Complex Pythagorean uncertain linguistic group decision-making model based on Heronian mean aggregation operator considering uncertainty, interaction and interrelationship

To effectively solve the mixed problem of considering the uncertainty of individuals and groups, the interaction between membership degree (MD) and non-membership (ND), and the interrelationship between attribute variables in complicated multiple attribute group decision-making (MAGDM) problems, in this paper, a concept of complex Pythagorean uncertain linguistic (CPUL) set (CPULS) is introduced, the interaction operational laws (IOLs) of CPUL variables (CPULVs) are defined. The CPUL interaction weighted averaging and geometric operators are presented. A new concept of CPUL rough number (CPULRN) is further constructed. The CPUL rough interaction weighted averaging and geometric aggregation operators (AOs) are extended. The ordering rules of any two CPULRNs are defined. The CPUL rough interaction Heronian mean (HM) (CPULRIHM) operator and its weighted form are advanced, related properties and special cases are explored. An MAGDM model based on CPUL rough interaction weighted HM (CPULRIWHM) operator is built. Lastly, we conduct a case study of location selection problem for logistics town project to show the applicability of the proposed methodology. The sensitivity and methods comparison are analyzed to verify the effectively and superiority.


Introduction
An attribute evaluation value usually embraces ambiguous and impermeable information in MAGDM problems. However, as the actual group decision-making (GDM) problem becomes more and more complex, scholars are faced with significant challenges in the expression of attribute variables. As an information representation method, fuzzy set (FS) [1] is widely applied to solve information modeling problems with vague and uncertain information in many fields, but the FS has only MD M (x) (0 ≤M(x) ≤ 1), so it is difficult to comprehensively describe and depict the uncertainty degree B Haolun Wang haolunwang@ncu.edu.cn 1 difference from the FSs. Alkouri et al. [5] extended the complex IFSs (CIFSs), where MDN αe iϕ and ND N βe iδ provided a wider decision-making range for DMs. However, the limitation of CIFSs is that the sum of MD and ND cannot be greater than one in the real and imaginary parts. For this reason, Ullah et al. [6] developed the CPFSs consisting of the MD M αe iϕ and ND N βe iδ , which satisfies the α 2 + β 2 ≤ 1 and ϕ 2π 2 + δ 2π 2 ≤ 1.
At present, many scholars have studied the AOs and decision technologies in CPFS context. Some researchers have paid attention to the aggregation methods for complex Pythagorean fuzzy numbers, and developed operational laws including Algebraic [7], Hamacher [8], Einstein [9], Dombi [10], Yager's operations [11] and operational laws based on confidential levels [12], which are the core of aggregation methods. Other scholars have studied ranking methods, such as TOPSIS and ELECTRE [13,14]. However, it is difficult for decision-makers (DMs) to express the "good" or "bad" evaluation degree of an object with MD and ND in CPFS in actual decision-making. In fact, DMs are more inclined to use linguistic variable set or uncertain linguistic variable set (ULVS) to evaluate objects [15][16][17], but uncertain linguistic terms have not additional phase term, and the change of the phase term cannot be described. To overcome their disadvantages and broad the scope of the supporting grade reaching out from the unit plate in the form of complex number belonging to unit disc in a complex plane, the new concept of CPULS is proposed, which combines CPFS with ULVS. The CPULV is divided into uncertain linguistic and complex Pythagorean fuzzy parts, in which the former represents the linguistic level of DM's qualitative evaluation of the object, and the latter describes the MD and ND to the uncertain linguistic part. The CPULS contains more information than CPFS and ULVS, so it is more suitable to deal with practical complex decision-making problems containing a large amount of uncertain, vague information with phase changes, such as medical research, government work reports, biological and face recognition, digital and image processing [18].
There may be various potential characteristics and correlations among variables in GDM. To better fit the objectivity and systematicness of actual decision-making problems, we need to carry out systematic research on the following three challenges in combination with the GDM problems in the literature [12][13][14][15][16]: (1) the interaction between MD and ND is not considered. (2) Individual uncertainty and group uncertainty cannot be dealt with simultaneously. (3) The interrelationship between attributes is ignored. Therefore, how to develop aggregation operators to deal with interaction, uncertainty and interrelationship is a meaning work, which is the main motivation of this paper. Specifically, for challenge (1), not matter in IFS, PFS, CIFS or CPFS environment, existing AOs adopted basic Algebraic, Einstein and Hamacher operational rules. When the variable's MD or ND is zero, the decision results may be counterintuitive [6,12]. However, there is no research on IOLs in CIFS and CPFS at present. Therefore, by considering the advantages of IOLs in IFS and PFS [19][20][21], we develop some new IOLs to use in modeling a problem involving CPULV. For challenge (2), in actual GDM process, the evaluation values given by a single DM can only express the vagueness and hesitation (it is called individual uncertainty) of the DM's individual evaluation, but cannot deal with the imprecision and subjectivity (it is called group uncertainty) of the DM's group evaluation. Recently, the integration of fuzzy theory and rough set theory can produce a more flexible and expressive potential framework for processing fuzzy evaluation information from a global perspective [22][23][24], which can reflect the integrity and rationality of the views of DMs, and also provides a research idea for this paper. For challenge (3), the interrelationship between attributes can affect the final decision result [25]. Among many existing AOs, the HM operator [26] is one of the information fusion tools capturing the interrelationship between input arguments, and it can overcome the disadvantage of computational redundancy and lower computational complexity compared with Bonferroni mean operator [27,28]. Recently, some scholars have done a lot of research on HM with PFS [29], q-rung orthopair fuzzy set (QROFS) [30], linguistic term set [31], neutrosophic ULVS [32] and complex q-rung orthopair linguistic (CQROL) set [33]. However, no scholar has extended HM operator to CPUL context. In addition, few scholars have studied the integration of IOLs and HM operator for picture fuzzy set [34,35] and QROFS [36], which can provide feasible ideas for this paper.
According to the above analysis, the main target of this work is to propose a GDM methodology in the CPUL environment, which comprehensively considers the uncertainty of individuals and groups, the interaction effect between membership functions and the interrelationship between input variables, and applies to solve the site selection of logistic town project. Thus, motivated by the above analysis, this paper has the following contributions: (1) We propose the CPULS, and develop the IOLs of CPULVs and the CPUL interaction AOs, such as the CPUL interactive weighted averaging (CPULIWA) and CPUL interactive weighted geometric (CPULIWG) operators. (2) We define the concept of CPULRN, and extend the IOLs of CPULRNs and the CPUL rough interaction AOs. The ordering rules of any two CPULRNs are defined. (3) We develop the CPULRIHM AOs integrating uncertainty, interaction and interrelationship, and discuss their desirable properties and some special cases.
(4) The MAGDM model based on CPULRIWHM operator is constructed to solve the location problem of logistics town project. The sensitivity and comparative analysis are performed.
To this end, several basic notions are briefly described in "Preliminaries", that is, CPFSs, ULVSs and HM operator. In the subsequent section, the new concept of CPULS is defined, the IOLs of CPULVs and two CUPL interaction AOs are developed followed by which the novel concept of CPULRN is defined, the IOLs of CPULRNs and CPULR interaction AOs are extended. In "CPULRIHM AOs", the CPULRIHM and CPULRIWHM operators are developed. Then the MAGDM model is established based on the CPULRIWHM operator with CPULVs. In "Case study", the MAGDM model is utilized to obtain the optimal location of logistic town project. The sensitivity and comparative analyses are performed. The conclusion is summarized in the last section.

Preliminaries
We briefly describe several related definitions in this section, such as the CPFSs, ULVSs and HM operator. Definition 1 [6,12] . Suppose X is a finite universe set, the mathematical form of CPFS Ã on X can be given as.
) be any three CPFNs, then the basic Algebraic operational laws (AOLs) of CPFNs are defined as.
The HM η,ρ is known as Heronian mean operator.

Complex Pythagorean uncertain linguistic sets
This article holds that CPULS is the combination of CPFS and ULVS, which can be used to handle vague, indeterminate, unreliable and periodic assessment information in the actual world. Therefore, we define the concept of CPULS, IOLs of CPULVs and related AOs in this section.

Definition 7
Suppose X is the finite universe set, the mathematical form of CPULSP on X is given as follows: where s α(x j ) ,s β(x j ) ∈Ṡ; the real part function MP (x j ), NP (x j ), respectively, represent the modules that x j belongs to and does not belong , respectively, represent the amplitude angles that x j belongs to and does not belong  [40].

Definition 8
Letp j be any CPULV, its score function sc(p j ) and accuracy function ac(p j ) are defined as.

Interaction operational laws of CPULVs
The operations of CPFNs in the literature [7] are AOLs. However, some counterintuitive phenomena may occur in these calculation results, because when the MD or ND in CPFN is zero, the zero can also appear in the calculation results. Therefore, the IOLs of CPULVs is proposed based on the literature [19][20][21].

CPUL interaction aggregation operators
We proposed the CPULIWA and CPULIWG operators based on the IOLs in this subsection, and we discuss their effective properties.
Theorem 2 Supposep j ( j 1, 2, . . . , n) is a set of CPULVs, the weight vector ofp j ( j 1, 2, . . . , n) is W (w 1 , w 2 , . . . , w n ) T , satisfies w j ∈ [0, 1], n j 1 w j 1. The aggregation results of CPULIWA and CPULIWG operators are still CPULV, and even C PU L I W G(p 1 ,p 2 , · · · ,p n ) ⎛ Proof It is easy to prove that the aggregation results of CPULIWA and CPULIWG operators are still CPULVs, the proof process is omitted here. The next proof is that Eqs. (9-10) are true. First, the Eq. (9) is proved by mathematical induction.
When n 2, according to Definition 10, we have So, Eq. (9) holds for n 2.
Obviously, when n k, Eq. (9) is also true, that is, When n k + 1, then we have Thus, Eq. (9) has been proved to be true. Similarly, Eq. (10) can be proved to be true.
Therefore, the proof of Theorem 2 is completed.

Concept of complex Pythagorean uncertain linguistic rough number
As an effective tool to deal with subjective and imprecise information, the rough number can mine the potential knowledge hidden under the data surface, and convert each value into the form of rough number composed of lower limit and upper limit. To effectively deal with the individual uncertainty and group uncertainty in DMs' assessment information in GDM, we combine the concepts of CPULV and rough number to construct a new concept of CPULRN.

CPULRN
Suppose that the finite non-empty set ℵ is the universe,∀ ∈ ℵ, the definition of CPULRN assume that t class is equivalent to CPLUV class to form a family of CPULV topological sets, expressed as {p j | j 1, 2, . . . , t}, there is a certain dominant ordering relationship between each equivalent CPULV class, namely,p 1 <p 2 < . . . <p t , then for any classp j ∈ , 1 ≤ j ≤ t, Y ⊆ ℵ, X ⊆ ℵ, the upper approximation ofp j can be defined: And the lower approximation ofp j : where (ℵ, R) forms approximate CPUL rough space. Based on the classical rough number construction, any CPUL classp j can be expressed by CPULRN, which consists of the CPUL rough lower limit (CPULRLL) C PU L RN(p j ) and the CPUL rough upper limit (CPUL-RUL) C PU L RN(p j ), and can be expressed as follows: C PU L RN(p j ) is obtained by utilizing the CPULIWA (Eq. (9)) or CPULIWG (Eq. (10)) operator to aggregate CPULVs of Y 1 , Y 2 ,…,Y QL as (Y 1 ), (Y 2 ), ..., (Y Q L ), and their weights are equal, namely, w 1 w 2 . . . w Q L 1 Q L . Similarly, C PU L RN( p j ) can be obtained.

Definition 12
Based on the CPULRLL C PU L RN(p j ) and CPULRUL C PU L RN(p j ) of CPUL classp j , C PU L RN(p j ) is defined as.
Theorem 4 Let [p j ](j 1, 2) be any two CPULRN, and they have the following operation properties: Proof Since CPULV(p j ) (1 ≤ j ≤ t) appears in the form of interval, its operation property is the same as the general interval number, so it is easy to prove Theorem 4.

Example 1.
Suppose that four DMs to evaluate an attribute, and the variables are expressed by CPULVs: 2) )), z 7. We can use Eq. (7) in Definition 8 and we can get: So, according to Definition 8, these CPULVs are ranked asp 1 <p 3 <p 2 <p 4 . Takingp 1 as an example, it can be seen from the CPULRN structure that there are Apr(p 1 ) {p 1 } and Apr(p 1 ) {p 1 , p 2 ,p 3 ,p 4 }, then it can be obtained by CPULIWA operation (Eq. The significance of the CPULVs conversion into the CPULRNs is: The initial CPULV is only given by individual DM, which ignored the interaction between DMs and cannot accurately express the group opinions of DMs. However, the CPULRN is derived from a holistic perspective and can reflect the integrity and rationality of the DMs' opinions. For example, the third DM's evaluation of the attribute isp 3 ([s 1 ,s 3 ], (0.7e i2π(0.6) , 0.5e i2π(0.3) )), but from overall perspective, the attribute value should be [  .298) )). Thus, the CPULRN should be all CPULVs between the lower limit and the upper limit. In addition, the CPULVs are transformed into CPULRNs by applying the CPULIWA or CPULIWG operator, which taking IOLs of CPULVs into account, so that the CPULRN is more reasonable, and it can avoid the counterintuitive dilemmas. Therefore, the CPULRNs can not only reflect the uncertainty of individuals and groups, but also avoid the loss or attenuation of information.

Interaction operational laws of CPULRNs
The following IOLs of CPULRNs can be extended based on the Definition 10 in this subsection.
Similar to Theorem 2, we can easily prove that the Definition 14 is true according to the IOLs of CPULRNs in Definition 13, and the CPULRIWA and CPULRIWG operators satisfy Idempotence, Boundedness and Monotonicity, thus the proof is omitted here.

Complex Pythagorean uncertain linguistic rough interaction Heronian mean aggregation operators
Considering uncertainty, interactivity and interrelationship, we define the CPULRIHM and CPULRIWHM operators based on the Definition 13 and HM operator, and then their some effective properties and special cases are explored.
Theorem 6 Suppose [p j ] (j 1, 2 ,…,n) is a family of CPULRNs, the aggregation result of above Eq. (24) is a CPULRN, that is, where A Proof According to Definition 13, we can get.
Then, 2 n(n + 1) And then we obtain, where the special forms of A, B, C, G C , D and G D can be seen from the Eq. (25) in Theorem 6.
Therefore, the proof is completed. Based on Definition 4, we can easily prove that the CPUL-RIHM operator has the below properties.  Case 3. If σ 0,τ 0, then the CPULRIHM reduces to CPUL rough interaction generalized linear ascending weighted operator, that is,

Case 5.
If σ + ∞, τ 0, then the CPULRIHM reduces to Case 6. If σ 0, τ 0, then the CPULRIHM reduces to CPUL rough interaction geometric averaging operator, that is (32)  Notably, the importance of each input CPULVs of the CPULRIHM operator in Definition 18 is not considered. However, the weight of attribute, as an important input parameter, has a crucial part in the aggregation process of attribute variables and can affect the result of AOs in many actual MAGDM problems. Therefore, we further develop the CPULRIWHM operator.
where Fig. 1 The flowchart of the proposed approach Similar to Theorems 6, 8 is true can be proved. We also can prove that the CPULRIWHM operator meets the Boundedness and Monotonicity.
Step 1: The individual CPUL evaluation matrixD ζ is constructed. It is necessary to guarantee that the types of attributes remain consistent in decision process, so the cost attribute is converted into benefit attribute. We can apply the conversion technology (Eq. (35)) to get the normalized individual CPULDMR ζ r ζ ιj m×n where I 1 and I 2 represent benefit and cost attribute respectively.
Step 2: According to Definition 11, the normalized individual CPULDMR ζ is transformed into CPUL rough Step 3: The CPULRIWA (Eq. (21)) or CPULRIWG (Eq. (22)) operator is used to aggregate the assessment information of alternative a ι under attribute c j , and the individual CPULRDM [R] ζ is aggregated into the group CPULRDM . (37) Step 4: The CPULRIWHM operator (Eq. (34)) is applied to get the comprehensive evaluation value of each alternative [x ι ].
Step 6: The alternatives are ranked according to the expected value and distance measure of each alternative.

Case study: site selection of logistics town project
Jiujiang is a prefecture-level city in northern Jiangxi Province, China. Located at the junction of the Yangtze River Economic Development Zone and Beijing-Kowloon Railway Economic Development Zone, Jiujiang is a shipping center city in the middle reaches of Yangtze River. It is about 240 km away from the central city of central China-Wuhan, and about 650 km away from the core city of Yangtze River Delta economic circle in eastern China-Shanghai. Jiujiang's highway, railway, waterway and other infrastructure layout is reasonable, and the comprehensive transportation system is perfect. In 2015, Jiujiang was designated as a regional circulation node city by the Ministry of commerce and other departments of China, becoming an important node in the national backbone circulation network.
To better join the Yangtze River Delta economic circle and play to the advantage of the regional logistics node city and channel network, the government of Jiujiang city wants to accelerate the development and upgrading of regional logistics and create characteristic logistics industry, and plans to introduce the logistics town project. Taking this as the carrier, the integration of warehousing, distribution, processing, centralization, procurement and retail store can be realized by building hardware facilities such as warehousing and freight center in the town and using modern information technology such as Internet of Things and cloud computing as soft support.
One of the challenges facing the government is how to select the best location for the logistics town project. Project location has a great influence on the function and productivity of logistics town. Jiujiang government invites three experts (DMs) to form an evaluation committee E {e 1 , e 2 , e 3 } to understand the basic characteristics of logistics town location, where e 1 is a senior executive of a consulting company, who is good at regional logistics industry planning; e 2 is a local government official in charge of land resource planning and utilization; e 3 is a senior professor in the field of logistics from Nanchang University in Jiangxi Province. After discussion, the DMs determine four sites as the alternative location for the logistics town project A {a 1 , a 2 , a 3 , a 4 }, which are scattered in Jiujiang and depicted in Fig. 2. According to the evaluation opinions of the DM panel, the location of the logistics town is related to the following key factors to maximize profits: project floor area (c 1 ), land cost (c 2 ), distance from port (c 3 ), distance from railway freight station (c 4 ), distance to highways (c 5 ) and impact on local economy (c 6 ). DMs give the evaluation value in the form of CPULV, with attributes c 1 and c 6 as benefit type attributes, and the linguistic term set S {s 0 extremely poor, s 1 very poor, s 2 poor, s 3 medium, s 4 good, s 5 very good, s 6 Table 1 show the evaluation matrices of individual CPULṼ D ζ (ζ 1, 2, 3) given by DMs.
We determine the best location of logistics town according to Sect. 6, and the specific process is as below: Step 1: Eq. [s 2 , s 3 ], (0.5e i2π (0.7) , 0.6e i2π (0.7) ) . We convert these CPULVs into individual CPULRNs, respectively: Step 3: Using CPULRIWA (Eq. (21)), the individual CPULRDM [R] ζ is aggregated into the group CPULRDM [G], the aggregation results are shown in Table 2. Step6: The ranking of all alternatives is determined by the expected value of the alternative: a 4 > a 2 > a 3 > a 1 . That is, a 4 is the optimal location of logistics town project.

Sensitivity analysis
To analyze the impact of parameters σ and τ in the CPULRI-WHM operator on alternative ranking, the expected values of alternatives are calculated to determine the ordering of alternatives when the parameters σ and τ take different values, the results are seen in Table 3.
From Table 3, on the whole, the expected value of each alternative decreases gradually with the increase of parameters σ andτ . In this process, a 4 and a 2 are the optimal and second best alternatives respectively and remain unchanged, while a 1 and a 3 have slight changes. It can be seen that the alternative ranking is generally relatively stable in the process of parameters σ andτ changes. It is worth noting that the parameter value is chosen by DMs based on their oven risk preferences. However, if the value of parameter is too small, the interrelationship between input arguments cannot be reflected. Therefore, considering the complexity

Comparative study
The proposed method is compared with the existing methods to verify the effectiveness of the method presented. There are existing methods, such as the PUL weighted averaging (PULWA) operator [38], PUL weighted geometric (PULWG) operator [38], PUL-VIKOR [39], complex intuitionistic uncertain linguistic weighted arithmetic HM (CIULWAHM) operator [41], complex intuitionistic uncertain linguistic weighted geometric HM (CIULWGHM) operator [41], complex q-rung orthopair uncertain linguistic weighted averaging (CQROULWA) operator [42], complex q-rung orthopair uncertain linguistic weighted geometric (CQROULWG) operator [42], complex q-rung orthopair uncertain linguistic VIKOR (CQ-VIKOR) [42] and CQROL weighted HM (CQROLWHM) operator [33]. It should be noted that the imaginary parts of MD and ND of CPULVs in Table 1 are omitted in the calculation process of PULWA, PULWG and PUL-VIKOR; the compromise coefficient η 0.5 in the calculation process of PUL-VIKOR and CQ-VIKOR methods; the CPULV is converted into complex Pythagorean linguistic number by averaging the upper and lower limits of the uncertain linguistic part of CPULVs in Table 1, and the parameters σ τ 1 are also used in the CQROLWHM operator calculation process. Thus, the comparison results are shown in Table 4. An intuitive comparison of ranking results is shown in Fig. 3.
From Table 4, the CIULWAHM and CIULWGHM operators are not applicable to this case, because these two operators cannot process CPULVs in Table 1 where the sum of MD and ND of real and imaginary parts is greater than one. The reason why the results of PULWA operator, PULWG operator and PUL-VIKOR method are different from the proposed method is that these methods do not consider the interaction between MD and ND in the real and imaginary parts of CPULVs shown in Table 1, the uncertainty of individuals and groups, and the interrelationship between variables. The CQROULWA and CQROULWG operators also do not consider the above interaction, uncertainty and interrelationship at the same time, but carry out aggregation of CPULVs more rigidly. Although the best alternative a 4 for ) >]   Ranking CQROLWHM operator which only considers the interrelationship between attributes, the results of other alternatives are slightly diverse from those of the proposed method. The CQ-VIKOR method is completely different from our method in terms of decision principle, but the obtained results are completely consistent, which demonstrates the rationality and effectiveness of the method in this work. Therefore, compared with the above-mentioned existing methods, the results of this method are more comprehensive, effective and scientific in terms of both the form expression of evaluation variables and the process of variable processing.

Advantage analysis
The existing AOs are compared with the CPUL interactive, CPUL rough interactive and CPULRIHM AOs proposed in this paper in terms of operational laws, data periodicity, interaction between membership functions, uncertainty of individuals and groups, interrelationship between variables and flexibility of DMs' preference. Table 5 shows the comparison of features of various AOs. It is not difficult to find that the proposed AOs are based on the IOLs and can deal with vague, uncertain and periodic evaluation data. Moreover, the CPULRIWHM operator can systematically and comprehensively consider the interaction, uncertainty and interrelationship, and can express the preferences of DMs through parameter changes. Therefore, some superiorities of the proposed method are summarized as below: (1) The CPULS cannot only accurately characterize vagueness and uncertainty, but also characterize the periodicity of data. It can help DMs improve their freedom to represent their opinions and more accurately describe preferences. Moreover, the CPULS has a certain generality. (2) The concept of CPULRN can describe the uncertainty of individual and group evaluation information, and can preserve the uncertainty in the decision-making process to avoid partial information loss. (3) The proposed CPULIWA and CPULIWG operators can avoid the counterintuitive phenomenon. The proposed CPULRWA and CPULRWG operators can not only avoid counterintuitive phenomena, but also express and preserve individual and group uncertainties. In addition, the CPULRIHM AOs, which integrates the interaction between membership functions in real and imaginary parts, individual and group uncertainty and interrelationship between variables, is more suitable for complex realistic decision problems. (4) The MAGDM model based on the CPULRIWHM operator can be effectively applied to settle the practical GDM problem. The proposed CPULRIWHM operator is more flexible than the existing PULWA, PULWG, CQROULWA and CQROULWG operators because its parameters σ and τ can be changed.

Conclusion
The CPULS theory has the combined characteristics of the CPFS and ULVS theories, which can accurately describe the vagueness, uncertainty and periodicity of evaluation information. In this paper, we have defined the CPULS and IOLs of CPULVs, and the CPULIWA and CPULIWG operators have been developed. Then, we have constructed the new concept of CPULRN, and have extended the IOLs of CPULRNs, the CPULRIWA and CPULRIWG operators. Further, the CPUL-RIHM and CPULRIWHM operators have been proposed. Additionally, we have applied the CPULRIWHM operator to handle the MAGDM issues. Finally, the plausibility, scientificity and superiority of the proposed are illustrated by a real case of location selection for logistics town project. Using the proposed model based on CPULRIWHM operator, we have integrated systematically uncertainty, interaction and interrelationship in CPULS context. The proposed operators not only enrich the fuzzy decision theories and have important enlightenment significance for the follow-up aggregation research, but also provide new ideas for solving practical decision-making problems. Directions for future research are described below. First, the use of the proposed improved AOs of CPULSs should be investigated in the different fuzzy climates, such as simplified Neutrosophic sets [43], hesitant fuzzy sets [44], probabilistic linguistic term sets [45] and linguistic q-rung orthopair fuzzy sets [46], etc. For the second future direction, we will focus on the integration of the proposed AOs with other ranking technologies (e.g., VIKOR [47], WASPAS [48], MARCOS [49], etc.), and extend and improve these technologies in the CPULSs environment. For the third future direction, to enhance more managerial implications, we need to utilize them to solve different decision problems in real world, such as business administration, environment protection, energy resources management, etc.