1. 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
because they hesitate to evaluate it. However, they believe that
is appropriate and
is less appropriate than the other values in the HFE. Therefore, although the HFE
cannot fully represent the evaluation, the HPFE
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.
3. 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 () be a collection of HPFEs; then, a hesitant probabilistic fuzzy Einstein weighted averaging (HPFEWA) operator is a mapping such thatwhere is the weight vector of () with and , and is the probability of in HPFE . In particular, if , 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.
Theorem 4. Let () be a collection of HPFEs; then, their aggregated value obtained using theHPFEWA
operator is also a HPFE, andwhere is the weight vector of () with and , and is the probability of in HPFE . Proof. We prove Equation (
8) by mathematical induction. For
, since
and
, then
If Equation (
8) holds for
, that is
then, when
, according to the Einstein operations of HPFEs, we have
i.e., Equation (
8) holds for
. Hence, Equation (
8) holds for all
T. Thus,
which completes the proof of theorem. □
Based on Theorem 4, we have basic properties of the HPFEWA operator, as follows:
Theorem 5. Let () be a collection of HPFEs, be the weight vector of () such that and , and be the corresponding probability of in HPFE ; then, we have the following:
(1) (Boundary):where and . (2) (Monotonicity): Let () be a collection of HPFEs with for , be the weight vector of (), such that and , and is the probability of in HPFE . If for each , ; then, Proof. (1) Let
,
, then
, i.e.,
is a decreasing function. Let
and
. For any
(
), since
, then
, and so
Since
is the weight vector of
(
) with
and
, we have
Since
and
, we get
i.e.,
Let
,
, where
,
and
; then, Equation (
11) is transformed into the following form:
for all
. Thus,
and
.
If and , then by Definition 3, we have . If , i.e., , then . In this case, in accordance with Definition 3, it follows that . If , then similarly, we have .
(2) Let
,
; then,
is a decreasing function. If
for each
,
; then,
, for each
,
, i.e.,
, for each
,
. For any
(
), since
is the weight vector of
(
) such that
,
and
, we have
Let
and
, where
#, and
is the number of possible elements in
and
, respectively, then the Equation (
12) is transformed into the form
(
). Thus,
.
If , then, according to Definition 3, we have . If , i.e., , then . In this case, based on Definition 3, it follows that . □
However, the HPFEWA operator does not satisfy the idempotency. To illustrate this, we give the following example.
Example 3. Let , and is the weight vector (); then,and thus . Based on the HPFWG operator and Einstein operation, we developed the hesitant probabilistic fuzzy Einstein weighted geometric operator as follows:
Definition 9. Let () be a collection of HPFEs; then, the hesitant probabilistic fuzzy Einstein weighted geometric (HPFEWG) operator is a mapping () such thatwhere is the weight vector of () with and , and is the probability of in HPFE . In particular, if , then the HPFEWG operator is reduced to the hesitant probabilistic fuzzy Einstein geometric (HPFEG) operator: Theorem 6. Let () be a collection of HPFEs; then, their aggregated value obtained using theHPFEWG
operator is also a HPFE andwhere is the weight vector of () with and , and is the probability of in HPFE . Proof. We prove Equation (
15) by mathematical induction on
T. When
, since
and
, we have
Assume that Equation (
15) holds for
, i.e.,
In accordance with the Einstein operational laws of HPFEs for
, we have
i.e., Equation (
15) holds for
. 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:
Theorem 7. Let () be a collection of HPFEs, be the weight vector of () such that and , and be the corresponding probability of in HPFE . Then, we have the following.
(1) (Boundary):where and . (2) (Monotonicity): Let () be a collection of HPFEs with for , be the weight vector of () such that and , and be the probability of in HPFE . If for each , , then Proof. (1) Let
,
; then,
, i.e.,
is a decreasing function. Let
and
. For any
(
), since
; then,
, and so
Since
is the weight vector of
(
) with
and
, we have
Since
and
, we obtain
i.e.,
Let
,
, where
,
and
. Then, Equation (
18) is transformed into the following forms:
for all
. Thus,
and
. If
and
. Then, based on Definition 3, we have
. If
, i.e.,
, then
. In this case, based on Definition 3, it follows that
. If
. Then, similarly, we have
.
(2) Let
,
; then,
is a decreasing function. If
for each
,
, then
, for each
,
, i.e.,
, for each
,
. For any
(
), since
is the weight vector of
(
) such that
,
and
, we have
Let
and
, where
#, and
is the number of possible elements in
and
, respectively. Then, the Equation (
19) is transformed into the form
(
). Thus,
. If
, then based on Definition 3,
. If
, i.e.,
, then
. In this case, based on Definition 3, it follows that
. □
If all probabilities of values in each HPFE are equal, i.e.,
(
), then the HPFE is reduced to the HFE. In this case, the score function of the HPFEWA (resp. HPFEWG) operator is consistent with that of the HFEWA (resp. HFEWG) operator [
21]. So, we can conclude that the HPFEWA (resp. HPFEWG) operator is reduced to the HFEWA (resp. HFEWG) operator [
21]. In order to analyze the relationship between the HPFEWA (resp. HPFEWG) operator and the HPFWA (resp. HPFWG) operator [
27], we introduce the following lemma.
Lemma 1. [
41,
42]
Let , , , and , then , with equality if and only if . Theorem 8. If () are a collection of HPFEs and is the weight vector of , with and , and is the probability of in HPFE , then
(1) ;
(2) .
Proof. (1) For any
(
), based on Lemma 1, we obtain the inequality
, and then
Hence, we can obtain the inequality
Let
and
,
, where
is the number of possible elements in
and
, respectively. Then, Equation (
20) is transformed into the form
(
). According to
, we have
.
(2) For any
(
), bsed on Lemma 1, we have
, and then
Hence, similarly to (1), we have .
Example 4. Let and be two HPFEs and be the weight vector of them. Then, based on Equation(
8)
, the aggregated value from the HPFEWA operator is If we use the HPFWA operator (Equation(
2)
) to aggregate two HPFEs, then we have Then, and , and thus, .
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 .
Theorem 9. If () are a collection of HPFEs, is the weight vector of with and , and is the probability of in HPFE . Then,
(1) ;
(2) .
Proof. Since (2) is similar (1), we only prove (1).
□
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.
Let () be a collection of HPFEs, be the tth largest of (), and be the probability of in the HPFE ; then, we have the following two aggregation operators, which are based on the mapping with an associated vector , such that and :
(1) The hesitant probabilistic fuzzy Einstein ordered weighted averaging (HPFEOWA) operator is
(2) The hesitant probabilistic fuzzy Einstein ordered weighted geometric (HPFEOWG) operator is
Example 5. Let and be two HPFEs, and suppose that the associated aggregated vector is . Based on Definition 3, the score values of and are and . Since ; then, Based on Equation (21), the aggregated values by the HPFEOWA operator are On the other hand, based on Equation(
22)
, the aggregated values by the HPFEOWG operator are In the following section, we look at the HPFEOWA and HPFEOWG operators for some special cases of the associated vector .
(1) If
, then
(2) If
, then
(3) If
, then
where
is the
sth largest
(
).
(4) If , then
i.e., the HPFEOWA (resp. HPFEOWG) operator is reduced to HPFEA (resp. HPFEG) operator.
Similar to Theorems 8 and 9, the above ordered weighted operators have the relationship below.
Theorem 10. If () is a collection of HPFEs, is the associated vector of the aggregation operator such that and . Then,
(1) ;
(2) .
Theorem 11. If () is a collection of HPFEs, is the associated vector of the aggregation operator, such that and . Then,
(1) ;
(2) .
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.
Let () be a collection of HPFEs, be the weight vector of with and , and be the probability of in the HPFE . Then, we have the following two aggregation operators which are based on the mapping with an associated vector , such that and :
(1) The hesitant probabilistic fuzzy Einstein hybrid averaging (HPFEHA) operator is
where
is the
tth largest of the weighted HPFEs
(
),
T is the balancing coefficient, and
be the probability of
in the HPFE
.
(2) The hesitant probabilistic fuzzy Einstein hybrid geometric (HPFEHG) operator is
where
is the
tth largest of the weighted HPFEs
(
),
T is the balancing coefficient, and
is the probability of
in the HPFE
.
Especially, if
, then
(
). In this case, the HPFEHA (resp. HPFEHG) operator is reduced to the HPFEOWA (resp. HPFEOWG) operator. If
, then since
and
, we have
i.e., the HPFEHA (resp. HPFEHG) operator is reduced to the HPFEWA (resp. HPFEWG) operator.
Example 6. Let and be two HPFEs. Suppose that the weight vector of them is , and the aggregation associated vector is . Then,and and . Since , we have From Equation(
23)
, we have On the other hand,and since , we have and . From Equation(
24)
, we have 4. 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.
Let
be a set of
n alternatives and
be a set of
m attributes whose weight vector is
, satisfying
(
) and
, where
denotes the importance degree of attribute
. Suppose the decision makers provide the evaluating values that the alternatives
(
) satisfy the attributes
(
) represented by the HPFEs
(
;
). All of these HPFEs are contained in the hesitant probabilistic fuzzy decision matrix
(see
Table 1).
The following steps can be used to solve the MADM problem under the hesitant probabilistic fuzzy environment and obtain an optimal alternative.
Step 1: Obtain the normalized hesitant probabilistic fuzzy decision matrix. In general, the attribute set (
G) can be divided two subsets,
and
, where
and
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
is the complement of
. Then, we obtain the normalized hesitant probabilistic fuzzy decision matrix
(see
Table 2).
Step 2: Compute the overall assessment of alternatives. Utilize the HPFEWA operator
or the HPFEWG operator
to aggregate all the evaluating values
(
) of the
jth column and get the overall rating value
corresponding to the alternative (
(
)).
Step 3: Rank the order of all alternatives. Utilize the method in Definition 3 to rank the overall rating values (). Rank all the alternatives( ()) in accordance with () in descending order, and finally, select the most desirable alternative(s) with the largest overall evaluation value(s).
Step 4: End.
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 . (1) is the conventional propeller and high lift rudder; (2) is the Z drive; and (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) is the investment cost; (2) is the operating cost; (3) is the manoeuvrability; (4) is the propulsive power requirement; (5) is the reliability.; (6) is the propulsive power requirement; and (7) is the propulsive arrangement requirement. Note that the attributes are cost attributes, except for attributes and , and the corresponding weight vector is .
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
Table 4.
Step 1: Based on
Table 3 and
Table 4, we can get the hesitant probabilistic fuzzy decision matrix
(see
Table 5).
Then, considering that the attributes are cost attributes, except for attributes
and
, based on Equation (
25), the hesitant probabilistic fuzzy decision matrix (
D) can be transformed into the following normalized hesitant probabilistic fuzzy decision matrix:
(see
Table 6).
Step 2: Utilize the decision information given in matrix
H and the HPFEWA operator (
26) to derive the overall rating values (
) of the alternative
(
):
Step 3: Calculate the score values of the overall rating values (
) of the alternatives (
(
)):
Since
, the ranking order of the alternatives
(
) is
Therefore, the best alternative is .
If we utilize the HPFEWG operator (
27) in Step 2 to get the overall rating values (
) of the alternatives (
(
)), we obtain
Then, we calculate the scores of the overall rating values
of the alternatives:
Since
, the ranking order of the alternatives
(
) is
Then, the best alternative is also .
In order to compare the performance with the existing operators, in the following text, the HPFWA operator (
2) and HPFWG operator (
3) proposed by Xu and Zhou [
27] are used to computing the overall rating values. If we first utilize the HPFWA operator (
2) presented in Step 2, then we get the overall rating values
of the alternatives (
(
)):
Then, the scores of the overall rating values ( ()) are , , and , and so, the ranking order of the alternatives ( ()) is . Thus, the best alternative is .
Next, if we utilize the HPFWG operator (
3) presented in Step 2, we get the overall rating values (
) of the alternatives
(
):
Then, the scores of the overall rating values ( ()) are , , and , and so the ranking order of the alternatives ( ()) is . Thus, the best alternative is .
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.
Consequently, the use of different hesitant probabilistic fuzzy aggregation operators reflects the decision maker’s pessimistic (or optimistic) attribute. For example, the proposed HPFEWA operator shows that the decision maker has a more pessimistic attribute than the HPFWA operator [
27], and the proposed HPFEWG operator shows that the decision maker has a more optimistic attribute than the HPFWG operator [
27] in the aggregation process.