1. Introduction
Multi-attribute decision-making (MADM), whose main purpose is to get the rank of candidates based on a set of principles and decision experts’ opinions, is a promising research field, which has extensively gained many interests [
1,
2,
3,
4,
5]. Nevertheless, decision makers (DMs) feel it is becoming increasingly difficult and complicated to evaluate the performance of all the possible alternatives and appropriately determine the most satisfactory one in MADM issues. One of the primary reasons is the extensive existence of uncertainty and indeterminacy in decision-making process. For the sake of more convenient expressions of evaluation information, a large number of researchers have made great efforts to probe theories and tools to assistant DMs. One of the most advantageous theories is hesitant fuzzy sets (HFSs) [
6], which remind scholars and scientists the importance and necessity to consider both vagueness and DMs’ hesitancy in one framework. The peculiarity and characteristics of HFS make it well-known and its applications in MADM approaches have soon been proven to be promising and potential [
7,
8,
9,
10]. Later on, scholars focused on extensions of the classical HFSs, and dual hesitant fuzzy set (DHFS) [
11] is one of the most representative. The superiorities of DHFSs are reflected in two aspects, viz., they interpret fuzzy information from both positive and negative points, and efficiently represent DMs’ high hesitancy. Soon afterwards, MADM methods based on aggregation operators (AOs) and information measures of DHFSs, as well as extensions of classical decision methods into dual hesitant fuzzy environment, have become an active research area.
Dual hesitant fuzzy MADM methods provide DMs convenient manners to choose a wise alternative, however, as discussed in many publications, they still have limitations and shortcomings. For example, Hao et al. [
12] pointed out one defect of DHFSs in depicting fuzzy information. In the opinions of Hao et al. [
12], each member in dual hesitant fuzzy element (DHFE) has the same importance, which is counterintuitive and inconsistent with actual situations to a certain degree. Actually, the importance or DMs’ preferred intensity of each element in evaluation values should be counted and some similar researches have been done on the basis of this perspective. For instance, by considering the frequency or probability of each linguistic term in hesitant fuzzy linguistic term set, Pang et al. [
13] introduced the probabilistic linguistic term sets. Analogously, Zhao et al. [
14] proposed the probabilistic hesitant fuzzy sets by adding the probabilistic value of each membership degrees (MDs) in hesitant fuzzy elements. To evade the flaw of DHFSs, Hao et al. [
12] continued to propose the probabilistic DHFSs (PDHFSs) by adding corresponding probability of each member in DHFEs. As an attractive extension of DHFSs, PDHFSs can describe DMs’ assessment values more accurately and comprehensively, as they denote not only the MDs and non-membership degrees (NMDs), but also the corresponding probabilistic information. In Hao et al.’s [
12] publication, authors investigated operations, comparison principle and AOs of PDHFSs as well as a novel decision method to facilitate their applications in realistic MADM problems. It is necessary to point out that Hao et al.’s [
12] MADM method still have drawbacks, which limit its use in solving practical MADM issues and these flaws are still existing, although some improved decision approaches have been proposed. Generally speaking, the shortcomings of Hao et al.’s [
12] method are two-folds. First, PDHFSs have drawbacks in presenting complex DMs’ evaluation information and there exist many situations that cannot be adequately handled by PDHFSs. For instance, the restriction of PDHFSs is that the sum of MD and NMD should be less than one and if such sum is greater than one, then PDHFS is powerless. The second drawback is that the information integration methods proposed by Hao et al. [
12] fail to handle complicated realistic situations, such as wherein attributes are correlated.
Based on the above analysis, our motivations and goals are to avert aforementioned shortcomings by proposing a novel MADM method. To this end, we first propose a new technique to overcome the drawback of PDHFSs in denoting fuzzy decision information. The
q-rung dual hesitant fuzzy sets (
q-RDHFSs) [
15], as a new extension of Yager’s [
16]
q-rung orthopair fuzzy sets (
q-ROFSs), allow multiple MDs and NMDs, which is similar to DHFSs. However,
q-RDHFSs are more powerful than DHFSs, as they inherit the remarkable advantage of
q-ROFSs, i.e., permitting the sum of
qth power of MD and
qth power of NMD to be less than or equal to one. This character makes
q-ROFSs and
q-RDHFSs to be promising theories or tools, which has been widely noticed by scientists [
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29]. Therefore, aiming at the drawback of PDHFSs, we extend
q-RDHFSs to
q-rung probabilistic dual hesitant fuzzy sets (
q-RPDHFSs) by taking the probability of each member in
q-rung dual hesitant fuzzy element (
q-RDHFE) into consideration. The
q-RPDHFSs are parallel to PDHFS but are more powerful and useful, as they have a much laxer constraint, making the describable information space larger. Additionally, owing to the ability of denoting MDs, NMDs as well as their probabilities simultaneously,
q-RPDHFSs also exhibit advantages over
q-RDHFSs. To circumvent the second defect of Hao et al.’s [
12] method, we provide a series of compound AOs of
q-rung probabilistic dual hesitant fuzzy elements (
q-RPDHFEs). Absorbing the advantages of power average operator [
30] and Muirhead mean [
31], the power Muirhead mean (PMM), originated by Li and her colleagues [
32], has been proved to have flexibility and advantages in information fusion process [
33,
34,
35]. Naturally, the characteristics of PMM motivate us to extend it to
q-RPDHFSs to introduce some novel powerful hybrid AOs. Hence, we generalize PMM into
q-rung probabilistic dual hesitant fuzzy environment to propose some new AOs for
q-RPDHFEs. In this paper, we further illustrate why our AOs can overcome the second flaw of Hao et al.’s [
12] method.
The novelties and contributions of this work are presented as follows. (1) A new information representation model, called
q-RPDHFSs, was proposed. This contribution makes it easier and more convenient to depict DMs’ complex and fuzzy assessment information in decision-making problems. (2) The operations, score function, accuracy function, comparison method and distance measure of
q-RPDHFSs were presented and discussed. (3) Novel efficient AOs were put forward, which effectively aggregate integrate attribute values under
q-RPDHFSs. (4) A new MADM method was developed to judge the best alternative in
q-RPDHFSs. (5) Some actual MADM examples were provided to show the effectiveness of our new method. The structure of the rest of this paper is as follows.
Section 2 recalls basic concepts.
Section 3 proposes the
q-RPDHFSs and introduces their related notions, such as operational rules, comparison method, distance measure, etc.
Section 4 presents some AOs of
q-RPDHFEs and discusses their properties.
Section 5 presents a new MADM method under
q-RPDHFSs.
Section 6 conducts numerical experiments to show the performance of the new MADM method. Conclusions are provided in
Section 7.
3. q-Rung Probabilistic Dual Hesitant Fuzzy Sets
In this section, we propose the concept of q-RPDHFSs. In order to do this, we first briefly introduce the motivations of proposing q-RPDHFSs and explain why we need q-RPDHFSs. Then the definition, operational rules, comparison method and distance measure of q-RPDHFSs are further introduced.
3.1. Motivations of Proposing q-RPDHFSs
In actual MADM problems, it is highly necessary to comprehensively express DMs’ evaluation information before determining the best alternatives. In other word, depicting DMs’ evaluation values accurately and appropriately is a precondition, which makes the final decision consequences reliable and reasonable. As fuzziness and vagueness extensively exist in realistic decision-making issues, DMs usually express their assessment with the help of fuzzy sets. In addition, sometimes it is also needful to consider the probabilities of fuzzy values to more precisely denote attribute values provided by decision experts. We provide the following example to better demonstrate this phenomenon.
Example 1. The library of a university plans to purchase a batch of books. The library invites three decision experts to evaluate the performance of a potential book vendor under the attribute “reputation”. Each DM is required to use several values to denote the MDs and NMDs of his/her evaluation value. The assessment information provided by the three DMs is listed in Table 1. If we integrate each DM’s evaluation values in the form of DHFEs, then it can be denoted as {{0.1, 0.2, 0.3, 0.4, 0.5, 0.6}, {0.1, 0.2, 0.3, 0.5}}. However, it is noted that the multiple appearances of the MDs 0.4 and 0.5, and the NMDs 0.1, 0.2, and 0.3 are ignored, which implies that some fundamental information is lost. If we denote the group’s overall evaluation value by PDHFE, then it can be expressed as {{0.1|0.125, 0.2|0.125, 0.3|0.125, 0.4|0.25, 0.5|0.25, 0.6|0.125}, {0.1|0.25, 0.2|0.375, 0.3|0.25, 0.5|0.125}}. It is noted that when using PDHFE to express the evaluation value of the group, not only each MD and NMD, but also their corresponding probabilistic information is taken into account, which indicates the superiority of PDHFE. This example reveals the advantage of PDHFSs over DHFS. Nevertheless, PDHFSs still have shortcomings. If the third DM would like to employ {0.4, 0.6} to denote his/her preferred MDs, then the overall evaluation value cannot be handled by PDHFSs as 0.6 + 0.5 = 1.1 > 1. This example reveals the shortcomings of PHFSs and PDHFSs is they fail to deal with situations in which the sum of MD and NMD is greater than one. Hence, to circumvent such drawback and more accurately describe groups’ evaluation opinions, it is necessary to propose a new fuzzy information expression tool. Motivated by the q-RDHFSs, which have the character that the sum of qth power of MD and qth power of NMD is greater than one, we extend q-RDHFSs to q-RPDHFSs, which consider both the multiple MDs and NMDs, and their probabilistic information. The definition as well as some related notions of q-RPDHFSs are presented in the following subsections.
3.2. The Definition of q-RPDHFSs
Motivated by DHFSs, PDHFSs and q-RDHFS, we present the definition of q-RPDHFSs.
Definition 6. Let X be a fixed set, a q-rung probabilistic dual hesitant fuzzy set (q-RPDHFS) D defined on X is given by the following mathematical symbolwhere and are two series of possible elements, and denote the possible MDs and NMDs of the element to the set D, respectively. and are the probabilistic information for the MDs and NMDs, respectively. In addition, the elements , , and satisfying the following conditions:andwhere , , , , and . The symbols #h and #g represent the total numbers of elements in and , respectively. For convenience, is called a q-rung probabilistic dual hesitant fuzzy element (q-RPDHFE), which can be denoted by for simplicity. Remark 1. Especially, if all the probability values are equal in p and t, then the q-RPDHFS reduces to the q-RDHFS. In addition, when q = 1, the q-RPDHFS reduces to PDHFS proposed by Hao et al. [12]. If q = 2, then the probabilistic dual Pythagorean hesitant fuzzy sets (PDPHFSs) are obtained. In other word, the PDHFSs and PDPHFSs are special cases of our proposed q-RPDHFSs and q-RPDHFS is a generalized form of PDHFS and PDPHFS.
In Example 1, when the third DM uses {0.4, 0.6} to denote his/her preferred MDs, then the overall evaluation values of the group can be expressed as d = {{0.2|0.125, 0.3|0.125, 0.4|0.25, 0.5|0.25, 0.6|0.25}, {0.1|0.25, 0.2|0.375, 0.3|0.25, 0.5|0.125}}, which is a q-RPDHFE, as 0.62 + 0.52 = 0.61 < 1. This example implies that the proposed q-RPDHFSs are more powerful and flexible and have a lager range of applications than PDHFSs. In addition, compared with the traditional q-RDHFSs, q-RPDHFSs can more comprehensively express DMs’ evaluation opinions.
3.3. Basic Operational Rules of q-RPDHFEs
In this subsection, we introduce some basic operations of q-RPDHFEs and discuss their properties.
Definition 7. Let , and be any three q-RPDHFEs, and be a possible real number, then
- (1)
;
- (2)
;
- (3)
;
- (4)
.
Example 2. Let , and be three q-RPDHFEs (q = 3), then Based on Definition 7, we can obtain the following theorem.
Theorem 1. Let , and be any three q-RPDHFEs, and , then
- (1)
;
- (2)
;
- (3)
;
- (4)
;
- (5)
;
- (6)
;
- (7)
.
Proof. - (1)
.
- (2)
.
- (3)
.
- (4)
.
- (5)
.
- (6)
.
- (7)
.
□
3.4. Comparison Method of q-RPDHFEs
Definition 8. Let be a q-RPDHFE, the score function of is defined asand the accuracy function of is defined asFor any two q-RPDHFEs and , - (1)
If , then ;
- (2)
If , then
if , then ;
if , then .
Example 3. Let = {{0.3|0.6, 0.5|0.4}, {0.2|0.3, 0.5|0.7}} and = {{0.3|0.1, 0.7|0.3, 0.8|0.6}, {0.4|0.6, 0.5|0.2, 0.6|0.2}} be two q-RPDHFEs (q = 2), then according to Definition 8, we have Hence, we can obtain .
3.5. Distance between Two q-RPDHFEs
In this subsection, we propose the distance between any two q-RPDHFEs and discuss its properties.
Definition 9. Letand be any two q-RPDHFEs, then the distance measure between and is defined aswhere , , , . and are the ith largest values of and , and are the jth largest values of and . The symbol #h denotes the number of values in h1 and h2, and #g represents the number of values in g1 and g2. Remark 2. From Definition 9, we can find out that when calculating the distance between two q-RPDHFEs, they must have the same numbers of MDs and NMDs. However, this requirement cannot be always met. Hence, to operate correctly, the shorter q-RPDHFEs should be extended by adding some values until the numbers of the MDs and NMDs of the two q-RPDHFEs are equal. In the following, we present a principle to extend the short q-RPDHFEs. LetandIf and , then we have two methods to extend and . First, we assume DMs are optimistic to their evaluations, then we can extend and toandrespectively, where and . If DMs are pessimistic to their evaluations, then we can extend and toandrespectively, where and . In addition, from Definition 8 we can find out that the score and accuracy values are invariable. In this paper, we assume DMs are optimistic to their evaluation values and we always take the first method to extend q-RPDHFEs. To better illustrate this method, we provide the following example. Let and be two q-RPDHFEs (q = 4), then we can extend and to and . Then, according to Equation (12), the distance between and is Theorem 2. Let and be two q-RPDHFEs, then the distance between and satisfies the following conditions:
- (1)
;
- (2)
;
- (3)
, if and only if.
Proof. - (1)
Since , then we have . Hence, we can further obtain . Similarly, we can get . Therefore, we can derive .
- (2)
From Definition 9, we have
and
Hence, .
- (3)
If , then it is easy to get . If , then from Definition 9, we can obtain and . From Definition 8, we can easily get and . Thus, we can derive □
4. Some Aggregation Operators for Q-RPDHFEs and Their Properties
In this section, we extend PMM and PDMM to q-RPDHFSs and propose new AOs for q-RPDHFEs. We also investigate desirable properties of the proposed AOs.
4.1. The q-Rung Probabilistic Dual Hesitant Fuzzy Power Muirhead Mean (q-RPDHFPMM) Operator
Definition 10. Let be a collection of q-RPDHFEs, and be a vector of parameters. Then, the q-rung probabilistic dual hesitant fuzzy power Muirhead mean (q-RPDHFPMM) operator is defined as followswhererepresents any permutation of , denotes all possible permutations of , n is the balancing coefficient, and denotes the support for from , satisfying the following properties - (1)
;
- (2)
;
- (3)
If , then , where is the distance between and .
In order to simplify Equation (19), we assume
then Equation (19) can be written as
where
and
.
Theorem 3. Let be a collection of q-RPDHFEs, and be a vector of parameters. The aggregated value using the q-RPDHFPMM operator is still a q-RPDHFE and Proof. According to Definition 7, we get
then,
In addition, the q-RPDHFPMM operator has the property of boundedness.
Theorem 4. (Boundedness) Let be a collection of q-RPDHFEs, ifandthen Proof. For each element in the
q-RPDHFE, we have
and
. Then
and
.
For the probabilities, it is easy to get
and
. In addition, according to Theorem 3, we have
According to the score function, we have
Similarly, we have
and so that the proof of Theorem 4 is completed. □
From Definition 10, we can find out that the proposed q-RPDHFPMM operator is a generalized AO. Hence, it is interesting and necessary to study the special cases of the q-RPDHFPMM operator with respect to its contained parameters, which are presented as follows.
Case 1. If , then the q-RPDHFPMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power average (q-RPDHFPA) operator, i.e., In this case, if
for all
, then the
q-RPDHFPMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy average (
q-RPDHFA) operator i.e.,
Case 2. If , then the q-RPDHFPMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power Bonferroni mean (q-RPDHFPBM) operator, i.e., In this case, if
for
, then the
q-RPDHFPMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy Bonferroni mean (
q-RPDHFBM) operator, i.e.,
Case 3. If , then q-RPDHFPMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power Maclaurin symmetric mean (q-RPDHFPMSM) operator, i.e., In this case, if
for
, then the
q-RPDHFPMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy Maclaurin symmetric mean (
q-RPDHFMSM) operator, i.e.,
Case 4. If or , then the q-RPDHFPMM operator reduces to the following form In this case, if
for
, then
q-RPDHFPMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy geometric (
q-RPDHFG) operator, i.e.,
Case 5. If q = 2, then the q-RPDHFPMM operator reduces to the following form probabilistic dual Pythagorean hesitant fuzzy power Muirhead mean (PDPHFPMM) operator, i.e., Case 6. If q = 1, then the q-RPDHFPMM reduces to the probabilistic dual hesitant fuzzy power Muirhead mean (PDHFPMM) operator i.e., 4.2. The q-Rung Probabilistic Dual Hesitant Fuzzy Power Weighted Muirhead Mean (q-RPDHFPWMM) Operator
Definition 11. Let be a collection of q-RPDHFEs, be a vector of parameters and be the corresponding weight vector, satisfying and . The q-rung probabilistic dual hesitant fuzzy power weighted Muirhead mean (q-RPDHFPWMM) operator is expressed aswhereis the distance between and , represents any permutation of , denotes all possible permutations of , n is the balancing coefficient, and denotes the support for from , satisfying the properties in Definition 10. Similarly, letthen Equation (35) can be written aswhere and . Theorem 5. Let be a collection of q-RPDHFEs, and be a vector of parameters. The aggregated value using q-RPDHFPWMM operator is still a q-RPDHFE and The proof of Theorem 5 is similar to that of Theorem 3, which is mitted here. In addition, it is easy to prove that the q-RPDHFPWMM operator has the property of boundedness, but does not have the properties of monotonicity and idempotency.
4.3. The q-Rung Probabilistic Dual Hesitant Fuzzy Power Dual Muirhead Mean (q-RPDHFPDMM) Operator
Definition 12. Let be a collection of q-RPDHFEs, and be a vector of parameters. Then q-rung probabilistic dual hesitant fuzzy power dual Muirhead mean (q-RPDHFPDMM) operator is expressedwhereand is any permutation of (1, 2, …, n), is the collection of all permutations of (1, 2, …, n), and n is the balancing coefficient. is the distance between and , and denotes the support for from , satisfying the properties presented in Definition 10. To simplify Equation (40), we denotethen (40) can be written aswhere and . Theorem 6. Let be a collection of q-RPDHFEs, and be a vector of parameters. The aggregated value using q-RPDHFPDMM operator is still a q-RPDHFE and The proof of Theorem 6 is similar to that of Theorem 3, which is mitted here. In addition, it is easy to prove that the q-RPDHFPDMM operator has the property of boundedness, but does not have the properties of monotonicity and idempotency.
In the followings, we discuss some special cases of the q-RPDHFPDMM operator with respect to its contained parameters.
Case 7. If , then the q-RPDHFPDMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power geometric (q-RPDHFPG) operator, i.e., In this case, if for all , then the q-RPDHFPDMM operator reduces to the q-RPDHFG operator, which is shown as Equation (32).
Case 8. If , then the q-RPDHFPDMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power geometric Bonferroni mean (q-RPDHFPGBM) operator, i.e., In this case, if
for
, then the
q-RPDHFPDMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy geometric Bonferroni mean (
q-RPDHFGBM) operator, i.e.,
Case 9. If , then q-RPDHFPDMM operator reduces to the q-rung probabilistic dual hesitant fuzzy power dual Maclaurin symmetric mean (q-RPDHFPDMSM) operator, i.e., In this case, if
for
, then the
q-RPDHFPDMM operator reduces to the
q-rung probabilistic dual hesitant fuzzy dual Maclaurin symmetric mean (
q-RPDHFDMSM) operator, i.e.,
Case 10. If or , then the q-RPDHFPDMM operator reduces to the following form In this case, if for , then q-RPDHFPDMM operator reduces to the q-RPDHFA operator, which is shown as Equation (25).
Case 11. If q = 2, then the q-RPDHFPDMM operator reduces to the probabilistic dual Pythagorean hesitant fuzzy power dual Muirhead mean (PDPHFPDMM) operator, i.e., Case 12. If q = 1, then the q-RPDHFPDMM reduces to the probabilistic dual hesitant fuzzy power dual Muirhead mean (PDHFPDMM) operator, i.e., 4.4. The q-Rung Probabilistic Dual Hesitant Fuzzy Power Weighted Dual Muirhead Mean (q-RPDHFPWDMM) Operator
Definition 13. Let be a collection of q-RPDHFEs, be a vector of parameters and be the corresponding weight vector, satisfying that and . The q-rung probabilistic dual hesitant fuzzy power weighted dual Muirhead mean (q-RPDHFPWDMM) operator is expressed aswhereand is any permutation of (1, 2, …, n), is the collection of all permutations of (1, 2, …, n), and n is the balancing coefficient. is the distance between and , and denotes the support for from , satisfying the properties presented in Definition 10. Similarly, we assumethus (53) can be written aswhere and . Theorem 7. Let be a collection of q-RPDHFEs, and be a vector of parameters. The aggregated value using q-RPDHFPWDMM operator is still a q-RPDHFE and The proof of Theorem 7 is similar to that of Theorem 3, which is mitted here. In addition, it is easy to prove that the q-RPDHFPWDMM operator has the property of boundedness, but does not have the properties of monotonicity and idempotency.
5. A Novel MADM Approach Based on Q-RPDHFEs
This section gives a MADM method under q-RPDHFSs on the basis of the aforementioned AOs. We assume the alternative set is denoted as , and the attribute set is denoted as . The weight vector of attributes is , satisfying and . When evaluating the performance of alternative under attribute , each DM provides his/her preferred MDs and NMDs and based on DMs’ preferred degrees and the probabilistic values, the overall evaluation value can be denoted by , which is a q-RPDHFE. Finally, a q-rung probabilistic dual hesitant fuzzy matrix can be obtained, which can be denoted as . Based on the proposed AOs, we put forward a new MADM method, which consists of the following steps
Step 1. Normalize the decision matrix. In real MADM problems, attributes can be generally divided into two types: benefit attribute and cost attribute. Therefore, the decision matrix should be normalized in the following method
where
and
represent the benefit-type attribute and the cost-type attribute respectively.
Step 2. Calculate the support
by
where
is the distance between the two
q-PRDHFEs
and
.
Step 3. Compute the overall supports
by
Step 4. Compute the power weight
associated with the
q-PRDHFE
by
Step 5. Utilize the
q-RPDHFPWMM operator
or the
q-RPDHFPWDMM operator
to determine the collective overall preference value
of alternatives
.
Step 6. According to Definition 8, calculate the score function and accuracy function of the overall preference value .
Step 7. Order the alternatives and select the optimal alternative(s).
7. Conclusions
This paper demonstrated a novel MADM method, which can be used to solve practical decision-making problems effectively. The main contributions of this paper are three-fold. Firstly, we proposed a novel tool, called q-RPDHFSs to more accurately and effectively depict DMs’ complicated evaluation information. Compared with q-RDHFSs, our proposed q-RPDHFSs more effectively deal with DMs’ fuzzy judgements as they not only describe the MD and NMD, but also depict their corresponding probabilistic information. Compared with the PDHFSs, the q-RPDHFSs are more powerful as they provide DMs more freedom to express their evaluation values. Due to this characteristic, in the framework of q-RPDHFSs, DMs can fully express their evaluations, which leads to less information loss. Secondly, a series of AOs of q-RPDHFEs were developed, which are useful to aggregate attribute values given in the form of q-rung probabilistic dual hesitant fuzzy information. The advantages of superiorities of our proposed AOs are obvious, as they not only reduce the negative effect of DMs’ unduly high or low evaluation values on the final decision results, but also reflect the interrelationship among any numbers of attributes. Thirdly, a new MADM method was originated to help DMs to choose the optimal alternatives. Through numerical examples, the effectiveness of our method has been clearly illustrated. By comparative analysis, the advantages of our method are that it not only provides DMs great freedom to express their decision information, but also produces reasonable and reliable decision results. These characteristics make our method more suitable to deal with MADM problems in actual life.
In the further, we plan to continue our research from three aspect. Firstly, we shall study new applications of our decision-making method in more practical MADM problems, such as selection real estate investment [
39], medicine selection [
40], best research topic selection [
41], evaluation of outsourcing for information systems [
42], etc. Secondly, we will study more AOs of
q-RPDHFEs and propose corresponding MADM methods. Thirdly, we shall continue to investigate extensions of
q-RPDHFSs, such as interval-valued
q-RPDHFSs, complex
q-RPDHFSs, complex interval-valued
q-RPDHFSs, etc.