Distance measures on circular interval-valued intuitionistic fuzzy sets

Abstract

The region encircled by the circle of radius $u$ around an interval-valued intuitionistic fuzzy set is known as the circular interval-valued intuitionistic fuzzy set. Since intervals are used to represent them, interval-valued intuitionistic fuzzy sets can deal with ambiguity and uncertainty in a variety of ways, however, the area encircling them is not taken into account. As the area around it is taken into consideration, a circular interval-valued intuitionistic fuzzy set is therefore more able to handle uncertainty than an interval-valued intuitionistic fuzzy set. Effective decision-making can result in progress, happiness, and prosperity, making it crucial for both professional and personal success. All of the fuzzy extension's distance measures have applications in a variety of fields, including pattern recognition, clustering, medical diagnosis, and more. As a result, distance measures for circular interval-valued intuitionistic fuzzy sets are presented, and the proposed distance measures are used to solve a group decision-making challenge that is selected from the literature. TOPSIS is the method used for tackling the decision-making problem. The same problem is assessed using the existing distance measures in order to further validate the suggested distance measures of circular interval-valued intuitionistic fuzzy sets, and the outcomes are contrasted. In addition, to verify the robustness of the proposed distance measures on circular interval-valued intuitionistic fuzzy environment, sensitivity analysis is done using the operators $\otimes_{m i n}$ and $\otimes_{m a x}$ .

Keywords

Circular interval-valued intuitionistic fuzzy set distance measure multi-criteria group decision making TOPSIS

1 Introduction

The fuzzy set (FS) theory was introduced by Zadeh in 1965 to address vague and imprecise information.¹ A membership value between 0 and 1 is assigned to each element in an FS. FSs have been used in many domains, such as clustering, medical diagnosis, pattern recognition, etc. Gorzalczancy examined interval-valued fuzzy sets (IVFSs) in 1987 because FS's approach to uncertainty would be better if its membership values are intervals.² In 1986, Atanassov developed the concept of intuitionistic fuzzy sets (IFSs) since it is impossible to identify the non-membership value of an element in a FS.³ The membership and non-membership values of each element in an IFS range from 0 to 1, and their sum is less than or equal to 1. Although IFSs perform better than FSs in uncertain conditions, they may perform even better in some circumstances if their values are expressed in terms of intervals. Therefore, interval-valued intuitionistic fuzzy set (IVIFS) which was introduced by Atanassov and Gargov in 1989,⁴ uses closed intervals to express the membership and non-membership values. Circular IFS (C-IFS), which Atanassov introduced in 2020 as an extension of IFS,⁵ represents each element as a circle encircling IFS. There are several researchers who are actively studying C-IFSs. Alimohammadlou et al. used C-IFSs in the manufacturing industry.⁶ C-IFSs were used by Alsattar et al. for developing IoT monitoring devices.⁷ Using the Tomada de Decisão Interativa Multicritério (TODIM) technique, Ashraf et al. discussed the application of C-IFSs.⁸ Higher order C-IFSs were employed in the forecasting process by Ashraf et al.⁹ Present worth analysis was discussed on IVIFS and C-IFS by Boltürk and Kahraman.¹⁰ Bozyiğit and Ünver used parametric C-IFSs in the multi-criteria decision-making (MCDM) process.¹¹ Çakir and Demircioğlu used Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) technique in MCDM problem.¹² The DM problem using C-IFSs are further discussed by Çakır and Taş,¹³ Çaloğlu Büyükselçuk and Sarı.¹⁴ Evaluation Based on Distance from Average Solution (EDAS) method was applied in MCDM using C-IFSs by Garg et al.¹⁵ C-IFS in evaluation of human capital was employed by Imanov and Aliyev.¹⁶ Kahraman and Otay extend Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method using C-IFSs.¹⁷ Divergence measure for C-IFSs was found by Khan et al.¹⁸ C-IF Analytic Hierarchy Process (AHP) and VIKOR method were employed by Otay and Kahraman.¹⁹ ELimination Et Choix Traduisant la REalité (ELECTRE) III method in C-IFSs was employed by Yusoff et al.²⁰ Circular IVIFSs (C-IVIFSs), introduced by Gajalaxmi et al. depict each element as a circle encircling an IVIFS.²¹ Gajalaxmi et al. introduced and used aggregation operators, score, and accuracy functions for C-IVIFSs in DM.²²

Since distance measures are very effective at comparing two objects based on their uneven content, they are vital in applications where ambiguity of information is a critical component, such as MCDM, pattern recognition, and clustering. For the past few decades, researchers have been actively working on distance measures of FS extensions. FS's distance measures are described by Xuecheng.²³ Ashraf et al. developed distance measures for IFSs based on difference sequence.²⁴ Distances measures on intuitionistic hesitant FS were introduced by Chen et al.²⁵ For IVIFSs, Dugenci created a distance measure.²⁶ Park et al. briefly examined distance measures for IVIFSs.²⁷ Furthermore, additional distance measures for IVIFSs were introduced by Qin et al. and used in DM.²⁸ For C-IFSs, Chen created distance measures.²⁹ Xu and Wen discussed the use of distance measures for C-IFSs in DM.³⁰

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is one of the most popular MCDM techniques. The principle of TOPSIS is that the best alternative is the one that has the shortest distance from the positive-ideal solution and the farthest distance from the negative-ideal solution. Sun et al. briefly extended the TOPSIS approach in Picture Fuzzy Sets (PFSs),³¹ whereas Sindhu et al. extended the TOPSIS method in bipolar PFSs and employed it to MCDM problems.³² Joshi et al. selected a renewable energy source using the TOPSIS approach in IFS. Using TOPSIS,³³ Kahraman et al. employed IFSs with ordered pairings in DM.³⁴ The TOPSIS method for single and IV hybrid enthalpy FSs was developed by Olgun et al.³⁵ The TOPSIS approach for IVIFSs in the MCDM problem was developed by Qin et al.²⁸ Alkan and Kahraman extend TOPSIS in C-IFS to select hospitals in pandemic situation.³⁶ In parametric C-IFS, Bozyiğit and Ünver expand TOPSIS.¹¹ Using TOPSIS and C-IFSs, Büyükselçuk assessed the Industrial IoT service.³⁷ Chen used the TOPSIS approach to apply C-IFSs in MCDM.²⁹ In order to solve the supplier selection problem, Kahraman and Alkan offered the C-IF TOPSIS approach.³⁸

The distance measure is a useful tool for TOPSIS method, which is one of the most effective approaches to apply in MCGDM situations. This work focuses on distance measures for C-IVIFSs and their application in the TOPSIS decision-making technique.

The objectives of this study are

To develop distance measures for C-IVIFSs

To apply the distance measures in MCGDM problem

To employ TOPSIS method in solving the MCGDM problem

To compare the result with existing results to validate the proposed technique

The outline of this study: Section 2 gives the preliminaries in which the definitions from literature are given. In Section 3, distance measures for C-IVIFSs are proposed and mathematically validated. The proposed distance measures are Normalized Hamming, Normalized Euclidean, Hausdorff-Normalized Hamming and Hausdorff-Normalized Euclidean distances. In Section 4, MCGDM problem is discussed and TOPSIS method is employed to solve the problem and the results obtained are compared with the results of existing distance measures. Sensitivity analysis is also done in this section to validate the proposed measure. Section 5 is the conclusion part.

2 Preliminaries

Definitions which are defined in the literature and used in this study are presented in this section.

Definition 21 ⁴

Let $T$ be a fixed non empty set. The set $P = {⟨ t, [ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)], [Ω_{P}^{-} (t), Ω_{P}^{+} (t)] ⟩ | t \in T}$ is an IVIFS over $T$ , where $[ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)] \subset [0, 1]$ and $[Ω_{P}^{-} (t), Ω_{P}^{+} (t)] \subset [0, 1]$ are respectively the membership and non-membership closed intervals over $T$ such that $0 \leq ϕ_{P}^{+} (t) + Ω_{P}^{+} (t) \leq 1$ .

Definition 2.2:
Let $T = {t_{1}, t_{2}, \dots, t_{n}} and P, Q$ be IVIFSs on $T$ where
$\begin{aligned} P & = {⟨ t_{i}, [ϕ_{P}^{-} (t_{i}), ϕ_{P}^{+} (t_{i})], [Ω_{P}^{-} (t_{i}), Ω_{P}^{+} (t_{i})] ⟩ | t_{i} \in T} & \end{aligned}$

$\begin{aligned} Q & = {⟨ t_{i}, [ϕ_{Q}^{-} (t_{i}), ϕ_{Q}^{+} (t_{i})], [Ω_{Q}^{-} (t_{i}), Ω_{Q}^{+} (t_{i})] ⟩ | t_{i} \in T}, then \end{aligned}$
(i)
The Hamming distance²⁷
$\begin{aligned} d_{h} (P, Q) = \frac{1}{4} \sum_{i = 1}^{n} [| ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) |] \end{aligned}$

(ii)
The normalized Hamming distance²⁷
$\begin{aligned} d_{n h} (P, Q) = \frac{1}{4 n} \sum_{i = 1}^{n} [| ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) |] \end{aligned}$

(iii)
The Euclidean distance²⁷
$\begin{aligned} d_{e} (P, Q) = {\frac{1}{4} \sum_{i = 1}^{n} [{(ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}))}^{2} + {(ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}))}^{2} + {(Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}))}^{2} + {(Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}))}^{2}]}^{\frac{1}{2}} \end{aligned}$

(iv)
The normalized Euclidean distance²⁷
$\begin{aligned} d_{n e} (P, Q) = {\frac{1}{4 n} \sum_{i = 1}^{n} [{(ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}))}^{2} + {(ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}))}^{2} + {(Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}))}^{2} + {(Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}))}^{2}]}^{\frac{1}{2}} \end{aligned}$

(v)
Hausdorff-Hamming Distance²⁷
$\begin{aligned} d_{H H} (P, Q) = \frac{1}{2} \sum_{i = 1}^{n} [\begin{array}{l} max {| ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) |, | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) |} \\ + max {| Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) |, | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) |} \end{array}] \end{aligned}$

(vi)
Hausdorff-Euclidean Distance²⁷
$\begin{aligned} d_{H E} (P, Q) = \sqrt{\frac{1}{2} \sum_{i = 1}^{n} [\begin{array}{l} {(max {| ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) |, | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) |})}^{2} \\ + {(max {| Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) |, | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) |})}^{2} \end{array}]} \end{aligned}$

(vii)
Distance measure based on the triangular divergence²⁸
$\begin{aligned} d_{I v} (P, Q) = \sqrt{\frac{1}{4 n} \sum_{i = 1}^{n} [\begin{array}{l} \frac{{(ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}))}^{2}}{ϕ_{P}^{-} (t_{i}) + ϕ_{Q}^{-} (t_{i})} + \frac{{(ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}))}^{2}}{ϕ_{P}^{+} (t_{i}) + ϕ_{Q}^{+} (t_{i})} \\ \frac{{(Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}))}^{2}}{Ω_{P}^{-} (t_{i}) + Ω_{Q}^{-} (t_{i})} + \frac{{(Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}))}^{2}}{Ω_{P}^{+} (t_{i}) + Ω_{Q}^{+} (t_{i})} \end{array}]} \end{aligned}$

Definition 23 ²¹

Let X be a non-empty set. The set $P = {⟨ t, [ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)], [Ω_{P}^{-} (t), Ω_{P}^{+} (t)], u ⟩ | t \in T}$ is called a circular interval-valued intuitionistic fuzzy set where $[ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)]$ , $[Ω_{P}^{-} (t), Ω_{P}^{+} (t)]$ are respectively the membership and non-membership intervals with $0 \leq ϕ_{P}^{+} (t) + Ω_{P}^{+} (t) \leq 1$ and $u \in [0, 1]$ is the radius of the circle around each element $t \in T$ .

\begin{aligned} Centre, (ϕ_{P} (t), Ω_{P} (t)) & = (\frac{ϕ_{P}^{-} (t) + ϕ_{P}^{+} (t)}{2}, \frac{Ω_{P}^{-} (t) + Ω_{P}^{+} (t)}{2}) \end{aligned}

\begin{aligned} Radius, u & = \sqrt{{(ϕ_{P}^{-} (t) - ϕ_{P} (t))}^{2} + {(Ω_{P}^{-} (t) - Ω_{P} (t))}^{2}} \end{aligned}

\begin{aligned} = \sqrt{{(ϕ_{P}^{+} (t) - ϕ_{P} (t))}^{2} + {(Ω_{P}^{+} (t) - Ω_{P} (t))}^{2}} \end{aligned}

The C-IVIFS,

P = {⟨ t, [ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)], [Ω_{P}^{-} (t), Ω_{P}^{+} (t)], u ⟩ | t \in T}

can be written as

P = ⟨ [ϕ_{P}^{-}, ϕ_{P}^{+}], [Ω_{P}^{-}, Ω_{P}^{+}], u ⟩

Definition 24 ²²

Let $P_{i} = ⟨ [ϕ_{P_{i}}^{-}, ϕ_{P_{i}}^{+}], [Ω_{P_{i}}^{-}, Ω_{P_{i}}^{+}], u_{i} ⟩, (i = 1, \dots, n)$ be a collection of C-IVIFSs. The aggregation operators are

\begin{aligned} W_{P_{q}} (P_{1}, \dots, P_{n}) & = ⟨ \begin{array}{ll} [1 - \prod_{i = 1}^{n} {(1 - ϕ_{P_{i}}^{-})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - ϕ_{P_{i}}^{+})}^{w_{i}}], \\ [\prod_{i = 1}^{n} Ω {_{P_{i}}^{-}}^{w_{i}}, \prod_{i = 1}^{n} Ω {_{P_{i}}^{+}}^{w_{i}}], \prod_{i = 1}^{n} {u_{i}}^{w_{i}} \end{array} ⟩ \\ W_{P_{p}} (P_{1}, \dots, P_{n}) & = ⟨ \begin{array}{l} [1 - \prod_{i = 1}^{n} {(1 - ϕ_{P_{i}}^{-})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - ϕ_{P_{i}}^{+})}^{w_{i}}], \\ [\prod_{i = 1}^{n} Ω {_{P_{i}}^{-}}^{w_{i}}, \prod_{i = 1}^{n} Ω {_{P_{i}}^{+}}^{w_{i}}], 1 - \prod_{i = 1}^{n} {(1 - u_{i})}^{w_{i}} \end{array} ⟩ \\ W_{G_{q}} (P_{1}, \dots, P_{n}) & = ⟨ \begin{array}{ll} [\prod_{i = 1}^{n} {(ϕ_{P_{i}}^{-})}^{w_{i}}, \prod_{i = 1}^{n} {(ϕ_{P_{i}}^{+})}^{w_{i}}], \\ [1 - \prod_{i = 1}^{n} {(1 - Ω_{P_{i}}^{-})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - Ω_{P_{i}}^{+})}^{w_{i}}], \prod_{i = 1}^{n} {u_{i}}^{w_{i}} \end{array} ⟩ \\ W_{G_{p}} (P_{1}, \dots, P_{n}) & = ⟨ \begin{array}{ll} [\prod_{i = 1}^{n} {(ϕ_{P_{i}}^{-})}^{w_{i}}, \prod_{i = 1}^{n} {(ϕ_{P_{i}}^{+})}^{w_{i}}], \\ [1 - \prod_{i = 1}^{n} {(1 - Ω_{P_{i}}^{-})}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - Ω_{P_{i}}^{+})}^{w_{i}}], 1 - \prod_{i = 1}^{n} {(1 - u_{i})}^{w_{i}} \end{array} ⟩ \end{aligned}

3 Distance measures on C-IVIFSs

Definition 3.1:
Let $T$ be a non-empty set. A mapping $d : C - I V I F S (T) \times C - I V I F S (T) \to [0, 1]$ is said to be a distance measure $d (P, Q)$ between two C-IVIFSs $P$ and $Q$ if it satisfies the following: I.
$0 \leq d (P, Q) \leq 1$
II.
$d (P, Q) = 0 \Leftrightarrow P = Q$
III.
$d (P, Q) = d (Q, P)$
IV.
if $P \subseteq Q \subseteq C$ then $d (P, Q) \leq d (P, C)$ , $d (Q, C) \leq d (P, C)$

where C is also a C-IVIFS in X.
Definition 3.2:
In $T = {t_{1}, t_{2}, \dots, t_{n}}$ ,

let $P = {⟨ t_{i}, [ϕ_{P}^{-} (t_{i}), ϕ_{P}^{+} (t_{i})], [Ω_{P}^{-} (t_{i}), Ω_{P}^{+} (t_{i})], u (t_{i}) ⟩ | t_{i} \in T}$ and

$Q = {⟨ t_{i}, [ϕ_{Q}^{-} (t_{i}), ϕ_{Q}^{+} (t_{i})], [Ω_{Q}^{-} (t_{i}), Ω_{Q}^{+} (t_{i})], v (t_{i}) ⟩ | t_{i} \in T}$ be two C-IVIFSs. The various distance measures between $P$ and $Q$ are defined below : (i)
Normalized Hamming Distance
$\begin{aligned} d_{H}^{'} (P, Q) = \frac{1}{5 n} \sum_{i = 1}^{n} {\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}} \end{aligned}$

(ii)
Normalized Euclidean Distance
$\begin{aligned} d_{E}^{'} (P, Q) = {\frac{1}{5 n} \sum_{i = 1}^{n} [\begin{array}{l} {(ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}))}^{2} + {(ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}))}^{2} \\ + {(Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}))}^{2} + {(Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}))}^{2} + {(u (t_{i}) - v (t_{i}))}^{2} \end{array}]}^{\frac{1}{2}} \end{aligned}$

(iii)
Hausdorff-Normalized Hamming Distance
$\begin{aligned} d_{H H}^{'} (P, Q) = \frac{1}{5 n} \sum_{i = 1}^{n} {max [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) |, | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (_{i}) |, \\ | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) |, | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) |, | u (t_{i}) - v (t_{i}) | \end{array}]} \end{aligned}$

(iv)
Hausdorff-Normalized Euclidean Distance
$\begin{aligned} d_{H E}^{'} (P, Q) = {\frac{1}{5 n} \sum_{i = 1}^{n} [max (\begin{array}{l} {(ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}))}^{2}, {(ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}))}^{2}, \\ {(Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}))}^{2}, {(Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}))}^{2}, \\ {(u (t_{i}) - v (t_{i}))}^{2} \end{array})]}^{\frac{1}{2}} \end{aligned}$

Proposition 3.3:
Normalized Hamming distance $d_{H}^{'}$ is a distance measure.
Proof:
Let $P$ , $Q$ and $C$ be C-IVIFSs in $T = {t_{1}, t_{2}, \dots, t_{n}}$ where
$\begin{aligned} P & = {⟨ t_{i}, [ϕ_{P}^{-} (t_{i}), ϕ_{P}^{+} (t_{i})], [Ω_{P}^{-} (t_{i}), Ω_{P}^{+} (t_{i})], u (t_{i}) ⟩ | t_{i} \in T}, \end{aligned}$

$\begin{aligned} Q & = {⟨ t_{i}, [ϕ_{Q}^{-} (t_{i}), ϕ_{Q}^{+} (t_{i})], [Ω_{Q}^{-} (t_{i}), Ω_{Q}^{+} (t_{i})], v (t_{i}) ⟩ | t_{i} \in T} and \end{aligned}$

$\begin{aligned} C & = {⟨ t_{i}, [ϕ_{C}^{-} (t_{i}), ϕ_{C}^{+} (t_{i})], [Ω_{C}^{-} (t_{i}), Ω_{C}^{+} (t_{i})], \; w (t_{i}) ⟩ | t_{i} \in T} \end{aligned}$

To prove $d_{H}^{'}$ is a distance measure, it should satisfy the following conditions: (a)
$0 \leq d_{H}^{'} (P, Q) \leq 1$
(b)
$d_{H}^{'} (P, Q) = 0 \Leftrightarrow P = Q$
(c)
$d_{H}^{'} (P, Q) = d_{H}^{'} (Q, P)$
(d)
If $P \subseteq Q \subseteq C$ then $d_{H}^{'} (P, Q) \leq d_{H}^{'} (P, C), d_{H}^{'} (Q, C) \leq d_{H}^{'} (P, C)$

We have,
$\begin{aligned} d_{H}^{'} (P, Q) = \frac{1}{5 n} \sum_{i = 1}^{n} {\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}} \end{aligned}$

a) We know that,
$\begin{aligned} 0 & \leq | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | \leq 1 \end{aligned}$

$\begin{aligned} 0 & \leq | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | \leq 1 \end{aligned}$

$\begin{aligned} 0 & \leq | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \leq 1 \end{aligned}$

$\begin{aligned} 0 & \leq | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | \leq 1 \end{aligned}$

$\begin{aligned} 0 & \leq | u (t_{i}) - v (t_{i}) | \leq 1 \end{aligned}$

On adding all these above equations,
$\begin{aligned} 0 & \leq [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}] \leq 5 \end{aligned}$

$\begin{aligned} 0 & \leq \sum_{i = 1}^{n} [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}] \leq 5 n \end{aligned}$

$\begin{aligned} 0 & \leq \frac{1}{5 n} \sum_{i = 1}^{n} [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}] \leq 1 \end{aligned}$

$\begin{aligned} 0 & \leq d_{H}^{'} (P, Q) \leq 1 \end{aligned}$

Hence (a) is proved

b) Now, $d_{H}^{'} (P, Q) = 0$
$\begin{aligned} \Leftrightarrow \frac{1}{5 n} \sum_{i = 1}^{n} [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \\ + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \end{array}] = 0 \end{aligned}$

$\begin{aligned} \Leftrightarrow | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | = 0, | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | = 0, | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | = 0, \\ | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | = 0 & | u (t_{i}) - v (t_{i}) | = 0, \forall i = 1 t o n \end{aligned}$

$\begin{aligned} \Leftrightarrow ϕ_{P}^{-} (t_{i}) = ϕ_{Q}^{-} (t_{i}), ϕ_{P}^{+} (t_{i}) = ϕ_{Q}^{+} (t_{i}), Ω_{P}^{-} (t_{i}) = Ω_{Q}^{-} (t_{i}), Ω_{P}^{+} (t_{i}) = Ω_{Q}^{+} (t_{i}) & \\ u (t_{i}) = v (t_{i}) \forall i = 1 t o n \end{aligned}$

$\begin{aligned} \Leftrightarrow P = Q \end{aligned}$

Hence (b) is proved

c) We have,
$\begin{aligned} d_{H}^{'} (P, Q) & = \frac{1}{5 n} \sum_{i = 1}^{n} [\begin{array}{l} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | \\ + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | \\ + | u (t_{i}) - v (t_{i}) | \end{array}] \\ = \frac{1}{5 n} \sum_{i = 1}^{n} [\begin{array}{l} | ϕ_{Q}^{-} (t_{i}) - ϕ_{P}^{-} (t_{i}) | + | ϕ_{Q}^{+} (t_{i}) - ϕ_{P}^{+} (t_{i}) | + | Ω_{Q}^{-} (t_{i}) - Ω_{P}^{-} (t_{i}) | \\ + | Ω_{Q}^{+} (t_{i}) - Ω_{P}^{+} (t_{i}) | + | v (t_{i}) - u (t_{i}) | \end{array}] \end{aligned}$

$\begin{aligned} = d_{H}^{'} (Q, P) \end{aligned}$

Hence (c) is proved

d) Let $P \subseteq Q \subseteq C$ . Then
$\begin{aligned} 0 & \leq ϕ_{P}^{-} (t_{i}) \leq ϕ_{Q}^{-} (t_{i}) \leq ϕ_{C}^{-} (t_{i}) \leq 1 \end{aligned}$
(1)

$\begin{aligned} 0 & \leq ϕ_{P}^{+} (t_{i}) \leq ϕ_{Q}^{+} (t_{i}) \leq ϕ_{C}^{+} (t_{i}) \leq 1 \end{aligned}$
(2)

$\begin{aligned} 0 & \leq Ω_{C}^{-} (t_{i}) \leq Ω_{Q}^{-} (t_{i}) \leq Ω_{P}^{-} (t_{i}) \leq 1 \end{aligned}$
(3)

$\begin{aligned} 0 & \leq Ω_{C}^{+} (t_{i}) \leq Ω_{Q}^{+} (t_{i}) \leq Ω_{P}^{+} (t_{i}) \leq 1 \end{aligned}$
(4)

$\begin{aligned} 0 & \leq u (t_{i}) \leq v (t_{i}) \leq w (t_{i}) \leq 1 \end{aligned}$
(5)

Now,
$\begin{aligned} (1) & \Rightarrow | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | \leq | ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | \end{aligned}$

$\begin{aligned} (2) & \Rightarrow | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | \leq | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | \end{aligned}$

$\begin{aligned} (3) & \Rightarrow | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | \leq | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | \end{aligned}$

$\begin{aligned} (4) & \Rightarrow | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | \leq | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | \end{aligned}$

$\begin{aligned} (5) & \Rightarrow | u (t_{i}) - v (t_{i}) | \leq | u (t_{i}) - w (t_{i}) | \end{aligned}$

On adding all these, we get,
$\begin{aligned} | ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) | \\ \leq | ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | u (t_{i}) - w (t_{i}) | \end{aligned}$

$\begin{aligned} \begin{array}{l} \Rightarrow \frac{1}{5 n} \sum_{i = 1}^{n} [| ϕ_{P}^{-} (t_{i}) - ϕ_{Q}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{Q}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{Q}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{Q}^{+} (t_{i}) | + | u (t_{i}) - v (t_{i}) |] \\ \leq \frac{1}{5 n} \sum_{i = 1}^{n} [| ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | u (t_{i}) - w (t_{i}) |] \\ \Rightarrow d_{H}^{'} (P, Q) \leq d_{H}^{'} (P, C) \end{array} \end{aligned}$

Now,
$\begin{aligned} (1) & \Rightarrow | ϕ_{Q}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | \leq | ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | \end{aligned}$

$\begin{aligned} (2) & \Rightarrow | ϕ_{Q}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | \leq | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | \end{aligned}$

$\begin{aligned} (3) & \Rightarrow | Ω_{Q}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | \leq | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | \end{aligned}$

$\begin{aligned} (4) & \Rightarrow | Ω_{Q}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | \leq | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | \end{aligned}$

$\begin{aligned} (5) & \Rightarrow | v (t_{i}) - w (t_{i}) | \leq | u (t_{i}) - w (t_{i}) | \end{aligned}$

On adding all these, we get,
$\begin{aligned} \begin{array}{l} | ϕ_{Q}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{Q}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{Q}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{Q}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | v (t_{i}) - w (t_{i}) | \\ \leq | ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | u (t_{i}) - w (t_{i}) | \end{array} \end{aligned}$

$\begin{aligned} \begin{array}{l} \frac{1}{5 n} \sum_{i = 1}^{n} [| ϕ_{Q}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{Q}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{Q}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{Q}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | v (t_{i}) - w (t_{i}) |] \\ \leq \frac{1}{5 n} \sum_{i = 1}^{n} [| ϕ_{P}^{-} (t_{i}) - ϕ_{C}^{-} (t_{i}) | + | ϕ_{P}^{+} (t_{i}) - ϕ_{C}^{+} (t_{i}) | + | Ω_{P}^{-} (t_{i}) - Ω_{C}^{-} (t_{i}) | + | Ω_{P}^{+} (t_{i}) - Ω_{C}^{+} (t_{i}) | + | u (t_{i}) - w (t_{i}) |] \\ d_{H}^{'} (Q, C) \leq d_{H}^{'} (P, C) \end{array} \end{aligned}$

Hence (d) is also proved

Since all the conditions for a distance measure are satisfied by $d_{H}^{'},$ it is a distance measure.

Note 3.4:

Normalized Euclidean distance $d_{E}^{'}$ , Hausdorff-Normalized Hamming distance $d_{H H}^{'}$ , Hausdorff-Normalized Euclidean distance $d_{H E}^{'}$ satisfies the properties defined in Definition 3.1, hence they are all distance measures.
4 Application in multi-criteria group decision making

In this section, we develop TOPSIS method to handle MCGDM problems using C-IVIFSs.

4.1 Solving MCGDM using C-IVIFS

TOPSIS is a technique for evaluating and arranging alternatives in a situation where several criteria are taken into account. The closeness ratio is used in TOPSIS to rank the options.

Let $P = {P_{i}} (i = 1, \dots, m)$ be the set of alternatives which has to be ranked. The criteria used for evaluation are $C = {C_{j}} (j = 1, \dots, n)$ and the weights of each criterion will be $W = {w_{i}} (j = 1, \dots, n)$ where each $w_{j} > 0$ and $\sum_{j = 1}^{n} w_{j} = 1$ . The evaluation of the alternatives will be done by the set of experts $Z = {Z_{p}} (p = 1, \dots, k)$ whose C-IVIF decision matrix will be in the form of $D_{Z_{p}} = (P_{i j})_{m \times n}$ in which each element, $P_{i j} = ⟨ [ϕ_{P_{i j}}^{-}, ϕ_{P_{i j}}^{+}], [Ω_{P_{i j}}^{-}, Ω_{P_{i j}}^{+}], u_{i j} ⟩$ is a C-IVFSs.

The steps of TOPSIS are employed below:

Step 1. Construction of a decision matrix $D_{Z_{p}} = (P_{i j})_{m \times n}$ by each expert in the form of C-IVIFSs

Step 2. Aggregate the decision matrices of all experts into a single C-IVIF decision matrix $D = (P_{i j})_{m \times n}$ by using the weighted arithmetic aggregation operators and weighted geometric aggregation operators

\begin{aligned} D = [\begin{array}{ccc} P_{11} & \dots & P_{1 n} \\ ⋮ & ⋱ & ⋮ \\ P_{m 1} & \dots & P_{m n} \end{array}] \end{aligned}

Step 3. Normalize the C-IVIF decision matrix $D$ and the normalized decision matrix will be $\tilde{D}$ . Criteria will be either cost or benefit type. If the criterion is of cost type, convert it into benefit type by taking complement for that particular C-IVIFS.

Step 4. Determine the C-IVIF positive ideal solution $P^{+}$ and negative ideal solution $P^{-}$ by using the following formulae where $i = 1, \dots, m$ and $j = 1, \dots, n$

\begin{aligned} P^{+} & = ⟨ t_{j}, [max_{i} ϕ_{P_{i}}^{-} (t_{j}), max_{i} ϕ_{P_{i}}^{+} (t_{j})], [min_{i} Ω_{P_{i}}^{-} (t_{j}), min_{i} Ω_{P_{i}}^{+} (t_{j})], max_{i} u_{i} (t_{j}) ⟩ \end{aligned}

\begin{aligned} P^{-} & = ⟨ t_{j}, [min_{i} ϕ_{P_{i}}^{-} (t_{j}), min_{i} ϕ_{P_{i}}^{+} (t_{j})], [max_{i} Ω_{P_{i}}^{-} (t_{j}), max_{i} Ω_{P_{i}}^{+} (t_{j})], min_{i} u_{i} (t_{j}) ⟩ \end{aligned}

Step 5. Compute the distance measures $(d)$ between the positive ideal solution and $P_{i} (i = 1, \dots, m)$ and the distance measures $(d)$ between the negative ideal solution and $P_{i} (i = 1, \dots, m)$

Step 6. Compute the relative closeness by using the following formula

\begin{aligned} V (P_{i}) = \frac{d (P_{i}, P^{+})}{d (P_{i}, P^{+}) + d (P_{i}, P^{-})} \cdot i = 1, \dots, m \end{aligned}

Step 7. Rank the alternatives based on the relative closeness $(V)$ . The alternative $P_{i}$ which has the least relative closeness is the better alternative among the alternatives.

Illustration 4.1:

We adapt this problem from literature.²⁶

Following market research and initial screening, four potential cloud services-SAP Sales on Demand $(P_{1})$ , Salesforce Sales Cloud $(P_{2})$ , Microsoft Dynamic CRM $(P_{3})$ , and Oracle Cloud CRM $(P_{4})$ - are available for further consideration.

Five criteria (attributes), including performance $(C_{1})$ , payment $(C_{2})$ , reputation $(C_{3})$ , scalability $(C_{4})$ , and security $(C_{5})$ , are used for evaluation. The selection panel is of four members $(Z_{p}, p = 1, 2, 3, 4)$ .

The objective is to determine which of the four cloud services is better. TOPSIS is the technique used in this instance.

Step 1: Construction of C-IVIF decision matrix of each expert

\begin{aligned} D_{Z_{1}} & = [\begin{array}{ccccc} ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.15, 0.25], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.55], \\ [0.35, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.75], \\ [0.15, 0.25], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.75], \\ [0.05, 0.15], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.40], \\ [0.30, 0.50], \\ 0.18 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.25, 0.35], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.35], \\ [0.15, 0.35], \\ 0.14 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.45, 0.55], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.25, 0.45], \\ [0.45, 0.55], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.70, 0.80], \\ [0.10, 0.20], \\ 0.07 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.15, 0.25], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.75, 0.85], \\ [0.05, 0.15], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.85], \\ [0.15, 0.15], \\ 0.15 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.45, 0.65], \\ [0.25, 0.35], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.50, 0.60], \\ [0.20, 0.30], \\ 0.07 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.35, 0.55], \\ [0.35, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.65, 0.75], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.55, 0.75], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.35, 0.55], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.20, 0.30], \\ [0.50, 0.60], \\ 0.07 \end{array} ⟩ \end{array}] \end{aligned}

\begin{aligned} D_{Z_{2}} & = [\begin{array}{ccccc} ⟨ \begin{array}{cc} [0.45, 0.55], \\ [0.25, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.30, 0.40], \\ [0.40, 0.60], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.10, 0.15], \\ 0.06 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.05, 0.25], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.35], \\ [0.25, 0.45], \\ 0.14 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.35, 0.55], \\ [0.30, 0.40], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.30], \\ [0.30, 0.70], \\ 0.22 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.25, 0.35], \\ [0.35, 0.45], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.25, 0.35], \\ [0.55, 0.65], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.85], \\ [0.05, 0.15], \\ 0.11 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.45, 0.65], \\ [0.25, 0.35], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.60, 0.80], \\ [0.10, 0.20], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.75], \\ [0.05, 0.15], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.45, 0.65], \\ [0.25, 0.35], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.15, 0.35], \\ 0.11 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.35, 0.55], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.20], \\ [0.60, 0.80], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.05, 0.15], \\ [0.65, 0.75], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.35, 0.55], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.25, 0.45], \\ [0.45, 0.55], \\ 0.11 \end{array} ⟩ \end{array}] \end{aligned}

\begin{aligned} D_{Z_{3}} & = [\begin{array}{ccccc} ⟨ \begin{array}{cc} [0.45, 0.75], \\ [0.15, 0.25], \\ 0.16 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.55], \\ [0.25, 0.35], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.60, 0.70], \\ [0.10, 0.20], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.05, 0.25], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.55], \\ [0.25, 0.45], \\ 0.14 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.45, 0.55], \\ [0.25, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.85], \\ [0.05, 0.15], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.40, 0.50], \\ [0.30, 0.40], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.65, 0.75], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.75], \\ [0.15, 0.25], \\ 0.07 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.25, 0.45], \\ [0.35, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.85], \\ [0.05, 0.15], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.50, 0.70], \\ [0.10, 0.30], \\ 0.14 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.75], \\ [0.15, 0.25], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.85], \\ [0.05, 0.15], \\ 0.11 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.25, 0.45], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25] \\ [0.55, 0.75], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.30], \\ [0.50, 0.70], \\ 0.14 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.25, 0.35], \\ [0.45, 0.65], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.55, 0.75], \\ 0.11 \end{array} ⟩ \end{array}] \end{aligned}

\begin{aligned} D_{Z_{4}} & = [\begin{array}{ccccc} ⟨ \begin{array}{cc} [0.65, 0.75], \\ [0.15, 0.25], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.30, 0.40], \\ [0.30, 0.40], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.75, 0.85], \\ [0.05, 0.15], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.50, 0.60], \\ [0.10, 0.30], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.45, 0.65], \\ 0.11 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.40, 0.50], \\ [0.40, 0.50], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.20], \\ [0.20, 0.30], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.45, 0.55], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.20, 0.30], \\ [0.40, 0.60], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.65, 0.75], \\ [0.05, 0.15], \\ 0.07 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.40, 0.50], \\ [0.30, 0.40], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.60, 0.70], \\ [0.10, 0.30], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.85], \\ [0.15, 0.15], \\ 0.15 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.40, 0.50], \\ [0.20, 0.30], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.55, 0.65], \\ [0.25, 0.35], \\ 0.07 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.30, 0.40], \\ [0.40, 0.50], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.10, 0.30] \\ [0.60, 0.70], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.25], \\ [0.55, 0.75], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.20, 0.30], \\ [0.40, 0.50], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.35, 0.45], \\ [0.45, 0.55], \\ 0.07 \end{array} ⟩ \end{array}] \end{aligned}

Step 2: Aggregation of decision matrices of all experts into a single matrix $D$ by using the weighted arithmetic aggregation operator $W_{P_{q}}$ ,

\begin{aligned} D = [\begin{array}{ccccc} ⟨ \begin{array}{cc} [0.53, 0.69], \\ [0.17, 0.29], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.33, 0.48], \\ [0.32, 0.44], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.64, 0.75], \\ [0.09, 0.18], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.54, 0.67], \\ [0.06, 0.23], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.20, 0.40], \\ [0.30, 0.50], \\ 0.14 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.39, 0.51], \\ [0.29, 0.42], \\ 0.09 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.15, 0.33], \\ [0.24, 0.43], \\ 0.13 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.34, 0.44], \\ [0.38, 0.48], \\ 0.07 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.21, 0.34], \\ [0.51, 0.63], \\ 0.09 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.66, 0.79], \\ [0.08, 0.18], \\ 0.08 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.42, 0.57], \\ [0.25, 0.35], \\ 0.09 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.66, 0.81], \\ [0.07, 0.19], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.56, 0.80], \\ [0.10, 0.18], \\ 0.12 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.47, 0.65], \\ [0.21, 0.31], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.57, 0.71], \\ [0.14, 0.27], \\ 0.09 \end{array} ⟩ \\ ⟨ \begin{array}{cc} [0.34, 0.47], \\ [0.33, 0.48], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.13, 0.25] \\ [0.60, 0.75], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.11, 0.24], \\ [0.56, 0.74], \\ 0.11 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.29, 0.39], \\ [0.39, 0.56], \\ 0.10 \end{array} ⟩ & ⟨ \begin{array}{cc} [0.29, 0.39], \\ [0.39, 0.56], \\ 0.10 \end{array} ⟩ \end{array}] \end{aligned}

Step 3: The criteria $C_{j} (j = 1, 2, 3, 4, 5)$ are beneficial, so the criteria values in $D$ do not need to be normalized.

Step 4: The positive and negative ideal solutions are as follows:

\begin{aligned} P^{+} = {\begin{matrix} ⟨ [0.53, 0.69], [0.17, 0.29], 0.10 ⟩, ⟨ [0.66, 0.81], [0.07, 0.19], 0.13 ⟩, \\ ⟨ [0.64, 0.80], [0.09, 0.18], 0.12 ⟩, ⟨ [0.54, 0.67], [0.06, 0.23], 0.11 ⟩, \\ ⟨ [0.66, 0.79], [0.08, 0.18], 0.14 ⟩ \end{matrix}} \end{aligned}

\begin{aligned} P^{-} = {\begin{matrix} ⟨ [0.34, 0.47], [0.33, 0.48], 0.09 ⟩, ⟨ [0.13, 0.25], [0.60, 0.75], 0.10 ⟩, \\ ⟨ [0.11, 0.24], [0.56, 0.74], 0.07 ⟩, ⟨ [0.21, 0.34], [0.51, 0.63], 0.09 ⟩, \\ ⟨ [0.20, 0.37], [0.49, 0.61], 0.08 ⟩ \end{matrix}} \end{aligned}

Step 5: Computation of the distance measures $(d)$ between the positive ideal solution and $P_{i} (i = 1, \dots, m)$ and the distance measures $(d)$ between the negative ideal solution and $P_{i} (i = 1, \dots, m)$ are presented in Table 1.

Step 6: Computation of the relative closeness is given in Table 2 and depicted in Figure 1.

Step 7: Ranking of alternatives

Table 1.
Distance measures between the alternatives and the ideal solutions.

Normalized Hamming Distance Normalized Euclidean Distance Hausdorff-Normalized Hamming Distance Hausdorff-Normalized Euclidean Distance

$d (P_{1}, P^{+})$ 0.11 0.19 0.03 0.15

$d (P_{2}, P^{+})$ 0.19 0.25 0.06 0.18

$d (P_{3}, P^{+})$ 0.05 0.07 0.02 0.05

$d (P_{4}, P^{+})$ 0.32 0.38 0.08 0.23

$d (P_{1}, P^{-})$ 0.23 0.29 0.07 0.19

$d (P_{2}, P^{-})$ 0.14 0.22 0.05 0.16

$d (P_{3}, P^{-})$ 0.29 0.35 0.08 0.22

$d (P_{4}, P^{-})$ 0.02 0.04 0.01 0.03

	Normalized Hamming Distance	Normalized Euclidean Distance	Hausdorff-Normalized Hamming Distance	Hausdorff-Normalized Euclidean Distance
$d (P_{1}, P^{+})$	0.11	0.19	0.03	0.15
$d (P_{2}, P^{+})$	0.19	0.25	0.06	0.18
$d (P_{3}, P^{+})$	0.05	0.07	0.02	0.05
$d (P_{4}, P^{+})$	0.32	0.38	0.08	0.23
$d (P_{1}, P^{-})$	0.23	0.29	0.07	0.19
$d (P_{2}, P^{-})$	0.14	0.22	0.05	0.16
$d (P_{3}, P^{-})$	0.29	0.35	0.08	0.22
$d (P_{4}, P^{-})$	0.02	0.04	0.01	0.03

Table 2.

Relative closeness of the alternatives.

	Normalized Hamming Distance	Normalized Euclidean Distance	Hausdorff-Normalized Hamming Distance	Hausdorff-Normalized Euclidean Distance
$V (P_{1})$	0.32	0.39	0.33	0.45
$V (P_{2})$	0.57	0.54	0.57	0.54
$V (P_{3})$	0.14	0.16	0.19	0.18
$V (P_{4})$	0.95	0.91	0.91	0.89

In all the distance measures, $V (P_{3}) < V (P_{1}) < V (P_{2}) < V (P_{4})$ .

Hence, Microsoft Dynamic CRM $(P_{3})$ is the better cloud service.

4.2 Validation

Sensitivity analysis:

Sensitivity analysis shows how the suggested distance measures of C-IVIFS are stable and robust in the presence of imprecise or uncertain data when the underlying membership or non-membership information are marginally perturbed.

The systematic procedure outlined in Section 4.1, which also ranks the various alternatives, successfully overcomes the difficulty of MCDM in the C-IVIF environment. However, if the decision-maker chooses to increase, lower, scale, or divide an assessment value for any of the alternatives based on specific criteria during the analytical process, it may or may not have an impact on the final ranking of the alternatives.

Consider the circumstance in which any or all of the evaluation values in illustration 4.1 are scaled by a specific factor. In this, we use two operators $\otimes_{m i n}$ and $\otimes_{m a x}$ , proposed by Gajalaxmi et al.²¹

Let $P = ⟨ [ϕ_{P}^{-} (t), ϕ_{P}^{+} (t)], [Ω_{P}^{-} (t), Ω_{P}^{+} (t)], u ⟩,$ and $Q = ⟨ [ϕ_{Q}^{-} (t), ϕ_{Q}^{+} (t)], [Ω_{Q}^{-} (t), Ω_{Q}^{+} (t)], v ⟩$ be two C-IVIFSs in a non-empty set $T$ , then

\begin{aligned} P \otimes_{m i n} Q & = ⟨ [ϕ_{P}^{-} ϕ_{Q}^{-}, ϕ_{P}^{+} ϕ_{Q}^{+}], [Ω_{P}^{-} + Ω_{Q}^{-} - Ω_{P}^{-} Ω_{Q}^{-}, Ω_{P}^{+} + Ω_{Q}^{+} - Ω_{P}^{+} Ω_{Q}^{+}], min (u, v) ⟩ \end{aligned}

\begin{aligned} P \otimes_{m a x} Q & = ⟨ [ϕ_{P}^{-} ϕ_{Q}^{-}, ϕ_{P}^{+} ϕ_{Q}^{+}], [Ω_{P}^{-} + Ω_{Q}^{-} - Ω_{P}^{-} Ω_{Q}^{-}, Ω_{P}^{+} + Ω_{Q}^{+} - Ω_{P}^{+} Ω_{Q}^{+}], max (u, v) ⟩ \end{aligned}

Case 1:
One of the criteria is scaled by any C-IVIFS. Criteria 5 in illustration 4.1 is security; in the present world, cloud service security needs to be prioritised more to protect an extensive amount of data. In light of this, the decision-maker wants to scale the evaluations for each alternative of the security criterion by $⟨ [0.5, 0.6], [0.2, 0.4], 0.11 ⟩$ (say).

The procedure is same as in section 4.1, but after step 2, the criterion $C_{5}$ is scaled by $⟨ [0.5, 0.6], [0.2, 0.4], 0.11 ⟩$ . The newly computed relative closeness is depicted in Table 3 and is graphically represented in Figure 2. Here the ranking of the alternatives in each of the distance measures is $V (P_{3}) < V (P_{1}) < V (P_{2}) < V (P_{4})$ and hence from this ranking we found that, Microsoft Dynamic CRM $(P_{3})$ is the better cloud service.

Figure 1.
Relative closeness of the alternatives.

Figure 2
New relative closeness obtained in Case 1.

Table 3.
New relative closeness of the alternatives (Case 1).

Normalized Hamming Distance Normalized Euclidean Distance Hausdorff-Normalized Hamming Distance Hausdorff-Normalized Euclidean Distance

$V (P_{1})$ 0.27 0.33 0.27 0.38

$V (P_{2})$ 0.63 0.61 0.60 0.60

$V (P_{3})$ 0.14 0.15 0.19 0.18

$V (P_{4})$ 0.93 0.88 0.86 0.83

Case 2:
In this case, each criterion is scaled by a separate factor.

After step 2, the criterion $C_{1}$ is scaled by $⟨ [0.65, 0.75], [0.16, 0.28], 0.08 ⟩$ , the criterion $C_{2}$ is scaled by $⟨ [0.7, 0.8], [0.1, 0.2], 0.07 ⟩$ , the criterion $C_{3}$ is scaled by $⟨ [0.6, 0.7], [0.18, 0.3], 0.08 ⟩$ , the criterion $C_{4}$ is scaled by $⟨ [0.5, 0.6], [0.2, 0.4], 0.11 ⟩$ , and the criterion $C_{5}$ is scaled by $⟨ [0.25, 0.3333], [0.5556, 0.6667], 0.07 ⟩$ . The newly computed relative closeness is mentioned in Table 4 and is graphically represented in Figure 3. Here the ranking of the alternatives in each of the distance measures is $V (P_{3}) < V (P_{1}) < V (P_{2}) < V (P_{4})$ and hence from this ranking we found that, Microsoft Dynamic CRM $(P_{3})$ is the better cloud service.

Figure 3.
New relative closeness obtained in Case 2.

Table 4.
New relative closeness of the alternatives (Case 2).

Normalized Hamming Distance Normalized Euclidean Distance Hausdorff-Normalized Hamming Distance Hausdorff-Normalized Euclidean Distance

$V (P_{1})$ 0.28 0.33 0.24 0.34

$V (P_{2})$ 0.62 0.40 0.55 0.48

$V (P_{3})$ 0.13 0.27 0.17 0.28

$V (P_{4})$ 0.86 0.68 0.70 0.62

Case 3:
In this case, each criterion is scaled by the unique factor.

After step 2, the criteria $C_{1} - C_{5}$ are scaled by $⟨ [0.7, 0.75], [0.05, 0.2], 0.08 ⟩$ . The newly computed relative closeness is given in Table 5 and is graphically represented in Figure 4. Here the ranking of the alternatives in each of the distance measures is $V (P_{3}) < V (P_{1}) < V (P_{2}) < V (P_{4})$ and hence from this ranking we found that, Microsoft Dynamic CRM $(P_{3})$ is the better cloud service.

Figure 4.
New relative closeness obtained in Case 3..

Table 5.
New relative closeness of the alternatives (Case 3).

Normalized Hamming Distance Normalized Euclidean Distance Hausdorff-Normalized Hamming Distance Hausdorff-Normalized Euclidean Distance

$V (P_{1})$ 0.33 0.33 0.29 0.35

$V (P_{2})$ 0.57 0.43 0.54 0.47

$V (P_{3})$ 0.14 0.22 0.18 0.25

$V (P_{4})$ 0.92 0.80 0.82 0.77

Discussion: In all the three cases, the operator used is $\otimes_{m i n}$ . The ranking of the alternatives remains the same in all the cases which is $V (P_{3}) < V (P_{1}) < V (P_{2}) < V (P_{4})$ and the conclusion is, Microsoft Dynamic CRM $(P_{3})$ is the better cloud service in each of the cases. The same analysis is done by using the operator $\otimes_{m a x}$ and the results are coinciding with the results obtained by using the operator $\otimes_{m i n}$ . Consequently, when some of the assessments need to be changed during the analysis and decision-making process, the suggested operation will be necessary to speed up the process. Hence, the proposed distance measures are robust and stable.

Comparison:

The same instance is evaluated using the already existing distance measures and the results are computed in Table 6.

Table 6.
Existing distance measures values.

$d_{h}$ ²⁷ $d_{n h}$ ²⁷ $d_{e}$ ²⁷ $d_{n e}$ ²⁷ $d_{H H}$ ²⁷ $d_{H E}$ ²⁷ $d_{I v}$ ²⁸

$V (P_{1})$ 0.32 0.32 0.39 0.39 0.33 0.35 0.39

$V (P_{2})$ 0.60 0.60 0.57 0.57 0.58 0.59 0.58

$V (P_{3})$ 0.15 0.15 0.17 0.17 0.16 0.05 0.21

$V (P_{4})$ 0.92 0.92 0.87 0.87 0.91 0.95 0.88

From the Table 6, it can be seen that the ranking of alternatives by the existing distance measures and the newly proposed distance measures are similar, hence our proposed distance measures are valid.
5 Conclusion

	Normalized Hamming Distance	Normalized Euclidean Distance	Hausdorff-Normalized Hamming Distance	Hausdorff-Normalized Euclidean Distance
$V (P_{1})$	0.27	0.33	0.27	0.38
$V (P_{2})$	0.63	0.61	0.60	0.60
$V (P_{3})$	0.14	0.15	0.19	0.18
$V (P_{4})$	0.93	0.88	0.86	0.83

	Normalized Hamming Distance	Normalized Euclidean Distance	Hausdorff-Normalized Hamming Distance	Hausdorff-Normalized Euclidean Distance
$V (P_{1})$	0.28	0.33	0.24	0.34
$V (P_{2})$	0.62	0.40	0.55	0.48
$V (P_{3})$	0.13	0.27	0.17	0.28
$V (P_{4})$	0.86	0.68	0.70	0.62

	Normalized Hamming Distance	Normalized Euclidean Distance	Hausdorff-Normalized Hamming Distance	Hausdorff-Normalized Euclidean Distance
$V (P_{1})$	0.33	0.33	0.29	0.35
$V (P_{2})$	0.57	0.43	0.54	0.47
$V (P_{3})$	0.14	0.22	0.18	0.25
$V (P_{4})$	0.92	0.80	0.82	0.77

	$d_{h}$ ²⁷	$d_{n h}$ ²⁷	$d_{e}$ ²⁷	$d_{n e}$ ²⁷	$d_{H H}$ ²⁷	$d_{H E}$ ²⁷	$d_{I v}$ ²⁸
$V (P_{1})$	0.32	0.32	0.39	0.39	0.33	0.35	0.39
$V (P_{2})$	0.60	0.60	0.57	0.57	0.58	0.59	0.58
$V (P_{3})$	0.15	0.15	0.17	0.17	0.16	0.05	0.21
$V (P_{4})$	0.92	0.92	0.87	0.87	0.91	0.95	0.88

The distance measures of circular interval-valued intuitionistic fuzzy sets are the primary objective of this study since they are important in many different application domains. The following distance measures are suggested: Normalized Hamming, Normalized Euclidean, Hausdorff-Normalized Hamming, and Hausdorff-Normalized Euclidean distances. The developed distance measures are applied to a group decision-making problem from the literature, and the TOPSIS approach is used to solve the problem. This led to the finding of the better cloud service. In order to verify the suggested distances, the same problem is examined using the literature's existing distance measures, and the results showed that they aligned with the proposed distances. To quantify and demonstrate the robustness of the proposed approach, sensitivity analysis is done by using the operators $\otimes_{m i n}$ and $\otimes_{m a x}$ . The distance measures on circular interval-valued intuitionistic fuzzy sets can be applied in multi-criteria decision making, pattern recognition and classification, image processing, control systems, forecasting and prediction. This manuscript devotes to the applicability of the proposed measure in multi-criteria group decision making and our future work is focussing on the other application which are pattern recognition and classification and image processing.

Footnotes

ORCID iDs

Gajalaxmi Pandaram

Jayanthi Duraisamy

Priyadharshini Manoharan

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Distance measures on circular interval-valued intuitionistic fuzzy sets

Abstract

Keywords

1 Introduction

2 Preliminaries

Definition 21 4

Definition 24 22

3 Distance measures on C-IVIFSs

4.1 Solving MCGDM using C-IVIFS

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

References

Definition 21 ⁴

Definition 24 ²²