#### Abstract

In this study, a new parametric method is proposed to rank intuitionistic fuzzy numbers in a general form. One of the advantages of the proposed method is that the decision maker’s idea is taken into account by selecting appropriate amounts of decision level and hesitation degree parameters. In some illustrative examples, the superiority of the proposed method over some other approaches is demonstrated. Furthermore, to show the ability of the method to solve intuitionistic fuzzy optimization problems, the proposed method is applied to solve intuitionistic fuzzy network data envelopment analysis (IFNDEA) problems. Also, in three appropriate examples, the validity of the suggested method and its capacity to solve real-world problems are illustrated.

#### 1. Introduction

Intuitionistic fuzzy sets (IFSs), first proposed by Atanassov [1, 2], are a generalization of Zadeh’s fuzzy sets [3, 4] to model noncrisp and uncertain sets. For these sets, we define both membership and nonmembership functions. In this case, we will be able to define a hesitation function that is the difference between the “membership function” and “one minus nonmembership function.” Thus, by IFSs, we can efficiently model imperfect information. Many researchers provided enormous works on IFSs in both theory and applications (see e.g., [5–8]). As an application, Akram et al. [9] proposed a novel decision-making method based on hypergraphs in the intuitionistic fuzzy environment and applied it to real-life problems.

In 1994, by proposing intuitionistic fuzzy numbers (IFNs), Burillo et al. [10] generalized a fuzzy numbers concept and paved the way for applying intuitionistic fuzzy logic in real-life problems such as decision-making and risk analysis. For example, Liu and Wang [11] and Wang and Liu [12] proposed new Schweizer–Sklar Maclaurin and Einstein operation rules for IFNs and applied them to solve real-life decision-making problems.

Ranking IFNs is one of the important subjects in IFS theory, and a considerable number of articles have been published in this field. In these papers, different approaches are employed to compare and rank IFNs. For example, Grzegorzewski [13] generalized the ranking approach of fuzzy numbers in Grzegorzewski [14] and suggested a new method for ranking IFNs by defining concepts of expected interval and expected value. Later, Ye [15] calculated the expected value of a TraIFN that was introduced by Grzegorzewski [13] and applied it to a trapezoidal intuitionistic fuzzy multicriteria decision-making problem. Using four pairwise ranking functions, Mitchel [16] defined the average intuitionistic fuzzy rank index and proposed an approach to compare each pair of IFNs. Using score and accuracy functions of trapezoidal intuitionistic fuzzy numbers (TraIFNs), Jianqiang and Zhong [17] proposed a stepwise ranking method and applied it to a multicriteria decision-making problem. Wang and Zhang [18] ranked TraIFNs by converting them into interval numbers. Nehi [19] proposed a generalization of the characteristic value first suggested by Chiao [20] and ranked IFNs. In the work of Nan et al. [21], by introducing average indexes of membership and nonmembership functions, TraIFNs are ranked, and then, the approach is applied to matrix games with payoffs. Later, in a note, Verma and Kumar [22] modified some mathematical incorrect assumptions in the work of Nan et al. [21]. Value and ambiguity concepts were first introduced by Delgado et al. [23], and employing these concepts is a famous approach for ranking IFNs (see e.g., [24–28]). For example, in the work of Li [26], value and ambiguity are used to rank nonnormal triangular intuitionistic fuzzy numbers (TriIFNs), and then, the proposed method is applied to a personnel selection problem. In addition, Chutia and Saikia [25] generalized value and ambiguity concepts for membership and nonmembership functions and proposed a ranking method for IFNs using *α* and *β* cuts. They also applied the proposed method in a risk analysis problem with intuitionistic fuzzy data. Wan [29], using possibility mean, possibility variance, standard deviation, and variance coefficient of TriIFNs, introduced possibility variance coefficients of membership and nonmembership functions to create a lexicographic ranking method. Das and Guha [30] ranked TraIFNs by calculating the centroid points and compared them. Li and Chen [31] proposed a distance index between two arbitrary TraIFNs based on *α*-cuts, which was, then, applied to the intuitionistic fuzzy multicriteria group decision-making problem. In the work of Lakshmana Gomathi Nayagam et al. [32], eight score functions are defined and applied to rank TraIFNs. By appropriate examples, the method is compared with some other approaches in this field. Also, Nayagam et al. [33] generalized a lexicographic ranking method to propose a total ordering of IFNs using *α*-cut and *β*-cut sets. In another work, Nayagam et al. [28] proposed a parametric score function using improved value and ambiguity indexes to rank TraIFNs and applied it to solve a multicriteria decision-making problem. Aggarwal and Gupta [34] defined an index to rank generalized symmetrical TraIFNs and solved an intuitionistic fuzzy solid transportation problem. Prakash et al. [35] introduced a ranking method by calculating the centroid of TraIFNs. Singh and Yadav [36] defined score and accuracy indices for the LR type of normal IFNs and proposed an order relation. Furthermore, by applying these indices, they proposed a method for solving a fully intuitionistic fuzzy linear-programming problem. Later, Canedo and Morales [37] analyzed this approach and modified it to obtain unique optimal solutions. Darehmiraki [38] extended the method by Shureshjani and Daremiraki [39] to rank IFNs and applied it to the partner selection problem. Table 1 summarizes the abovementioned studies based on the relevant themes. Although many methods have been proposed, there is still not a method to rank different kinds of IFNs, which can also be easily employed in optimization problems such as Data Envelopment Analysis (DEA) under intuitionistic fuzzy environment via the decision-maker idea.

DEA is a nonparametric and linear-programming-based approach for evaluating the efficiency of decision-making units (DMUs) with multiple inputs and outputs data. From the first work by Charnes et al. [48], there are many studies on both theory and applications in this field. In DEA models, a DMU is considered as a black box with its interior structures not taken into account. However, in many applications, DMUs have important interior structures, and ignoring them has led to inappropriate results. Network DEA is an attempt to consider the interior structures of DMUs. Among all the proposed structures, the simplest and most basic network DEA model is two-stage network DEA models. Also, in conventional DEA models, all the data should be crisp and certain which is not always possible in real applications. To solve these problems, fuzzy numbers and IFNs, fuzzy DEA, and intuitionistic fuzzy DEA (IFDEA) models are proposed, respectively (see Emrouznejad et al. [49] for a literature review on fuzzy DEA models and its applications until 2013). Fuzzy DEA can efficiently model DEA problems with noncrisp data, but when we face both noncrisp and uncertain data, IFDEA models are the best alternative. Although many papers have been published on fuzzy DEA, the proposed works in the IFDEA field are limited. For example, Xu et al. [40] introduced a fuzzy superefficient cross-DEA model to measure the efficiency of logistics enterprises using an interval-valued intuitionistic fuzzy Bayesian network. Using a weighted aggregation operator, Razavi Hajiagha et al. [41] proposed an approach to evaluate the efficiency of DMUs in an intuitionistic fuzzy BCC model. Puri and Yadav [42], by applying the expected interval and expected value (Grzegorzewski [13]), proposed an index to estimate triangular intuitionistic fuzzy input and output data in an IFDEA model. Similar to the work of Puri and Yadav [42], by applying the expected value (Grzegorzewski [13]), Singh [43] transformed an intuitionistic fuzzy DEA/AR model into a conventional DEA/AR model. Using *α*-cut and *β*-cut concepts in IFS theory, Arya and Yadav [44] introduced lower and upper bounds for the efficiency of DMUs in intuitionistic fuzzy SBM and superefficiency intuitionistic fuzzy SBM models. In another paper, Arya and Yadav [45] proposed appropriate models based on *α*-cuts and *β*-cuts to provide lower and upper bounds of efficiency measures in IFDEA, and then, by applying the index proposed by Chen and Klein [50], they ranked DMUs. Also, their method was applied to evaluate and rank 16 hospitals in India. Moreover, Arya and Yadav [46] developed intuitionistic fuzzy BCC and intuitionistic fuzzy superefficiency BCC models with triangular IF inputs and outputs data. They developed the proposed approach to determine the efficiencies and rankings of DMUs in the presence of infeasibility. Finally, in Ameri et al. [47], by substituting the IF inputs and IF outputs of a parallel IFNDEA model with their assigned expected values (Grzegorzewski [13]), the parallel IFNDEA model transforms into a linear program. In a case study, their method was applied for the self-assessment of an Iranian hospital (See Table 1).

Two important limitations of the abovementioned approaches in IFDEA are that, almost all of them work for a specific group of IFNs (triangular IFNs) only and the decision maker does not play any role in the decision-making process. In addition, they are not easily generalizable to intuitionistic fuzzy network DEA (IFNDEA) models. Despite a large number of papers published in the IFDEA field, there has been one published paper (by Ameri et al. [47]) in the IFNDEA field.

The main motive of this study is to overcome the shortcomings listed above by assigning an appropriate parametric index to IFNs considering two main concepts in IFSs, i.e., alpha cut and hesitation degree. In this case, by proper selection of the decision level (alpha cut) and the hesitation degree, the decision maker’s idea is accounted for in the decision-making process, and less information is lost in IFNs. This parametric index can be applied to compare and rank IFNs.

Also, by substituting the IF data with their assigned parametric indexes, an intuitionistic fuzzy decision-making problem transforms into a parametric decision-making problem that can be easily solved by proper selection of the parameters. In this study, a new parametric method is developed to rank IFNs. Also, some reasonable properties to rank IFNs are examined based on the work of Wang and Kerre [51]. To show the ability of the method in real-life problems, it is applied to solve the IFDEA problem in single-stage, series, and parallel structures. As will be seen, the proposed method is also easily generalizable to other kinds of IFNDEA problems.

The remainder of this research is organized as follows. In Section 2, some preliminaries of IFSs, IFNs, and the related arithmetic operators are introduced. In Section 3, using decision level and hesitation degree concepts, a new parametric method is proposed to rank IFNs in a general form, by appropriate examples, the advantages of the proposed method are illustrated, and the obtained results are finally compared with some other approaches in this field. In Section 4, we applied our proposed method to solve the IFNDEA problem. In this method, IFDEA and IFNDEA models are transformed into parametric DEA and parametric NDEA models, respectively. Three examples illustrate the method. Finally, Section 5 concludes the paper.

#### 2. Preliminaries

This section includes some basic concepts and notions. To easily understand the equations in this paper, some mathematical symbols are summarized in Table 2.

*Definition 1. *(see [1, 2]). Let be a fixed universe. An IFS in is given aswhere are the degrees of membership and nonmembership functions from to , respectively. Also, and .

*Definition 2 (see [1, 2]). *The hesitation function of an IFS is defined asIt is clear that . This function indicates the hesitation degree of an element (indeterminacy) belonging or not belonging to .

In fuzzy sets, we do not have a hesitation degree (hesitation degree is zero). Thus, we can consider each fuzzy set as a special case of an IFS as

*Definition 3 (see [5]). *An IFN is an IFS in the set of real numbers with the membership function asand the nonmembership function aswhere and are nondecreasing continuous functions from to and and are nonincreasing continuous functions from to . Also, . In this definition, is the maximum degree of the membership function and is the minimum degree of the nonmembership function. So, according to the Definition 1, and (Figure 1).

In Definition 3, if and , then, we have a normal IFN.

Considering the abovementioned definition (Definition 3), *α*-cut, *β*-cut, and -cut sets of an IFN are defined as follows.

*Definition 4 (see [2]). **α*-cut, *β*-cut, and -cut sets of an IFN are crisp subsets of that are defined as follows:*α*-cut set: , , *β*-cut set: , , and -cut set:Li et al. [27] proposed a new approach for defining a TriIFN, and here, we generalized their approach for TraIFNs as follows:

*Definition 5. *A TraIFN is an IFN with membership and nonmembership functions as follows:We denote a TraIFN by (Figure 2).

In Definition 5, if and , then, we have a normal TraIFN. Also, TriIFN is a special case of TraIFN in which and .

Figure 3 shows the hesitation degree of belonging or not belonging an element to TraIFN .

By Definition 4, *α*-cut, *β*-cut, and -cut sets of a TraIFN are the following closed intervals, respectively,We can see that the (*α*)-cut set of a TraIFN can be calculated from the intersection of the closed intervals obtained from the -cut and -cut sets of . These three cut sets (-cut, -cut, and -cut sets) remove some of the elements of a given TraIFN, whose degrees of its membership and nonmembership functions do not satisfy the levels given as arguments of the corresponding operators.

*Definition 6. *(see [25]). Let and be two arbitrary TraIFNs and ; then, the arithmetic operations are stipulated as follows:Ma et al. [52] proposed a new approach for defining a fuzzy number. In this approach, the variable on the vertical axis in Figure 1 is considered as an independent variable. By generalizing this approach, we obtain the following definition for IFNs.

*Definition 7. * is an IFN, in which is a pair of functions which satisfy the following requirements:(1) is a bounded monotonic increasing left continuous function(2) is a bounded monotonic decreasing left continuous function(3)and is a pair of functions which satisfy the following requirements:(1) is a bounded monotonic decreasing left continuous function(2) is a bounded monotonic increasing left continuous function(3)So, by Definition 7, we can represent a TraIFN as , where , and (Figure 4).

As mentioned above, in Definition 7, the variable on the vertical axis is considered as an independent variable. It provides the ability to obtain the location of the left and right parts of the membership and nonmembership functions of an IFN on the horizontal axis for different amounts of the independent variable. In the following, we will use this property to assign an appropriate parametric index to IFNs.

#### 3. Ranking Intuitionistic Fuzzy Numbers (IFNs)

In real-world applications, dealing with imprecise information is a common problem, and the uncertainty is unavoidably involved in every real-world problem. To model these problems, we should correctly incorporate the uncertainty concept into the problem description. Because of the ability of the intuitionistic fuzzy numbers to express imprecise information (by applying the hesitation concept), they are a useful tool to model these problems. One of the first important issues after the definition of intuitionistic fuzzy numbers is how to compare and rank them. The comparison and ranking of intuitionistic fuzzy numbers are complicated, and different methods have been proposed to address this problem.

Engaging the decision maker in the decision process provides flexibility and is a fascinating phenomenon [24]. Using *α*-cuts, Shureshjani and Darehmiraki [39] proposed a parametric method to rank fuzzy numbers based on the decision maker’s opinion. In addition to *α*-cuts, we have another important concept in IFSs, namely, hesitation. Hesitation in an IFN, as shown in Figure 3, is an area between “one minus nonmembership function” and “membership function.” It is clear that, optimally, this area will be added to the membership function, and in the most pessimistic view, we should consider this area as a part of the nonmembership function. Thus, in ranking IFNs, it would be appropriate to consider a degree of optimism (or pessimism) on the part of the decision maker in the decision-making process.

In this section, to consider the abovementioned points, we generalize Shureshjani and Darehmiraki’s method [39] and propose a new parametric method for ranking IFNs based on the decision level and hesitation degree parameters.

##### 3.1. The Proposed Index

For an arbitrary IFN , we assign the following index:where and belong to .

The selection of *α* and

*k*parameters depend on the decision maker’s idea. From the index definition, we can see that, after the selection of

*α*parameter, only the elements of an IFN with the membership and hesitation values of larger than or equal to

*α*will be important. Therefore, if we choose

*α*parameter close to one, we have a “high-level decision,” and a “low-level decision” is made when the selected

*α*is close to zero.

If we rewrite the abovementioned formula, we will have

It is clear that if the decision maker sets , we have a pessimistic decision because we have not considered the hesitation area in our decision (hesitation area is added to the nonmembership function). Also, if the decision maker sets , an optimistic decision has been made because all the hesitation areas are added to the membership function. It should be noted that any choice of *k* reflects the desirable hesitation degree that is considered by the decision maker in ranking IFNs process.

For example, for a TraIFN , , and are represented in Figure 5.

Let be a TraIFN; then, we have

So, the assigned index to a TraIFN will be as follows:

Figure 6 shows the geometric meaning of the index for an arbitrary TraIFN with and , respectively. We can see that the value of the index is the summation of the marked areas with solid and empty dots.

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The following theorems demonstrate the reasonable behavior of the proposed index for specific types of IFNs.

Theorem 1. *If and , then, , we have*

*Proof. *

Theorem 2. *Let and be two arbitrary TraIFNs where and ; then, we have*

*Proof. *From Definition 6, we have

Theorem 3. *Let and ; then, we have*

*Proof. *From Definition 6, we haveLet ; then, we haveNow, if , then, we have

Corollary 1. *If and , then, we have*

##### 3.2. Our Proposed Ranking Approach

Based on the proposed index, we can give the following definitions to rank IFNs.

*Definition 8. *Let and be two arbitrary IFNs; then, for a decision level higher than *α* and a selected hesitation degree *k*, we have(1),(2),(3).

*Definition 9. *Let and be two arbitrary IFNs; then, by considering , we have(1),(2),(3).Wang and Kerre [51] suggested seven reasonable properties to evaluate the rationality of the proposed indices for ranking fuzzy numbers. Here, we examine the validity of our proposed index for ranking IFNs based on Wang and Kerre’s [51] approach.

Let *I* be the set of intuitionistic fuzzy quantities and be a finite subset of *I*.

*Property 1. *For an arbitrary finite subset of *I* and , ( has at least the same ranking as ) by the proposed ranking method on .

*Proof. *After calculating the , it is clear that we have . So, according to Definition 9, .

*Property 2. *For an arbitrary finite subset of I and , and by the proposed ranking method on , we should have by the proposed ranking method on .

*Proof. *Let and ; then, simultaneously, we have and So, and according to Definition 9, .

*Property 3. *For an arbitrary finite subset of I and , and by the proposed ranking method on , we should have by the proposed ranking method on .

*Proof. *From and , we have and . These imply that that means .

*Property 4. *Let and be two arbitrary IFNs of an arbitrary finite subset of I with and , and , we should have by the proposed ranking method on .

*Proof. *Using the index, we have where is the mean value of *α*-cut and is the mean value of *β*-cut .

Similarly, Therefore, , i.e., .

*Property 5. *Let I and be two arbitrary finite sets of intuitionistic fuzzy quantities in which the proposed ranking method can be applied and and are in . By the proposed ranking method on iff on I.

*Proof. *Trivial.

*Property 6. *Let , and are three arbitrary TraIFNs of an arbitrary finite subset of I and with and . If , we should have by the proposed ranking method on .

*Proof. *Let , so . From Theorem 2, we have , which means .

*Property 7. *Let and are two arbitrary TraIFNs of an arbitrary finite subset of I and , by the proposed ranking method on and , for we should have and for we should have by the proposed ranking method on .

*Proof. * implies that . From Theorem 3, if , we have that means . Also, for , that means .

##### 3.3. The Advantages of the Proposed Method

We demonstrate the advantages of the proposed method over some other approaches by illustrative examples.

In Example 1, we will see that the proposed method can efficiently compare intersected TraIFNs, and the decision maker can apply his/her preferences by choosing appropriate amounts of decision level (*α*) and hesitation degree (*k*). Similar to our proposed method, Darehmiraki [38] tried to extend Shureshjani and Daremiraki’s [39] method to rank IFNs. However, Example 2 shows that there are some situations where this method led to inappropriate results. As mentioned in the introduction section, various methods have been proposed for ranking IFNs. All these methods can be classified into two categories: parametric and nonparametric. In the parametric approach, using value and ambiguity concepts is the most popular (see e.g. [25–28]). Among these papers, two of the newest are that of Nayagam et al. [28] and Chutia and Saikia [25]. In Example 3, our parametric method is compared with that proposed by Nayagam et al. [28] and Chutia and Saikia [25] and two other nonparametric methods (by Ye [15] and Nayagam et al. [32]).

*Example 1. *Consider two intersected TraIFNs and . index of and are calculated as follows: and , .

In Figure 7, functions of TraIFNs and are plotted. As can be seen, different decision levels (*α*-levels) and hesitation degrees (*k*) led to different ranking results. For example, in an intermediate-level decision , if a low hesitation degree (*k* = 0) is selected by a decision maker, we have , but for a high hesitation degree (*k* = 1), we obtain . Also, for a relatively high hesitation degree (*k* = 0.7), if a low-level decision is chosen, , but for an intermediate-level decision , . Equality can happen for different amounts of and *k* too (see Table 3).

*Example 2. *Consider two TraIFNs and . It is clear that the acceptable result will be (see Figure 8).

In an attempt to generalize Shureshjani and Darehmiraki’s [39] method to rank IFNs, Darehmiraki [38] proposed an index using *α*- and *β*-cuts. Practically, however, it is evident that this index leads to inappropriate results.

The obtained results from Darehmiraki’s [38] method for different amounts of the parameters are presented in Table 4. As observed, the results are not correct.

But from our proposed method, we haveIt is clear that, for all decision levels of and hesitation degrees of , we have(See Figure 9). So from Definition 9, we obtain that is a reasonable result.

*Example 3. *Consider three intersected TraIFNs , and .

From the proposed index, we haveThese parametric functions are compared and plotted at different decision levels of and hesitation degrees of *k* in Figure 10.

As can be seen from Figure 10, for different amounts of and , we obtain 5 different ranking results as follows:(1)If , then, , and from Definition 8, we have (2)If , then, , and so (3)If , then, , and so (4)If : , and so (5)If : , and so By applying Chutia and Saikia’s [25] method, after plotting value function of and , it can be seen that, for most values of *α* and *β*, we obtain (see Figure 11), so for these values, we have . For specific amounts of *α* and *β* such as , we have , so (see Figure 11). Also, for all amounts of *α* and *β*, we obtain , so we have (see Figure 11). Moreover, for different amounts of *α* and *β*, we obtain different ranking results between and (see Figure 11). So, by appropriate amounts of *α* and *β*, we can obtain the following ranking results:Ambiguity function of and is plotted in Figure 12 too. We can see that the equality condition between these IFNs will not occur by Chutia and Saikia’s [25] method. Therefore, unlike our method, the ranking results of or are not obtained by Chutia and Saikia’s [25] method. Besides, unlike Chutia and Saikia’s [25] method that apply a two-level approach to compare and rank IFNs, we assign a parametric index to IFNs which is dependent on the decision maker’s idea. Therefore, we can easily apply our method for solving optimization problems under intuitionistic fuzzy environments.

Using the parametric score function of Nayagam et al. [28], we haveWe can see that, for all amounts of *λ* parameter , we obtain (see Figure 13), so from the work of Nayagam et al. [28], . This result is the same as our obtained ranking results for .

If we use Ye [15] index, we obtain , , and , so we have , which agrees with our results for and .

In another paper, Lakshmana Gomathi Nayagam et al. [32] proposed a nonparametric index for ranking IFNs. By considering Lakshmana Gomathi Nayagam et al.’s [32] index, we obtain , , and . Thus, based on their index, we have which is not achieved by Ye [15], Chutia and Saikia [25], and our proposed method.

From the abovementioned examples, we can see that the proposed parametric method has appropriate discrimination power to rank IFNs based on the decision level (*α*) and hesitation degree (*k*).

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