Development of Bi-Objective Fuzzy Data Envelopment Analysis Model to Measure the Efficiencies of Decision-Making Units

Abstract

The proposed bi-objective fuzzy data envelopment analysis (BOFDEA) model is a new approach to assess the performance efficiency of decision-making units (DMUs) in uncertain environments using $\alpha$-cuts. The model is based on fuzzy data envelopment analysis (FDEA) and considers two objectives, and a solution method and ranking system are provided. Generally, the efficiency score obtained for a DMU using the $\alpha$-cut approach is an interval. Intervals are partially ordered sets, due to which ranking intervals is a challenging task. The proposed BOFDEA model with $\alpha$-cuts provides the efficiency of DMUs in the crisp form, not in the form of intervals. Due to this, ranking DMUs with the proposed method’s help becomes very easy and less computationally. The proposed model has been validated through numerical examples, and a real-world application in the education sector has been shown to demonstrate its practicality.

1. Introduction

The data envelopment analysis (DEA) is a tool that may be utilized to quantify the relative performance efficiency of decision-making units (DMUs). Charnes, Cooper, and Rhodes developed a model known as the CCR DEA Model in 1978 [1] to measure how well DMUs work. To investigate the effects of scale, Banker, Charnes, and Cooper [2] extended the CCR model to the BCC model. DMUs are a type of unit that generates multiple numbers of outputs from multiple numbers of inputs. Key examples that qualify as DMUs include higher education institutions, libraries, hospitals, banks, airlines, and other similar businesses. The output-to-input ratio of the DMU is the best way to express efficiency, which can be written as efficiency equals output divided by input. The definition of what is meant by the term “relative efficiency” is the ratio of the efficiency of a DMU to the efficiency that has been determined to be the highest efficiency being considered. On the scale, it has a value between 0 and 1. If a DMU has a score of 1 for efficiency, then we consider it to be efficient; if it does not have a score of 1, we consider it inefficient.

It is impossible to retrieve accurate input and output data in real-world settings because of certain ambiguity and fuzziness. The ambiguity or fluctuation of data can manifest itself in various forms, including intervals, ordinal relations, fuzzy numbers, and so on. Numerous studies have been carried out to make sense of this kind of ambiguous data [3,4,5,6,7,8]. This research proposes treating inputs and outputs as fuzzy numbers to extend the crisp DEA model into a fuzzy DEA (FDEA) model. This would involve treating the inputs and outputs as fuzzy numbers. This expansion is motivated by FDEA models providing a more accurate perspective of real-world applications than traditional DEA models. A plethora of information is available regarding the FDEA and the ranks of DMU. There are a variety of DEA models (input-oriented, output-oriented, slack-based measure, and New slack model) and their applications in the literature. Some of these domains include education, health, and banking [9,10,11,12,13].

Arya and Yadav [3] developed the $\alpha$-cut approach to measuring the performance efficiencies of DMUs in a fuzzy environment. A multi-objective DEA model was developed by Singh et al. [8] to address the issue of dealing with the performance efficiencies of DMUs in an uncertain environment. This research solved a value-based fuzzy multi-objective DEA model using the $\alpha$-cuts. Several researchers have proposed a multi-objective DEA model to measure the performance of DMUs [8,14,15]. To investigate the connection between the prosperous operations of bank holding companies (BHCs) and their intellectual capital level, Wang et al. [16] constructed a DEA model with two stages. Research into the innovation ratio allowed for the achievement of this goal. After that, a fuzzy multi-objective programming approach was implemented to determine the effectiveness ratings. Using a fuzzy multi-objective multi-period network DEA model, Tavana et al. [17] reviewed the dynamic performance of oil refineries in the presence of unwanted outputs. They developed and implemented a fuzzy multi-objective multi-period network DEA model to do this. Using this model, Tavana et al. could evaluate the dynamic performance of the refineries. They utilized a typical fuzzy operator to determine the levels of efficiency that were present. The term “having maximum potential” refers to a singular framework for maximization that deftly integrates the numerous objectives and periods contained within it. In a different piece of research, Boubaker et al. [18] investigated, with the help of a fuzzy multi-objective two-stage DEA technique, how the levels of productivity of banks that are affiliated with single-bank holding companies and those that are affiliated with multi-bank holding companies are distinct from one another. This was accomplished by examining the similarities and differences between the affiliated banks with single-bank holding companies and those affiliated with multi-bank holding companies.

In this study, we created a bi-objective model based on $\alpha$-cuts to assess the effectiveness of DMUs in a fuzzy setting. With the help of $\alpha$-cuts, Arya and Yadav [3] created lower and upper-bound efficiency models to evaluate the effectiveness of DMUs. An efficiency interval per DMU is formed by the determined minimum and maximum efficiencies using their method. Due to partial ordering in intervals, ranking these intervals is once again difficult. However, the efficiency achieved with our proposed bi-objective approach is razor-sharp. As a result, establishing a hierarchy is a breeze. Further, the time and effort needed to calculate the results of the suggested bi-objective method for assessing DMU performance are reduced. Despotis and Smirlis [4] constructed a model to compute the upper and lower bounds of interval efficiency for each DMU to handle ambiguous situations. Despotis and Smirlis accounted for $j=k$ and $j\ne k$ in their model proposal. Once $\alpha$-cut bounds for efficiency (both lower and upper) are obtained, the intervals can be ranked. However, this problem can be solved by replacing the traditional FDEA model with the proposed bi-objective model that uses $\alpha$-cuts to calculate two alternative DMU efficiencies. As a result, the suggested bi-objective method for assessing DMU performance necessitates less work. The proposed bi-objective FDEA model’s efficiency scores are compared with the geometric average efficiency scores computed by Wang et al. [16]. The proposed model worked very efficiently, and it required very less computational effort for the efficiency evaluation of DMUs. Moreover, a real-life application to the education sector is presented.

The rest of the sections of the paper are: Important definitions, multi-objective optimization problems, and bi-objective optimization problems are given in Section 2. These definitions and concepts are used in the whole paper where needed. Section 3 provides an overview of the proposed bi-objective DEA model in a fuzzy environment and further information. In this section, the complete methodology is presented for the development of a bi-objective DEA model in a fuzzy environment. In Section 4, the author unveils the ranking algorithm, which uses a ranking approach and fuzzy parameters. Section 5 demonstrates how the proposed approach can be used in education. A comparison of the proposed methodology is also presented in this section to validate the methodology. This study summarizes everything in Section 6.

2. Bi-Objective Optimization Problem (BOOP)

The multi-objective optimization problem (MOOP) refers to a challenge in which several objectives must be optimized simultaneously. In its most basic form, MOOP is expressed as follows:

$$\begin{array}{c}\hfill \left(\mathbf{MOOP}\right)MinimizeZ\left(X\right)=[z_1\left(X\right),z_2\left(X\right),\dots ,z_k\left(X\right)]\end{array}$$

(1)

$$\mathrm{subject}\mathrm{to},$$

$$\begin{array}{c}\hfill c_i\left(X\right)=0\forall i=1,2,\dots ,q_1;\\ \hfill c_i\left(X\right)\ge 0\forall i=q_1+1,q_1+2,\dots ,q_2;\\ \hfill c_i\left(X\right)\le 0\forall i=q_2+1,q_2+2,\dots ,q;\\ \hfill x_j^{lower}\le x_j\le x_j^{upper}\forall j=1,2,\dots ,n;\end{array}$$

where $k\ge 2$, $X={[x_1,x_2,...,x_n]}^T$ is n dimensional decision vector from the decision space S defined by

$$\begin{array}{c}\hfill S=\{X\in {\mathbb{R}}^n:\{c_i\left(X\right)=0,i=1,2,...,q_1\}\cap \{c_i\left(X\right)\ge 0,i=q_1+1,q_1+2,...,q_2\}\\ \hfill \cap \{c_i\left(X\right)\le 0,i=q_2+1,q_2+2,\dots ,q\}\}\end{array}$$

(2)

where $z_p:S\to \mathbb{R},p=1,2,...,k$ = objective function,

• $c_i:S\to \mathbb{R},i=1,2,...,q$ = constraint functions,
• $F\subset {\mathbb{R}}^k$ = feasible objective space which is defined by $F=\{Z\left(X\right)\in {\mathbb{R}}^k|X\in S\}$
• The elements of F = objective vectors;
• $x_j^{lower}$ = lower bound of the decision variables
• $x_j^{upper}=$ upper bound of the decision variables.

Definition 1.

If $\overline{X}\in S$ optimizes all the k-objectives simultaneously, i.e., $z_p\left(\overline{X}\right)\le z_p\left(X\right)\forall p=1,2,...,k\&\forall X\in S$ for MOOP, then $\overline{X}$ is called the perfect solution or ideal solution.  However, we rarely have such solutions because the goals are often at odds with one another. The Pareto solution is the optimal compromise solution when this occurs.

Definition 2.

In Equation (1), if $k=2$, then MOOP becomes BOOP.

3. Proposed Bi-Objective Fuzzy DEA Model

Our objective is to evaluate the performance efficiencies of n homogeneous DMUs ($DM{U}_j;j=1,2,3,...,n$). Suppose that $DM{U}_j$ consumes m inputs $x_{ij},i=1,2,3,...,m$ to produce s outputs $y_{rj},r=1,2,3,...,s$. Let $u_{ik}$ and $v_{rk}$ be the weights corresponding to the $i^{th}$ input $x_{ik}$ and $r^{th}$ output $y_{rk}$ of $DM{U}_k(k=1,2,3,...,n).$ Then the efficiency ($E_k$) of $DM{U}_k$ is given by

Model 1 

(CCR DEA model [19]): For $k=1,2,3,...,n,$

$$\begin{array}{ccc}& Max{E}_k=\frac{{\displaystyle \sum _{r=1}^s}{y}_{rk}{v}_{rk}}{{\displaystyle \sum _{i=1}^m}{x}_{ik}{u}_{ik}}\hfill & \\ & \mathrm{subject}\mathrm{to}{\displaystyle \frac{{\displaystyle \sum _{r=1}^s}{y}_{rj}{v}_{rk}}{{\displaystyle \sum _{i=1}^m}{x}_{ij}{u}_{ik}}}\le 1\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0.\hfill \end{array}$$

(3)

Model 1, given in the fractional form. Model 1 can be converted into a linear form in two ways: (1) by normalizing the numerator of the objective function and (2) by normalizing the denominator of the objective function. In this paper, Model 1 is converted into a linear form by normalizing the denominator of the objective function given by Equation (3) as follows:

Model 2 

: For $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max{E}_k={\displaystyle \sum _{r=1}^s}{y}_{rk}{v}_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}{\displaystyle \sum _{i=1}^m}{x}_{ik}{u}_{ik}=1\hfill & \\ & \sum _{r=1}^s{y}_{rj}{v}_{rk}-\sum _{i=1}^m{x}_{ij}{u}_{ik}\le 0\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0.\hfill \end{array}$$

Model 2 is in linear form. Model 2 can easily be solved using linear programming problems assumptions. The efficiency scores $E_k$ of $DM{U}_k$ are obtained by solving Model 2. DMUs are categorized based on the efficiency scores obtained by solving Model 2.

Definition 3.

([20]). Let the optimal values of the DEA model for $DM{U}_k$ be $E_k^{*}$. Then, $DM{U}_k$ is said to be efficient if $E_k^{*}=1;$ otherwise  in-efficient.

Model 2 uses input-output data in crisp form. However, in real-world problems, the input and output data cannot be obtained accurately due to vagueness/fluctuation. Fuzzy numbers can represent such vagueness. Suppose ${\tilde{x}}_{ij}$ and ${\tilde{y}}_{rj}$ are the fuzzy inputs and fuzzy outputs for the $DM{U}_j$, respectively [21]. Then, the fuzzy DEA model is given as follows:

Model 3 

(Fuzzy DEA Model): For $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& max{\tilde{E}}_k=\sum _{r=1}^s{\tilde{y}}_{rk}{v}_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m{\tilde{x}}_{ik}{u}_{ik}=\tilde{1}\hfill & \\ & \sum _{r=1}^s{\tilde{y}}_{rj}{v}_{rk}-\sum _{i=1}^m{\tilde{x}}_{ij}{u}_{ik}\le \tilde{0}\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0.\hfill \end{array}$$

Assume that the fuzzy input ${\tilde{x}}_{ij}=(x_{ij}^L,x_{ij}^M,x_{ij}^U)$, fuzzy output ${\tilde{y}}_{rj}=(y_{rj}^L,y_{rj}^M,y_{rj}^U)$ for the $DM{U}_j$ and, $\tilde{1}=(1,1,1)$ are taken as TFNs [21]. Then triangular fuzzy DEA (TFDEA) model is presented as follows:

Model 4 

(TFDEA Model): For $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& max{\tilde{E}}_k=\sum _{r=1}^s(y_{rk}^L,y_{rk}^M,y_{rk}^U)v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m(x_{ik}^L,x_{ik}^M,x_{ik}^U)u_{ik}=(1,1,1)\hfill & \\ & \sum _{r=1}^s(y_{rj}^L,y_{rj}^M,y_{rj}^U)v_{rk}-\sum _{i=1}^m(x_{ij}^L,x_{ij}^M,x_{ij}^U)u_{ik}\le (0,0,0)\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0.\hfill \end{array}$$

Now, we will propose a methodology to solve the TFDEA model. In this methodology, we apply the $\alpha$-cut [21] approach to transforming the fuzzy input and output data into intervals [22]. $\alpha -$cuts of the fuzzy input ${\tilde{x}}_{ij}=(x_{ij}^L,x_{ij}^M,x_{ij}^U)$ and fuzzy output ${\tilde{y}}_{rj}=(y_{rj}^L,y_{rj}^M,y_{rj}^U)$ are defined as follows:

$$\begin{array}{c}{\tilde{x}}_{ij}=[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ,x_{ij}^U-(x_{ij}^U-x_{ij}^M)\alpha ],i=1,2,3,\dots ,m;\hfill \\ {\tilde{y}}_{rj}=[y_{rj}^L+(y_{rj}^M-y_{rj}^L)\alpha ,y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ],r=1,2,3,\dots ,s.\hfill \end{array}$$

On applying the $\alpha$-cut on the TFDEA model, we obtain an interval DEA (IDEA) model [3]. The IDEA represents the left and right-end efficiencies of $DM{U}_k$. The IDEA model is given as follows:

Model 5 

( IDEA model): For $\alpha \in (0,1]$ and $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max[E_k^L\left(\alpha \right),E_k^U\left(\alpha \right)]=\sum _{r=1}^s[y_{rk}^L+(y_{rk}^M-y_{rk}^L)\alpha ,y_{rk}^U-(y_{rk}^U-y_{rk}^M)\alpha ]v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ,x_{ik}^U-(x_{ik}^U-x_{ik}^M)\alpha ]u_{ik}=[1,1],\hfill & \\ & \sum _{r=1}^s[y_{rj}^L+(y_{rj}^M-y_{rj}^L)\alpha ,y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\hfill & \\ & \sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ,x_{ij}^U-(x_{ij}^U-x_{ij}^M)\alpha ]u_{ik}\le [0,0]\forall j,\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Using arithmetic operations of intervals [21], Model 5 can be converted into Model 6 as follows:

Model 6 

: For $\alpha \in (0,1]$ and $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max[E_k^L\left(\alpha \right),E_k^U\left(\alpha \right)]=\sum _{r=1}^s[y_{rk}^L+(y_{rk}^M-y_{rk}^L)\alpha ,y_{rk}^U-(y_{rk}^U-y_{rk}^M)\alpha ]v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ,x_{ik}^U-(x_{ik}^U-x_{ik}^M)\alpha ]u_{ik}=[1,1],\hfill & \\ & [\sum _{r=1}^s[y_{rj}^L+(y_{rj}^M-y_{rj}^L)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^U-(x_{ij}^U-x_{ij}^M)\alpha ]u_{ik},\hfill & \hfill \\ \hfill & \sum _{r=1}^s[y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ]u_{ik}]\le [0,0]\forall j,\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Intervals are partially ordered sets [21]. So, it is always difficult to do the ordering of intervals. However, a few special relations between the intervals are defined in the literature. Let $I_1=[a_1,b_1]$ and $I_2=[a_2,b_2]$ be two intervals then,

• $I_1=I_2\mathrm{iff}{a}_1=a_2,b_1=b_2$
• $I_1\le I_2\mathrm{iff}{a}_1\le a_2,b_1\le b_2$
• $I_1\ge I_2\mathrm{iff}{a}_1\ge a_2,b_1\ge b_2$

Now, using the properties of intervals, Model 6 can be simplified into Model 7 as follows:

Model 7 

: For $\alpha \in (0,1]$ and $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max[E_k^L\left(\alpha \right),E_k^U\left(\alpha \right)]=\sum _{r=1}^s[y_{rk}^L+(y_{rk}^M-y_{rk}^L)\alpha ,y_{rk}^U-(y_{rk}^U-y_{rk}^M)\alpha ]v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ]u_{ik}=1\hfill & \\ & \sum _{r=1}^s[y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ]u_{ik}\le 0\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Model 7 is decomposed into Model 8 (left-end efficiency model ) and Model 9 (right-end efficiency model) as follows:

Model 8 

(Left-end efficiency model): For $\alpha \in (0,1]$ and $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max{E}_k^L\left(\alpha \right)=\sum _{r=1}^s[y_{rk}^L+(y_{rk}^M-y_{rk}^L)\alpha ]v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ]u_{ik}=1\hfill & \\ & \sum _{r=1}^s[y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ]u_{ik}\le 0\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Model 9 

(Right-end efficiency model): For $\alpha \in (0,1]$ and $k=1,2,3,...,n,$

$$\begin{array}{ccc}& Max{E}_k^U\left(\alpha \right)=\sum _{r=1}^s[y_{rk}^U-(y_{rk}^U-y_{rk}^M)\alpha ]v_{rk}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ]u_{ik}=1\hfill & \\ & \sum _{r=1}^s[y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ]u_{ik}\le 0\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Model 8 and Model 9 maximize two objective functions for the same set of constraints. So, it will be better to solve both models simultaneously instead of separately. Solving both models will be less time-consuming and will not require much calculation. Hence, a bi-objective model can be developed with the help of Model 8 and Model 9. The proposed bi-objective model to evaluate the performance efficiencies of DMUs is given by Model 10 as follows:

Model 10 

(Bi-objective FDEA model): For $\alpha \in (0,1]$ and $k=1,2,3,\dots ,n,$

$$\begin{array}{ccc}& Max\left\{\sum _{r=1}^s[y_{rk}^L+(y_{rk}^M-y_{rk}^L)\alpha ]v_{rk},\sum _{r=1}^s[y_{rk}^U-(y_{rk}^U-y_{rk}^M)\alpha ]v_{rk}\right\}\hfill & \\ & \mathrm{subject}\mathrm{to}\sum _{i=1}^m[x_{ik}^L+(x_{ik}^M-x_{ik}^L)\alpha ]u_{ik}=1\hfill & \\ & \sum _{r=1}^s[y_{rj}^U-(y_{rj}^U-y_{rj}^M)\alpha ]v_{rk}-\sum _{i=1}^m[x_{ij}^L+(x_{ij}^M-x_{ij}^L)\alpha ]u_{ik}\le 0\forall j\hfill & \\ & u_{ik}\ge \epsilon \forall i,v_{rk}\ge \epsilon \forall r,\epsilon >0\hfill \end{array}$$

Model 10 is the proposed bi-objective model to evaluate the performance efficiencies of DMUs based on the $\alpha$-cuts approach. Generally, the efficiency score obtained for a DMU using the $\alpha$-cut approach is an interval. Intervals are partially ordered sets, due to which ranking intervals is a challenging task. The proposed BOFDEA model with $\alpha$-cuts provides the efficiency of $DM{U}_k$ in the crisp form, not in the form of intervals. Due to this, ranking DMUs with the proposed method’s help becomes very easy and less computationally. How to solve the proposed Model 10 and rank the DMUs is shown in Section 4.

4. Ranking Approach

The ranking strategy is introduced within DEA to rank DMUs according to the relative efficiency scores that each one possesses. The linear programming method is used to generate the efficiency scores. This method involves comparing the inputs and outputs of the DMU in question to those of a set of other efficient DMUs (the “envelope” of efficient DMUs). The DMUs with the highest efficiency scores are considered the most efficient and ranked first. The DMUs with the lowest efficiency scores are considered the least efficient and ranked last. This method is called the simple ranking method for DMUs. The ranking approach can be used to identify best practices and areas for improvement among the DMUs [11,19,20].

4.1. Algorithm for Solving the Proposed Bi-Objective FDEA Model

The algorithm for solving the reduced BOFDEA model based on $\alpha$-cuts can be outlined as follows:

Step 1:

The weighted sum method is a technique used to transform a bi-objective optimization problem into a single-objective problem. This method involves assigning weights to each objective, then combining the objectives into a single scalar value using a weighted sum. The weighted sum is calculated by multiplying each objective’s value by its corresponding weight and then summing the results. In the context of the BOFDEA model, the weighted sum method can be used to reduce the bi-objective problem for each DMU to a single-objective problem for a range of $\alpha$-cuts. This is achieved by assigning weights to the two objectives and using the weighted sum method to combine the objectives into a single scalar value. The weighted sum method is a useful tool for simplifying the evaluation of DMUs in the BOFDEA model and is commonly used in multi-objective optimization problems. The method is described in detail in the reference [23].

Step 2:

To produce a sample of random weights in the BOFDEA model, the number of optimization goals (D) and the total number of variables (B) in the problem must first be determined. Then, a set of D weights can be generated, equal to ($10\times$ B). This computation uses ten times the amount of actual variables in the problem to ensure that a sufficient number of weights are generated for evaluation. In regards to selecting a population, there is no set rule or guideline. The choice of the population is entirely up to the person making the decision.

Step 3:

In Step 2 of the BOFDEA model, weights are generated for different $\alpha$-cuts. These weights can then be used in Step 3 to resolve the single-objective problem associated with each DMU created in Step 1. Using the weights obtained in Step 2 for the $\alpha$-cuts, the multi-objective optimization problem for each DMU can be transformed into a single-objective problem. This process allows for a direct evaluation of the performance of each DMU in terms of the defined objectives.

Step 4:

From Step 3 of the BOFDEA model, 10 solutions are obtained for each of the (B) weight vectors. To determine the best solution among these, it is necessary to evaluate each solution based on the objective function and criteria defined in the model. The solution that performs best according to the objectives set in the model can be considered the best among the ($10\times$ B) solutions obtained.

Step 5:

The optimal solution obtained in Step 4 of the BOFDEA model, represented by the best weights, can be used to calculate the efficiency of each decision-making unit (DMU). The DMUs can then be ranked in decreasing order of their efficiency scores, determined by the $\alpha$- values. The ranking is performed from 1 to n, with 1 being the most efficient DMU and n being the least efficient . Schematic diagram for efficiency evalution of DMUs is presented in Figure 1.

4.2. Advantages of the Proposed Bi-Objective FDEA Model over Existing Methods

• Generally, the efficiency score obtained for a DMU by using the $\alpha$-cut approach is in the form of an interval. Intervals are partially ordered sets due to which ranking of intervals is a challenging task. The proposed bi-objective FDEA model with $\alpha$-cuts provides the efficiency of $DM{U}_k$ in the crisp form, not in the form of intervals. Due to this ranking of DMUs with the help of the proposed method becomes very easy and less computational.
  Arya and Yadav [3] developed lower and upper-bound efficiency models to measure the performance efficiency of DMUs using $\alpha$-cuts. In their approach, the calculated lower and upper bounds of efficiencies form an efficiency interval for each DMU. To rank, these intervals is again a challenging thing due to the presence of partial ordering in intervals. However, in our proposed bi-objective model efficiency obtained will be crisp in nature. So the ranking can be performed very easily. Moreover, the proposed bi-objective method for the performance evaluation of DMUs requires less time and calculation.
• Despotis and Smirlis [4], to deal with uncertain situations, developed a model to generate the upper and lower bounds of interval efficiency for each DMU. In the proposed model, Despotis and Smirlis considered different constraints for $j=k$ and $j\ne k.$ After obtaining lower and upper bounds of efficiencies based on $\alpha$-cuts ranking of intervals is performed. However, the proposed bi-objective FDEA model with $\alpha$-cuts is able to overcome the situation of calculating two different efficiencies for a DMU. Hence, the proposed bi-objective method for the performance evaluation of DMUs requires less time and calculation.

5. Numerical Illustraions

We take an example from Guo and Tanaka’s research [24], which consists of five DMUs with two fuzzy inputs and two fuzzy outputs, to illustrate the efficacy of our proposed models. Using this data, we can check the reliability of our models and ranking methodology.

5.1. Numerical Illustration: An Example

Table 1. Fuzzy input-output data for 5 DMUs (Source: Guo and Tanaka [24]).

DMUs	Fuzzy Inputs		Fuzzy Outputs
	Input 1	Input 2	Output 1	Output 2
DMU1	(3.5, 4.0, 4.5)	(1.9, 2.1, 2.3)	(2.4, 2.6, 2.8)	(3.8, 4.1, 4.4)
DMU2	(2.9, 2.9, 2.9)	(1.4, 1.5, 1.6)	(2.2, 2.2, 2.2)	(3.3, 3.5, 3.7)
DMU3	(4.4, 4.9, 5.4)	(2.2, 2.6, 3.0)	(2.7, 3.2, 3.7)	(4.3, 5.1, 5.9)
DMU4	(3.4, 4.1, 4.8)	(2.1, 2.3, 2.5)	(2.5, 2.9, 3.3)	(5.5, 5.7, 5.9)
DMU5	(5.9, 6.5, 7.1)	(3.6, 4.1, 4.6)	(4.4, 5.1, 5.8)	(6.5, 7.4, 8.3)

contains the fuzzy input-output data for the problem that was analyzed by Guo and Tanaka [24]. This problem involves five DMUs, each of which has two fuzzy inputs and two fuzzy outputs. Fuzzy input-output data may be found in Table 1.

Now, the proposed bi-objective FDEA model is applied to the input and output data given in Table 1 to evaluate the efficiency scores of DMUs based on $\alpha$- values. To rank the DMUs according to their efficiency scores, the proposed algorithm is applied. The ranking of DMUs so obtained for different $\alpha$- values is given in

Table 2. Efficiency scores of DMUs and ranks.

DMUs	$\mathit{\alpha }=0.1$		$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$		$\mathit{\alpha }=1.0$
	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathbf{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank
DMU1	0.894152	5	0.913696	5	0.908735	5	0.884486	5	0.854815	5
DMU2	0.978102	4	0.999463	1	1	1	0.999999	1	1	1
DMU3	0.998977	1	0.997751	4	0.953414	4	0.904858	4	0.860793	4
DMU4	0.996905	3	0.999413	2	0.998539	2	0.999385	2	0.999998	3
DMU5	0.997929	2	0.998431	3	0.998474	3	0.999283	3	0.999999	2

Where PE_k = The proposed efficiency score.

.

Comparison of Proposed Efficiency Technique Scores with Existing Method

Wang et al. [7] proposed a geometric average ranking approach based on $\alpha$-cuts to rank the DMUs in the Fuzzy environment (Figure 2). The proposed study compared with the efficiencies calculated by Wang et al.’s [7] geometric average technique efficiency approach of DMUs. The comparison of efficiency scores and rankings is presented in

Table 3. Comparison of the proposed efficiency (PE) and geometric average efficiency(GeAE) and ranks.

DMUs	$\mathit{\alpha }=0.1$				$\mathit{\alpha }=0.25$				$\mathit{\alpha }=0.5$				$\mathit{\alpha }=0.75$				$\mathit{\alpha }=1.0$
	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{GeAE}}_{\mathit{k}}$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathbf{\alpha }\right)$	Rank	${\mathit{GeAE}}_{\mathit{k}}$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{GeAE}}_{\mathit{k}}$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{GeAE}}_{\mathit{k}}$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{GeAE}}_{\mathit{k}}$	Rank
DMU1	0.894152	5	0.81	4	0.913696	5	0.84	4	0.908735	5	0.87	4	0.884486	5	0.86	3	0.854815	5	0.85	3
DMU2	0.978102	4	0.93	2	0.999463	1	0.99	1	1	1	0.994	2	0.999999	1	1	1	1	1	1	1
DMU3	0.998977	1	0.77	5	0.997751	4	0.799	5	0.953414	4	0.85	2	0.904858	4	0.86	4	0.860793	4	0.86	2
DMU4	0.996905	3	0.94	1	0.999413	2	0.97	2	0.998539	2	1	1	0.999385	2	1	1	0.999998	3	1	1
DMU5	0.997929	2	0.82	3	0.99843	3	0.86	3	0.998474	3	0.92	3	0.999283	3	0.98	2	0.999999	2	1	1

Where PE_k = Proposed efficiency score and GeAE_k = Geometric average efficiency score.

. It can be observed that the rankings of DMUs obtained from both methods are different. In this case, it is difficult to compare both models. In order to compare both models, the coefficient of variation method [25] is applied. The coefficient of variation (CV) is a relative measure of dispersion that does not take into account dimensions and is calculated as the ratio of the standard deviation to the mean. The CV allows comparisons to be made between different data sets that are not possible using any other method. A coefficient of variation (CV) that is high indicates that there is a greater degree of data dispersion around the mean. A distribution with a lower coefficient of variation (CV) is considered to be more trustworthy than one with a larger CV. A CV comparison is presented in

Table 4. Comparison of the proposed method with Wang et al.’s [7] geometric average efficiency method with the help of Coefficient of Variation (CV).

DMUs	$\mathit{\alpha }=0.1$		$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$		$\mathit{\alpha }=1.0$
	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	${\mathit{GeAE}}_{\mathit{k}}$	${\mathit{PE}}_{\mathit{k}}\left(\mathbf{\alpha }\right)$	${\mathit{GeAE}}_{\mathit{k}}$	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	${\mathit{GeAE}}_{\mathit{k}}$	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	${\mathit{GeAE}}_{\mathit{k}}$	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	${\mathit{GeAE}}_{\mathit{k}}$
DMU1	0.894152	0.81	0.913696	0.84	0.908735	0.87	0.884486	0.86	0.854815	0.85
DMU2	0.978102	0.93	0.999463	0.99	0.994	0.999999	1	1	1	1
DMU3	0.998977	0.77	0.997751	0.799	0.953414	0.85	0.904858	0.86	0.860793	0.86
DMU4	0.996905	0.94	0.999413	0.97	0.998539	1	0.999385	1	0.999998	1
DMU5	0.997929	0.82	0.99843	0.86	0.998474	0.92	0.999283	0.98	0.999999	1
Std ($\sigma$)	0.04681	0.07635	0.04719	0.08376	0.0525	0.069	0.0495	0.0735	0.05270.0795
Mean ($\mu$)	0.9686	0.854	0.9684	0.8919	0.9667	0.9268	0.9668	0.94	0.968	0.942
CV($\sigma /\mu$)	0.0483	0.0894	0.0487	0.0939	0.0543	0.0744	0.0512	0.0782	0.0545	0.0844

Where PEk = Proposed efficiency score and GeAEk = Geometric average efficiency score; σ = Standard deviation, μ = Mean and CV = $\frac{\sigma }{\mu }$ = Coefficient of variation.

. From Table 4, it is concluded that the proposed methodology is more powerful and effective in ranking DMUs.

5.2. Numerical Illustration: An Education Sector Application

To prove the practicality and usefulness of the proposed BOFDEA model, it is demonstrated in the education sector. Thirteen Indian Institutes of Management (IIMs) are evaluated over a four-year period from 2015-16 to 2018-19 using two inputs and two outputs. The input-output data variables used in this application are as follows:

(i)

Input 1: Total number of students ($x_1$)

(ii)

Input 2: Total number of faculty members ($x_2$)

(iii)

Output 1: Total number of students who went for placements and higher studies ($y_1$)

(iv)

Output 2: Total number of publications ($y_2$)

The data utilized in this study is sourced from the NIRF website, established on 29 September 2015 by the Minister of Human Resource Development. Additional information was taken from the institute’s yearly report. The data collected was transformed into TFNs from input-output data. This transformation process involved converting four years of crisp data into TFNs ($\tilde{a}=(a^L,a^M,a^U)$) where $a^L$ represents the minimum value of data for a particular IIM, $a^M$ represents the average, and $a^U$ represents the maximum value of data for the same IIM.

The input-output data for IIMs for the period 2015-16 to 2018-19 in the form of TFNs is given in

Table 5. Fuzzy input and output data for IIMs.

DMUs	IIM Name	State	Inputs		Outputs
			$\widetilde{{\mathit{x}}_{\mathbf{1}}}$	$\widetilde{{\mathit{x}}_{\mathbf{2}}}$	$\widetilde{{\mathit{y}}_{\mathbf{1}}}$	$\widetilde{{\mathit{y}}_{\mathbf{2}}}$
$D_1$	IIM Bangalore	Karnataka	(424, 682, 955)	(91, 104, 113)	(393, 410, 449)	(72, 136, 212)
$D_2$	IIM Ahmadabad	Gujarat	(461, 715, 992)	(91, 112, 128)	(411, 421, 427)	(30, 109, 217)
$D_3$	IIM Calcutta	West Bengal	(487, 803, 1042)	(86, 94, 105)	(483, 505, 535)	(29, 109, 207)
$D_4$	IIM Lucknow	Uttar Pradesh	(455, 725, 990)	(81, 88, 95)	(440, 456, 506)	(8, 65, 126)
$D_5$	IIM Indore	Madhya Pradesh	(549, 1020, 1657)	(73, 94, 104)	(508, 593, 634)	(16, 66, 141)
$D_6$	IIM Kozhikode	Kerala	(370, 593, 806)	(58, 69, 77)	(347, 360, 382)	(28, 74, 97)
$D_7$	IIM Udaipur	Rajasthan	(120, 260, 419)	(21, 46, 101)	(120, 136, 171)	(14, 37, 67)
$D_8$	IIM Tiruchirapalli	Tamilnadu	(108, 228, 387)	(25, 37, 52)	(102, 121, 172)	(5, 16, 24)
$D_9$	IIM Raipur	Chhatisgarh	(160, 270, 438)	(21, 33, 46)	(111, 140, 193)	(12, 35, 52)
$D_{10}$	IIM Rohtak	Haryana	(158, 276, 428)	(20, 28, 34)	(137, 146, 155)	(31, 52, 69)
$D_{11}$	IIM Shillong	Meghalaya	(155, 263, 365)	(27, 27, 28)	(118, 146, 172)	(3, 15, 30)
$D_{12}$	IIM Kashipur	Uttarakhand	(125, 262, 472)	(15, 28, 38)	(101, 123, 164)	(7, 21, 36)
$D_{13}$	IIM Ranchi	Jharkhand	(189, 300, 452)	(16, 29, 40)	(156, 169, 178)	(11, 22, 51)

Source: www.nirfindia.org.

. The proposed BOFDEA Model (Model Section 3) is applied to the data and results are calculated for each IIM based on $\alpha$- values. The results so obtained are presented in

Table 6. Efficiency scores of IIMs and ranks.

DMUs	$\mathit{\alpha }=0.1$		$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$		$\mathit{\alpha }=1.0$
	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathit{\alpha }\right)$	Rank	${\mathit{PE}}_{\mathit{k}}\left(\mathbf{\alpha }\right)$	Rank
IIM Bangalore	0.9100105	8	0.9856132	7	0.9972293	3	0.9958669	6	0.9999985	5
IIM Ahmadabad	0.8877189	10	0.9126174	10	0.9160096	13	0.9361728	9	0.9521349	8
IIM Calcutta	0.9782908	6	0.9998370	1	0.9999993	1	0.9999992	1	0.9999999	2
IIM Lucknow	0.8203216	11	0.9014835	11	0.9932412	7	0.9987330	4	1	1
IIM Indore	0.8968479	9	0.9578101	9	0.9967247	4	0.9976602	5	0.9999987	4
IIM Kozhikode	0.7915698	13	0.8725351	13	0.9571886	10	0.9652226	7	0.9673308	6
IIM Udaipur	0.9977421	3	0.9947059	3	0.9897305	8	0.9174692	11	0.8502580	10
IIM Tiruchirapalli	0.9905848	4	0.9934330	5	0.9946371	6	0.9420763	8	0.8436714	12
IIM Raipur	0.9309199	7	0.9895158	6	0.9580962	9	0.8990768	12	0.8454402	11
IIM Rohtak	0.9981651	2	0.9821789	8	0.9959387	5	0.9994340	3	0.9999996	3
IIM Shillong	0.8062730	12	0.8741184	12	0.9398288	12	0.9330522	10	0.9228445	9
IIM Kashipur	0.9853705	5	0.9968505	2	0.9461943	11	0.8552886	13	0.7703081	13
IIM Ranchi	0.9988746	1	0.9935265	4	0.9999389	2	0.9976703	2	0.9548352	7

Where PE_k = The proposed efficiency score.

. Table 6 shows that there is no significant pattern in the rankings of DMUs for different $\alpha$- values. The graphical representation of efficiencies for different $\alpha$- values is given in Figure 3.

6. Conclusions

In this paper, we have developed a bi-objective FDEA (BOFDEA) model to evaluate the performance of DMUs. An algorithm is also proposed to solve the developed BOFDEA model. The optimal solution of the proposed model gives the efficiency score of a DMU based on $\alpha$- values. The rankings of DMUs are performed based on the efficiency scores obtained. To validate the efficacy of the proposed model two numerical examples are considered in this study. The example is from Wang et al.’s [7] study. The results obtained from the proposed model are compared with Wang et al.’s study and shown in Table 4. It can be observed that the rankings of DMUs obtained from both methods are different. In this case, it is difficult to compare both models. In order to compare both models, the coefficient of variation (CV) method [25] is applied. A high CV denotes a larger degree of data dispersion around the mean. In comparison to a distribution with a higher CV, one with a smaller CV is more reliable. From Table 4, it is quite visible that the CV calculated from the proposed BOFDEA model is lower than the CV calculated from Wang et al.’s [7] geometric average efficiency. Hence, it is concluded the proposed methodology is more powerful and effective in ranking DMUs. An education sector application, a real-life application, is also presented in Section 5.2. Validation of the proposed methodology is performed in Section 5.1 and for comparison of the efficiencies for the education sector application, we do not have any available study. So no comparison of efficiencies for education sector application is presented here. In this article, the application efficiencies of 13 IIMs are calculated for different $\alpha$- values, and ranking is performed based on the efficiency scores calculated. No significant pattern in the rankings of IIMs for different $\alpha$- values is obtained. The rankings of IIMs for different $\alpha$- values obtained from the proposed BOFDEA approach indicate that the rankings of IIMs are changing with the change in $\alpha$- values.

The current work aims to provide new perspectives on the BOFDEA model’s solution. However, several of the fundamental presumptions of LPP solution methodologies restrict the suggested technique. One major limitation of the proposed BOFDEA models is that they only account for TFNs. The uncertainty in input-output data and decision parameters can be described by decision makers (DM) using trapezoidal fuzzy numbers (TrFNs), L-R-type fuzzy numbers, or any other kind of fuzzy numbers. In this research, we focus on the BOFDEA framework’s application in constant returns to scale (CRS) setting, but it is also applicable in a VRS setting.