Classifying fuzzy flexible measures in data envelopment analysis

Abstract

In classical models of data envelopment analysis, the identification of factors as input or output factors has been assumed. In the last decade, research on the impact of flexible factors on the assessment of the efficiency of decision-making units has been expanded. Flexible factors are factors that cannot be categorized precisely as input or output factors. In this paper, unlike former studies, the classification of fuzzy flexible measures is addressed. Here, two kinds of absolute and fuzzy classifications are introduced for fuzzy flexible factors. In fact, the proposed classifier fuzzy models, unlike the absolute input/output classifications in former models, is such that part of the flexible measures can play input role for all DMUs and the rest are treated as output. The advantage of a fuzzy classification is the ability to make a unique decision on defining the type of each flexible measure. This feature prevents an inhomogeneous status of decision making units.

Keywords

Data envelopment analysis input and output factors flexible measures fuzzy numbers

1 Introduction

Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the relative efficiency of a set of Decision-Making units (DMUs) that generate multiple outputs using multiple inputs with each unit. DEA was presented by Charnes et al. [5]. In the initial model, efficiency is presented as a ratio between a weighted sum of outputs and a weighted sum of inputs. An important feature of DEA is the ability to provide efficiency scores by calculating input and output weights. Due to the capabilities of DEA, this method is used in a wide range of applications, including hospitals, banks, service organizations, higher education institutions and so on (see [3 , 8]). Classical DEA models assume that the inputs and outputs are completely given. However, in recent models, it is specified that there are some factors that are called flexible measures. These factors can sometimes have an input role and, in some cases, an output role. For example, institutions that give universities grants based on having good performance in this regard are more willing to give more scholarships to such universities. From one perspective, whatever the university is doing well in terms of research, institutions are more willing to give more grant to these universities. It can be considered as an output in such a way that the university considers it as the result of its performance (output). On the other hand, this grant enters the scientific system and is used only for further scientific work. In this case, it is considered as the input of the system from another perspective. The preceding example expresses the importance of studying flexible measures attentively. In the models that have been introduced so far, choosing flexible measures are unit-oriented, in the concept that each decision maker unit determines the type of the flexible measures independently of others. Cook and Zhu [7] presented a fractional model (Cook and Zhu model hereafter) for classifying these factors. After that, Toloo [17], Amirteimoori et al. [2], Toloo [18] presented some modifications and interpretations on Cook and Zhu [7] study.

In recent years, several researchers have suggested different fuzzy DEA models that are used to solve problems in which some of the data are imprecise or vague. Using fuzzy sets, such as those introduced by Zadeh [21] allows experts to use distinct types of incomplete information in the decision model. Since then, numerous studies have been applied fuzzy sets in DEA and decision making methods in practical cases [9–16 , 22].

In this study, we investigate the Cook and Zhu model in the presence of fuzzy flexible measures, with a fuzzy number obtained at the domain [0, 1] related to each measures. The presented models can be used for issues where flexible measures may apply, such as customer satisfaction, the impact of the brand on performance, etc. The novelty of this study is that the proposed model, unlike the absolute input/output classifications in former models (e.g. Nabahat & Esmaeeli-Sangari [14] and Kordrostami et al. [12]), is such that part of the flexible measures for all DMUs can have an input role and the rest considered as output. For example, in a manufacturing company, the quality of the firm’s brand can be considered as a fuzzy flexible measure. This factor is considered in some ways as an input to the system and in some ways as an output. At the beginning of the business, the company is trying to increase its credibility and create a quality brand, and in such, it is the output of the system. After the company has reached a higher stage of production and credibility, the brand as the input of the system can generate revenue and produce more output for the institution. So, at this stage, it can be considered as an input of the system. On the other hand, part of this factor can be considered as the output of the system and part of it as an input. Therefore, in this case, the strict classification is not achieved in two independent classes of inputs and outputs. Another important advantage of a fuzzy classification in this study is the ability to obtain a unique decision on the type of each flexible measure. This feature prevents an inhomogeneous status of decision making units.

The remainder of the paper is structured as follows. The second section involves the introduction of Cook and Zhu method in determining the classification of the flexible measures. The third and fourth sections introduce proposed fuzzy-models. Section five uses the proposed models in numerical examples, then followed by the conclusion in the sixth section.

2 Preliminaries

This section contains brief definitions of fuzzy numbers that are necessary for this study and afterward preliminaries on flexible measures and DEA models.

2.1 Fuzzy numbers

Fuzzy numbers are a special type of fuzzy sets. The fuzzy number should be convex and a normal set with limited support. A wide variety of fuzzy numbers are presented and used with different names and attributes, but an important principle in the application of fuzzy theory is its computational efficiency.

Definition 1. A triangular fuzzy number $\tilde{A}$ can be defined by triplet real number (a ^l, a ^m, a ^u), where a ^u ≥ a ^m ≥ a ^l. The membership function $μ \tilde{A} (x)$ is given by the following expression. $μ \tilde{A} (x) = {\begin{matrix} \frac{x - a^{l}}{a^{m} - a^{l}} & a^{l} \leq x \leq a^{m} \\ \frac{a^{u} - x}{a^{u} - a^{m}} & a^{m} \leq x \leq a^{u} \\ 0 & otherwise \end{matrix}$

Figure 1, displays the triangular fuzzy number $\tilde{A} = (α^{l}, α^{m}, α^{u})$ in geometric space.

Fig. 1.

Illustration of a Triangular fuzzy number.

Definition 2. A triangular fuzzy number $\tilde{A} = (α^{l}, α^{m}, α^{u})$ is called positive (negative) where α ^l > 0 (α ^u < 0).

2.1.1 Operations of positive triangular fuzzy numbers

Suppose positive scalar c and positive triangular fuzzy numbers $\tilde{A} = (α^{l}, α^{m}, α^{u})$ and $\tilde{B} = (b^{l}, b^{m}, b^{u})$ , then following operations including addition ⊕, subtraction ⊖, multiplication ⊗ and scalar multiplication are defined as follows: $\begin{matrix} \tilde{A} \oplus \tilde{B} = (a^{l} + b^{l}, a^{m} + b^{m}, a^{u} + b^{u}) \\ \tilde{A} ⊖ \tilde{B} = (a^{l}, - b^{u}, a^{m} - b^{m}, a^{u} - b^{l}) \\ \tilde{A} \otimes \tilde{B} = (a^{l} b^{l}, a^{m} b^{m}, a^{u} b^{u}) \\ c \tilde{A} = ({ca}^{l}, {ca}^{m} {ca}^{u}) \end{matrix}$

2.2 Fuzzy ranking method

Many common methods for solving fuzzy linear programming problems are based on the comparison of fuzzy numbers and, in particular, the use of ranking functions. A convenient method for ranking fuzzy numbers is to use a ranking function $T : F (R) \to R$ where $F (R)$ is denoted as the set of all triangular fuzzy numbers. The ordering in $F (R)$ is defined as follows:

$\tilde{A} ≽ \tilde{B} \Leftrightarrow T (\tilde{A}) \geq T (\tilde{B})$

$\tilde{A} \approx \tilde{B} \Leftrightarrow T (\tilde{A}) = T (\tilde{B})$

$\tilde{A} ≻ \tilde{B} \Leftrightarrow T (\tilde{A}) > T (\tilde{B})$

Where $\tilde{A}$ and $\tilde{B}$ are triangular fuzzy numbers. Also, $\tilde{A} ≼ \tilde{B}$ if and only if $\tilde{A} ≽ \tilde{B}$ .

One of the most commonly used ranking functions, first proposed by Yager [20], is defined as follows: $T (\tilde{A}) = 0.5 (a^{m} + 0.5 (a^{l} + a^{u}))$ (1)

Throughout this article, we use the above ranking function.

2.3 Flexible measures

Suppose that there are n DMUs that each DMU_j, j = 1, …, n, produces s different output y _rj (r = 1, …, s) using m input of x _ij (i = 1, …, m). The CCR fractional model proposed by Charnes et al. [5] is as follows:

$\begin{matrix} e_{o} = max \sum_{r = 1}^{s} μ_{r} y_{ro} \\ s . t . \\ \sum_{i = 1}^{m} v_{i} x_{io} = 1 \\ \sum_{r = 1}^{s} μ_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} \leq 0 \forall j \\ μ_{r}, v_{i} \geq 0 \forall r, i \end{matrix}$ (2)

Furthermore, assume that there are K flexible factors w _kj (k = 1, …, K) that we would like to determine the input or output status of. Suppose the values of these flexible factors are determined. Cook and Zhu [7] presented the fractional model (3) to determine the nature of the flexible measures in the evaluation of DMU_o, o ∈ {1, …, n}. For each kth flexible measure, d _k ∈ {0, 1} is introduced as a binary variable where d _k = 1 represents the output nature and d _k = 0 indicates the input type of this factor.

$\begin{matrix} max \sum_{r = 1}^{s} μ_{r} y_{ro} + \sum_{k = 1}^{K} d_{k} γ_{k} w_{ko} \\ s . t . \\ \sum_{r = 1}^{s} μ_{r} y_{rj} + \sum_{k = 1}^{K} d_{k} γ_{k} w_{kj} - \sum_{i = 1}^{m} v_{i} x_{ij} - \\ \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} w_{kj} \leq 0, \forall j \\ \sum_{i = 1}^{m} v_{i} x_{io} + \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} w_{ko} = 1, \\ d_{k} \in {0, 1}, μ_{r,} v_{i}, γ_{k} \geq 0, \forall r, i, k \end{matrix}$ (3)

Using model (3), the nature of each flexible measure is determined by each decision making unit.

3 Fuzzy flexible measures

In some applications, we face qualitative factors. These factors are, in some cases, considered as inputs to the system and, in other cases, as outputs. For example, in a manufacturing company, consider the quality of the firm’s brand, which is a qualitative factor. This factor is considered in some ways as an input to the system and in some ways as an output. At the beginning of the business, the company is trying to increase its credibility and create a quality brand, and in such, it is the output of the system. After the company has reached a higher stage of production and credibility, the brand as the input of the system can generate revenue and produce more output for the institution. So, at this stage, it can be considered as an input of the system. On the other hand, part of this factor can be considered as the output of the system and part of it as an input. Therefore, in this case, the strict classification is not achieved in two independent classes of inputs and outputs.

In this section, we firstly define a fuzzy triangular number corresponding to each of the flexible measures (qualitative factors) of each DMU. We then examine two methods concerning the problem of input/output classification. In the first method, it is assumed that the variables are binary and in the second method, unlike the former models, instead of assuming the variable d _k as binary, we consider it as a fuzzy number in the range of 0 to 1; then, by modeling, we decide to what extent these factors are placed in the input and output category. In both methods, a fuzzy number ranking method is used to create an equivalent definite model.

Suppose that the values of the inputs, outputs and flexible measures are defined as triangular fuzzy numbers: $\begin{matrix} {\tilde{w}}_{kj} = (w_{kj}^{l}, w_{kj}^{m}, w_{kj}^{u}), \forall j, k \\ {\tilde{y}}_{rj} = (y_{rj}^{l}, y_{rj}^{m}, y_{rj}^{u}), \forall j, r \\ {\tilde{x}}_{ij} = (x_{ij}^{l}, x_{ij}^{m}, x_{ij}^{u}), \forall j, i \end{matrix}$

Therefore, according to model (3), the fuzzy model in the first method and assuming the binary variables of the classification is as follows: $\begin{matrix} max \sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{ro} \oplus \sum_{k = 1}^{K} d_{k} γ_{k} {\tilde{w}}_{ko} \\ s . t . \\ \sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{rj} \oplus \sum_{k = 1}^{K} d_{k} γ_{k} {\tilde{w}}_{kj} ⊖ \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij} ⊖ \\ \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} {\tilde{w}}_{kj} \leq (0, 0, 0), \forall j \\ \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} \oplus \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} {\tilde{w}}_{ko} = (1, 1, 1) \\ d_{k} \in {0, 1}, μ_{r}, v_{r}, γ_{k} \geq 0, \forall r, i, k \end{matrix}$ (4)

In order to extract the corresponding definitive model, the ranking function is used. Then, model (4) can be modified to the following model: $\begin{matrix} max T (\sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{ro} \oplus \sum_{k = 1}^{K} d_{k} γ_{k} {\tilde{w}}_{ko}) \\ s . t . \\ T (\sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{rj} \oplus \sum_{k = 1}^{K} d_{k} γ_{k} {\tilde{w}}_{kj} ⊖ \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij} \\ ⊖ \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} {\tilde{w}}_{kj}) \leq 0, \forall j \\ T (\sum_{i = 1}^{m} v_{i} {\tilde{x}}_{io} \oplus \sum_{k = 1}^{K} (1 - d_{k}) γ_{k} {\tilde{w}}_{ko}) = 1, \\ d_{k} \in {0, 1}, μ_{r}, v_{i} γ_{k} \geq 0, \forall r, i, k \end{matrix}$ (5)

According to fuzzy number ranking method in expression (1) and change of variable, δ _k = d _k y _k, model (5) converts to the deterministic model (6). Where in model (6) M is a very large positive number. Model (6) is a binary linear programming model that identifies the type of flexible measures for each unit.

As it is mentioned in the second method, instead of assuming the variable d _k as binary, is considered to be a fuzzy number in the domain of 0 and 1. So assume that this variable is expressed as a triangular fuzzy variable ${\tilde{d}}_{k} = (d_{k}^{l}, d_{k}^{m}, d_{k}^{u})$ . Therefore, the model (3) is converted to the following proposed form: $\begin{matrix} max 0.5 \sum_{r = 1}^{s} μ_{r} (y_{ro}^{m} + 0.5 (y_{ro}^{l} + y_{ro}^{u})) + 0.5 \sum_{k = 1}^{K} δ_{k} (w_{ko}^{m} + 0.5 (w_{ko}^{l} + w_{ko}^{u})) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r} (y_{rj}^{m} + 0.5 (y_{rj}^{l} + y_{rj}^{u})) + 2 \sum_{k = 1}^{K} δ_{k} (w_{kj}^{m} + 0.5 (w_{kj}^{l} + w_{kj}^{u})) - \\ \sum_{i = 1}^{m} v_{i} (x_{ij}^{m} + 0.5 (x_{ij}^{l} + x_{ij}^{u})) - \sum_{k = 1}^{K} γ_{k} (w_{kj}^{m} + 0.5 (w_{kj}^{l} + w_{kj}^{u})) \leq 0, \forall j \\ \sum_{i = 1}^{m} v_{i} (x_{io}^{m} + 0.5 (x_{io}^{l} + x_{io}^{u})) + \sum_{k = 1}^{K} (γ_{k} - δ_{k}) (w_{ko}^{m} + 0.5 (w_{ko}^{l} + w_{ko}^{u})) = 2, \\ 0 \leq δ_{k} \leq {Md}_{k}, \forall k \\ δ_{k} \leq γ_{k} \leq δ_{k} + M (1 - d_{k}), \forall k \\ d_{k} \in {0, 1}, μ_{r}, v_{i}, γ_{k}, δ_{k}, γ_{k} \geq 0, \forall r, i, k \end{matrix}$ (6) $\begin{matrix} max \sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{ro} \oplus \sum_{k = 1}^{K} γ_{k} ({\tilde{d}}_{k} \otimes {\tilde{w}}_{ko}) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r} {\tilde{y}}_{rj} \oplus \sum_{k = 1}^{K} γ_{k} ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kj}) ⊖ \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij} ⊖ \\ \sum_{k = 1}^{K} (γ_{k} {\tilde{w}}_{kj} - γ_{k} ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kj})) \leq (0, 0, 0) \forall j \\ \sum_{i = 1}^{m} (v_{i} {\tilde{x}}_{io} \oplus \sum_{k = 1}^{K} (γ_{k} {\tilde{w}}_{ko} - ({\tilde{d}}_{k} \otimes {\tilde{w}}_{ko})) = (1, 1, 1), \\ 0 \leq d_{k}^{l} \leq d_{k}^{m} \leq d_{k}^{u} \leq 1, \forall r, k \\ μ_{r}, v_{i} γ_{k} \geq 0, \forall r, i, k \end{matrix}$ (7)

Given the existing ranking method in expression (1), multiplication operation and the change of variables $δ_{k}^{u} = γ_{k} d_{k}^{u}$ , $δ_{k}^{m} = γ_{k} d_{k}^{m}$ and $δ_{k}^{l} = γ_{k} d_{k}^{l}$ the definitive model corresponding to model (8) is obtained as model (8).

By solving model (8) and finding optimal solutions, the value of ${\tilde{d}}_{k}$ is obtained. If $γ_{k}^{*} \neq 0$ , set: $d_{k}^{l *} = δ_{k}^{l *} / γ_{k}^{*}, d_{k}^{u *} = δ_{k}^{u *} / γ_{k}^{*} and d_{k}^{m *} = δ_{k}^{m *} / γ_{k}^{*} .$

Therefore, ${\tilde{d}}_{k}^{*} = (d_{k}^{l *}, d_{k}^{m *}, d_{k}^{u *})$ is obtained as a triangular fuzzy number. If $γ_{k}^{*} = 0$ , we set ${\tilde{d}}_{k}^{*} = (0, 0, 0)$ , (the sign * represents the optimal solution). All components of this fuzzy number are between 0 and 1. If $d_{k}^{u *} < 0.5$ and $d_{k}^{l *} < 0.5$ then it can be said that this flexible factor is inclined to the input (output) state. The following is a numerical example of how the results are interpreted.

$\begin{matrix} max 0.5 \sum_{r = 1}^{s} μ_{r} (y_{ro}^{m} + 0.5 (y_{ro}^{l} + y_{ro}^{u})) + 0.5 \sum_{k = 1}^{K} (δ_{k}^{m} w_{ko}^{m} + 0.5 (δ_{k}^{l} w_{ko}^{l} + δ_{k}^{u} w_{ko}^{u})) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r} (y_{rj}^{m} + 0.5 (y_{rj}^{l} + y_{rj}^{u})) + 2 \sum_{k = 1}^{K} (δ_{k}^{m} w_{kj}^{m} + 0.5 ((δ_{k}^{l} w_{kj}^{l} + (δ_{k}^{u} w_{kj}^{u})) - \\ \sum_{i = 1}^{m} v_{i} (x_{ij}^{m} + 0.5 (x_{ij}^{l} + x_{ij}^{u})) - \sum_{k = 1}^{K} γ_{k} (w_{kj}^{m} + 0.5 + (w_{kj}^{l} + w_{kj}^{u})) \leq 0, \forall j (8) \\ \sum_{i = 1}^{m} v_{i} (x_{io}^{m} + 0.5 (x_{io}^{l} + x_{io}^{u})) + \sum_{k = 1}^{K} γ_{k} (w_{ko}^{m} + 0.5 (w_{ko}^{l} + w_{ko}^{u})) - \\ \sum_{k = 1}^{K} (δ_{k}^{m} w_{ko}^{m} + 0.5 (δ_{k}^{l} δ_{ko}^{l} + δ_{k}^{u} w_{ko}^{u})) = 2, \\ 0 \leq δ_{k}^{l} \leq δ_{k}^{m}, \forall k \\ 0 \leq δ_{k}^{m} \leq δ_{k}^{u}, \forall k \\ 0 \leq δ_{k}^{u} \leq γ_{k}, \forall k \\ μ_{r}, v_{i}, γ_{k} \geq 0, \forall r, i, k \end{matrix}$

4 Uniquely identification of fuzzy flexible measures

Two models were presented for the categorization of fuzzy flexible measures in the previous section. Model (6) is in line with the previous methods in the literature of binary decision making for categorizing the flexible measures in the input or output groups, while in model (8) a new method was adopted in which part of the flexible measures can be categorized in the input category and the other part in the output category. Both (6) and (8) models are unit-oriented, meaning that the ways in which flexible measures are categorized vary from one decision making unit to another. This change in the classification in particular causes the creation of units in model (6) with inhomogeneous efficiency factors, whereas the common bundle of efficiency measures for all decision-making units is one of the fundamental principles of data envelopment analysis. Therefore, according to Toloo [17], the decision making to categorize a flexible measure with the maximum votes obtained (majority rule). However, in the case that the difference between the votes is not high, for example when 9 DMUs have confirmed the input nature and 11 units have confirmed the output nature of a flexible measure among 20 DMUs, considering this measure as output essentially disregards the opinion of a large number of units. In result, it does not seem reasonable in all circumstances to comprehensively decide on the type of flexible measures by model (6). It is slightly different in the model (8) by which in the average decision can be calculated. Although the average criterion is not always an appropriate decision, especially in cases where the standard deviation is high, it can be better than the votes’ maximum decision. Therefore, the following equation can be used for a unique decision in model (8): $d_{k} = (\frac{1}{n} \sum_{j = 1}^{n} d_{kj}^{l} *, \frac{1}{n} \sum_{j = 1}^{n} d_{kj}^{m} *, \frac{1}{n} \sum_{j = 1}^{n} d_{kj}^{u} *)$

Where, $(d_{kj}^{l *}, d_{kj}^{m *}, d_{kj}^{u *})$ represents the optimal fuzzy decision related to under evaluation DMU_j and kth flexible measure.

To solve the problem of making DMUs inhomogeneous, Cook and Zhu [7] presented an aggregated model by which a unique classification is given to determine the nature of the flexible measures. One failure of this model is a lack of logical justification and a method that is influenced by major units which, as Cook and Zhu put it in their paper, cannot be justified in practical problems.

In the following, an approach differing from models (6) and (8) is presented which directly provided the comprehensive decision model on each flexible measure, no longer requiring the use of a maximum or average vote.

Our approach to solve the aforementioned problem uses an aggregate version of models (6) and (8). To do so, consider model (9) as the aggregate form of model (2), in which only one model is solved instead of implementing n mathematical programming models.

$\begin{matrix} max \sum_{p = 1}^{n} \sum_{r = 1}^{s} μ_{r}^{p} y_{rp} \\ s . t . \\ \sum_{i = 1}^{m} v_{i}^{p} x_{ip} = 1, \forall p \\ \sum_{r = 1}^{s} μ_{r}^{p} y_{rj} - \sum_{i = 1}^{m} v_{i}^{p} x_{ij} \leq 0, \forall j, p \\ μ_{r}^{p}, v_{i}^{p} \geq 0, \forall r, i, p \end{matrix}$ (8)

Where $v$ ^p and μ ^p; p ∈ {1, …, n}, are input and output weights related to DMU_p.

The following theorem states that an optimal solution obtained from model (9) results in a set of optimal solutions for each model (2) corresponding to different DMUs under evaluation. Therefore, model (9) can be replaced by the set of models (2).

Theorem 1. Let $(μ^{*}, v^{*}) \in ℝ^{n (m + s)}$ be the optimal solution of model (9) and $θ_{p}^{*} = \sum_{r = 1}^{s} {μ_{r}}^{p *} y_{rp}$ , then $θ_{p}^{*} = e_{p}$ , where e _p is the optimal objective value of model (2), when DMU_p is under evaluation.

Proof. Model (9) can be decomposed into n independent model (t = 1, …, n) as following, $\begin{matrix} max \sum_{r = 1}^{s} μ_{r}^{t} y_{rt} \\ s . t . \\ \sum_{r = 1}^{m} v_{r}^{t} x_{it} = 1 \\ \sum_{r = 1}^{s} μ_{r}^{t} y_{rj} - \sum_{i = 1}^{m} v_{i}^{t} x_{ij} \leq 0 \forall j \\ μ_{r}^{t}, v_{i}^{t} \geq 0 \forall r, i \end{matrix}$

Each which are the related model (2) to DMU_t. This proves the theorem. ■

At this point, the unique decision maker model can be obtained by adding a decision maker variable along with flexible measures to model (9) and other considerations similar to model (3). Here again, the problem has been examined from two perspectives. Firstly, similar to model (6), we assumed decision variables as binary, and secondly, fuzzy decisions were considered similar to model (8). The model corresponding to binary decisions is as follows: $\begin{matrix} max \sum_{p = 1}^{n} (\sum_{r = 1}^{s} μ_{r}^{p} {\tilde{y}}_{rp} \oplus \sum_{k = 1}^{K} d_{k} γ_{k}^{p} {\tilde{w}}_{kp}) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r}^{p} {\tilde{y}}_{rj} \oplus \sum_{k = 1}^{K} d_{k} γ_{k}^{p} {\tilde{w}}_{kj} ⊖ \sum_{i = 1}^{m} v_{i}^{p} {\tilde{x}}_{ij} ⊖ \\ \sum_{k = 1}^{K} (1 - d_{k}) γ_{k}^{p} {\tilde{w}}_{kj} \leq (0, 0, 0), \forall j, p \\ \sum_{i = 1}^{m} v_{i}^{p} {\tilde{x}}_{ip} \oplus \sum_{k = 1}^{K} (1 - d_{k}) γ_{k}^{p} {\tilde{w}}_{kp} = (1, 1, 1), \forall p \\ d_{k} \in {0, 1}, μ_{r}^{p}, v_{i}^{p}, γ_{k}^{p} \geq 0, \forall r, i, k, p \end{matrix}$ (9) $\begin{matrix} max 0.5 \sum_{p = 1}^{n} (\sum_{r = 1}^{s} μ_{r}^{p} (y_{rp}^{m} + 0.5 (y_{rp}^{l} + y_{rp}^{u})) \sum_{k = 1}^{K} δ_{k}^{p} (w_{kp}^{m} + 0.5 (w_{kp}^{l} + w_{kp}^{u}))) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r}^{p} (y_{rj}^{m} + 0.5 (y_{rj}^{l} + y_{rj}^{u})) + 2 \sum_{k = 1}^{K} δ_{k}^{p} (w_{kj}^{m} + 0.5 (w_{kj}^{l} + w_{kj}^{u})) - \\ \sum_{i = 1}^{m} v_{i}^{p} (x_{ij}^{m} + 0.5 (x_{ij}^{l} + x_{ij}^{u})) - \sum_{k = 1}^{K} γ_{k}^{p} (w_{kj}^{m} + 0.5 (w_{kj}^{l} + w_{kj}^{u})) \leq 0, \forall j, p \\ \sum_{i = 1}^{m} v_{i}^{p} (x_{ip}^{m} + 0.5 (x_{ip}^{l} + x_{ip}^{u})) + \sum_{k = 1}^{K} (γ_{k}^{p} - δ_{k}^{p}) (w_{kp}^{m} + 0.5 (w_{kp}^{l} + w_{kp}^{u})) = 2, \forall p \\ 0 \leq δ_{k}^{p} \leq M d_{k}, \forall k \\ δ_{k}^{p} \leq γ_{k}^{p} \leq δ_{k}^{p} + M (1 - d_{k}), \forall k \\ d_{k} \in {0, 1}, μ_{r}^{p}, v_{i}^{p}, γ_{k}^{p}, δ_{k}^{p} \geq 0, \forall r, i, k \end{matrix}$ (10)

Here, the decision variable d _k is considered independent from p, meaning none of the DMUs can affect the classification. By changing the variable $δ_{k}^{p} = d_{k} γ_{k}^{p}$ and using the ranking function, according to the previous section, model (11), was obtained.

Note that in the presented aggregated model, model (11), $d_{k}^{*} = 0$ results in the Kth flexible measure being chosen as input (or output if $d_{k}^{*} = 0$ ), similar to previous sections. Now, by fuzzifying the flexible measures, we obtain the following model in which part of the flexible measures can be considered as input and the remainder as output for all units.

$\begin{matrix} max \sum_{p = 1}^{n} (\sum_{r = 1}^{s} μ_{r}^{p} {\tilde{y}}_{p} \oplus \sum_{k = 1}^{K} γ_{k}^{p} ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kp})) \\ s . t . \\ \sum_{r = 1}^{s} μ_{r}^{p} {\tilde{y}}_{rj} \oplus \sum_{k = 1}^{K} γ_{k}^{p} ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kj})) ⊖ \sum_{i = 1}^{m} v_{i}^{p} {\tilde{x}}_{ij} ⊖ \\ \sum_{k = 1}^{K} (γ_{k}^{p} {\tilde{w}}_{kj} ⊖ ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kj})) \leq (0, 0, 0), \forall j, p \\ \sum_{i = 1}^{m} v_{i}^{p} {\tilde{x}}_{ip} \oplus \sum_{k = 1}^{K} (γ_{k}^{p} {\tilde{w}}_{kp} ⊖ ({\tilde{d}}_{k} \otimes {\tilde{w}}_{kp})) = (1, 1, 1), \forall p \\ 0 \leq d_{k}^{l} \leq d_{k}^{m} \leq d_{k}^{u} \leq 1, \forall k \\ μ_{r}^{p}, v_{i}^{p}, γ_{k}^{p} \geq 0, \forall r, i, k, p \end{matrix}$ (11)

Where model (12) results in the deterministic model (13) by the change of the variables $δ_{k}^{p, l} = γ_{k}^{p}, d_{k}^{l}, δ_{k}^{p, m} = γ_{k}^{p} d_{k}^{m}$ and $δ_{k}^{p, u} = γ_{k}^{p} d_{k}^{u}$

$\begin{matrix} max 0.5 \sum_{p = 1}^{n} (\sum_{r = 1}^{s} μ_{r}^{p} (y_{rp}^{m} + 0.5 (y_{rp}^{l} + y_{rp}^{u})) + \sum_{k = 1}^{K} (δ_{k}^{p, m} w_{kp}^{m} + 0.5 (δ_{k}^{p, l} w_{kp}^{l} + δ_{k}^{p, u} w_{kp}^{u}))) \\ s . t . \\ \sum_{i = 1}^{s} μ_{r}^{p} (y_{rj}^{m} + 0.5 (y_{rj}^{l} + y_{rj}^{u})) + 2 \sum_{k = 1}^{K} (δ_{k}^{p, m} w_{kj}^{m} + 0.5 (δ_{k}^{p, l} w_{kj}^{l} + δ_{k}^{p, u} w_{kj}^{u})) - \\ \sum_{i = 1}^{m} v_{i}^{p} (x_{ij}^{m} + 0.5 (x_{ij}^{l} + x_{ij}^{u})) - \sum_{k = 1}^{K} γ_{k}^{p} (w_{kj}^{m} + 0.5 (w_{kj}^{l} + w_{kj}^{u})) \leq 0, \forall j, p \\ \sum_{i = 1}^{m} v_{i}^{p} (x_{ip}^{m} + 0.5 (x_{ip}^{l} + x_{ip}^{u})) + \sum_{k = 1}^{K} γ_{k}^{p} (w_{kp}^{m} + 0.5 (w_{kp}^{l} + w_{kp}^{u})) - \\ \sum_{k = 1}^{K} (δ_{k}^{p, m} w_{kp}^{m} + 0.5 (δ_{k}^{p, l} w_{kj}^{l} + δ_{k}^{p, u} w_{kp}^{u})) = 2, \forall p \\ 0 \leq δ_{k}^{p, l} \leq δ_{k}^{p, m}, \forall k, p \\ 0 \leq δ_{k}^{p, m} \leq δ_{k}^{p, u}, \forall k, p \\ 0 \leq δ_{k}^{p, u} \leq γ_{k}^{p}, \forall k, p \\ μ_{r}^{p}, v_{i}^{p}, γ_{k}^{p} \geq 0, \forall r, i, k, p \end{matrix}$ (12)

Since $0 \leq d_{k}^{l} \leq d_{k}^{m} \leq d_{k}^{u} \leq 1$ , the three-to-five constraints sets can be obtained by multiplying these inequalities in $γ_{k}^{p}$ .

Now, ${\tilde{d}}_{k}^{*}$ , which represents the optimal value of the unique fuzzy classifier of the kth flexible measure, is obtained by solving model (13) as follows: $\sum_{p = 1}^{n} \sum_{k = 1}^{K} d_{k}^{l} γ_{k}^{p} w_{kp}^{l} = \sum_{p = 1}^{n} \sum_{k = 1}^{K} δ_{k}^{p, l} w_{kp}^{l}$

So, $\sum_{k = 1}^{K} d_{k}^{l} \sum_{p = 1}^{n} γ_{k}^{p} w_{kp}^{l} = \sum_{k = 1}^{K} \sum_{p = 1}^{n} δ_{k}^{p, l} w_{kp}^{l}$

Since each d _k is independent of each other, $d_{k}^{l} = \frac{\sum_{p = 1}^{n} δ_{k}^{p, l} w_{kp}^{l}}{\sum_{p = 1}^{n} γ_{k}^{p} w_{kp}^{l}}$

Thus $d_{k}^{u}$ and $d_{k}^{m}$ are found to be similar. Thus, the corresponding fuzzy variable for each k = 1, …, K is: ${\tilde{d}}_{k}^{*} = (\frac{\sum_{p = 1}^{n} δ_{k}^{p, l} w_{kp}^{l}}{\sum_{p = 1}^{n} γ_{k}^{p, *} w_{kp}^{l}}, \frac{\sum_{p = 1}^{n} δ_{k}^{p, m *} w_{kp}^{m}}{\sum_{p = 1}^{n} γ_{k}^{p, *} w_{kp}^{m}}, \frac{\sum_{p = 1}^{n} δ_{k}^{p, u *} w_{kp}^{m}}{\sum_{p = 1}^{n} γ_{k}^{p, *} w_{kp}^{u}})$

Furthermore, ${\tilde{d}}_{k}^{*} = (d_{k}^{l *} d_{k}^{m *}, d_{k}^{u *})$ is obtained as a triangular fuzzy number. All components of this fuzzy number are between zero and one. If $d_{k}^{l *} > 0.5 (d_{k}^{u *} < 0.5)$ , we can say that this flexible measure tends to the output (input) state.

5 Numerical example

Consider the 25 DMUs with an input, an output, and a flexible qualitative factor which, according to experts, is a fuzzy triangle. The corresponding data is given in Table 1. Here inputs and outputs are precise values; then we included them as triangular fuzzy numbers with equal values in the proposed models.

Table 1
Input, Output and Flexible Factor values of the example.

# DMU Input Outputs Flexible factor

x y ₁ y ₂ w ^l w ^m w ^u

1 18 14 8 1 1 1

2 26 38 16 1 2 3

3 30 44 3 7 8 9

4 16 28 3 4 5 6

5 49 91 8 1 2 3

6 26 35 4 7 8 9

7 42 70 12 3 4 5

8 98 20 2 2 3 4

9 52 60 1 1 1 1

10 58 80 17 1 2 3

11 27 39 19 6 7 8

12 43 18 23 2 3 4

13 32 37 17 1 2 3

14 57 19 11 8 9 10

15 19 23 9 8 9 10

16 14 25 15 1 2 3

17 29 34 12 5 6 7

18 28 16 18 5 6 7

19 37 49 32 3 4 5

20 63 38 23 2 3 4

21 42 33 14 6 7 8

22 34 74 19 4 5 6

23 44 49 21 8 9 10

24 39 53 13 1 2 3

25 15 22 17 2 3 4

# DMU	Input	Outputs	Flexible factor
1	18	14	8	1	1	1
2	26	38	16	1	2	3
3	30	44	3	7	8	9
4	16	28	3	4	5	6
5	49	91	8	1	2	3
6	26	35	4	7	8	9
7	42	70	12	3	4	5
8	98	20	2	2	3	4
9	52	60	1	1	1	1
10	58	80	17	1	2	3
11	27	39	19	6	7	8
12	43	18	23	2	3	4
13	32	37	17	1	2	3
14	57	19	11	8	9	10
15	19	23	9	8	9	10
16	14	25	15	1	2	3
17	29	34	12	5	6	7
18	28	16	18	5	6	7
19	37	49	32	3	4	5
20	63	38	23	2	3	4
21	42	33	14	6	7	8
22	34	74	19	4	5	6
23	44	49	21	8	9	10
24	39	53	13	1	2	3
25	15	22	17	2	3	4

The results of the implementation of models (6) and (8) are presented in Table 2. The binary column is related to model (6) and the fuzzy column is related to model (8). The efficiency is also calculated by assuming that the flexible measure for all units as input (d = 0) or as output (d = 1). In the last row of Table 2, the mean of the flexible factor is obtained. $\bar{d} = (0.28, 0.34, 0.37)$ and since the range of this fuzzy number is less than 0.5, the flexible factor tends to zero; in result, the flexible factor tends toward being an input on average. The comparison of the average performance at the end of this table also confirms the same. In the last row of Table 2 we can see that when the flexible factor is considered as input, the average efficiency is 0.77, and this value is larger than 0.69, when compared with the flexible factor as output. This is also confirming that the flexible measure tends to be input.

Table 2

Computational results of model (7) and (9)

#DMU	Flexible factor				Efficiency
	Binary	Fuzzy
	d	d ^l	d ^m	d ^u	d = 1	d = 0	$\tilde{d} = \bar{d}$
1	0	0.5	0.5	0.5	0.43	0.94	0.99
2	0	0	0	0	0.74	1	1
3	1	0.5	0.5	0.5	0.84	0.67	0.68
4	1	0	0.53	1	1	0.8	0.8
5	0	0	0	0	0.85	1	1
6	1	0.5	0.5	0.5	0.85	0.62	0.64
7	0	0.5	0.5	0.5	0.77	0.82	0.86
8	0	0.5	0.5	0.5	0.11	0.14	0.17
9	0	0	0	0	0.53	1	1
10	0	0	0	0	0.63	1	1
11	1	0.5	0.5	0.5	0.91	0.76	0.76
12	0	0.5	0.5	0.5	0.47	0.92	0.96
13	0	0	0	0.24	0.6	1	1
14	1	0.5	0.5	0.5	0.36	0.18	0.28
15	1	0	1	1	1	0.6	0.6
16	0	0	0	0	1	1	1
17	1	0.5	0.5	0.5	0.7	0.57	0.66
18	1	0.5	0.5	0.5	0.72	0.57	0.7
19	0	0	0	0	0.79	1	1
20	0	0.5	0.5	0.5	0.34	0.9	0.96
21	1	0.5	0.5	0.5	0.51	0.4	0.63
22	0	0	0	0	1	1	1
23	1	0.5	0.5	0.5	0.69	0.57	0.67
24	0	0.5	0.5	0.5	0.62	0.88	0.92
25	1	0	0	0	1	1	1
Mean		0.28	0.34	0.37	0.69	0.77	0.83

Comparison of the values of (d ^l, d ^m, d ^u) of each unit with binary d value shows that when the domain of the fuzzy number (d ^l, d ^m, d ^u) is less than 0.5, the value of binary d is zero, and vice versa; when the domain of the fuzzy number (d ^l, d ^m, d ^u) is greater than 0.5, the binary value of d is equal to one (units 4 and 15). In the case where the domain of the fuzzy number (d ^l, d ^m, d ^u) is not less than 0.5 or greater than 0.5, the binary value of d may be 0 or 1. For example, the flexible measure for unit 1 has (d ^l, d ^m, d ^u) = (0.5, 0.5, 0.5) and is considered as input binary mode, whereas for unit 3 with the fuzzy value equal to unit 1, the output is evaluated. Finally, with the unique decision $\bar{d}$ , the type of fuzzy flexible measure can be uniformly determined for all units. By putting the obtained values $\bar{d}$ for the variables δ and γ in model (9), the efficiency of each unit is obtained in accordance with the last column of Table 2.

The results of the implementation of model (11) show that d ^* = 1, i.e. the flexible measure is considered generally as an output, while the result of model (13) is ${\tilde{d}}_{k}^{*} = (0.17, 0.56, 0.80)$ , indicating that there is only a small difference between the input or output nature of the variable in fuzzy terms.

It must be note that in previous studies [12, 14], decisions are based on majority rule. Here, majority rule indicates an input nature for the flexible measure, since the number of units that tend to consider the flexible measure as an output is 11 while the number of units that tend to consider the flexible measure as input is 14, according to Table 2. Binary-based decisions prove to be difficult due to the nearly insignificant difference between these two values. However, aggregate decision making is based on the output-orientation of this measure, according to model (11). Nevertheless, there is only a minor difference between the input and output nature of this measure according to the fuzzy aggregate model (13), which is similar to the binary results of model (6). Of course, it is not an appropriate approach to compare the efficiency of the units resulting from aggregate and individual models because in the individual mode, as stated above, the choice of the input or output nature of each unit is self-determined, which ultimately leads to the creation of inhomogeneous units. Therefore, aggregate models are suitable for solving this problem and the decision to categorize the measure entirely in the distinct categories of input-input, or to categorize a part of the measure as input and the other as output, is a managerial decision that can be made according to the structure of the problem.

6 Conclusion

In some practical cases, we are faced with factors with unclear natures in terms of being input or output. These factors are known as flexible measures. In this article, the condition of these fuzzy factors was studied. Following this, a model was presented in which some of these factors could be considered as output and the other part as input. Therefore, in this case, the strict classification is not achieved in two independent inputs and outputs. For this purpose, the triangular fuzzy decision variable has been used instead of the binary variables, and the corresponding deterministic model has been obtained that uses a unique decision about the type of flexible measures that can be obtained by combining the decisions of all DMUs. The results of these models in a numerical example illustrate the improvement of score efficiencies by assuming the fuzziness of the classification variable.

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