been developed. However, there are still some important issues need to be solved. (1) The concepts of additive and multiplicative consistent IHFPR. The existing concepts are developed based on the concepts of additive and multiplicative consistent HFPR. As the contents discuss in front, optimistic consistency, normalized consistency and average consistency have some shortcomings. When the consistency of IHFPR are developed based on them, they still have the defects discussed above (Zhang et al. 2018c). (2) The MCDM methods with IHFPR seldom consider the satisfaction degree of decision makers. Almost MCDM methods with IHFPR focus on checking and improving the consistency and consensus (Meng et al. Normalized consistency Based on ordered FPRs derived from normalized HFPRs (Zhang et al. 2015a;Liu et al. 2016) Average consistency Based on all possible FPRs including in HFPRs (Zhang et al. 2018c(Zhang et al. , 2018b Partial average consistency Based on the derived FPRs for each value in HFEs Tang et al. 2017) 2020a). The priority vector follows the consistent IHFPR can obtain a reason ranking. However, for improving the consistency and consensus level may lead to destroy the original evaluation information (Ren et al. 2021). Moreover, the disobedience, non-cooperation behaviors and satisfaction degree of decision makers may have an impact on the results of decision-making (Zhang et al. 2018a).
To eliminate above-mentioned defects, the consistency measures, respectively, from the perspectives of additive and multiplicative consistent IHFPR are defined based on the relationships of the IHPFRs and their corresponding weight vector. And two decision-making models are developed taking into account decision makers' satisfaction degree. The primary contributions of this study are summarized as follows.
(1) To overcome the shortcoming of additive and multiplicative consistent IHFPR developed in considering one FPR derived from IHFPR, ordered FPRs derived from normalized IHFPR, and all possible FPRs including in IHFPR, new concepts of additive and multiplicative consistent IHFPR are proposed, respectively.
(2) To consider the satisfaction degree of decision makers, two decision-making models are developed based on the proposed additive and multiplicative consistency measures. The main characteristic of the constructed models is that the objective functions of the models are obtained by maximizing the parameter of satisfaction degree.
The remainder of the paper is organized as follows. In Sect. 2, basic concepts and operations related to FPR, HFS and HFPR are reviewed. In Sect. 3, the concepts of additive and multiplicative consistent IHFPR are presented, and two decision-making models are developed in view of the proposed additive and multiplicative consistency measures. In Sect. 4, a square programming model is developed to obtain the decision makers' weights, and a procedure for MCDM problems with IHFPR is provided. 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 FPR, HFS, and HFPR.

FPR
Let X ¼ x 1 ; x 2 ; . . .; x n f gdenotes a finite set of alternatives, where x i represents the ith alternate. Orlovsky (1978) introduced the concept of FPR to represent fuzzy judgement value.
Definition 1 (Orlovsky 1978). A FPR on a set of alternatives X is represented by a matrix R ¼ r ij À Á nÂn & X Â X, where r ij is interpreted as the degree to which alternative x i is preferred to x j . Furthermore, r ij should satisfy the following conditions: To measure the rationality of FPR, the concepts of additive and multiplicative consistent FPR were developed.
Definition 2 (Tanino 1984). For a FPR R ¼ r ij À Á nÂn , suppose that W ¼ w 1 ; w 2 ; . . .; w n ð Þis the priority weight vector derived from R, where w i 2 0; 1 ½ and P n i¼1 w i ¼ 1. Then FPR is called to be additive consistency if And FPR is multiplicative consistency if

HFS
To express the hesitant preference information, Torra and Narukawa (2009) introduced the concept of HFS.
Definition 3 (Torra and Narukawa 2009). Let X be a fixed set. Accordingly, a HFS E on X is defined in terms of a function h E x ð Þ that when applied to X returns a finite subset of [0, 1].
To be easily understood, Xia and Xu (2011) expressed the HFS as the following mathematical symbol: where h E x ð Þ is a set of values in [0, 1] representing the possible membership degrees of element x in X to E, and h E x ð Þ is named hesitant fuzzy element (HFE) and denoted as h ¼ c s s ¼ 1; 2; . . .; #h j f g , #h is the number of elements including in h.
The number of elements in different HFEs may be different. Any two HFEs are required to have the same length when develop the MCDM methods which the evaluation values with HFEs. To this end, a normalization process is necessary.
Definition 4 (Zhu and Xu 2014b). Let h ¼ c s s ¼ 1; 2; . . .; #h j f gbe a HFE, c À and c þ denote the minimum and maximum values in h, respectively. Then, 1 À n ð Þc À þ nc þ is called an adding value in h, where n is an optimized parameter determined by decision makers' Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation 3131 risk preference. Especially, the decision makers are pessimistic when n ¼ 0 and decision makers are optimistic when n ¼ 1.
Evidently, different forms of normalized HFEs (NHFEs) will be derived with respect to decision makers with different risk preferences. That is, the normalization process is influenced by the subjectivity of the decision makers, and different ranking values will be derived with respect to MCDM methods with different forms of NHFEs. These shortcomings have been studied by several scholars. For more detail, the readers turn to Meng and An 2017;Zhang et al. 2018c). Meanwhile, an issue on obtaining ranking values that are not relied on NHFEs arises. It will be discussed in Sect. 3. Xia and Xu (2013) first proposed the concept of HFPR. However, the sequence relationships of the elements are needed, this leads to some complexity in actual application. To overcome this weaknesses, Xu et al. (2017) developed a new definition of HFPR that does not need to arrange the elements in descending or ascending sequence.

HFPR
Definition 5 (Xu et al. 2017). Let X ¼ x 1 ; x 2 ; . . .; x n f gbe a fixed set, HFPRs on X is represented by a matrix is a HFE indicating the possible values of the preference degrees to which alternative x i is preferred to alternative x j . For all i; j 2 N, h ij should satisfy: where c s ij refers to the sth element in h ij . Incomplete evaluations sometimes occur for many reasons, including time pressure or lack of decision maker background knowledge. Xu et al. (2016) developed the concept of IHFPR as follows.
Definition 6 . Let X ¼ x 1 ; x 2 ; . . .; x n f gbe a fixed set, then an IHFPR on X is represented by a matrix indicating the possible values of the preference degrees to which alternative x i is preferred to x j . For all i; j 2 N, h ij should satisfy the following conditions: where c s ij refers to the sth element in h ij . Integrating the concepts of HFPR with additive and multiplicative consistency into IHFPR, Xu et al. (2016) developed the concepts of additive and multiplicative consistent IHFPR.
Definition 7 If R satisfies the following condition: then R is called additive consistent IHFPR. Where W ¼ w 1 ; w 2 ; . . .; w n ð Þis the priority weight vector derived from R.
And IHFPR is multiplicative consistency if 3 Deriving priority weight vectors from IHFPRs In this section, we first introduce the concepts of additive and multiplicative consistent IHFPR, and then introduce several programming models for deriving priority weight vectors from IHFPRs, which considers the satisfaction degree of decision makers.

Additive and multiplicative consistent IHFPR
To further consider Definition 7, the concepts of additive and multiplicative consistent IHFPR are, respectively, defined on the basis of the relationships between the formula consisting of priority weights and the values including in HFEs. However, the relationships present in Eqs. (7) and (8) only consider the relationships of one priority weight formula and all the values including in IHFPR but cannot reflect the hesitancy of decision makers. It is reasonable that for every value including in IHFPR has a relationship to one priority weight formula. That is to say, the additive and multiplicative consistent IHFPR are in accordance with the derived FPRs with respect to each fixed values. On the basis of this consideration, new concepts for additive and multiplicative consistent IHFPR are defined as follows.
Then, R is called additive consistent IHFPR if all known elements including in R satisfying the following condition: . . .; n are the priority weights such that w k i ! 0 and P n i¼1 . . .; #h ij has a relationship to one priority weight formula, we set a s In a similarly way, the concept of multiplicative consistent IHFPR is developed as follows.
Then, R is called multiplicative consistent IHFPR if all known elements including in R satisfying the following condition: for all i; j ¼ 1; 2; . . .; n, with i\j. The meanings of symbols w k i , d ij and a s ij are the same as those shown in Eq. (9).
Remark 1 From Definition 8 and Definition 9 can be easily found that, there are #h ij equations including in Eqs. (9)

Deriving priority weight vectors from IHFPRs
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 IHFPR at present. Therefore, in this section, several mathematical programming models are proposed to obtain acceptable consistent IHFPR which considering the satisfaction degree of decision makers. There are two cases including, namely deriving priority weight vectors from IHFPRs based on the additive consistency and multiplicative consistency, respectively. Case 1. Deriving priority weight vectors from IHFPRs with additive consistency According to the definition of additive consistent . . .; n indicates whether c ij is a missing value or not. The priority weights of complete additive consistent IHFPR can be derived by solving a list of equations d ij . . .; n, i\j, k ¼ 1; 2; . . .; l. However, the above-mentioned equations do not constantly hold in general given a deviation between approaches to 0, the more valid and reasonable the priority weights are.
In this regard, we try to obtain the priority weights of IHFPR by constructing a satisfaction degree function for the decision makers with respect to the priority weights. If the IHFPR is complete consistency then the decision maker completely satisfies the priority weights w k i and w k j , and the satisfaction degree of the decision maker is 1; otherwise, the satisfaction degree of the decision maker reduces. Motivated by these studies (Ren et al. , 2018Zhu and Xu 2014a), a membership function for the satisfaction degree in incomplete hesitant fuzzy environment can be constructed as: that the decision makers completely satisfy the priority weights; If 0\M ij w k À Á \1, means the absolute value indicates that the decision makers partly satisfy the priority weights; If M ij w k À Á 0, means the absolute value indicates that the decision makers does not satisfy the priority weights. We can utilize a regular n À 1-simplex to present the membership functions of M ij w k À Á , i; j ¼ 1; 2; . . .; n, which can be denoted as follows: A function can be provided to synthesize all satisfaction degrees of the priority weights w k i and w k j , i; j ¼ 1; 2; . . .; n in SX nÀ1 simplex The theory of maximum minimization is utilized to guarantee the minimum satisfaction degree be not too low. Therefore, the objective function is denoted as: max i;j¼1;2;...;n min w k 2SX nÀ1 M ij w k À Á i; j ¼ 1; 2; . . .; n È É , for all k ¼ 1; 2; . . .; l, which can be represented by the following programming model: Substituting Eq. (11) into Eq. (14), for each possible value c s i o j o 2 h ij , with i 0 \j 0 for each s ¼ 1; 2; . . .; #h ij , the mathematical programming model can be expressed in detail as: Suppose the decision makers provide the preference degree 1 ij with the equal significance, that is, 1 ij ¼ 1, for all i; j ¼ 1; 2; . . .; n, in this case, the mathematical programming model can be further simplified to: max z k ¼ u s:t: Substituting deviation variable value 1 into the above model, it can be easily proved that Eq. (16) is a 0-1 programming model. Therefore, at least one priority weight vector can be derived with the maximum satisfaction degree u.
Solving Eq. (16), a list of priority weight vectors w k , k ¼ 1; 2; . . .; l can be derived. Since w k can be viewed as the possible priority weight vector of R. And based on the ideas of Zhang et al. (2018c) and Wu et al. (2019). The distance between w k and R is developed to select the best priority weight vector of R.
nÂn & X Â X be an IHFPR, and w k ¼ w k 1 ; w k 2 ; . . .; w k n À Á , k ¼ 1; 2; . . .; l be a list of priority weight vectors derived from Eq. (16). Then, the distance between w k and R is developed as follows: where in upper triangle of IHFPR. It can be easily found that the distance d 1 w k ; R À Á reflects the average of total the square deviation between 1 2 w k i À w k j þ 0:5 and P #h ij s¼1 a s ij c s ij for all known elements including in R. It is natural that the optimal priority weight vector is the one that minimizes the deviation d 1 w k ; R À Á . For the distance measure presents in Eq. (17), it can be proven that it satisfy the axiom of the distance measure.
Property 1 For the distance measure presents in Eq. (17), we have: (1) non-negativity: 0 d 1 w k ; R À Á 1; (2) reflexivity: a s ij c s ij ; (3) commutativity: Proof The proof of (2) and (3) is obvious, they do not appear in this study. In the following section, we only provide the proof of (1) and (4).
(1) non-negativity: for each known elements including in R, we have 0 1 2 w k i À w k j þ 0:5 À a s ij c s ij 1, and

) triangle inequality:
Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation 3135 As a consequence, the priority weight vector of R is developed as follows.
nÂn & X Â X be an IHFPR, and w k ¼ w k 1 ; w k 2 ; . . .; w k n À Á , k ¼ 1; 2; . . .; l be a list of priority weight vectors derived from Eq. (16). Then, the priority weight vector of R is developed as follows: The symbol arg presents in Eq. (18) means the priority weight vector is derived from the minimum distance calculate from d 1 w k ; R À Á .
Remark 2 There are may be more than one priority weight vectors including in min w k d 1 w k ; R À Á , that is to say, sometimes the solution of Eq. (18) is not unique. In this case, the priority weight vector of R is developed as average of multiple priority weight vectors: where w k i , k ¼ 1; 2; . . .; n, and it indicates that the number of priority weight vectors including in min w k d 1 w k ; R À Á is l 0 .

Case 2. Deriving priority weight vectors from IHFPRs with multiplicative consistency
Similar to the idea of additive consistent IHFPR presents in case 1, the following membership function with the satisfaction degree can be constructed when we consider multiplicative consistent IHFPR.
where k ¼ 1; 2; . . .; l. The meanings of symbols w k i , e ij , a s ij and F ij w k À Á are the same as those given in Eq. (11). Similar to the idea of additive consistent IHFPR presents in case 1, the mathematical programming model for deriving priority weight vectors can be expressed in detail as: Suppose e ij ¼ e, for all i; j ¼ 1; 2; . . .; n, then, the mathematical programming model can be further simplified to: max z k ¼ p Substituting deviation variable value e into the above model, it can be easily proved that Eq. (22) is a nonlinear programming model. In this case, the optimal solution of it can be obtained by utilizing the optimization software, such as LINGO 11.0, Matlab and Mathematica.
Solving Eq. (22), a list of weight vectors w k , k ¼ 1; 2; . . .; l can be derived. Since w k can be viewed as the possible priority weight vector of R. Similar to additive consistent IHFPR, the distance between w k and R is developed to select the best priority weight vector.
nÂn & X Â X be an IHFPR, and w k ¼ w k 1 ; w k 2 ; . . .; w k n À Á , k ¼ 1; 2; . . .; l be a list of priority weight vectors derived from Eq. (22). Then, the distance between w k and R is developed as follows: The meanings of symbols w k i , d ij , a s ij , l and d 2 w k ; R À Á are the same as those given in Eq. (17).
Remark 3 Similar to the idea of Eq. (17), the proof of the axiom of the distance measure presents in Eq. (23) can be developed in a similar way.
Similarly, the priority weight vector of R is developed as follows.
nÂn & X Â X be an IHFPR, and w k ¼ w k 1 ; w k 2 ; . . .; w k n À Á , k ¼ 1; 2; . . .; l be a list of priority weight vectors derived from Eq. (22). Then, the priority weight vector of R is developed as follows: Remark 4 Similar to the idea of additive consistent IHFPR presents in case 1, there are may be more than one priority weight vectors including in min w k d 2 w k ; R À Á . In this case, the priority weight vector of R is developed according to Eq. (19).

Framework of MCDM procedure with IHFPRs
In this section, the MCDM problems with IHFPRs are firstly introduced, and then an optimization model is constructed for determining the weights of decision makers. Finally, a framework of MCDM procedure with IHFPRs is introduced.

The MCDM problems with IHFPRs
Hesitant MCDM problems involve m alternatives denoted as A ¼ a 1 ; a 2 ; . . .; a m f g . Each alternative is assessed based on several feature criteria. E ¼ e 1 ; e 2 ; . . .; e n f gis a set of decision makers and k ¼ k 1 ; k 2 ; . . .; k n ð Þ T is the decision makers' weight vector. We assume that the weights of decision makers are completely unknown. The evaluation of the alternative a i ,i ¼ 1; 2; . . .; m with respect to the feature criterion is provided by decision maker e j , j ¼ 1; 2; . . .; n, and denotes as h¼ c s s ¼ 1; 2; . . .; #h j f g , which is an IHFPR. Suppose IHFPRs provide by decision makers with additive consistency, then the priority weight vectors of each individual IHFPRs are derived according to Eq. (18)

Calculate the weight vectors of the decision makers
In this subsection, an optimization model is constructed to derive the weights of decision makers with complete unknown information. Considering that decision makers in the MCDM process typically construct from different knowledge backgrounds and have varied expertise in the domain area, each decision makers has different judgment values, which influences the solution differently. Therefore, each decision makers has a different importance weight when collecting the priority weight vectors. Given that the decision maker whose judgment values are far away from the collected judgment values indicate that the judgment values which he/she provided are the least reliable, and the decision maker should endow a smaller weight value. By comparison, the decision maker whose judgment values are close to the collect judgment values indicates that the judgment values he/she provides are the most reliable, and the decision maker should endow a larger weight value. On this basis, the optimization model is constructed as follows: As seen, Eq. (25) is a square programming model. The optimal solution of it can be obtained by utilizing the optimization software. Moreover, it can be easily found that Eq. (25) is constructed from the algorithm of least square method, according to the Lagrange multiplier method, the Lagrange function of Eq. (25) can be constructed as follows,L k; l , where l is a Lagrange multiplier. The solution of Eq. (25) can be obtained by solving the partial derivatives of Lagrange function L k; l ð Þ, and the result is listed as follows: where A ¼

Framework of MCDM procedure with IHFPRs
The proposed decision-making procedure is summarized in the following steps.
Step 1 Form individual IHFPR matrices. According to the determine criteria and alternatives, the decision makers respectively provide their judgement matrices, and denotes as R k ¼ h ij;k À Á mÂm & X Â X, k ¼ 1; 2; . . .; n.
Step 3 Derive the optimal priority weight vector. First, Utilize Eq. (17) or Eq. (23) to calculate the distance between w k and R k , if there is only one priority weight vector including in Eq. (18) or Eq. (24), then the optimal priority weight vector is determined according to Eq. (18) or Eq. (24), otherwise, the optimal priority weight vector is determined according to Eq. (19).
Step 4 Determine the weights of decision makers. The weights of decision makers are determined according to Eq. (26).
Step 5 Compute the collective optimal priority weight vector.
The collective optimal priority weight vector is determined by the following formula: where k k is the weight of decision maker, and w kÃ i is the optimal priority weight vector determines in Step 3.
Step 6: Rank the alternatives. The ranking order of all alternatives is obtained by the value of collective optimal priority weight vector # i , i ¼ 1; 2; . . .; m.
The proposed decision-making procedure is depicted in Fig. 1.

Illustrative example
In this section, selection of the most important project to invest problem (adapted from Xu et al. (2016) and Parreiras et al. (2010)) is provided to illustrate the use of the proposed method, and conjunction with comparative analysis is conducted.
Step 4 0 : Determine the weights of decision makers.
Since # 3 [ # 2 [ # 1 , the ranking order of all alternatives is obtained as a 3 1a 2 1a 1 . Thus, the most important project to invest is cold chain intelligent logistics.

Comparative analysis and discussion
To validate the feasibility of the proposed method, we conducted a comparative study with other method based on the same illustrative example. Xu et al. (2016) first proposed the concept of IHFPR, and then introduced the concept of additive consistent IHFPR and multiplicative consistent IHFPR. Moreover, to obtain the priority vector of an IHFPR, two goal programming models are developed based on additive consistency and multiplicative consistency, respectively. Finally, these two goal programming models have been extended to obtain the collective priority vector of several IHFPRs. To better comparison, the results obtained by Xu et al. (2016)'s methods and the proposed methods are summarized in Table 2. The detailed calculation process of Xu et al. (2016)'s methods can be found in Xu et al. (2016).
As shown in Table 2, it can be seen that there are some differences in the ranking results. In Xu et al. (2016)'s methods, the ranking results are the same, but the ranking values are different when considering different consistency. In the proposed methods, both the ranking results and ranking values are different when considering different consistency. It can be easily found that the best alternative obtained from Xu et al. (2016)'s methods are the same as the proposed method with multiplicative consistency, but the ranking values are different. The possibility reasons for the inconsistency are explained as follows. The consistency definitions are different. In Xu et al. (2016)'s methods, the consistency definitions based on one FPR derived from IHFPR, that is, optimistic consistency, while the proposed methods the consistency definitions based on each value in HFEs, they considering some possible FPRs derived from IHFPR. The consistency considers all the evaluation information, and neither adds values into HFEs nor removes values from HFEs. It can then avoid information loss and distort. Compare to Xu et al. (2016)'s method, they only consider some evaluation information, based on this fact, the ranking result obtained from the proposed method seems more reasonable. Moreover, the objective functions are different. In Xu et al. (2016)'s methods, the objective functions constructed based on minimizing the deviation from the target of the goal. However, the proposed methods focus on maximizing the parameter of satisfaction degree. The different perspectives for solving the problems lead to different decision-making results, but the proposed methods take the decision makers' satisfaction degree into account, this is more suitable for solving decision-making problems in some backgrounds (Zhang and Chen 2021b).
It is shown in Table 2 that the best alternative is a 1 if R 1 , R 2 and R 3 with additive consistency in the proposed approaches, while the best alternative is a 3 if R 1 , R 2 and R 3 with multiplicative consistency. Obviously, the best alternative may vary relying on individual IHFPRs with different consistency. Therefore, decision makers can flexibly choose different approaches to derive priority weights of alternatives based on different indexes, such as, the satisfaction degree of decision makers, average consistency level of R 1 , R 2 and R 3 , and the consensus level among R 1 , R 2 and R 3 .
To verify the advantages of our approaches, we compare them with several representative models under the MCDM environment with IHFPRs. Table 3 presents the performances of these approaches regarding several indexes.
(1) Zhang et al. (2015b) and Zhang (2016): These approaches derive priority weights of alternatives based on two stages strategy. The first stage is estimating the missing elements in IHFPR based on the properties of consistent IHFPR, and the second stage is deriving the priority weights based on complete HFPR. Compared with Zhang et al. (2015b) and Zhang (2016)'s approaches, the proposed approaches develop the concepts of consistency based on partial average consistency, and obtain the priority weights rely on only one stage strategy. Based on this fact, the proposed approaches have advantage in avoiding the loss of information and the calculation seems simpler. (2) Xu et al. (2016) and Zhang et al. (2018c). These approaches first defined the concepts of additive and multiplicative consistent IHFPR, and then developed two programming models to derive the priority weights from IHFPR based on additive and multiplicative consistency, respectively. Compared with Xu et al. (2016) and Zhang et al. (2018c)'s approaches, the proposed approaches develop the concepts of consistency based on partial average consistency, and obtain the priority weights taking into account decision makers' satisfaction degree. On account of these, the proposed approaches have advantage in avoiding the loss of information and considering the psychological characteristics of decision makers. (3) Meng et al. (2020a) and Meng et al. (2020b). These approaches first defined the concepts of acceptably additive and multiplicative consistent IHFPR, and then a series of optimization models to acquire complete HFPR. Finally, the priority weights are derived based on certain indexes. Compared with Meng et al. (2020a) and Meng et al. (2020b)'s approaches, the proposed approaches do not focus on checking and improving the consistency and consensus, for improving the consistency and consensus level may lead to destroy the original evaluation information and sometimes several iteration rounds are required. In view of these, the proposed approaches have advantage in avoiding the loss of information and the calculation seems simpler.
According to the comparison analysis, the approaches proposed in this study have the following advantages over other existing approaches.
(1) Two decision-making models are developed in view of the proposed additive and multiplicative consistency measures. These approaches directly utilize the original judgement of evaluation information to make decision without adjusting them. It can then avoid information loss and distort, and the ranking result obtained from the proposed approaches seems more reasonable.
(2) The proposed approaches taking into account the decision makers' satisfaction degree. They are superior to the decision-making approaches based on acceptably consistent preference relations because it can omit the procedure to repair the inconsistent preference relations. (3) The method of determining the weights of decision makers is developed. This method makes full use of the evaluation information of decision makers.

Conclusion
This paper develops decision-making approaches based on decision makers' satisfaction degree with IHFPR. First, the consistency measures from the perspectives of additive and multiplicative consistent IHFPR are defined. Then, two decision-making models are developed in view of the proposed additive and multiplicative consistency measures. Second, a square programming model is developed to obtain the decision makers' weights. Finally, a procedure for MCDM problems with IHFPR is given, and an illustrative example in conjunction with comparative analysis is conducted.
The present study provides several significant contributions for MCDM problems with IHFPR. They are summarized as follows: (1) a new concept of additive and multiplicative consistent IHFPR is proposed, respectively. The main feature of them is that they consider all the evaluation information including in HFEs, and neither add values into HFEs nor remove values from HFEs. They can avoid information loss and distort. (2) Two decision-making models are developed based on the proposed additive and multiplicative consistency measures. The main characteristic of the constructed models is that these approaches consider the decision makers' satisfaction degree. (3) A square programming model is developed to obtain the decision makers' weights, which is utilized the optimal priority weight vectors information derived from individual IHFPR matrices. In our future research, the proposed approaches are extended to hesitant fuzzy linguistic preference relation and applied them to solve other practical MCDM problems.

Declarations
Conflict of Interest Authors Jian Li, Li-li Niu, Jianping Ye, Qiongxia Chen and Zhong-xing Wang declare that they have no conflict of interest.
Ethical approval This article does not contain any studies with human participants performed by any of the authors. Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation 3143