A New Entropy Measurement for the Analysis of Uncertain Data in MCDA Problems Using Intuitionistic Fuzzy Sets and COPRAS Method

Abstract

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support methods, where dealing with uncertain data grows in importance. The Complex Proportional Assessment (COPRAS) method identifies the preferences and ranking of decisional variants. It also allows for a more comprehensive analysis of complex decision-making problems, where many opposite criteria are observed. This approach allows us to minimize cost and maximize profit in the finally chosen decision (alternative). This paper presents a new entropy measurement for fuzzy intuitionistic sets and an application example using the IFS COPRAS method. The new entropy method was used in the decision-making process to calculate the objective weights. In addition, other entropy methods determining objective weights were also compared with the proposed approach. The presented results allow us to conclude that the new entropy measure can be applied to decision problems in uncertain data environments since the proposed entropy measure is stable and unambiguous.

1. Introduction

Multi-criteria decision analysis (MCDA) is an approach to dealing more easily with complex decision problems, which applies to decision-making related to professional activity and everyday life problems. Many authors have used MCDA methods for a wide range of problems. One of the fields in which multi-criteria decision problems arise is economics. For example, Stević et al. presented research on sustainable production assessment for forestry companies in the Eastern Black Sea region [1]. This research is essential for the circular economy, which aims to reduce the waste of natural resources. The evaluation process used a Rough PIvot Pairwise RElative Criteria Importance Assessment (PIPRECIA) method belonging to the stream of multi-criteria decision analysis methods. Another example of an emerging problem in economics is the problem of installing dams on the Drina River [2]. This problem is essential for the economy of Yugoslavia because of the extracted benefits from the use of a renewable energy source.

The field in which the approach of multi-criteria decision analysis is used is sustainable transport. The problem that arises in the evaluation of urban transport is the selection of relevant criteria. Shekhovtsov et al. [3] proposed an approach based on a reference ranking that determines the relevance of criteria using four approaches. For the supplier selection problem, studies have also been conducted on selecting an appropriate weight selection method [4]. As a result, objective methods of selecting weights, such as entropy, equal, or standard deviation methods, were selected. Pamucar et al. [5] used an approach combining the Best Worst Method (BWM) and Compressed Proportional Assessment (COPRAS) techniques to evaluate off-road vehicles for the Serbian Armed Forces (SAF). Choosing the appropriate off-road vehicle to transport units has a massive impact on their safety. Therefore, this problem is also essential to fill the research gap on defining a proper vehicle selection methodology for the military.

One of the most important fields in which MCDA methods are used is the healthcare field. La Scalia et al. [6] proposed using Technique Ordered Preference by Similarity to the Ideal Solution (TOPSIS) to evaluate information related to pancreatic islet transplantation. It was decided to use a multi-criteria decision analysis method because of the many variables involved in this problem. The TOPSIS method was also modified in this study to fit the medical problem better. Multi-criteria decision analysis was also used in the problem of supplier selection in the healthcare industry in Bosnia and Herzegovina [7]. A new method, Measurement of Alternatives and Ranking according to COmpromise Solution (MARCOS), was used in this work. The sensitivity analysis of the proposed approach in the study showed that this method could be applied in a supplier selection problem in the healthcare industry.

MCDA methods are also applied to solid waste management problems. Muhammad et al. presented a study using the GREY-EDAS hybrid model to evaluate alternative waste treatment methods in Nigeria [8]. Due to Nigeria being a developing country, it is still struggling with the problem of municipal solid waste treatment. The application of the proposed approach, i.e., the GREY-EDAS method, yielded consistent results, i.e., the most suitable method for municipal solid waste treatment was achieved.

Another field in which multi-criteria decision analysis techniques are applied is renewable energy. Bączkiewicz et al. presented an approach in which they used the COMET-TOPSIS and SPOTIS methods to create a decision support system (DSS) for the solar panel selection problem [9]. Proper selection of photovoltaic systems for companies and households involves the possibility of discontinuing the use of energy obtained through conventional sources. Therefore, the creation of a decision support system for this problem proves its great practical usefulness. The problem of selecting the development of hydrogen buses is also an example of the application of MCDA methods in renewable energy. An approach to this problem was proposed by Pamucar et al. using a model composed of Best-Worst Method (BWM) and Measurement Alternatives and Ranking according to COpromise Solution (MARCOS) [10]. The results obtained in the study show that the best solution is co-generated electricity from a municipality cogeneration power plant.

Multi-criteria decision analysis is also applied in E-commerce. For example, Bączkiewicz et al. [11] proposed a consumer decision support system based on five multi-criteria decision analysis methods for selecting the most suitable cell phone. The results obtained from the methods were the components of the trade-off ranking obtained using the Copeland method.

One of the most significant challenges of multi-criteria decision-making is to express the uncertainty in the data correctly. The source of such uncertainty can be unreliable information, information noise, or conflicting information. Greis et al. presented a study on the source of uncertain information from user inputs, considering that their input is not mainly controlled [12]. Decisions made with uncertainty in the data are more challenging to analyze and often require specialized tools. One such tool is interval fuzzy sets, which can be considered as an extension of the concept of real numbers [13]. Shekhovtsov et al. proposed a new approach in multi-criteria decision analysis, where they combined interval fuzzy sets with the Stable Preference Ordering Towards Ideal Solution (SPOTIS) method. The proposed approach aims to deal with the rank reversal paradox with uncertain data [14].

Fuzzy sets are another popular tool used to deal with uncertainty in data. Zadeh proposed a fuzzy set as a class of objects with some continuum of degrees of membership [15]. An example application of fuzzy sets in MCDA methods is the problem of selecting the best solution for the business balance of a passenger rail carrier. Vesković et al. used a combination of the fuzzy PIvot Pairwise RElative Criteria Importance Assessment (F-PIPRECIA) weight selection method with the fuzzy Evaluation based on Distance from Average Solution (F-EDAS) method [16]. The results obtained were also compared with other methods based on fuzzy sets, i.e., Fuzzy Measurement Alternatives and Ranking according to the COmpromise Solution (F-MARCOS), Fuzzy Simple Additive Weighing (F-SAW) and the Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (F-TOPSIS). The following example of using fuzzy sets in multi-criteria decision analysis is the housing selection problem. Kizielewicz and Bączkiewicz proposed using four MCDA methods extended with fuzzy sets to select the best housing alternative [17]. The results have been compared using the similarity coefficients of the rankings and indicate high similarity between the obtained rankings of the decision alternatives.

A tool that is also used in multi-criteria decision analysis to deal with uncertainty in data is Pythagorean fuzzy sets (PFSs). They were introduced to deal with ambiguity by Yager, where the degrees of membership are represented as pairs [18]. The proposed tool is close to intuitionistic fuzzy sets (IFS) [19]. One of the applications of Pythagorean fuzzy sets in MCDA is to extend the VIKOR method to it. This method was used to assess the occupational risk in constructing a natural gas pipeline through Mete et al. [20]. The built DSS system aims to improve the situation in the energy industry. Methods such as AHP and TOPSIS also have extensions in the form of Pythagorean fuzzy sets. Both methods were applied in the problem of evaluating the quality of hospital services [21]. In this approach, three hospitals were evaluated, of which one was private and two were public. Furthermore, Pythagorean fuzzy sets are also used in hybrid approaches, both in European school methods and in American school methods. An example of such an approach is the risk assessment problem in failure modes and effects analysis (FMEA) [22]. The Pythagorean fuzzy hybrid Order of Preference by Similarity to an Ideal Solution (PFSH-TOPSIS) method from the American school and the Pythagorean fuzzy hybrid ELimination and Choice Translating REality I (PFH-ELECTRE I) method from the European school were used for the evaluation. The effectiveness of the proposed approach is demonstrated using the color super-twisted nematic (CSTN) as an example.

Hesitant fuzzy sets are another tool used to deal with uncertain data. Torr proposed it because of the difficulty of determining the membership of elements to a set in some cases [23]. It is used, for example, in decision-making problems concerning the selection of electric city buses. An approach in which the method of characteristic objects using hesitant fuzzy sets was presented for this problem by Salabun et al. [24]. Another example of HFS application in MCDA is the use of the hesitant fuzzy TOPSIS method in the problem of site selection for a hospital [25]. Selecting a location for a new hospital in Istanbul is difficult because many important factors, such as transportation, demographics and infrastructure, influence the location. Therefore, it was decided to use a multi-criteria decision analysis method.

Decision-making only in certain exceptional situations can be performed with crisp data. More and more frequently, MCDA methods support decision-makers in complex systems in an uncertain environment. In these cases, a decision-maker generally has uncertain or incomplete data at his disposal, and choosing the best alternative with often conflicting criteria is even more challenging to achieve. In the case of crisp data, sensitivity analysis is performed to determine how much impact a change in attribute values has on the final ranking positions of the alternatives. However, to properly deal with uncertainty in the data, an appropriate tool must be developed to model this kind of uncertainty well. This paper is focused on intuitionistic fuzzy sets (IFSs), which have repeatedly proven their usefulness in the MCDA domain [26,27]. Intuitionistic fuzzy sets are an essential generalization of the fuzzy sets [15], where the main concept supposes that, for each value in a defined domain, we assign a membership value and a non-membership value. Using such tools in combination with multi-criteria decision analysis allows for better identification of data uncertainties and enables their proper modelling in the decision-making process. Therefore, many approaches in decision-making use fuzzy sets and their generalizations, including IFSs [28,29].

Intuitionistic fuzzy sets are still being developed, and there are discussions about their implementation. Szmidt and Kacprzyk proposed new definitions concerning the distance between intuitionistic fuzzy sets [30]. They aim to address the issue that the degree to which an element does not belong to an intuitionistic fuzzy set will not always be 1. Turanli et al. presented new types of fuzzy connectedness in intuitionistic fuzzy topological spaces [31]. Bustince presented several theorems that allow intuitionistic fuzzy relations to be formed on a set with specific properties [32]. Ciftcibasi and Altunay presented the concept of two-sided fuzzy reasoning and developed its mathematical structure [33].

One of the key features of IFS is to better account for uncertainty. Many MCDA methods have used this concept to solve problems with uncertain data, and we can mention here such methods as Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [34,35], Preference Ranking Organization METHod for Enrichment of Evaluations (PROMETHEE) [36,37], VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) [38,39], Analytic Hierarchy Process (AHP) [40,41], ÉLimination et Choix Traduisant la REalité (ELECTRE) [42,43], Multi-Attributive Border Approximation area Comparison (MABAC) [44], COmbinative Distance-based ASsessment (CODAS) [45,46], COmplex PRoportional ASsessment (COPRAS) [47], Multi-Objective Optimization Method by Ratio Analysis (MOORA) [48,49], Analytic Network Process (ANP) [50,51], Decision Making Trial and Evaluation Laboratory (DEMATEL) [52,53], Stepwise Weight Assessment Ratio Analysis (SWARA) [47,54] and so on. All of these mentioned approaches mainly use attributes stored as uncertain data. A separate issue is the use of entropy IFSs to determine significance weights for criteria [55,56], which we address in this paper. Classical Information Entropy defines the average amount of information per single message from an information source. Entropy can be interpreted as a measure of uncertainty, and when the uncertainty increases, the entropy value increases also [57,58].

Entropy describes the uncertainty information, and it is an important research topic in the decision-making domain [59]. In 1996, the notion of intuitionistic fuzzy entropy was proposed for measuring the degree of an intuitionistic fuzzy set. Xu [60] examined the intuitionistic fuzzy Multiple Attribute Decision Making (MADM) with a study concerning attribute weights. Many well-known MCDA methods have been extended by using the entropy concept. One example is the COPRAS method [61]. According to these advantages, the COPRAS method has been applied by numerous scholars from distinctive features of composition in most modern times [62]. Banaitiene et al. [63] considered the applicability of a methodology for the multivariate perspective and multiple criteria interpretation of the life cycle of a building based on COPRAS. Ghorabaee et al. [64] considered the COPRAS method in interval type-2 fuzzy sets for supplier selection. Mishra et al. [47] studied the COPRAS method for the sustainability evaluation of the bioenergy production process. Jurgis et al. [65] considered the COPRAS method to assess city compactness. Zavadskas and Kaklauskas [66] considered the COPRAS method for the selection of effective dwelling house walls.

This work proposes a novel entropy measure for intuitionistic fuzzy sets compared with other existing approaches. We prove that the proposed measure satisfies conditions of minimality, maximality, resolution and symmetry according to axioms of valid entropy. We also present an illustrative example of using the proposed entropy in decision-making analysis using the COPRAS method, where new entropy is used. The results are compared with existing approaches, and their analysis is conducted using WS and weighted Spearman coefficient. This shows that novel entropy can be easily used instead of the old one. However, more extensive research to compare them is required.

The rest of the paper is organized as follows. In Section 2, basic definitions and notations of the IFSs are described. The novel proposed entropy measure is presented and verified in Section 3. Section 4 recalls the algorithm on the COPRAS method using the new entropy measurement. In Section 5, a simple example is presented, and the results are compared with various existing entropy measures defined on an intuitionistic set. Finally, the benefits and conclusions regarding the new entropy measure are discussed.

2. Preliminaries

An intuitionistic fuzzy set M in the classical a non-empty and finite set S is determined as the set of standardized triplets of the following Equation (1).

$$M=\left\{\left(x,{\mu }_M\left(x\right),{\vartheta }_M\left(x\right)\right)|x\in S\right\}$$

(1)

where ${\mu }_M\left(x\right):S\to [0,1]$ and ${\vartheta }_M\left(x\right):S\to [0,1]$ are the membership degree and non-membership degree, sequentially with the condition $0\le {\mu }_M\left(x\right)+{\vartheta }_M\left(x\right)\le 1$.

The numbers ${\mu }_M\left(x\right)\in \left[0,1\right]$ and ${\vartheta }_M\left(x\right)\in \left[0,1\right]$ determine the degree of membership of the element $x\in S$ to the intuitionistic fuzzy set M and the degree of non-membership of the element $x\in S$ to the intuitionistic fuzzy set M, sequentially. The compilation of all intuitionistic fuzzy sets in the classical set S of the formal values is signified by IFS(S).

${\pi }_M\left(x\right)$ is determined by the following Equation (2).

$${\pi }_M\left(x\right)=1-{\mu }_M\left(x\right)-{\vartheta }_M\left(x\right)$$

(2)

is called the hesitancy degree (or the degree of indeterminacy of information) of the element $x\in S$ to the set M, and ${\pi }_M\left(x\right)\in \left[0,1\right],\forall x\in S$.

3. Entropy for Intuitionistic Fuzzy Set

The fuzzy set theory uses entropy to measure the degree of fuzziness in a fuzzy set, called fuzzy entropy. Fuzzy entropy is used to express the mathematical values of the fuzziness of fuzzy sets. Classical Shannon entropy concerns probabilistic uncertainties, whereas fuzzy entropy concerns randomness, vagueness, fuzziness and ambiguous uncertainties [67]. Axiomatic entropy of fuzzy sets is continued as the intuitionistic fuzzy sets.

Definition 1.

A real function E: intuitionistic fuzzy set $IFS\left(M\right)\to [0,1]$ is characterized as entropy of intuitionistic fuzzy sets (M) if E satisfies the latter axioms [68].

• Minimality: $E\left(A\right)=0$, if A is crisp set;
• Maximality: $E\left(A\right)=1,$ if ${\mu }_A\left(x\right)={\vartheta }_A\left(x\right)=\frac{1}{3},\forall x$;
• Resolution: $E\left(A\right)\le E\left(B\right)$, if A is less fuzzy than B,
  i.e., ${\mu }_A\left(x\right)\le {\mu }_B\left(x\right)\le \frac{1}{3}$ and ${\vartheta }_B\left(x\right)\le {\vartheta }_A\left(x\right)\le \frac{1}{3}$ for ${\mu }_B\left(x\right)\le {\mu }_A\left(x\right)$
• Symmetry: $E\left(A\right)=E\left(A^c\right)$, where $A^c$ is the complement of A.

In a sense, with intuitionistic fuzzy knowledge, we suggest the latter intuitionistic fuzzy entropy analogous to criteria such as:

Let $M=x_1,x_2,x_3,\dots ,x_m$ be the universal set. Let $S=\left\{\left(x_i,{\mu }_s\left(x_i\right),{\vartheta }_s\left(x_i\right)|x_i\in M\right)\right\}$ be a intuitionistic fuzzy set on M.

$$E_p\left(S\right)=\frac{1}{n}\sum _{i=1}^n{E}_p^{\#}\left(S\right)$$

(3)

where

$$E_p^{\#}\left(S\right)=\frac{sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}$$

(4)

where

$${\pi }_s\left(x_i\right)=1-{\mu }_s\left(x_i\right)-{\vartheta }_s\left(x_i\right),\mathrm{for}\mathrm{all}i=1,2,3,\dots ,m.$$

(5)

Theorem 1.

The measure defined in Equation (3) is a valid entropy.

Proof. 

To prove that the proposed measure is valid, we must show that it satisfies the properties as provided in Definition 1.

• Minimality: if S is a crisp set, i.e., ${\mu }_s\left(x_i\right)=1,{\vartheta }_s\left(x_i\right)=0$ or ${\mu }_s\left(x_i\right)=0,{\vartheta }_s\left(x_i\right)=1$ for all $x_i\in M$, then,

  $$\begin{array}{c}\frac{sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}=0\end{array}$$

  (6)
  Therefore, $E_p\left(S\right)=0$
• Maximality: for all $x_i\in M$, if ${\mu }_s\left(x_i\right)={\vartheta }_s\left(x_i\right)=\frac{1}{3}$ for all $x_i\in M$, then,

  $$\begin{array}{c}\frac{sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}=1\end{array}$$

  (7)
• Resolution: in order to prove the fourth property, consider the function $f\left(\mu ,\vartheta \right)$ such that

  $$\begin{array}{c}f\left(\mu ,\vartheta \right)=\frac{sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}\end{array}$$

  (8)

  where $\mu ,\vartheta \in [0,1]$. The partial derivatives with respect to $\mu$ and $\vartheta$ are obtained as

  $$\begin{array}{c}\frac{\partial f}{\partial \mu }=-\frac{135\pi \left(\frac{2}{3}-2\mu \left(x\right)\right)\left(\left|\frac{2}{3}-2\mu \left(x\right)\right|-\frac{7}{3}\right)tan\left(\frac{1}{7}\pi \parallel \frac{2}{3}-2\mu \left(x\right)\left|-\frac{7}{3}\right|\right)sec\left(\frac{1}{7}\pi \parallel \frac{2}{3}-2\mu \left(x\right)\left|-\frac{7}{3}\right|\right)}{{\left.721\right|}_3^2-2\mu \left(x\right)\left|{\parallel }_3^2-2\mu \left(x\right)\right|-\frac{7}{3}\mid }\\ \frac{\partial f}{\partial \vartheta }=-\frac{135\pi \left(\frac{2}{3}-2\vartheta \left(x\right)\right)\left(\left|\frac{2}{3}-2\vartheta \left(x\right)\right|-\frac{7}{3}\right)tan\left(\frac{1}{7}\pi \parallel \frac{2}{3}-2\vartheta \left(x\right)\left|-\frac{7}{3}\right|\right)sec\left(\frac{1}{7}\pi \parallel \frac{2}{3}-2\vartheta \left(x\right)\left|-\frac{7}{3}\right|\right)}{721\left|\frac{2}{3}-2\vartheta \left(x\right)\right|{\parallel }_3^2-2\vartheta \left(x\right)\left|-\frac{7}{3}\right|}\end{array}$$

  (9)
  We obtain that $\frac{\partial f\left(\mu ,\vartheta \right)}{\partial \mu }\ge 0$ when $\mu \le \vartheta$ and $\frac{\partial f\left(\mu ,\vartheta \right)}{\partial \mu }\le 0$ when $\mu \ge \vartheta$, whereas $\frac{\partial f\left(\mu ,\vartheta \right)}{\partial \vartheta }\le 0$ when $\mu \le \vartheta$ and $\frac{\partial f\left(\mu ,\vartheta \right)}{\partial \vartheta }\ge 0$ when $\mu \ge \vartheta$. Thus, f is increasing with respect to $\mu$ when $\mu \le \vartheta$ and decreasing when $\mu \ge \vartheta$. Moreover, f is decreasing with respect to $\vartheta$ when $\mu \le \vartheta$ and increasing when $\mu \ge \vartheta$.
  Now, by using this property of the function, we can conclude that $E\left(S\right)\le E\left(\dot{S}\right)$, if A is less fuzzy than B, i.e., ${\mu }_A\left(x\right)\le {\mu }_B\left(x\right)\le \frac{1}{3}$ and ${\vartheta }_B\left(x\right)\le {\vartheta }_A\left(x\right)\le \frac{1}{3}$ for ${\mu }_B\left(x\right)\le {\mu }_A\left(x\right)$ or ${\mu }_A\left(x\right)\ge {\mu }_B\left(x\right)\ge \frac{1}{3}$ and ${\vartheta }_B\left(x\right)\ge {\vartheta }_A\left(x\right)\ge \frac{1}{3}$ for ${\mu }_B\left(x\right)\ge {\mu }_A\left(x\right)$.
• Symmetry: for the property, we have $S=\left({\mu }_s,{\vartheta }_s\right)$ as $S^c=\left({\vartheta }_s,{\mu }_s\right)$. Thus, we have

  $$\begin{array}{c}{E}_p^{\#}\left(S^c\right)=\frac{sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}=\\ =\frac{sec\left(\frac{1}{7}\parallel 3-2{\vartheta }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)+sec\left(\frac{1}{7}\parallel 3-2{\mu }_s\left(x_i\right)-\frac{7}{3}\left|-\frac{7}{3}\right|\pi \right)-\frac{334}{135}}{\frac{206}{135}}=E_p^{\#}\left(S\right)\end{array}$$

  (10)

□

4. Intuitionistic Fuzzy Multi-Criteria Decision-Making Based on COPRAS Approach

The “Complex Proportional Assessment”, commonly known as COPRAS, was first introduced in 1994 by Zavadskas and Kaklauskas [69]. This method is used to estimate the uniqueness of one alternative over another and presents it as reasonable to equate alternatives [70]. In addition, this method can be implemented to maximize and minimize criteria in an assessment where more than one criterion should be considered [66]. The COPRAS technique ranks and estimates alternatives step-by-step for their relevance and utility degree [71]. The development and extensions of this technique, e.g., in uncertainty problems [72,73], make it an advanced approach among the methods of multi-criteria decision-making. The method considers that the importance and advantage of the reviewed versions depend directly on and are comparable to a system of criteria appropriately specifying the alternatives and the values and weights of criteria [74]. The method determines a clarification and the ratio to the ideal solution and the ratio to the worst-ideal solution and consequently can be considered a compromising method. The method is applied for solving numerous difficulties by its exhibitors and their associates. Figure 1 can represent the stages of the COPRAS method.

Explanation of Figure 1:

Step 1:

Establishment of intuitionistic fuzzy decision matrix.

The initial move is relevant to the establishment of the intuitionistic fuzzy decision matrix. We determine the decision matrix $M={\left(H_i,J_j\right)}_{r\times n}$, where $H_i$ are the alternatives concerning the criteria $J_j$, such that $M_i=\left\{M_1,M_2,\dots ,M_m\right\},i=1,2,\dots ,m$ and $J_j=\left\{J_1,J_2,\dots ,J_n\right\},j=1,2,\dots ,n$.

$$\begin{array}{c}a{J}_1{J}_2\cdots J_n\\ \begin{array}{l}{M}_1\\ M_2\\ ⋮\\ M_m\end{array}\left[\begin{array}{cccc}\left({\mu }_{11},{\vartheta }_{11}\right)& \left({\mu }_{12},{\vartheta }_{12}\right)& \cdots & \left({\mu }_{1n},{\vartheta }_{1n}\right)\\ \left({\mu }_{21},{\vartheta }_{21}\right)& \left({\mu }_{22},{\vartheta }_{22}\right)& \cdots & \left({\mu }_{2n},{\vartheta }_{2n}\right)\\ ⋮& ⋮& \ddots & ⋮\\ \left({\mu }_{m1},{\vartheta }_{m1}\right)& \left({\mu }_{m2},{\vartheta }_{m2}\right)& \cdots & \left({\mu }_{mn},{\vartheta }_{mn}\right)\end{array}\right]\end{array}$$

(11)

Step 2:

Judgment of the weights of criteria.

Criteria weights symbolize a significant part of the clarification of MCDM issues. The weights can be obtained in different ways [75]. In a decision-making system, the experience concerning criteria weights is seldom entirely unknown or imperfectly known and somewhat known at specific times.

• For unknown criteria weights:
  If weights of criteria are entirely unknown, then we determine the weights by utilizing the following equation:

  $$w_j=\frac{1-E_p\left(S\right)}{{\sum }_{i=1}^m\left(1-E_p\left(S\right)\right)},\left(t=1,2,\dots ,m\right)$$

  (12)

  where $E_p\left(S\right)=\frac{3}{4\sqrt{3}+3}{\sum }_{i=1}^m{E}_p^{\#}\left(S\right)$.
• For partially known criteria weights:
  Because of the increasing intricacy of decision-making issues, it may not frequently be reasonable for the decision-makers to describe their viewpoint as exact numbers. In case the criteria weights are imperfectly known for MCDM issues,

  $$minA=\sum _{t=1}^m{w}_j{E}_p\left(S\right)$$

  (13)

Step 3:

Calculate the weighted decision matrix $D={\left[d_{ij}\right]}_{m\times n}$:

$$\mathrm{where},d_{ij}=w_j{n}_{ij}=\left(\sqrt{1-{\left(1-{\mu }_{ij}^2\right)}^{w_u}},{\vartheta }_{ij}^{w_j}\right)$$

(14)

where $(j=1,2,\dots ,m)$.

Step 4:

Calculate the score function:

$$g\left(d_{ij}\right)={\mu }_{ij}^2-{\vartheta }_{ij}^2$$

(15)

where $i=1,2,\dots ,m;j=1,2,\dots ,n$.

Step 5:

Determine the maximizing and minimizing index:

$$g\left(O_i\right)=\frac{1}{\left|B\right|}\sum _{j\in B}g\left(d_{ij}\right)$$

(16)

and

$$g\left(R_i\right)=\frac{1}{\left|NB\right|}\sum _{j\in NB}g\left(d_{ij}\right)$$

(17)

where, B is the set of benefit criteria and $NB$ is the set of non-benefit criteria, for all $(i=1,2,\dots ,m)$.

Step 6:

Determine the relative significance value of each alternative:

$$T_i=g\left(O_i\right)+\frac{{\sum }_{i=1}^m{e}^{g\left(R_i\right)}}{e^{g\left(R_i\right)}{\sum }_{i=1}^m\frac{1}{e^{g\left(R_i\right)}}}$$

(18)

where $i=1,2,\dots ,m$.

Step 7:

Determine the priority order:

$$V_i=\frac{T_i}{max{T}_i}\times 100$$

(19)

where $i=1,2,\dots ,m$.

Step 8:

Ranking of the alternatives:

The ranking of the alternatives is regulated in declining order based on the values of priority order. Thus, the highest final value has the highest rank.

5. Numerical Example

Let us consider determining the best private hospital in a large city. First, we have to survey four hospitals named A, B, C and D, with different facilities. Next, we select the five essential criteria by which the hospitals are evaluated, which are management ($J_1$), waiting lines ($J_2$), services ($J_3$), charges ($J_4$) and developed area ($J_5$). Here, {$J_1,J_3,J_5$} are taken as benefit criteria, whereas {$J_2,J_4$} are taken as cost criteria. Finally, the intuitionistic fuzzy decision matrix is shown in

Table 1. Intuitionistic fuzzy decision matrix.

Hospitals	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$	${\mathit{J}}_5$
A	0.41, 0.38	0.48, 0.57	0.36, 0.43	0.33, 0.37	0.28, 0.34
B	0.46, 0.37	0.48, 0.39	0.37, 0.41	0.35, 0.44	0.51, 0.39
C	0.36, 0.39	0.21, 0.37	0.41, 0.38	0.28, 0.34	0.32, 0.46
D	0.51, 0.39	0.37, 0.45	0.32, 0.46	0.37, 0.57	0.37, 0.45
${\mathbf{E}}_{\mathbf{p}}\left(\mathbf{S}\right)$	0.7409	0.6374	0.7783	0.7979	0.7525

. This study used the uncertain decisions library available at https://gitlab.com/kiziub/uncertain-decisions (accessed on 10 November 2021).

Using Equation (3), we calculate the entropies: $e_1=0.7409$, $e_2=0.6374$, $e_3=0.7783$, $e_4=0.7979$ and $e_5=0.7525$ (also shown in Table 1).

Corresponding to Equation (12), we compute the weights of each criterion across all the alternatives, utilizing the values of E_p(S): $w_1=0.2004$, $w_2=0.2804$, $w_3=0.1714$, $w_4=0.1563$ and $w_5=0.1914$.

Using Equation (15), we calculate the weighted decision matrix, shown in

Table 2. Weighted decision matrix.

	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$	${\mathit{J}}_5$
A	0.19028, 0.82374	0.26609, 0.85416	0.15334, 0.86528	0.13365, 0.85604	0.12452, 0.81342
B	0.21569, 0.81935	0.26609, 0.76794	0.15787, 0.85825	0.14220, 0.87954	0.23671, 0.83506
C	0.16562, 0.82804	0.11210, 0.75668	0.17623, 0.84714	0.11261, 0.84479	0.14306, 0.86187
D	0.24202, 0.82804	0.20110, 0.79938	0.13546, 0.87535	0.15084, 0.91587	0.16669, 0.85825

.

After calculating the weighted decision matrix, we obtain the value of the score function $g\left(d_{ij}\right)$, maximizing $g\left(O_i\right)$ and minimizing $g\left(R_i\right)$ index, relative significance value $T_i$ and priority order $V_i$. In the final step, we rank the alternatives in declining order based on the values of priority order. The highest final value has the highest rank.

The order of the alternatives of the proposed entropy is given in

Table 3. The score function and ranking of alternatives.

g	${\mathit{J}}_1$	${\mathit{J}}_2$	${\mathit{J}}_3$	${\mathit{J}}_4$	${\mathit{J}}_5$	${\mathit{O}}_{\mathit{i}}$	${\mathit{R}}_{\mathit{i}}$	${\mathit{T}}_{\mathit{i}}$	${\mathit{V}}_{\mathit{i}}$	Rank
A	−0.64235	−0.65879	−0.72521	−0.71494	−0.64614	−0.67123	−0.68686	−0.14581	100	$\mathrm{IV}$
B	−0.62482	−0.51892	−0.71167	−0.75338	−0.64130	−0.65926	−0.63615	−0.15982	109.61	$\mathrm{II}$
C	−0.65823	−0.56000	−0.68659	−0.70100	−0.72236	−0.68906	−0.63050	−0.19243	131.97	$\mathrm{I}$
D	−0.62708	−0.59856	−0.74789	−0.81606	−0.70881	−0.69459	−0.70731	−0.15832	108.57	$\mathrm{III}$

. Here, we have taken the hesitant intuitionistic fuzzy set to describe the uncertainty in the actual world. This entropy consists of a membership function, non-membership function and hesitant function. Therefore, the obtained results are more precise.

6. Comparative Analysis

In this section, we compare the results of the entropy introduced in this paper with those entropies introduced in earlier works [76,77,78,79]. We have used the same example for the comparison. Those entropies that we have taken for this comparison are listed below.

• Burillo and Bustince (1996) intuitionistic fuzzy set [76]:

  $$E\left(B\right)=\frac{1}{n}\sum _{i=1}^n\left[1-\left({\mu }_s\left(x_i\right)+{\vartheta }_s\left(x_i\right)\right)\right]$$

  (20)
• Szmidt and Kacprzyk (2001) intuitionistic fuzzy entropy [77]:

  $$E\left(K\right)=\frac{1}{n}\sum _{i=1}^n\frac{min\left({\mu }_s\left(x_i\right),{\vartheta }_s\left(x_i\right)\right)+{\pi }_s\left(x_i\right)}{max\left({\mu }_s\left(x_i\right),{\vartheta }_s\left(x_i\right)\right)+{\pi }_s\left(x_i\right)}$$

  (21)
• Liu and Ren (2014) introduced intuitionistic fuzzy entropy [78]:

  $$E\left(L\right)=\frac{1}{n}\sum _{i=1}^{n}cot\left(\frac{\pi }{4}+\frac{\left|{\mu }_s^2\left(x_i\right)-{\vartheta }_s^2\left(x_i\right)\right|}{4}\pi \right)$$

  (22)
• Ye (2010) intuitionistic fuzzy set [79]:

  $$\begin{array}{c}E\left(Y\right)=\frac{1}{n}{\displaystyle \sum _{i=1}^n}\left[\left(sin\frac{{\mu }_s\left(x_i\right)+1-{\vartheta }_s\left(x_i\right)}{4}\pi +\right.\right.\\ \left.\left.+sin\frac{{\vartheta }_s\left(x_i\right)+1-{\mu }_s\left(x_i\right)}{4}\pi -1\right)\times \frac{1}{\sqrt{2}-1}\right]\end{array}$$

  (23)

We have analyzed the priority order and rank of the alternatives. The entropies (20)–(23) that we have taken for the resemblance contain only the membership function $\left(\mu \right)$ and non-membership function $\left(\vartheta \right)$, whereas the entropy that we have introduced involves a membership function $\left(\mu \right)$, non-membership function $\left(\vartheta \right)$ and the hesitant function $\left(\pi \right)$. Therefore, due to the hesitancy measure, the alternative order that we have obtained is slightly different. The results obtained are presented using

Table 4. Comparison of $V_i$ and rank with different entropies.

	Proposed Entropy		Burillo and Bustince [76]		Szmidt and Kacprzyk [77]		Liu and Ren [78]		Ye [79]
	${\mathit{V}}_{\mathit{i}}$	Rank	${\mathit{V}}_{\mathit{i}}$	Rank	${\mathit{V}}_{\mathit{i}}$	Rank	${\mathit{V}}_{\mathit{i}}$	Rank	${\mathit{V}}_{\mathit{i}}$	Rank
A	100	IV	100	IV	100	IV	105.77	III	100	IV
B	109.61	II	103.74	III	103.152	III	100	IV	103.23	III
C	131.97	I	129.67	I	128.631	I	126.03	I	129.43	I
D	108.57	III	105.25	II	104.78	II	120.17	II	104.61	II

. However, the analysis shows that the most likely preferred alternative is C, unanimously selected by all the entropy measures. Therefore, hospital C is the best choice among the ranking of the top hospitals. Figure 2 shows a comparison of the obtained rankings using WS and weighted Spearman coefficients [80], which once again shows the high similarity of the obtained results.

The advantages are as follows: in a comparable study, it is apparent that the $V_i$ for suggested entropy is more well-defined than existing intuitionistic fuzzy entropies. Therefore, the proposed novel entropy measure is more reliable and feasible to solve complex multi-criteria decision-making problems.

7. Conclusions

In this paper, we have introduced a novel intuitionistic fuzzy entropy with hesitancy measure. The proposed entropy measure develops a new approach of the COPRAS method based on the MCDM problem. We have taken an example to compare our entropy to some current entropies of intuitionistic fuzzy sets. The results achieved from the example show that the submitted entropy is more reliable as it has a hesitancy measure. The results of the new entropy measure are examined and interpreted favorably with the selected entropy measure on an intuitionistic fuzzy set. The intuitionistic fuzzy set is very efficient and suitable for handling uncertainty in complex MCDA problems. The proposed measure can be used in future research on multi-criteria decision analysis as a component of new objective methods for determining attribute weights using intuitionistic fuzzy sets. In addition, taking part in the decision-making process can help rank and select problems in various domains as a standard method for uncertain decision analysis.