Comprehensive intuitionistic fuzzy network data envelopment analysis incorporating undesirable outputs and shared resources

Graphical abstract

Abstract

The economic growth rate is intricately linked to the efficiency and effectiveness of the banking industry. A widely applicable mathematical technique for such assessments is Data Envelopment Analysis (DEA), which evaluates the relative efficiency of Decision-Making Units (DMUs) by comparing their inputs and outputs. Traditional DEA treats DMUs as black boxes, neglecting internal processes that contribute to inefficiencies in individual DMUs. Additionally, it assumes precise values for inputs and outputs that do not apply to real-world problems. This study introduces a comprehensive network series of two-stage DEA, incorporating shared inputs and intermediate measures, undesirable outputs, external inputs and outputs, initial inputs, and terminal outputs. The network two-stage DEA is extended to intuitionistic fuzzy circumstances to address uncertainty. In this extension, a non-linear intuitionistic fuzzy number, namely a parabolic intuitionistic fuzzy number, represents higher-order imprecise datasets. An illustrative example validates the proposed methodology, and comparisons with existing methods are conducted. Moreover, the methodology is applied to assess the efficiency of Indian public sector banks, demonstrating its applicability and showcasing the efficacy of the procedures and algorithms used. Decision-makers can make better choices using optimal efficiency values to gain insights into inputs, intermediate measures, and outputs.

• •The research study focused on a network two-stage DEA model, incorporating undesirable outputs and shared resources in the presence of uncertainty.
• •The methodology involves solving the network two-stage DEA model using parabolic intuitionistic fuzzy numbers.
• •The experimental analysis involves assessing the efficiency of Indian public sector banks.

Specifications table

Subject area:	Mathematics
More specific subject area:	Network Data Envelopment Analysis and Intuitionistic Fuzzy Set Theory
Name of your method:	Comprehensive Intuitionistic Fuzzy Network Data Envelopment Analysis
	Incorporating Undesirable Outputs and Shared Resources
Name and reference of original method:	A novel two-stage DEA production model with freely distributed
	initial inputs and shared intermediate outputs.
	https://doi.org/10.1016/j.eswa.2017.11.005
Resource availability:	N/A

Method details

This section is divided into four subsections: introduction, literature review, pertinent definitions, and the proposed methodology.

Introduction

In the field of DEA, the configuration of decision-making units (DMUs) and the level of certainty or uncertainty, along with desirability and undesirability associated with inputs and outputs, offers a diverse spectrum for research exploration. Within the framework of DMU structures, DEA is categorized into two main approaches: black-box and network DEA. Black-box DEA utilizes initial inputs to generate final outputs without delving into internal processes, thereby overlooking inefficiencies (see [1], [2], [3], [4], [5]). In contrast, the network’s DEA considers internal processes to identify the sources of inefficiency for each DMU individually. [6] pioneered the introduction of a network structure within DEA to address the limitations of the black-box DEA model. This enhanced model, known as network DEA (NDEA), incorporates network structures to capture more nuanced efficiency considerations. Specifically, when NDEA comprises precisely two network systems or two internal processes, it is termed a two-stage DEA model. The network structure can manifest in various forms, including series (see [7], [8], [9], [10]) and parallel (see [11], [12]). In a two-stage DEA model, the first stage employs inputs to generate intermediate measures, which then serve as inputs for the second stage to produce final outputs.

In a two-stage DEA, the inputs can be internal inputs used for the first stage, shared inputs partially used in the first and second stages, and external or free inputs directly used in the second stage. Intermediate measures could be divided into two categories: internal inputs for the second stage and shared intermediate measures partially used as inputs for the second stage and partially as the first stage’s outputs. Similarly, the outputs could be categorized as undesirable outputs produced by preceding stages that neglect to enter as inputs to subsequent stages. The first stage produces the external or free output, and the second stage produces the final output. The existing literature on the hybrid of the topics mentioned above based on uncertainty includes undesirable outputs into the two-stage DEA by ignoring shared inputs and intermediate measures such as [13], [14], [15] or considering shared inputs and intermediate measures by neglecting the accessible inputs and outputs (see [16], [17], [18], [19], [20])

In response to these shortcomings, our paper introduces a comprehensive two-stage network DEA model. This model considers shared inputs/intermediate measures, accessible inputs, terminal outputs of the first stage, and undesirable outputs, as illustrated in Fig. 1 with precise or crisp values. The efficiency model utilized in our study is calculated by taking the arithmetic mean of the first and second stages, following the methodologies outlined in the works of [7] and [9]. Despite numerous extensions to network DEA models, current approaches still exhibit some limitations. A significant drawback is the vagueness of inputs and outputs that need consideration in practical situations. In response, we propose an extension to our comprehensive two-stage network DEA model by introducing a parabolic intuitionistic fuzzy number to represent imprecise datasets in an intuitionistic fuzzy environment. We term this extended model the comprehensive intuitionistic fuzzy two-stage network DEA model. To address the extended models, we define a new center of gravity method for intuitionistic fuzzy sets applied to the illustrative intuitionistic fuzzy dataset. We conduct a comparative analysis against existing models to validate our proposed models, employing a comparative analysis. Furthermore, we apply our models to the empirical context of Indian public sector banks to demonstrate their applicability in real-world scenarios.

Fig. 1.

Proposed two-stage network DEA.

The main contribution of the paper is outlined as follows:

• 1.Develop a comprehensive two-stage network DEA model that considers shared inputs/intermediate measures, accessible inputs, the first stage’s terminal outputs, and undesirable outputs.
• 2.A parabolic intuitionistic fuzzy number has been introduced to handle imprecise datasets in an intuitionistic fuzzy environment, resulting in the comprehensive intuitionistic fuzzy two-stage network DEA model.
• 3.Definition of a novel center of gravity method for intuitionistic fuzzy sets to convert illustrative intuitionistic fuzzy numbers into corresponding crisp values.
• 4.Validation of the proposed models through a comparative analysis against existing models employing comparative analysis.
• 5.Application of the models to the Indian public sector banks as an empirical case study, showcasing their practical relevance and applicability.

Literature review

Ever since [1] introduced the DEA, there has been an exponential growth in DEA literature. For instance, [3], [21] proposed a fuzzy DEA model with undesirable outputs and an intuitionistic fuzzy DEA to assess the Indian banking sector in a fuzzy and intuitionistic fuzzy environment.

Two-stage DEA proposed by Färe and Grosskopf [6] is extended and applied in several domains, including the banking industry. In the context of the bank industry, [17] proposed a two-stage DEA model that used fixed assets, labor, and IT investments as inputs to generate deposits as outputs in the first stage. In the second stage, the generated deposits were used as inputs to produce loans and profits as the final outputs. Similarly, [22] aimed to measure the efficiency of 40 central Brazilian banks using the network DEA centralized efficiency model to optimize cost and product efficiency. In the first stage, they used the number of branches and employees to measure administrative and personal expenses as outputs. They used them as inputs to generate equity and permanent assets in the second stage. A recent study by Ferreira et al. [23] conducted a comprehensive review of two-stage DEA in Banks, focusing on disputed terminology and future directions. In the context of modeling bank performance [24], [25] proposed a network DEA and dynamical DEA approaches to assess the performance of the Chinese banking industry.

Although traditional network DEA models offer several benefits, they have two significant drawbacks regarding inputs and outputs. First, they lack shared inputs and outputs and may include undesirable outputs. Second, these models consider inputs and outputs as precise quantities, which is only sometimes the case due to several unknown factors, such as unquantifiable results and fluctuations that cause imprecision and vagueness in the dataset. In some work problems, specific inputs are shared resources between stages. For instance, in a study proposed by Liu et al. [26], the labor force and fixed asset investment are considered shared inputs between the first and second stages. The total water resources and land area are first-stage inputs to produce water consumption and agriculture, and the land construction area is the output. These outputs are then taken as second-stage inputs to construct the total production as terminal outputs. The purpose of this is to evaluate the measurement efficiency of the water and resources system in China. Ma et al. [11] analyzed the efficiency of Stackelberg game scenarios for shared inputs using additive decomposition in Two-stage DEA models. Furthermore, a novel two-stage DEA production model was proposed by Izadikhah et al. [9] based on shared input/intermediate measures. In their model, some accessible inputs are consumed directly to the second stage, while some outputs of the first stage are the final outputs.

Real-world problems can be complex and result in undesirable inputs and outputs. There are two methods in the literature to handle the incorporation of these undesirable outputs in DEA models: the direct and indirect approaches. In the indirect approach, the undesirable outputs are transformed into desirable inputs using a monotone decreasing function [27]. On the other hand, the direct method directly incorporates undesirable outputs by assigning them negative weights without using any transformation functions [4], [28], [29]. In literature, hybrid studies involve shared resources and undesirable outputs. For example, [13] developed a network DEA method based on bargaining, considering shared inputs and undesirable outputs. Zhou and Hu [14] developed an additive network DEA model to assess the overall sustainability performance of China’s railway transportation, considering shared inputs and undesirable outputs. [15] proposed a slack-based measure for a two-stage DEA model incorporating shared inputs and undesirable feedback.

The second limitation of DEA and network DEA is that these models consider inputs/outputs as crisp numbers. At the same time, real-world problems possess a certain degree of fluctuation, leading to uncertainty and vagueness in the data set. In such cases, intuitionistic fuzzy set theory with its membership and not membership grade in capturing vagueness is a more applicable method [30]. IFS has been utilized in various fields, such as DEA and Network DEA models. In a study conducted by Puri and Yadav [21], the Indian banking sector’s optimistic and pessimistic efficiency was assessed using intuitionistic fuzzy inputs and outputs. The supper efficiency method was employed to rank the DMUs comprehensively. Sahil et al. [5] proposed parabolic intuitionistic fuzzy DEA models to calculate the parametric efficiency of DUMs based on the $\alpha -$ cut and $\beta -$ cut approaches. Mohanta and Sharanappa [31] presented a new method to solve an intuitionistic fuzzy DEA model, which was applied to the agricultural sector in India. Mohammadi Ardakani et al. [32] developed the best and worst relative returns two-stage DEA models within IF inputs/outputs based on $\alpha -$cut and $\beta -$cut approaches. Shariatmadari Serkani et al. [33] proposed a hybrid intuitionistic fuzzy analytic and network DEA model to measure the relative efficiency of the Faculty of Basic Sciences of Islamic Azad University. Only a few research papers examine network two-stage DEA in an intuitionistic fuzzy environment. Therefore, this article is the first to address the incorporation of undesirable outputs with shared resources in network two-stage DEA within such an environment. The aim is to represent vague datasets using parabolic intuitionistic fuzzy numbers.

Pertinent definition

We establish the pertinent mathematical definitions of intuitionistic fuzzy set theory and network data envelopment analysis, which are subsequently applied in the methodology part of the study.

Intuitionistic fuzzy set

An intuitionistic fuzzy set (IFS) $\tilde{A}$ is a set defined on a universe discourse set $X$, represented as $\tilde{A}=\left\{(x,{\mu }_{\tilde{A}}\left(x\right),{\nu }_{\tilde{A}}\left(x\right))\right|x\in X\}$. The functions ${\mu }_{\tilde{A}}:X⟶[0,1]$ and ${\nu }_{\tilde{A}}:X⟶[0,1]$, known as the membership function and non-membership function respectively, determine the degree of membership and non-membership for each $x\in X$, subject to the condition $0\le {\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)\le 1$. Additionally, the function${\pi }_{\tilde{A}}\left(x\right)=1-{\mu }_{\tilde{A}}\left(x\right)-{\nu }_{\tilde{A}}\left(x\right)$ is referred to as the degree of hesitation [30].

Intuitionistic fuzzy number

Let $b_1\le a_1\le b_2\le a_2\le a_3\le b_3\le a_4\le b_4$ then $\tilde{A}=(a_1,a_2,a_3,a_4;b_1,b_2,b_3,b_4)$ is called an intuitionistic fuzzy number (IFN) if its membership and non-membership functions are given below [30]:

$${\mu }_{\tilde{A}}\left(x\right)=\{\begin{array}{cc}{f}_A\left(x\right)\hfill & ,a_1\le x<a_2\hfill \\ 1\hfill & ,a_2\le x\le a_3\hfill \\ g_A\left(x\right)\hfill & ,a_3<x\le a_4\hfill \\ 0\hfill & ,otherwise\hfill \end{array}$$ (1)

$${\nu }_{\tilde{A}}\left(x\right)=\{\begin{array}{cc}{h}_A\left(x\right)\hfill & ,b_1\le x<b_2\hfill \\ 0\hfill & ,b_2\le x\le b_3\hfill \\ k_A\left(x\right)\hfill & ,b_3<x\le b_4\hfill \\ 1\hfill & ,otherwise\hfill \end{array}$$ (2)

Here, the functions $f_A,k_A$ are non-decreasing piecewise continuous functions, while $g_A,h_A$ are non-increasing piecewise continuous functions on their respective defined domain.

Parabolic intuitionistic fuzzy number

The IFN, $\tilde{A}=(a_1,a_2,a_3;b_1,b_2,b_3)$ is called a Parabolic Intuitionistic Fuzzy Number (PIFN) if its membership and non-membership functions are defined as follows [5]:

$${\mu }_{\tilde{A}}\left(x\right)=\{\begin{array}{cc}\frac{(x-a_1)(x-2a_2+a_1)}{(a_2-a_1)(a_1-a_2)}\hfill & ,a_1<x\le a_2\hfill \\ \\ \frac{(x-a_3)(x-2a_2+a_3)}{(a_3-a_2)(a_2-a_3)}\hfill & ,a_2\le x<a_3\hfill \\ \\ 0\hfill & ,otherwise\hfill \end{array}$$ (3)

$${\nu }_{\tilde{A}}\left(x\right)=\{\begin{array}{cc}\frac{(x-b_2)(x-b_2)}{(b_2-b_1)(b_2-b_1)}\hfill & ,b_1<x\le b_2\hfill \\ \\ \frac{(x-b_2)(x-b_2)}{(b_2-b_3)(b_2-b_3)}\hfill & ,b_2\le x<b_3\hfill \\ \\ 1\hfill & ,otherwise\hfill \end{array}$$ (4)

Here $b_1\le a_1\le a_2=b_2\le a_3\le b_3$.

A graphical illustration of PIFN, which can be symmetric or asymmetric, is shown in Fig. 2.

Fig. 2.

Parabolic intuitionistic fuzzy number.

A two-stage data envelopment analysis

Consider a network of two-stage DEA with $n$ DMUs, where each $(DM{U}_k,k=1,2,3,\dots ,n)$ has to be measured with $m$ inputs, $p$ intermediate measures and $r$ outputs. Let $x_{ik}(i=1,2,3,\dots ,m),z_{tk}(t=1,2,3,\dots ,p$, and $y_{rk}(r=1,2,3,\dots s)$ denote the $i$th input, $t$thintermediate measure, and $r$th output of the $k$th DMU, respectively. The following models represent the first, second, and overall efficiency models [7]

The first stage efficiency model

Let $x_{ik}(i=1,2,3,\dots ,m)$ and $z_{tk}(t=1,2,3,\dots ,p$ denote the $i$th input and $t$th output of the $k$th DMU, respectively. Its mathematical formulation is given below [1]:

$$\begin{array}{cc}& E_k^1=\max \frac{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}{{\sum }_{k=1}^n{\sum }_{i=1}^m{v}_{ik}{x}_{ik}}s.t\frac{{\sum }_{j=1}^n{\sum }_{t=1}^p{w}_{tj}{z}_{tj}}{{\sum }_{j=1}^n{\sum }_{i=1}^m{v}_{ij}{x}_{ij}}\le 1\hfill \\ & w_{tk}\ge ϵ\forall t,v_{ik}\ge ϵ\forall i\hfill \end{array}$$ (5)

Here, $w_{tk}$ and $v_{ik}$ represent the $r$th weight of output and input, respectively, for the $k$th DMU in the first stage, and $ϵ$ is a non-Archimedean infinitesimal number.

The second stage efficiency model

Let $z_{tk}(t=1,2,3,\dots ,p$ and $y_{rk}(r=1,2,3,\dots s)$ denote the $t$th input and $r$th output of the $k$th DMU, respectively. Its mathematical formulation is given below [1]:

$$\begin{array}{cc}& E_k^2=\max \frac{{\sum }_{k=1}^n{\sum }_{r=1}^s{u}_{rk}{y}_{rk}}{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}s.t\frac{{\sum }_{j=1}^n{\sum }_{r=1}^s{u}_{rj}{y}_{rj}}{{\sum }_{j=1}^n{\sum }_{t=1}^p{w}_{tj}{z}_{tj}}\le 1\hfill \\ & w_{tk}\ge ϵ\forall t,u_{rk}\ge ϵ\forall r\hfill \end{array}$$ (6)

Here $u_{rk}$ and $w_{tk}$ represent the $r$th and $t$th weights of the output and input, respectively, of the $k$th DMU for the second stage, and $ϵ$ is a non-Archimedean infinitesimal number.

The overall efficiency model

Let $x_{ik}(i=1,2,3,\dots ,m),z_{tk}(t=1,2,3,\dots ,p$, and $y_{rk}(r=1,2,3,\dots s)$ denote the $i$th input, $t$thintermediate measure, and $r$th output of the $k$th DMU, respectively. Its mathematical model is as follows [7]:

$$\begin{array}{cc}& E_k=\max {\omega }_1\frac{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}{{\sum }_{k=1}^n{\sum }_{i=1}^m{v}_{ik}{x}_{ik}}+{\omega }_2\frac{{\sum }_{k=1}^n{\sum }_{r=1}^s{u}_{rk}{y}_{rk}}{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}s.t\frac{{\sum }_{j=1}^n{\sum }_{t=1}^p{w}_{tj}{z}_{tj}}{{\sum }_{j=1}^n{\sum }_{i=1^m}{v}_{ij}{x}_{ij}}\le 1,\frac{{\sum }_{j=1}^n{\sum }_{r=1}^s{u}_{rj}{y}_{rj}}{{\sum }_{j=1}^n{\sum }_{t=1}^p{w}_{tj}{z}_{tj}}\le 1\hfill \\ & u_{rk}\ge ϵ,\forall r,w_{tk}\ge ϵ\forall t,u_{rk}\ge ϵ\forall i\hfill \end{array}$$ (7)

Here $u_{rk}$, $w_{tk}$, and $v_{ik}$ represent the weights of output, intermediate measure, and input of $k$th DMU, respectively, and $ϵ$ is a non-Archimedean infinitesimal number. Moreover

$$\begin{array}{c}\hfill {\omega }_1=\frac{{\sum }_{k=1}^n{\sum }_{i=1}^m{v}_{ik}{x}_{ik}}{{\sum }_{k=1}^n{\sum }_{i=1}^m{v}_{ik}{x}_{ik}+{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}\end{array}$$ (8)

$$\begin{array}{c}\hfill {\omega }_2=\frac{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}}{{\sum }_{k=1}^n{\sum }_{t=1}^p{w}_{tk}{z}_{tk}+{\sum }_{k=1}^n{\sum }_{r=1}^s{u}_{rk}{y}_{rk}}\end{array}$$ (9)

are the arithmetic mean with the property ${\omega }_1+{\omega }_2=1$.

Proposed methodology

We elucidate the theoretical part of the study, encompassing problem description, model derivation, and solution procedure.

Problem description

Decision analysis proves to be a potent method for tackling real-world problems, and one of its notable applications is Data Envelopment Analysis (DEA), which manifests in diverse forms, including network DEA. Network DEA grapples with issues surrounding undesirable outputs, shared resources (inputs/intermediate measures), and external inputs/outputs, all contributing to the efficiency scores of individual Decision-Making Units (DMUs). To illustrate, within the banking sector, non-performing assets such as bad loans significantly impact a bank’s profitability, raising concerns among lenders and policymakers. In a study proposed by Puri and Yadav [3], non-performing assets are identified as undesirable outputs. Notably, these undesirable outputs must exit antecedent stages. They cannot serve as inputs in subsequent stages, as highlighted in a study by [34], which identifies four approaches to address undesirable outputs in DEA:

• •First, disregarding undesirable outputs from the production function.
• •Second, treating these undesirable outputs as regular inputs.
• •Third, consider these undesirable outputs as normal outputs.
• •Fourth, transforming undesirable outputs using a non-decreasing function in DEA models.

In network DEA models, shared resources are partially integrated into sub-stages. In a study by Izadikhah et al. [9], profitability and marketability were designated as the first and second stages, respectively. The study utilized expenses and services as shared inputs and intermediate measures, while employees and facilities served as external inputs and documents as external outputs. This study amalgamates all these components into a sequential network DEA model, as depicted in Fig. 1.

Model derivation

Given the problem description, we present a two-stage DEA model structured in a series configuration of the network DEA model. In this model, each Decision Making Unit (DMU) incorporates the dataset with the following characteristics:

• (1)Three categories of inputs: internal, shared, and external.
• (2)Two categories of intermediate measures: internal and shared intermediate measures.
• (3)Three categories of outputs: undesirable, external, and final.

Internal and external inputs directly impact the first and second stages, respectively, while both partially utilize shared inputs. Similarly, internal intermediate measures directly contribute to the second stage, whereas the second stage partially uses shared intermediate measures. Undesirable outputs are incorporated into the model with a coefficient of $(-1)$, while external and final outputs are produced by the first and second stages, respectively. The diagram in Fig. 3 illustrates the multi-level structure of a two-stage DEA model with an extension of series configuration. The first level is a simple series two-stage DEA structure encompassing inputs $X$, intermediate measures $Z$, and outputs $Y$. The second level is an extended series two-stage DEA structure that includes undesirable outputs $UY1$ and $UY2$, generated by the first and second stages. The third level is also an extension of series two-stage DEA that displays final outputs $Y1$ and $Y2$, produced by the first and second stages, along with external inputs $X2$, directly employed by the second stage. The fourth level is a complex structure of a two-stage DEA that illustrates the allocation of shared resources (inputs/intermediate measures) between the first and second stages based on convexity theory. Shared inputs $X_2$ are divided into $\alpha X_2$ for the first stage and $(1-\alpha )x_2$ for the second stage. Similarly, shared intermediate measures $Z_2$ are split into $\beta Z_2$ for the second stage and $(1-\beta )Z_2$ as outputs of the first stage. The parameters $\alpha$ and $\beta$ are constrained between 0 and 1 (refer to [9], [17]). This paper introduces level 5 of Fig. 3, which encompasses shared inputs, intermediate measures, undesirable outputs, and free outputs and inputs. Building upon the foundational DEA framework proposed by [1] and the additive two-stage DEA model presented by Chen et al. [7], we develop a comprehensive two-stage network DEA model that integrates undesirable outputs, shared resources, and external inputs/outputs, as depicted in level 5 of Fig. 3. In level 5, we have three kinds of inputs known as $X_1$, $X_2$, and $X_3$ such that $X_1$ are internal inputs that are directly used for the first stage, $X_2$ are shared inputs which are divided as $\alpha X_2$ and $(1-\alpha )X_2$ that are partially consumed for the first and second stages, respectively, and $X_3$ is external or free inputs directly consumed by the second stage. Two intermediate measures say $Z_1$ and $Z_2$, are used in level 5 in which $Z_1$ is directly inter into the second stage while $Z_2$ are divided into $\beta Z_2$ and $(1-\beta )Z_2$ to partially use for the second stage and first stage’s outputs. Consequently, three distinct outputs produced in level 5, say external outputs ($Y_1$), Undesirable outputs ($U{Y}_1$), and final outputs $\left(Y_2\right)$

Fig. 3.

Two-stage DEA with different level process.

A comprehensive two-stage network DEA model

Suppose we have a group of decision-making units (DMUs) with the same characteristics and use the data in Table 1. We can derive the following mathematical models to calculate each DMU’s first-stage, second-stage, and overall efficiency scores. These models are based on the CCR approach, which assumes constant returns to scale.

Table 1.

Datasets (Inputs, Intermediate Measures, and Outputs).

$DM{U}_j$	$j=1,2,3,\dots ,n$	$j$th DMU
$x_{ij}^1$	$i=1,2,3,\dots ,I$	$i$th internal input which are directly consumed by the first stage
$z_{dj}^1$	$d=1,2,3,\dots ,D$	$d$th internal intermediate measure directly consumed by the second stage.
$y_{hj}^1$	$h=1,2,3,\dots ,H$	$h$th external output that is produced by the first stage.
$x_{tj}^2$	$t=1,2,3,\dots ,T$	$t$th shared input that is partially consumed by the first and second stages
		${\alpha }_t{x}_{tj}^2$ is used for the first stage and $(1-{\alpha }_t)x_{tj}^2$ is used for the second stage.
${\alpha }_t$	$t=1,2,3,\dots ,T$	is a parameter bounded between 0 and 1.
$z_{sj}^2$	$s=1,2,3,\dots ,S$	$s$th shared intermediate measure that is partially utilized by the second stage.
		${\beta }_s{z}_{sj}^2$ is used for the second stage and $(1-{\beta }_s)z_{sj}^2$ is avoided by the second stage.
${\beta }_s$	$s=1,2,3,\dots ,S$	is a parameter located in the unit interval [0,1]
$y_{rj}^2$	$r=1,2,3,\dots ,R$	$r$th final output that is produced by the second stage.
$x_{mj}^3$	$m=1,2,3,\dots ,M$	$m$th external input that is directly consumed by the second stage.
$y_{lj}^b$	$l=1,2,3,\dots ,L$	$l$th undesirable output is produced by the first stage and avoids entering the second stage.
${\theta }_p^1$	$p=1,2,3,\dots ,n$	The efficiency score of the $p$th DMU evaluated by the first-stage network DEA model.
${\theta }_p^2$	$p=1,2,3,\dots ,n$	The efficiency score of the $p$th DMU evaluated by the second-stage network DEA model.
${\theta }_p$	$p=1,2,3,\dots ,n$	The overall efficiency score of the $p$th DMU evaluated
		by the arithmetic mean of the first and second stages of the network DEA model.

Model1: The first-stage efficiency model

Recall the initial CCR proposed by Charnes et al. [1] involving the datasets outlined in Table 1. The below model measures the efficiency score of the first-stage network DEA model.

$$\begin{array}{cc}\hfill {\theta }_p^1& =\max \frac{{\sum }_{p=1}^n{\sum }_{d=1}^D{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^S{w}_{sp}^2{z}^{2}sp+{\sum }_{p=1}^n{\sum }_{h=1}^H{u}_{hp}^1{y}_{hp}^1-{\sum }_{p=1}^n{\sum }_{l=1}^L{u}_{lp}^b{Y}_{lp}^b}{{\sum }_{p=1}^n{\sum }_{i=1}^I{v}_{ip}^1{x}_{ip}^1+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2{\alpha }_t{x}_{tp}^2}\hfill \end{array}$$ (10)

$$\begin{array}{cc}\hfill s.t& \frac{{\sum }_{j=1}^n{\sum }_{d=1}^D{w}_{dj}^1{z}_{dj}^1+{\sum }_{j=1}^n{\sum }_{s=1}^S{w}_{sj}^2{z}^{2}sj+{\sum }_{j=1}^n{\sum }_{h=1}^H{u}_{hj}^1{y}_{hj}^1-{\sum }_{j=1}^n{\sum }_{l=1}^L{u}_{lj}^b{Y}_{lj}^b}{{\sum }_{j=1}^n{\sum }_{i=1}^I{v}_{ij}^1{x}_{ij}^1+{\sum }_{j=1}^n{\sum }_{t=1}^T{v}_{tj}^2{\alpha }_t{x}_{tj}^2}\le 1.\hfill \\ & w_{dp}^1\ge ϵ,w_{sp}^2\ge ϵ,v_{ip}^1\ge ϵ,v_{tp}^2\ge ϵ,u_{hp}^1\ge ϵ,u_{lp}^b\ge ϵ.\hfill \end{array}$$ (11)

Here $w_{dp}^1,w_{sp}^2{v}_{ip}^1,v_{tp}^2,u_{hp}^1,u_{lp}^b.$ all are positive weights and $ϵ$ is non-Archimedean small number.

Model2: The second-stage efficiency modelReconsider the concept of the initial CCR model proposed by Charnes et al. [1] involving the dataset given in Table 1. The following mathematical model evaluates the efficiency score of the second-stage network DEA model.

$$\begin{array}{cc}\hfill {\theta }_p^2=& \max \frac{{\sum }_{p=1}^n{\sum }_{r=1}^R{u}_{rp}^2{y}_{rp}^2}{{\sum }_{p=1}^n{\sum }_{d=1}{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2(1-{\alpha }_t)x_{tp}^2+{\sum }_{p=1}^n{\sum }_{m=1}^M{v}_{mp}^3{x}_{mp}^3}\hfill \end{array}$$ (12)

$$\begin{array}{cc}\hfill s.t& \frac{{\sum }_{j=1}^n{\sum }_{r=1}^R{u}_{rj}^2{y}_{rj}^2}{{\sum }_{j=1}^n{\sum }_{d=1}{w}_{dj}^1{z}_{dj}^1+{\sum }_{j=1}^n{\sum }_{s=1}^2{w}_{sj}^2{\beta }_s{z}_{sj}^2+{\sum }_{j=1}^n{\sum }_{t=1}^T{v}_{tj}^2(1-{\alpha }_t)x_{tj}^2+{\sum }_{j=1}^n{\sum }_{m=1}^M{v}_{mj}^3{x}_{mj}^3}\le 1.\hfill \\ & w_{dp}^1\ge ϵ,w_{sp}^2\ge ϵ,v_{mp}^3\ge ϵ,v_{tp}^2\ge ϵ,u_{rp}^2\ge ϵ.\hfill \end{array}$$ (13)

Here $w_{dp}^1,w_{sp}^2{v}_{mp}^3,v_{tp}^2,u_{rp}^2$ all are positive weights and $ϵ$ is non-Archimedean small number.

Model3: The overall efficiency model

Reconsider the additive two-stage DEA model proposed by Chen et al. [7] consisting of the data set provided in Table 1. Hence, we can derive the following mathematical formulation that measures the overall efficiency score of the first and second stages.

$$\begin{array}{cc}& {\theta }_p=\max {\omega }_1\frac{{\sum }_{p=1}^n{\sum }_{d=1}^D{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^S{w}_{sp}^2{z}^{2}sp+{\sum }_{p=1}^n{\sum }_{h=1}^H{u}_{hp}^1{y}_{hp}^1-{\sum }_{p=1}^n{\sum }_{l=1}^L{u}_{lp}^b{Y}_{lp}^b}{{\sum }_{p=1}^n{\sum }_{i=1}^I{v}_{ip}^1{x}_{ip}^1+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2{\alpha }_t{x}_{tp}^2}\hfill \\ & +{\omega }_2\frac{{\sum }_{p=1}^n{\sum }_{r=1}^R{u}_{rp}^2{y}_{rp}^2}{{\sum }_{p=1}^n{\sum }_{d=1}{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2(1-{\alpha }_t)x_{tp}^2+{\sum }_{p=1}^n{\sum }_{m=1}^M{v}_{mp}^3{x}_{mp}^3}\hfill \end{array}$$ (14)

$$\begin{array}{cc}\hfill s.t& \frac{{\sum }_{j=1}^n{\sum }_{d=1}^D{w}_{dj}^1{z}_{dj}^1+{\sum }_{j=1}^n{\sum }_{s=1}^S{w}_{sj}^2{z}^{2}sj+{\sum }_{j=1}^n{\sum }_{h=1}^H{u}_{hj}^1{y}_{hj}^1-{\sum }_{j=1}^n{\sum }_{l=1}^L{u}_{lj}^b{Y}_{lj}^b}{{\sum }_{j=1}^n{\sum }_{i=1}^I{v}_{ij}^1{x}_{ij}^1+{\sum }_{j=1}^n{\sum }_{t=1}^T{v}_{tj}^2{\alpha }_t{x}_{tj}^2}\le 1.\hfill \end{array}$$ (15)

$$\begin{array}{cc}& \frac{{\sum }_{j=1}^n{\sum }_{r=1}^R{u}_{rj}^2{y}_{rj}^2}{{\sum }_{j=1}^n{\sum }_{d=1}{w}_{dj}^1{z}_{dj}^1+{\sum }_{j=1}^n{\sum }_{s=1}^2{w}_{sj}^2{\beta }_s{z}_{sj}^2+{\sum }_{j=1}^n{\sum }_{t=1}^T{v}_{tj}^2(1-{\alpha }_t)x_{tj}^2+{\sum }_{j=1}^n{\sum }_{m=1}^M{v}_{mj}^3{x}_{mj}^3}\le 1.\hfill \\ \hfill & w_{dp}^1\ge ϵ,w_{sp}^2\ge ϵ,v_{ip}^1\ge ϵ,v_{tp}^2\ge ϵ,v_{mp}^3\ge ϵ,u_{hp}^1\ge ϵ,u_{lp}^b\ge ϵ,u_{rp}^2\ge ϵ.\hfill \end{array}$$ (16)

Here $w_{dp}^1,w_{sp}^2{v}_{ip}^1,v_{tp}^2,v_{mp}^3,u_{hp}^1,u_{lp}^b,u_{rp}^2.$ all are positive weights and $ϵ$ is non-Archimedean small number. Additionally, we calculate the ${\omega }_1$ and ${\omega }_2$ as below that preserve the property ${\omega }_1+{\omega }_2=1$

$$\begin{array}{cc}& {\omega }_1=\frac{{\sum }_{p=1}^n{\sum }_{i=1}^I{v}_{ip}^1{x}_{ip}^1+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2{\alpha }_t{x}_{tp}^2}{{\sum }_{p=1}^n{\sum }_{i=1}^I{v}_{ip}^1{x}_{ip}^1+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2{x}_{tp}^2+{\sum }_{p=1}^n{\sum }_{d=1}{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+{\sum }_{p=1}^n{\sum }_{m=1}^M{v}_{mp}^3{x}_{mp}^3}\hfill \end{array}$$ (17)

$$\begin{array}{cc}& {\omega }_2=\frac{{\sum }_{p=1}^n{\sum }_{d=1}{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2(1-{\alpha }_t)x_{tp}^2+{\sum }_{p=1}^n{\sum }_{m=1}^M{v}_{mp}^3{x}_{mp}^3}{{\sum }_{p=1}^n{\sum }_{i=1}^I{v}_{ip}^1{x}_{ip}^1+{\sum }_{p=1}^n{\sum }_{t=1}^T{v}_{tp}^2{x}_{tp}^2+{\sum }_{p=1}^n{\sum }_{d=1}{w}_{dp}^1{z}_{dp}^1+{\sum }_{p=1}^n{\sum }_{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+{\sum }_{p=1}^n{\sum }_{m=1}^M{v}_{mp}^3{x}_{mp}^3}\hfill \end{array}$$ (18)

By substituting Eqs. (17) and (18) into the objective function of Model3 and implementing [1] linear transformation, we obtain the below Model4 is a linear form of the two-stage network DEA model.

Model4: The linear overall efficiency model

Model4 is a linear two-stage network DEA form, evaluating the overall efficiency score of the first and second stages assuming constant returns to scale.

$$\begin{array}{cc}& {\theta }_p=\max \sum _{p=1}^n\sum _{d=1}^D{w}_{dp}^1{z}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^S{w}_{sp}^2{z}^{2}sp+\sum _{p=1}^n\sum _{h=1}^H{u}_{hp}^1{y}_{hp}^1+\sum _{p=1}^n\sum _{r=1}^R{u}_{rp}^2{y}_{rp}^2-\sum _{p=1}^n\sum _{l=1}^L{u}_{lp}^b{Y}_{lp}^b\hfill \end{array}$$ (19)

$$\begin{array}{cc}& s.t\sum _{p=1}^n\sum _{i=1}^I{v}_{ip}^1{x}_{ip}^1+\sum _{p=1}^n\sum _{t=1}^T{v}_{tp}^2{x}_{tp}^2+\sum _{p=1}^n\sum _{d=1}{w}_{dp}^1{z}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^2{w}_{sp}^2{\beta }_s{z}_{sp}^2+\sum _{p=1}^n\sum _{m=1}^M{v}_{mp}^3{x}_{mp}^3=1\hfill \end{array}$$ (20)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{d=1}^D{w}_{dj}^1{z}_{dj}^1+\sum _{j=1}^n\sum _{s=1}^S{w}_{sj}^2{z}^{2}sj+\sum _{j=1}^n\sum _{h=1}^H{u}_{hj}^1{y}_{hj}^1-\sum _{j=1}^n\sum _{l=1}^L{u}_{lj}^b{Y}_{lj}^b-\sum _{j=1}^n\sum _{i=1}^I{v}_{ij}^1{x}_{ij}^1-\sum _{j=1}^n\sum _{t=1}^T{v}_{tj}^2{\alpha }_t{x}_{tj}^2\le 0.\hfill \end{array}$$ (21)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{r=1}^R{u}_{rj}^2{y}_{rj}^2-\sum _{j=1}^n\sum _{d=1}{w}_{dj}^1{z}_{dj}^1-\sum _{j=1}^n\sum _{s=1}^2{w}_{sj}^2{\beta }_s{z}_{sj}^2-\sum _{j=1}^n\sum _{t=1}^T{v}_{tj}^2(1-{\alpha }_t)x_{tj}^2-\sum _{j=1}^n\sum _{m=1}^M{v}_{mj}^3{x}_{mj}^3\le 0.\hfill \\ & w_{dp}^1\ge ϵ,w_{sp}^2\ge ϵ,v_{ip}^1\ge ϵ,v_{tp}^2\ge ϵ,v_{mp}^3\ge ϵ,u_{hp}^1\ge ϵ,u_{lp}^b\ge ϵ,u_{rp}^2\ge ϵ.\hfill \end{array}$$ (22)

Here $w_{dp}^1,w_{sp}^2{v}_{ip}^1,v_{tp}^2,v_{mp}^3,u_{hp}^1,u_{lp}^b,u_{rp}^2.$ all are positive weights and $ϵ$ is non-Archimedean small number.

The fourth model is a linear and comprehensive two-stage network DEA model that considers undesirable outputs, shared resources (inputs/intermediate measures), and external inputs/outputs, as shown in Fig. 1. The dataset used in Model4 consists of precise or crisp numbers. However, real-life problems often have some degree of fluctuation, causing imprecision or vagueness. Therefore, Model4 cannot be used for imprecise datasets like the one outlined in Table 1. To address uncertain situations, we extend Model4 to the intuitionistic fuzzy environment and represent the imprecise dataset using parabolic intuitionistic fuzzy numbers, as defined in Eqs. (3) and (4).

A comprehensive intuitionistic fuzzy two-stage network DEA model

Let’s consider we have a group of decision-making units (DMUs) that share the same characteristics and use the parabolic intuitionistic fuzzy (PIF) data presented in Table 2. In this case, we can formulate a mathematical model to calculate the overall efficiency score for each DMU in the intuitionistic fuzzy environment. The model is based on the CCR approach, which assumes constant returns to scale.

Table 2.

PIF datasets (Inputs, Intermediate measures, and Outputs).

$\mathit{DM}{U}_j$	$j=1,2,3,\cdots ,n$	$j$th DMU
${\tilde{x}}_{\mathit{ij}}^1$	$i=1,2,3,\cdots ,I$	$i$th internal PIF input, which is directly consumed by the first stage
${\tilde{z}}_{\mathit{dj}}^1$	$d=1,2,3,\cdots ,D$	$d$th internal PIF intermediate measure which is directly consumed by the second stage.
${\tilde{y}}_{\mathit{hj}}^1$	$h=1,2,3,\cdots ,H$	$h$th external PIF output that is produced by the first stage.
${\tilde{x}}_{\mathit{tj}}^2$	$t=1,2,3,\cdots ,T$	$t$th shared PIF input that is partially consumed by the first and second stages
		${\alpha }_t{\tilde{x}}_{\mathit{tj}}^2$ is used for the first stage and $(1-{\alpha }_t){\tilde{x}}_{\mathit{tj}}^2$ is used for the second stage.
${\alpha }_t$	$t=1,2,3,\cdots ,T$	is a parameter bounded between 0 and 1.
${\tilde{z}}_{\mathit{sj}}^2$	$s=1,2,3,\cdots ,S$	$s$th shared PIF intermediate measure that is partially utilized by the second stage.
		${\beta }_s{\tilde{z}}_{\mathit{sj}}^2$ is used for the second stage and $(1-{\beta }_s){\tilde{z}}_{\mathit{sj}}^2$ is consumed as the final output.
${\beta }_s$	$s=1,2,3,\cdots ,S$	is a parameter located in the unit interval [0,1]
${\tilde{y}}_{\mathit{rj}}^2$	$r=1,2,3,\cdots ,R$	$r$th final PIF output that is produced by the second stage.
${\tilde{x}}_{\mathit{mj}}^3$	$m=1,2,3,\cdots ,M$	$m$th external PIF input that is directly consumed by the second stage.
${\tilde{y}}_{lj}^b$	$l=1,2,3,\dots ,L$	$l$th undesirable PIF output is produced by the first stage and avoids entering the second stage.

The datasets are in the following form:

• •Parabolic intuitionistic fuzzy inputs: ${\tilde{x}}_{ij}^1=(x_{ij}^{1l},x_{ij}^{1m},x_{ij}^{1u};x_{ij}^{1l^{\prime }},x_{ij}^{1m},x_{ij}^{1u^{\prime }})$, ${\tilde{x}}_{tj}^2=(x_{tj}^{2l},x_{tj}^{2m},x_{tj}^{2u};x_{tj}^{2l^{\prime }},x_{tj}^{2m},x_{tj}^{2u^{\prime }})$, ${\tilde{x}}_{mj}^3=(x_{mj}^{3l},x_{mj}^{3m},x_{mj}^{3u};x_{mj}^{3l^{\prime }},x_{mj}^{3m},x_{mj}^{3u^{\prime }})$.
• •Parabolic intuitionistic fuzzy intermediate measures: ${\tilde{z}}_{dj}^1=(z_{dj}^{1l},z_{dj}^{1m},z_{dj}^{1u};z_{dj}^{1l^{\prime }},z_{dj}^{1m},z_{dj}^{1u^{\prime }})$, ${\tilde{z}}_{sj}^2=(z_{sj}^{2l},z_{sj}^{2m},z_{sj}^{2u};z_{sj}^{2l^{\prime }},z_{sj}^{2m},z_{sj}^{2u^{\prime }})$.
• •Parabolic Intuitionistic fuzzy outputs: ${\tilde{y}}_{hj}^1=(y_{hj}^{1l},y_{hj}^{1m},y_{hj}^{1u};y_{hj}^{1l^{\prime }},y_{hj}^{1m},y_{hj}^{1u^{\prime }})$, ${\tilde{y}}_{rj}^2=(y_{rj}^{2l},y_{rj}^{2m},y_{rj}^{2u};y_{rj}^{2l^{\prime }},y_{rj}^{2m},y_{rj}^{2u^{\prime }})$, ${\tilde{y}}_{lj}^b=(y_{lj}^{bl},y_{lj}^{bm},y_{lj}^{bu};y_{lj}^{b{l}^{\prime }},y_{hj}^{lm},y_{lj}^{b{u}^{\prime }})$

Model5: The parabolic intuitionistic fuzzy overall efficiency model

Model5 is an extension of Model4 that represents the entire dataset with PIFN, defined by membership and non-membership functions in Eqs. (3) and (4), respectively.

$$\begin{array}{cc}& {\tilde{\theta }}_p=\max \sum _{p=1}^n\sum _{d=1}^D{\tilde{w}}_{dp}^1{\tilde{z}}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^S{\tilde{w}}_{sp}^2{\tilde{z}}^{2}sp+\sum _{p=1}^n\sum _{h=1}^H{\tilde{u}}_{hp}^1{\tilde{y}}_{hp}^1+\sum _{p=1}^n\sum _{r=1}^R{\tilde{u}}_{rp}^2{\tilde{y}}_{rp}^2-\sum _{p=1}^n\sum _{l=1}^L{\tilde{u}}_{lp}^b{\tilde{Y}}_{lp}^b\hfill \end{array}$$ (23)

$$\begin{array}{cc}& s.t\sum _{p=1}^n\sum _{i=1}^I{\tilde{v}}_{ip}^1{\tilde{x}}_{ip}^1+\sum _{p=1}^n\sum _{t=1}^T{\tilde{v}}_{tp}^2{\tilde{x}}_{tp}^2+\sum _{p=1}^n\sum _{d=1}{\tilde{w}}_{dp}^1{\tilde{z}}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^2{\tilde{w}}_{sp}^2{\beta }_s{\tilde{z}}_{sp}^2+\sum _{p=1}^n\sum _{m=1}^M{\tilde{v}}_{mp}^3{\tilde{x}}_{mp}^3=1\hfill \end{array}$$ (24)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{d=1}^D{\tilde{w}}_{dj}^1{\tilde{z}}_{dj}^1+\sum _{j=1}^n\sum _{s=1}^S{\tilde{w}}_{sj}^2{\tilde{z}}^{2}sj+\sum _{j=1}^n\sum _{h=1}^H{\tilde{u}}_{hj}^1{\tilde{y}}_{hj}^1-\sum _{j=1}^n\sum _{l=1}^L{\tilde{u}}_{lj}^b{\tilde{Y}}_{lj}^b-\sum _{j=1}^n\sum _{i=1}^I{\tilde{v}}_{ij}^1{\tilde{x}}_{ij}^1-\sum _{j=1}^n\sum _{t=1}^T{\tilde{v}}_{tj}^2{\alpha }_t{\tilde{x}}_{tj}^2\le 0.\hfill \end{array}$$ (25)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{r=1}^R{\tilde{u}}_{rj}^2{\tilde{y}}_{rj}^2-\sum _{j=1}^n\sum _{d=1}{\tilde{w}}_{dj}^1{\tilde{z}}_{dj}^1-\sum _{j=1}^n\sum _{s=1}^2{\tilde{w}}_{sj}^2{\beta }_s{\tilde{z}}_{sj}^2-\sum _{j=1}^n\sum _{t=1}^T{\tilde{v}}_{tj}^2(1-{\alpha }_t){\tilde{x}}_{tj}^2-\sum _{j=1}^n\sum _{m=1}^M{\tilde{v}}_{mj}^3{\tilde{x}}_{mj}^3\le 0.\hfill \\ & {\tilde{w}}_{dp}^1\ge ϵ,{\tilde{w}}_{sp}^2\ge ϵ,{\tilde{v}}_{ip}^1\ge ϵ,{\tilde{v}}_{tp}^2\ge ϵ,{\tilde{v}}_{mp}^3\ge ϵ,{\tilde{u}}_{hp}^1\ge ϵ,{\tilde{u}}_{lp}^b\ge ϵ,{\tilde{u}}_{rp}^2\ge ϵ.\hfill \end{array}$$ (26)

Here ${\tilde{w}}_{dp}^1,{\tilde{w}}_{sp}^2{\tilde{v}}_{ip}^1,{\tilde{v}}_{tp}^2,{\tilde{v}}_{mp}^3,{\tilde{u}}_{hp}^1,{\tilde{u}}_{lp}^b,{\tilde{u}}_{rp}^2.$ all are positive weights and $ϵ$ is non-Archimedean small number.

Solution procedure

Model5 is a two-stage network DEA model that uses linear and comprehensive intuitionistic fuzzy methodology. It considers shared PIF resources, external PIF inputs/outputs, and undesirable PIF outputs given in Table 2. The corresponding crisp numbers for each PIF dataset must be obtained using a defuzzification method to solve Model 5. The centroid method is the most popular and applicable defuzzification method, which can be extended from a fuzzy set into the intuitionistic fuzzy set. This paper aims to present the extension of the centroid method to the intuitionistic fuzzy set.

A new center of gravity method

Let $\tilde{A}=\{(x,{\mu }_{\tilde{A}}\left(x\right),{\nu }_{\tilde{A}}\left(x\right)|x\in X\}$ denotes an IFS. Then, the mathematical formulation below is defined as a new center of gravity method for an IFS.

$$\begin{array}{c}\hfill Co{G}_{\tilde{A}}\left(x\right)=\frac{{\displaystyle {\int }_{-\infty }^{\infty }|{\mu }_{\tilde{A}}\left(x\right)-{\pi }_{\tilde{A}}\left(x\right)|·xdx}}{{\displaystyle {\int }_{-\infty }^{\infty }|{\mu }_{\tilde{A}}\left(x\right)-{\pi }_{\tilde{A}}\left(x\right)|dx}}\end{array}$$ (27)

The equation defined in Eq. (27) equals the initial center of gravity method when the IFS becomes FS. Therefore, Eq. (27) is well defined.

Remark 1

Suppose $\tilde{A}=(a_1,a_2,a_3;b_1,b_2,b_3)$ is the PIFN whose membership and non-membership values are defined by (3) and (4), respectively. Then notion $|{\mu }_{\tilde{A}}\left(x\right)-{\pi }_{\tilde{A}}\left(x\right)|$ is modified into the following equation:

$$|{\mu }_{\tilde{A}}\left(x\right)-{\pi }_{\tilde{A}}\left(x\right)|=\{\begin{array}{cc}|{\nu }_{\tilde{A}}\left(x\right)-1|,\hfill & b_1\le x<a_1\hfill \\ |2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|,\hfill & a_1\le x\le a_2\hfill \\ |2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|,\hfill & a_2\le x\le a_3\hfill \\ |{\nu }_{\tilde{A}}\left(x\right)-1|,\hfill & a_3\le x\le b_3\hfill \\ 0,\hfill & Otherwise\hfill \end{array}$$ (28)

Based on Eq. (28), the new center of gravity method that is defined by Eq. (27) can be modified in the below equation:

$$x^{★}=\frac{{\displaystyle {\int }_{b_1}^{a_1}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{a_1}^{a_2}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{a_2}^{a_3}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{a_3}^{b_3}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{b_1}^{a_1}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{a_1}^{a_2}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{a_2}^{a_3}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{a_3}^{b_3}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (29)

By applying Eq. (29) to all datasets in Table 2, the PIF datasets can be converted to their corresponding crisp datasets, as follows:

$${\bar{x}}_{ij}^1=\frac{{\displaystyle {\int }_{x_{ij}^{1l^{\prime }}}^{x_{ij}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{ij}^{1l}}^{x_{ij}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{ij}^{1m}}^{x_{ij}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{ij}^{1u}}^{x_{ij}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{x_{ij}^{1l^{\prime }}}^{x_{ij}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{ij}^{1l}}^{x_{ij}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{ij}^{1m}}^{x_{ij}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{ij}^{1u}}^{x_{ij}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (30)

$${\bar{x}}_{tj}^2=\frac{{\displaystyle {\int }_{x_{tj}^{2l^{\prime }}}^{x_{tj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{tj}^{2l}}^{x_{tj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{tj}^{2m}}^{x_{tj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{tj}^{2u}}^{x_{tj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{x_{tj}^{2l^{\prime }}}^{x_{tj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{tj}^{2l}}^{x_{tj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{tj}^{2m}}^{x_{tj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{tj}^{2u}}^{x_{tj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (31)

$${\bar{x}}_{mj}^3=\frac{{\displaystyle {\int }_{x_{mj}^{3l^{\prime }}}^{x_{mj}^{3l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{mj}^{3l}}^{x_{mj}^{3m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{mj}^{3m}}^{x_{mj}^{3u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{x_{mj}^{3u}}^{x_{mj}^{3u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{x_{mj}^{3l^{\prime }}}^{x_{mj}^{3l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{mj}^{3l}}^{x_{mj}^{3m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{mj}^{3m}}^{x_{mj}^{3u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{x_{mj}^{3u}}^{x_{mj}^{3u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (32)

$${\bar{z}}_{dj}^1=\frac{{\displaystyle {\int }_{z_{dj}^{1l^{\prime }}}^{z_{dj}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{dj}^{1l}}^{z_{dj}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{dj}^{1m}}^{z_{dj}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{dj}^{1u}}^{z_{dj}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{z_{dj}^{1l^{\prime }}}^{z_{dj}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{dj}^{1l}}^{z_{dj}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{dj}^{1m}}^{z_{dj}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{dj}^{1u}}^{z_{dj}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (33)

$${\bar{z}}_{sj}^2=\frac{{\displaystyle {\int }_{z_{sj}^{2l^{\prime }}}^{z_{sj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{sj}^{2l}}^{z_{sj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{sj}^{2m}}^{z_{sj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{z_{sj}^{2u}}^{z_{sj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{z_{sj}^{2l^{\prime }}}^{z_{sj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{sj}^{2l}}^{z_{sj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{sj}^{2m}}^{z_{sj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{z_{sj}^{2u}}^{z_{sj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (34)

$${\bar{y}}_{hj}^1=\frac{{\displaystyle {\int }_{y_{hj}^{1l^{\prime }}}^{y_{hj}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{hj}^{1l}}^{y_{hj}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{hj}^{1m}}^{y_{hj}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{hj}^{1u}}^{y_{hj}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{y_{hj}^{1l^{\prime }}}^{y_{hj}^{1l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{hj}^{1l}}^{y_{hj}^{1m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{hj}^{1m}}^{y_{hj}^{1u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{hj}^{1u}}^{y_{hj}^{1u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (35)

$${\bar{y}}_{rj}^2=\frac{{\displaystyle {\int }_{y_{rj}^{2l^{\prime }}}^{y_{rj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{rj}^{2l}}^{y_{rj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{rj}^{2m}}^{y_{rj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{rj}^{2u}}^{y_{rj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{y_{rj}^{2l^{\prime }}}^{y_{rj}^{2l}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{rj}^{2l}}^{y_{rj}^{2m}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{rj}^{2m}}^{y_{rj}^{2u}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{rj}^{2u}}^{y_{rj}^{2u^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (36)

$${\bar{y}}_{lj}^b=\frac{{\displaystyle {\int }_{y_{lj}^{b{l}^{\prime }}}^{y_{lj}^{bl}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{lj}^{bl}}^{y_{lj}^{bm}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{lj}^{bm}}^{y_{lj}^{bu}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|·xdx+{\int }_{y_{lj}^{bu}}^{y_{lj}^{b{u}^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|·xdx}}{{\displaystyle {\int }_{y_{lj}^{b{l}^{\prime }}}^{y_{lj}^{bl}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{lj}^{bl}}^{y_{lj}^{bm}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{lj}^{bm}}^{y_{lj}^{bu}}|2{\mu }_{\tilde{A}}\left(x\right)+{\nu }_{\tilde{A}}\left(x\right)-1|dx+{\int }_{y_{lj}^{bu}}^{y_{lj}^{b{u}^{\prime }}}|{\nu }_{\tilde{A}}\left(x\right)-1|dx}}$$ (37)

In the Eqs. (30)–(37), the membership and non-membership grades are defined by (3) and (4), respectively. Replacing the dataset of Model5 with the dataset produced by Eqs. (30)–(37), the following modified Model 6 is obtained.

Model6: A deterministic comprehensive two-stage network overall efficiency model

Model6 is a modified and deterministic version of Model5, which uses a new center of gravity method to transform the PIF dataset provided in Table 2 into a precise or crisp dataset. This is done using the Eqs. (30)–(37).

$$\begin{array}{cc}& {\bar{\theta }}_p=\max \sum _{p=1}^n\sum _{d=1}^D{\bar{w}}_{dp}^1{\bar{z}}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^S{\bar{w}}_{sp}^2{\bar{z}}^{2}sp+\sum _{p=1}^n\sum _{h=1}^H{\bar{u}}_{hp}^1{\bar{y}}_{hp}^1+\sum _{p=1}^n\sum _{r=1}^R{\bar{u}}_{rp}^2{\bar{y}}_{rp}^2-\sum _{p=1}^n\sum _{l=1}^L{\bar{u}}_{lp}^b{\bar{Y}}_{lp}^b\hfill \end{array}$$ (38)

$$\begin{array}{cc}& s.t\sum _{p=1}^n\sum _{i=1}^I{\bar{v}}_{ip}^1{\bar{x}}_{ip}^1+\sum _{p=1}^n\sum _{t=1}^T{\bar{v}}_{tp}^2{\bar{x}}_{tp}^2+\sum _{p=1}^n\sum _{d=1}{\bar{w}}_{dp}^1{\bar{z}}_{dp}^1+\sum _{p=1}^n\sum _{s=1}^2{\bar{w}}_{sp}^2{\beta }_s{\bar{z}}_{sp}^2+\sum _{p=1}^n\sum _{m=1}^M{\bar{v}}_{mp}^3{\bar{x}}_{mp}^3=1\hfill \end{array}$$ (39)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{d=1}^D{\bar{w}}_{dj}^1{\bar{z}}_{dj}^1+\sum _{j=1}^n\sum _{s=1}^S{\bar{w}}_{sj}^2{\bar{z}}^{2}sj+\sum _{j=1}^n\sum _{h=1}^H{\bar{u}}_{hj}^1{\bar{y}}_{hj}^1-\sum _{j=1}^n\sum _{l=1}^L{\bar{u}}_{lj}^b{\bar{Y}}_{lj}^b-\sum _{j=1}^n\sum _{i=1}^I{\bar{v}}_{ij}^1{\bar{x}}_{ij}^1-\sum _{j=1}^n\sum _{t=1}^T{\bar{v}}_{tj}^2{\alpha }_t{\bar{x}}_{tj}^2\le 0.\hfill \end{array}$$ (40)

$$\begin{array}{cc}& \sum _{j=1}^n\sum _{r=1}^R{\bar{u}}_{rj}^2{\bar{y}}_{rj}^2-\sum _{j=1}^n\sum _{d=1}{\bar{w}}_{dj}^1{\bar{z}}_{dj}^1-\sum _{j=1}^n\sum _{s=1}^2{\bar{w}}_{sj}^2{\beta }_s{\bar{z}}_{sj}^2-\sum _{j=1}^n\sum _{t=1}^T{\bar{v}}_{tj}^2(1-{\alpha }_t){\bar{x}}_{tj}^2-\sum _{j=1}^n\sum _{m=1}^M{\bar{v}}_{mj}^3{\bar{x}}_{mj}^3\le 0.\hfill \\ & {\bar{w}}_{dp}^1\ge ϵ,{\bar{w}}_{sp}^2\ge ϵ,{\bar{v}}_{ip}^1\ge ϵ,{\bar{v}}_{tp}^2\ge ϵ,{\bar{v}}_{mp}^3\ge ϵ,{\bar{u}}_{hp}^1\ge ϵ,{\bar{u}}_{lp}^b\ge ϵ,{\bar{u}}_{rp}^2\ge ϵ.\hfill \end{array}$$ (41)

Here ${\bar{w}}_{dp}^1,{\bar{w}}_{sp}^2{\bar{v}}_{ip}^1,{\bar{v}}_{tp}^2,{\bar{v}}_{mp}^3,{\bar{u}}_{hp}^1,{\bar{u}}_{lp}^b,{\bar{u}}_{rp}^2.$ all are positive weights and $ϵ$ is non-Archimedean small number.

The relative efficiency can be represented as a matrix. Several software or programming languages can be used to compute the results. For this study, we have chosen to use MATLAB software to calculate the relative efficiency of Model4 and Model6. Three distinct Matlab files evaluate the relative efficiency with the following file links (Model4’s link, Model6’s link, Tables link). The following Algorithm 1 outlines the step-by-step solution procedure for each proposed model.

Algorithm 1

Solution Procedure for Proposed Models

• •
  Inputs:

  • 1.
    Provide the dataset outlined in Tables 1 or 2.
  • 2.
    Specify whether the dataset is in crisp or intuitionistic fuzzy form.
• •
  Process:

  • 1.
    If the dataset is in crisp form, proceed to step 2. If the dataset is in intuitionistic fuzzy form, proceed to step 3.
  • 2.
    Calculate the relative efficiency of the proposed comprehensive two-stage network DEA model using Model4.
  • 3.
    Calculate the relative efficiency of the proposed comprehensive intuitionistic fuzzy two-stage network DEA model using Model5.
  • 4.
    Transform each PIF dataset into a crisp dataset using the proposed new center of gravity method defined in Eq. (29).
  • 5.
    Calculate the relative efficiency of the deterministic comprehensive two-stage network DEA model using Model6.
• •
  Outputs:

  • 1.
    Efficiency result of Table 10.
  • 2.
    Efficiency result of Table 12.

Table 10.

Efficiency result of Model4 for Indian public sector banks.

Banks	$\alpha =0.2,\beta =0.3$	$\alpha =0.5,\beta =0.2$	$\alpha =0.7,\beta =0.1$	$\alpha =0.5,\beta =0.5$	$\alpha =0.2,\beta =0.7$	$\alpha =0.9,\beta =0.7$
BOB	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
BOI	0.6498	0.6553	0.7078	0.6414	0.6326	0.6982
BOMH	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
CB	0.8387	0.8486	0.8619	0.8486	0.8279	0.8658
CBOI	0.7289	0.7506	0.7747	0.6923	0.6693	0.6693
IB	0.7847	0.7957	0.8177	0.7938	0.7850	0.8143
IOB	0.6859	0.6924	0.7447	0.6796	0.6689	0.7251
PASB	0.8531	0.9067	0.9440	0.8881	0.8451	0.9129
PNB	0.7925	0.8201	0.8502	0.7559	0.7558	0.7558
SBOI	0.9845	0.9845	0.9895	0.9845	0.9942	0.9942
UB	0.6790	0.7480	0.8214	0.7179	0.6475	0.7678
UBI	0.8767	0.8914	0.9080	0.8548	0.8374	0.8604

Table 12.

Efficiency result of Model6 for Indian public sector banks.

Banks	$\alpha =0.2,\beta =0.3$	$\alpha =0.5,\beta =0.2$	$\alpha =0.7,\beta =0.1$	$\alpha =0.5,\beta =0.5$	$\alpha =0.2,\beta =0.7$	$\alpha =0.9,\beta =0.7$
BOB	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
BOI	0.7253	0.7314	0.8421	0.7314	0.7253	0.9484
BOMH	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000
CB	0.8301	0.8491	0.8622	0.8491	0.8301	0.8662
CBOI	0.7289	0.7506	0.7747	0.6923	0.6693	0.6693
IB	0.7847	0.7957	0.8178	0.7938	0.7825	0.8143
IOB	0.6859	0.6924	0.7430	0.6796	0.6689	0.7251
PASB	0.8530	0.9064	0.9439	0.8881	0.8451	0.9129
PNB	0.7925	0.8201	0.8501	0.7558	0.7558	0.7558
SBOI	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
UB	0.6790	0.7480	0.8214	0.7164	0.6475	0.7678
UBI	0.8767	0.8913	0.9080	0.8548	0.8554	0.8604

Method validation

This section encompasses the method’s validation, including a comparative analysis of the proposed method and existing methods across network structure and uncertainty scenarios and an empirical application to the assessment of Indian public-sector banks.

Comparative analysis

Several new studies regarding two-stage network DEA have been proposed in the literature to measure the relative efficiency of DMUs. In Table 3, we provide a comparative overview of the related studies that are theoretically innovative and experimentally applicable.

Table 3.

Comparison between the proposed method and the existing methods.

Method	Shared resources	External input/output	Undesirable output	Uncertain dataset
Wang [37]	$\times$	$\times$	$\times$	$\times$
Wu [38]	$✓$	$\times$	$✓$	$\times$
Izadikhah [9]	$✓$	$✓$	$\times$	$\times$
Wang [17]	$✓$	$\times$	$\times$	$\times$
Zeng [39]	$\times$	$✓$	$✓$	$\times$
Chen [40]	$\times$	$✓$	$✓$	$\times$
Liu [26]	$✓$	$\times$	$\times$	$\times$
Zhou [14]	$✓$	$\times$	$✓$	$\times$
Arya [12]	$\times$	$\times$	$\times$	$✓$
Mohammadi [32]	$\times$	$\times$	$\times$	$✓$
Shariatmadari [33]	$✓$	$\times$	$\times$	$✓$
Izadikhah [41]	$\times$	$✓$	$\times$	$✓$
Proposed Method	$✓$	$✓$	$✓$	$✓$

In the current methods, two significant limitations need to be addressed. Firstly, no methods incorporate shared resources (inputs, intermediate measures), external inputs/outputs, and undesirable outputs. Secondly, several novel methods consider the dataset (inputs, intermediate measures, and outputs) as precise values or crisp numbers. However, in real-world scenarios, due to unquantifiable and incomplete information, the dataset possesses some degree of fluctuation, which causes impreciseness. This article proposes a model combining shared resources, external inputs/outputs, and undesirable outputs. Moreover, we extend our proposed model in the IFS environment to deal with uncertainty. Therefore, our model is more general than other existing methods. We highlight some current novel methods that apply to the proposed model to support this statement. For instance, in their experimental study, [17] used deposit and loan systems as the first and second stages, respectively. They utilized fixed assets, IT budget, and employees as shared inputs, deposits as an intermediate measure, and profit and fraction of loan recovery as outputs. However, this approach is not applicable in cases with shared intermediate measures, external inputs/outputs, and undesirable outputs. [9] utilized the Philadelphia National Bank (PNB) as a case study and used profitability and marketability as the first and second stages, respectively. They considered location and employees as direct inputs for the first stage, which produced deposits, loans, and services. These outputs were then used as inputs for the second stage to produce net profits as final outputs. Additionally, expenses were used as a shared input and services as a shared intermediate measure. On the other hand, employees, facilities, and documents were used as external inputs and outputs, respectively. It is worth noting that undesirable outputs should have been considered in this approach. In a study by Zhou and Hu [14], undesirable outputs and shared inputs were considered. Their empirical study utilized production and service as the first and second stages. They used capital and land as inputs for the first stage to produce railway mileage and railway density as outputs. These outputs were then used for the second stage to generate passenger turnover, freight turnover, and wage as final desirable outputs. Despite considering dust an undesirable output, this approach lacked shared intermediate measures and external inputs/outputs. In the realm of literature, numerous innovative approaches have emerged to address uncertainty through the two-stage network DEA model. However, these methods face a notable deficiency in effectively integrating shared resources, external inputs/outputs, and undesirable outputs (see [32], [35], [36]).

To compare the effectiveness of the proposed method with the existing one, we will use a numerical example from a previous study of [9]. The example involves evaluating the efficiency score of 15 branches of the Philadelphia National Bank, as shown in Table 4. Table 4 uses location and Employees 1 as the first-stage inputs. The expenses are the shared input for both the first and second stages. Deposits and loans are the intermediate measures directly used as inputs for the second stage. On the other hand, services are shared intermediate measures that are partially consumed by the second stage. Documents are used as the first stage’s external output. Meanwhile, Employees 2 and facilities are external inputs freely consumed by the second stage. Finally, net profits are the final outputs produced by the second stage. Compared to the existing method, we present the efficiency results of the proposed method in Table 5 and Fig. 4. The results indicate the validity and superiority of the proposed method by the following points.

• •The numerical example provided validates the proposed method.
• •The proposed method produces larger efficiency values for DMUs and identifies more efficient DMUs in the first stage, indicating superiority.

Table 4.

The datasets of Philadelphia National Banks.

DMUs	Location	Employees 1	Expenses	Employees 2	Facilities	Deposits	Loans	Services	Documents	Net profits
B1	4	33	6053950	31	45	88805870	9930070	1586	202402	886700
B2	5	42	2568000	41	48	80331040	10558070	2587	130134	12504200
B3	3	31	4937460	20	39	54096730	26908410	1928	157240	3883930
B4	5	38	13770420	51	58	159553230	23552210	4334	316914	369000
B5	5	34	16406000	52	47	176834230	48220890	5051	334180	13600000
B6	3	44	5602820	19	54	78190480	22086900	3417	162561	482030
B7	4	46	2961270	28	56	64369880	27478490	2533	263767	3388480
B8	4	33	2566360	41	37	59637190	18286690	3141	236540	118910
B9	4	43	2231200	63	57	116929340	13285005	4558	246474	16330000
B10	3	34	6102760	30	41	60218920	16808870	3161	240654	122670
B11	4	41	11888260	47	53	124104230	32805910	3196	221537	311640
B12	3	31	5122280	29	34	66308610	17785790	2548	162928	19708980
B13	4	34	3943260	34	43	69911600	43754650	2518	286015	2500000
B14	4	31	4451080	22	44	46755500	19889450	2266	212533	102780
B15	4	40	4075000	16	51	28471320	31604330	1286	131854	11610000

Table 5.

The result of comparative analysis.

Banks	Proposed Method			Izadikhah et al. [9] Method
	${\theta }_p$	${\theta }_p^1$	${\theta }_p^2$	${\theta }_p$	${\theta }_p^1$	${\theta }_p^2$
B1	0.7051	0.8249	0.0793	0.0320	0.7730	0.0420
B2	0.8088	0.6985	1.0000	0.6180	0.6900	0.8960
B3	0.6765	0.8681	0.2780	0.2030	0.7290	0.2780
B4	0.4407	1.0000	0.0109	0.0100	0.9170	0.0110
B5	0.5997	1.0000	0.3761	0.4990	1.0000	0.4990
B6	0.6736	1.0000	0.0362	0.0360	1.0000	0.0360
B7	0.8985	1.0000	0.2577	0.2630	1.0000	0.2630
B8	0.8385	1.0000	0.0097	0.0100	1.0000	0.0100
B9	0.9105	1.0000	1.0000	1.0000	1.0000	1.0000
B10	0.9921	1.0000	0.0068	0.0070	1.0000	0.0070
B11	0.4392	0.9028	0.0102	0.0100	0.8440	0.0120
B12	1.0000	0.8438	1.0000	0.7910	0.7910	1.0000
B13	0.9871	1.0000	0.1528	0.1520	1.0000	0.1520
B14	0.7624	0.8322	0.0069	0.0060	0.8340	0.0070
B15	0.9934	0.7212	1.0000	0.4720	0.4720	1.0000

Fig. 4.

The result of comparative analysis.

An empirical application

We have selected Indian Public Sector Banks to assess their efficiency utilizing the two-stage network DEA model. This model comprises four key stages: input/output selection, data collection, fuzzification, and defuzzification. It is applied in both deterministic and uncertain scenarios for comprehensive evaluation.

Input and output selection

Selecting the input and output is a crucial aspect of DEA. Usually, the inputs and outputs are chosen based on the research objective. In the banking industry, two approaches are used for selecting inputs and outputs: the production and intermediate processes. There is general agreement on the categories of inputs and outputs, except for deposits. The production process approach considers deposits as outputs while the intermediate approach counts deposits as inputs [3], [42], [43]. In Table 6, we have compiled some inputs and outputs of DEA based on the banking industry.

Table 6.

Inputs and outputs of banks.

Inputs	Outputs
Total assets	Net interest income
Number of employees	Net income after taxes
Operating expenses	Return on assets
Capital expenditure	Return on equity
Interest expended	Interest earned
Total deposits	Other incomes
Non-interest incomes	Total loans and receivable
	Non-performing loans
	Customer satisfaction

The DEA has extended into the two-stage network DEA, and this extension has also been applied to the banking industry. For example, in a study proposed by Chen and Zhu [44], fixed assets, employees, and IT investments are considered inputs to produce deposits. The produced deposits are then used as inputs to produce profit and recover loans. The recent application of two-stage DEA to the banking industry can be addressed in the study proposed by Fukuyama and Matousek [24], Fukuyama et al. [25]. Seiford and Zhu [45] also investigated the efficiencies of U.S. commercial banks, treating them as two-stage processes. Stage 1 focused on assessing profitability performance, while Stage 2 concentrated on marketability performance. A study [46] assessed the profitability and marketability efficiency of a broad range of banks. Building on this foundation, [9] applied the two-stage process framework (profitability and marketability) to evaluate Philadelphia National Banks, incorporating shared inputs/intermediate measures and external inputs/outputs. In the present study, inspired by the profitability and marketability framework, we assess the performance efficiency of Indian Public Sector Banks, incorporating shared resources, external inputs/outputs, and undesirable outputs.

Data collection

We have conducted an empirical study to evaluate the efficiency of Indian Public Sector Banks using two-stage network DEA models. The study is performed for the financial year 2023, and the data for inputs, intermediate measures, and outputs used in the model were collected from the Reserve Bank of India (RBI: https://www.rbi.org.in/Scripts/AnnualPublications.aspx?head=Statistical%20Tables%20Relating%20to%20Banks%20in%20India) for the same year. The inputs, intermediate measures, and outputs are described in Table 7 and illustrated in Fig. 5.

Table 7.

Descriptive table of collected datasets.

Dataset	Description
Inputs:	1) Fixed Assets: The total of premises, fixed assets under construction,
	and other fixed assets.
	2) Employees: The total number of officers, clerks, and sub-staff working in the banks.
Shared Input	Expenses: The total annual operating expenses of the banks
Shared Intermediate Measure:	Investment: The total investments in India and investments outside India
Intermediate Measures	1) Deposits: The total of demand deposits, saving bank deposits, and term deposits
	which are equivalent to the total of deposits branches inside and outside India
	2) Loans: These are performing loans calculated by subtracting the gross NPAs from
	the advances, where NPAs are the non-performing assets.
External Input	Capital Market Sector: is a free input that is directly consumed by the second stage
External Output	Documents: The total of cash in hand and balance with RBI
Undesirable Output	NPLs: The total of standard assets during the year, sub-standard assets during the year,
	and doubtful assets during the year.
Final Output	Net Profits: is the final output that is produced by the second stage.

Fig. 5.

Proposed two-stage structure of Indian public sector banks.

The collected data for all variables except employees are in Rs. Crores. Table 8 presents the names and abbreviations of Indian public sector banks in 2023. Meanwhile, Table 9 provides the numerical values of the input, intermediate measure, and output variables collected from Reserve Bank of India (RBI:https://www.rbi.org.in/Scripts/AnnualPublications.aspx?head=Statistical%20Tables%20Relating%20to%20Banks%20in%20India). In Table 9, $I_1$ and $I_2$ denote fixed assets and employees, respectively, both directly consumed by the first stage as inputs. $ShI$ represents expenses, serving as a shared input utilized partially by the first and second stages. Investments, abbreviated as $ShIM$, are shared intermediate measures. Deposits and loans are intermediate measures denoted by $IM1$ and $IM2$, respectively. The Capital Market Sector is a free or external input directly consumed by the second stage, identified as $EI$. External output is defined as documents and denoted by $EO$. Undesirable outputs, specifically bad loans or non-performing assets generated by the first stage, are avoided in the second stage and labeled as $UO$. Lastly, the second stage produces Net Profits, represented by $FO$.

Table 8.

Indian public sector banks in 2023.

Abbreviation	Name of the Bank	Abbreviation	Name of the Bank
BOB	BANK OF BARODA	IB	INDIAN BANK
BOI	BANK OF INDIA	IOB	INDIAN OVERSEAS BANK
BOMH	BANK OF MAHARASHTRA	PASB	PUNJAB AND SIND BANK
CB	CANARA BANK	PNB	PUNJAB NATIONAL BANK
CBOI	CENTRAL BANK OF INDIA	SBOI	STATE BANK OF INDIA
UB	UCO BANK	UBI	UNION BANK OF INDIA

Table 9.

The datasets of Indian public sector banks, 2023.

DMUs	I1	I2	ShI	ShIM	IM1	IM2	EI	EO	UO	FO
BOB	8,706.57	77,167	24,518.31	3,62,485.36	12,03,687.79	810578.6898	3,997.06	54882.6314	9,736.82	14,109.62
BOI	9,961.00	52,209	13,982.17	2,04,397.88	6,69,585.77	378751.1952	3,864.21	44034.5054	0	4,022.94
BOMH	2,156.71	12,977	3,921.84	68,866.95	2,34,082.68	157306.811	170.46	18507.7206	0	2,602.04
CB	10,230.67	84,978	22,481.48	3,19,038.45	11,79,218.61	695410.5524	3,340.63	54988.4454	19,565.00	10,603.76
CBOI	4,776.28	30,770	8,886.74	1,36,583.48	3,59,296.47	138520.4265	1,450.24	27432.9198	15,725.30	1,582.20
IB	7,459.04	40,797	12,097.90	1,85,988.25	6,21,165.76	364782.7737	2,680.61	32692.6301	23,329.31	5,281.70
IOB	3,709.98	22,052	6,421.46	94,170.41	2,60,883.29	138510.471	1,569.20	17148.35616	5,824.93	2,098.79
PASB	1,519.42	9,068	2,463.40	44,838.42	1,09,665.49	57872.8689	435.71	6225.4008	2,718.34	1,313.03
PNB	12,051.07	1,04,120	24,105.41	3,95,996.72	12,81,163.10	613880.0613	3,752.37	78176.5752	8,077.05	2,507.20
SBOI	42,381.80	2,35,858	97,743.14	15,70,366.23	44,23,777.78	2938381.277	27,494.19	247087.5752	0	50,232.45
UB	3,509.51	21,746	5,510.92	95,169.35	2,49,337.74	131203.2587	441.00	10300.0144	4,606.28	1,862.34
UBI	8,825.61	75,594	21,931.33	3,39,299.05	11,17,716.32	577634.1177	4,910.97	50254.2741	15,233.73	8,433.28

Fuzzification

Real-world challenges, particularly within the banking sector, are frequently characterized by non-quantifiable information, resulting in fluctuations around specific data points and uncertain or vague data. Following an extensive literature review and consultations with industry experts, it was observed that inputs and outputs display a variation of approximately $\%2$ to $\%3$. Consequently, this paper employs a fuzzification approach to introduce a $\%2$ to $\%3$ variation to the dataset obtained from the Reserve Bank of India (RBI).

We propose using a parabolic intuitionistic fuzzy number to represent the uncertain dataset. A parabolic intuitionistic fuzzy number can be defined as $(x_1^{\prime },x_2,x_3^{\prime };x_1,x_2,x_3)$, where $x_2$ is a number with the largest membership and smallest non-membership values. $x_1<x_2$ represents the left spread, and $x_3<x_3^{\prime }$ represents the right spread. The $x_2$ value represents the data set collected from RBI, encompassing input/output, intermediate measures, shared resources (inputs, intermediate measures), external input/output, and undesirable output. These values can be calculated as follows:

• 1.
  Let the fixed assets be denoted by $I_1$, then its left and right spread can be calculated as follows:

  • •
    Left and right spread based on membership function: $X_1=0.98·I_1,X_3=1.25·I_1$
  • •
    Left and right spread based on non-membership function: $X_1^{\prime }=0.97·I_1,X_3^{\prime }=1.3·I_1$
• 2.
  The number of employees is shown by $I_2$ therefore, its left and right spread can be obtained as follows:

  • •
    Left and right spread based on membership function: $X{X}_1=0.98·I_2,X{X}_3=1.25·I_2$
  • •
    Left and right spread based on non-membership function: $X{X}_1^{\prime }=0.972·I_2,X{X}_3^{\prime }=1.32·I_2$
• 3.
  The shared input which is given the expenses are described by $ShI$; therefore, its left and right spread can be calculated as follows:

  • •
    Left and right spread based on membership function: $S{X}_1=0.97·ShI,S{X}_3=1.20·ShI$
  • •
    Left and right spread based on non-membership function: $S{X}_1^{\prime }=0.96·ShI,S{X}_3^{\prime }=1.23·ShI$
• 4.
  The shared intermediate measure (investments) is denoted by $ShIM$; therefore, its left and right spread based on membership and non-membership functions can be calculated as below:

  • •
    Left and right spread based on membership function: $S{Z}_1=0.98·ShIM,S{Z}_3=1.20·ShIM$
  • •
    Left and right spread based on non-membership function: $S{Z}_1^{\prime }=0.985·ShIM,S{Z}_3^{\prime }=1.32·ShIM$
• 5.
  The first intermediate measure (deposits) is shown by $I{M}_1$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $Z_1=0.98·I{M}_1,Z_3=1.28·I{M}_1$
  • •
    Left and right spread based on non-membership function: $Z_1^{\prime }=0.96·I{M}_1,Z_3^{\prime }=1.36·I{M}_1$
• 6.
  The second intermediate measure (loans) is shown by $I{M}_2$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $Z{Z}_1=0.97·I{M}_2,Z{Z}_3=1.28·I{M}_2$
  • •
    Left and right spread based on non-membership function: $Z{Z}_1^{\prime }=0.968·I{M}_2,Z{Z}_3^{\prime }=2.32·I{M}_2$
• 7.
  The external/free input (capital market sector) is shown by $EI$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $E{X}_1=0.96·EI,E{X}_3=1.5·EI$
  • •
    Left and right spread based on non-membership function: $E{X}_1^{\prime }=0.94·EI,E{X}_3^{\prime }=1.55·EI$
• 8.
  The external/free output (documents) is shown by $EO$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $E{Y}_1=0.97·EO,E{Y}_3=1.25·EO$
  • •
    Left and right spread based on non-membership function: $E{Y}_1^{\prime }=0.95·EO,E{Y}_3^{\prime }=1.35·EO$
• 9.
  The undesirable output (non-performing loans) is shown by $UO$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $U{Y}_1=0.97·UO,U{Y}_3=1.40·UO$
  • •
    Left and right spread based on non-membership function: $U{Y}_1^{\prime }=0.95·UO,U{Y}_3^{\prime }=1.45·EO$
• 10.
  The final or terminal output (net profits) is shown by $FO$ whose left and right spread based on membership and non-membership functions can be evaluated as follows:

  • •
    Left and right spread based on membership function: $F{Y}_1=0.975·FO,F{Y}_3=1.35·FO$
  • •
    Left and right spread based on non-membership function: $F{Y}_1^{\prime }=0.96·FO,F{Y}_3^{\prime }=1.4·FO$

Defuzzification

Defuzzification methods are required to obtain precise crisp values for solving fuzzy equations. Several methods in the literature are used to convert a fuzzy number to its corresponding crisp value. The center of gravity method is among the most popular and widely used. This paper proposes a new center of gravity method for intuitionistic fuzzy sets. This method is defined in Eq. (27).

Using the fuzzification method described in the preceding subsection, we obtain the lower and upper boundaries for each data set given in Table 9. As a result, with the use of the proposed centroid method and utilizing Eqs. (30)–(36), each parabolic intuitionistic fuzzy input, intermediate measure, and output are transformed to their related crisp numbers as shown in Table 11. The symbols used in Table 11 have the same meaning as those utilized in Table 9.

Table 11.

Defuzzified datasets of Indian public sector banks, 2023.

DMUs	I1	I2	ShI	ShIM	IM1	IM2	EI	EO	UO	FO
BOB	9462.63	84466.94	26038.70	399840.43	1328557.70	885139.13	4652.26	60408.45	11016.85	19329.073
BOI	10825.99	57147.93	14849.21	225461.61	739048.23	413590.32	4497.63	48468.08	0.00	5510.98
BOMH	2343.99	14204.61	4165.03	75963.87	258366.29	171776.55	198.40	20371.15	0.00	3564.44
CB	11119.08	93016.86	23875.57	351916.20	1301550.10	759377.34	3888.23	60524.91	22137.07	14526.27
CBOI	5191.04	33680.82	9437.81	150658.76	396569.69	151262.11	1687.96	30194.98	17792.59	2167.32
IB	8106.76	44656.36	12848.09	205154.83	685605.15	398337.03	3120.01	35984.26	26396.25	7235.40
IOB	4032.14	24138.10	6819.65	103874.92	287947.18	151251.24	1826.42	18874.92	6590.69	2875.02
PASB	1651.36	9925.82	2616.15	49459.14	121042.12	63196.25	507.13	6852.20	3075.70	1798.58
PNB	13097.56	113969.68	25600.20	436805.22	1414070.26	670347.33	4367.46	86047.72	9138.88	3434.52
SBOI	46062.15	258170.01	103804.25	1732196.60	4882698.08	3208665.98	32001.08	271965.41	0.00	68814.98
UB	3814.26	23803.15	5852.65	104976.80	275203.90	143271.88	513.28	11337.06	5211.83	2551.10
UBI	9592.00	82745.14	23291.30	374264.71	1233667.60	630767.34	5715.98	55314.09	17236.40	11552.85

Results and discussion

Models 1, 2, and 3 assess the relative efficiency of the first stage, second stage, and overall efficiency, respectively, applied to Philadelphia National banks with the methodology outlined in the study by Izadikhah et al. [9], as depicted in Table 4 and Fig. 4. The table demonstrates that the proposed method is valid and yields more efficient DMUs, presenting larger efficiency values for inefficient DMUs. This superiority of the method over existing approaches is evident.

The proposed methodology is implemented in the context of Indian public sector banks. Fig. 5 illustrates the network structure of Indian public sector banks, constructed based on the proposed methodology. In this structure, profitability and marketability factors are considered in the first and second stages, respectively, utilizing datasets obtained from the Reserve Bank of India (RBI) and organized in Table 9. Subsequently, Model4 is employed to assess the overall efficiency of Indian public sector banks across different $\alpha$ and $\beta$ values. Here, $\alpha$ and $\beta$ represent the coefficients of shared inputs and intermediate measures, respectively, facilitating the segmentation of the dataset into the first and second stages using a convexity approach. The efficiency outcomes for Indian public sector banks, according to Model4, are presented in Table 10 and Fig. 6, showcasing various values for $\alpha$ and $\beta$. Table 10 identifies two efficient banks, BOB and BOMH, and emphasizes that the optimal efficiency score is attained with higher $\alpha$ and smaller $\beta$, as indicated in column 4.

Fig. 6.

Efficiency result of Model4.

Since the datasets are not available in the crisp form, we fuzzified the collected datasets as per discussion with bank experts and review of the relevant literature by parabolic intuitionistic fuzzy number. Moreover, the IFS datasets of Indian public sector banks are defuzzified by the defined centroid method for IFS, and the result is given in Table 11. The efficiency outcomes of Indian public sector banks, following Model6, are showcased in Table 12 and Fig. 7, featuring identical values of $\alpha$ and $\beta$ as utilized in Table 10. Table 12 yields increased efficiency scores for each DMU, depending on varying values of $\alpha$ and $\beta$, and preserves two efficient banks, BOB and BOMH. Consequently, these results empower decision-makers to determine the optimal levels of inputs, intermediate measures, and outputs.

Fig. 7.

Efficiency result of Model6.

We calculate the average value of relative efficiency using the data presented in Tables 10 and 12. The average value of $E_k=\{x_1,x_2,x_3,\dots ,x_n\}$ is defined as follows:

$$Ave\left[E_k\right]=\frac{x_1+x_2+x_3+\dots +x_n}{n}$$ (42)

Each $x_i$, where $i=1,2,3,\dots ,n$, represents a column in Tables 10 and 12. The outcome is depicted in Fig. 8, demonstrating the optimality of the proposed methodology based on the ascending order of Eq. (42) for each inefficient DMU. It is noteworthy that the average value remains one for efficient DMUs. Within Fig. 8, the blue color bar represents the average value of Table 10, while the red color bar illustrates the average value of Table 12. The red color bar exhibits a more significant value than the blue one for bank numbers 2, 10, and 12, indicating optimal datasets for the decision-makers and policymakers of the respective banks.

Fig. 8.

Average efficiency result of Model4 and Model6.

Limitations and wider applications

A two-stage network DEA can adopt various structures such as series, parallel, hierarchical, and mixture forms combining the first and second stages. However, our study specifically focuses on the series configuration of the two-stage network DEA. Consequently, one limitation of our paper is its inability to address real-world applications in parallel, hierarchical, or mixed structures. Furthermore, we acknowledge that uncertainty exists in intuitionistic fuzzy scenarios, wherein the sum of membership and non-membership grades must not exceed one. However, in specific real-life scenarios, this sum may not be less than or equal to one. Consequently, the proposed methodology may need to be revised in such contexts.

The two-stage network DEA is used in various real-life applications, such as banks, insurance companies, educational institutes, water and land, and many others studied by researchers in the literature [35], [47], [48]. Therefore, our proposed methodology applies to several real-life problems, considering shared inputs and intermediate measures, undesirable outputs produced by each stage, accessible inputs consumed by the second stage, and free outputs produced by the first stage.

Conclusion

The present study introduces a comprehensive network data envelopment analysis (DEA) that incorporates shared resources (inputs and intermediate measures), external inputs and outputs, and undesirable outputs to assess the relative efficiency of the banking industry. Initially, we develop a two-stage DEA model for cases with precise or crisp data sets. However, real-world applications often exhibit fluctuations, introducing uncertainty and vagueness to the data set. Subsequently, we extend our precise model to accommodate imprecise situations and introduce a non-linear fuzzy number known as a parabolic intuitionistic fuzzy number, which effectively represents vague data sets. An illustrative example is provided to demonstrate the methodology’s validation. To apply the theoretical framework, we focus on Indian public sector banks for the financial year 2023. Firstly, we assess the overall efficiency based on crisp values. Secondly, we convert these precise numbers into parabolic intuitionistic fuzzy numbers. We utilize a newly proposed centroid method for intuitionistic fuzzy sets to obtain corresponding crisp values, enabling the measurement of the overall efficiency of banks. Results from the intuitionistic environment yield optimal efficiency values, surpassing those derived from the initial precise situation. These findings empower decision-makers by providing insights into determining optimal levels of inputs, intermediate measures, and outputs.

A trajectory for future research involves expanding our framework to incorporate additional variations of the network DEA method, encompassing parallel structures and combinations of series and parallel configurations. Moreover, exploring the development of hybrid approaches by integrating network DEA methods with various extensions of fuzzy sets, such as type-2 and Pythagorean fuzzy sets, holds promise for advancing the field.

Ethics statements

Ethical considerations were deemed unnecessary.

CRediT authorship contribution statement

Mohammad Aqil Sahil: Conceptualization, Investigation, Writing – original draft. Q.M. Danish Lohani: Supervision, Investigation, Writing – original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary materials

Supplementary Data S1

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

Supplementary Data S2

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

Supplementary Data S3

Supplementary Raw Research Data. This is open data under the CC BY license http://creativecommons.org/licenses/by/4.0/

Data availability

Data will be made available on request.