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
-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
-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
-cuts to assess the effectiveness of DMUs in a fuzzy setting. With the help of
-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
and
in their model proposal. Once
-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
-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.
3. Proposed Bi-Objective Fuzzy DEA Model
Our objective is to evaluate the performance efficiencies of n homogeneous DMUs (). Suppose that consumes m inputs to produce s outputs . Let and be the weights corresponding to the input and output of Then the efficiency () of is given by
- Model 1
(CCR DEA model [
19]): For
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
Model 2 is in linear form. Model 2 can easily be solved using linear programming problems assumptions. The efficiency scores of 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 be . Then, is said to be efficient if 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
and
are the fuzzy inputs and fuzzy outputs for the
, respectively [
21]. Then, the fuzzy DEA model is given as follows:
- Model 3
(Fuzzy DEA Model): For
Assume that the fuzzy input
, fuzzy output
for the
and,
are taken as TFNs [
21]. Then triangular fuzzy DEA (TFDEA) model is presented as follows:
- Model 4
(TFDEA Model): For
Now, we will propose a methodology to solve the TFDEA model. In this methodology, we apply the
-cut [
21] approach to transforming the fuzzy input and output data into intervals [
22].
cuts of the fuzzy input
and fuzzy output
are defined as follows:
On applying the
-cut on the TFDEA model, we obtain an interval DEA (IDEA) model [
3]. The IDEA represents the left and right-end efficiencies of
. The IDEA model is given as follows:
- Model 5
( IDEA model): For
and
Using arithmetic operations of intervals [
21], Model 5 can be converted into Model 6 as follows:
- Model 6
: For
and
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
and
be two intervals then,
Now, using the properties of intervals, Model 6 can be simplified into Model 7 as follows:
- Model 7
: For
and
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
and
- Model 9
(Right-end efficiency model): For
and
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
and
Model 10 is the proposed bi-objective model to evaluate the performance efficiencies of DMUs based on the
-cuts approach. Generally, the efficiency score obtained for a DMU using the
-cut approach is an interval. Intervals are partially ordered sets, due to which ranking intervals is a challenging task. The proposed BOFDEA model with
-cuts provides the efficiency of
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.
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
- 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
- values, and ranking is performed based on the efficiency scores calculated. No significant pattern in the rankings of IIMs for different
- values is obtained. The rankings of IIMs for different
- values obtained from the proposed BOFDEA approach indicate that the rankings of IIMs are changing with the change in
- 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.