Hesitant Probabilistic Fuzzy Information Aggregation Using Einstein Operations

In this paper, a hesitant probabilistic fuzzy multiple attribute group decision making is studied. First, some Einstein operations on hesitant probability fuzzy elements such as the Einstein sum, Einstein product, and Einstein scalar multiplication are presented and their properties are discussed. Then, several hesitant probabilistic fuzzy Einstein aggregation operators, including the hesitant probabilistic fuzzy Einstein weighted averaging operator and the hesitant probabilistic fuzzy Einstein weighted geometric operator and so on, are introduced. Moreover, some desirable properties and special cases are investigated. It is shown that some existing hesitant fuzzy aggregation operators and hesitant probabilistic fuzzy aggregation operators are special cases of the proposed operators. Further, based on the proposed operators, a new approach of hesitant probabilistic fuzzy multiple attribute decision making is developed. Finally, a practical example is provided to illustrate the developed approach.


Introduction
Decision making problems typically consist of finding the most desirable alternative(s) out of a given set of alternatives.So far, there are applications of decision making into different disciplines, such as railroad container terminal selection, pharmaceutical supplying, hospital service quality, and so on [1][2][3].Due to the increasing ambiguity and complexity of the socio-economic environment, it is difficult to obtain accurate and sufficient data for practical decision making.Therefore, uncertainty data needs to be addressed in the actual decision making process, and several other methodologies and theories have been proposed.Among them, the fuzzy set theory [4] is excellent and has been widely used in many areas of real life [5][6][7][8].Since Zadeh [4] introduced the fuzzy set (FS) in 1965, many researchers have developed extended forms of FS, such as the intuitive fuzzy set (IFS) [9], the type-2 fuzzy set [10], the type-n fuzzy set [10], the fuzzy multiset [11] and the fuzzy hesitant set (HFS) [12].Among these, the HFS was broadly applied to the practical decision making process.In fact, the HFS is widely used in decision making problems with the aim of resolving the difficulty of explaining hesitation in the actual assessment.The main reason is that experts may face situations in which people are hesitant to provide their preferences in the decision making process by allowing them to prefer several possible values between 0 and 1. Torra [12] introduced some basic operations of HFSs.Xia and Xu [13] defined the hesitant fuzzy element (HFE), which is the basic component of the HFS, and proposed and investigated the score function and comparison law of HFEs as the basis for its calculation and application.Li et al. [14] and Meng and Chen [15] proposed various distance measures and some correlation coefficients for HFSs.They also investigated applications based on the distance measures and correlation coefficients.Over the past decade, there many researchers [16][17][18][19][20][21][22][23] have studied the aggregation operators, one of the core issues of HFSs.Thus, many researchers have worked hard to develop the HFS theory and have helped to develop it in uncertain decision making problems [24][25][26].
However, there is one obvious weakness in the current approaches; namely, each of the possible values in the HFE provided by the experts has the same weight.To overcome this weakness, Xu and Zhou [27] proposed the hesitant probabilistic fuzzy set (HPFS) and hesitant probabilistic fuzzy element (HPFE) developed by introducing probabilities to HFS and HFE respectively.For example, experts evaluate a house's "comfort" using an HFE (0.3, 0.4, 0.5) because they hesitate to evaluate it.However, they believe that 0.4 is appropriate and 0.3 is less appropriate than the other values in the HFE.Therefore, although the HFE (0.3, 0.4, 0.5) cannot fully represent the evaluation, the HPFE (0.3|0.2, 0.4|0.5, 0.5|0.3)can present this issue vividly and is more convenient than HFE.Consequently, the HPFS can overcome the defect of HFS to great extent, so it can remain the experts' evaluation information and describe their preferences better.In Ref. [27], the HPFE was combined with weighted operators to develop basic weighted operators, such as hesitant probabilistic fuzzy weighted average/geometric (HPFWA or HPFWG) operators and the hesitant probabilistic fuzzy ordered weighted averaging/geometric (HPFOWA or HPFOWG) operators.Based on the perspective of the aggregation operators, they established the consensus among decision makers in group decision making.Zhang and Wu [28] investigated some operations of HPFE and applied them to multicriteria decision making (MCDM).In another way, some scholars recently tried to solve the problem of HFSs.Bedregal et al. [29] tried to use fuzzy multisets to improve the HFSs.This method has been worked out to some extent.Wang and Li [30] proposed the picture hesitant fuzzy set to express the uncertainty and complexity of experts' opinions and applied them to solve diverse situations during MCDM processes.Interval-valued HFSs have been used in the applications of group decision making in [31].Multiple attribute decision making (MADM) using the trapezoidal valued HFSs is discussed in [32].Yu [33] gave the concept of triangular hesitant fuzzy sets and used it for the solution of decision making problems.Mahmood et al. [34] introduced the cubic hesitant fuzzy set and applied it to MCDM.
The study on aggregation operators to fuse hesitant probabilistic information is one of the core issues in HPFS theory.The all aggregation operators introduced previously, such as the HPFWA, HPFWG, HPFOWA, and HPFOWG operators, are based on the algebraic product and algebraic sum of HPFEs, which are a pair of the special dual t-norm and t-conorm [35].Although the algebraic product and algebraic sum are the basic algebraic operations of HPFEs, they are not the only ones.The Einstein product and Einstein sum are good alternatives to the algebraic product and algebraic sum for structuring aggregation operators, respectively, and they have been used to aggregate the intuitionistic fuzzy values or the HFEs by many researchers [21][22][23][36][37][38].However, it seems that in the literature, there has been little investigation on aggregation techniques using the Einstein operations to aggregate hesitant probabilistic fuzzy information.Thus, it is meaningful to research the hesitant probabilistic fuzzy information aggregation methods based on the Einstein operations.In this paper, motivated by the works of Xu and Zhou [27] and Yu [21], we propose the hesitant probabilistic fuzzy Einstein weighted aggregation operators with the help of Einstein operations, and apply them to MADM under a hesitant probabilistic fuzzy environment.To do this, the remainder of this paper is organized as follows: The following section recalls briefly some basic concepts and notions related to the HPFSs and HPFEs.In Section 3, based on the hesitant probabilistic fuzzy weighted aggregation operator and the Einstein operations, we propose the hesitant probabilistic fuzzy Einstein weighted aggregation operators including the hesitant probabilistic fuzzy Einstein weighted averaging/geometric (HPFEWA or HPFEWG) operators and the hesitant probabilistic fuzzy Einstein ordered weighted averaging/geometric (HPFEOWA or HPFEOWG) operators.Section 4 develops an approach to MADM with hesitant probabilistic fuzzy information based on the proposed operators.An example is given to demonstrate the practicality and effectiveness of the proposed approach in Section 4. Section 5 gives some concluding remarks.

HPFS and HPFE
The HPFS and HPFE represent hesitant fuzzy information with the following probabilities.Definition 1. [27] Let R be a fixed set, then an HPFS on R is expressed by a mathematical symbol: where h(γ i |p i ) is a set of some elements (γ i |p i ) denoting the hesitant fuzzy information with probabilities to the set H P , is the hesitant probability of γ i , and For convenience, Xu and Zhou [27] called h(γ i |p i ) a HPFE, and H P the set of HPFSs.In addition, they gave the following score function, deviation function, and comparison law to compare different HPFEs.

Some HPFE Weighted Aggregation Operators Based on Einstein Operation
One important issue is the question of how to extend Einstein operations to aggregate the HPFE information provided by the decision makers.The optimal approach is weighted aggregation operators, in which the widely used technologies are the weighted averaging (WA) operator, the ordered weighted averaging (OWA) operator, and their extended forms [39,40].Yu [21] proposed the hesitant fuzzy Einstein weighted averaging (HFEWA) operator, the hesitant fuzzy Einstein ordered weighted averaging (HFEOWA) operator, the hesitant fuzzy Einstein weighted geometric (HFEWG) operator, and the hesitant fuzzy Einstein ordered weighted geometric (HFEOWG) operator based on those operators.Similar to these hesitant fuzzy information aggregation operators, we propose the corresponding hesitant probabilistic fuzzy Einstein weighted and ordered operators to aggregate the HPFEs.Definition 8. Let ht (t = 1, 2, . . ., T) be a collection of HPFEs; then, a hesitant probabilistic fuzzy Einstein weighted averaging (HPFEWA) operator is a mapping H T P → H P such that where w = (w 1 , w 2 , . . ., w T ) T is the weight vector of ht (t = 1, 2, . . ., T) with w t ∈ [0, 1] and ∑ T t=1 w t = 1, and p t is the probability of γ t in HPFE ht .In particular, if w = 1 T , 1 T , . . ., 1 T T , then the HPFEWA operator is reduced to the hesitant probabilistic fuzzy Einstein averaging (HPFEA) operator: From Definitions 7 and 8, we can get the following result by using mathematical induction.
However, the HPFEWA operator does not satisfy the idempotency.To illustrate this, we give the following example.
Proof.We prove Equation ( 15) by mathematical induction on T.
Assume that Equation (15) holds for T = k, i.e., In accordance with the Einstein operational laws of HPFEs for T = k + 1, we have i.e., Equation ( 15) holds for T = k + 1.Then, Equation ( 15) holds for all T. Hence, we complete the proof of the theorem.
Based on Theorem 6, we have basic properties of the HPFEWG operator, as follows: . ., T) be a collection of HPFEs, w = (w 1 , w 2 , . . ., w T ) T be the weight vector of ht (t = 1, 2, . . ., T) such that w t ∈ [0, 1] and ∑ T t=1 w t = 1, and p t be the corresponding probability of γ (t) i in HPFE ht .Then, we have the following.
On the other hand, based on Equation (15), the aggregated value by HPFEWG operator is If we use the HPFWG operator (Equation (3)) to aggregate two HPFEs, then we get It is clear that HPFEWG( h1 , h2 ) > HPFWG( h1 , h2 ).
Theorem 8 shows that (1) the values aggregated by the HPFEWA operator are not larger than those obtained by the HPFWA operator.That is to say, the HPFEWA operator reflects the decision maker's pessimistic attitude rather than the HPFWA operator in the aggregation process; and (2) the values aggregated by the HPFWG operator are not larger than those obtained by the HPFEWG operator.Thus, the HPFEWG operator reflects the decision maker's optimistic attitude rather than the HPFWG operator in the aggregation process.Moreover, we developed the following ordered weighted operators based on the HPFOWA operator [27] and the HPFOWG operator [27] to aggregate the HPFEs.
In the following section, we look at the HPFEOWA and HPFEOWG operators for some special cases of the associated vector ω.
Similar to Theorems 8 and 9, the above ordered weighted operators have the relationship below.
Clearly, the fundamental characteristic of the HPFEWA and HPFEWG operators is that they consider the importance of each given HPFE, whereas the fundamental characteristic of the HPFEOWA and HPFEOWG operators is the weighting of the ordered positions of the HPFEs instead of weighting the given HPFEs themselves.By combining the advantages of the HPFEWA (resp.HPFEWG) and HPFEOWA (resp.HPFEOWG) operators, in the following text, we develop some hesitant probabilistic fuzzy hybrid aggregation operators that weight both the given HPFEs and their ordered positions.

An Approach to MADM with Hesitant Probabilistic Fuzzy Information
In this section, we utilize the proposed aggregation operators to develop an approach for MADM with hesitant probabilistic fuzzy information.
Step 1: Obtain the normalized hesitant probabilistic fuzzy decision matrix.In general, the attribute set (G) can be divided two subsets, G 1 and G 2 , where G 1 and G 2 are the set of benefit attributes and cost attributes, respectively.If all of the attributes are of the same type, then the evaluation values do not need normalization, whereas if there are benefit attributes and cost attributes in MADM, in such cases, we may transform the evaluation values of cost type into the evaluation values of the benefit type by the following normalization formula: where Then, we obtain the normalized hesitant probabilistic fuzzy decision matrix H = rij (β ij |p ij ) m×n (see Table 2).
Step 3: Rank the order of all alternatives.Utilize the method in Definition 3 to rank the overall rating values rj (j = 1, 2, . . ., n).Rank all the alternatives( x j (j = 1, 2, . . ., n)) in accordance with rj (j = 1, 2, . . ., n) in descending order, and finally, select the most desirable alternative(s) with the largest overall evaluation value(s).
In the above-mentioned procedure, the HPFEWA (or HPFEWG) operator is utilized to aggregate the evaluating values of each alternative with respect to a collection of the attributes to rank and select the alternative(s).So we give a detail illustration of the decision making procedure with a propulsion/manoeuvring system selection problem.
Example 7. The propulsion/manoeuvring system selection is based on a study that was conducted for the selection of propulsion/manoeuvring system of a double ended passenger ferry to operate across the Bosphorus in Istanbul with the aim of reducing the journey time in highly congested seaway traffic (adopted from Ölçer and Odabaşi [43] and Wang and Liu [37]).
The propulsion/manoeuvring system alternatives are given as the set of alternatives X = {x 1 , x 2 , x 3 }.(1) x 1 is the conventional propeller and high lift rudder; (2) x 2 is the Z drive; and (3) x 3 is the cycloidal propeller.The selection decision is made on the basis of one objective and seven subjective attributes, which are the following: (1) g 1 is the investment cost; (2) g 2 is the operating cost; (3) g 3 is the manoeuvrability; (4) g 4 is the propulsive power requirement; (5) g 5 is the reliability.;(6) g 6 is the propulsive power requirement; and (7) g 7 is the propulsive arrangement requirement.Note that the attributes are cost attributes, except for attributes g 3 and g 5 , and the corresponding weight vector is w = (0.15, 0.2, 0.3, 0.2, 0.15) T .
Assume that the decision makers use the linguistic terms shown in Table 3 to represent the evaluating values of the alternatives with respect to different attributes, respectively, and they provide their linguistic decision matrices (D) as listed in Tables 4.
Table 3. Linguistic terms and their corresponding hesitant probabilistic fuzzy elements (HPFEs).
The relative comparison of the methods using different operators proposed by Xu and Zhou [27] is shown in Table 7. From Table 7, we can see that the obtained overall rating values of the alternatives are different with each of the four operators, respectively, and then, the ranking orders of the alternatives also are different.Each of the methods using different hesitant probabilistic fuzzy operators has its advantages and disadvantages, and none of them always perform better than the others in any situation.It depends on how we look at things, and not on how they are themselves.

Conclusions
The hesitant probabilistic fuzzy MADM is an important research topic in HPFS theory and decision science with uncertain information.Information aggregation is one of the core issues.Based on the Einstein operational rules of HPFEs, in this paper, we developed a series of hesitant probabilistic fuzzy Einstein aggregation operators, including the HPFEWA, HPFEWG, HPFEOWA, HPFEOWG, HPFEHA, and HPFEHG operators.Some basic properties of the proposed aggregation operators, such as boundedness and monotonicity, and the relationships between them were investigated.We compared the proposed operators with the existing hesitant probabilistic fuzzy aggregation operators proposed by Xu and Zhou [27] and presented corresponding relations.These proposed hesitant probabilistic Einstein aggregation operators provide a fine supplement to the existing work on HPFSs.Based on the HPFEWA and HPFEWG operators, a new method for MADM was developed in hesitant probabilistic fuzzy environments.A practical example was provided to illustrate the hesitant probabilistic fuzzy MADM process.Through a comparison between the proposed method with the previously proposed hesitant probabilistic fuzzy MADM method [27], we showed some advantages of the proposed hesitant probabilistic fuzzy MADM method.
This paper only considered decision makers with equl weights in the decision making process, but further studies on unequal weights are needed.Moreover, research using other operations, such as Hamacher and Frank t-conoms and t-norms instead of the Einstein t-conorm and t-norm, should be discussed in future studies.

Table 7 .
Comparison of overall rating values and ranking orders of alternatives.