Complex intuitionistic fuzzy Maclaurin symmetric mean operators and its application to emergency program selection

Abstract

Due to the indeterminacy and uncertainty of the decision-makers (DM) in the complex decision making problems of daily life, evaluation and aggregation of the information usually becomes a complicated task. In literature many theories and fuzzy sets (FS) are presented for the evaluation of these decision tasks, but most of these theories and fuzzy sets have failed to explain the uncertainty and vagueness in the decision making issues. Therefore, we use complex intuitionistic fuzzy set (CIFS) instead of fuzzy set and intuitionistic fuzzy set (IFS). A new type of aggregation operation is also developed by the use of complex intuitionistic fuzzy numbers (CIFNs), their accuracy and the score functions are also discussed in detail. Moreover, we utilized the Maclaurin symmetric mean (MSM) operator, which have the ability to capture the relationship among multi-input arguments, as a result, CIF Maclarurin symmetric mean (CIFMSM) operator and CIF dual Maclaurin symmetric mean (CIFDMSM) operator are presented and their characteristics are discussed in detail. On the basis of these operators, a MAGDM method is presented for the solution of group decision making problems. Finally, the validation of the propounded approach is proved by evaluating a numerical example, and by the comparison with the previously researched results.

Keywords

Intuitionistic fuzzy set complex intuitionistic fuzzy set multi-attribute group decision making emergency management program evaluation Maclaurin symmetric mean operator

1 Introduction

In daily life, decision making is a complicated common task which requires sophistication and intelligence. Multiple attribute decision making (MADM) is the selection of a suitable alternative from a group of quantitative and qualitative attributes. Among a variety of membership grading (MG) and non-membership grading (NMG) values, a decision-maker hesitates to select the preferred value; this hesitation leads to vagueness and uncertainties. These assessments and selection problems are termed as DM (Decision Making) problems. DM problem is constrained by a group of external and internal factors. To address this problem, decision-makers have to present their preferred information in terms of numerical values, which is, however, not possible due to many practical complications and the periodicity of the information. So these complications and some limitations, such as lack of knowledge, shortage of time and difficulties in handling information, attracted the attention and consideration of many researchers [1 –3]. To handle the problem of decision-making, researchers have suggested that their cognitions can be possibly presented in numerical terms by a group of linguistic fuzzy variables. These linguistic variables gained the concentration of many practitioners as well as researchers. For the effective representation of indeterminate and vague assessment data, fuzzy set (FS) is a significant method to solve DM problems in tentative environments.

To address these problems, Zadeh (1965) led the inventive idea of FS, which recalculated the components of the Universal set to a real number, known as the membership degree (MD) [4]. Since being reported in 1965, FS has fascinated many researchers and achieved research results in various aspects [5-7]. However, as the range of the FS is restricted [0, 1], it failed to express the assessment evaluation data. Therefore, complex fuzzy set (CFS) was presented by Ramot et al. [8], which is the expansion of membership grade to complex value from the real value in the unit disc. CFS is successfully implemented in distinct fields like decision science and fuzzy logic [9, 10]. Nonetheless, FS and CFS theory fails to present the non-membership grade (NMG) of the member which is related to the given objective. This is the common drawback of both the theories.

Atanassov [11] advanced the FS concept to the Intuitionistic Fuzzy Set (IFS) by attaching NMG there by fulfilling the defect of both the theories. The basic theory and its applications gained huge concentration in different fields such as decision making [12], information entropy [13], distance measure [14], aggregation operator [15] and many others. Later, Alkouri et al. [16] provided a framework of complex intuitive fuzzy sets (CIFS), to explain imprecise, uncertain information and judgment in daily life issues. The CIFS constitutes a complex valued MG and complex-valued NMG in polar coordinates, and is more generalized in DM from CFS. Rani and Garg [17] developed a MADM method by describing the basic algorithm of CIFS using different powers operators. Attanassov [18] presented the interval-valued intuitionistic fuzzy set, which was further studied by Garg and Rani [19]. Based on this study, Garg and Rani proposed the complex interval-valued intuitionistic fuzzy set and researched on aggregation operators and operational rules associated with it. To incorporate the complex intuitionistic fuzzy information, Garg and Rani [20] presented the generalized Bonferroni mean (BM) operators based on the Archimedean operations [21, 22].

Some kinds of information cannot be described efficiently by IFS in the real decision making problems such as (0.3, 0.8) for MG and NMG because of 0.3 + 0.8 > 1. For the solution of this type of DM problems, Yager [23] initially propounded Pythagorean fuzzy set (PFS) as a useful tool. As the 0 . 3² + 0 .8² = 0.73 < 1, it shows that the PFS theory is indispensable and better than both the IFS and FS theories. On the basis of the PFS theory, some ordered weighted distance measures were presented by Qin et al. [24] which are associated to the decision making issues. Garg [25] introduced many logarithmic operators in Pythagorean fuzzy environment and also presented new logarithmic operations.

In some specific real situations, both the theories, IFS and PFS are incompetent to evaluate the information, for example, if a decision maker selects 0.8 and 0.9 as MG and NMG respectively; it gives a sum of square is 1.45, which is greater than 1. To overcome this defect, Yager [26] initiated the idea of a linguistic variable known as q-rung orthropair fuzzy set, which successfully evaluates the information because MG^q + NMG^q<1. It is clear that q-ROFS is more universal than IFS and PFS because it can solve the DM problems more precisely than both of them. Liu and Wang [27] propounded many q-rung orthopair fuzzy Archimedean BM operators using the q-ROFS. Li et al. [28] resolved the DM approach and extended the EDAS methodology to q-rung orthopair fuzzy framework. For the establishment of decision model, Liu et al. [29] extended many Cq-ROFL Heronian Mean (HM) operators to construct a decision model and portray the idea of complex q- rung orthropair fuzzy linguistic set (Cq-ROFLS) and complex q-ROFS.

The fuzzy sets deliberated above can only evaluate the information in quantitative form and it is very hard for the decision makers to describe their opinion in accurate numerical values. Therefore, to overcome this limitation and to prevent the loss of data, Zadeh [30] introduced the linguistic variable (LV) to express the data in qualitative form. By the combination of fuzzy sets with the LV, many diverse novel concepts were presented by many researchers like, Linguistic q-rung orthopair fuzzy number [31], single-valued neutrosophic linguistic set [32] and intuitionistic linguistic numbers [33]. In decision making, for the prevention of information wastage, Herrera and Martłnez [34] offered the idea of 2-tuple fuzzy linguistic through the combination of a numerical with a LV. The 2- tuple linguistic PFS (2TLPFS) [35], intuitionistic 2-tuple linguistic model [36] and many other techniques like these were propounded later by several researchers. These techniques are very effective in the expression of the fuzzy and undetermined information in decision making problems.

In many everyday MCDM situations due to the time pressure, the lack of data and many other factors, DM or experts may irresolute and hesitate in choosing favorites. For these situations, the notion of the hesitant fuzzy linguistic term sets (HFLTSs) was propounded by Rodriguez et al. [37] in 2012 to address difficult qualitative information like multiple linguistic terms and sentences. Lin et al. [38], suggested an effectual distance measures and comparison technique for HFLTSs, and then utilized them to develop a new TODIM method for the solution of MCDM issues by using hesitant fuzzy linguistic information. Furthermore, Lin et al. [39] presented a Pythagorean fuzzy MULTIMOORA technique based on new score function and distance measure. By the use of score function and dice distance the MULTIMOORA technique is improved and is used for the solution of MCDM issues in PFS environment. A practical situation for assessing solid-state disk productions is solved utilizing the established Pythagorean fuzzy MULTIMOORA technique. Similarly, an innovative picture fuzzy MCDM algorithm based on the improved MULTIMOORA approach is propounded to handle the site selection of car sharing station [40]. As we have observed that HFLTS has failed in evaluating various qualitative data, Pang et al. [41] devised the concept of probabilistic linguistic term set (PLTS) to overcome the above mentioned defect and effectively handle the complex qualitative data. Due to this PLTS, the DMs are able to express their preferred information in the form of a set of various linguistic terms related with probabilities. In addition, a new score-entropy-based ELECTRE II technique for the PLTSs is presented by Lin et al. [42] to deal with the edge node selection issue in the edge computing network. On the basis of TODIM approach, Lin et al. [43] established an innovative aggregated probabilistic linguistic MCDM algorithm for the ranking of Internet of Things (IoT) platforms.

It is obvious that the aggregation operator is a significant method in the field of information and it has gained special attention from the researchers in different aspects. For the aggregation of the intuitionistic fuzzy information, Xu [44] propounded many geometric operators. For the initiation of MAGDM approach, Liu and Wang [45] presented geometric operators for Cq-ROFS and weighted average but these operators were incapable to consider the interrelationship of the discussed attribute and decision making issues. To fulfill this deficiency, the HM and BM operators are propounded to take into consideration the significance of any two dimensional data. Therefore, Lin et al. [46] fused HM operator with linguistic q-rung orthopair fuzzy sets (LqROFSs) and innovated linguistic q-rung orthopair fuzzy interactional weighted Pythagorean geometric HM (LqROFIWPGHM) operator. Likewise, Lin et al. [47] integrated LPFSs with partitioned BM (PBM) and proposed some new aggregation operators for assessing the MAGDM problems. Moreover, Lin et al. [48] presented probability density based ordered weighted averaging (PDOWA) operator and suggested an MCDM technique, based on the probability density approach, and implemented it for assessing the smart phones. But unfortunately, HM and BM operators were unsuccessful in considering the interrelationship between the multi input information. So Maclaurin [49] introduced the Maclaurin symmetric mean (MSM) to eliminate the above mentioned deficiencies. Subsequently, for the aggregation of Intuitionistic fuzzy information, dual MSM operator was presented by Qin and Liu [50]. Moreover many linguistic fuzzy MSM operators for the establishment of MAGDM method were propounded by Liu and Qin [51]. To deal with the DM problems efficiently, the MSM operators were expanded to Pythagorean fuzzy environment by Wei and Liu [52]. As we have seen in the literature that MSM operators are not generalized to complex fuzzy sets, so it will be helpful to expand MSM operators to CIF set to introduce many new operators.

According to the above discussion, the summary of this paper is as follows: (a) the previous studies about complex fuzzy sets were unsuccessful in depicting the vague and ill-defined information. As the DM problem is difficult to be solved in a single dimension, the CIF proves to be helpful by solving the issue in two-dimensions. Moreover, this CIF also prevents the loss of information, while dealing with the linguistic information. Hence, CIFS is considered to be much more generalized than the presented fuzzy sets theories. So we initially present the CIF set and the related basic concepts which will help to describe the evaluation information. (b) The information synthesis has a vital role in combining preferred information of the DM experts. Moreover many practical problems are needed for the association of the identified attributes. Considering the advantages of CIF and the dominance of the MSM operators, many CIFMSM operators are presented for the solution of 2-dimensional fuzzy information. (c) In management science, the emergency program selection and evaluation remained a serious topic for researchers. Due to the irregularity and complication of the emergency situation, different methods are required to solve the emergency program problems in the best manner.

As mentioned previously, our contribution in this study is to construct novel models by the fusion of the CIFS with MSM and DMSM aggregation operators, for the evaluation of MCDM and MAGDM problems. And then the supremacy of the proposed operator is validated by the implementation for the selection of optimal emergency program. To achieve this goal, first we have to search a tool suitable for the expression of information. Then we have to establish a decision making algorithm, which will be helpful from different aspects. Our basic goal is discussed as follows:

To propound complex intuitionistic fuzzy averaging operator (CIFA) and complex intuitionistic fuzzy geometric operator (CIFG) on the basis of CIFMSM and CIFDMSM.

To introduce many MSM operators like the CIF Maclaurin symmetric mean (CIFMSM) operator and the CIF dual Maclaurin symmetric mean (CIFDMSM) operator with detailed basic characteristics.

To establish a MAGDM method based on the proposed operators.

To illustrate the performance and authentication of the established method by a universal numerical example for the assessment of emergency program.

To achieve these goals, the summary of this paper is illustrated as follows: In section 2, we successfully reviewed the basic definitions and concepts like IF, CIF operators. Section 3 explains the idea of CIF, comparison method, fundamental operation rules, basic operators and MSM. Section 4 propounds CIFMSM, CIFDMSM operators and includes the study of particular characteristics. Section 5 is related with the new MAGDM approach based on the CIFMSM and CIFDMSM. Section 6 evaluates the emergency program problem and shows the effectiveness of the method. Finally, some conclusion remarks are enlisted at the end.

2 Preliminaries

The aim of this section is to review succinctly essential concepts associated to IFS, CFS and CIFS as well as some other important notations.

Definition 1. [11, 53] let U be a non-empty universal set. For each element of U, IFS I is an ordered pair given as $I = {(x, μ_{I} (x), υ_{I} (x)) | x \in U},$ (1) Where μ_I, ν_I: U → [0,1] is denoting the membership function (MF) and Non-membership function (NMF) respectively, also satisfying each element of U such that MF+NMF ∈ [0, 1] And h_I(x)=1 - μ_I(x) - ν_I (x) denotes the degree of hesitation of x in I. For convenience, the IF number is the pair of μ and ν where μ+ ν ⩽ 1 represented by ρ i.e ρ = (μ, ν) and μ+ ν ∈ [0, 1].

Definition 2. [53] If given an IFN ρ = (μ, ν) the score and the accuracy index are defined as follows: $S (ρ) = μ + ν$ (2) $H (ρ) = μ - ν$ (3)

An order relationship based on these functions, between the various IFNs ρ andσ, can be defined as follows:

If S(ρ)>S(σ), then ρ is superior over σ and is represented by ρ> σ.

If S(ρ)=S(σ), then

If H(ρ)>H(σ), then ρ> σ

If H(ρ)=H(σ) then ρ and σ denote the similar numbers, represented as ρ ∼ σ.

Ramot et al. [8] extended the concept of FS to the theory of CFS by integrating the phase angle with the analysis, which is defined as follows:

Definition 3. [8] A CFS I over U is stated as below: $I = {(x, μ_{I} (x)) | x \in U},$ (4)

Where μ_I: U →{ b : b ∈ C, |b| ⩽ 1 } is a MF and μ_I(x) is denoted as: μ_I(x)=r_I(x)e^{i2πw_{r
_I}(x)} for any x belongs to U and here $i = \sqrt{- 1}$ , 0 ⩽r_I (x), w_{r
_I} (x)⩽1.

Later on, Alkouri and Salleh [16] advanced the concept of CFS to theory of CIFS, by taking the degree of non-membership function into consideration as follows:

Definition 4. [16] A CIFS I on set U is stated as a set defined as follows: $I = {(x, μ_{I} (x), υ_{I} (x)) : x \in U},$ (5)

Where, μ_I, υ_I: U →{ b : b ∈ C, |b| ⩽ 1 } are complex valued membership and NMFs respectively and are denoted as: $μ_{I} (x) = r_{I} (x) e^{i 2 π w_{r_{I}} (x)}, υ_{I} (x) = k_{I} (x) e^{i 2 π w_{k_{I}} (x)}$

Here, the restriction on r_I (x) and k_I (x) is 0⩽r_I (x), k_I (x)⩽1, similarly on w_{r
_I} (x) and w_{k
_I} (x) is 0 ⩽w_{r
_I} (x), w_{k
_I} (x)⩽ 1 for any x∈ U. If U contain just single element then, for convenience CIFS I on U we write ρ = 〈re^{iw
_r}, ke^{iw
_k}〉 as ρ = ((r, w_r) , (k, w_k)) and is termed as CIFN, where r, k and its sum will be restricted to [0, 1] and in the same way 0⩽w_r, w_k⩽1.

Definition 5. [16] If X=((r_x, w_{r
_x}) , (k_x, w_{k
_x})) and Y=((r_y, w_{r
_y}) , (k_y, w_{k
_y})) are two CIFNs, we have

X ⊆ Y if r_x ⩽ r_y, k_x ⩾ k_y and w_{r
_x} ⩽ w_{r
_y}, w_{k
_x} ⩾ w_{k
_y}.

X = Y if X ⊆ Y and X ⊇ Y.

X^c=((k_x, w_{k
_x}) , (r_x, w_{r
_x})) is the complement of CIFN X.

3 Complex intuitionistic fuzzy set

3.1 Operational laws of CIFNs

Definition 6. [17] Let ρ = ((r_ρ, w_{r
_ρ}) , (k_ρ, w_{k
_ρ})) is a CIFN the score function and accuracy function are defined as follows: $S (ρ) = (r_{ρ} - k_{ρ}) + \frac{1}{2 π} (w_{r_{ρ}} - w_{k_{ρ}})$ (6) $H (ρ) = (r_{ρ} + k_{ρ}) + \frac{1}{2 π} (w_{r_{ρ}} + w_{k_{ρ}})$ (7)

It is obviously seen in the above defined functions, S(ρ)∈ [–2, 2] and H(ρ)∈ [0, 2].

Definition 7. Suppose that ρ = ((r_ρ, w_{r
_ρ}) , (k_ρ, w_{k
_ρ})) and σ = ((r_σ, w_{r
_σ}) , (k_σ, w_{k
_σ})) are two CIFNs, and the order relation between them is stated as:

If S(ρ)> S(σ) then ρ > σ

If S(ρ) = S(σ) then

If H(ρ)> H(σ) then ρ > σ

If H(ρ) = H(σ) thenρ ∼ σ, represents the same numbers.

Definition 8. Let us consider σ₁ = ((r₁, w_{r
₁}) , (k₁, w_{k
₁})) and σ₂ = ((r₂, w_{r
₂}) , (k₂, w_{k
₂})) are two CIFNs. With γ⩾ 1 the basic operational laws on these numbers are given as:

(1) $σ_{1} \oplus σ_{2} = ((1 - \prod_{j = 1}^{2} (1 - r_{j}), 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 1}^{2} k_{j}, 2 π (\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π})))$

(2) $σ_{1} \otimes σ_{2} = ((\prod_{j = 1}^{2} r_{j}, 2 π (\prod_{j = 1}^{2} \frac{w_{r_{j}}}{2 π})), (1 - \prod_{j = 1}^{2} (1 - k_{j}), 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{k_{j}}}{2 π}))))$

(3) $γ σ_{1} = . ((1 - (1 - r_{1})^{γ}, 2 π (1 - (1 - \frac{w_{r_{1}}}{2 π})^{γ})), (k_{1}^{γ}, 2 π (\frac{w_{k_{1}}}{2 π})^{γ}))$

(4) $σ_{1}^{γ} = . ((r_{1}^{γ}, 2 π (\frac{w_{r_{1}}}{2 π})^{γ}), (1 - (1 - k_{1})^{γ}, 2 π (1 - (1 - \frac{w_{k_{1}}}{2 π})^{γ})))$

Theorem 1. If σ₁ and σ₂ are CIFNs and also with γ⩾ 1, then σ₁ ⊕ σ₂, σ₁ ⊗ σ₂, γσ₁, $σ_{1}^{γ}$ are also CIFNs.

Proof. Given that σ₁ and σ₂ are CIFNs, so we can write as: σ₁ = ((r₁, w_{r
₁}) , (k₁, w_{k
₁})) and

σ₂ = ((r₂, w_{r
₂}) , (k₂, w_{k
₂})) such that r_j, k_j, r_j+ k_j ∈ [0, 1] and also w_{r
_j}, w_{k
_j}, w_{r
_j}+ w_{k
_j} ∈ [0, 1]

Let σ₃ = σ₁ ⊕ σ₂ = ((r₃, w_{r
₃}) , (k₃, w_{k
₃})), where $r_{3} = 1 - \prod_{j = 1}^{2} (1 - r_{j})$ , $w_{r_{3}} = 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{r_{j}}}{2 π}))$ , $k_{3} = \prod_{j = 1}^{2} k_{j}$ , $w_{k_{3}} = 2 π (\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π})$ .

Now as r₁, r₂∈ [0, 1] which implies that 1 - r_j∈ [0, 1] and therefore $1 - \prod_{j = 1}^{2} (1 - r_{j}) \in$ [0, 1].

Hence, r₃∈ [0, 1] and also k₁, k₂, ∈ [0, 1] which implies that $\prod_{j = 1}^{2} k_{j} \in$ [0, 1] therefore k₃∈ [0, 1]. Further r_j+ k_j ⩽1 for j = 1, 2, which implies that $\prod_{j = 1}^{2} k_{j} ⩽ \prod_{j = 1}^{2} (1 - r_{j})$ and

Therefore $r_{3} + k_{3} = 1 - \prod_{j = 1}^{2} (1 - r_{j}) + \prod_{j = 1}^{2} k_{j} ⩽ 1 - \prod_{j = 1}^{2} (1 - r_{j}) + \prod_{j = 1}^{2} (1 - r_{j}) = 1$ .

As r₃⩾ 0 and k₃⩾ 0 implies that r₃+k₃⩾ 0. Hence 0 ⩽r₃+ k₃ ⩽ 1. Similarly, we can attain that 0 ⩽w_{r
₃}, w_{k
₃}⩽ 1 such that w_{r
₃}+ w_{k
₃} ∈ [0, 1]. So, we acquire that σ₁ ⊕ σ₂ is a CIFN. Correspondingly, it can be showed that σ₁ ⊗ σ₂, γσ₁, $σ_{1}^{γ}$ are also CIFNs.

Theorem 2. Given σ₁, σ₂andσ₃ are three CIFNs, we have

σ₁ ⊕ σ₂ = σ₂ ⊕ σ₁

σ₁ ⊗ σ₂ = σ₂ ⊗ σ₁

(σ₁ ⊕ σ₂) ⊕ σ₃ = σ₁ ⊕ (σ₂ ⊕ σ₃)

(σ₁ ⊗ σ₂) ⊗ σ₃ = σ₁ ⊗ (σ₂ ⊗ σ₃)

Proof. In this, we proved only the first and third part, as the remaining parts are similar.

(1) Because, given that σ₁andσ₂ are CIFNs, so we can write as: σ₁ = ((r₁, w_{r
₁}) , (k₁, w_{k
₁})) and σ₂ = ((r₂, w_{r
₂}) , (k₂, w_{k
₂})), then we have $σ_{1} \oplus σ_{2} = ((1 - \prod_{j = 1}^{2} (1 - r_{j}), 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 1}^{2} k_{j}, 2 π (\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π}))) .$ $= ((r_{1} + r_{2} - r_{1} r_{2}, 2 π (\frac{w_{r_{1}}}{2 π} + \frac{w_{r_{2}}}{2 π} - \frac{w_{r_{1}}}{2 π} \frac{w_{r_{2}}}{2 π})), (k_{1} k_{2}, 2 π (\frac{w_{k_{1}}}{2 π} \frac{w_{k_{2}}}{2 π})))$ $= ((r_{2} + r_{1} - r_{2} r_{1}, 2 π (\frac{w_{r_{2}}}{2 π} + \frac{w_{r_{1}}}{2 π} - \frac{w_{r_{2}}}{2 π} \frac{w_{r_{1}}}{2 π})), (k_{2} k_{1}, 2 π (\frac{w_{k_{2}}}{2 π} \frac{w_{k_{1}}}{2 π})))$ $= σ_{2} \oplus σ_{1}$

(2) Since σ₁ = ((r₁, w_{r
₁}) , (k₁, w_{k
₁})), σ₂ = ((r₂, w_{r
₂}) , (k₂, w_{k
₂})) and σ₃ = ((r₃, w_{r
₃}) , (k₃, w_{k
₃})) are CIFNs, then we have (σ₁ ⊕ σ₂) ⊕ σ₃ $= ((1 - \prod_{j = 1}^{2} (1 - r_{j}), 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 1}^{2} k_{j}, 2 π (\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π}))) \oplus ((r_{3}, w_{r_{3}}), (k_{3}, w_{k_{3}}))$ $= ((1 - \prod_{j = 1}^{3} (1 - r_{j}), 2 π (1 - \prod_{j = 1}^{3} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 1}^{3} k_{j}, 2 π (\prod_{j = 1}^{3} \frac{w_{k_{j}}}{2 π})))$ $= ((r_{1}, w_{r_{1}}), (k_{1}, w_{k_{1}})) \oplus ((1 - \prod_{j = 2}^{3} (1 - r_{j}), 2 π (1 - \prod_{j = 2}^{3} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 2}^{3} k_{j}, 2 π (\prod_{j = 2}^{3} \frac{w_{k_{j}}}{2 π})))$ $= σ_{1} \oplus (σ_{2} \oplus σ_{3}) .$

Theorem 3. Given two CIFNs σ₁ = ((r₁, w_{r
₁}) , (k₁, w_{k
₁})) and σ₂ = ((r₂, w_{r
₂}) , (k₂, w_{k
₂})) with γ, γ₁, γ₂⩾ 1 then

γ (σ₁ ⊕ σ₂) = γσ₁ ⊕ γσ₂,

${(σ_{1} \otimes σ_{2})}^{γ} = σ_{1}^{γ} \otimes σ_{2}^{γ}$ ,

(γ₁ + γ₂) σ₁ = γ₁σ₁ ⊕ γ₂σ₁,

$σ_{1}^{γ_{1} + γ_{2}} = σ_{1}^{γ_{1}} \otimes σ_{2}^{γ_{2}}$ .

Proof. Here, we are proving only (1) and (3), while others are similar.

(1) As σ₁ and σ₂ are two CIFNs, then we have $γ (σ_{1} \oplus σ_{2}) = γ ((1 - \prod_{j = 1}^{2} (1 - r_{j}), 2 π (1 - \prod_{j = 1}^{2} (1 - \frac{w_{r_{j}}}{2 π}))), (\prod_{j = 1}^{2} k_{j}, 2 π (\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π})))$ $= ((1 - \prod_{j = 1}^{2} {(1 - r_{j})}^{γ}, 2 π (1 - \prod_{j = 1}^{2} {(1 - \frac{w_{r_{j}}}{2 π})}^{γ})), ({(\prod_{j = 1}^{2} k_{j})}^{γ}, 2 π {(\prod_{j = 1}^{2} \frac{w_{k_{j}}}{2 π})}^{γ}))$ $= ((1 - {(1 - r_{1})}^{γ}, 2 π (1 - {(1 - \frac{w_{r_{1}}}{2 π})}^{γ})), (k_{1}^{γ}, 2 π {(\frac{w_{k_{1}}}{2 π})}^{γ}))$ $\oplus ((1 - {(1 - r_{2})}^{γ}, 2 π (1 - {(1 - \frac{w_{r_{2}}}{2 π})}^{γ})), (k_{2}^{γ}, 2 π {(\frac{w_{k_{2}}}{2 π})}^{γ}))$ $= γ σ_{1} \oplus γ σ_{2}$

(3) As σ₁ is a CIFN, then we have $(γ_{1} + γ_{2}) σ_{1} = ((1 - {(1 - r_{1})}^{γ_{1} + γ_{2}}, 2 π (1 - {(1 - \frac{w_{r_{1}}}{2 π})}^{γ_{1} + γ_{2}})), (k_{1}^{γ_{1} + γ_{2}}, 2 π {(\frac{w_{k_{1}}}{2 π})}^{γ_{1} + γ_{2}}))$ $= ((1 - {(1 - r_{1})}^{γ_{1}}, 2 π (1 - {(1 - \frac{w_{r_{1}}}{2 π})}^{γ_{1}})), (k_{1}^{γ_{1}}, 2 π {(\frac{w_{k_{1}}}{2 π})}^{γ_{1}}))$ $\oplus ((1 - {(1 - r_{1})}^{γ_{2}}, 2 π (1 - {(1 - \frac{w_{r_{1}}}{2 π})}^{γ_{2}})), (k_{1}^{γ_{2}}, 2 π {(\frac{w_{k_{1}}}{2 π})}^{γ_{2}}))$ $= γ_{1} σ_{1} \oplus γ_{2} σ_{1}$

3.2 Maclaurin symmetric mean

The Maclaurin symmetric mean (MSM) is initiated basically by Maclaurin [19]. MSM is categorized due to the ability of capturing the relationship between the multi input data. The definition for the MSM is stated as follows:

Definition 9. [19] Suppose that b_r⩾ 1, be a family and k = 1, 2 ... m, r₁, r₂ ... ,r_k, where k is a parameter and takes its values from the collection {1, 2… , m }, then the mathematical form of MSM is defined as: ${MSM}^{(k)} (b_{1}, b_{2}, \dots, b_{m}) = {(\frac{\sum_{\begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix}} (\prod_{j = 1}^{k} b_{rj})}{C_{m}^{k}})}^{\frac{1}{k}}$ (8)

Where, $C_{m}^{k} = \frac{m!}{k! (m - k)!}$ , is the binomial coefficient.

The obvious properties of MSM are as follows:

MSM^(k) (0, 0, ... , 0) = 0

MSM^(k) (b, b, ... , b) = b

MSM^(k) (b₁, b₂, ... , b_m)⩽MSM^(k) (a₁, a₂, ... , a_m), if b_i⩽ a_i ∀ i

$min_{i} {b_{i}} ⩽ {MSM}^{(k)}$ (b₁, b₂, ... , b_m)⩽ $max_{i} {b_{i}}$

3.2 Dual Maclaurin symmetric mean

Qin and Liu [50] propounded the definition of dual MSM (DMSM) on the basis of MSM definition, defined as follows:

Definition 10. Suppose that b_r⩾ 1, be a collection and k = 1, 2 ... m, r₁, r₂ ... , r_k. $If {DMSM}^{(k)} (b_{1}, b_{2}, \dots, b_{m})$ $= \frac{1}{k} (\prod_{1 ⩽ r_{1} < \dots < r_{k} ⩽ m} {(\sum_{j = 1}^{k} b_{r_{j}})}^{\frac{1}{C_{m}^{k}}})$ (9)

then DMSM^(k) termed as dual Maclaurin symmetric mean

The characteristics of DMSM are similar as the MSM.

4 Complex intuitionistic fuzzy Maclaurin symmetric meanoperator for CIFNs

In this part, inspired by the concept of complex intuitionistic fuzzy set, we shall generalize the MSM operator to CIF environment to build up the CIFMSM operator and CIFDMSM operators. Furthermore, some of their fundamental properties are studied in detail.

Definition 11. Suppose Q_i = (r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) (i = 1, 2, …, m) is a family of CIFNs and k = 1, 2, ... , m, r₁, r₂, ... , r_k, takes its values from the collection {1, 2, … , m }. A mapping: ℵ→ ℵ is called CIFMSM operator and is defined by ${CIFMSM}^{(k)} (Q_{1}, Q_{2}, \dots, Q_{m}) =$ ${(\frac{\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{k} Q_{r_{i}})}{C_{m}^{k}})}^{\frac{1}{k}}$ (10)

Where, ℵ is the collection of CIFNs and $C_{m}^{k} = \frac{m!}{k! (m - k)!}$ , is the binomial coefficient.

Theorem 4. Suppose Q_i = (r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) (i = 1, 2, …, m) is a collection of CINFs, the synthesis result of {Q₁, Q₂, … , Q_m } by using CIFMSM operator is stated as: ${CIFMSM}^{(k)} (Q_{1}, Q_{2}, \dots, Q_{m}) =$ $| \begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}}, \\ 1 - \begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{r_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}, \end{matrix} \end{matrix} |$

Where Ω used for the convenience represents the subscript (1 ⩽ r₁ < … < r_k ⩽ m).

Proof. According to the basic algorithms of CIFMSMNs, we get $\otimes_{i = 1}^{k} Q_{r_{i}} = [(\otimes_{i = 1}^{k} r_{i} e^{i 2 π (\otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π})}), (1 - \otimes_{i = 1}^{k} (1 - k_{i}) e^{i 2 π (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π}))})]$ and $\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{k} Q_{r_{i}})$ $= [\begin{matrix} (1 - \prod_{Ω} (1 - (\otimes_{i = 1}^{k} r_{i}))) e^{i 2 π (1 - \prod_{Ω} (1 - (\otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π})))}, \\ \prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})) e^{i 2 π \prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π}))} \end{matrix}]$ Then $\frac{1}{C_{m}^{k}} (\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{k} Q_{r_{i}}))$ $= [\begin{matrix} (1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}}) e^{i 2 π (1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}, \\ {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})))}^{\frac{1}{C_{m}^{k}}} e^{i 2 π ({(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})} \end{matrix}]$ Hence ${(\frac{1}{C_{m}^{k}} (\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{k} Q_{r_{i}})))}^{\frac{1}{k}}$ $= [\begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}}, \\ 1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})} \end{matrix}]$

4.1 Properties of CIFMSM operator

Suppose Q_i = (r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) and $\overset{s}{Q_{i}} quo$ = $(\overset{s}{r_{i}} quo e^{i 2 π \overset{s}{w_{r_{i}}} quo}, \overset{s}{k_{i}} quo e^{i 2 π \overset{s}{w_{k_{i}}} quo})$ is a collections of CIFNs

where (i = 1, 2, ... , m), then the CIFMSM possesses the following properties.

(1) Idempotentency If Q₁=Q₂= ... =Q_m=Q=(re^i2πw_r, ke^i2πw_k), then

CIFMSM(Q₁, Q₂, …, Q_m)=Q.

(2) Monotonicity For Q_i and $\overset{s}{Q_{i}} quo (i = 1, 2, \dots, m)$ , if $Q_{i} ⩽ \overset{s}{Q_{i}} quo \forall$ i, then

CIFMSM (Q₁, Q₂, … , Q_m) ⩽

CIFMSM $(\overset{s}{Q_{1}} quo, \overset{s}{Q_{2}} quo, \dots, \overset{s}{Q_{m}} quo)$ .

(3) Boundedness Assume Q_i (i = 1, 2, … , m) be a setof CINFs, $\overset{˘}{Q}$ = $min_{i} {Q_{i}}$ and $\hat{Q}$ = $max_{i} {Q_{i}}$ , then, $min_{i} {Q_{i}} ⩽$ CIFMSM $(Q_{1}, Q_{2}, \dots, Q_{m}) ⩽ max_{i} {Q_{i}}$ .

Proof 1. CIFMSM^(k) (Q, Q, …, Q) $= [\begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}}, \\ 1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})} \end{matrix}]$ $= [\begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}}, \\ 1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k)))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})} \end{matrix}]$ $= [\begin{matrix} {(1 - {({(1 - r^{k})}^{C_{m}^{k}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {({(1 - {(\frac{w_{r}}{2 π})}^{k})}^{C_{m}^{k}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}}, \\ 1 - {(1 - {({(1 - {(1 - k)}^{k})}^{C_{m}^{k}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {({(1 - {(1 - \frac{w_{k}}{2 π})}^{k})}^{C_{m}^{k}})}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})} \end{matrix}]$ $= [\begin{matrix} {(1 - (1 - r^{k}))}^{\frac{1}{k}} e^{i 2 π {(1 - (1 - {(\frac{w_{r}}{2 π})}^{k}))}^{\frac{1}{k}}}, \\ {ke}^{i 2 π (\frac{w_{k}}{2 π})} \end{matrix}] = ({re}^{{iw}_{r}}, {ke}^{{iw}_{k}}) .$

Proof 2. As $Q_{i} ⩽ \overset{s}{Q_{i}} quo$ , then we have $r_{i} ⩽ \overset{s}{r_{i}} quo$ , $w_{r_{i}} ⩽ \overset{s}{w_{r_{i}}} quo$ and $k_{i} ⩾ \overset{s}{k_{i}} quo$ , $w_{k_{i}} ⩾ \overset{s}{w_{k_{i}}} quo$ for ∀ i (i = 1, 2, …, m), then CIFMSMNs. The real valued membership degrees is, $r_{i} ⩽ \overset{s}{r_{i}} quo$ , one has $\otimes_{i = 1}^{k} r_{i} ⩽ \otimes_{i = 1}^{k} \overset{s}{r_{i}} quo \Rightarrow \prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}) ⩾ \prod_{Ω} (1 - \otimes_{i = 1}^{k} \overset{s}{r_{i}} quo)$ $\Rightarrow 1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}} ⩽ 1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \overset{s}{r_{i}} quo))}^{\frac{1}{C_{m}^{k}}}$ $\Rightarrow {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} ⩽ {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \overset{s}{r_{i}} quo))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} .$

Similarly, we can get the imaginary valued membership degree ${(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} ⩽ {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{\overset{s}{w_{r_{i}}} quo}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} .$

Consequently, we attain $\begin{array}{l} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} \\ ⩽ {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k}^{‵} r_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{^{‵} w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} \end{array}$

In the same way, we can get the nonmember ship degree, $1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - k_{i})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{k_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})} ⩾$ $= 1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \overset{s}{k_{i}} quo)))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{\overset{s}{w_{k_{i}}} quo}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}$

Proof 3. Because of the above two proved properties of the presented CIFMSM operator, we get the following results:

For $Q_{i} ⩾ \overset{˘}{Q}$ = $min_{i} {Q_{i}}$ , one has

CIFMSM (Q₁, Q₂, … , Q_m) ⩾ CIFMSM $(\overset{˘}{Q}, \overset{˘}{Q}, \dots, \overset{˘}{Q})$ = $\overset{˘}{Q}$

For $Q_{i} ⩽ \hat{Q}$ = $max_{i} {Q_{i}}$ , one has

CIFMSM (Q₁, Q₂, … , Q_m) ⩽ CIFMSM $(\hat{Q}, \hat{Q}, \dots, \hat{Q})$ = $\hat{Q}$

Hence, $min_{i} {Q_{i}} ⩽$ CIFMSM $(Q_{1}, Q_{2}, \dots, Q_{m}) ⩽ max_{i} {Q_{i}}$ .

Moreover some special operator shall be formed by changing the values of k.

Case 1. When k = 1, the CIFMSM operator is reduced to CIF arithmetic averaging (CIFAA) operator, which is presented as follows: ${CIFMSM}^{(1)} (Q_{1}, Q_{2}, \dots, Q_{m}) = \frac{1}{m} (\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} ⩽ m \end{matrix} \end{matrix} Q_{r_{i}}) = \frac{1}{m} (\otimes_{r = 1}^{n} Q_{r}) (suppose r_{1} = r)$ $= [\begin{matrix} 1 - \otimes_{i = 1}^{m} {(1 - r_{i})}^{\frac{1}{m}} e^{i 2 π (1 - \otimes_{i = 1}^{m} {(1 - \frac{w_{r_{i}}}{2 π})}^{\frac{1}{m}})}, \\ \otimes_{i = 1}^{m} k_{i}^{\frac{1}{m}} e^{i 2 π (\otimes_{i = 1}^{m} {(\frac{w_{k_{i}}}{2 π})}^{\frac{1}{m}})} \end{matrix}] .$ (11)

Case 2. When k = 2, then, the CIFMSM is reduced to a particular operator CIF Bonferroni mean (CIFBM), presented as below: ${CIFMSM}^{(2)} (Q_{1}, Q_{2}, \dots, Q_{m}) = {(\frac{\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < r_{2} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{2} Q_{r_{i}})}{C_{m}^{2}})}^{\frac{1}{2}} = {(\frac{1}{n (n - 1)} \otimes_{\begin{matrix} r_{1}, r_{2} = 1 \\ r_{1} \neq r_{2} \end{matrix}}^{k} (Q_{r_{1}}^{1} \otimes Q_{r_{2}}^{1}))}^{\frac{1}{2}}$ (12) $= | \begin{matrix} {(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - r_{r_{1}} r_{r_{2}}))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}} e^{i 2 π ({(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - \frac{{wr}_{r 1} {wr}_{r 2}}{2 π 2 π}))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}}),} \\ 1 - \begin{matrix} {(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - (1 - k_{r_{1}}) (1 - k_{r_{2}})))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}} e^{i 2 π (1 - {(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - (1 - \frac{w_{{kr}_{1}}}{2 π}) (1 - \frac{w_{{kr}_{2}}}{2 π})))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}})} \end{matrix} \end{matrix} |$

= CPIBM⁽²⁾ (Q₁, Q₂, …, Q_m).

Case 3. When k = 3, then, the CIFMSM is reduced to a particular operator CIF generalized Bonferroni mean (CIFGBM) operator, presented as below: ${CIFMSM}^{(3)} (Q_{1}, Q_{2}, \dots, Q_{m}) = {(\frac{\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < r_{2} < r_{3} ⩽ m \end{matrix} \end{matrix} (\otimes_{i = 1}^{3} Q_{r_{i}})}{C_{m}^{3}})}^{\frac{1}{3}}$ (13) $= {(\frac{1}{n (n - 1) (n - 2)} \otimes_{\begin{matrix} r_{1}, r_{2}, r_{3} = 1 \\ r_{1} \neq r_{2} \neq r_{3} \end{matrix}}^{k} (Q_{r_{1}}^{1} \otimes Q_{r_{2}}^{1} \otimes Q_{r_{3}}^{1}))}^{\frac{1}{3}}$

Case 4. When k = m, then, the CIFMSM is reduced to a particular operator CIF geometric average mean (CIFGAM) operator, presented as below: ${CIFMSM}^{(m)} (Q_{1}, Q_{2}, \dots, Q_{m}) = {(\frac{\otimes_{i = 1}^{m} Q_{i}}{C_{m}^{m}})}^{\frac{1}{m}}$ (14) $= [\begin{matrix} {(\otimes_{i = 1}^{m} r_{i})}^{\frac{1}{m}} e^{i 2 π {(\otimes_{i = 1}^{m} \frac{w_{r_{i}}}{2 π})}^{\frac{1}{m}}}, \\ 1 - {(\otimes_{i = 1}^{m} (1 - k_{i}))}^{\frac{1}{m}} e^{i 2 π (1 - {(\otimes_{i = 1}^{m} (1 - \frac{w_{k_{i}}}{2 π}))}^{\frac{1}{m}})} \end{matrix}]$

4.2 Complex intuitionistic fuzzy dual Maclaurin symmetric mean

CIFDMSM is attained through the fusion of the CIF and the DMSM operator.

Definition 12. Suppose Q_i = (r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) (i = 1, 2 … , m) be a set of CIFNs and k = 1, 2 ... m, r₁, r₂, ... ,r_k, takes its values from the collection {1, 2, … , m }. A mapping: ℵ→ ℵ is called CIFDMSM operator as below: ${CIFDMSM}^{(k)} (Q_{1}, Q_{2}, \dots, Q_{m}) = \frac{1}{k} (\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} {(\otimes_{i = 1}^{k} Q_{r_{i}})}^{\frac{1}{C_{m}^{k}}}),$ (15)

Where, ℵ is the collection of CIFNs.

Theorem 5. Suppose Q_i=(r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) (i = 1, 2 … , m) be a set of CIFNs, then the result from the above definition (12) is also a CIFNs and is expressed as: ${CIFDMSM}^{(k)} (Q_{1}, Q_{2}, \dots, Q_{m})$ $= | 1 - \begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - r_{i})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π (1 - {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} (1 - \frac{w_{r_{i}}}{2 π})))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}})}, \\ \begin{matrix} {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} k_{i}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}} e^{i 2 π {(1 - {(\prod_{Ω} (1 - \otimes_{i = 1}^{k} \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{C_{m}^{k}}})}^{\frac{1}{k}}} \end{matrix} \end{matrix} |$ (16)

Where Ω used for the convenience represents the subscript (1 ⩽ r₁ < … < r_k ⩽ m)

The proof of this theorem omitted as it is easy and similar to the Theorem 4.

4.3 Properties of CIFDMSM operator

Suppose Q_i=(r_ie^{i2πw_{r
_i}}, k_ie^{i2πw_{k
_i}}) and $\overset{s}{Q_{i}} quo$ = $(\overset{s}{r_{i}} quo e^{i 2 π \overset{s}{w_{r_{i}}} quo}, \overset{s}{k_{i}} quo e^{i 2 π \overset{s}{w_{k_{i}}} quo})$ be two collections of CIFNs

Where (i = 1, 2, ... , m), then the CIFDMSM possesses the following properties.

(1) Idempotentency If Q₁=Q₂= ... =Q_m=Q=(re^i2πw_r, ke^i2πw_k), then

CIFDMSM (Q₁, Q₂, … Q_m)=Q.

(2) Monotonicity For Q_i and $\overset{s}{Q_{i}} quo (i = 1, 2, \dots, m)$ , if $Q_{i} ⩽ \overset{s}{Q_{i}} quo \forall$ i, then

CIFDMSM (Q₁, Q₂, … , Q_m) ⩾ CIFDMSM $(\overset{s}{Q_{1}} quo, \overset{s}{Q_{2}} quo, \dots, \overset{s}{Q_{m}} quo)$ .

(3) Boundedness Assume Q_i (i = 1, 2, …, m) be a setof CIFNs, $\overset{˘}{Q}$ = $min_{i} {Q_{i}}$ and

$\hat{Q}$ = $max_{i} {Q_{i}}$ , then, $min_{i} {Q_{i}} ⩽$ CIFMSM $(Q_{1}, Q_{2}, \dots Q_{m}) ⩽ max_{i} {Q_{i}}$

The proof of these properties is easy and similar to the properties of CIFMSM.

Furthermore, in the same way some special operator will be achieved by changing the values of k.

Case 1. When k = 1, the CIFDMSM operator shall be reduced to CIF arithmetic averaging (CIFAA) operator, and is presented as follows: ${CIFDMSM}^{(1)} (Q_{1}, Q_{2}, \dots, Q_{m}) = {(\otimes_{i = 1}^{m} Q_{i})}^{\frac{1}{m}}$ (17) $= [\begin{matrix} \otimes_{i = 1}^{m} r_{i}^{\frac{1}{m}} e^{i 2 π (\otimes_{i = 1}^{m} {(\frac{w_{r_{i}}}{2 π})}^{\frac{1}{m}})}, \\ 1 - \otimes_{i = 1}^{m} {(1 - k_{i})}^{\frac{1}{m}} e^{i 2 π (1 - \otimes_{i = 1}^{m} {(1 - \frac{w_{k_{i}}}{2 π})}^{\frac{1}{m}})} \end{matrix}] .$

Case 2. When k = 2, then, the CIFDMSM is reduced to a particular operator CIF geometric Bonferroni mean (CIFGBM), presented as below: ${CIFDMSM}^{(2)} (Q_{1}, Q_{2}, \dots, Q_{m}) = \frac{1}{k} (\begin{matrix} \otimes \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} {(\oplus_{i = 1}^{k} Q_{r_{i}})}^{\frac{1}{C_{m}^{k}}}) = \frac{1}{2} (\otimes_{\begin{matrix} r_{1}, r_{2} = 1 \\ r_{1} \neq r_{2} \end{matrix}}^{k} {(Q_{r_{1}} \oplus Q_{r_{2}})}^{\frac{2}{n (n - 1)}})$ (18) $= | 1 - \begin{matrix} {(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - (1 - r_{r_{1}}) (1 - r_{r_{2}})))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}} e^{i 2 π (1 - {(1 - {(\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - (1 - \frac{w_{{rr}_{1}}}{2 π}) (1 - \frac{w_{{rr}_{2}}}{2 π})))}^{\frac{2}{n (n - 1)}})}^{\frac{1}{2}})}, \\ {(1 - (\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - k_{r_{1}} k_{r_{2}})) \frac{2}{n (n - 1)})}^{\frac{1}{2}} e^{i 2 π ({(1 - (\otimes_{r_{1} \neq r_{2} r_{1}, r_{2} = 1}^{k} (1 - \frac{w_{{kr}_{1}}}{2 π} \frac{w_{{kr}_{2}}}{2 π})) \frac{2}{n (n - 1)})}^{\frac{1}{2}})} \end{matrix} |$

Case 3. When k = 3, Then, the CIFDMSM is reduced to a particular operator CIF generalized geometric Bonferroni mean (CIFGGBM) operator, and is presented as below: ${CIFDMSM}^{(3)} (Q_{1}, Q_{2}, \dots, Q_{m}) = \frac{1}{k} (\begin{matrix} \otimes \\ \begin{matrix} 1 ⩽ r_{1} < \dots < r_{k} ⩽ m \end{matrix} \end{matrix} {(\oplus_{i = 1}^{k} Q_{r_{i}})}^{\frac{1}{C_{m}^{k}}})$ (19) $= \frac{1}{3} (\otimes_{\begin{matrix} r_{1}, r_{2}, r_{3} = 1 \\ r_{1} \neq r_{2} \neq r_{3} \end{matrix}}^{k} {(Q_{r_{1}} \oplus Q_{r_{2}} \oplus Q_{r_{3}})}^{\frac{3}{n (n - 1) (n - 2)}})$

Case 4. When k = m, then, the CIFDMSM is reduced to a particular operator CIF geometric average mean (CIFGAM) operator, presented as below: ${CIFDMSM}^{(m)} (Q_{1}, Q_{2}, \dots, Q_{m}) = \frac{1}{m} (\begin{matrix} \oplus \\ \begin{matrix} 1 ⩽ r_{1} ⩽ m \end{matrix} \end{matrix} Q_{r_{i}}) = \frac{1}{m} (\oplus_{r = 1}^{m} Q_{r}) (set r_{1} = r)$ (20) $= [\begin{matrix} 1 - {(\otimes_{i = 1}^{m} (1 - r_{i}))}^{\frac{1}{m}} e^{i 2 π (1 - {(\otimes_{i = 1}^{m} (1 - \frac{w_{r_{i}}}{2 π}))}^{\frac{1}{m}})}, \\ \otimes_{i = 1}^{m} k_{i}^{\frac{1}{m}} e^{i 2 π {(\otimes_{i = 1}^{m} \frac{w_{k_{i}}}{2 π})}^{\frac{1}{m}}} \end{matrix}]$

5 MAGDM method using suggested operators

This section solves the conventional MAGDM problem and presents a new MAGDM algorithm, on the basis of the suggested CIFMSM operators to address the real decision making problems.

In this section, we will establish the CIFMSM and CIFDMSM operators to handle MAGDM issues by using the information of CIFNs. Assume G = {G₁, G₂, …, G_m} be a set of alternatives and Å={Å₁, Å ₂, ... , Å _m} be a collection of attributes. Let suppose E={E₁, E₂, ... , E_q} be the group of evaluators. The expert E_s (s = 1, 2, … , q) provides his/her judgment information for each alternative G i.e G_i (i = 1, 2, … , n) regarding the attribute Å_p (p = 1, 2, … , m) in terms of the CIFNs which is denoted by $Q_{ip}^{s}$ = $(r_{ip}^{s} e^{i 2 π w_{r_{ip}}^{s}}, k_{ip}^{s} e^{i 2 π w_{k_{ip}}^{s}})$ . The DM matrices are denoted by D^s= ${(Q_{ip}^{s})}_{n \times m}$ . So, as to achieve the most optimal alternative, we propound a new MAGDM approach to rank the alternatives using the proposed CIFMSM operator, which includes the following steps:

Step 1. Establish the CIF decision matrices through the transformation method proposed by the expert: $Q_{ip}^{s} = {\begin{matrix} (r_{ip}^{s} e^{i 2 π w_{r_{ip}}^{s}}, k_{ip}^{s} e^{i 2 π w_{k_{ip}}^{s}}), Å_{p} for benefit attribute, \\ (k_{ip}^{s} e^{i 2 π w_{k_{ip}}^{s}}, r_{ip}^{s} e^{i 2 π w_{r_{ip}}^{s}}), Å_{p} for cost attribute . \end{matrix}$ (21)

Step 2. Use the CIFMSM operator or the CIFDMSM to combine all of the single matrices

D^s= ${(Q_{ip}^{s})}_{n \times m} (s = 1, 2, \dots, q)$ into another single matrix Z=(Q_ip) _n×m: $Q_{ip} = CIFMSM (Q_{ip}^{1}, Q_{ip}^{2}, \dots, Q_{ip}^{q})$ (22) or $Q_{ip} = CIFDMSM (Q_{ip}^{1}, Q_{ip}^{2}, \dots, Q_{ip}^{q})$ (23)

Step 3. Use the CIFMSM or the CIFDMSM operator to integrate linguistic information

Q_ip (p = 1, 2, … , m) into the inclusive assessment value of the alternatives G_i (i = 1, 2, …, n): $Q_{i} = CIFMSM (Q_{ip}, Q_{ip}, \dots, Q_{ip})$ (24) or $Q_{i} = CIFDMSM (Q_{ip}, Q_{ip}, \dots, Q_{ip})$ (25)

Step 4. Calculate the score function S(Q_i) for each evaluation value G_i (i = 1, 2, …, n) on the basis of definition 6, if the different alternative’s score value becomes equal, we shall further compute the accuracy function H(Q_i) of them.

Step 5. Rank all of the alternatives

G_i (i = 1, 2, …, n) using definition 7, and then select the best optimum alternative.

Step 6. The end.

6 Illustrative example

In this part, a daily life example is presented to validate the effectiveness of the propounded method. Moreover, the sensitivity of the k is studied. A comparative analysis is conducted to show the distinction of the presented method.

Emergency management is an appropriate phenomenon for handling major accidents and disaster risks. It refers to the establishment of necessary response mechanisms and the adoption of emergency measures by the government and other public authorities for process of preventing, responding, handling and restoring emergency situation. It performs a sequence of important measures, use technology, science, management methods and planning, for the conformation of activities related to emergency programing, health public life, property safety, and to promote the healthy and harmonious development of society. In the few recent years, natural disasters have occurred frequently, which have caused a tremendous loss to the global economy and human lives. In order to efficiently optimize the damage, caused by the disasters and accidents, various emergency alternatives are articulated according to the type of accidents by the emergency management center, depending on the type of accidents. Moreover, experts are also invited for the evaluation of alternative emergency plans in different fields. Evaluation of this emergency alternative is an integral part of the emergency management program. Its base is the traditional decision making issue, which attracted the attention of many researchers and scholars. So, in this part, we will implement the presented approach to deal and evaluate the problem of selection of the desirable emergency alternative for the emergency management program. Four best alternatives are selected for the further assessment after a series of screening. The four alternatives {G₁, G₂, G₃, G₄} are assessed by the three evaluators {E₁, E₂, E₃}. After the discussion with experts, for useful modeling of the characteristics of the alternatives, four attributes are taken into consideration. The four attributes are Å₁, taken for preparation ability, Å₂, for the rescue ability, Å₃, for the restoring ability and Å₄, for the reaction capacity. According to the view point and knowledge level, the individual decision matrices are given as E₁, E₂ and E₃, stated in Tables 1 –3.

Table 1
The preferences input values given by expert E₁ are

Å₁ Å₂ Å₃ Å₄

G ₁ $(\begin{matrix} 0.30 e^{i 2 π (0.50)} \\ 0.20 e^{i 2 π (0.40)} \end{matrix})$ $(\begin{matrix} 0.25 e^{i 2 π (0.56)} \\ 0.35 e^{i 2 π (0.40)} \end{matrix})$ $(\begin{matrix} 0.20 e^{i 2 π (0.70)} \\ 0.45 e^{i 2 π (0.28)} \end{matrix})$ $(\begin{matrix} 0.25 e^{i 2 π (0.65)} \\ 0.15 e^{i 2 π (0.20)} \end{matrix})$

G ₂ $(\begin{matrix} 0.45 e^{i 2 π (0.55)} \\ 0.28 e^{i 2 π (0.42)} \end{matrix})$ $(\begin{matrix} 0.26 e^{i 2 π (0.46)} \\ 0.32 e^{i 2 π (0.36)} \end{matrix})$ $(\begin{matrix} 0.31 e^{i 2 π (0.31)} \\ 0.43 e^{i 2 π (0.48)} \end{matrix})$ $(\begin{matrix} 0.30 e^{i 2 π (0.58)} \\ 0.18 e^{i 2 π (0.32)} \end{matrix})$

G ₃ $(\begin{matrix} 0.40 e^{i 2 π (0.29)} \\ 0.28 e^{i 2 π (0.22)} \end{matrix})$ $(\begin{matrix} 0.29 e^{i 2 π (0.64)} \\ 0.35 e^{i 2 π (0.29)} \end{matrix})$ $(\begin{matrix} 0.35 e^{i 2 π (0.41)} \\ 0.51 e^{i 2 π (0.28)} \end{matrix})$ $(\begin{matrix} 0.37 e^{i 2 π (0.39)} \\ 0.19 e^{i 2 π (0.42)} \end{matrix})$

G ₄ $(\begin{matrix} 0.43 e^{i 2 π (0.65)} \\ 0.20 e^{i 2 π (0.12)} \end{matrix})$ $(\begin{matrix} 0.29 e^{i 2 π (0.56)} \\ 0.30 e^{i 2 π (0.29)} \end{matrix})$ $(\begin{matrix} 0.38 e^{i 2 π (0.61)} \\ 0.44 e^{i 2 π (0.33)} \end{matrix})$ $(\begin{matrix} 0.33 e^{i 2 π (0.48)} \\ 0.10 e^{i 2 π (0.39)} \end{matrix})$

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.30 e^{i 2 π (0.50)} \\ 0.20 e^{i 2 π (0.40)} \end{matrix})$	$(\begin{matrix} 0.25 e^{i 2 π (0.56)} \\ 0.35 e^{i 2 π (0.40)} \end{matrix})$	$(\begin{matrix} 0.20 e^{i 2 π (0.70)} \\ 0.45 e^{i 2 π (0.28)} \end{matrix})$	$(\begin{matrix} 0.25 e^{i 2 π (0.65)} \\ 0.15 e^{i 2 π (0.20)} \end{matrix})$
G ₂	$(\begin{matrix} 0.45 e^{i 2 π (0.55)} \\ 0.28 e^{i 2 π (0.42)} \end{matrix})$	$(\begin{matrix} 0.26 e^{i 2 π (0.46)} \\ 0.32 e^{i 2 π (0.36)} \end{matrix})$	$(\begin{matrix} 0.31 e^{i 2 π (0.31)} \\ 0.43 e^{i 2 π (0.48)} \end{matrix})$	$(\begin{matrix} 0.30 e^{i 2 π (0.58)} \\ 0.18 e^{i 2 π (0.32)} \end{matrix})$
G ₃	$(\begin{matrix} 0.40 e^{i 2 π (0.29)} \\ 0.28 e^{i 2 π (0.22)} \end{matrix})$	$(\begin{matrix} 0.29 e^{i 2 π (0.64)} \\ 0.35 e^{i 2 π (0.29)} \end{matrix})$	$(\begin{matrix} 0.35 e^{i 2 π (0.41)} \\ 0.51 e^{i 2 π (0.28)} \end{matrix})$	$(\begin{matrix} 0.37 e^{i 2 π (0.39)} \\ 0.19 e^{i 2 π (0.42)} \end{matrix})$
G ₄	$(\begin{matrix} 0.43 e^{i 2 π (0.65)} \\ 0.20 e^{i 2 π (0.12)} \end{matrix})$	$(\begin{matrix} 0.29 e^{i 2 π (0.56)} \\ 0.30 e^{i 2 π (0.29)} \end{matrix})$	$(\begin{matrix} 0.38 e^{i 2 π (0.61)} \\ 0.44 e^{i 2 π (0.33)} \end{matrix})$	$(\begin{matrix} 0.33 e^{i 2 π (0.48)} \\ 0.10 e^{i 2 π (0.39)} \end{matrix})$

Table 2

The preferences input values provided by expert E₂ are

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.20 e^{i 2 π (0.80)} \\ 0.26 e^{i 2 π (0.17)} \end{matrix})$	$(\begin{matrix} 0.15 e^{i 2 π (0.55)} \\ 0.42 e^{i 2 π (0.35)} \end{matrix})$	$(\begin{matrix} 0.20 e^{i 2 π (0.37)} \\ 0.69 e^{i 2 π (0.48)} \end{matrix})$	$(\begin{matrix} 0.22 e^{i 2 π (0.56)} \\ 0.24 e^{i 2 π (0.39)} \end{matrix})$
G ₂	$(\begin{matrix} 0.37 e^{i 2 π (0.34)} \\ 0.39 e^{i 2 π (0.28)} \end{matrix})$	$(\begin{matrix} 0.26 e^{i 2 π (0.38)} \\ 0.24 e^{i 2 π (0.19)} \end{matrix})$	$(\begin{matrix} 0.43 e^{i 2 π (0.19)} \\ 0.32 e^{i 2 π (0.47)} \end{matrix})$	$(\begin{matrix} 0.20 e^{i 2 π (0.47)} \\ 0.17 e^{i 2 π (0.28)} \end{matrix})$
G ₃	$(\begin{matrix} 0.10 e^{i 2 π (0.27)} \\ 0.37 e^{i 2 π (0.38)} \end{matrix})$	$(\begin{matrix} 0.36 e^{i 2 π (0.28)} \\ 0.46 e^{i 2 π (0.55)} \end{matrix})$	$(\begin{matrix} 0.41 e^{i 2 π (0.46)} \\ 0.37 e^{i 2 π (0.29)} \end{matrix})$	$(\begin{matrix} 0.28 e^{i 2 π (0.59)} \\ 0.36 e^{i 2 π (0.37)} \end{matrix})$
G ₄	$(\begin{matrix} 0.39 e^{i 2 π (0.64)} \\ 0.18 e^{i 2 π (0.25)} \end{matrix})$	$(\begin{matrix} 0.17 e^{i 2 π (0.60)} \\ 0.27 e^{i 2 π (0.34)} \end{matrix})$	$(\begin{matrix} 0.18 e^{i 2 π (0.49)} \\ 0.55 e^{i 2 π (0.38)} \end{matrix})$	$(\begin{matrix} 0.24 e^{i 2 π (0.45)} \\ 0.57 e^{i 2 π (0.41)} \end{matrix})$

Table 3

The preferences input values provided by expert E₃ are

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.61 e^{i 2 π (0.40)} \\ 0.33 e^{i 2 π (0.38)} \end{matrix})$	$(\begin{matrix} 0.37 e^{i 2 π (0.65)} \\ 0.24 e^{i 2 π (0.22)} \end{matrix})$	$(\begin{matrix} 0.29 e^{i 2 π (0.57)} \\ 0.59 e^{i 2 π (0.32)} \end{matrix})$	$(\begin{matrix} 0.22 e^{i 2 π (0.42)} \\ 0.21 e^{i 2 π (0.39)} \end{matrix})$
G ₂	$(\begin{matrix} 0.47 e^{i 2 π (0.36)} \\ 0.36 e^{i 2 π (0.27)} \end{matrix})$	$(\begin{matrix} 0.32 e^{i 2 π (0.38)} \\ 0.23 e^{i 2 π (0.40)} \end{matrix})$	$(\begin{matrix} 0.17 e^{i 2 π (0.58)} \\ 0.30 e^{i 2 π (0.44)} \end{matrix})$	$(\begin{matrix} 0.29 e^{i 2 π (0.49)} \\ 0.15 e^{i 2 π (0.46)} \end{matrix})$
G ₃	$(\begin{matrix} 0.37 e^{i 2 π (0.36)} \\ 0.30 e^{i 2 π (0.39)} \end{matrix})$	$(\begin{matrix} 0.28 e^{i 2 π (0.29)} \\ 0.45 e^{i 2 π (0.45)} \end{matrix})$	$(\begin{matrix} 0.21 e^{i 2 π (0.57)} \\ 0.43 e^{i 2 π (0.31)} \end{matrix})$	$(\begin{matrix} 0.19 e^{i 2 π (0.11)} \\ 0.27 e^{i 2 π (0.74)} \end{matrix})$
G ₄	$(\begin{matrix} 0.22 e^{i 2 π (0.64)} \\ 0.10 e^{i 2 π (0.25)} \end{matrix})$	$(\begin{matrix} 0.36 e^{i 2 π (0.19)} \\ 0.29 e^{i 2 π (0.56)} \end{matrix})$	$(\begin{matrix} 0.29 e^{i 2 π (0.53)} \\ 0.56 e^{i 2 π (0.83)} \end{matrix})$	$(\begin{matrix} 0.38 e^{i 2 π (0.56)} \\ 0.57 e^{i 2 π (0.26)} \end{matrix})$

6.1 DM process

Step 1. Since all of the attributes are of the same kind, therefore the DM does not require any standardization.

Step 2. Use the CIFMSM operator or the CIFDMSM to combine all of the single matrices into a collective one matrix, which are shown in Tables 4 and 5 (takek = 2).

Table 4
Integrate the DM by using CIFMSM operator

Å₁ Å₂ Å₃ Å₄

G ₁ $(\begin{matrix} 0.3508 e^{i 2 π (0.5581)} \\ 0.2631 e^{i 2 π (0.3180)} \end{matrix})$ $(\begin{matrix} 0.2492 e^{i 2 π (0.5862)} \\ 0.3372 e^{i 2 π (0.3239)} \end{matrix})$ $(\begin{matrix} 0.2280 e^{i 2 π (0.5425)} \\ 0.5812 e^{i 2 π (0.3605)} \end{matrix})$ $(\begin{matrix} 0.2298 e^{i 2 π (0.5411)} \\ 0.1997 e^{i 2 π (0.3277)} \end{matrix})$

G ₂ $(\begin{matrix} 0.4292 e^{i 2 π (0.4121)} \\ 0.3436 e^{i 2 π (0.3233)} \end{matrix})$ $(\begin{matrix} 0.2793 e^{i 2 π (0.4060)} \\ 0.2632 e^{i 2 π (0.3175)} \end{matrix})$ $(\begin{matrix} 0.2950 e^{i 2 π (0.3431)} \\ 0.2511 e^{i 2 π (0.4634)} \end{matrix})$ $(\begin{matrix} 0.2615 e^{i 2 π (0.5125)} \\ 0.1666 e^{i 2 π (0.3537)} \end{matrix})$

G ₃ $(\begin{matrix} 0.2766 e^{i 2 π (0.3056)} \\ 0.3167 e^{i 2 π (0.3307)} \end{matrix})$ $(\begin{matrix} 0.3090 e^{i 2 π (0.3872)} \\ 0.4206 e^{i 2 π (0.4328)} \end{matrix})$ $(\begin{matrix} 0.3187 e^{i 2 π (0.4783)} \\ 0.4374 e^{i 2 π (0.2933)} \end{matrix})$ $(\begin{matrix} 0.2757 e^{i 2 π (0.3418)} \\ 0.2731 e^{i 2 π (0.5170)} \end{matrix})$

G ₄ $(\begin{matrix} 0.3418 e^{i 2 π (0.6434)} \\ 0.1593 e^{i 2 π (0.2061)} \end{matrix})$ $(\begin{matrix} 0.2683 e^{i 2 π (0.4392)} \\ 0.2867 e^{i 2 π (0.3981)} \end{matrix})$ $(\begin{matrix} 0.2778 e^{i 2 π (0.5426)} \\ 0.5178 e^{i 2 π (0.5245)} \end{matrix})$ $(\begin{matrix} 0.3143 e^{i 2 π (0.4959)} \\ 0.4291 e^{i 2 π (0.3540)} \end{matrix})$

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.3508 e^{i 2 π (0.5581)} \\ 0.2631 e^{i 2 π (0.3180)} \end{matrix})$	$(\begin{matrix} 0.2492 e^{i 2 π (0.5862)} \\ 0.3372 e^{i 2 π (0.3239)} \end{matrix})$	$(\begin{matrix} 0.2280 e^{i 2 π (0.5425)} \\ 0.5812 e^{i 2 π (0.3605)} \end{matrix})$	$(\begin{matrix} 0.2298 e^{i 2 π (0.5411)} \\ 0.1997 e^{i 2 π (0.3277)} \end{matrix})$
G ₂	$(\begin{matrix} 0.4292 e^{i 2 π (0.4121)} \\ 0.3436 e^{i 2 π (0.3233)} \end{matrix})$	$(\begin{matrix} 0.2793 e^{i 2 π (0.4060)} \\ 0.2632 e^{i 2 π (0.3175)} \end{matrix})$	$(\begin{matrix} 0.2950 e^{i 2 π (0.3431)} \\ 0.2511 e^{i 2 π (0.4634)} \end{matrix})$	$(\begin{matrix} 0.2615 e^{i 2 π (0.5125)} \\ 0.1666 e^{i 2 π (0.3537)} \end{matrix})$
G ₃	$(\begin{matrix} 0.2766 e^{i 2 π (0.3056)} \\ 0.3167 e^{i 2 π (0.3307)} \end{matrix})$	$(\begin{matrix} 0.3090 e^{i 2 π (0.3872)} \\ 0.4206 e^{i 2 π (0.4328)} \end{matrix})$	$(\begin{matrix} 0.3187 e^{i 2 π (0.4783)} \\ 0.4374 e^{i 2 π (0.2933)} \end{matrix})$	$(\begin{matrix} 0.2757 e^{i 2 π (0.3418)} \\ 0.2731 e^{i 2 π (0.5170)} \end{matrix})$
G ₄	$(\begin{matrix} 0.3418 e^{i 2 π (0.6434)} \\ 0.1593 e^{i 2 π (0.2061)} \end{matrix})$	$(\begin{matrix} 0.2683 e^{i 2 π (0.4392)} \\ 0.2867 e^{i 2 π (0.3981)} \end{matrix})$	$(\begin{matrix} 0.2778 e^{i 2 π (0.5426)} \\ 0.5178 e^{i 2 π (0.5245)} \end{matrix})$	$(\begin{matrix} 0.3143 e^{i 2 π (0.4959)} \\ 0.4291 e^{i 2 π (0.3540)} \end{matrix})$

Table 5

Integrate the DM by using CIFDMSM operator

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.3706 e^{i 2 π (0.5775)} \\ 0.2608 e^{i 2 π (0.3095)} \end{matrix})$	$(\begin{matrix} 0.2558 e^{i 2 π (0.5876)} \\ 0.3332 e^{i 2 π (0.3195)} \end{matrix})$	$(\begin{matrix} 0.2296 e^{i 2 π (0.5546)} \\ 0.5748 e^{i 2 π (0.3554)} \end{matrix})$	$(\begin{matrix} 0.2299 e^{i 2 π (0.5471)} \\ 0.1982 e^{i 2 π (0.3216)} \end{matrix})$
G ₂	$(\begin{matrix} 0.4304 e^{i 2 π (0.4179)} \\ 0.3420 e^{i 2 π (0.3200)} \end{matrix})$	$(\begin{matrix} 0.2799 e^{i 2 π (0.4069)} \\ 0.2619 e^{i 2 π (0.3111)} \end{matrix})$	$(\begin{matrix} 0.3036 e^{i 2 π (0.3607)} \\ 0.3478 e^{i 2 π (0.4632)} \end{matrix})$	$(\begin{matrix} 0.2633 e^{i 2 π (0.5141)} \\ 0.1664 e^{i 2 π (0.3496)} \end{matrix})$
G ₃	$(\begin{matrix} 0.2945 e^{i 2 π (0.3067)} \\ 0.3156 e^{i 2 π (0.3262)} \end{matrix})$	$(\begin{matrix} 0.3100 e^{i 2 π (0.4050)} \\ 0.4189 e^{i 2 π (0.4251)} \end{matrix})$	$(\begin{matrix} 0.3240 e^{i 2 π (0.4812)} \\ 0.4352 e^{i 2 π (0.2932)} \end{matrix})$	$(\begin{matrix} 0.2799 e^{i 2 π (0.3688)} \\ 0.2692 e^{i 2 π (0.4995)} \end{matrix})$
G ₄	$(\begin{matrix} 0.3479 e^{i 2 π (0.6433)} \\ 0.1572 e^{i 2 π (0.2025)} \end{matrix})$	$(\begin{matrix} 0.2732 e^{i 2 π (0.4618)} \\ 0.2865 e^{i 2 π (0.3890)} \end{matrix})$	$(\begin{matrix} 0.2833 e^{i 2 π (0.5442)} \\ 0.5158 e^{i 2 π (0.4923)} \end{matrix})$	$(\begin{matrix} 0.3169 e^{i 2 π (0.4973)} \\ 0.3955 e^{i 2 π (0.3507)} \end{matrix})$

Step 3. Use the CIFMSM or the CIFDMSM operator to integrate linguistic information in Table 4 or Table 5 and get the collection of the alternatives G_i (i = 1, 2, … , n), which are presented in Table 6 (takek = 2)

Table 6

The collective assessment information attained through using CIFMSM and CIFDMSM operator

Alternative	CIFMSM operator	CIFDMSM operator
G ₁	$(\begin{matrix} 0.2631 e^{i 2 π (0.5570)} \\ 0.3409 e^{i 2 π (0.3325)} \end{matrix})$	$(\begin{matrix} 0.2703 e^{i 2 π (0.5668)} \\ 0.3336 e^{i 2 π (0.3263)} \end{matrix})$
G ₂	$(\begin{matrix} 0.3143 e^{i 2 π (0.4175)} \\ 0.2548 e^{i 2 π (0.3641)} \end{matrix})$	$(\begin{matrix} 0.3184 e^{i 2 π (0.4248)} \\ 0.2769 e^{i 2 π (0.3597)} \end{matrix})$
G ₃	$(\begin{matrix} 0.2948 e^{i 2 π (0.3770)} \\ 0.3614 e^{i 2 π (0.3930)} \end{matrix})$	$(\begin{matrix} 0.3021 e^{i 2 π (0.3902)} \\ 0.3582 e^{i 2 π (0.3840)} \end{matrix})$
G ₄	$(\begin{matrix} 0.3002 e^{i 2 π (0.5298)} \\ 0.3455 e^{i 2 π (0.3696)} \end{matrix})$	$(\begin{matrix} 0.3052 e^{i 2 π (0.5373)} \\ 0.3326 e^{i 2 π (0.3552)} \end{matrix})$

Step 4. Calculate the score function S(Q_i) for each evaluation value G_i (i = 1, 2, … , n) on the basis of definition 6, which are displayed in Table 7

Table 7

The score index of alternatives G_i (i = 1, 2, … , n) are as follows

Operator	G ₁	G ₂	G ₃	G ₄
CIFMSM operator	S(Q₁) = 0.1467	S(Q₂) = 0.1129	S(Q₃) = 0.0826	S(Q₄) = 0.1149
CIFDMSM operator	S(Q₁) = 0.1772	S(Q₂) = 0.1066	S(Q₃) = 0.0499	S(Q₄) = 0.1547

Steps 5. Sort all of the alternatives G_i (i = 1, 2, …, n) using definition 7, and the best optimal alternatives are obtained, which are listed below:

6.2 Sensitivity analysis

In the above stated calculations, the parameter has an essential role in DM procedure and directly affects the last decision results. So in this section, we will conduct an analysis for the parameter k. For ease, we have used CIFMSM operator to solve the stated example, and to get the analysis for this section. To show the sensitivity of parameter k in the final decision result, we again used the presented.

CIFMSM operator with different values of parameter k to incorporate with the above mentioned real example and achieve the sorting results, shown in Table 9 (if k = 2).

Table 8
The ranking of all alternatives are

Operator Order relation of G_i (i = 1, … , 4) Most optimal

CIFMSM operator G₁ > G₄ > G₂ > G₃ G ₁

CIFDMSM operator G₁ > G₄ > G₂ > G₃ G ₁

Operator	Order relation of G_i (i = 1, … , 4)	Most optimal
CIFMSM operator	G₁ > G₄ > G₂ > G₃	G ₁
CIFDMSM operator	G₁ > G₄ > G₂ > G₃	G ₁

Table 9

The values of the score index and ranking of alternatives are based on the different values of k are

Parameter value	S(G₁)	S(G₂)	S(G₃)	S(G₄)	Ranking relation	Optimal selection
K=1	–1.2861	–1.2881	–1.5155	–1.362	G₁ > G₂ > G₄ > G₃	G ₁
K =2	0.8752	0.8474	0.8368	0.8720	G₁ > G₄ > G₂ > G₃	G ₁
K =3	1.6992	1.6446	1.6263	1.716	G₄ > G₁ > G₂ > G₃	G ₄

Table 9 shows that if the decision makers take the different values of parameter k, the calculation results of the alternatives will be slightly changed. The different values of k show the relationship between the various attributes in the DM procedure. For example, when we have taken k = 3 in CIFMSM operator, the ranking of alternative is G₄ > G₁ > G₂ > G₃ which is different from others. Because the CIFMSM operator will change into CIFGBM operator, if k = 3 is taken then, the relationship between the related attributes are unsuccessful in dealing with the decision making problems. When the decision making problem needs to be solved, the connection between the input data in the assessment procedure, and to aggregate the assessment information, experts can take k = 1 or k = 2 in CIFMSM operator. Decision makers can choose the desired parameter values according to their preferred attitude and can achieve a satisfactory decision outcome.

6.3 Generalization analysis

After the comparison analysis, numerical case is used to explain the standardization of the proposed technique. Table 10 shows the assessment information, proposed by the experts regarding the complex intuitionistic linguistic numbers, and then the cases are managed by the approach presented in this essay, and the results are shown in Table 11.

Table 10
The DM matrix from example 2

Å₁ Å₂ Å₃ Å₄

G ₁ $(\begin{matrix} 0.70 e^{i 2 π (0.80)} \\ 0.36 e^{i 2 π (0.95)} \end{matrix})$ $(\begin{matrix} 0.55 e^{i 2 π (0.55)} \\ 0.62 e^{i 2 π (0.35)} \end{matrix})$ $(\begin{matrix} 0.89 e^{i 2 π (0.37)} \\ 0.69 e^{i 2 π (0.48)} \end{matrix})$ $(\begin{matrix} 0.78 e^{i 2 π (0.56)} \\ 0.84 e^{i 2 π (0.39)} \end{matrix})$

G ₂ $(\begin{matrix} 0.67 e^{i 2 π (0.34)} \\ 0.69 e^{i 2 π (0.28)} \end{matrix})$ $(\begin{matrix} 0.56 e^{i 2 π (0.38)} \\ 0.59 e^{i 2 π (0.85)} \end{matrix})$ $(\begin{matrix} 0.69 e^{i 2 π (0.59)} \\ 0.78 e^{i 2 π (0.47)} \end{matrix})$ $(\begin{matrix} 0.87 e^{i 2 π (0.47)} \\ 0.97 e^{i 2 π (0.73)} \end{matrix})$

G ₃ $(\begin{matrix} 0.90 e^{i 2 π (0.27)} \\ 0.96 e^{i 2 π (0.67)} \end{matrix})$ $(\begin{matrix} 0.68 e^{i 2 π (0.28)} \\ 0.85 e^{i 2 π (0.79)} \end{matrix})$ $(\begin{matrix} 0.65 e^{i 2 π (0.46)} \\ 0.76 e^{i 2 π (0.48)} \end{matrix})$ $(\begin{matrix} 0.81 e^{i 2 π (0.59)} \\ 0.77 e^{i 2 π (0.37)} \end{matrix})$

G ₄ $(\begin{matrix} 0.67 e^{i 2 π (0.64)} \\ 0.78 e^{i 2 π (0.25)} \end{matrix})$ $(\begin{matrix} 0.57 e^{i 2 π (0.60)} \\ 0.48 e^{i 2 π (0.88)} \end{matrix})$ $(\begin{matrix} 0.33 e^{i 2 π (0.49)} \\ 0.66 e^{i 2 π (0.38)} \end{matrix})$ $(\begin{matrix} 0.24 e^{i 2 π (0.45)} \\ 0.87 e^{i 2 π (0.81)} \end{matrix})$

	Å₁	Å₂	Å₃	Å₄
G ₁	$(\begin{matrix} 0.70 e^{i 2 π (0.80)} \\ 0.36 e^{i 2 π (0.95)} \end{matrix})$	$(\begin{matrix} 0.55 e^{i 2 π (0.55)} \\ 0.62 e^{i 2 π (0.35)} \end{matrix})$	$(\begin{matrix} 0.89 e^{i 2 π (0.37)} \\ 0.69 e^{i 2 π (0.48)} \end{matrix})$	$(\begin{matrix} 0.78 e^{i 2 π (0.56)} \\ 0.84 e^{i 2 π (0.39)} \end{matrix})$
G ₂	$(\begin{matrix} 0.67 e^{i 2 π (0.34)} \\ 0.69 e^{i 2 π (0.28)} \end{matrix})$	$(\begin{matrix} 0.56 e^{i 2 π (0.38)} \\ 0.59 e^{i 2 π (0.85)} \end{matrix})$	$(\begin{matrix} 0.69 e^{i 2 π (0.59)} \\ 0.78 e^{i 2 π (0.47)} \end{matrix})$	$(\begin{matrix} 0.87 e^{i 2 π (0.47)} \\ 0.97 e^{i 2 π (0.73)} \end{matrix})$
G ₃	$(\begin{matrix} 0.90 e^{i 2 π (0.27)} \\ 0.96 e^{i 2 π (0.67)} \end{matrix})$	$(\begin{matrix} 0.68 e^{i 2 π (0.28)} \\ 0.85 e^{i 2 π (0.79)} \end{matrix})$	$(\begin{matrix} 0.65 e^{i 2 π (0.46)} \\ 0.76 e^{i 2 π (0.48)} \end{matrix})$	$(\begin{matrix} 0.81 e^{i 2 π (0.59)} \\ 0.77 e^{i 2 π (0.37)} \end{matrix})$
G ₄	$(\begin{matrix} 0.67 e^{i 2 π (0.64)} \\ 0.78 e^{i 2 π (0.25)} \end{matrix})$	$(\begin{matrix} 0.57 e^{i 2 π (0.60)} \\ 0.48 e^{i 2 π (0.88)} \end{matrix})$	$(\begin{matrix} 0.33 e^{i 2 π (0.49)} \\ 0.66 e^{i 2 π (0.38)} \end{matrix})$	$(\begin{matrix} 0.24 e^{i 2 π (0.45)} \\ 0.87 e^{i 2 π (0.81)} \end{matrix})$

Table 11

The value of the score function and ranking of alternatives of example 2 are

Approaches	Value of score function	Sorting
CIFWA operator propounded in [20]	unable to calculate	unable to calculate
CIFBM operator [21] (p = q = 1)	unable to calculate	unable to calculate
CPFS in [54]	unable to calculate	unable to calculate
Cq -ROF2TLMSM operator propounded	S(G₁)=0.8924, S(G₂)=0.7290	G₁ > G₂ > G₃ > G₄
in [55] (k = 2 andq = 3)	S(G₃)=0.7036, S(G₄)=0.6228
CIFMSM operator propounded in this	S(G₁)=0.1223, S(G₂)=-0.2182	G₁ > G₂ > G₃ > G₄
manuscript (k = 2)	S(G₃)=-0.2674, S(G₄)=-0.3051

Table 11 shows that previous methods such as CIFBM presented by Garg [21], CIFWA propounded by Garg [20] and distance measure [54] operators are unable to handle the example 2, whereas the method presented here can be effectively resolve it. This shows that the method suggested here is more useful than the approaches described previously, as CPFS and CIF are the specific (cases) of our approach.

6.4 Further contrastive analyzing

Now in this section after the generalization analysis, a detail contrastive analysis will be carried out between the propounded approach and the previous works. The dominancy of designed method is described as follows:

(1) Compare the approach proposed by Garg and Rani [20] known as CIFWA operator. The CIFWA operator is a basic aggregation operator for the combination of complex intuitionistic fuzzy information. The CIFWA proposes that the attributes (considered) in daily life issues are non-related. It also shows the relevance of attributes, because of that the decision becomes undefined and vague. The CIFMSM operator can affectively fulfill the above mentioned defect and also takes the relationship of the attributes into consideration. Moreover, the CIFMSM also shows the trend of the order relation of alternatives by the help of adjustable parameters, in addition it also shows the DM’s personal favorites. Over all, the CIFMSM operators are more effective and universal for the solution of decision analysis issues.

(2) Compare the presented approach with the CIFBM operator suggested by Garg and Rani [21]. Although the CIFBM operator is used for the aggregation of the complex intuitionistic fuzzy information but it only considers the relationship between two attributes. The CIFMSM operator used in the manuscript not only reduces the computational difficulty in the aggregation process of the information but it also capture the relationship between the attributes. The CIFMSM solves the problem of decision making in a qualitative manner and has the ability to resolve several issue which are not solved by CIFBM operator. So, the CIFMSM is more general and gives a rational result in decision making problems.

(3) Liu et al. [29] propounded an operator known as Cq-ROFLHM. The HM operator catches the relationship between the attributes but gives an irrational decision results. The CIFMSM operator fulfills the defect and gives a rational result along with the correlation of the attributes. Due to the presence of two parameter values in the HM operator, it makes the computational process more complex and becomes difficult for the DMs to select a satisfied parameter value. The MSM has a single parameter value, so, it becomes easy for the DMs to select an optimum value according to the desire and hence proves helpful and flexible in the addressing of DM issues.

By the above mentioned detailed analysis, we showed the marked characteristics of our propounded method. These characteristics are shown in Table 12. From the Table 12, we conclude that CIPFS and CIFs are the special cases of CIFMSM. This shows that the CIFMSM will be more universal for the solution of DM problem. Following are the main properties of the CIFMSM operator. (i) The information loss is reduced with the outcome of a ration result and relationship between the multiple input argument is also taken in consideration, while the use of CIFMSM. (ii) The CIFMSM increases the flexibility of decision making process by the use of diverse parameters. (iii) By changing the parameter values, many different operators are propounded, which shows that the CIFMSM is more suitable to solve the DM issues.

Table 12
Characteristic comparison with existing approaches

Approaches Ability to integrate data Capture the relationship between any two attributes Capture the relationship amongst multiple attributes Ability to work on 2 dimensional data Ability to convey data with complex numbers suppleness of technique for decision making

CIFPA Method [17] ✓ ✓ × ✓ ✓ ×

GCIFBM Method [20] ✓ × × ✓ ✓ ×

The method proposedin [21] ✓ ✓ × ✓ ✓ ✓

The method proposed in [54] × × × ✓ ✓ ×

The method proposed in [31] ✓ × × × × ×

Cq -ROHLM method [29] ✓ ✓ × ✓ ✓ ✓

The proposed method in [53] ✓ ✓ × × × ✓

The presented method ✓ ✓ ✓ ✓ ✓ ✓

Approaches	Ability to integrate data	Capture the relationship between any two attributes	Capture the relationship amongst multiple attributes	Ability to work on 2 dimensional data	Ability to convey data with complex numbers	suppleness of technique for decision making
CIFPA Method [17]	✓	✓	×	✓	✓	×
GCIFBM Method [20]	✓	×	×	✓	✓	×
The method proposedin [21]	✓	✓	×	✓	✓	✓
The method proposed in [54]	×	×	×	✓	✓	×
The method proposed in [31]	✓	×	×	×	×	×
Cq -ROHLM method [29]	✓	✓	×	✓	✓	✓
The proposed method in [53]	✓	✓	×	×	×	✓
The presented method	✓	✓	✓	✓	✓	✓

5 Conclusion

In this study, to deal with MAGDM issues, we used MSM operators, which are based on the CIFNs. The values of the attributes are in the form of CIFNs, which is the generalized form of IFNs. Previously, some aggregation operators were defined in the environment of IFNs, whereas, the range of consistent membership and non-membership grading is a subset of R (real numbers). Here, by utilizing the CIFSs, we protracted the ranges of the MG and NMG from real numbers to complex numbers within the unit disc. So, environment models are better for the solution of DM problems, because they can solve the problem in two dimensions as well as presents the information for periodic issues. First we have proposed the CIFMSM operator and then CIFDMSM operator.

Afterwards, the operational procedures are discussed in detail. Moreover, under the CIFS environment, the best alternative is found for the decision making approach. The most conspicuous property of this operator is that, they can capture the relationship among the multi input data. By changing the parameter value, a MAGDM process is easier to solve in the CIF environment. In addition, by changing the parameter value, these operators can be changed to other operators. This is the simplification of some previous propounded aggregation operators. Furthermore, the CIFN handles the information more efficiently in a quantitative manner. By the study, we decide that the presented approach can be used effectively, where the information is presented in two dimensions. The present operators under the IFS environment are considered to be a specific case of the propounded measure. The different parameters cause various rankings, so, the main challenge is to find the best appropriatevalue for the parameter of the proposed algorithm. Finally, a numerical example is illustrated to demonstrate the decision making steps and to validate its efficiency by comparing the result with previously existing research.

In the future study, it will be useful to implement the propounded approach for real issues, like evaluations about environment and resources in distinct fields [56, 57]. At the same time, we will also study the theories about complex fuzzy set like fuzzy logic, decision analysis, and basic operational and other. The spherical fuzzy set [58, 59] is also proposed as a universal picture fuzzy in order to solve fuzziness and uncertainty. So the study of the spherical fuzzy set becomes an important point in the following phase.

Footnotes

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 19-SCI-1-01-0041.

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Complex intuitionistic fuzzy Maclaurin symmetric mean operators and its application to emergency program selection

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 Operational laws of CIFNs

Table 8 The ranking of all alternatives are Operator Order relation of G i (i = 1, … , 4) Most optimal CIFMSM operator G1 > G4 > G2 > G3 G 1 CIFDMSM operator G1 > G4 > G2 > G3 G 1

Footnotes

Acknowledgments

References

Table 8
The ranking of all alternatives are

Operator Order relation of G_i (i = 1, … , 4) Most optimal

CIFMSM operator G₁ > G₄ > G₂ > G₃ G ₁

CIFDMSM operator G₁ > G₄ > G₂ > G₃ G ₁