Hesitant Probabilistic Fuzzy Preference Relations in Decision Making

Preference of an alternative over another alternative is a useful way to express the opinion of decisionmaker. In the process of group decisionmaking, preference relations are used in preference modelling of the alternatives under given criteria.The probability is an important tool to deal with uncertainty; in many scenarios of decision making probabilities of different events affect the decision making process directly. In order to deal with this issue, in this paper, hesitant probabilistic fuzzy preference relation (HPFPR) is defined. Furthermore, consistency of HPFPR and consensus among decision makers are studied in the hesitant probabilistic fuzzy environment. In this respect, many novel algorithms are developed to achieve consistency of HPFPRs and reasonable consensus between decision makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision making. Lastly, we present a case study with complete illustration of the proposed model and discussed the effects of probabilities on decision making validating the importance of the introduction of probability in hesitant fuzzy preference relation.


Introduction
Fuzzy set theory was initially introduced by Zadeh [1] in 1965 as an extension of the classical set theory.In classical set theory, an element either belongs to or does not belong to the set.In fuzzy set theory, the gradual assessment of elements of set is described by the membership function that is in [0, 1].Fuzzy set theory can be used in which information is vague, incomplete, or imprecise and it is successfully used in decision making problems [2].After the popularity of this extension in set theory, several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, trapezoidal-valued intuitionistic fuzzy sets, type-2 fuzzy sets, and fuzzy multisets.These extensions have been successfully used in several practical applications of real life problems and scientific problems.Applications of these extensions can be found in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics.Torra felt that there are some limitations or deficiencies in these extensions.He proposed another extension of fuzzy set theory which is named as hesitant fuzzy set (HFS) theory [3].This extension permits the several possible membership degrees of an element in [0, 1].HFS provides a much better description than the other extensions of fuzzy sets where the difficulty of establishing the membership degree and there is a specific set of possible values.Many studies on HFS have been conducted, such as extensions of HFS (see [4][5][6][7][8][9]).
Group decision making (GDM) is a procedure to find the best/optimal alternative from a set of alternatives under the basis of certain criteria and the alternatives are evaluated by the group of decision makers for the criteria [10][11][12][13][14].The opinion of DMs may also be in the form of preference of each pair of alternatives and provides a comparison of one alternative over another and this comparison is a preference relation; for some basics of preference relation see [15].Preference relations have been developed and investigated in different modes, like multiplicative preference relations [16,17], fuzzy preference relations [18], multiplicative fuzzy preference relation [10], incomplete fuzzy preference relation [11,19], linguistic preference relations [20,21], intuitionistic fuzzy preference relations [22,23], and interval-valued hesitant preference relations [6].Due to several external nonfeasible circumstances, like shortage of time, lack of knowledge, and available data resources, DMs provided their preferences opinion over the alternatives in the form of several possible numerical values.All the discussed preference relations do not handle such kind of situations.To overcome this problem, Zhu and Xu [24] introduced hesitant fuzzy preference relation (HFPR).Here the preference of an alternative over another alternative is a HFE; this HFE shows all possible preference values that denote the hesitant degree between two alternatives.HFPRs provide a better framework for the description of the DMs' hesitation while providing their preferences among the alternatives [25][26][27][28].
In preference relations based decision making problems, the concept of consistency plays an important role.It is the level or degree of satisfaction among the values in preference relation and these values are given by each DM [29].Consensus is also an important and valuable concept in decision making problems based on preference relations.Consensus measure is used for the mutual understanding of DMs on the finally obtained alternatives [29].Until now, several researchers of this area successfully made some progress to convert the preference relations as consistent and generate a certain level of consensus in decision making problems [30][31][32][33].Nowadays, hesitant probabilistic fuzzy sets received good attention in multicriteria decision making.So, it is important to discuss the consistency and consensus measure for these preference relations.In this paper, we target to develop a group decision making model for HPFPRs where the consistency of the model and consensus among the DMs are under consideration.Our proposed model is efficient and practical; furthermore, it is strictly based on theoretical foundations.Pang et al. introduce the idea of probabilistic linguistic term set [34].They also develop the aggregation operator of this set and proposed the extended TOPSIS version to handle the multicriteria group decision making problems.By getting the motivation from the hesitant fuzzy set and probabilistic linguistic term set, Xu and Zhou proposed the concept of hesitant probabilistic fuzzy set [35].They investigated several aggregation operators with properties for hesitant probabilistic fuzzy set.A novel algorithm was developed to handle the multicriteria group decision making problems for hesitant probabilistic fuzzy set [36].This concept was further extended to the continuous form of hesitant probabilistic fuzzy set [37].Distance measures were also discussed for the continuous form and it was also applied to automotive industry safety evaluation problem.In these no one discusses the preference relations of the hesitant probabilistic fuzzy set.It is worth defining the preference relations for this set and discussing the consistency and consensus measure of the decision model.
To accomplish these goals, this paper is structured in the following way.In Section 2, some preliminary concepts are discussed to understand our proposal.Section 3 is devoted to design the basic structure of hesitant probabilistic fuzzy preference relations.A distance measured between the preference relations is developed, based on it, consistency measure of preference relations is derived.Also, two novel algorithms are designed to achieve acceptable consistency.In Section 4, consensus measure is defined for the group decision making and a novel algorithm is presented for reaching acceptable consensus among decision makers.Section 5 is dedicated to presenting a complete group decision making model dealing with both issues of consistency and consensus.In Section 6, numerical analysis of the developed model through a case study is performed to understand the importance of our proposal.Section 7 is dedicated for comparison between proposed model and existing ones.Section 8 ends the paper with some concluding remarks.
The notion of the hesitant fuzzy set given by Torra [3] is well known and has been successfully used to model vagueness of real life.It allows decision makers to give multiple membership values, but has the deficiency to deal with probabilities of preference degrees.To make hesitant fuzzy sets more compatible with real life, Xu and Zhou [35] defined hesitant probabilistic fuzzy element (HPFE) and hesitant probabilistic fuzzy set (HPFS).Definition (see [35]).Consider a fixed set .The HPFS on  is defined as a mathematical symbol: where ℎ(  The score and deviation functions are similar to the expectation and variance of the random variable, respectively, and, thus, the comparison laws for two HPFEs ℎ 1 and ℎ 2 can be presented as follows:

Hesitant Probabilistic Fuzzy Preference Relation
In order to build a complete model for group decision making first some operations are defined for HPFEs of the same length.Let ℎ(  |   ), ℎ 1 (  To allow decision makers to provide the preferences in hesitant probabilistic environment, we define hesitant probabilistic fuzzy preference relation (HPFPR).
Definition (HPFPR).Let  = { 1 ,  2 ,  3 , . . .,   } be the set of alternatives.The HPFPR is a matrix  = (ℎ  ) × , where ℎ  = {ℎ   |    :  = 1, 2, 3, . . ., #ℎ  } is the HPFE expressing the possible preference degrees of the alternative   over   with probabilities and with  >  satisfying the following conditions: But if one adopted the technique of -normalization, that is, making the length of ℎ  the same by adding elements to HPFEs of shorter length, it implicates numerous errors and difficulties when dealing with consistency of HPFPRs and consensus among decision makers based on -normalization.So fixing the length of diagonal elements to one does not match with the condition ∑ #ℎ =1  ()  = 1.So by allowing variation, the probabilities of diagonal HPFEs helps us to maintain the spirit of HPFE in discussing consistency and consensus in the context of -normalization.The diversity in probabilities of diagonal preference degrees does not cause any harm; the net impact remains the same as the sum of all probabilities is 1.
Often the length of HPFEs has been different, but to apply the above defined operations (3), (4), and (7), the length is needed to be equal for all HPFEs.Some elements will be added to HPFE who has less elements, but the information it provides that should not be changed.Now, the definition of normalized hesitant probabilistic fuzzy preference relation (NHPFPR) is proposed.
Let ℎ = {ℎ  |   :  = 1, 2, 3, . . ., #ℎ} be a HPFE.For preference degrees ℎ  Zhu et al. [28] define a way to add elements in HFE; for an optimized parameter 0 ≤  ≤ 1 the preference degree that will be added to ℎ  is ℎ + + (1 − )ℎ − , where ℎ + is the largest and ℎ − is the smallest among ℎ  .The decision maker can choose the value of  according to his risk preferences.The added element will be ℎ + and ℎ − for  = 1 and  = 0, respectively, which demonstrate the optimistic and pessimistic approach of decision maker proposed by Xu and Xia [42].In hesitant probabilistic fuzzy environment, some way is needed to assign probability to the added preference degree such that the information of HPFPR is not changed.There are many ways to do it; one option is to assign 0 to added preference degree ℎ + + (1 − )ℎ − , but for the extreme cases pessimistic approach, that is,  = 0, and optimistic approach, that is,  = 1, the added element in HPFE is ℎ − |  − /( − #ℎ + 1) and ℎ + |  + /( − #ℎ + 1), respectively, where  is the required length of HPFEs and  − and  + are the probabilities of ℎ − and ℎ + , respectively.

Example . Let
( Then  1 and  2 are HPFPRs.Now by taking optimized parameters 0, 1 for  1 and  2 , respectively, the following NHPFPR are obtained.) . ( . .Consistency Measure of Hesitant Probabilistic Fuzzy Preference Relation.In order to obtain valuable decision from preference relations they should be consistent in a sense that, let us say,  1 is preferable to  2 and  2 is preferable to  3 then  1 must be preferable to  3 .Several authors have pursued consistency issues for preference relations [1,21,32,33,43].
Additive consistency for fuzzy preference degrees is well known; actually Tanino [44] defined the additive consistency for fuzzy preference relation based on moderate stochastic transitivity, well known in the probabilistic choice theory [45, page 27].Furthermore, many kinds of transitivity are proposed and studied for probabilities in comparing the preferences in the choice theory.
But many times preference relations are not consistent and, for meaningful decision making, some level of consistency is required in the least if it is not fully consistent.For preference degrees, take the summation of ( 16) for all  ℎ therefore, Thus, (19) (22) then obtained probability will increase.These observations lead to the following novel algorithm producing a sequence of HPFPRs convergent to fully consistent HPFPR.
Output.NHPFPR , consistent HPFPR H, and number of iterations .
This result also gives the following theorem.] .
The area graphs are made of the above matrices by using Matlab drawing tool bar.Figures 1, 2, and 3 and Figures 4,  5, and 6 show the comparison of area graphs for fuzzy preference degrees, probability values, and score values between  1 , H1 and  2 , H2 , respectively.The areas are more smooth for consistent HPFPRs H1 and H2 .
Once the consistent HPFPR is found, we are in a position to make HPFPR acceptably consistent by defining consistency measure.In order to define consistency measure, first distance between two HPFEs is defined.Let Based on the above equation the distance between two HPFPRs is defined as follows.

Definition
(distance).Consider two HPFPRs  1 = (ℎ ,1 ) × and  2 = (ℎ ,2 ) × and their NHPFPRs  1 = (ℎ ,1 ) × and  2 = (ℎ ,2 ) × .The distance is defined as It is clear that the following properties are satisfied for ( 1 ,  2 ): (1) 0 ≤ ( 1 ,  2 ) ≤ 1; (2) ( 1 ,  2 ) = ( 2 ,  1 ); While the decision maker provides its preference in form of HPFPR, it should be noted that it can be used for decision making with good results only if it has sufficient consistency.Definition (consistency index).For a given HPFPR  = (ℎ  ) × , its NHPFPR  = (ℎ  ) × and consistent HPFPR H = ( h ) × with optimized parameter 0 ≤  ≤ 1 obtained from Algorithm 11.The consistency index of  is defined to be the distance between  and H, denoted as It is clear that CI() = 0 if and only if  is consistent.Ideally, the decision maker should provide consistent HPFPR  so that it can be used for meaningful decision making.However, some margin of error should be provided to decision maker relative to the practical problems.

Definition
(acceptably consistent HPFPR).Consider a HPFPR  = (ℎ  ) × .For a given tolerance value CI  HPFPR  is said to be acceptably consistent HPFPR if If a HPFPR is not even acceptably consistent, then the decision maker should revisit and modify it.A novel algorithm is proposed to make a HPFPR acceptably consistent.
The choice of parameters , ,  directly affects the performance of Algorithm 19.For performance measurement of Algorithm 19, 1000 random HPFPRs are generated and their acceptably consistent HPFPRs are computed by Algorithm 19, the average iteration in Algorithm 19 with respect to different parameters shown in Table 2.It is apparent that Algorithm 19 is quite efficient and, as for the effects of parameters, the increase in value  leads to more iteration in Algorithm 19 to compute consistent HPFPR; therefore, it is suggested to choose  small.Also, the number of iterations in Algorithm 19 is inversely proportional to the consistency index CI  .
The next result shows that the aggregated group HPFPR   obtained by ( 48 , ) For the inductive step, suppose the following: Then ( p(+1) , ) Hence, by principle of mathematical induction for all  ∈ N Equations ( 51) and (57) imply thus by ( 30) Now, the consensus is another very important aspect of group decision making.To deal with the consensus issue among all the individual decision makers, the following consensus index is defined.

Mathematical Problems in Engineering
The agreement between individual decision maker with group decision is measured by the distance between individual HPFPR   and group HPFPR   .Therefore, GCI(  ) = 0 means  th decision maker has full agreement with group decision; otherwise, the smaller the value of GCI(  ) is, the better the consensus will be.In many real life scenarios, it is important to have consensus among all decision makers, although we have to live with difference of opinion and it is hard to reach complete consensus; for this reason a threshold value can be decided based on practical nature of the problem to allow the difference of opinion to some extent.

Definition
(acceptably consensus HPFPRs).Let   = (ℎ , ) × ( = 1, 2, 3, . . ., ) be  HPFPRs provided by decision makers and their NHPFPRs   are obtained with optimized   ( = 1, 2, . . ., ).Furthermore compute group HPFPR   and GCI(  ) ( = 1, 2, . . ., ).Consider GCI  a tolerance value of consensus measure; then HPFPR   is said to be acceptably consensus with group HPFPR   if But it is possible that a decision maker has an unacceptable difference of opinion with group decision; in this regard a novel algorithm is proposed to modify his HPFPR to reach acceptable consensus.
The convergence of the above algorithm follows from the following result.(64) Proof.By (62), (3), and ( 7), Now, 1000 random sets of  HPFPRs are generated and Algorithm 29 is applied to develop consensus.The average value of iterations of Algorithm 29 is presented in Table 3 for different values of parameters.The readings of Table 3 suggest that the increase in value of parameter  has adverse effects on the number of iterations of Algorithm 29.So, the value of parameter  must be small.The more iterations are needed to develop consensus when GCI  is nearer to 1. Furthermore, Algorithm 29 does not disturb the consistency of HPFPRs; that is, if the individual HPFPRs   = (ℎ , ) × ( = 1, 2, 3, . . ., ) are consistent, then output of Algorithm 29 and the adjusted HPFPRs  ()   = (ℎ () , ) × are also consistent.

Decision Support Model for Group Decision Making with HPFPRs
Now, the issues of consistency and consensus are addressed.Algorithms 11-29 will provide the consistent individual HPF-PRs and the group HPFPR with agreement among decision makers.To comprehend final standing of the alternatives, first for the alternative   the  th row of group HPFPR is aggregated by HPFA operator (44) and secondly the aggregated HPFEs are ordered according to their score and deviation [35].In form of the following algorithm a complete decision model is presented.
Step .End Flowchart of the decision making model is presented in Figure 7.

Case Study
The proposed decision making model will be applied to a practical problem of investment in Forex.
Example .The Flagship Investment Company (FIC) is known for investment plans in different commodities in Forex.The investors hire FIC for making profitable plans of investments.An investor is interested to invest in four commodities: oil, gold, wheat, and copper denoted as  1 ,  2 ,  3 , and  4 , respectively.He approached FIC to indicate which commodity is more profitable to invest in.FIC makes a committee of four economic experts  1 ,  2 ,  3 , and  4 .Each expert will provide its preferences in terms of HPFPR depending upon many factors like previous market rates, market trends, possible future decisions of different regulatory bodies, economic stability, global peace situation, and so on.The complex nature of the Forex required tools that will model the vagueness and produce results closer to real life.HPFPR will allow the economic experts to express their  Remark .Inspection of Table 4 will reveal why it is important to consider the probabilities of different preference degrees.Many a time, decision makers are taking decisions related to future scenarios like our case study.The future is a mystery that cannot be predicted completely; probability is a useful tool to deal with uncertainties.In real life scenarios probability of different events can alter the decisions we make today.

Comparison with Existing Models
Preference relations are one of the important tools that are used for practical decision making in different complex scenarios.A wide number of researchers developed different kinds of preference relations models as fuzzy [10,11,32,40,43], intuitionistic fuzzy [23,26,27], and hesitant fuzzy [6,24,28,33,38].Of course fuzzy preference relations are subcase of hesitant fuzzy preference relations, as we have discussed the occurrence probability is considered to be the same which is impractical as Table 4 reflects the impact of probabilities on decision making.So, for this reason the proposed model has provided better modelling of real world problems.In hesitant probabilistic fuzzy environment Zhou and Xu [41] proposed and studied probabilistic fuzzy preference relations.The drawback and difference between their definition of preference relation and proposed Definition 6 have already been discussed in Remark 7. First, they proposed expected consistency based upon multiplicative consistency of fuzzy preference relation [44]; then based on expected consistency multiobjective goal programming models are developed to compute occurrence probability and priority weight vector.Also, an iterative optimization algorithm to improve consistency of nonconsistent preference relations was designed.The comparison is summarized as follows.
(i) Zhou and Xu's definition [41,Definition 4] is not suitable for decision making based on -normalization principle, whereas proposed Definition 6 is consistent with -normalization; in fact HPFPRs are normalized following -normalization principle.(ii) They proposed expected consistency, taking a unifying approach to handle both preference degrees and their occurrence probabilities, whereas additive consistency [44] and weak stochastic transitivity [45,Page 27] are adopted for preference degrees and probabilities, respectively, in our proposal.(iii) They defined expected consistency index based on priority weights computed from multiobjective goal programming model made from expected consistency, whereas for HPFPR  Algorithm 11 (based on convex combination technique) will provide its corresponding NHPFPR  and consistent HPFPR H and consistency index is defined to be the distance between  and H, in our proposal.
(iv) They developed an algorithm which will replace the preference degrees by optimal ones iteratively and occurrence probabilities are computed accordingly to improve consistency, whereas Algorithm 19 (based on convex combination technique) will improve consistency, in our proposal.
(v) Both proposals use different techniques and ideas to define and deal with consistency as the above points reflect.However, Zhou and Xu make no effort for group decision making; the main advantage of our proposal is that it is a complete model, not only dealing with group decision making but also providing solution to consensus problem among decision maker, which is a key issue in group decision making.
The HFPR is a special case of HPFPR to provide experimental comparison with existing models of group decision making models with HFPRs [33,38]; we solved their case studies by our proposed model, in a way that preference degrees are the same and their occurrence probabilities are considered to be equal.The comparison is provided in Table 5; clearly our proposed model is quite efficient as it provides the correct results.

Conclusion
In this paper, we have developed a complete group decision support model based on HPFPRs; it consists of three parts: a consistency improving process, a consensus reaching process, and the selection process.The consistency measure of a HPFPR has been defined in the consistency improving process.For HPFPRs that have unacceptable consistency, an optimization method is proposed to improve the consistency until the HPFPRs have an acceptable consistency.In the consensus reaching process, a consensus index is defined to measure the consensus level.For HPFPRs that have an unacceptable consensus, an optimization method is designed to assist them in achieving a predefined consensus level.The proposed model can be used to address GDM problems with HPFPRs.Optimization methods are also developed to help the individual HPFPRs to achieve a predefined consistency level and consensus level with fewer interactions of the DMs.As a consequence, our model is time saving, efficient, and convenient for practical applications.The consistency improving process is performed to ensure that the DMs are neither random nor illogical in their pairwise comparisons.Procedure to reach a consensus level ensures that the adjusted HPFPRs not only achieve the predefined level of consensus but also maintain acceptable consistency.Proposed model also ensures that the consistent HPFPRs do not change at each iteration.This property can retain the DMs' original decision making information to the greatest extent possible.Here we did not discuss the effects of the application of different controlling parameters and distance functions in the developed model.This model is not useful for the incomplete preference relation.In the future, we will pay attention to addressing these problems.Future research can also focus on the extension of the proposed model to other types of preference relations.

Figure 1 :
Figure 1: Area graphs of fuzzy preference degrees for  1 .

Theorem 20 .
Consider a HPFPR , consistency tolerance value   , and the sequence { () } of HPFPRs generated by Algorithm .en
|   ) is HPFE comprising the elements of the form   |   , expressing the hesitant fuzzy information with Mathematical Problems in Engineering 3 probabilities to the set () |   () and   () |   () are  th elements of ℎ 1 (   |    ) and ℎ 2 (   |    ), respectively.For simplicity ℎ(  |   ) will be written as ℎ throughout this paper.

Table 1 :
Average value of iterations in Algorithm 11.

Table 2 :
Average iterations values in Algorithm 19.

Table 3 :
Average values of iterations in Algorithm 29.

Table 3 :
Continued. , group consensus tolerance value GCI  , the maximum number of iterations allowed  max , and the parameters 0 < ,  < 1 for modification.