Abstract
As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed. Secondly, several aggregation operators including complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA), complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG), complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) as well as their weighted forms that can reflect the interrelationship among all decision attributes are proposed, which is of great use in multi-attribute decision making(MADM) or in multi-attribute group decision making(MAGDM) issues when we want to synthesize multiple evaluation values into one. Besides, we also present a number of valuable properties of these provided aggregation operators. Thirdly, a MADM process is built in the CPHF environment step by step based on the aggregation operators we have just established. Furthermore, we provide an example about purchasing a most suitable car to demonstrate the effectiveness and practicability of the proposed decision process. Finally, we point out the superiorities of MADM in working under the CPHF environment through comparing it with the existing method and also offer some possible prospects for further research. The method displayed in this work can effectively solve the MADM problems where the decision-making information is described by CPHFEs and the attributes are interactive.
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Introduction
With the continuous progress of science and technology, our living environment is becoming more and more informationalized, intelligentized and diversified. As a result, we usually have multiple possible alternatives when faced with something we need to make a decision. Moreover, in order to maximize the global benefit, we should manage to pick out the best alternative based on our personal experience and preferences. However, one of the difficulties is that there is often no such alternative that satisfies all the preferences perfectly in our real life, that is to say, some alternatives may meet certain preferences and others maybe good in some preferences else. Therefore, in order to solve this kind of practical problems, many scientists have carried out research and proposed the framework of multi-attribute decision making(MADM)1,2,3. It turns out that all the alternatives can be sorted by using the MADM method, and then we can make the optimal decision. On the other hand, owing to the complexity of the real world and the limitations of human cognition, scientists have realized that many phenomena are not deterministic. In other words, the emergence of many things is often accompanied by some uncertainty in our real life. Consequently, considering it is necessary to estabilish a new tool to express imprecise, uncertain or incomplete information before implementing a decision-making process, Zadeh4 initiated fuzzy set(FS) to characterize the fuzzy relationship between an object and the reference set in 1965.
Later, scholars have extended the FS from different perspectives. Detailedly, FS cannot cope with the membership grades in the shape of groups because it can only provide one value at a time. To overcome this disadvantage, Torra and Narukawa5 presented the conception of hesitant fuzzy set(HFS) in which the membership grade of each object to the reference set is a finite subset of unit interval [0, 1].To deal with three aspects of opinions in a voting process, Zhu et al.6,7 investigated dual hesitant fuzzy set(DHFS) to present people’s hesitation state when making decisions, that is, there are multiple possible values for both the membership and nonmembership grade in such a fuzzy set. However, both the HFS and DHFS ignore the randomness that exists everywhere. Therefore, Xu and Zhou proposed the concept of probabilistic hesitation fuzzy sets (PHFS) and defined corresponding operations for the convenience of application8. Later, probabilistic dual hesitant fuzzy set(PDHFS) was investigated by Hao et al.9 in consideration of the hesitant fuzzy membership and non-membership degree often accompied by different possibilities in reality. It is proved that PDHFS can better describe the uncertain information and could be used to many occasions of MADM10,11,12,13,14. Cuong and Kreinovich15,16 provided the notion of picture fuzzy set(PFS),which is expressed by three distinct maps consisting of a positive, a neutral, and a negative membership function. Wu et al.17 and Lin et al.18 explored the application of PFS in MADM. Gradually, it turns out that FS as well as its generalization forms indeeed have a wide range of applications in resolving ambiguous issues on many occasions, such as decision-making problems19,20,21,22,23,24,25,26,27,28,29, medical diagnosis30,31,32,33,34,35 and pattern recognition36,37,38,39,40,41.
However, all of these above fuzzy sets can only handle one-dimension fuzzy information. From this point of view, Ramot et al.42 explored the complex fuzzy set(CFS), which expresses the grade of membership in the shape of a complex number lying in unit disc. CFS have received extensive attention during last twenty years for its capability to simultaneously contain two-dimension data information in one set. However, it is gradually found that CFS fails to seize the psychological feature of one’s hesitation in a decision-making process simlar to the shortcoming of aforementioned FS. Garg et al.43 then offered complex hesitant fuzzy set(CHFS) to quantify complex and ambiguous information in reality, which is a combination of the HFS and the CFS as mentioned before. Although CHFS could better express the hesitation of humanlity and the vagueness of real-world, it negelect the randomness that widely exists in human socity. To present the randomicity and fuzziness at once and contains more information in one framework, we provide a completely new kind of generalized fuzzy set named as complex probabilistic hesitant fuzzy set(CPHFS) and investigate the necessary opreations for complex probabilistic hesitant fuzzy element(CPHFE) in this work. In fact, CPHFS are more powerful and general than existing FS, HFS, PHFS, CFS and CHFS in dealing with awkward and complex information. Since these aforementioned fuzzy sets are all special cases of CPHFS, the advantages of CPHFS are pointed below:
-
(1)
If we let the imaginary of each element in a CPHFS be zero, then the CPHFS would convert into PHFS.
-
(2)
If we neglect the probability information in a CPHFS, then the CPHFS would convert into CHFS.
-
(3)
If we let the imaginary of each element in a CPHFS be zero and overlook the probability information, then the CPHFS would further convert into HFS.
-
(4)
If there is only one membership grade for each x to the reference set X, then the CPHFS would degenerate into CFS because the probability of this elment is 1 and could be omitted.
-
(5)
If there is only one element in a CPHFS and the imaginary of it is zero, then the CPHFS would become FS.
In a multi-attribute decision making problem, the expression of information is only the first step. As we have just pointed out, using CPHFSs to express decision information is more in line with people’s cognition. What needs to be noted next is the establishment of the decision-making process, and an important issue involved in this stage is to develop effective decision-making methods for acquiring better results. At present, many decision-making methods are built on the basis of distance or similarity measures, such as EDAS,TOPSIS,TODIM and so on44,45,46. Therefore, the research of distance and similarity measures are important topics in fuzzy mathematics theory. It is universally acknowledged that distance measurement is an important quantity for identifying the difference between two objects and is extensively used in many MADM problems47,48,49,50. The similarity measure is a useful tool to express the degree of similarity between two objects. In view of the significance of similarity measures, many scholars have come up with a variety of similarity based on different perspectives51,52. Recently, this theory has been widely used in many MADM problems such as medical diagnosis, pattern recognition, etc53,54,55. Moreover, in the general axiomatic definition, the sum of normalized distance and normalized similarity is one, which indicates that the similarity can be obtained by the distance, so the distance-based decision method and the similarity-based decision method are essentially equivalent in many occasions.On the other hand, the decision-making method based on aggregation operator is also a hot topic in the field of MADM. In such method, fuzzy aggregation operators directly affect the decision result, so many scholars devoted themselves to construct a variety of fuzzy fusion operators under various fuzzy environments56,57,58,59,60. As two kinds of elementary operators, arithmetic and geometric mean operator are widely studied by researchers. Therefore, in this manuscript, we first study the arithmetic and geometric average operator under CPHF environment. In addition, the Heronian mean is also a kind of important aggregation operator with practical value because it can reflect the interrelation among a set of fuzzy numbers. Hence, in the following part of the paper, we also bring this type of operator to the CPHF environment and propose the Heronian mean operator based on the arithmetic and geometric mean. Finally, as an application of CPHF aggregation operators, we propose a MADM method and prove its reliability by a numerical example.
The rest of this work is arranged as follows. First of all, we simply recall some terminology, concepts of FS, HFS, CFS ,PHFS and CHFS in Section “Preliminaries”. Then in Section “Complex probabilistic hesitant fuzzy element and related arithmetic operations”, we first present the concept of CPHFS and CPHFE. Moreover, we introduce four essential operations for CPHFE and some useful properties of these operations are estabilished. Next, we build several aggregation operators for a collection of CPHFEs in Section “Aggregation operators of CPHFEs and their properties”. Meanwhile, we further study the good properties of aggregation operators we have built in the same section. Then, in Section “A novel MADM approach in the environment of CPHF information”, a framework of MADM in CPHF environment is structured. Furthermore, we select an example of decision making in practice to examine the credibility and validity of the MADM procedure we built. Finally, the main results of this research are summarized in the last section.
Preliminaries
We will, in this section, comb and summarize some basic knowledge of several types of fuzzy sets which are essential for the development of this manuscript.
To characterize vagueness appearing in industrial production and daily life, Zadeh4 pioneered the concept of FS, which was a huge breakthrough in the field of applied mathematics.
Definition 2.1
A FS A is a map from the fixed set X to a real number included in unit interval [0, 1]:
where A(x) is the grade of memebership of \(x \in X\) in the set A. Usually we express a FS in the form of set, i.e. \(A=\{<x,\mu _{A}(x)>|x\in X\},\) in which \(\mu _{A}(x)\) indicate the membership grade of \(x \in X\) belonging to A.
Considering that people often hesitate about making a definite decision, the conception of HFS was provided by Torra and Narukawa5 to reflect this psychological state of individuals.
Definition 2.2
Let X be the universe of discourse, then a HFS H on X is defined as:
where \(h_{E}(x)\) is a finite set of real numbers taken from unit interval [0, 1], desicribing the possible memebership degree of \(x \in X\) in the set A. For the sake of simplicity in practical applications, Xu and Xia62 called the set of possible membership degree of x as the hesitant fuzzy element(HFE), expressed as \(h=h_{E}(x)\).
Essentially, Xia and Xu62 provided several operations of HFE for acquiring the arithmetic results of a real number and an HFE as well as one HFE and another.
Definition 2.3
Assuming that \(h,h_{1},h_{2}\) are any three HFEs, then Xu and Xia raised four elementary algebraic operations of HFEs as shown below:
-
1.
\(h^{\lambda }=\underset{\gamma \in h}{\cup }\{{\gamma }^{\lambda }\};\)
-
2.
\(\lambda h=\underset{\gamma \in h}{\cup }\{1-(1-\gamma )^{\lambda }\},\lambda >0;\)
-
3.
\(h_{1}\oplus h_{2}=\underset{\gamma _{1} \in h_{1},\gamma _{2} \in h_{2}}{\cup }\{\gamma _{1}+\gamma _{2}-\gamma _{1}\gamma _{2}\};\)
-
4.
\(h_{1}\otimes h_{2}=\underset{\gamma _{1} \in h_{1},\gamma _{2} \in h_{2}}{\cup }\{\gamma _{1}\gamma _{2}\}.\)
Whether FS or HFS can only cope with one dimension fuzzy information at a time, which maybe result in a loss of information sometimes. To broaden the FS theory from the perspective of dimension, Ramot et al.42 proposed the notion of CFS, in which previous real membership degree was replaced by a complex number in the exponential form. Furthermore, HFS was also be generalized to PHFS by Xu and CHFS by Garg et al.8,61. Next, we will briefly review the definitions of CFS,PHFS and CHFS.
Definition 2.4
The CFS C on X is presented by:
in which \(\mu _{C}(x)=\mu (x)e^{i2\pi (\omega (x))}\) represents complex-valued membership grade of \(x \in X\) in the set C with \(0\le \mu {(x)},\omega {(x)}\le 1\). Moreover, the complex fuzzy element(CFE) is denoted by \(C=<\mu _{C}e^{i2\pi (\omega _{C})}>\).
Definition 2.5
The PHFS \(H_{P}\) on X is expressed by:
where \(\overline{h}(\gamma _{i}|p_{i})\) represents the hesitant fuzzy information along with probabilities to \(H_{P}\), \(0\le \gamma _{i},p_{i}\le 1\), \(i=1,2,\ldots ,\#\overline{h}\), and \(\sum \limits _{i=1}^{\#\overline{h}}p_{i}= 1\).
Definition 2.6
Let X be the universe of discourse, then a CHFS Q on X is recorded as:
where \(i=\sqrt{-1}\) and \(\mu (x)=\{\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}|j=1,2,\ldots ,\#\mu \}\) is the complex-valued membership degree(CMD) of x which is a finite nonempty subset of unit disc in the complex plane. Besides, \(\gamma _{j}(x),j=1,2,\ldots ,\#\mu\) and \(\omega _{j}(x),j=1,2,\ldots ,\#\mu\) denote the possible amplitude and phase membership degrees of x,respectively. Meanwhile, they must satisfy the following conditions: \(\forall x\in X,0 \le \mathop {max}\limits _{j}\{\gamma _{j}(x)\}\le 1, 0 \le \mathop {max}\limits _{j} \{\omega _{j}(x)\}\le 1,\gamma _{j}(x),\omega _{j}(x)\in [0,1].\) In addition, \(Q(x)=\{\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}|j=1,2,\ldots ,\#\mu \}\) is known as a complex hesitant fuzzy element (CHFE). And symbol \(\#\mu\) represent the cardinality of \(\mu (x)\).
Complex probabilistic hesitant fuzzy element and related arithmetic operations
In the definition of CHFE, each complex-valued membership degree has the same possibility. However, it is difficult to achieve this goal because of the randomness in reality. To make up this shortcoming reasonably, we introduce probabilities into CHFS and therefore propose the notion of CPHFS and CPHFE in this section. After that, we furthermore provide the opreational laws of CPHFE and investigate the useful properties of them.
Definition 3.1
An CPHFS Q on X have the following form:
The components \(\widehat{\mu _j(x)}|p_{j}(x)\) are some possible elements of CMD and correspondingly probability information of \(x \in X\), where \(\widehat{\mu _j(x)}=\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}\) is the CMD and \(p_{j}(x)\) is the probability value of it. Moreover, \(\gamma _{j}(x),j=1,2,\ldots ,\#\mu\) and \(\omega _{j}(x),j=1,2,\ldots ,\#\mu\) denote the possible amplitude and phase membership degrees of x, respectively. Meanwhile, \(\forall x \in X, 0 \le \mathop {max}\limits _{j} \{\gamma _{j}(x)\}\le 1, 0 \le \mathop {max}\limits _{j} \{\omega _{j}(x)\}\le 1,0 \le \mathop {max}\limits _{j} \{p_{j}(x)\}\le 1,\gamma _{j}(x),\omega _{j}(x),p_{j}(x)\in [0,1]\) and \(\sum \limits _{j=1}^{\#\mu |p}p_{j}(x) = 1\). For simplicity and convenience, \(Q_{p}(x)=\{\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}|p_{j}(x),j=1,2,\ldots ,\#\mu |p\}\) is called as a CPHFE. Besides, the symbol \(\#\mu |p\) is the cardinality of set \(\mu (x)|p(x)\).
Definition 3.2
The empty CPHFS and the full CPHFS are respectively presented as:
The fundamental elements \(Q_{p}^{\circ }(x)=\{0|1\}\),\(Q_{p}^{*}(x)=\{1|1\}\) in \(Q_{p}^{\circ }\), \(Q_{p}^{*}\) are called as the empty CPHFE and the full CPHFE, respectively. Furthermore, we can omit the probability information and express them as \(Q_{p}^{\circ }(x)=\{0\}\),\(Q_{p}^{*}(x)=\{1\}\) due to the elimination of hesitation.
Now, to compare the size of two CPHFEs, we demonstrate the score function of a CPHFE in the following definition.
Definition 3.3
Suppose \(Q_{p}(x)=\{\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}|p_{j}(x),j=1,2,\ldots ,\#\mu |p\}\) is a CPHFE, then
is called the score of \(Q_{p}(x)\). For two CPHFEs \(Q_{p}^{1}(x)\) and \(Q_{p}^{2}(x)\), if \(s(Q_{p}^{1}(x))>s(Q_{p}^{2}(x))\), then we consider \(Q_{p}^{1}(x)\) is greater than \(Q_{p}^{2}(x)\), denoted by \(Q_{p}^{1}(x)>Q_{p}^{2}(x)\); if \(s(Q_{p}^{1}(x))<s(Q_{p}^{2}(x))\), then \(Q_{p}^{1}(x)\) is smaller than \(Q_{p}^{2}(x)\), denoted by \(Q_{p}^{1}(x)<Q_{p}^{2}(x)\); and if \(s(Q_{p}^{1}(x))=s(Q_{p}^{2}(x))\), then we say \(Q_{p}^{1}(x)\) is equal to \(Q_{p}^{2}(x)\), denoted by \(Q_{p}^{1}(x)= Q_{p}^{2}(x)\).
Example 3.1
Suppose \(Q_{p}^{1}(x)=\{0.6\cdot e^{i2\pi (0.2)}|0.2,0.4\cdot e^{i2\pi (0.2)}|0.1,0.7\cdot e^{i2\pi (0.5)}|0.3,0.3\cdot e^{i2\pi (0.3)}|0.4,\}\), \(Q_{p}^{2}(x)=\{0.4\cdot e^{i2\pi (0.7)}|0.4,0.3\cdot e^{i2\pi (0.4)}|0.6\}\) are two CPHFEs. Then according to above definition
Therefore, \(Q_{p}^{2}(x)>Q_{p}^{1}(x)\).
Next, we introduce some opreations of CPHFE, which is vital in practical applications.
Definition 3.4
Given any three CPHFEs \(Q_{p}(x)=\{\gamma _{j}(x)e^{i2\pi (\omega _{j}(x))}|p_{j}(x),j=1,2,\ldots ,\#\mu |p\},Q_{p}^{1}(x)=\{\gamma _{j_{1}}^{1}(x)e^{i2\pi (\omega _{j_{1}}^{1}(x))}\)\(|p_{j_{1}}^{1}(x),j_{1}=1,2,\ldots ,\#\mu ^{1}|p^{1}\},Q_{p}^{2}(x)=\{\gamma _{j_{2}}^{2}(x)e^{i2\pi (\omega _{j_{2}}^{2}(x))}|p_{j_{2}}^{2}(x), j_{2}=1,2,\ldots ,\#\mu ^{2}|p^{2}\}\) and \(\lambda >0\) is a real number. Then the fundamental operations of CPHFE are presented as the following way:
-
(1)
\(\lambda Q_{p}(x)=\underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x), p_{j}\in p_{j}(x)}{\cup }\left\{ \left( 1-(1-\gamma _{j})^{\lambda }\right) e^{i2\pi \left( 1-(1-\omega _{j})^{\lambda }\right) }\bigg |\frac{p_{j}^{\lambda }}{\sum \limits _{j}p_{j}^{\lambda }}\right\} ;\)
-
(2)
\((Q_{p}(x))^{\lambda }=\underset{\gamma _{j}\in \gamma _{j}(x), \omega _{j}\in \omega _{j}(x), p_{j}\in p_{j}(x)}{\cup }\left\{ \gamma _{j}^{\lambda }e^{i2\pi \omega _{j}^{\lambda }}\bigg |\frac{p_{j}^{\lambda }}{\sum \limits _{j}p_{j}^{\lambda }}\right\} ;\)
-
(3)
\(Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x), \omega _{j_{1}}^{1}\in \omega _{j}^{1}(x), p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x), \omega _{j_{2}}^{2}\in \omega _{j}^{2}(x), p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \Big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )e^{i2\pi \Big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\}\)
-
(4)
\(Q_{p}^{1}(x)\otimes Q_{p}^{2}(x)=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x), \omega _{j_{1}}^{1}\in \omega _{j}^{1}(x), p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}e^{i2\pi \big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\}\)
-
(5)
\((Q_{p}(x))^{c}=\underset{\gamma _{j}\in \gamma _{j}(x), \omega _{j}\in \omega _{j}(x), p_{j}\in p_{j}(x)}{\cup }\left\{ \left( 1-\gamma _{j}\right) e^{i2\pi \left( 1-\omega _{j}\right) }\bigg |p_{j}\right\} .\)
Example 3.2
Assume that \(Q_{p}(x),Q_{p}^{1}(x),Q_{p}^{2}(x)\) are three CPHFEs, which are defined as \(Q_{p}^{1}(x)=\{0.6\cdot e^{i2\pi (0.2)}|0.2,0.4\cdot e^{i2\pi (0.2)}|0.1,0.7\cdot e^{i2\pi (0.5)}|0.3,0.3\cdot e^{i2\pi (0.3)}|0.4\}\),\(Q_{p}^{2}(x)=\{0.4\cdot e^{i2\pi (0.7)}|0.4,0.3\cdot e^{i2\pi (0.4)}|0.6\}\), and \(Q_{p}(x)=Q_{p}^{1}(x)\), \(\lambda =2\) is a positive real number. Then, the following results can be derived by Definition 3.4.
Theorem 3.1
Provided that \(Q_{p}(x),Q_{p}^{1}(x),Q_{p}^{2}(x),Q_{p}^{3}(x)\) are arbitrary four CPHFEs, \(\lambda ,\lambda _1,\lambda _2\) are three positive real numbers. Then the following operational laws hold:
-
(1)
\(Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)=Q_{p}^{2}(x)\oplus Q_{p}^{1}(x);\)
-
(2)
\((Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)) \oplus Q_{p}^{3}(x)=Q_{p}^{1}(x)\oplus (Q_{p}^{2}(x)\oplus Q_{p}^{3}(x));\)
-
(3)
\(\lambda (Q_{p}^{1}(x)\oplus Q_{p}^{2}(x))=\lambda Q_{p}^{1}(x)\oplus \lambda Q_{p}^{2}(x)\);
-
(4)
\(\lambda _1 Q_{p}(x)\oplus \lambda _2 Q_{p}(x)=(\lambda _1 +\lambda _2) Q_{p}(x)\);
-
(5)
\(Q_{p}^{1}(x)\otimes Q_{p}^{2}(x)=Q_{p}^{2}(x)\otimes Q_{p}^{1}(x);\)
-
(6)
\((Q_{p}^{1}(x)\otimes Q_{p}^{2}(x)) \otimes Q_{p}^{3}(x)=Q_{p}^{1}(x)\otimes (Q_{p}^{2}(x)\otimes Q_{p}^{3}(x));\)
-
(7)
\((Q_{p}^{1}(x)\otimes Q_{p}^{2}(x))^{\lambda }= (Q_{p}^{1}(x))^{\lambda }\otimes (Q_{p}^{2}(x))^{\lambda }\);
-
(8)
\((Q_{p}(x))^{\lambda _1}\otimes (Q_{p}(x))^{\lambda _2}= (Q_{p}(x))^{\lambda _1 +\lambda _2}\);
-
(9)
\(((Q_{p}(x))^{c})^{c}=Q_{p}(x)\).
Proof
-
(1)
According to Definition 3.4(3), we have
$$\begin{aligned} \begin{aligned}&Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)\\&=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \Big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )e^{i2\pi \Big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \\&=\underset{\begin{array}{c} \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x);\\ \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x) \end{array}}{\cup }\left\{ \Big (\gamma _{j_{2}}^{2}+\gamma _{j_{1}}^{1}-\gamma _{j_{2}}^{2}\gamma _{j_{1}}^{1}\Big )e^{i2\pi \Big (\omega _{j_{2}}^{2}+\omega _{j_{1}}^{1}-\omega _{j_{2}}^{2}\omega _{j_{1}}^{1}\Big )}\bigg |\frac{p_{j_{2}}^{2}p_{j_{1}}^{1}}{\sum \limits _{j_2,j_1}p_{j_2}^{2}p_{j_1}^{1}}\right\} \\&=Q_{p}^{2}(x)\oplus Q_{p}^{1}(x); \end{aligned} \end{aligned}$$ -
(2)
Taking advantage of (1) we have just proved, we find
$$\begin{aligned} & \big (Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)\big ) \oplus Q_{p}^{3}(x)\\ & \quad =\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2 \end{array}}{\cup }\left\{ \Big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )e^{i2\pi \Big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \\ & \quad \oplus \underset{\gamma _{j_{3}}^{3}\in \gamma _{j}^{3}(x),\omega _{j_{3}}^{3}\in \omega _{j}^{3}(x),p_{j_{3}}^{3}\in p_{j}^{3}(x)}{\cup }\Big \{\gamma _{j_{3}}^{3}e^{i2\pi \omega _{j_{3}}^{3}}|p_{j_{3}}^{3}\Big \}\\ & \quad =\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2,3 \end{array}}{\cup }\left\{ \left( \begin{array}{c} \big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\big )+\gamma _{j_{3}}^{3} -\\ \big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\big )\gamma _{j_{3}}^{3} \end{array}\right) e^{i2\pi \left( \begin{array}{c} \big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\big )+\omega _{j_{3}}^{3} -\\ \big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\big )\omega _{j_{3}}^{3} \end{array}\right) }\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )p_{j_{3}}^{3}}{\sum \limits _{j_1,j_2,j_3}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )p_{j_3}^{3}}\right\} \\ & \quad =\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2,3 \end{array}}{\cup }\left\{ \left( \begin{array}{c} \gamma _{j_{1}}^{1}+\big (\gamma _{j_{2}}^{2}+\gamma _{j_{3}}^{3}-\gamma _{j_{2}}^{2}\gamma _{j_{3}}^{3}\big ) -\\ \gamma _{j_{1}}^{1}\big (\gamma _{j_{2}}^{2}+\gamma _{j_{3}}^{3}-\gamma _{j_{2}}^{2}\gamma _{j_{3}}^{3}\big ) \end{array}\right) e^{i2\pi \left( \begin{array}{c} \omega _{j_{1}}^{1}+\big (\omega _{j_{2}}^{2}+\omega _{j_{3}}^{3}-\omega _{j_{2}}^{2}\omega _{j_{3}}^{3}\big ) -\\ \omega _{j_{1}}^{1}\big (\omega _{j_{2}}^{2}+\omega _{j_{3}}^{3}-\omega _{j_{2}}^{2}\omega _{j_{3}}^{3}\big ) \end{array}\right) }\bigg |\frac{p_{j_{1}}^{1}\Big (p_{j_{2}}^{2}p_{j_{3}}^{3}\Big )}{\sum \limits _{j_1,j_2,j_3}p_{j_1}^{1}\Big (p_{j_2}^{2}p_{j_3}^{3}\Big )}\right\} \\ & \quad =Q_{p}^{1}(x)\oplus \big (Q_{p}^{2}(x)\oplus Q_{p}^{3}(x)\big ); \end{aligned}$$ -
(3)
Based on Definition 3.4(3) and (1) we have just demonstrated, it is not difficult to discover the following process is correct.
$$\begin{aligned} & \lambda \big (Q_{p}^{1}(x)\oplus Q_{p}^{2}(x)\big )\\ & \quad =\lambda \left( \underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \Big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )e^{i2\pi \Big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \right) \\ & \quad =\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \begin{array}{c} \left( 1-\Big (1-\Big (\gamma _{j_{1}}^{1}+\gamma _{j_{2}}^{2}-\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )\Big )^{\lambda }\right) e^{i2\pi \left( 1-\Big (1-\Big (\omega _{j_{1}}^{1}+\omega _{j_{2}}^{2}-\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )\Big )^{\lambda }\right) }\\ \bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_1,j_2}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )^{\lambda }} \end{array}\right\} \\ & \quad =\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \left( 1-\Big (\Big (1-\gamma _{j_{1}}^{1}\Big )\Big (1-\gamma _{j_{2}}^{2}\Big )\Big )^{\lambda }\right) e^{i2\pi \left( 1-\Big (\Big (1-\omega _{j_{1}}^{1}\Big )\Big (1-\omega _{j_{2}}^{2}\Big )\Big )^{\lambda }\right) }\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_1,j_2}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )^{\lambda }}\right\} \\ & \quad =\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \left( \begin{array}{c} 1-(1-\gamma _{j_{1}}^{1})^{\lambda }+1-(1-\gamma _{j_{2}}^{2})^{\lambda }-\\ (1-(1-\gamma _{j_{1}}^{1})^{\lambda })(1-(1-\gamma _{j_{2}}^{2})^{\lambda }) \end{array}\right) e^{i2\pi \left( \begin{array}{c} 1-(1-\omega _{j_{1}}^{1})^{\lambda }+1-(1-\omega _{j_{2}}^{2})^{\lambda }-\\ (1-(1-\omega _{j_{1}}^{1})^{\lambda })(1-(1-\omega _{j_{2}}^{2})^{\lambda }) \end{array}\right) }\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_1,j_2}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )^{\lambda }}\right\} \\ & \quad =\underset{\gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x) }{\cup }\left\{ 1-\Big (1-\gamma _{j_{1}}^{1}\Big )^{\lambda } e^{i2\pi \left( 1-\Big (1-\omega _{j_{1}}^{1}\Big )^{\lambda }\right) }\bigg |\frac{\Big (p_{j_{1}}^{1}\Big )^{\lambda }}{\sum \limits _{j_1}\Big (p_{j_1}^{1}\Big )^{\lambda }}\right\} \\ & \quad \oplus \underset{\gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) }{\cup }\left\{ 1-\Big (1-\gamma _{j_{2}}^{2}\Big )^{\lambda } e^{i2\pi \left( 1-(1-\omega _{j_{2}}^{2})^{\lambda }\right) }\bigg |\frac{\Big (p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_2}\Big (p_{j_2}^{2}\Big )^{\lambda }}\right\} \\ & \quad =\lambda Q_{p}^{1}(x)\oplus \lambda Q_{p}^{2}(x) \end{aligned}$$ -
(4)
By using Definitions 3.4(3) and 3.4(1), the following computation could be conducted.
$$\begin{aligned} & \lambda _1 Q_{p}(x)\oplus \lambda _2 Q_{p}(x)\\ & \quad =\underset{\gamma _j \in \gamma _{j}(x),\omega _j\in \omega _{j}(x),p_j \in p_{j}(x) }{\cup }\left\{ (1-(1-\gamma _j)^{\lambda _1}) e^{i2\pi \left( 1-(1-\omega _j)^{\lambda _1}\right) }\bigg |\frac{(p_j)^{\lambda _1}}{\sum \limits _{j}(p_j)^{\lambda _1}}\right\} \\ & \quad \oplus \underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x),p_{j}\in p_{j}(x) }{\cup }\left\{ (1-(1-\gamma _{j})^{\lambda _2}) e^{i2\pi \left( 1-(1-\omega _{j})^{\lambda _2}\right) }\bigg |\frac{(p_j)^{\lambda _2}}{\sum \limits _{j}(p_j)^{\lambda _2}}\right\} \\ & \quad =\underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x),p_{j}\in p_{j}(x)}{\cup }\left\{ \left( \begin{array}{c} 1-(1-\gamma _{j})^{\lambda _1}+1-(1-\gamma _{j})^{\lambda _2}-\\ (1-(1-\gamma _{j})^{\lambda _1})(1-(1-\gamma _{j})^{\lambda _2}) \end{array}\right) e^{i2\pi \left( \begin{array}{c} 1-(1-\omega _{j})^{\lambda _1}+1-(1-\omega _{j})^{\lambda _2}-\\ (1-(1-\omega _{j})^{\lambda _1})(1-(1-\omega _{j})^{\lambda _2}) \end{array}\right) }\bigg |\frac{p_{j}^{\lambda _1}p_{j}^{\lambda _2}}{\sum \limits _{j}p_j^{\lambda _{1}}p_j^{\lambda _{2}}}\right\} \\ & \quad =\underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x),p_{j}\in p_{j}(x)}{\cup }\left\{ \left( 1-(1-\gamma _{j})^{\lambda _1+\lambda _2}\right) e^{i2\pi \left( 1-(1-\omega _{j})^{\lambda _1}(1-\omega _{j})^{\lambda _2}\right) }\bigg |\frac{p_{j}^{\lambda _1+\lambda _2}}{\sum \limits _{j}p_{j}^{\lambda _1+\lambda _2}}\right\} \\ & \quad =(\lambda _1 +\lambda _2) Q_{p}(x) \end{aligned}$$ -
(5)
Directly, by Definition 3.4(3)
$$\begin{aligned} \begin{aligned} Q_{p}^{1}(x)\otimes Q_{p}^{2}(x)&=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}e^{i2\pi \Big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \\&=\underset{\begin{array}{c} \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x);\\ \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x) \end{array}}{\cup }\left\{ \gamma _{j_{2}}^{2}\gamma _{j_{1}}^{1}e^{i2\pi \Big (\omega _{j_{2}}^{2}\omega _{j_{1}}^{1}\Big )}\bigg |\frac{p_{j_{2}}^{2}p_{j_{1}}^{1}}{\sum \limits _{j_1,j_2}p_{j_2}^{2}p_{j_1}^{1}}\right\} \\&=Q_{p}^{2}(x)\otimes Q_{p}^{1}(x); \end{aligned} \end{aligned}$$ -
(6)
Combining 3.4(3) and (5) in this theorem, it is obviously that
$$\begin{aligned} \begin{aligned} (Q_{p}^{1}(x)\otimes Q_{p}^{2}(x)) \otimes Q_{p}^{3}(x)&=\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2 \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}e^{i2\pi \Big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \\&\otimes \underset{\gamma _{j_{3}}^{3}\in \gamma _{j}^{3}(x),\omega _{j_{3}}^{3}\in \omega _{j}^{3}(x),p_{j_{3}}^{3}\in p_{j}^{3}(x)}{\cup }\left\{ \gamma _{j_{3}}^{3}e^{i2\pi \omega _{j_{3}}^{3}}|p_{j_{3}}^{3}\right\} \\&=\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2,3 \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\gamma _{j_{3}}^{3}e^{i2\pi \Big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\omega _{j_{3}}^{3}\Big )}\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )p_{j_{3}}^{3}}{\sum \limits _{j_1,j_2,j_3}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )p_{j_3}^{3}}\right\} \\&=\underset{\begin{array}{c} \gamma _{j_{i}}^{i}\in \gamma _{j}^{i}(x),\omega _{j_{i}}^{i}\in \omega _{j}^{i}(x),\\ p_{j_{i}}^{i}\in p_{j}^{i}(x);i=1,2,3 \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\Big (\gamma _{j_{2}}^{2}\gamma _{j_{3}}^{3}\Big )e^{i2\pi \Big (\omega _{j_{1}}^{1}\Big (\omega _{j_{2}}^{2}\omega _{j_{3}}^{3}\Big )\Big )}\bigg |\frac{p_{j_{1}}^{1}\Big (p_{j_{2}}^{2}p_{j_{3}}^{3}\Big )}{\sum \limits _{j_1,j_2,j_3}p_{j_1}^{1}\Big (p_{j_2}^{2}p_{j_3}^{3}\Big )}\right\} \\&=Q_{p}^{1}(x)\times \big (Q_{p}^{2}(x)\otimes Q_{p}^{3}(x)\big ); \end{aligned} \end{aligned}$$ -
(7)
The following results can be gotten through making use of Defintion 3.4(3) and 3.4(2).
$$\begin{aligned} & \begin{aligned} (Q_{p}^{1}(x)\oplus Q_{p}^{2}(x))^{\lambda }&=\left( \underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}e^{i2\pi \big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\big )}\bigg |\frac{p_{j_{1}}^{1}p_{j_{2}}^{2}}{\sum \limits _{j_1,j_2}p_{j_1}^{1}p_{j_2}^{2}}\right\} \right) ^{\lambda } \\&=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \Big (\gamma _{j_{1}}^{1}\gamma _{j_{2}}^{2}\Big )^{\lambda }e^{i2\pi \Big (\Big (\omega _{j_{1}}^{1}\omega _{j_{2}}^{2}\Big )^{\lambda }\Big )}\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_1,j_2}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )^{\lambda }}\right\} \\&=\underset{\begin{array}{c} \gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x);\\ \gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) \end{array}}{\cup }\left\{ \big (\gamma _{j_{1}}^{1}\big )^{\lambda }\big (\gamma _{j_{2}}^{2}\big )^{\lambda }e^{i2\pi \big (\big (\omega _{j_{1}}^{1}\big )^{\lambda }\big (\omega _{j_{2}}^{2}\big )^{\lambda }\big )}\bigg |\frac{\Big (p_{j_{1}}^{1}p_{j_{2}}^{2}\Big )^{\lambda }}{\sum \limits _{j_1,j_2}\Big (p_{j_1}^{1}p_{j_2}^{2}\Big )^{\lambda }}\right\} \\ \end{aligned}\\ & \begin{aligned}&=\underset{\gamma _{j_{1}}^{1}\in \gamma _{j}^{1}(x),\omega _{j_{1}}^{1}\in \omega _{j}^{1}(x),p_{j_{1}}^{1}\in p_{j}^{1}(x) }{\cup }\left\{ \big (\gamma _{j_{1}}^{1}\big )^{\lambda } e^{i2\pi \left( \big (\omega _{j_{1}}^{1}\big )^{\lambda }\right) }\bigg |\frac{(p_{j_{1}}^{1})^{\lambda }}{\sum \limits _{j_1}(p_{j_1}^{1})^{\lambda }}\right\} \\&\otimes \underset{\gamma _{j_{2}}^{2}\in \gamma _{j}^{2}(x),\omega _{j_{2}}^{2}\in \omega _{j}^{2}(x),p_{j_{2}}^{2}\in p_{j}^{2}(x) }{\cup }\left\{ \big (\gamma _{j_{2}}^{2}\big )^{\lambda } e^{i2\pi \left( \big (\omega _{j_{2}}^{2}\big )^{\lambda }\right) }\bigg |\frac{(p_{j_{1}}^{1})^{\lambda }}{\sum \limits _{j_1}(p_{j_1}^{1})^{\lambda }}\right\} \\&=\big (Q_{p}^{1}(x)\big )^{\lambda }\otimes \big (Q_{p}^{2}(x)\big )^{\lambda } \end{aligned} \end{aligned}$$ -
(8)
Employing Defintion 3.4(4) and 3.4(2), we acquire
$$\begin{aligned} & \begin{aligned} \big (Q_{p}(x)\big )^{\lambda _1}\otimes \big (Q_{p}(x)\big )^{\lambda _2}&=\underset{\gamma _j \in \gamma _{j}(x),\omega _j\in \omega _{j}(x),p_j \in p_{j}(x) }{\cup }\left\{ \gamma _j^{\lambda _1} e^{i2\pi \left( \omega _j^{\lambda _1}\right) }\bigg |\frac{(p_j)^{\lambda _1}}{\sum \limits _{j}(p_j)^{\lambda _1}}\right\} \\ \end{aligned}\\ & \begin{aligned}&\quad \oplus \underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x),p_{j}\in p_{j}(x) }{\cup }\left\{ \gamma _{j}^{\lambda _2} e^{i2\pi \left( \omega _{j}^{\lambda _2}\right) }\bigg |\frac{(p_j)^{\lambda _2}}{\sum \limits _{j}(p_j)^{\lambda _2}}\right\} \\&\quad =\underset{\gamma _{j}\in \gamma _{j}(x),\omega _{j}\in \omega _{j}(x),p_{j}\in p_{j}(x)}{\cup }\left\{ \gamma _{j}^{\lambda _1+\lambda _2}e^{i2\pi \left( \omega _{j}^{\lambda _1+\lambda _2}\right) }\bigg |\frac{p_{j}^{\lambda _1+\lambda _2}}{\sum \limits _{j}p_{j}^{\lambda _1+\lambda _2}}\right\} \\&\quad =\big (Q_{p}(x)\big )^{\lambda _1 +\lambda _2} \end{aligned} \end{aligned}$$ -
(9)
The proof is straightforward by using Definition 3.4(5). \(\square\)
Aggregation operators of CPHFEs and their properties
In this section, we investigate several aggregation operators of a collection of CPHFEs based on the operation laws we have proposed previously, which are useful in MADM and MAGDM.
CPHF weighted averaging operator and CPHF weighted geometric operator
Fuzzy weighted averaging operator and fuzzy weighted geometric operator are two classes of typical and powerful aggregation opreators in fuzzy environment. To make a better decision when faced with multiple possible choices in increasingly complex practical problems, in this section, we promote them to situations where the fuzzy message are represented by CPHFEs.
Definition 4.1
Suppose \(Q_{p}^{i}(x)=\{\gamma _{j_{i}}^{i}(x)e^{i2\pi (\omega _{j_{i}}^{i}(x))}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\}(i=1,2,\ldots , n)\) is an accumulation of CPHFEs, and \(w=(w_1,w_2,\ldots ,w_n)\) is the weight vector of \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\) with \(0<w_i<1,i=1,2,\ldots ,n\) and \(\sum \limits _{i=1}^{n} w_i=1.\) Then the complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA) and the complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG) are defined as:
Example 4.1
Assume that \(Q_{p}^{1}(x),Q_{p}^{2}(x),Q_{p}^{3}(x)\) are three CPHFEs, which are defined as \(Q_{p}^{1}(x)=\{0.5\cdot e^{i2\pi (0.6)}|1\}\),\(Q_{p}^{2}(x)=\{0.5\cdot e^{i2\pi (0.3)}|0.4,0.2\cdot e^{i2\pi (0.1)}|0.6\}\), \(Q_{p}^{3}(x)=\{0.5\cdot e^{i2\pi (0.7)}|0.7, 0.6\cdot e^{i2\pi (0.8)}|0.3\}\),and \(w=(0.3,0.4,0.3)\) is the weight vector of \(Q_{p}^{i}(x)(i=1,2,3)\). Then by Definition 3.4,
Theorem 4.1
Let \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi (\omega _{j_{i}}^{i}(x))}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \}(i=1,2,\ldots ,n)\) be a group of CPHFEs, then the fused value by using CPHFWA and CPHFWG are still CPHFEs, and
Proof
We prove this theorem by mathematical induction method shown as follows. When \(n=2\), according to Definition 3.4, we have
Now we assume that the result is true for \(n=k\), then when \(n=k+1\),
where,
Therefore,
which completes the proof. \(\square\)
Similarity, the fused value by using CPHFWG can also be proved by mathematical induction method.
Next, we would point out that CPHFWA and CPHFWG possess some useful properties including idempotency, commutativity, monotonicity and boundness.
Theorem 4.2
(Idempotency) If all \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\) are the same as each other, i.e. for any \(i,Q_{p}^{i}(x)=Q_{p}(x)\), then
Proof
According to Definition 4.1 and Theorem 3.1(4), we can easily obtain
Taking advantage of Definition 4.1 and Theorem 3.1(8), we have
Hence, the result follows. \(\square\)
Theorem 4.3
(Commutativity) If \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi \Big (\omega _{j_{i}}^{i}(x)\Big )}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \} (i=1,2,\ldots ,n)\) and \(Q_{p}^{i^{\prime }}(x)=\Bigg \{\gamma _{j_{i^{\prime }}}^{i^{\prime }}(x)e^{i2\pi \Big (\omega _{j_{i^{\prime }}}^{i^{\prime }}(x)\Big )}|p_{j_{i^{\prime }}}^{i^{\prime }}(x),j_{i^{\prime }}=1,2,\ldots ,\#\mu ^{i^{\prime }}|p^{i^{\prime }}\Bigg \}(i^{\prime }=1^{\prime },2^{\prime },\ldots ,n^{\prime })\) are two sets of CPHFEs and \(Q_{p}^{i^{\prime }}(x)\) is an arbitrary permutation of \(Q_{p}^{i}(x)(i=1, 2,\ldots ,n)\), then
Theorem 4.4
(Monotonicity) If \(Q_{p}^{1i}(x)\le Q_{p}^{2i}(x)(i=1,2,\ldots ,n)\), then
Proof
Based on the assumption \(Q_{p}^{1i}(x)<Q_{p}^{2i}(x)(i=1,2,\ldots ,n)\), we have for each \(1\le i \le n\),
then,
which indicates that
Similarly, we can also prove another inequality hold. \(\square\)
Theorem 4.5
(Boundness) The aggregation results of CPHFWA and CPHFWG are bounded. That is, for any collection of CPHFEs \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi \Big (\omega _{j_{i}}^{i}(x)\Big )}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \} (i=1,2,\ldots ,n)\)
Proof
We have known the aggregation results of a group of CPHFEs through CPHFWA and CPHFWG are still a CPHFE. Then, we have the following results according to the definition of the score of CPHFE \(Q_{p}(x)\)
On the other hand,
Meanwhile,\(s(Q_{p}(x))=0 \Leftrightarrow Q_{p}(x)=Q_{p}^{\circ }(x)\) and \(s(Q_{p}(x))=1\Leftrightarrow Q_{p}(x)=Q_{p}^{*}(x).\) Hence the results follows. \(\square\)
CPHF Heronian Mean operator and CPHF Geometric Heronian Mean operator
Heronian Mean operator and Geometric Heronian Mean operator are another commonly used fuzzy aggregation operators beacuse of their good capability to capture interrelationships between input arguments18. Therefore, in this subsection, we take Heronian Mean and Geometric Heronian Mean into CPHF environment.
Definition 4.2
Let \(Q_{p}^{i}(x)=\{\gamma _{j_{i}}^{i}(x)e^{i2\pi (\omega _{j_{i}}^{i}(x))}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\}(i=1,2,\ldots ,n)\) be a group of PCHFEs, then the complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and the complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) are defined as:
Example 4.1
Assume that \(Q_{p}^{1}(x),Q_{p}^{2}(x)\) are two CPHFEs, which are defined as \(Q_{p}^{1}(x)=\Big \{0.5\cdot e^{i2\pi (0.3)}|0.4,0.2\cdot e^{i2\pi (0.1)}|0.6\Big \}\), \(Q_{p}^{2}(x)=\Big \{0.5\cdot e^{i2\pi (0.7)}|0.7,0.6\cdot e^{i2\pi (0.8)}|0.3\Big \}\).Then by Definition 4.2,
Denote
According to Definition 3.4,
Hence,
Finally,
Next,we compute \(\text {CPHFGHM}(Q_{p}^{1}(x),Q_{p}^{2}(x))\) by Definition 4.2 and Definition 3.4. For the sake of convenience, we adopt the following notations. Denote
Next,
Therefore,
Theorem 4.6
Let \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi \Big (\omega _{j_{i}}^{i}(x)\Big )}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \}(i=1,2,\ldots ,n)\) be a group of CPHFEs, then the fused value by using CPHFHM and CPHFGHM are also a CPHFE, and
Proof
Referring to the proof of Theorem 3.1(7), we find that
Then we take advantange of Theorem 3.1(3) and Definition 4.1,
Next, let us consider the second formula. Based on the proof of Theorem 3.1(3) and Definition 3.3(2), we obtain
We then have the following result according to Definition 4.1,
The next three theorems are all obvious, and we omit their proofs here. \(\square\)
It is not difficult to derive that the idempotency does not hold for CPHFHM and CPHFGHM except two special cases where the CPHFEs being fused are all empty CPHFEs or full CPHFEs. Detailedly, Theorems 4.7 and 4.8 conclude that the result of aggregation is also a CPHFE of the same type when the CPHFEs being integrated are all empty CPHFEs or full CPHFEs.
Theorem 4.7
If \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\) is a group of empty CPHFEs, i.e. \(Q_{p}^{i}(x)=Q_{p}^{\circ }(x)=\{0\}\), then
Theorem 4.8
If \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\) is a group of full CPHFEs, i.e. \(Q_{p}^{i}(x)=Q_{p}^{*}(x)=\{1\}\), then
Theorem 4.9
(Commutativity) If \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi \Big (\omega _{j_{i}}^{i}(x)\Big )}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \}(i=1,2,\ldots ,n)\) and \(Q_{p}^{i^{\prime }}(x)=\{\gamma _{j_{i^{\prime }}}^{i^{\prime }}(x)e^{i2\pi (\omega _{j_{i^{\prime }}}^{i^{\prime }}(x))}|p_{j_{i^{\prime }}}^{i^{\prime }}(x),j_{i^{\prime }}=1,2,\ldots ,\#\mu ^{i^{\prime }}|p^{i^{\prime }}\}(i^{\prime }=1^{\prime },2^{\prime },\ldots ,n^{\prime })\) are two sets of CPHFEs and \(Q_{p}^{i^{\prime }}(x)\) is an arbitrary permutation of \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\), then
More propreties of CPHFHM and CPHFGHM can be explored for readers, we omit them. It is well known that weight information is espically important in a MADM issue, we now introduce them into CPHFHM and CPHFGHM for the needs of application.
Definition 4.3
Let \(Q_{p}^{i}(x)=\Bigg \{\gamma _{j_{i}}^{i}(x)e^{i2\pi \Big (\omega _{j_{i}}^{i}(x)\Big )}|p_{j_{i}}^{i}(x),j_{i}=1,2,\ldots ,\#\mu ^{i}|p^{i}\Bigg \}(i=1,2,\ldots ,n)\) be an accumulation of CPHFEs, and \(w=(w_1,w_2,\ldots ,w_n)\) be the weight vector of \(Q_{p}^{i}(x)(i=1,2,\ldots ,n)\) with \(0<w_i<1,i=1,2,\ldots ,n\) and \(\sum \limits _{i=1}^{n} w_i=1.\) Then the complex probabilistic hesitant fuzzy weighted Heronian Mean operator(CPHFWHM) and the complex probabilistic hesitant fuzzy weighted Geometric Heronian Mean operator(CPHFWGHM) are respectively defined as:
Theorem 4.10 gives the explicit expression for the aggregation results,which can be used in MADM directly when we want to fuse several attribute values to one.
Theorem 4.10
Let \(Q_{p}^{i}(x)\) be an accumulation of CPHFEs with weight vector \(w=(w_1,w_2,\ldots , w_n)^{T}\) satisfying \(0<w_i<1,i=1,2,\ldots ,n\) and \(\sum \limits _{i=1}^{n} w_i=1.\) Then the aggregated value acquired by CPHFWHM and CPHFWGHM are also a CPHFE, and
Proof
Through replacing \(\frac{2}{n(n+1)}\) in the proof of Theorem 4.4 with \(\frac{2w_{k}w_{l}}{1+w_{k}}\), we can immediately deduce this assert is ture.
There are also some desirable properties for CPHFWHM and CPHFWGHM analogously to the general form we have discussed just now. We omit them here due to these properties could be easily obtained by readers. \(\square\)
A novel MADM approach in the environment of CPHF information
In this section, a novel MADM approach in the environment of CPHF information is established. Then, an expample is presented to explain the proposed decision process.
Decision process
As mentioned above, CPHFS can better express fuzzy information with randomness in many cases. At the same time, MADM is a problem we often encounter in our daily life. In this section, we set the MADM process in the CPHF setting for the needs of practical application.
Assume there are n alternatives denoted by \(A_l(l=1,2,\ldots ,n)\) and m attributes recorded as \(x_j(j=1,2,\ldots ,m)\). And \(w=(w_1,w_2,\ldots ,w_m)\) is the attribute weight vector provided that \(0<w_j<1,j=1,2,\ldots ,m,\sum \limits _{j=1}^{m} w_j=1.\) Now we invite an authority to determine all the possible grades that the alternative \(A_l\) meets the attribute \(x_j\) in the form of CPHFE, denoted by \(g_{lj}(l=1,2,\ldots ,n;j=1,2,\ldots ,m)\). We then intend to find out the best alternative based on CPHFWA & CPHFWG operation and the score function of CPHFEs. The following algorithm present this process in detail:
Step 1. Build the CPHF decision matrix.
Suppose the original information matrix is \((g_{lj})_{n\times m}\). Generally, there are two categories of attributes named as benefit attributes and cost attributes. When facing such kind of MADM issues, we usually should exchange all the attributes to one type before making decisions. Therefore, considering that the bigger preference values of benefit type is better and the smaller preference values of cost type is better, we transform the cost attribute to the benefit type as presented in the following formula:
Then we obtain the normalized CPHF decision matrix \(Q=(q_{lj})_{n\times m}\).
Step 2. Determine the overall CPHF information of each alternative.
Using the aggregation operator of CPHFEs, we can compute the total evaluation information of alternative \(A_l\) through the following expression. For example, if we take advantage of CPHFWA opreator, the following formula can be approached.
Surely, we can also employ other aggregation opreators that we have estabilished in Section 4 to work out the final evaluation information of those alternatives.
Step 3. Calculate the score value of each alternative.
We can take advantage of the following formula to gain the score of each alternative \(A_l(l=1,2,\ldots ,n)\).
Step 4. Rank all the alternatives according to their score values in descending order.
Next,we summarize the entire process of determining the ranking of n alternatives as Algorithm 1.
Determine the ranking of n alternatives
Instance analysis
Application. (Adapted from63) Assume that a customer intends to purchase a car. After pre-screening, five cars \(A_l(l=1,2,\ldots ,5)\) have been designated for further consideration. Moreover, this customer pays special attention to three attributes of these cars, which are \(x_1:\) quality, \(x_2:\) overall cost and \(x_3:\) the appearance of the car with weight vector \(W=(0.3,0.4,0.3)\). Now, an expert is invited to make an evaluation of each car corresponding to every attribute based on his/her experience. Then, we obtain the following original information matrix shown in Table 1.
Due to \(x_2\) is cost attribute but \(x_1,x_3\) are benefit attributes, we shall first transform \(x_2\) to the benefit type according to above procedures. Hence, following normalized CPHF decision matrix shown in Table 2 could be obtained.
Next, according to Step 2, if we adopt CPHFWA operator, then the total evaluation of each alternative \(A_l(l=1,2,3)\) could be worked out as follows.
-
\(q_1=\{0.5611\cdot e^{i2\pi (0.4438)}|0.2148,0.5075\cdot e^{i2\pi (0.3043)}|0.1827,0.5974\cdot e^{i2\pi (0.5170)}|0.3256,0.5483\cdot e^{i2\pi (0.3958)}|0.2769\};\)
-
\(q_2=\{0.2950\cdot e^{i2\pi (0.3402)}|0.1705,0.3268\cdot e^{i2\pi (0.3133)}|0.1705,0.2314\cdot e^{i2\pi (0.3881)}|0.3295,0.2661\cdot e^{i2\pi (0.3631)}|0.3295\};\)
-
\(q_3=\{0.5000\cdot e^{i2\pi (0.5410)}|0.2588,0.5324\cdot e^{i2\pi (0.5936)}|0.2007,0.3966\cdot e^{i2\pi (0.4925)}|0.3044,0.4357\cdot e^{i2\pi (0.5506)}|0.2361\};\)
-
\(q_4=\{0.6188\cdot e^{i2\pi (0.6634)}|0.1494,0.7104\cdot e^{i2\pi (0.6331)}|0.1158,0.6759\cdot e^{i2\pi (0.6041)}|0.1494,0.7538\cdot e^{i2\pi (0.5685)}\)\(|0.1158, 0.7485\cdot e^{i2\pi (0.6077)}|0.1322,0.8089\cdot e^{i2\pi (0.5723)}|0.1026,0.7862\cdot e^{i2\pi (0.5386)}|0.1322, 0.8375\cdot e^{i2\pi (0.4970)}|0.1026\};\)
-
\(q_5=\{0.2714\cdot e^{i2\pi (0.4000)}|0.3712,0.3316\cdot e^{i2\pi (0.4319)}|0.1920,0.3413\cdot e^{i2\pi (0.3459)}|0.2879,0.3958\cdot e^{i2\pi (0.3807)}|0.1489\};\)
Next, we compute the score value of alternative \(A_l(l=1,2,\ldots ,5)\) in accordance with Step 3.
Finally, it is found that \(s(q_{4})>s(q_{1})>s(q_{3})>s(q_{5})>s(q_{2})\), or in other words, \(A_4\succ A_1\succ A_3\succ A_5\succ A_2,\) which indicates that car \(A_{4}\) is the best choice and we suggest he/her to consider \(A_{4}\) at first.
Besides, we can also employ other operators we have built to conduct the MADM process. Now, we calculate the final evaluation information of each alternative by adopting CPHFWG as follows.
-
\(q_1=\{0.4590\cdot e^{i2\pi (0.3821)}|0.2148,0.4315\cdot e^{i2\pi (0.2896)}|0.1827,0.4807\cdot e^{i2\pi (0.5030)}|0.3256,0.4520\cdot e^{i2\pi (0.3812)}|0.2769\};\)
-
\(q_2=\{0.2781\cdot e^{i2\pi (0.2647)}|0.1705,0.3031\cdot e^{i2\pi (0.2344)}|0.1705,0.2259\cdot e^{i2\pi (0.3680)}|0.3295,0.2462\cdot e^{i2\pi (0.3259)}|0.3295\};\)
-
\(q_3=\{0.5000\cdot e^{i2\pi (0.4762)}|0.2588,0.5281\cdot e^{i2\pi (0.4957)}|0.2007,0.3466\cdot e^{i2\pi (0.3069)}|0.3044,0.3661\cdot e^{i2\pi (0.3194)}|0.2361\};\)
-
\(q_4=\{0.6042\cdot e^{i2\pi (0.6581)}|0.1494,0.6957\cdot e^{i2\pi (0.6284)}|0.1158,0.6373\cdot e^{i2\pi (0.5596)}|0.1494,0.7339\cdot e^{i2\pi (0.5343)}\)\(|0.1158, 0.6823\cdot e^{i2\pi (0.5950)}|0.1322,0.7857\cdot e^{i2\pi (0.5681)}|0.1026,0.7198\cdot e^{i2\pi (0.5059)}|0.1322, 0.8288\cdot e^{i2\pi (0.4830)}|0.1026\};\)
-
\(q_5=\{0.2656\cdot e^{i2\pi (0.4000)}|0.3712,0.3270\cdot e^{i2\pi (0.4277)}|0.1920,0.3096\cdot e^{i2\pi (0.3249)}|0.2879,0.3812\cdot e^{i2\pi (0.3474)}|0.1489\};\)
Similarly, we further compute the score value of each alternative \(A_l(l=1,2,\ldots ,5)\) based on Step 3.
Finally, it is found that \(s(q_{4})>s(q_{1})>s(q_{3})>s(q_{5})>s(q_{2})\). Therefore, the best selection is \(A_{4}\), which is the same as the outcome we have just worked out.
Table 3 indicates that the score values gained by CPHFWA is always bigger than the values computed by CPHFWG for the same collection of CPHFEs. Furthermore, there are no difference betweeen the ranking results acquired by CPHFWA and CPHFWG. Concretely, \(A_{4}\) is always the best choice and then \(A_{1}\) follows, while the \(A_{2}\) is the worst choice among three alternatives.
Comparative analysis
To prove the reliability and rationality of the MADM process we estabilished, now we take advantage of other operators and MADM methods to compute the ranking results of example given in section 5.2.
Compare with the operators proposed in61,64,65
Now we adopt other operators including probabilistic hesitant fuzzy weighted averaging operator (PHFWA) and probabilistic hesitant fuzzy weighted geometric operator (PHFWG)(Phase term \(=\)0), complex hesitant fuzzy weighted averaging operator (CHFWA) and complex hesitant fuzzy weighted geometric operator (CHFWG) (Probabilistic information is overlooked), hesitant fuzzy weighted averaging operator (HFWA) and hesitant fuzzy weighted geometric operator (HFWG)(Phase term \(=\)0 and probabilistic information is also overlooked) to achieve the sorting results of three alternatives61,64,65. The outcomes are presented in Table 4.
All Results obtained by eight operators are summarized in Fig. 1 in a visual way. The comparison results between the proposed opreators and existing operators are summarized in Table 4. On one hand, we discover that \(A_4\) is always the most favourable alternative and \(A_2\) is always the worst one in comparative results regardless of which operator was adopted. Furthermore, the sorting results derived by most operators are exactly the same as ours, which indicates that our method is effective and reliable. On the other hand, the phase information used for expressing periodicity was overlooked by PHFWG and the probabilistic information for displaying randomness was overlooked by CHFWA, which result in the ranking order of another three alternatives is different from ours. Sincerely speaking, in view of the comprehensiveness of the fuzzy information description, it is reasonable to believe that these operators may lead to imprecise result and our novel operators are more appropriate. Overall, we insist that the operators built in this manuscript are more impactful in MADM when the fuzzy information were characterized by CPHFEs based on above analysis.
Compare with MADM methods proposed in48,50
In this subsection, the proposed MADM method is compared with the existing distance-based decision methods under the CHFS environment provided in48,50. Garg et al. proposed a series of innovative distance measures and built a MADM method in48. We take the weighted generalized complex hesitant normalized distance as an example to compute the distance between each alternative and the positive ideal alternative given as \(S_{+}^{*}=\{<x_1,\{1\cdot e^{i2\pi (1)}\},<x_2,\{1\cdot e^{i2\pi (1)}\},<x_3,\{1\cdot e^{i2\pi (1)}\}>\}\). Moreover, it is obvious that the smaller the distance from the positive ideal alternative, the better the alternative. In50, Khan et al. constructed a closeness coefficient based on distance measure to resolve MADM issue. To compute the closeness coefficient, we employ their proposed generalized complex hesitant weighted hybrid normalized distance to calculate the distance between each alternative and the positive ideal alternative given as \(S_{+}^{*}=\{<x_1,\{1\cdot e^{i2\pi (1)}\},<x_2,\{1\cdot e^{i2\pi (1)}\},<x_3,\{1\cdot e^{i2\pi (1)}\}>\}\) as well as the distance between each alternative and the negative ideal alternative given as \(S_{-}^{*}=\{<x_1,\{0\cdot e^{i2\pi (0)}\},<x_2,\{0\cdot e^{i2\pi (0)}\},<x_3,\{0\cdot e^{i2\pi (0)}\}>\}\). Afterwards, we can find that the smaller the closeness coefficient, the better the alternative. Next, we compare the proposed MADM method with the distance based MADM methods under complex hesitant fuzzy settings established in48,50. The results are shown in Tables 5 and 6.
From Table 5, we observe that the best alternative is \(A_3\), which is different from the result of our proposed method. In addition, it can also be found that the ranking position of \(A_4\) becomes more and more advanced as the parameter \(\alpha _{cc}\) decreases. on the whole, there are two main reasons lead to these consequences. Firstly, the amplitude and phase membership degrees of A3 are larger than those of the other alternatives when the probability information was overlooked, thereby its distance from the positive ideal alternative is smaller and the ranking is relatively higher. Secondly, the influence of elements’ number is considered in the distance function presented by Garg et al. It can be seen from the table that the elements’ number of alternative \(A_4\) is more than other alternatives, and at the same time the influence is more significant when the weight coefficient \(\alpha\) in front of it is larger. However, the influence is weakened when \(\alpha\) become smaller, which improves the ranking of \(A_4\). Although the results are slightly different, our method considers both hesitancy and probability and is more comprehensive in information expression. In addition, arithmetic average is used to calculate the score function of the total evaluation value of each alternative, which equally considers the contribution of each element, making the ranking result relatively objective and insensitive to the number of elements.
We noted that the best alternative is always \(A_4\) and the worst one is \(A_2\) from Table 6, which is consistent with the results we obtain. However, the ranking result of another three alternatives is slightly different. Specifically, our method can completely distinguish three of them, while the closeness coefficient proposed by Khan et al. fails to distinguish alternative \(A_1\) and \(A_5\). Therefore, we conclude that the proposed method is more advantageous in identifying the subtle differences among alternatives.
Score values obtained by eight operators.
Conclusions
In order to simultaneously describe randomness and ambiguity appeared in reality, we pioneer a new mathematical tool referred as CPHFS as an expansion of several types of common fuzzy set in this work. Next, we present four operations of CPHFE and several properties of these operations are estabilished. More importantly, we present several aggregation operators for a collection of CPHFEs, which is useful in MADM. Meanwhile, we also point out some useful properties of these proposed aggregation operators. At the end, a framework of MADM in CPHF environment is structured and an example is employed to examine the credibility and validity of the proposed MADM method. Overall, the main contributions of this article are concluded as follows: (1) The CPHFS we provided can better describe two-dimensional fuzzy information with different importance. (2) The CPHF aggregation operator we established can reflect the interaction between attributes and provide powerful support for fusing evaluation information. (3) We take CPHFWA operator as an example to establish a MADM method, which can deal with MADM problems under CPHF settings in a more reasonable way.
However, this research may still exist some limitations: firstly, the MADM technique we presented here cannot handle the MADM problems in the CDHF and CPDHF environments. Secondly, we do not think about the MAGDM problems in the presence of multiple experts, which limits the range of application of our approach. Thirdly, in present work, we only employed subjective weighting method to determine the weights of attribute set, but did not consider the objective weighting method such as entropy weighting method, CRITIC method and FUCOM method in decision-making methods. Therefore, the scientific and objective weighting method is also the requirement for effective application of this MADM technology.
As for the subsequent research, we intend to integrate probability information into other forms of fuzzy set. Such as we can insert probability into complex spherical fuzzy set for better describing ambiguous and stochastic information from three viewpoints. Besides, the distance, similarity and entropy measure of CPHFS is also worthwhile to be explored by researchers for their wide use in MADM.
Data availability
All data generated or analysed during this study are included in this published article.
References
Cochrane, J. L. & Zeleny, M. Multiple Criteria Decision Making (University of South Carolina Press, 1973).
Churchman, C. W., Ackoff, R. L. & Arnoff, E. L. Introduction to Operations Research (Wiley, 1957).
Hwang, C. L. & Yoon, K. Multiple Attribute Decision Making and Applications (Springer-Verlag, 1981).
Zadeh, L. A. Fuzzy sets. Inf. Control 8(3), 338–353 (1965).
Torra, V., & Narukawa, Y. On hesitant fuzzy sets and decision. In The 18th IEEE international Conference on Fuzzy Systems, Jeju Island, Korea 1378–1382 (2009).
Zhu, B., Xu, Z. S. & Xia, M. M. Dual hesitant fuzzy sets. J. Appl. Math. Comput. 2012, 2607–2645 (2012).
Zhu, B. & Xu, Z. S. Some results for dual hesitant fuzzy sets. J. Intell. Fuzzy Syst. 26, 1657–1668 (2014).
Xu, Z. S. & Zhou, W. Consensus building with a group of decision makers under the hesitant probabilistic fuzzy environment. Fuzzy Optim. Decis. Making 16, 481–503 (2017).
Hao, Z. N. et al. Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl. Based Syst. 127, 16–28 (2017).
Ning, B. Q. & Wei, G. W. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures. Demonstr. Math. 56, 20220239 (2023).
Ning, B. Q. et al. Probabilistic dual hesitant fuzzy MAGDM method based on generalized extended power average operator and its application to online teaching platform supplier selection. Eng. Appl. Artif. Intell. 125, 106667 (2023).
Liu, P. Q. et al. MAGDM method based on generalized hesitant fuzzy TODIM and cumulative prospect theory and application to recruitment of university researchers. J. Intell. Fuzzy Syst. 45(1), 1863–1880 (2023).
Ning, B. Q. et al. EDAS method for multiple attribute group decision making with probabilistic dual hesitant fuzzy information and its application to suppliers selection. Technol. Econ. Dev. Econ. 29, 326–352 (2023).
Ning, B. Q., Wei, G. W. & Guo, Y. F. Some novel distance and similarity measures for probabilistic dual hesitant fuzzy sets and their applications to MAGDM. Int. J. Mach. Learn. Cybern. 13(12), 3887–3907 (2022).
Cuong, B. C. Picture fuzzy sets. J. Comput. Sci. Cybern. 30(4), 409–420 (2014).
Cuong, B. C. & Kreinovich, V. Picture fuzzy sets-a new concept for computational intelligence problems. In 2013 Third World Congress on Information and Communication Technologies (WICT 2013) 1–6 (IEEE, 2013).
Wu, X. X., Zhu, Z. Y., & Çayli, G. D. Picture fuzzy interactional aggregation operators via strict triangular norms and applications to multi-criteria decision making. arXiv:2204.03878 (2022).
Lin, M. W. et al. Picture fuzzy interaction-al partitioned Heronian mean aggregation operators: An application to MADM process. Artif. Intell. Rev. 55(2), 1171–1208 (2022).
Ning, B. Q., Lei, F. & Wei, G. W. CODAS method for multi-attribute decision-making based on some novel distance and entropy measures under probabilistic dual hesitant fuzzy sets. Int. J. Fuzzy Syst. 24(8), 3626–3649 (2022).
Özlü, Ş. New q-rung orthopair fuzzy Aczel-Alsina weighted geometric operators under group-based generalized parameters in multi-criteria decision-making problems. Comput. Appl. Math. 43(3), 122 (2024).
Ning, B. Q. et al. Several similarity measures of probabilistic dual hesitant fuzzy sets and their applications to new energy vehicle charging station location. Alex. Eng. J. 71, 371–385 (2023).
Özlü, Ş & Aktaş, H. Correlation coefficient of r, s, t-spherical hesitant fuzzy sets and MCDM problems based on clustering algorithm and technique for order preference by similarity to ideal solution method. Comput. Appl. Math. 43(8), 429 (2024).
Yang, G. F., Ren, M. & Hao, X. M. Multi-criteria decision-making problem based on the novel probabilistic hesitant fuzzy entropy and TODIM method. Alex. Eng. J. 68, 437–451 (2023).
Ali, J. & Garg, H. On spherical fuzzy distance measure and TAOV method for decision-making problems with incomplete weight information. Eng. Appl. Artif. Intell. 119, 105726 (2023).
Li, D. Q., Zeng, W. Y. & Li, J. H. New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Eng. Appl. Artif. Intell. 40(apr.), 11–16 (2015).
Li, X. N. et al. Three-way decision on information tables. Inf. Sci. 545, 25–43 (2021).
Yao, Y. Y. Tri-level thinking: Models of three-way decision. Int. J. Mach. Learn. Cybern. 11(5), 947–959 (2020).
Li, X. N., Sun, Q. Q., Chen, H. M. & Yi, H. J. Three-way decision on two universes. Inf. Sci. 515, 263–279 (2020).
Xu, W. Y., Jia, B. & Li, X. N. A two-universe model of three-way decision with ranking and reference tuple. Inf. Sci. 581, 808–839 (2021).
Smets, P. Medical diagnosis: Fuzzy sets and degrees of belief. Fuzzy Sets Syst. 5(3), 259–266 (1981).
Liu, X. D. et al. Novel correlation coefficient between hesitant fuzzy sets with application to medical diagnosis. Expert Syst. Appl. 183, 115393 (2021).
Mjka, B. et al. Bi-parametric distance and similarity measures of picture fuzzy sets and their applications in medical diagnosis. Egypt. Inform. J. 22, 201–212 (2021).
Xing, J. M., Gao, C. & Zhou, J. Weighted fuzzy rough sets-based tri-training and its application to medical diagnosis. Appl. Soft Comput. 124, 109025 (2022).
Chen, Y. F. et al. Three-way decision support for diagnosis on focal liver lesions. Knowl. Based Syst. 127, 85–99 (2017).
Lin, M. W. et al. Directional correlation coefficient measures for Pythagorean fuzzy sets: Their applications to medical diagnosis and cluster analysis. Complex Intell. Syst. 7(2), 1025–1043 (2021).
Mitra, S. & Pal, S. K. Fuzzy sets in pattern recognition and machine intelligence. Fuzzy Sets Syst. 156(3), 381–386 (2005).
Ashraf, Z., Khan, M. S. & Lohani, Q. D. New bounded variation based similarity measures between Atanassov intuitionistic fuzzy sets for clustering and pattern recognition. Appl. Soft Comput. 85, 105529 (2019).
Zeng, W. Y., Li, D. Q. & Yin, Q. Distance and similarity measures between hesitant fuzzy sets and their application in pattern recognition. Pattern Recogn. Lett. 84, 267–271 (2016).
Chu, C. H., Hung, K. C. & Julian, P. A complete pattern recognition approach under Atanassov’s intuitionistic fuzzy sets. Knowl. Based Syst. 66, 36–45 (2014).
Jiang, Q., Jin, X., Lee, S. J. & Yao, S. A new similarity/distance measure between intuitionistic fuzzy sets based on the transformed isosceles triangles and its applications to pattern recognition. Expert Syst. Appl. 116, 439–453 (2019).
Cheng, S. H., Chen, S. M. & Lan, T. C. A novel similarity measure between intuitionistic fuzzy sets based on the centroid points of transformed fuzzy numbers with applications to pattern recognition. Inf. Sci. 343–344, 15–40 (2016).
Ramot, D. et al. Complex fuzzy sets. IEEE Trans. Fuzzy Syst. 10(2), 171–186 (2002).
Garg, H. et al. CHFS: Complex hesitant fuzzy sets-and their applications to decision making with novel distance measures. CAAI Trans. Intell. Technol. 3, 93–122 (2020).
Torkayesh, A. E. et al. A state-of-the-art survey of evaluation based on distance from average solution (EDAS): Developments and applications. Expert Syst. Appl. 221, 119724 (2023).
Wu, X. X. et al. A monotonous intuitionistic fuzzy TOPSIS method under general linear orders via admissible distance measures. IEEE Trans. Fuzzy Syst. 31(5), 1552–1565 (2022).
Liu, D., Xu, J. & Du, Y. F. An integrated HPF-TODIM-MULTIMOORA approach for car selection through online reviews. Ann. Oper. Res. 1–40 (2024)
Özlü, Ş & Faruk, K. Some distance measures for type 2 hesitant fuzzy sets and their applications to multi-criteria group decision-making problems. Soft. Comput. 24(13), 9965–9980 (2020).
Garg, H. et al. CHFS: Complex hesitant fuzzy sets-their applications to decision making with different and innovative distance measures. CAAI Trans. Intell. Technol. 6(1), 93–122 (2021).
Özlü, Ş & İrfan, D. Novel distance measures over SVTN-numbers and their application to multi-criteria decision making problems. In Neutrosophic Operational Research: Methods and Applications 103–126 (Springer, 2021).
Khan, M. S. A. et al. Priority degrees and distance measures of complex hesitant fuzzy sets with application to multi-criteria decision making. IEEE Access 11, 13647–13666 (2022).
Özlü, Ş. Multi-criteria decision making based on vector similarity measures of picture type-2 hesitant fuzzy sets. Granul. Comput. 8(6), 1505–31 (2023).
Özlü, Ş. Generalized Dice measures of single valued neutrosophic type-2 hesitant fuzzy sets and their application to multi-criteria decision making problems. Int. J. Mach. Learn. Cybern. 14(1), 33–62 (2023).
Mahmood, T., Rehman, U. U. & Ali, Z. Exponential and non-exponential based generalized similarity measures for complex hesitant fuzzy sets with applications. Fuzzy Inf. Eng. 12(1), 38–70 (2020).
Mahmood, T. et al. Hybrid vector similarity measures based on complex hesitant fuzzy sets and their applications to pattern recognition and medical diagnosis. J. Intell. Fuzzy Syst. 40(1), 625–646 (2021).
Mahmood, T. et al. Jaccard and dice similarity measures based on novel complex dual hesitant fuzzy sets and their applications. Math. Probl. Eng. 2020(1), 5920432 (2020).
Jana, C., Pal, M. & Liu, P. D. Multiple attribute dynamic decision making method based on some complex aggregation functions in CQROF setting. Comput. Appl. Math. 41(3), 103 (2022).
Özlü, Ş. Bipolar-valued complex hesitant fuzzy Dombi aggregating operators based on multi-criteria decision-making problems. Int. J. Fuzzy Syst. 1–28 (2024)
Jana, C. et al. MABAC framework for logarithmic bipolar fuzzy multiple attribute group decision-making for supplier selection. Complex Intell. Syst. 10(1), 273–288 (2024).
Komal, T. P. Picture fuzzy power Muirhead mean operators and their application to multi attribute decision making. J. Ind. Manag. Optim. 19(6), 4321–4349 (2023).
Jana, C. et al. Hybrid multi-criteria decision-making method with a bipolar fuzzy approach and its applications to economic condition analysis. Eng. Appl. Artif. Intell. 132, 107837 (2024).
Talafha, M. et al. Complex hesitant fuzzy sets and its applications in multiple attributes decision-making problems. J. Intell. Fuzzy Syst. 41(6), 7299–7327 (2021).
Xia, M. M. & Xu, Z. S. Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52(3), 395–407 (2011).
Talafha, M. et al. Complex hesitant fuzzy sets and its applications in multiple attributes decision-making problems. J. Intell. Fuzzy Syst. Appl. Eng. Technol. 6, 41 (2021).
Shen, Z., Xu, Z. S. & He, Y. Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf. Fusion 38, 1–11 (2017).
Xia, M. M. & Xu, Z. S. Studies on the aggregation of intuitionistic fuzzy and hesitant fuzzy information. Int. J. Intell. Syst. 26 (2011)
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The authors are very grateful to the Editor-in-Chief and anonymous reviewers for their valuable comments and useful suggestions, which improved the quality of this paper.
Funding
This work is supported by The Scientific Research and Cultivation Project of Liupanshui Normal University (LPSSY2023KJZDPY08), Discipline Cultivation Team of Liupanshui Normal University(LPSSY2023XKPYTD04), First-Class Undergraduate Course Cultivation Project of Liupanshui Normal University in 2022(2022-03-012) and The Science Research Project for Ordinary Undergraduate Universities (Youth Project) of Guizhou Provincial Department of Education in 2022 “Research on the Measurement of the Construction Level of Green and Low carbon cycle Development Economic System in Guizhou Province ” (QJJ [2022] No. 336).
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Yang, Y., Ning, B., Tian, F. et al. Some complex probabilistic hesitant fuzzy aggregation operators and their applications to multi-attribute decision making. Sci Rep 15, 4342 (2025). https://doi.org/10.1038/s41598-024-84582-y
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DOI: https://doi.org/10.1038/s41598-024-84582-y
