Fuzzy efficiency measures in data envelopment analysis with interactive fuzzy variables

1 Introduction

DEA initially proposed by Charnes et al. [1] is a non-parametric technique for measuring and evaluating the relative efficiencies of a set of DMUs with multi-inputs and multi-outputs, then it was further developed by Banker et al. [2]. DEA combines and transforms multiple inputs and outputs into a single efficiency index. Efficiency will be calculated relative to a homogeneous collection of decision-making units (DMUs). Nowadays, a great variety of DEA efficiency models [1–4] have been reported and applied to various industrial and non-industrial contexts, such as bank [5–7], education [8, 9], hospital [10, 11], energy [12], etc.

Recently, the fuzzy set theory has been used to quantify imprecise and vague data in DEA models. We can find several fuzzy approaches to assess efficiency in the DEA literature. The fuzzy DEA(FDEA) methods can be classified into the following five approaches: (1) The tolerance approach [13]; (2) The α-level based approach [14–16]; (3) The fuzzy ranking approach [17, 18]; (4) The possibility approach [19, 20]; (5) Other developments [21–23].

In classical DEA (fuzzy DEA) models, the way to combine multiple variables (or fuzzy variables) is to use a linear weighted sum. It requires an assumption that there are no interactions among the contributions from each individual information source and the joint contribution is simple weighted sum of contributions from each individual information source. But in practice, the variables (fuzzy variables) are usually strongly related and there are interactions among the variables (fuzzy variables). In classical DEA, Some researchers have proposed several models to solve this problem. Adler and Golany [24, 25] suggested to use the principal component analysis(PCA) to produce uncorrelated linear combinations of original inputs and outputs, and construct a PCA based DEA model. Adler and Yazhemsky [26] concluded that PCA– DEA outperforms classical DEA by comparing their discrimination performance in a simulation exercise. Independent component analysis (ICA) is another information aggregation tool to resolve the issue of correlated variables. Using ICA to extract independent variables (inputs or outputs) in DEA efficiency measurement [27] can partially overcome the effect of correlated variables. But PCA based DEA and ICA based DEA both have high computation complexities, and the variables (inputs, outputs) in the two DEA models are not practical variables of the DMUs, the results of efficiency evaluations have fewer guides for production activities.

Taking the inherent interaction among the variables (inputs or outputs) into account, Ai-bing Ji and his colleagues [28] introduced data envelopment analysis with interactive variables (inputs or outputs), it uses Choquet integral to aggregate the multiple inputs and outputs into a single efficiency index, it is a generalization of the classical DEA. But for classical fuzzy DEA, there are no researches on DEA with interactive fuzzy variables(fuzzy inputs or fuzzy outputs).

In this paper, we extend fuzzy DEA to DEA with interactive fuzzy variables (inputs or outputs). Using fuzzy Choquet integral as an aggregating tool, we provide a novel fuzzy efficiency evaluation model with interactive fuzzy inputs (fuzzy outputs), the classical fuzzy DEA model [14] is a special case of our proposed fuzzy model. At the end of the paper, we apply the proposed fuzzy DEA model in efficiency evaluation.

2 Preliminary

In this section, preliminaries for DEA and Choquet integral are presented.

In order to reduce the impact of the interactions between variables(inputs or outputs) on the efficiency evaluation, Ai-bing Ji and his colleagues [28] introduced data envelopment analysis with interactive variables (inputs or outputs), it uses Choquet integral to aggregate the multiple inputs and outputs into a single efficiency index, it is a generalization of the classical DEA.

For a collection of DMUs, the attribute set of inputs is X = {x₁,  x₂, …,  x _s }, the attribute set of outputs is Y = {y₁,  y₂, …,  y _m }. Fuzzy measure μ (A) represent the joint efficiency of attribute set A ⊂ X, fuzzy measure ν (C) represent the joint efficiency of attribute set C ⊂ Y. The efficiency of any DMU is obtained as the maximum of the ratio of the aggregate inputs to aggregate outputs, where the aggregate inputs(or outputs) are calculated by the Choquet integral. The fuzzy measure μ and ν can be determined by the following optimization problem ((CH-CCR)^I Model).

Max { h 0 = ( C ) ∫ g 0 d ν ( C ) ∫ f 0 d μ } . s . t . : { ( C ) ∫ g k d ν ( C ) ∫ f k d ν ≤ 1 , k = 1 , 2 , … , n ; 0 ≤ μ ( A ) ≤ μ ( B ) , A ⊆ B , A , B ∈ P ( X ) ; 0 ≤ ν ( C ) ≤ ν ( D ) , C ⊆ D , C , D ∈ P ( X ) ;(4)

The efficiency ratio ranges from zero to one, DMU _k is considered relatively efficient if it receives a score of one. The result of the (CH-CCR)^I is the determination of the hyper-planes that define an envelope surface. DMUs that lie on the surface determine the envelope and are deemed efficient, whilst those that do not are deemed inefficient.

We utilize the following transformations: t = 1 ( C ) ∫ f k d μ , ( t > 0 ) , ω = t ν , λ = t μ(5)

The set functions λ : X → [0, + ∞), ω : Y → [0, + ∞) are all also nonnegative monotone fuzzy measures.

By using the transformations (5), the (CH-CCR)^I model (4) can be changed into the following equivalent CH-CCR model: { E 0 ( f k , g k ) = max { ( C ) ∫ g 0 d ω } s . t . ( C ) ∫ g k ( y ) d ω - ( C ) ∫ f k ( x ) d λ ≤ 0 , k = 1 , 2 , … , N ; ( C ) ∫ f 0 ( x ) d λ = 1 ; 0 ≤ ω ( A ) ≤ ω ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ λ ( A ) ≤ λ ( B ) , for C ⊆ D , C , D ∈ P ( X ) ;(6)

When the fuzzy measures λ : X → [0, + ∞), ω : Y → [0, + ∞) are additive fuzzy measures, the CH-CCR model (6) degrade into the CCR model. CH-CCR model (6) are in essence a linear programming by introducing an alternate calculation formula [29].

Definition 4. The performance of DMU₀ is efficient iff there exist the optimal solution μ on P(X), v on P(Y) in (CH-CCR)^I Model, such that h 0 = ( C ) ∫ g 0 d ν ( C ) ∫ f 0 d μ = 1 .

The Definition 4 is equivalent to the following definition.

Definition 4’ The performance of DMU0 is efficient iff there exist the optimal solution λ on P(X), ω on P(Y), such that h₀ = (C) ∫g₀dω = 1, (C) ∫f₀dλ = 1.

For a collection of DMUs DMU = {DMU₁,  DMU₂, …,  DMU _N }, the attribute set of inputs is x = {x₁,  x₂, …,  x _s }, the attribute set of outputs is y = {y₁,  y₂, …,  y _m }. Taking into account the interaction among the multiple inputs, the fuzzy measure μ ({x _i }) represents the weight of input index x _i , the fuzzy measure μ (A) (A isn’t a single point set) represents the joint weight of attribute set A ⊂ X; And it is similar to the fuzzy measure ν on P (Y). Let F ˜ k ( x i ) denotes the fuzzy numerical information that k-th DMU obtained from input attribute x _i , G ˜ k ( y j ) denotes the fuzzy numerical information that k-th DMU obtained from output attribute y _j . The k-th DMU uses the input bundle F ˜ k ( x ) = ( F ˜ k ( x 1 ) , F ˜ k ( x 2 ) , … , F ˜ k ( x s ) ) to produce the output bundle G ˜ k ( y ) = ( G ˜ k ( y 1 ) , G ˜ k ( y 2 ) , … , G ˜ k ( y m ) ) . The proposed measure of the fuzzy efficiency of DMU _{k 0} is obtained as the maximum of the ratio of its aggregate fuzzy outputs to its aggregate fuzzy inputs, where the aggregate inputs (or outputs) are calculated by the fuzzy Choquet integral. The fuzzy measure μ and ν can be determined by the following optimization problem ((CH-FDEA) Model). { E ˜ 0 ( F ˜ k , G ˜ k ) = max ( ( C ) ∫ G ˜ 0 ( y ) d ν ) ; s . t . ( C ) ∫ G ˜ k ( y ) d ν - ( C ) ∫ F ˜ k ( x ) d μ < ˜ 0 ˜ , k = 1 , 2 , … , N ; ( C ) ∫ F ˜ 0 ( x ) d μ = ˜ 1 ; 0 ≤ ν ( A ) ≤ ν ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ μ ( C ) ≤ μ ( D ) , for C ⊆ D , C , D ∈ P ( X ) ;(7)

For a given level α (0 ≤ α ≤ 1), the α - cuts of F ˜ k ( x i ) and G ˜ k ( y j ) are defined as ( F ˜ k ( x i ) ) α = { f k ( x i ) | μ F ˜ k ( x i ) ( f k ( x i ) ) ≥ α } = [ f k α L ( x i ) , f k α U ( x i ) ] ( G ˜ k ( y j ) ) α = { g k ( y j ) | μ G ˜ k ( x j ) ( g k ( y j ) ) ≥ α } = [ g k α L ( y j ) , g k α U ( y j ) ]

Based on Zadeh’s extension principle [30], the membership function of the fuzzy efficiency of DMU₀ is defined as μ E ˜ 0 ( z ) = sup f k ( x ) , g k ( y ) min { μ F ˜ k ( x i ) ( f k ( x i ) ) , μ G ˜ k ( y j ) ( g k ( y j ) ) , ∀ i , j , k | z = E 0 ( f k , g k ) }

Where E₀ (f _k ,  g _k ) is Defined in (6). We can construct the membership function of the fuzzy efficiency E ˜ 0 by finding the lower and upper bounds of the α - cut of E ˜ 0 . The lower and upper bounds of the α - cut of E ˜ 0 can be obtained by solving the following optimization

( E ˜ 0 ) α L = min f k α L ( x ) ≤ f k ( x ) ≤ f k α U ( x ) g k α L ( y ) ≤ g k ∀ , k ( y ) ≤ g k α U ( y ) { E 0 ( f k , g k ) = max { ( C ) ∫ g 0 ( y ) d ν } ; s . t . ( C ) ∫ g k ( y ) d ν - ( C ) ∫ f k ( x ) d μ ≤ 0 , k = 1 , 2 , … , N ; ( C ) ∫ f 0 ( x ) d μ = 1 ; 0 ≤ ν ( A ) ≤ ν ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ μ ( C ) ≤ μ ( D ) , for C ⊆ D , C , D ∈ P ( X ) ;(8a)

( E ˜ 0 ) α U = max f k α L ( x ) ≤ f k ( x ) ≤ f k α U ( x ) g k α L ( y ) ≤ g k ∀ , k ( y ) ≤ g k α U ( y ) { E 0 ( f k , g k ) = max { ( C ) ∫ g 0 ( y ) d ν } ; s . t . ( C ) ∫ g k ( y ) d ν - ( C ) ∫ f k ( x ) d μ ≤ 0 , k = 1 , 2 , … , N ; ( C ) ∫ f 0 ( x ) d μ = 1 ; 0 ≤ ν ( A ) ≤ ν ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ μ ( C ) ≤ μ ( D ) , for C ⊆ D , C , D ∈ P ( X ) ;(8b)

It is obvious that models (8a) and (8b) can be transformed into the following equivalent CH-CCR-like model

{ ( E ˜ 0 ) α L = max { ( C ) ∫ g 0 α L ( y ) d ν } ; s . t . ( C ) ∫ g k α U ( y ) d ν - ( C ) ∫ f k α L ( x ) d μ ≤ 0 ; ( C ) ∫ f 0 α L ( x ) d μ ≤ 1 ; ( C ) ∫ f 0 α U ( x ) d μ ≥ 1 ; 0 ≤ ν ( A ) ≤ ν ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ μ ( C ) ≤ μ ( D ) , for C ⊆ D , C , D ∈ P ( X ) ;(9a)

{ ( E ˜ 0 ) α U = max { ( C ) ∫ g 0 α L ( y ) d ν } ; s . t . ( C ) ∫ g k α L ( y ) d ν - ( C ) ∫ f k α U ( x ) d μ ≤ 0 ; ( C ) ∫ f 0 α L ( x ) d μ ≤ 1 ; ( C ) ∫ f 0 α U ( x ) d μ ≥ 1 ; 0 ≤ ν ( A ) ≤ ν ( B ) , for A ⊆ B , A , B ∈ P ( Y ) ; 0 ≤ μ ( C ) ≤ μ ( D ) , for C ⊆ D , C , D ∈ P ( X ) ;(9b)

Theorem 1. ( E ˜ 0 ) α L ( E ˜ 0 ) α U satisfy the following conditions (1) ( E ˜ 0 ) α L ≤ ( E ˜ 0 ) α U (2) For every level α (0 ≤ α ≤ 1), ( E ˜ 0 ) α L ↑ α ; ( E ˜ 0 ) α U ↓ α .

Proof. (1) For simplicity, the feasible solution set of (9a) is denoted by D α L , the feasible solution set of (9b) is denoted by D α U . By the monotonity of the Choquet integral [29], for every α (0 ≤ α ≤ 1), k (k = 1,  2, …,  N), ( c ) ∫ g k α L ( y ) d ν ≤ ( c ) ∫ g k α U ( y ) d ν and ( c ) ∫ f k α L ( y ) d μ ≤ ( c ) ∫ f k α U ( y ) d μ , and we can easily prove that D α L ⊂ D α U , therefore ( E ˜ 0 ) α L ≤ ( E ˜ 0 ) α U .

(2) We first prove ( E ˜ 0 ) α L ↑ α. For any α₁, α₂, 0 ≤ α₁ ≤ α₂ ≤1, f k α 1 L (x) ≤ f k α 2 L (x), g k α 1 L (x) ≤ g k α 2 L (x), f k α 1 U (y) ≥ f k α 2 U (y), g k α 1 U (y) ≥ g k α 2 U (y), then ∫ g k α 1 U (y) dμ ≥ ∫ g k α 2 U (y) dμ, ∫ f k α 1 L (x) dν ≤ ∫ f k α 2 L (x) dν, Since for every (ν,  μ) ∈ D α 1 L , ( C ) ∫ g k α 2 U (y) dν - ( C ) ∫ f k α 2 L (x) dμ ≤ ( C ) ∫ g k α 1 U (y) dν - ( C ) ∫ f k α 1 L (x) dμ ≤ 0 ; ( C ) ∫ f 0 α 2 L (x) dμ ≤ ( C ) ∫ f 0 α L (x) dμ ≤ 1, ( C ) ∫ f 0 α 2 U (y) dν ≥ ( C ) ∫ f 0 α 1 U (y) dν ≥ 1, then (ν,  μ) ∈ D α 2 L , thus D α 1 L ⊂ D α 2 L . Therefore ( E ˜ 0 ) α 1 L = max D α 1 L { ∫ g 0 α 1 L ( x ) d ν } ≤ max D α 2 L { ∫ g 0 α 2 L ( x ) d ν } = ( E ˜ 0 ) α 2 L , that is ( E ˜ 0 ) α L ↑ α.

The same to prove that ( E ˜ 0 ) α U ↓ α .

Following from Theorem 1, ( E ˜ 0 ) α L and ( E ˜ 0 ) α U can form a closed interval [ ( E ˜ 0 ) α L , ( E ˜ 0 ) α U ] , and [ ( E ˜ 0 ) α L , ( E ˜ 0 ) α U ] is a nested structures with respect to α (0 ≤ α ≤ 1). So E ˜ 0 = ∪ 0 ≤ α ≤ 1 α · [ ( E ˜ 0 ) α L , ( E ˜ 0 ) α U ] is a fuzzy number.

If all inputs and outputs are triangular fuzzy numbers, for level α = 1, Equation (8a) and (8b) degrade into CH-CCR model (6).

It is obvious that the efficiency measure is less than 1, that is, the fuzzy efficiency measures E ˜ i satisfies μ E ˜ i ( 1 ) ≤ 1 (i = 1,  2, …,  N).

Definition 5. For the evaluated DMU₀, if μ E ˜ 0 ( 1 ) = 1 , then DMU₀ is efficient; if μ E ˜ 0 ( 1 ) < 1 , then DMU₀ is inefficient.

Theoretically, we derive fuzzy efficiency measures for every DMUs, a subsequent task is to rank the fuzzy efficiency measures to determine the better ones. There are several methods for ranking fuzzy numbers [31, 32]. However, most of them require the membership functions of the fuzzy numbers to be ranked. The method of Chen and Klein [32] rank the fuzzy efficiency measures based on the α _i  - cut, but this order isn’t a total ordering. In this paper, we utilize the total ordering [33] to rank the DMUs based on α _i  - cut of the fuzzy efficiency measures.

According to the ranking method of Wang [33], we select an upper dense sequence S ={ α _i |i = 1,  2, …  } in [0,1] and use them as the level to solve the programming (9a) and (9b), we can derive the α _i  - cut of the fuzzy efficiency measure E ˜ i , denoted by ( E ˜ i ) α i = [ l i , r i ] . Let c_2i-1 = l _i  + r _i ,  c_2i = r _i  - l _i , the ranking of E ˜ i can be determined by c _i [33]. In practice, we can only select a subset of S to compute c _i and rank E ˜ i according to c _i . The flowchart of the proposed method is given in Fig. 1.

OPEN IN VIEWER

Fig.1

The flowchart of CH-FDEA model.

In the following, we provide a numerical example which have frequently appeared in the related literature to illustrate the performance of the proposed models.

Example 1. A numerical experiment is presented to illustrate the proposed approach. The example with two fuzzy inputs and two fuzzy outputs is illustrated in Table 1. These fuzzy inputs and fuzzy outputs have symmetrical triangular membership functions, a special case of trapezoidal membership functions. The membership functions are denoted by (c; d) where c is the center and d is the spread.

In this example, we use λ-fuzzy measure μ ({ x _i  }) to represent the weight of input index x _i , the fuzzy measure μ (A) (A isn’t a single point set) represents the joint weight of attribute set A ⊂ X.

We select an upper dense sequence S = { 1 , 1 2 , 1 3 , 2 3 , 1 4 , 3 4 , 1 5 , 2 5 , 3 5 , 4 5 , 1 6 , 5 6 , … … } in [0,1] and use α in S as level to solve the programming (9a) and (9b), the results are calculated by LINGO10. We can derive the α - cut of the fuzzy efficiency measure. For simplicity, we only list the α - cut of the fuzzy efficiency measure (α∈ { α|α = 0,  0.1,  0.2,  0.3, …,  0.9,  1  }), the results are given in Table 2.

When λ = 0, the λ-fuzzy measure degenerate into measure and CH-FDEA model degenerate into classical FDEA [14]. The α - cut of the fuzzy efficiency measures are given in Table 3.

From Tables 2 and 3, DMU2, DMU4 and DMU5 are efficient, DMU3 is inefficient, they are the same in CH-FDEA and FDEA [14]. But because the interactions among the fuzzy variables, DMU 1 is inefficient in CH-FDEA and is efficient in FDEA [14]. And the ranking of fuzzy efficiencies [33] are different, the ranking of fuzzy efficiency E ˜ i ( i = 1 , 2 , … , 5 ) in CH-FDEA is as follows E ˜ 4 ≻ E ˜ 5 ≻ E ˜ 2 ≻ E ˜ 3 ≻ E ˜ 1 , but in FDEA [14] is E ˜ 4 ≻ E ˜ 5 ≻ E ˜ 1 ≻ E ˜ 2 ≻ E ˜ 3 .

Example 2. In this section, we use the numerical example of Dia, Mohamed [34] to demonstrate the applicability and efficiency of the proposed CH-FCCR model. Consider Dia, Mohamed’s example presented in Table 4 with 10 DMUs that consume three fuzzy inputs to obtain two fuzzy outputs.

Based on the fuzzy training data given in Table 4, we select an upper dense sequence S = { 1 , 1 2 , 1 3 , 2 3 , 1 4 , 3 4 , 1 5 , 2 5 , 3 5 , 4 5 , 1 6 , 5 6 , … … } in [0,1] and use the level α in S as level to solve the programming (9a) and (9b), we can derive the α - cut of the fuzzy efficiency measure. For simplicity, we only list the α - cut of the fuzzy efficiency measure (α∈ { α|α = 0,  0.1,  0.2,  0.3, …,  0.9,  1  }), the results are given in Table 5.

As shown in Table 2, DMU₄, DMU₆ and DMU₇ are efficient, the other DMUs are inefficient. In the following, we rank them according to the ranking method of Wang [33], the ranking of fuzzy efficiency E ˜ i ( i = 1 , 2 , … , 10 ) is as follows E ˜ 4 ≻ E ˜ 7 ≻ E ˜ 6 ≻ E ˜ 10 ≻ E ˜ 1 ≻ E ˜ 2 ≻ E ˜ 5 ≻ E ˜ 3 ≻ E ˜ 8 ≻ E ˜ 9 .

To compare our proposed fuzzy approach with the classical fuzzy DEA [14], we apply the same data set and ranking method, the ranking of fuzzy efficiency [14] is as follows E ˜ 7 ≻ E ˜ 6 ≻ E ˜ 4 ≻ E ˜ 10 ≻ E ˜ 1 ≻ E ˜ 2 ≻ E ˜ 5 ≻ E ˜ 3 ≻ E ˜ 8 ≻ E ˜ 9 . Because the interactions among fuzzy inputs(fuzzy outputs), the ranking is slightly different.

Evaluating the performance of DMU by conventional DEA models requires deterministic or crisp input-output data. In recent years, fuzzy set theory has been utilized as a way to quantify imprecise and vague data in DEA models.

In classical fuzzy DEA models, the way to combine multiple fuzzy variables (fuzzy inputs, fuzzy outputs) is to use a linear weighted sum, it requires that all fuzzy variables (fuzzy inputs, fuzzy outputs) are independent. But in practice, the fuzzy variables (fuzzy inputs, fuzzy outputs) are usually strongly correlated, there are interactions among fuzzy variables.

In order to aggregate the related fuzzy variables, we first give a non-linear aggregating tool: fuzzy Choquet integral, then use fuzzy Choquet integral as an aggregating tool, we developed a novel fuzzy DEA model (CH-FDEA) to evaluate the efficiency of the DMUs with interactive fuzzy variables, the fuzzy efficiency measure, i.e., the solution of CH-FDEA, is a fuzzy number. The classical fuzzy DEA is a special form of the CH-FDEA.

The solution procedure is to transform the proposed CH-FDEA model to CH-DEA model [28] and obtain α - cut of the fuzzy efficiencies at level α (0 ≤ α ≤ 1).

We use a total ordering [33] defined on the set of fuzzy numbers to rank the DMUs, in which ranking DMUs only use the given α - cut of the fuzzy efficiencies.

This study has provided a theoretical FDEA framework with the interactive fuzzy variables, but model of CH-FDEA has high computational complexity. In the future, the proposed model can be extended to other FDEA models and intelligent algorithms for CH-FDEA will be developed.