Evaluating the Efficiency of Decision Making Units in Fuzzy two-stage DEA Models

Abstract

Data envelopment analysis (DEA) is an optimization method to assess the efficiency of decision-making units with multiple-inputs/multiple-outputs assumption. Most real-life issues contain more than one stage unit which needs multiple-stage data envelopment analysis models to be solved. Moreover, the inputs and outputs of the units are rarely measured accurately in real-life problems, hence fuzzy data envelopment analysis approaches can be significantly helpful in calculating efficiency scores. In this study, an approach for evaluating the performance of decision-making units (DMUs) in fuzzy two-stage DEA models is developed. The developed model is a parametric program based on alpha-cuts. The dependence on alpha allows the manager to compare and rank DMUs based on his/her degree of certainty and after the selection of alpha, our proposed model becomes linear. Furthermore, a theorem is proposed and proved for conventional multiplicative two-stage DEA models with the assumption of Variable Returns to Scale. This theorem can be used to evaluate the correctness of the results. Finally, by two illustrative examples, the ability of the proposed approach to solve fuzzy two-stage DEA models is shown, and the obtained results are compared to that of some other methods in this field.

KEYWORDS:

• Data envelopment analysis (DEA)
• two-stage DEA
• fuzzy network DEA
• fuzzy numbers
• ranking

1. Introduction

Data envelopment analysis (DEA) is a non-parametric approach that uses linear programming to assess the performance of decision-making units (DMUs) with multiple inputs and multiple outputs (Kiaei and Nasseri [1]). This approach treats a DMU as a ‘black box’, so its interior structures are not taken into consideration. In 1978, Charnes et al. [2] introduced DEA and extensive research has been performed since then. DEA is a powerful tool for comparing and ranking DMUs, and so far, various methods have been proposed in this field (Nasseri and Kiaei [3,4]). In real-world applications, such as supply chains, some DMUs have important interior structures; considering the system as a black-box ignores these interior structures when evaluating the efficiency of DMUs. Therefore, some important information may be disregarded, and consequently, the results will not be accurate or meaningful. In order to consider the interior structures of the DMUs, network DEA models have been developed. The network DEA models are capable of reflecting accurately the DMUs’ interior operations as well as incorporating their relationships and interdependences (Koronakos [5]). Studies carried out by Färe and Grosskopf [6,7] were some of the first studies in this field. Since then, many researchers have studied the network DEA. Numerous real applications of Network DEA models have been reported such as performance evaluation of baseball teams [8], banks [9], universities [10,11], non-life insurance companies [12,13] and electricity distribution network [14]. Kao [15] provided a review on the proposed approaches in this field until 2014 and organized them into seven categories for interior structures of DMUs in network DEA: basic two-stage structures, general two-stage structures, series structures, parallel structures, mixed structures, hierarchical structures and dynamic structures.

The simplest structure in the network DEA is the two-stage DEA. Most of the proposed approaches for solving general network DEA models are inspired by presented methods for solving two-stage DEA models. Hence, the high accuracy and meaningfulness of the proposed methods for solving two-stage DEA models are vital (Kiaei et al. [16]). Different approaches have been proposed to evaluate the efficiency of DMUs in two-stage DEA models (Guo et al. [17]). Cook et al. [18] presented a review on the proposed methods for evaluating the efficiency of two-stage DMUs. The most notable methods for evaluating the efficiency of DMUs in two-stage DEA models can be classified into four categories: first, the conventional model which uses the typical DEA model separately in the two stages without considering any interactions among them [19,20]. Second, the efficiency decomposition models are considering the interactions among the stages and also are able to yield efficiency results for the overall model and each stage. This category contains both the multiplicative model of Kao and Hwang [12] and the additive model of Chen et al. [21]. Third, the network DEA models which focus on the structure of the model (serial, parallel, etc.) [6,22]. Last, leader-follower or Stackelberg approaches [23–25].

In usual DEA models, inputs and outputs must be measured by accurate values and these DEA models are not applicable when the data are imprecise or vague (Olfati et al. [26]). In real-world problems, finding exact values for data is not usually possible and therefore novel methods need to be developed. One of the appropriate methods to overcome this weakness is to use fuzzy numbers as the inputs and outputs. From the original study by Sengupta [27], many papers have been written about fuzzy DEA. For example, Kao and Liu [28] proposed a pair of mathematical models to obtain the upper and lower bounds of efficiency measure of DMUs in a fuzzy DEA model. Emrouznejad et al. [29] provided a literature review of the published papers in this field until 2014. Foroughi and Shureshjani [30] proposed a parametric approach to evaluate the efficiency of DMUs in conventional fuzzy DEA models. Also, A significant number of papers have been published to solve DEA models in fuzzy stochastic environments (for example, see Izadikhah et al. [31], Nasseri et al. [32], Nasseri et al. [33], Ebrahimnejad et al. [34], Nasseri and Khatir [35] and Ebrahimnejad et al. [36]).

Despite many works on the fuzzy DEA models, few methods are offered for solving fuzzy network DEA models. First, Kao and Liu [37] developed a fuzzy two-stage DEA model. In this method, they try to obtain the lower and upper bounds of efficiency of DMUs in a fuzzy multiplicative two-stage DEA model for each α-cut by considering the appropriate parts of L-U representation of fuzzy number inputs and outputs. Lozano [38] proposed two models to assess the variation of stage efficiencies in the proposed method by Kao and Liu [37]. Tavana and Khalili-Damghani [39] applied trapezoidal fuzzy numbers into the Kao and Liu [37] models and converted their nonlinear models to linear ones which are independent of the α-cut variables. In another study, Liu [40] extended the Kao and Liu [37] method for fuzzy two-stage DEA models with assurance regions Since we cannot easily use the obtained bounds to compare and rank DMUs, Liu [41] used the Chen [42] index to combine bounds and rank DMUs. However, their proposed models for calculating the lower bounds are nonlinear. Using a bargaining game model, Tavana et al. [43] designed a fuzzy two-stage Game-DEA model to evaluate the efficiency of DMUs. Nasseri and Khatir [35] developed a two-stage DEA model with undesirable outputs in a fuzzy random environment and applied it in a case study for the banking industry. Nosrat et al. [44] applied the credibility theory to solve the fuzzy two-stage DEA model. Although their method needs to solve a linear programming problem for each credibility level, it is not easily generalizable to other network DEA structures. Inspired by Kao and Liu’s [37] approach, Arya and Singh [45] developed a method to solve a fuzzy parallel-series two-stage DEA model. They determined the lower and upper bounds of DMUs’ efficiencies and ranked them by applying Chen and Klein’s [46] index. In a similar work, Ostovan et al. [47] obtained lower, upper and average efficiency of two-stage DEA and DEA-R models with fuzzy data. Considering the optimistic-pessimistic viewpoints and confidence levels, Peykani [48] applied adjustable possibilistic and chance-constrained programs to present an approach to evaluate and rank DMUs in a fuzzy two-stage network structure. Most of these methods are generalizations of the Kao and Liu [28] approach in which the upper and lower bounds of efficiency scores are obtained.

In this study, however, a new approach is proposed to solve fuzzy two-stage DEA problems. This approach is basically inspired by the Foroughi and Shureshjani [30] method and transforms the fuzzy two-stage DEA model into a parametric programming model. The value of the parameter depends on the decision maker’s preferences. This dependence allows the manager to compare and rank DMUs based on his/her degree of certainty, and after the selection of the parameter, our proposed model becomes linear. This method results in the exact value of the overall efficiency rather than the lower–upper bounds, which makes it more applicable to compare and rank DMUs. Also, Compared to similar methods presented so far, our method is computationally less expensive and works for both Constant and Variable Returns to Scale (CRS and VRS) assumptions. Here, we apply this method to the fuzzy multiplicative two-stage DEA cases. However, unlike similar proposed approaches, this method can easily apply to other kinds of fuzzy network DEA models such as fuzzy additive two-stage DEA models, fuzzy weighted additive two-stage DEA models, fuzzy multi-stage DEA models, etc. In the last section of the paper, two examples are presented to explain the suggested method and compare our results with the results obtained from two other approaches in the literature.

2. Two-stage DEA Models

Consider a two-stage DEA network structure (Figure 1). Let $DM{U}_j,(j=1,2,\dots ,n)$ has m inputs $x_{ij},(i=1,2,\dots ,m)$ to the stage 1 and D outputs $z_{dj},(d=1,2,\dots ,D)$ from stage 1. These D outputs referred as intermediate measures become the inputs to stage 2. Also, the outputs from stage 2 are $y_{rj},(r=1,2,\dots ,s)$. With the assumption of Constant Returns to Scale (CRS), we can calculate the input-oriented efficiency of DMUs at each stage as below [2]: (1) $\begin{array}{rl}& max{e}_o^1=\frac{{\sum }_{d=1}^D{w}_d{z}_{do}}{{\sum }_{i=1}^m{v}_i{x}_{io}}\\ & s.t.\frac{{\sum }_{d=1}^D{w}_d{z}_{dj}}{{\sum }_{i=1}^m{v}_i{x}_{ij}}\le 1,\mathrm{\forall }j,\\ & v_i,w_d\ge 0,\mathrm{\forall }i,d\end{array}$(1) (2) $\begin{array}{rl}& max{e}_o^2=\frac{{\sum }_{r=1}^s{u}_r{y}_{ro}}{{\sum }_{d=1}^D{\hat{w}}_d{z}_{do}}\\ & s.t.\frac{{\sum }_{r=1}^s{u}_r{y}_{rj}}{{\sum }_{d=1}^D{\hat{w}}_d{z}_{dj}}\le 1,\mathrm{\forall }j,\\ & {\hat{w}}_d,u_r\ge 0,\mathrm{\forall }d,r\end{array}$(2)

Figure 1. The generic two-stage process.

$e_o^1$ and $e_o^2$ are the efficiency measures of $DM{U}_o$ for stage 1 and stage 2, respectively.

Consider a situation in which the stage 1 is DEA efficient and stage 2 is not; if stage 2 improves its efficiency by reducing the inputs $z_{dj}$, the reduced $z_{dj}$ may make stage 1 inefficient.

The most famous methods for evaluating the efficiency of DMUs in two-stage DEA models can be classified into four categories: multiplicative approaches, leader-follower approaches, additive approaches and weighted additive approaches. In order to calculate the efficiency of the two-stage DEA model in all these methods, it is generally accepted that the weights of the intermediate measures are equal (i.e. $w=\hat{w}$). $\begin{array}{rl}DM{U}_j,j& =1,\dots ,n\end{array}$ $\begin{array}{rl}{x}_{ij},i& =1,\dots ,m{z}_{dj},d=1,\dots ,D{y}_{rj},r=1,\dots ,s\end{array}$

By using the square geometric mean of the stage efficiencies (i.e. $e_j=e_j^1.e_j^2$), Kao and Hwang [12] introduced a centralized model (multiplicative model) to calculate the overall efficiency of DMUs (model 3). In this case, two stages jointly assign a set of optimal weights for the intermediate data to maximize the efficiency score of the stages. (3) $\begin{array}{l}max{e}_o={\displaystyle \frac{{\sum }_{d=1}^D{w}_d{z}_{do}}{{\sum }_{i=1}^m{v}_i{x}_{io}}}.{\displaystyle \frac{{\sum }_{r=1}^s{u}_r{y}_{ro}}{{\sum }_{d=1}^D{w}_d{z}_{do}}}={\displaystyle \frac{{\sum }_{r=1}^s{u}_r{y}_{ro}}{{\sum }_{i=1}^m{v}_i{x}_{io}}}\\ s.t.\sum _{r=1}^s{u}_r{y}_{rj}-\sum _{d=1}^D{w}_d{z}_{dj}\le 0,\mathrm{\forall }j,\\ \sum _{d=1}^D{w}_d{z}_{dj}-\sum _{i=1}^m{v}_i{x}_{ij}\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r\end{array}$(3)

Using the Charnes–Cooper transformation, model (3) can be transformed into model (4): (4) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{y}_{ro}\\ s.t.\sum _{i=1}^m{v}_i{x}_{io}=1,\\ \sum _{d=1}^D{w}_d{z}_{dj}-\sum _{i=1}^m{v}_i{x}_{ij}\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{y}_{rj}-\sum _{d=1}^D{w}_d{z}_{dj}\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r\end{array}$(4)

The dual of the above model is: (5) $\begin{array}{l}min{e}_o=\theta \\ s.t.\theta x_{io}\ge \sum _{j=1}^n{\lambda }_j{x}_{ij},\mathrm{\forall }i,\\ \sum _{j=1}^n{\lambda }_j{z}_{dj}\ge \sum _{j=1}^n{\mu }_j{z}_{dj},\mathrm{\forall }d,\\ \sum _{j=1}^n{\mu }_j{y}_{rj}\ge y_{ro},\mathrm{\forall }r,\\ {\lambda }_j,{\mu }_j\ge 0\end{array}$(5)

We can define the input, intermediate and output slacks of model (5) as below: (6) $\begin{array}{l}{s}_i^{-}=\theta x_{io}-\sum _{j=1}^n{\lambda }_j{x}_{ij},i=1,\dots ,m\\ s_d^L=\sum _{j=1}^n({\lambda }_j-{\mu }_j)z_{dj},d=1,\dots ,D\\ s_r^{+}=\sum _{j=1}^n{\mu }_j{y}_{rj}-y_{ro},r=1,\dots ,s\end{array}$(6)

According to the presented slacks, the following definitions for the efficiency of a DMU in the multiplicative two-stage DEA model under input oriented and the CRS assumption can be provided.

Definition 2.1:

$DM{U}_o$ is overall two-stage DEA efficient (input-oriented & CRS) if and only if ${\theta }^{\ast }=1,s_i^{-\ast }=0,s_r^{+\ast }=0,\mathrm{\forall }i,r$.

Definition 2.2:

$DM{U}_o$ is overall two-stage DEA weak efficient (input-oriented & CRS) if and only if ${\theta }^{\ast }=1$, while at least one of the related input and output slacks $\left(s_i^{-\ast },s_r^{+\ast }\right)$ is not zero.

Intermediate data are a link between two stages, so we can provide the following definition related to the measure of $s_d^L$.

Definition 2.3:

$DM{U}_o$ is overall two-stage DEA efficient (input-oriented & CRS) with a ‘fixed’ link if ${\theta }^{\ast }=1,s_i^{-\ast }=0,s_r^{+\ast }=0,s_d^{L\ast }=0,\mathrm{\forall }i,r,d$ and $DM{U}_o$ is overall two-stage DEA efficient (input-oriented & CRS) with a ‘free’ link if ${\theta }^{\ast }=1,s_i^{-\ast }=0,s_r^{+\ast }=0,\mathrm{\forall }i,r$ and there is at least one $d\in D$so that $s_d^{L\ast }\ne 0$.

Fixed and free link phrases in two-stage DEA models were first used by Tone and Tsutsui [49].

We can provide the same definitions for an output-oriented two-stage DEA model under the CRS assumption analogously.

By adding two constraints ${\sum }_{j=1}^n{\lambda }_j=1$ and ${\sum }_{j=1}^n{\mu }_j=1$ to model (5), the model converts to a multiplicative two-stage DEA case with input oriented and the VRS assumption [50]: (7) $\begin{array}{l}min{e}_o=\theta \\ s.t.\theta x_{io}\ge \sum _{j=1}^n{\lambda }_j{x}_{ij},\mathrm{\forall }i,\\ \sum _{j=1}^n{\lambda }_j{z}_{dj}\ge \sum _{j=1}^n{\mu }_j{z}_{dj},\mathrm{\forall }d,\\ \sum _{j=1}^n{\mu }_j{y}_{rj}\ge y_{ro},\mathrm{\forall }r,\\ \sum _{j=1}^n{\lambda }_j=1\\ \sum _{j=1}^n{\mu }_j=1\\ {\lambda }_j,{\mu }_j\ge 0\end{array}$(7)

We can define the input, intermediate and output slacks of model (7) as below: (8) $\begin{array}{l}{s}_i^{-}=\theta x_{io}-\sum _{j=1}^n{\lambda }_j{x}_{ij},i=1,\dots ,m\\ s_d^L=\sum _{j=1}^n({\lambda }_j-{\mu }_j)z_{dj},d=1,\dots ,D\\ s_r^{+}=\sum _{j=1}^n{\mu }_j{y}_{rj}-y_{ro},r=1,\dots ,s\end{array}$(8)

The dual of model (7) is as below: (9) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{y}_{ro}-u_0-u_1\\ s.t.\sum _{i=1}^m{v}_i{x}_{io}=1\\ \sum _{d=1}^D{w}_d{z}_{dj}-\sum _{i=1}^m{v}_i{x}_{ij}-u_0\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{y}_{rj}-\sum _{d=1}^D{w}_d{z}_{dj}-u_1\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r,u_0,u_{1}arefree\end{array}$(9)

We can similarly define the overall efficiency, weak efficiency, overall efficiency with fixed link and overall efficiency with free link for DMUs in a multiplicative two-stage DEA model with the assumption of VRS for both input and output oriented cases.

3. Some Sufficient Conditions for a DMU in a Multiplicative two-stage DEA Model with the Assumption of VRS to be Efficient

In conventional DEA model with the assumption of VRS (BCC model), we have following sufficient conditions for a DMU to be efficient.

Theorem 3.1:

[51] A DMU that has a minimum input value for any input item, or a maximum output value for any output item, is BCC-efficient.

Here we generalize this theorem for multiplicative two-stage DEA model with the assumption of VRS.

Theorem 3.2:

[52] We have the following sufficient conditions for efficiency of a DMU in a multiplicative two-stage DEA model under input oriented and the VRS assumption:

1. A $DM{U}_o$ that has a minimum input value for any input item is overall two-stage DEA weak efficient.

2. A $DM{U}_o$ that has a minimum input value for any input item and a minimum intermediate value for any intermediate item is overall two-stage DEA efficient with fixed link.

3. A $DM{U}_o$ that has a minimum input value for any input item and a maximum output value for any output item is overall two-stage DEA efficient with fixed link.

4. A $DM{U}_o$ that has a maximum intermediate value for any intermediate item and a maximum output value for any output item is overall two-stage DEA efficient with fixed link.

Proof

1. Suppose that $DM{U}_o$ has a minimum input value for input i, i.e. $x_{io}<x_{ij}(\mathrm{\forall }j\ne o)$. By considering assumptions $x_{io}<x_{ij}(\mathrm{\forall }j\ne o)$ and ${\sum }_{j=1}^n{\lambda }_j=1$ at constraint $\theta x_{io}\ge {\sum }_{j=1}^n{\lambda }_j{x}_{ij}$ (model (7)), we obtain ${\theta }^{\ast }=1,{\lambda }_o^{\ast }=1,{\lambda }_j^{\ast }=0,\mathrm{\forall }j\ne o,s_i^{-\ast }=0,\mathrm{\forall }i$. However, $\mathrm{\forall }r,s_r^{+\ast }$ is not necessarily zero. So $DM{U}_o$ is overall two-stage DEA weak efficient (input-oriented & VRS).

2. Suppose that $DM{U}_o$ has a minimum input value for input i, i.e. $x_{io}<x_{ij}(\mathrm{\forall }j\ne o)$, and has a minimum intermediate value for intermediate k, i.e. $z_{ko}<z_{kj}(\mathrm{\forall }j\ne o)$. By considering assumptions $x_{io}<x_{ij}(\mathrm{\forall }j\ne o)$ and ${\sum }_{j=1}^n{\lambda }_j=1$ at constraint $\theta x_{io}\ge {\sum }_{j=1}^n{\lambda }_j{x}_{ij}$ (model (7)), we obtain ${\theta }^{\ast }=1,{\lambda }_o^{\ast }=1,{\lambda }_j^{\ast }=0,\mathrm{\forall }j\ne o,s_i^{-\ast }=0,\mathrm{\forall }i$. As a result, constraint 2 of (7) converts to $z_{do}\ge {\sum }_{j=1}^n{\mu }_j{z}_{dj}$. By considering assumptions $z_{ko}<z_{kj}(\mathrm{\forall }j\ne o)$ and ${\sum }_{j=1}^n{\mu }_j=1$ at $z_{do}\ge {\sum }_{j=1}^n{\mu }_j{z}_{dj}$, we obtain ${\mu }_o^{\ast }=1,{\mu }_j^{\ast }=0,\mathrm{\forall }j\ne o,s_d^{L\ast }=s_r^{+\ast }=0,\mathrm{\forall }d,r$. So $DM{U}_o$ is overall two-stage DEA efficient (input-oriented & VRS) with fixed link.

We can similarly prove the cases (3) and (4). □

We will use this theorem to study the validity of the obtained results from the proposed approach to assess the efficiency of DMUs in fuzzy multiplicative two-stage DEA model with the assumption of VRS.

4. Our Proposed Method

By considering fuzzy input $\left({\tilde{x}}_{ij}\right)$, intermediate $\left({\tilde{z}}_{dj}\right)$ and output $\left({\tilde{y}}_{rj}\right)$ data, models (4) and (9) convert to the following fuzzy multiplicative two-stage DEA models with the CRS and VRS conditions, respectively: (10) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{\tilde{y}}_{ro}\\ s.t.\sum _{i=1}^m{v}_i{\tilde{x}}_{io}=1,\\ \sum _{d=1}^D{w}_d{\tilde{z}}_{dj}-\sum _{i=1}^m{v}_i{\tilde{x}}_{ij}\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{\tilde{y}}_{rj}-\sum _{d=1}^D{w}_d{\tilde{z}}_{dj}\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r\end{array}$(10) (11) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{\tilde{y}}_{ro}-u_0-u_1\\ s.t.\sum _{i=1}^m{v}_i{\tilde{x}}_{io}=1\\ \sum _{d=1}^D{w}_d{\tilde{z}}_{dj}-\sum _{i=1}^m{v}_i{\tilde{x}}_{ij}-u_0\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{\tilde{y}}_{rj}-\sum _{d=1}^D{w}_d{\tilde{z}}_{dj}-u_1\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r,u_0,u_{1}arefree\end{array}$(11)

We can categorize fuzzy DEA approaches into six groups and among them, α-level based approach is the most well-known [29]. In the α-level based approach, we solve the fuzzy DEA model by parametric programming using α-cuts [53].

In this section, we try to present a new method to evaluate the efficiency of the DMUs in fuzzy two-stage DEA models based on α-cuts. This method can be considered in the α-level based approach category.

Since there is not straightforward method to solve a fuzzy two-stage DEA model, we need to use simple practical tricks. For this reason, the $Q_{\alpha }$ transformation function suggested by Shureshjani and Darehmiraki [54] is applied to assign a function to the fuzzy input, intermediate and output data. Shureshjani and Darehmiraki [54] suggested this function to compare fuzzy numbers. The definition of the $Q_{\alpha }$ transformation function will be discussed in the following paragraphs.

Fuzzy numbers can be defined in various mathematical ways. In this study, we have used one of the famous definitions of fuzzy numbers in the literature [54]. This definition will be employed to build $Q_{\alpha }$ transformation function.

Definition 4.1:

A normal fuzzy number $\tilde{A}$ in parametric type is an arranged pair $\left(\underset{\_}{A}\right(r),\overline{A}(r\left)\right)$ of functions $\underset{\_}{A}\left(r\right)$ and $\overline{A}\left(r\right),0\le r\le 1$, which satisfy the following conditions:

1. $\underset{\_}{A}\left(r\right)$ is a bounded monotonic increasing left continuous function on [0,1],

2. $\overline{A}\left(r\right)$ is a bounded monotonic decreasing left continuous function on [0,1],

3. $\underset{\_}{A}\left(r\right)\le \overline{A}\left(r\right),0\le r\le 1$.

Suppose $\tilde{A}=\left(\underset{\_}{A}\right(r),\overline{A}(r\left)\right),0\le r\le 1$ is a normal fuzzy number (Figure 2). Shureshjani and Darehmiraki [54] assigned the following transformation function $Q_{\alpha }\left(\tilde{A}\right)$ to fuzzy number $\tilde{A}$: (12) $Q_{\alpha }\left(\tilde{A}\right)={\int }_{\alpha }^1\left(\underset{\_}{A}\right(r)+\overline{A}(r\left)\right)dr,0\le \alpha <1$(12)

Figure 2. $\tilde{A}=\left(\underset{\_}{A}\right(r),\overline{A}(r\left)\right)$.

The amount of function $Q_{\alpha }\left(\tilde{A}\right)$ is graphically shown in Figure 2. This value is the summation of the dotted area and the cross-hatched area.

By using this transformation function, Shureshjani and Darehmiraki [54] proposed two definitions to rank normal fuzzy numbers for each α-cut as below:

Definition 4.2:

Suppose $\tilde{A}$ and $\tilde{B}$ are two arbitrary normal fuzzy numbers, so: $\begin{array}{l}\tilde{A}\le \tilde{B}\leftrightarrow \mathrm{\forall }\alpha \in \left[0,1\right],Q_{\alpha }\left(\tilde{A}\right)\le Q_{\alpha }\left(\tilde{B}\right)\\ \tilde{A}=\tilde{B}\leftrightarrow \mathrm{\forall }\alpha \in \left[0,1\right],Q_{\alpha }\left(\tilde{A}\right)=Q_{\alpha }\left(\tilde{B}\right)\\ \tilde{A}\ge \tilde{B}\leftrightarrow \mathrm{\forall }\alpha \in \left[0,1\right],Q_{\alpha }\left(\tilde{A}\right)\ge Q_{\alpha }\left(\tilde{B}\right)\end{array}$

Definition 4.3:

suppose we compare two arbitrary normal fuzzy numbers including $\tilde{A}$ and $\tilde{B}$ at a decision level higher than α and $\alpha \in [0,1]$, so: $\begin{array}{l}\tilde{A}{\le }_{\alpha }\tilde{B}\leftrightarrow Q_{\alpha }\left(\tilde{A}\right)\le Q_{\alpha }\left(\tilde{B}\right)\\ \tilde{A}{=}_{\alpha }\tilde{B}\leftrightarrow Q_{\alpha }\left(\tilde{A}\right)=Q_{\alpha }\left(\tilde{B}\right)\\ \tilde{A}{\ge }_{\alpha }\tilde{B}\leftrightarrow Q_{\alpha }\left(\tilde{A}\right)\ge Q_{\alpha }\left(\tilde{B}\right)\end{array}$ where $\tilde{A}{\le }_{\alpha }\tilde{B}$ means that, at decision level higher than α, $\tilde{B}$ is greater than or equal to $\tilde{A}$.

In order to have a better idea about the proposed $Q_{\alpha }$ transformation function, we calculate this function for two famous and widely used types of fuzzy numbers which are trapezoidal and triangular fuzzy numbers (Figures 3 and 4). $\tilde{A}=\left(\underset{\_}{A}\right(r),\overline{A}(r\left)\right)=(x_0-\delta +\delta .r,x_0+\text{β}-\text{β}.r)$ and $\tilde{B}=\left(\underset{\_}{B}\right(r),\overline{B}(r\left)\right)=(x_0-\delta +\delta .r,y_0+\text{β}-\text{β}.r)$ are the representations of triangular and trapezoidal fuzzy numbers based on definition 4.1. So the proposed $Q_{\alpha }$ functions are: (13) $\begin{array}{l}{Q}_{\alpha }^{Tri}\left(\tilde{A}\right)={\int }_{\alpha }^1\left(\underset{\_}{A}\right(r)+\overline{A}(r)dr=2x_0(1-\alpha )+{\displaystyle \frac{\text{β}-\delta }{2}}{(1-\alpha )}^2,\\ Q_{\alpha }^{Tra}\left(\tilde{B}\right)={\int }_{\alpha }^1\left(\underset{\_}{B}\right(r)+\overline{B}(r)dr=(x_0+y_0\left)\right(1-\alpha )+{\displaystyle \frac{\text{β}-\delta }{2}}{(1-\alpha )}^2\end{array}$(13)

Figure 3. A triangular fuzzy number.

Figure 4. A trapezoidal fuzzy number.

$Q_{\alpha }$ function is based on α-cuts. In this function both left and right parts of the L-R fuzzy numbers is considered. Hence, it is sensitive to changes in both left and right part of L-R fuzzy numbers. This property makes $Q_{\alpha }$ transformation function an excellent representative of the fuzzy inputs, intermediates and outputs in fuzzy DEA models. Consider the fuzzy two-stage DEA models (models 10 and 11); by replacing fuzzy inputs, intermediates and outputs with their assigned $Q_{\alpha }$ functions, the fuzzy two-stage DEA models transform into parametric programming problems which are dependent on α-levels. We can write the transformed fuzzy multiplicative two-stage DEA model with the assumption of CRS as below: (14) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{Q}_{\alpha }\left({\tilde{y}}_{ro}\right)\\ s.t.\sum _{i=1}^m{v}_i{Q}_{\alpha }\left({\tilde{x}}_{io}\right)=1,\\ \sum _{d=1}^D{w}_d{Q}_{\alpha }\left({\tilde{z}}_{dj}\right)-\sum _{i=1}^m{v}_i{Q}_{\alpha }\left({\tilde{x}}_{ij}\right)\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{Q}_{\alpha }({\tilde{y}}_{rj})-\sum _{d=1}^D{w}_d{Q}_{\alpha }({\tilde{z}}_{dj})\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r\end{array}$(14)

Moreover, the transformed fuzzy multiplicative two-stage DEA model with the assumption of VRS as below: (15) $\begin{array}{l}max{e}_o=\sum _{r=1}^s{u}_r{Q}_{\alpha }\left({\tilde{y}}_{ro}\right)-u_0-u_1\\ s.t.\sum _{i=1}^m{v}_i{Q}_{\alpha }\left({\tilde{x}}_{io}\right)=1\\ \sum _{d=1}^D{w}_d{Q}_{\alpha }({\tilde{z}}_{dj})-\sum _{i=1}^m{v}_i{Q}_{\alpha }\left({\tilde{x}}_{ij}\right)-u_0\le 0,\mathrm{\forall }j,\\ \sum _{r=1}^s{u}_r{Q}_{\alpha }({\tilde{y}}_{rj})-\sum _{d=1}^D{w}_d{Q}_{\alpha }({\tilde{z}}_{dj})-u_1\le 0,\mathrm{\forall }j,\\ v_i,w_d,u_r\ge 0,\mathrm{\forall }i,d,r,u_0,u_{1}arefree\end{array}$(15)

The efficiency values obtained by these two models are functions of α-cuts. Hence, by selecting a suitable α (determined by the decision maker), we obtain the efficiency of DMUs. As can be seen from Figure 2, if the value of α is selected close to one, it means that a low-risk decision is adopted because just the elements with high membership values of fuzzy numbers are considered. If the value of α is selected close to zero, it means that a high-risk decision is adopted because the elements with low membership values are important as well. Another advantage of this method is that it works for the general form of fuzzy numbers (not only triangular and trapezoidal forms). Also, after determining the amount of α, an ordinary linear program needs to be solved to evaluate the efficiency of DMUs. So, it is easily possible to obtain the efficiency of DMUs for different values of α and obtain their efficiency functions. This feature facilitates the procedure of comparing and ranking DMUs and makes it more practical.

Almost all the proposed approaches for solving fuzzy two-stage DEA models based on α-cuts are developments of Kao and Liu’s [28] method, such as Kao and Liu [37], Lozano [38], Tavana and Khalili-Damghani [39], Liu [40,41], Tavana et al. [55] and Hatami-Marbini and Saati [56]. Kao and Liu [28] proposed a pair of mathematical programming to evaluate the lower and upper bounds of DMU’s efficiencies in a fuzzy DEA.

However, the above-mentioned approaches have some important disadvantages. For example, because of considering only left or right part of fuzzy data in their proposed models, the above-mentioned approaches just suggested a boundary for efficiency measure of a DMU at different α-cuts. This boundary includes a wide range of efficiency measures and especially for ranking DMUs, it can lead to inaccurate results but our proposed approach considers simultaneously left and right parts of the fuzzy data by applying $Q_{\alpha }$ transformation function. So, we can easily compare and rank DMUs. Also, their approaches just developed for fuzzy multiplicative two-stage DEA model under CRS assumption but our proposed approach works for both CRS and VRS assumptions.

Our proposed method is computationally less expensive compared to the above-mentioned methods. Moreover, unlike the above-mentioned methods that just work for multiplicative form of two-stage DEA models, our proposed approach can be easily generalized to other kinds of fuzzy network DEA models such as fuzzy additive two-stage DEA models, fuzzy weighted additive two-stage DEA models, fuzzy multi-stage DEA models, etc.

In the next section, two illustrations are utilized to demonstrate how the proposed method works.

5. Two Illustrative Examples

This section presents two different examples. The first is a simple hypothetical example that helps us to have a better understanding about the method and the second is a practical example which was investigated in the literature. In example 2, we will compare our results with the results obtained from Kao and Liu [37] and Liu [41].

Example 1.

Consider the data set in Table 1. This example was first presented by Sahoo et al. [50] in the crisp version. We have nine DMUs as A, B, C, D, E, F, G, H and I. Each DMU has one fuzzy input $\left(\tilde{x}\right)$, one fuzzy intermediate $\left(\tilde{z}\right)$ and one fuzzy output $\left(\tilde{y}\right)$. All the input, intermediate and output data are triangular fuzzy numbers $(x_0,\delta ,\text{β})$ in which $x_0$ is the center and $\delta$ and $\text{β}$ are left and right spreads, respectively. Using the Equation (13) the $Q_{\alpha }$ transformation function for each fuzzy number is found and presented in Table 2.

Table 1. The data set for a simple hypotehetical fuzzy two-stage DEA example.

DMUs	$\tilde{x}$	$\tilde{z}$	$\tilde{y}$
A	(1.5,0,1.5)	(1,1,2)	(1,0.5,1)
B	(4,1,0.5)	(6,1,2)	(4,1,3)
C	(6,1,0)	(7,3,1)	(7,2,0.5)
D	(2,2,0)	(2.5,1,0.5)	(3,2,1)
E	(5,0,2)	(4,1,2)	(6,1,0.5)
F	(4,1,1)	(3.5,0.5,0.5)	(2,1,1)
G	(5.5,0.5,1.5)	(5,1,0.5)	(5,3,1)
H	(4.5,0.5,0.5)	(6,1,1)	(6,1,2)
I	(7,4,1)	(6.5,0.5,1)	(6.5,0.5,2.5)

Table 2. $Q_{\alpha }$ function of input, intermediate and output fuzzy data in example 1.

Firms	$Q_{\alpha }\left(\tilde{X}\right)$	$Q_{\alpha }\left(\tilde{Z}\right)$	$Q_{\alpha }\left(\tilde{Y}\right)$
A	3(1−α) + 0.75(1−α)^2	2(1−α) + 0.5(1−α)^2	2(1−α) + 0.25(1−α)^2
B	8(1−α) − 0.25(1−α)^2	12(1−α) + 0.5(1−α)^2	8(1−α) + (1−α)^2
C	12(1−α) − 0.5(1−α)^2	14(1−α) − (1−α)^2	14(1−α) − 0.75(1−α)^2
D	4(1−α) − (1−α)^2	5(1−α) − 0.25(1−α)^2	6(1−α) − 0.5(1−α)^2
E	10(1−α)+ (1−α)^2	8(1−α) + 0.5(1−α)^2	12(1−α)−0.25(1−α)^2
F	8(1−α)	7(1−α)	4(1−α)
G	11(1−α) + 0.5(1−α)^2	10(1−α) − 0.25(1−α)^2	10(1−α) −  (1−α)^2
H	9(1−α)	12(1−α)	12(1−α) + 0.5(1−α)^2
I	14(1−α) − 1.5(1−α)^2	13(1−α) + 0.25(1−α)^2	13(1−α)+(1−α)^2

The efficiency values of the DMUs under the CRS assumption for different α-levels are presented in Table 3. As can be seen, the efficiency scores change by increasing the amount of α. The efficiency variations are graphed in Figure 5. These efficiency variations cause the change in the ranking of the DMUs for various α’s. The DMUs are ranked for three different values of α and given in Table 4.

Figure 5. Efficiency graph of DMUs under CRS assumption for different α.

Table 3. Efficiencies of DMUs in two-stage DEA model with the assumption of CRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version
A	0.269106	0.271331	0.273647	0.276060	0.278576	0.281203	0.283948	0.286821	0.289829	0.292984	0.295958	0.296296
B	0.520851	0.512999	0.505194	0.497436	0.489726	0.482062	0.474446	0.466875	0.459352	0.451875	0.445185	0.444444
C	0.516762	0.517026	0.517270	0.517494	0.517699	0.517885	0.518051	0.518198	0.518324	0.518431	0.518511	0.518519
D	0.822270	0.802339	0.783625	0.766015	0.749411	0.733727	0.718885	0.704817	0.691460	0.678760	0.667850	0.666667
E	0.479091	0.484129	0.489247	0.494447	0.499732	0.505102	0.510561	0.516111	0.521755	0.527495	0.532745	0.533333
F	0.224255	0.224077	0.223893	0.223703	0.223509	0.223308	0.223102	0.222891	0.222674	0.222451	0.222245	0.222222
G	0.351008	0.356174	0.361371	0.366598	0.371855	0.377143	0.382461	0.387810	0.393190	0.398600	0.403495	0.404040
H	0.622931	0.619946	0.616949	0.613942	0.610924	0.607895	0.604855	0.601805	0.598745	0.595674	0.592901	0.592593
I	0.502332	0.492437	0.482769	0.473319	0.464079	0.455043	0.446205	0.437557	0.429094	0.420809	0.413502	0.412698

Table 4. Ranking DMUs under CRS assumption for three different α’s.

α	A	B	C	D	E	F	G	H	I
0.1	8	4	3	1	6	9	7	2	5
0.5	8	5	3	1	4	9	7	2	6
0.9	8	5	4	1	3	9	7	2	6

With the assumption of VRS, the efficiency values of the DMUs for different α-levels are presented in Table 5. As can be seen in Table 1, DMUs A and D have the smallest fuzzy input value among all the DMUs. However, they are two intersected fuzzy numbers. According to their assigned $Q_{\alpha }$ functions (Table 2), for α less than 0.43 the input of DMU D is less than that of DMU A. So, according to Theorem 3.2 part (1), for α less than 0.43, DMU D is overall two-stage DEA weak efficient. For α more than 0.43, DMU A has smaller input than DMU D. Also, DMU A has the smallest fuzzy intermediate value among all the DMUs (Table 1). So, according to Theorem 3.2 part (2), for α more than 0.43, DMU A is overall two-stage DEA efficient with fixed link. These results can be observed in Table 5.

Table 5. Efficiencies of DMUs in two-stage DEA model with the assumption of VRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version
A	0.800000	0.843537	0.888889	0.936170	0.985507	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000
B	0.546306	0.545116	0.544113	0.543297	0.542669	0.542228	0.541975	0.541911	0.542036	0.542351	0.542798	0.542857
C	0.629500	0.645892	0.664065	0.756693	0.883761	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000
D	1.000000	1.000000	1.000000	1.000000	1.000000	0.994636	0.986584	0.978002	0.968990	0.959633	0.950974	0.950000
E	0.481672	0.489772	0.498051	0.506513	0.515164	0.524011	0.533061	0.542320	0.551796	0.561496	0.570425	0.571429
F	0.375000	0.387500	0.400000	0.412500	0.425000	0.428743	0.428656	0.428214	0.427442	0.426364	0.425149	0.425000
G	0.368163	0.376840	0.385561	0.394329	0.403144	0.412009	0.420926	0.429896	0.438920	0.448002	0.456226	0.457143
H	0.696535	0.695823	0.694503	0.692453	0.689520	0.685499	0.674886	0.664514	0.654390	0.644523	0.635869	0.634921
I	1.000000	1.000000	0.906250	0.860750	0.826336	0.799057	0.693497	0.608231	0.569700	0.549821	0.532503	0.530612

Furthermore, as can be seen in Table 1, DMU C and DMU I have the largest fuzzy intermediate and output data. However, these are also intersected fuzzy numbers. According to their $Q_{\alpha }$ functions for α less than 0.1, the intermediate and output of DMU I are the biggest among all the intermediates and outputs. So, according to Theorem 3.2 part (4), for α less than 0.1, DMU I is overall two-stage DEA efficient with a fixed link. Moreover, for α more than 0.5, DMU C has the biggest intermediate and output among all the intermediates and outputs. Therefore, according to Theorem 3.2 part (4), for α more than 0.5, DMU C is overall two-stage DEA efficient with a fixed link. These results can also be observed in Table 5.

From Table 5, we see that the efficiency scores change by increasing the amount of α. The efficiency variations are graphed in Figure 6. These efficiency variations cause the change in ranking of the DMUs for different α’s. The rankings of the DMUs for three different values of α are provided in Table 6.

Figure 6. Efficiency graph of DMUs under VRS assumption for different α.

Table 6. Ranking DMUs under VRS assumption for three different ’s.

α	A	B	C	D	E	F	G	H	I
0.1	2	5	4	1	6	7	8	3	1
0.5	1	5	1	2	6	7	8	4	3
0.9	1	6	1	2	4	8	7	3	5

Example 2:

In order to explain our method with a more practical example and compare the achieved results with the other studies in the literature, we used the example which is first presented by Kao and Liu [37] in the fuzzy environment. They proposed their method when applied to the non-life insurance companies’ dataset. In this example, there are two fuzzy inputs (Operating expenses, Insurance expenses), two fuzzy intermediates (Direct written premiums, Reinsurance premiums) and two fuzzy outputs (Underwriting profit, Investment profit). All the input, intermediate and output data are triangular fuzzy numbers presented here as $(x_0,\delta ,\text{β})$ in which $x_0$ is the center and $\delta$ and $\text{β}$ are left and right spreads (Table 7).

Table 7. Triangular fuzzy input, intermediate and output data of 24 insurance companies in Taiwan.

Co.	First input $\left({\tilde{X}}_1\right)$	Second input $\left({\tilde{X}}_2\right)$	First intermediate $\left({\tilde{Z}}_1\right)$	Second intermediate $\left({\tilde{Z}}_2\right)$	First output $\left({\tilde{Y}}_1\right)$	Second output $\left({\tilde{Y}}_2\right)$
1	(1178,65,78)	(673,37,44)	(7451,410,492)	(856,47,56)	(984,54,65)	(681,37,45)
2	(1381,76,91)	(1352,74,89)	(10020,551,661)	(1812,100,120)	(1228,68,81)	(834,46,55)
3	(1177,65,78)	(592,33,39)	(4776,263,315)	(560,31,37)	(293,16,19)	(658,36,43)
4	(601,33,40)	(594,33,39)	(3174,175,209)	(371,20,24)	(248,14,16)	(177,10,12)
5	(6699,368,442)	(3351,184,221)	(37362,2027,2318)	(1753,96,116)	(7851,432,518)	(3925,216,259)
6	(2627,144,173)	(668,37,44)	(9747,536,643)	(952,52,63)	(1713,94,113)	(415,23,27)
7	(1942,89,105)	(1443,66,78)	(10685,492,577)	(643,30,35)	(2239,103,121)	(439,20,24)
8	(3789,174,205)	(1873,86,101)	(17267,794,932)	(1134,52,61)	(3899,179,211)	(622,29,34)
9	(1567,72,85)	(950,44,51)	(11473,528,620)	(546,25,29)	(1043,48,56)	(264,12,14)
10	(1303,60,70)	(1298,60,70)	(8210,378,443)	(504,23,27)	(1697,78,92)	(554,25,30)
11	(1962,90,106)	(672,31,36)	(7222,332,390)	(643,30,35)	(1486,68,80)	(18,1,1)
12	(2592,119,140)	(650,30,35)	(9434,434,509)	(1118,51,60)	(1574,72,85)	(909,42,49)
13	(2609,128,130)	(1368,67,68)	(13921,682,696)	(811,40,41)	(3609,177,180)	(223,11,11)
14	(1396,68,70)	(988,48,49)	(7396,362,370)	(465,23,23)	(1401,69,70)	(332,16,17)
15	(2184,107,109)	(651,32,33)	(10422,511,521)	(749,37,37)	(3355,164,168)	(555,27,28)
16	(1211,59,61)	(415,20,21)	(5606,275,280)	(402,20,20)	(854,42,43)	(197,10,10)
17	(1453,71,73)	(1085,53,54)	(7695,377,385)	(342,17,17)	(3144,154,157)	(371,18,19)
18	(757,37,38)	(547,27,27)	(3631,178,182)	(995,49,50)	(692,34,35)	(163,8,8)
19	(159,8,8)	(182,9,9)	(1141,58,55)	(483,25,23)	(519,26,25)	(46,2,2)
20	(145,7,7)	(53,3,3)	(316,16,15)	(131,7,6)	(355,18,17)	(26,1,1)
21	(84,4,4)	(26,1,1)	(225,11,11)	(40,2,2)	(51,3,2)	(6,0,0)
22	(15,1,1)	(10,1,0)	(52,3,2)	(14,1,1)	(82,4,4)	(4,0,0)
23	(54,3,3)	(28,1,1)	(245,12,12)	(49,2,2)	(1,0,0)	(18,1,1)
24	(163,8,8)	(235,12,11)	(476,24,23)	(644,33,31)	(142,7,7)	(16,1,1)

After employing our method, the efficiency values under the CRS assumption are found and presented in Table 8. The obtained efficiency scores are within the ranges obtained by Kao and Liu [37], Lozano [38] and Tavana et al. [55] methods. However, in almost all the cases, the obtained efficiencies by Hatami-Marbini and Saati [56] are smaller than that of Kao and Liu [37], Lozano [38], Tavana et al. [55] and our proposed method. Table 9 presents the obtained efficiency scores from different approaches in the case of α = 0. Since our method leads to a number rather than a range, it is easier for decision-makers to evaluate and rank DMUs.

Table 8. Efficiency scores of 24 insurance companies in Taiwan with the assumption of CRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version	Rank	Rank from Liu [41]
1	0.699671	0.699656	0.699642	0.699627	0.699613	0.699599	0.699584	0.699570	0.699555	0.699541	0.699528	0.699526	3	3
2	0.625627	0.625638	0.625650	0.625661	0.625672	0.625683	0.625695	0.625706	0.625717	0.625728	0.625738	0.625740	5	5
3	0.690428	0.690408	0.690388	0.690369	0.690349	0.690329	0.690310	0.690290	0.690270	0.690251	0.690233	0.690231	4	4
4	0.304236	0.304244	0.304251	0.304259	0.304267	0.304275	0.304283	0.304291	0.304299	0.304307	0.304315	0.304315	15	15
5	0.791964	0.791981	0.791999	0.792016	0.792034	0.792052	0.792069	0.792087	0.792105	0.792122	0.792138	0.792140	1	1
6	0.388887	0.388934	0.388980	0.389027	0.389073	0.389120	0.389166	0.389213	0.389259	0.389306	0.389348	0.389352	12	12
7	0.277049	0.277072	0.277094	0.277116	0.277139	0.277161	0.277183	0.277206	0.277228	0.277250	0.277270	0.277273	17	17
8	0.274688	0.274710	0.274733	0.274755	0.274778	0.274800	0.274823	0.274845	0.274868	0.274890	0.274917	0.274920	18	18
9	0.223579	0.223595	0.223611	0.223627	0.223644	0.223660	0.223676	0.223692	0.223708	0.223724	0.223738	0.223740	20	20
10	0.467387	0.467406	0.467425	0.467445	0.467464	0.467483	0.467502	0.467522	0.467541	0.467560	0.467577	0.467579	9	9
11	0.159349	0.159359	0.159369	0.159379	0.159388	0.159398	0.159408	0.159418	0.159428	0.159438	0.159447	0.159448	23	23
12	0.759372	0.759404	0.759436	0.759468	0.759500	0.759532	0.759564	0.759596	0.759629	0.759661	0.759690	0.759693	2	2
13	0.206355	0.206388	0.206421	0.206454	0.206487	0.122230	0.206553	0.206586	0.206619	0.206652	0.206682	0.206685	21	21
14	0.289089	0.289101	0.289113	0.289125	0.289137	0.289149	0.289161	0.289173	0.289185	0.289197	0.289207	0.289209	16	16
15	0.610980	0.611048	0.611116	0.611184	0.611252	0.611320	0.611388	0.611456	0.611524	0.611592	0.611654	0.611661	6	6
16	0.318516	0.318559	0.318602	0.318645	0.318688	0.318731	0.318774	0.318817	0.318860	0.318903	0.318941	0.318946	14	14
17	0.360319	0.360365	0.360410	0.360456	0.360501	0.360547	0.360592	0.360638	0.360684	0.360729	0.360770	0.360775	13	13
18	0.258324	0.258345	0.258366	0.258386	0.258407	0.258428	0.258448	0.258469	0.258490	0.258511	0.258529	0.258531	19	19
19	0.411984	0.412108	0.412232	0.412356	0.412480	0.412605	0.412729	0.412853	0.412977	0.413101	0.413213	0.413226	10	11
20	0.538344	0.538417	0.538491	0.538564	0.538637	0.538711	0.538784	0.538858	0.538931	0.539005	0.539071	0.539078	8	8
21	0.190364	0.190431	0.190499	0.190566	0.190634	0.190701	0.190769	0.190837	0.190904	0.190972	0.191033	0.191040	22	22
22	0.618817	0.617519	0.616227	0.614941	0.613660	0.612385	0.611115	0.609850	0.608591	0.607338	0.606214	0.606089	7	7
23	0.404265	0.404255	0.404244	0.404234	0.404224	0.404214	0.404204	0.404194	0.404184	0.404174	0.404165	0.404164	11	10
24	0.130450	0.130481	0.130511	0.130542	0.130572	0.130603	0.130634	0.130664	0.130695	0.130725	0.130753	0.130756	24	24

Table 9. Efficiency scores of 24 insurance companies from different approaches with the assumption of CRS (α = 0).

α = 0
Co.	Our proposed approach	Kao and Liu [37]	Lozano [38]	Tavana et al. [55]	Hatami-Marbini and Saati [56]
1	0.699671	(0.493,0.906)	(0.500,0.904)	(0.679,0.9)	(0.437,0.559)
2	0.625627	(0.439,0.798)	(0.447,0.795)	(0.604,0.8)	(0.331,0.422)
3	0.690428	(0.487,0.762)	(0.549,0.861)	(0.669,0.86)	(0.406,0.501)
4	0.304236	(0.213,0.426)	(0.213,0.426)	(0.284,0.43)	(0.160,0.203)
5	0.791964	(0.562,0.957)	(0.563,1.000)	(0.771,1.00)	(0.499,0.637)
6	0.388887	(0.279,0.514)	(0.279,0.510)	(0.369,0.51)	(0.214,0.273)
7	0.277049	(0.202,0.377)	(0.203,0.375)	(0.26,0.38)	(0.217,0.250)
8	0.274688	(0.202,0.374)	(0.203,0.371)	(0.258,0.37)	(0.201,0.246)
9	0.223579	(0.162,0.296)	(0.163,0.294)	(0.206,0.29)	(0.157,0.191)
10	0.467387	(0.335,0.638)	(0.337,0.636)	(0.449,0.64)	(0.272,0.332)
11	0.159349	(0.122,0.221)	(0.116,0.217)	(0.147,0.22)	(0.081,0.099)
12	0.759372	(0.553,0.945)	(0.559,0.942)	(0.773,0.94)	(0.392,0.484)
13	0.206355	(0.153,0.280)	(0.153,0.276)	(0.192,0.28)	(0.169,0.206)
14	0.289089	(0.211,0.394)	(0.212,0.392)	(0.273,0.39)	(0.207,0.253)
15	0.610980	(0.449,0.797)	(0.460,0.794)	(0.598,0.79)	(0.383,0.466)
16	0.318516	(0.233,0.436)	(0.233,0.433)	(0.304,0.43)	(0.205,0.250)
17	0.360319	(0.263,0.488)	(0.266,0.484)	(0.344,0.48)	(0.287,0.349)
18	0.258324	(0.189,0.352)	(0.189,0.350)	(0.243,0.35)	(0.189,0.227)
19	0.411984	(0.300,0.513)	(0.309,0.497)	(0.395,0.5)	(0.280,0.307)
20	0.538344	(0.405,0.735)	(0.395,0.725)	(0.532,0.72)	(0.378,0.458)
21	0.190364	(0.148,0.273)	(0.145,0.247)	(0.189,0.25)	(0.127,0.146)
22	0.618817	(0.438,0.651)	(0.510,0.783)	(0.483,0.78)	(0.563,0.679)
23	0.404265	(0.302,0.578)	(0.291,0.556)	(0.409,0.56)	(0.234,0.285)
24	0.130450	(0.097,0.183)	(0.095,0.176)	(0.12,0.18)	(0.072,0.089)

Using the proposed bounds in Kao and Liu’s [37] paper, it is not easily possible to compare and rank DMUs. In an attempt to solve this problem, Liu [41] used Chen [42] index to combine the bounds and ranked DMUs. From Table 8, we can see that our obtained ranking coincides with the ranking in Liu [41]. In Kao and Liu [37] method, the proposed program to obtain the upper bound of the efficiency measure of a DMU is linear but their program for lower bound is non-linear. Also, all the above-mentioned approaches just developed for fuzzy multiplicative two-stage DEA model under CRS assumption but our proposed approach is linear and works for both CRS and VRS assumptions.

The obtained efficiency scores with the assumption of VRS are also found and presented in Table 10. From Table 7, DMU 5 has a minimum input value for any input item and a minimum intermediate value for any intermediate item, so according to Theorem 3.2 part (2) DMU 5 is overall two-stage DEA efficient with fixed link. Also, DMU 22 has a maximum intermediate value for any intermediate item and a maximum output value for any output item, so from Theorem 3.2 part (4) DMU 22 is overall two-stage DEA efficient with fixed link. The obtained results from our approach are coincident with Theorem 3.2 (see Table 10).

Table 10. Efficiency scores of 24 insurance companies in Taiwan with the assumption of VRS.

Co.	α = 0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 0.99	Crisp version	Rank
1	0.735957	0.735976	0.735995	0.736002	0.736009	0.736016	0.736024	0.736031	0.736038	0.736045	0.736052	0.736053	5
2	0.710066	0.710091	0.710116	0.710141	0.710166	0.710191	0.710216	0.710241	0.710266	0.710291	0.710313	0.710316	7
3	0.690765	0.690743	0.690721	0.690699	0.690677	0.690654	0.690632	0.690610	0.690587	0.690565	0.690545	0.690543	8
4	0.311879	0.311873	0.311867	0.311860	0.311854	0.311848	0.311841	0.311835	0.311829	0.311822	0.311817	0.311816	21
5	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1
6	0.490102	0.490125	0.490148	0.490170	0.490193	0.490216	0.490239	0.490262	0.490285	0.490308	0.490329	0.490331	14
7	0.446903	0.446903	0.446904	0.446904	0.446905	0.446905	0.446906	0.446907	0.446907	0.446908	0.446908	0.446908	15
8	0.526460	0.526385	0.526310	0.526235	0.526160	0.526085	0.526009	0.525934	0.525859	0.525783	0.525715	0.525708	13
9	0.278449	0.278467	0.278484	0.278502	0.278519	0.278537	0.278555	0.278572	0.278590	0.278608	0.278624	0.278626	22
10	0.626336	0.626341	0.626345	0.626350	0.626354	0.626359	0.626364	0.626368	0.626373	0.626378	0.626382	0.626382	11
11	0.332613	0.332608	0.332603	0.332598	0.332593	0.332588	0.332582	0.332577	0.332572	0.332567	0.332562	0.332562	19
12	0.782873	0.782912	0.782952	0.782991	0.783031	0.783070	0.783110	0.783149	0.783189	0.783229	0.783264	0.783268	4
13	0.612840	0.612877	0.612915	0.612952	0.612989	0.144899	0.613064	0.613102	0.613139	0.613177	0.613210	0.613214	12
14	0.403446	0.403452	0.403459	0.403465	0.403472	0.403478	0.403484	0.403491	0.403497	0.403503	0.403509	0.403510	17
15	0.880752	0.880768	0.880785	0.880802	0.880819	0.880835	0.880852	0.880869	0.880885	0.880902	0.880917	0.880919	3
16	0.382897	0.382942	0.382987	0.383032	0.383077	0.383122	0.383167	0.383213	0.383258	0.383303	0.383343	0.383348	18
17	0.724138	0.724136	0.724135	0.724133	0.724132	0.724130	0.724129	0.724127	0.724126	0.724124	0.724123	0.724123	6
18	0.324558	0.324566	0.324574	0.324582	0.324591	0.324599	0.324607	0.324616	0.324624	0.324632	0.324640	0.324641	20
19	0.674994	0.674969	0.674944	0.674920	0.674895	0.674870	0.674846	0.674821	0.674796	0.674771	0.674749	0.674747	9
20	0.894859	0.895292	0.895726	0.896159	0.896593	0.897027	0.897461	0.897895	0.898329	0.898763	0.899154	0.899198	2
21	0.412761	0.413720	0.414680	0.415640	0.416600	0.417560	0.418520	0.419480	0.420440	0.421399	0.422263	0.422359	16
22	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1
23	0.629541	0.630058	0.630575	0.631091	0.631608	0.632125	0.632641	0.633158	0.633675	0.634192	0.634657	0.634708	10
24	0.180399	0.180393	0.180387	0.180381	0.180374	0.180368	0.180362	0.180356	0.180350	0.180343	0.180338	0.180337	23

6. Conclusion

DEA is a field of study that helps us to solve problems that require techniques to identify the best practices among peer decision-making units (DMUs). This method treats a DMU as a ‘black box’ and its interior structures are not taken into consideration. However, in order to avoid ignoring important data in the interior structure of DMUs, a network DEA is developed. Two-stage DEA is the simplest and most important of the network DEA models.

In real-life performance measurement problems, the data are mostly imprecise and uncertain. In these cases, fuzzy numbers are a better representative for the data sets. Hence, to solve these problems we use fuzzy DEA models.

In this study, we proposed a novel method to assess the efficiency of DMUs in fuzzy two-stage DEA models. This method can be considered in the α-level based approach. The proposed model is a parametric program based on α-cuts and after selection of α becomes linear. This feature helps the decision maker to easily compare and rank DMUs based on the degree of certainty by appropriate selection of α. Moreover, the efficiency functions of DMUs can be plotted for all amounts of alpha $(0\le \alpha <1)$. Our proposed method is computationally less expensive compared to the previous methods. Here we used our proposed method in the multiplicative fuzzy two-stage DEA model. However, unlike similar proposed approaches, this method can easily apply to other kinds of fuzzy network DEA models, such as fuzzy additive two-stage DEA models, fuzzy weighted additive two-stage DEA models, fuzzy multi-stage DEA models and so on with both CRS and VRS assumptions.

Finally, we illustrated the method with two different examples which confirm the workability of the proposed methodology to evaluate the efficiency of DMUs and rank them in fuzzy multiplicative two-stage DEA model. Also, the obtained results are compared with two other approaches in the literature. We can see that since our method leads to a number rather than a range, it is easier for decision-makers to evaluate and rank DMUs.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Roohollah Abbasi Shureshjani

Roohollah Abbasi Shureshjani is Assistant Professor of Applied Mathematics at Hazrat-e Masoumeh University, Qom, Iran. He received his Ph.D. degree in Applied Mathematics from University of Qom, Qom, Iran, in 2017. His research interests include fuzzy logic, operational research, optimization and data envelopment analysis.

Sara Askarinejad

Sara Askarinejad graduated student of Amirkabir University of Technology, Tehran, Iran.

Ali Asghar Foroughi

Ali Asghar Foroughi is Associate Professor of Applied Mathematics at University of Qom, Qom, Iran. He received his Ph.D. degree in Applied Mathematics from Kharazmi University, Tehran, Iran, in 2003. His research interests include operational research, optimization and data envelopment analysis.