Two non-radial measures of super-efficiency in DEA with data uncertainty

Abstract

Measuring errors or uncertainties in inputs and outputs create difficulties for performance evaluation in data envelopment analysis (DEA). The literature deals with the uncertainty using fuzzy or stochastic approaches. However, specifying the membership function or probability distribution is not always easy. This paper proposes a new method by assuming the inputs and outputs vary within a bounded interval and using interval analysis to extend the classic radial DEA models to two non-radial DEA models with bounded uncertainty, respectively. One is used to obtain efficiencies on the basis of slacks-based measurement (SBM) of super-efficiency DEA model, and the other is used to identify specific inefficiencies on the basis of additive super-efficiency DEA model for the decision making units (DMU) under evaluation. To solve the interval non-radial DEA models, the paper adopts the optimization theory to transform the uncertain two-level programs into deterministic one-level programs, and an acceptability index to compare and rank any of the resulting interval efficiencies. Numerical analysis illustrates the advantage of this new approach against conventional methods.

Keywords

Data envelopment analysis slacks additive model super-efficiency interval efficiency

1 Introduction

Data envelopment analysis developed by Charnes et al. in 1978 is a very useful tool to measure the relative efficiency of decision making units (DMUs) [1]. The original DEA models assume that all input and output data are deterministic. However, uncertainties abound in real life situation which can show up in the form of measurement error, data noisy, incomplete information and randomness of economic phenomena and laws and so on. Fuzzy, stochastic approaches are some of familiar tools to specify the uncertainty [2 –6]. When it is difficult to obtain the membership function or probability distribution, decision makers can direct their attention at the interval analysis to represent the impreciseness, as it can be a useful technique to deal with such uncertain problems. The current literature considers three main kinds of DEA problems. One focuses on how to solve the DEA model with imprecise data such as interval data, ordinal data, ratio bounded data or mix of them. The resulting model is treated by adopting scale transformations and variable alternations on the data and the efficiency score is a real number [7 –10]. In the second place, the efficiency score is an interval number by applying transformations only on the variables [11 –16]. Also, some classic radial DEA models with interval data have been formulated, such as the CCR or BCC model with interval data [11, 12], and the FDH model with interval data [13]. The last concentrates on the stability and sensitivity of the decision making units with uncertainty [17 –20] and the improvement of the interval efficiency by adjusting its given inputs and outputs [21].

This paper extends two non-radial DEA models to include data uncertainty. One is on the basis of slacks-based measurement (SBM) of super-efficiency DEA model from the efficiency perspective, and the other is on the basis of additive super-efficiency DEA model from inefficiency perspective. A pair of two-level mathematical programming problems are proposed to obtain the lower and upper bounds for each method when some of the inputs and outputs appear in the form of ranges. In turn, the resulting two-level mathematical programming problems can be transformed into one-level programs. All DMUs are categorized with regards to the variability of super-efficiency scores. 20 commercial bank branches in Iran is applied to illustrate the method.

The rest of this paper unfolds as follows: Section 2 provides preliminary information and useful for the succeeding sections. Section 3 presents the transformation techniques that we use to obtain the lower and upper bounds of super-efficiency scores. The section also introduces an effective approach for comparing and ranking interval efficiencies of all DMUs. Section 4 applies the approach for evaluating the performance of the 20 commercial bank branches in Iran. Section 5 concludes.

2 Preliminaries

This section lays out the assumptions and notations used in this paper. The study considers n DMUs for evaluation. Each DMU produce s outputs by consuming m inputs. All output and input data are assumed to be non-negative. Each DMU has at least one strictly positive input and output.

The following notations will be used throughout this paper.

Nomenclature

DMU _j

is the jth decision making unit,

DMU _j0

is the decision making unit under evaluation,

x_j = (x_1j, x_2j, …, x_mj) ^T

\in N_{+}^{m} \times 1

is the column vector of inputs,

y_j = (y_1j, y_2j, …, y_sj) ^T

\in N_{+}^{s} \times 1

is the column vector of outputs,

x_{ij}^{I} = [x_{ij}^{l}, x_{ij}^{u}] (y_{rj}^{I}

= [y_{rj}^{l}, y_{rj}^{u}])

is an interval number,

x_{ij}^{l} (y_{rj}^{l})

and

x_{ij}^{u} (y_{rj}^{u})

are the lower and upper bounds of the DMU_j,

λ = (λ₁, λ₂, …, λ_n) ^T

is the column vector of a linear combination of n DMUs,

N_k = 1, 2, …, k

is the index set,

τ_{i}^{I} (β_{i}^{I})

is the level of interval efficiency score,

t_{ij}^{-}

is represented as input excess,

t_{rj}^{+}

is represented as output shortfall.

3 Non-radial DEA models with data uncertainty

3.1 Interval slacks-based super-efficiency DEA model

Suppose that the evaluated unit is efficient, and the slacks-based measurement of super-efficiency score is defined in [22] as the optimal value of the following problem (1).

$\begin{matrix} \min ρ_{j 0} = \frac{\frac{1}{m} \sum_{i = 1}^{m} \frac{{\bar{x}}_{i}}{x_{ij 0}}}{\frac{1}{s} \sum_{r = 1}^{s} \frac{{\bar{y}}_{r}}{y_{rj 0}}} \\ s . t . {\begin{matrix} \bar{x} \geq \sum_{j = 1, \neq j 0}^{n} η_{j} x_{ij} & i \in N_{m} \\ \bar{y} \leq \sum_{j = 1, \neq j 0}^{n} η_{j} y_{rj} & r \in N_{s} \\ \bar{x} \geq {tx}_{ij 0} & i \in N_{m} \\ 0 \leq \bar{y} \leq {ty}_{rj 0} & r \in N_{s} \\ η_{j} \geq 0 & j \in N_{n} \end{matrix} \end{matrix}$ (1)

Since the model (1) is a fractional programming, it can produce an infinite number of solutions. Following the idea in Charnes and Cooper [23], Tone [24] solved this problem by (i) multiplying a scalar variable t (>0) to both denominator and the numerator of model (1), (ii) assuming that the objective function denominator becomes 1 and moving it to constraints, and (iii) applying variable alternation to transform it into a linear program. Through this procedure, model (1) can be recast into an equivalent linear program (2):

$\begin{matrix} \min τ_{j 0} = \frac{1}{m} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{x_{ij 0}} \\ s . t . {\begin{matrix} \frac{1}{s} \sum_{r = 1}^{s} \frac{{\tilde{y}}_{r}}{y_{rj 0}} = 1 & r \in N_{s} \\ \tilde{x} \geq \sum_{j = 1, \neq j 0}^{n} ω_{j} x_{ij} & i \in N_{m} \\ \tilde{y} \leq \sum_{j = 1, \neq j 0}^{n} ω_{j} y_{rj} & r \in N_{s} \\ \tilde{x} \geq t x_{ij 0} & i \in N_{m} \\ 0 \leq \tilde{y} \leq t y_{rj 0} & r \in N_{s} \\ ω_{j} \geq 0, t > 0 & j \in N_{n} \end{matrix} \end{matrix}$ (2) where $ρ_{j 0}^{*} = τ_{j 0}^{*}$ , η^* = ω^*/t^*, ${\bar{x}}^{*} = {\tilde{x}}^{*} / t^{*}$ , ${\bar{y}}^{*} = {\tilde{y}}^{*} / t^{*}$ . Due to the existence of uncertainty in real performance evaluation, one may not know the exact values of the input and output data. The model (2) allows that all output and input data can take from an interval. Since the modality of interval programming is similar to the modality of multi-objective program, the DEA model with interval data can be dealt with by solving a series of linear programs with deterministic data for each unit. Therefore, all efficiency scores can establish an interval number. The upper and lower bounds of the relative efficiency for DMU j₀ are obtained by the following pair of two-level mathematical programs, respectively: $τ_{rj 0}^{l} = \underset{\binom{x_{ij}^{l} \leq x_{ij}^{I} \leq x_{ij}^{u}}{\binom{y_{rj}^{l} \leq y_{rj}^{I} \leq y_{rj}^{u}}{\forall i, j, r}}}{minimize} {\begin{matrix} \min & \frac{1}{m} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{x_{ij 0}} \\ s . t . & \frac{1}{s} \sum_{r = 1}^{s} \frac{{\tilde{y}}_{r}}{y_{rj 0}} = 1 & r \in N_{s} \\ \tilde{x} \geq \sum_{j = 1, \neq j 0}^{n} ω_{j} x_{ij} & i \in N_{m} \\ \tilde{y} \leq \sum_{j = 1, \neq j 0}^{n} ω_{j} y_{rj} & r \in N_{s} \\ \tilde{x} \geq t x_{ij 0} & i \in N_{m} \\ 0 \leq \tilde{y} \leq t y_{rj 0} & r \in N_{s} \\ ω_{j} \geq 0, t > 0 & j \in N_{n} \end{matrix}$ (3) where the objective function value $τ_{j 0}^{l}$ represents the lower bound of the slacks-based measurement of super-efficiency score for the evaluated unit j₀. And $τ_{rj 0}^{u} = \underset{\binom{x_{ij}^{l} \leq x_{ij}^{I} \leq x_{ij}^{u}}{\binom{y_{rj}^{l} \leq y_{rj}^{I} \leq y_{rj}^{u}}{\forall i, j, r}}}{maximize} {\begin{matrix} \min & \frac{1}{m} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}}{x_{ij 0}} \\ s . t . & \frac{1}{s} \sum_{r = 1}^{s} \frac{{\tilde{y}}_{r}}{y_{rj 0}} = 1 & r \in N_{s} \\ \tilde{x} \geq \sum_{j = 1, \neq j 0}^{n} ω_{j} x_{ij} & i \in N_{m} \\ \tilde{y} \leq \sum_{j = 1, \neq j 0}^{n} ω_{j} y_{rj} & r \in N_{s} \\ \tilde{x} \geq t x_{ij 0} & i \in N_{m} \\ 0 \leq \tilde{y} \leq t y_{rj 0} & r \in N_{s} \\ ω_{j} \geq 0, t > 0 & j \in N_{n} \end{matrix}$ (4) where the objective function value $τ_{j 0}^{u}$ represents the upper bound of the slacks-based measurement of super-efficiency score for the evaluated unit j₀. Models (3) and (4) are both two-level mathematical models. The inner program calculates the super-efficiency score for each given set of (x_ij, y_rj) defined by the outer program. The outer program determines the set of (x_ij, y_rj) that can produce the best lower bound of super-efficiency score. Together with Models (3) and (4), $τ_{j 0}^{l}$ and $τ_{j 0}^{u}$ constitutes an interval that covers all possible efficiency scores for the evaluated unit j₀ and we define it as $τ_{j 0}^{I} = [τ_{j 0}^{l}, τ_{j 0}^{u}]$ .

Next our idea is to transform models (3) and (4) into one-level mathematical programming models. Based on Pareto’s optimization method, we can obtain the best lower bound and best upper bound of SBM super-efficiency scores for the evaluated unit j₀ by adjusting the levels of the inputs and outputs within the limits of the bounded intervals. Therefore, the two-level models (3) and (4) can be separately transformed to the following pair of one-level linear program:

$\begin{matrix} \min τ_{j 0}^{l} = \frac{1}{m} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}^{u}}{x_{ij 0}^{u}} \\ s . t . {\begin{matrix} \frac{1}{s} \sum_{r = 1}^{s} \frac{{\tilde{y}}_{r}^{l}}{y_{rj 0}^{l}} = 1 & r \in N_{s} \\ {\tilde{x}}_{i}^{u} \geq \sum_{j = 1, \neq j 0}^{n} ω_{j} x_{ij}^{l} & i \in N_{m} \\ {\tilde{y}}_{r}^{l} \leq \sum_{j = 1, \neq j 0}^{n} ω_{j} y_{rj}^{u} & r \in N_{s} \\ {\tilde{x}}_{i}^{u} \geq t x_{ij 0}^{u} & i \in N_{m} \\ 0 \leq {\tilde{y}}_{r}^{l} \leq t y_{{rj}_{0}}^{l} & r \in N_{s} \\ ω_{j} \geq 0, t > 0 & j \in N_{n} \end{matrix} \end{matrix}$ (5)

Model (5) is a DEA model with exact data, where the levels of inputs and outputs are adjusted unfavorably against the evaluated unit j₀ and in favor of the other units j. This way, we can find a data set that produces the smallest efficiency score for the evaluated unit. For the evaluated unit j₀, the inputs are adjusted at the upper bounds and the outputs at the lower bounds. For the other units j, the inputs are favorably adjusted at their lower bounds and the outputs at their upper bounds. Thus, the optimal value of model (5) can be served as a lower bound of its possible super-efficiency scores.

$\begin{matrix} \min τ_{j 0}^{u} = \frac{1}{m} \sum_{i = 1}^{m} \frac{{\tilde{x}}_{i}^{l}}{x_{ij 0}^{l}} \\ s . t . {\begin{matrix} \frac{1}{s} \sum_{r = 1}^{s} \frac{{\tilde{y}}_{r}^{u}}{y_{rj 0}^{u}} = 1 & r \in N_{s} \\ {\tilde{x}}_{i}^{l} \geq \sum_{j = 1, \neq j 0}^{n} ω_{j} x_{ij}^{u} & i \in N_{m} \\ {\tilde{y}}_{r}^{u} \leq \sum_{j = 1, \neq j 0}^{n} ω_{j} y_{rj}^{l} & r \in N_{s} \\ {\tilde{x}}_{i}^{l} \geq t x_{ij 0}^{l} & i \in N_{m} \\ 0 \leq {\tilde{y}}_{r}^{u} \leq t y_{{rj}_{0}}^{u} & r \in N_{s} \\ ω_{j} \geq 0, t > 0 & j \in N_{n} \end{matrix} \end{matrix}$ (6)

Model (6) is also a DEA model with exact data. Contrary to model (5), however, the levels of inputs and outputs are adjusted in favor of the evaluated unit j₀ and aggressively against the other units j. For the evaluated unit j₀, the inputs are adjusted at the lower bounds and the outputs at the upper bounds of the intervals. Unfavorably for the other units j, the inputs are contrarily adjusted at their upper bounds and the outputs at their lower bounds. Thus the optimal value of model (6) can be served as an upper bound of its possible super-efficiency score. So, the models (5) and (6) provide for each unit a bounded interval, in which its possible efficiency scores lie, from the worst to the best case.

Based on the idea of the previous classification [11, 15] and considering the obtained interval SBM super-efficiency of any DMU lies in an interval, all DMUs can be divided into one of the three following classes:

Class 1: Include all DMUs which are SBM super-efficient both in their best and worst situation; in other words, $E^{+ +} = {{DMU}_{j}, j \in N_{n} | τ_{j}^{l} \geq 1} .$

Class 2: Consists of all DMUs which are efficient in their best situation, but inefficient in their worst situation; in other words, $E^{+} = {{DMU}_{j}, j \in N_{n} | τ_{j}^{l} < 1 and τ_{j}^{u} \geq 1} .$

Class 3: Consists of all DMUs which are inefficient in their best situation. It goes without saying that such DMUs are, also, inefficient in their worst situation; in other words, $E^{-} = {{DMU}_{j}, j \in N_{n} | τ_{j}^{u} < 1} .$

3.2 Interval additive super-efficiency DEA model

The previous section establishes the interval SBM super-efficiency model from the efficiency perspective. This section proposes an interval additive super-efficiency model from the inefficiency perspective. In other words, the proposed approach can find out which variable causes a specific DMU to be inefficient. Considering the following additive super-efficiency model suggested in [25]:

$\begin{matrix} \min & β_{j_{0}} = \frac{1}{m + s} (\sum_{i = 1}^{m} \frac{t_{ij 0}^{-}}{x_{ij 0}} + \sum_{r = 1}^{s} \frac{t_{rj 0}^{+}}{y_{rj 0}}) \\ s . t . & {\begin{matrix} x_{ij 0} + t_{ij 0}^{-} \geq \sum_{j = 1, \neq j 0}^{n} λ_{j} x_{ij} & i \in N_{m} \\ y_{rj 0} - t_{rj 0}^{+} \leq \sum_{j = 1, \neq j 0}^{n} λ_{j} y_{rj} & r \in N_{s} \\ λ_{j} \geq 0, j \neq j 0 & j \in N_{n} \\ t_{ij 0}^{-}, t_{rj 0}^{+} \geq 0 \end{matrix} \end{matrix}$ (7)

Let an optimal solution of model (7) be ${β_{j 0}^{*}; λ_{j}^{*}, j \in N_{s}, j \neq j_{0}; t_{ij 0}^{-} *, i \in N_{m}; t_{rj 0}^{+} *, r \in N_{s}}$ and then the additive super-efficiency score for the unit j₀ can be defined as $δ_{j 0}^{*} = \frac{\frac{1}{m} \sum_{i = 1}^{m} (x_{ij 0} + t_{ij 0}^{-} *) / x_{ij 0}}{\frac{1}{s} \sum_{r = 1}^{s} (y_{rj 0} - t_{rj 0}^{+} *) / y_{rj 0}} \geq 1 .$

For our purpose of taking uncertainty into account, considering the following additive super-efficiency model when the inputs and outputs are in the form of ranges:

$\begin{matrix} \min & β_{j_{0}}^{I} = \frac{1}{m + s} (\sum_{i = 1}^{m} \frac{t_{ij 0}^{-}}{x_{ij 0}} + \sum_{r = 1}^{s} \frac{t_{rj 0}^{+}}{y_{rj 0}}) \\ s . t . & {\begin{matrix} x_{ij 0} + t_{ij 0}^{-} \geq \sum_{j = 1, \neq j 0}^{n} λ_{j} x_{ij} & i \in N_{m} \\ y_{rj 0} - t_{rj 0}^{+} \leq \sum_{j = 1, \neq j 0}^{n} λ_{j} y_{rj} & r \in N_{s} \\ x_{ij} \in [x_{ij}^{l}, x_{ij}^{u}] \\ y_{rj} \in [y_{rj}^{l}, y_{rj}^{u}] \\ λ_{j} \geq 0, j \neq j 0 & j \in N_{n} \\ t_{ij 0}^{-}, t_{rj 0}^{+} \geq 0 \end{matrix} \end{matrix}$ (8)

The additive super-efficiency score obtained by the model (8) for the evaluated unit is not worse (less) than any other additive super-efficiency scores that the DMU might attain, by adjusting the levels of the inputs and outputs within the limits of the bounded intervals. For detailed transformations, refer to the Section 3.1. The upper and lower bounds of the relative efficiency score are obtained by the following pair of programming for the evaluated unit j₀, respectively.

$\begin{matrix} \min & β_{j_{0}}^{l} = \frac{1}{m + s} (\sum_{i = 1}^{m} \frac{(t_{ij 0}^{-})^{u}}{x_{ij 0}^{u}} + \sum_{r = 1}^{s} \frac{(t_{rj 0}^{+})^{l}}{y_{rj 0}^{l}}) \\ s . t . & {\begin{matrix} x_{ij 0}^{u} + (t_{ij 0}^{-})^{u} \geq \sum_{j = 1, \neq j 0}^{n} λ_{j} x_{ij}^{l} & i \in N_{m} \\ y_{rj 0}^{l} - (t_{rj 0}^{+})^{l} \leq \sum_{j = 1, \neq j 0}^{n} λ_{j} y_{rj}^{u} & r \in N_{s} \\ λ_{j} \geq 0, j \neq j_{0} & j \in N_{n} \\ (t_{ij 0}^{-})^{u}, (t_{rj 0}^{+})^{l} \geq 0 \end{matrix} \end{matrix}$ (9) and

$\begin{matrix} \min & β_{j_{0}}^{u} = \frac{1}{m + s} (\sum_{i = 1}^{m} \frac{(t_{ij 0}^{-})^{l}}{x_{ij 0}^{l}} + \sum_{r = 1}^{s} \frac{(t_{rj 0}^{+})^{u}}{y_{rj 0}^{u}}) \\ s . t . & {\begin{matrix} x_{ij 0}^{l} + (t_{ij 0}^{-})^{l} \geq \sum_{j = 1, \neq j 0}^{n} λ_{j} x_{ij}^{u} & i \in N_{m} \\ y_{rj 0}^{u} - (t_{rj 0}^{+})^{u} \leq \sum_{j = 1, \neq j 0}^{n} λ_{j} y_{rj}^{l} & r \in N_{s} \\ λ_{j} \geq 0, j \neq j_{0} & j \in N_{n} \\ (t_{ij 0}^{-})^{l}, (t_{rj 0}^{+})^{u} \geq 0 \end{matrix} \end{matrix}$ (10) where $β_{j 0}^{l}$ and $β_{j 0}^{u}$ separately stand for the optimal value of the models (9) and (10). $δ_{j 0}^{l}$ and $δ_{j 0}^{u}$ are denoted as the lower and upper bound of the additive super-efficiency scores by substituting $β_{j 0}^{l}$ and $β_{j 0}^{u}$ into the $δ_{j 0}^{*}$ for the evaluated unit j₀. $δ_{j 0}^{l}$ and $δ_{j 0}^{u}$ constitute an interval that covers all possible efficiency scores for the evaluated unit j₀ and we define it as $δ_{j 0}^{I} = [δ_{j 0}^{l}, δ_{j 0}^{u}]$ . By the same token, considering the obtained additive super-efficiency scores $δ_{j}^{*}$ of any DMU lies in an interval, all DMUs can be divided into one of the three following classes:

Class 4: Include all DMUs which are additive super-efficient both in their best and worst situation; in other words, $E^{+ +} = {{DMU}_{j}, j \in N_{n} | δ_{j}^{l} > 1} .$

Class 5: Consists of all DMUs which are efficient in their best situation, but inefficient in their worst situation; in other words, $E^{+} = {{DMU}_{j}, j \in N_{n} | δ_{j}^{l} = 1 and δ_{j}^{u} > 1} .$

Class 6: Consists of all DMUs which are inefficient in their best situation. It goes without saying that such DMUs are, also, inefficient in their worst situation; in other words, $E^{-} = {{DMU}_{j}, j \in N_{n} | δ_{j}^{u} = 1} .$

3.3 Comparing and ranking intervals

An important issue with interval data is now to compare and rank any two interval efficiencies. The literature has considered many approaches to rank interval numbers, each with its own advantages, drawbacks and applied situations [26, 27]. This paper introduces an acceptability index firstly proposed in [28]. This approach can not only tell whether an interval is superior (or inferior) to another, but also let an optimistic decision maker know the grade of satisfaction.

Definition 1. Let IR be the set of all closed intervals on the real line R. The acceptability function λ : IR × IR → [0, ∞) for the intervals A, B ∈ IR with is denoted by λ (A ≺ B) or λ_≺ (A, B) and is defined by $λ (A ≺ B) = \frac{m (B) - m (A)}{w (B) + w (A)}, w (B) + w (A) \neq 0 and m (A) \leq m (B)$ .

λ (A ≺ B) is considered as the grade of acceptability of “the interval A to be inferior to the interval B”. Here, the terms “inferior” and “superior” are equivalent to the terms “smaller” and “greater” respectively.

The value of the grade of acceptability of the intervals A and B are given by $λ (A ≺ B) {\begin{matrix} = 0 & if m (A) = m (B) \\ = m (0 < m < 1) & if m (A) < m (B) \\ and a^{u} > b^{l} \\ \geq 1 & if m (A) < m (B) \\ and a^{u} \leq b^{l} \end{matrix}$

Example 1. Let A = [5, 15] =<10, 5 >, B = [16, 20] =<18, 2 >, then λ (A ≺ B) =1.1429. λ (A ≺ B) ≥1. For minimization problems, the interval A is accepted with full satisfaction. And for maximization problem, the interval B is accepted with full satisfaction. Also, we denoted it as $A \underset{1.1429}{≺} B$ .

Example 2. Let A = [5, 15] =<10, 5 >, B = [12, 18] =<15, 3 >, then λ (A ≺ B) =0.625 or $A \underset{1.1429}{≺} B$ , 0 < λ (A ≺ B) <1. For minimization problems, the interval A is accepted with the grade 0.625. And for maximization problem, the interval B is accepted with the grade 0.625.

Example 3. Let A = [5, 15] =<10, 5 >, B = [8, 12] =<10, 2 >, then λ (A ≺ B) =0, then neither the interval A nor B is accepted.

4 An example application

Table 1
Inputs and Outputs

Inputs Outputs

x1 Payable interest y1 The total sum of four main deposits

x2 Personnel y2 Other deposits

x3 Non-performing loans y3 Loans granted

y4 Received interest

y5 Fee

Inputs	Outputs
x1	Payable interest	y1	The total sum of four main deposits
x2	Personnel	y2	Other deposits
x3	Non-performing loans	y3	Loans granted
		y4	Received interest
		y5	Fee

Table 2

Input-data for the 20 bank branches

Bank	$x_{1}^{l}$	$x_{1}^{u}$	$x_{2}^{l}$	$x_{2}^{u}$	$x_{3}^{l}$	$x_{3}^{u}$
1	5007.37	9613.37	36.29	36.86	87243	87243
2	2926.81	5961.55	18.8	20.16	9945	12120
3	8732.7	17752.5	25.74	27.17	47575	50013
4	945.93	1966.39	20.81	22.54	19292	19753
5	8487.07	17521.66	14.16	14.8	3428	3911
6	13759.35	27359.36	19.46	19.46	13929	15657
7	587.69	1205.47	27.29	27.48	27827	29005
8	4646.39	9559.61	24.52	25.07	9070	9983
9	1554.29	3427.89	20.47	21.59	412036	413902
10	17528.31	36297.54	14.84	15.05	8638	10229
11	2444.34	4955.78	20.42	20.54	500	937
12	7303.27	14178.11	22.87	23.19	16148	21353
13	9852.15	19742.89	18.47	21.83	17163	17290
14	4540.75	9312.24	22.83	23.96	17918	17964
15	3039.58	6304.01	39.32	39.86	51582	55136
16	6585.81	13453.58	25.57	26.52	20975	23992
17	4209.18	8603.79	27.59	27.95	41960	43103
18	1015.52	2037.82	13.63	13.93	18641	19354
19	5800.38	11875.39	27.12	27.26	19500	19569
20	1445.68	2922.15	28.96	28.96	31700	32061

Table 3

Output-data for the 20 bank branches

Bank	$y_{1}^{l}$	$y_{1}^{u}$	$y_{2}^{l}$	$y_{2}^{u}$	$y_{3}^{l}$	$y_{3}^{u}$	$y_{4}^{l}$	$y_{4}^{u}$	$y_{5}^{l}$	$y_{5}^{u}$
1	2696995	3126798	263643	382545	1675519	1853365	108634.76	125740.28	956.97	6957.33
2	340377	440355	95978	117659	377309	390203	32396.65	37836.56	304.67	749.4
3	1027546	1061260	37911	503089	1233548	1822028	96842.33	108080.01	2285.03	3174
4	1145235	1213541	229646	268460	468520	542101	32362.8	39273.37	207.98	510.93
5	390902	395241	4924	12136	129751	142873	12662.71	14165.44	63.32	92.3
6	988115	1087392	74133	111324	507502	574355	53591.3	72257.28	480.16	869.52
7	144906	165818	180530	180617	288513	323721	40507.97	45847.48	176.58	370.81
8	408163	416416	405396	486431	1044221	1071812	56260.09	73948.09	4654.71	5882.53
9	335070	410427	337971	449336	1584722	1802942	176436.81	189006.12	560.26	2506.67
10	700842	768593	14378	15192	2290745	2573512	662725.21	791463.08	58.89	86.86
11	641680	696338	114183	241081	1579961	2285079	17527.58	20773.91	1070.81	2283.08
12	453170	481943	27196	29553	245726	275717	35757.83	42790.14	375.07	559.85
13	553167	574989	21298	23043	425886	431815	45652.24	50225.75	438.43	836.82
14	309670	342598	20168	26172	124188	126930	8143.79	11948.04	936.62	1468.45
15	286149	317186	149183	270708	787959	810088	106798.63	111962.3	1203.79	4335.24
16	321435	347848	66169	80453	360880	379488	89971.47	165524.22	200.36	399.8
17	618105	835839	244250	404579	9136507	9136507	33036.79	41826.51	2781.24	4555.42
18	248125	320974	3063	6330	26687	29173	9525.6	10877.78	240.04	274.7
19	640890	679916	490508	684372	2946797	3985900	66097.16	95329.87	961.56	1914.25
20	119948	120208	14943	17495	297674	308012	21991.53	27934.19	282.73	471.22

Table 4

Interval efficiencies for the 20 bank branches

Bank	Interval CCR		Interval SBM		Class	Interval Additive		Class
			Super-efficiency			Super-efficiency
	Lower	Upper	Lower	Upper		Lower	Upper
1	1	1	1.0458	1.5038	E++	1.0458	1.5041	E++
2	0.3604	1	0.1325	1.0072	E+	1	1.0082	E+
3	0.5226	1	0.1441	1.1849	E+	1	1.1853	E+
4	1	1	1.0817	1.8448	E++	1.11	2.1	E++
5	0.6333	0.769	0.0259	0.1064	E–	1	1	E–
6	0.9081	1	0.1661	1.0475	E+	1	1.0475	E+
7	0.7367	1	0.2147	1.7009	E+	1	1.7	E+
8	1	1	1.0968	1.5661	E++	1.0968	1.6683	E++
9	1	1	1.0764	2.1435	E++	1.1006	2.1435	E++
10	1	1	1.2822	1.4211	E++	1.2904	1.4269	E++
11	1	1	2.0937	4.3257	E++	3.3078	4.5357	E++
12	0.3288	0.500	0.0688	0.1842	E–	1	1	E–
13	0.4439	0.706	0.0766	0.2209	E–	1	1	E–
14	0.2691	0.729	0.0430	0.1120	E–	1	1	E–
15	0.4180	1	0.1740	1.3229	E+	1	1.3229	E+
16	0.2227	1	0.0664	1.0534	E+	1	1.0534	E+
17	1	1	1.1375	1.5793	E++	1.1375	1.5793	E++
18	0.2651	0.9548	0.0207	0.1160	E–	1	1	E–
19	0.9914	1	0.3823	1.4198	E+	1	1.4217	E+
20	0.1841	0.9789	0.0672	0.3425	E–	1	1	E–

Table 5

Data adjusted for the 20th bank branches

The upper bound of SBM super-efficiency score for the evaluated unit 20 is 0.3425
Bank	Inputs			Outputs
	$\tilde{x_{1}}$	$\tilde{x_{2}}$	$\tilde{x_{3}}$	$\tilde{y_{1}}$	$\tilde{y_{2}}$	$\tilde{y_{3}}$	$\tilde{y_{4}}$	$\tilde{y_{5}}$
1	9613.37	36.86	87243	1734217.19	263643	1675519	108634.76	956.97
2	5961.55	17.14	5379.68	340377	233656.66	858491.36	32762.03	2634.87
3	8657.13	27.17	22360.47	1027546	324141.03	1233548	96842.33	2285.03
4	1966.39	22.54	19753	207720.82	153581.32	468520	32362.80	207.98
5	4239.8	14.80	2359.35	390902	132962.08	969963.94	19272.77	1409.43
6	27359.36	19.46	15657	923737.44	74133	507502	53591.30	480.16
7	1205.47	27.48	29005	144906	130086.21	288513	24376.96	176.58
8	9559.61	25.07	9983	408163	297304.59	1044221	56260.09	1074.28
9	3427.89	21.59	413902	335070	235192.56	1584722	32723.66	560.26
10	36297.54	15.05	10229	474672.7	14378	2290745	19436.91	58.89
11	4955.78	20.54	2366.55	98715.54	93324.88	255995.59	17527.58	1070.81
12	7930.10	23.19	21353	453170	285629.58	4089171.26	39372.93	3260.92
13	6709.47	21.83	11427.14	553167	318438.15	793680.06	45652.24	3041.78
14	6804.12	12.44	17964	309670	97700.94	3819324.14	75295.09	1102.66
15	6304.01	39.86	55136	1015688.65	369986.70	924142.97	106798.63	1203.79
16	11592.35	26.52	10765.53	453843.47	417754.36	1185191.23	89971.47	4791.46
17	8603.79	27.95	43103	618105	244250	1859654.52	33036.79	2781.24
18	2037.82	8.26	11212.3	248125	75463.38	1999888.3	10338.89	615.41
19	11875.39	27.26	19569	640890	352366.63	2946797	66097.16	961.56
20	1445.68	15.67	24526.15	120208	127076.83	308012	27934.19	471.22

Table 6

Results of 7 efficient banks by interval additive super-efficiency DEA model

Bank	Additive super-efficiency	Slacks
$δ_{j}^{*}$			$t_{1}^{- *}$	$t_{2}^{- *}$	$t_{3}^{- *}$	$t_{1}^{+ *}$	$t_{2}^{+ *}$	$t_{3}^{+ *}$	$t_{4}^{+ *}$	$t_{5}^{+ *}$
1	L	1.0458	0	0	0	590714.7	0	0	0	0
	U	1.5041	0	0.5	0	1445544.4	0	0	36576.5	6311.5
4	L	1.11	613.7	0	0	31088.2	0	0	0	0
	U	2.1	639.13	0	1398.634	1086471.802	110591.7	269268.6	8705.797	0
8	L	1.0968	0	0	0	0	0	0	0	2053.576
	U	1.6683	101.821	0	0	0	406429.3	0	0	5579.807
9	L	1.1006	527.456	0	0	0	0	0	39527.12	0
	U	2.1435	837.1	0	0	0	286779.4	0	160437.1	1909.025
10	L	1.2904	0	1.893	0	0	0	0	638003.2	0
	U	1.4269	0	1.476	0	307019.4824	0	0	775806.7	0
11	L	3.3078	0	0	0	600932.3	86274.2	1374474	0	992
	U	4.5357	0	0	0	662080.383	240378.2	2173106	0	2280.201
17	L	1.1375	0	0	0	0	0	5521622	0	0
	U	1.5793	0	0	0	0	82308.56	8381661	0	3249.004

This section applies our approach to a real world example for illustration. The data include 20 commercial bank branches in Iran with 3 inputs and 5 outputs. The 3 inputs include the payable interest, personnel and non-performing loans. The 5 outputs include the total sum of four main deposits, other deposits, loans granted, received interest and fee. The evaluation index set and the data set are taken from Jahanshahloo et al. [15] and shown in Tables 1, 2 and 3. Table 4 reports the results respectively solved by Interval CCR, Interval Slacks-based measurement of super-efficiency model and Interval Additive Super-efficiency DEA model. The proposed methods allow to rank 20 commercial bank branches based on their different characteristics. For efficient bank branches, the one with better performance can be taken as a benchmark. For inefficient bank branches, the cause of inefficiency can be found out and the inputs and/or outputs can be adjusted accordingly to enhance performance.

4.1 Interval slacks-based super-efficiency DEA model

The slacks-based measurement of super-efficiency scores for all bank branches obtained by the transformed models (5) and (6) are shown in the fourth and fifth column of Table 4. According to the class 1-3 in Section 3.1, there are 7 DEA fully-efficient banks under Interval Super SBM model: 1, 4, 8, 9, 10, 11 and 17. These are all classified in E⁺⁺ due to the lower and upper bound of efficiency scores of 7 banks are both larger than 1. Likewise, it is apparently that there are 6 banks are relatively inefficient: 5, 12, 13, 14, 18, 20 due to their upper and lower bound efficiency scores are less than 1, so it belongs to E^-. Banks 2, 3, 6, 7, 15, 16 and 19 are evaluated to be weakly-DEA efficient, and their upper bound efficiency scores larger than 1 and their lower bound efficiency scores less than 1. So these banks belong to E⁺. The fully-efficient banks are usually thought to be better than any other banks that are evaluated to be weakly-DEA efficient or non-DEA efficient. The classifications of all banks are shown in the sixth column of Table 4.

As we have seen from Table 4, the fully-efficient Banks, i.e. banks 1, 4, 8, 9, 10, 11 and 17, cannot be ranked under the Interval CCR model. However, the interval slacks-based measurement super-efficiency model can distinguish between them. In addition, the Interval SBM super-efficiency score is no larger than Interval CCR efficiency score and has better discriminating power than Interval CCR model. The adjusted inputs and outputs for the evaluated unit 20 are given in Table 5. The corresponding data considered are the data in favor of the evaluated unit for unit 20 and unfavorably for the other units 1-19, that is, for the evaluated unit 20, the inputs are adjusted at the lower bounds and the outputs at the upper bounds of the intervals, and for the other units 1-19, the inputs are contrarily adjusted at their upper bounds and the outputs at their lower bounds.

4.2 Interval additive super-efficiency DEA model

The Interval Additive Super-efficiency scores for all bank branches obtained by the transformed models (9) and (10) are shown in Table 4. According to the class 4-6 in Section 3.2, 14 banks are efficient in the best situation, 7 banks are efficient due to their upper bound are all larger than zero, i.e., 7 banks (banks 1, 4, 8, 9, 10, 11 and 17) have been lied in E⁺⁺.

In addition, we can obtain inefficiency in each input and output of any DMU under interval additive super-efficiency model, and thus have a direct understanding of which variable causes a specific DMU to be inefficient. Table 6 lists the result of the 7 efficient banks identified from Table 4. It reports the interval additive super-efficiency scores and the corresponding optimal slacks obtained through models (9) and (10).

Table 7
Rank results by different approaches

Approach Bank1 Bank2 Bank3 Bank4 Bank5 Bank6 Bank7 Bank8 Bank9 Bank10

I-CCR Rank 1 13 11 1 15 9 10 1 1 1

I-SBM-SE Rank 7 13 11 3 20 12 8 6 2 5

I-additive-SE Rank 7 14 11 3 15 13 8 6 2 4

Bank11 Bank12 Bank13 Bank14 Bank15 Bank16 Bank17 Bank18 Bank19 Bank20

I-CCR Rank 1 20 18 19 12 14 1 16 8 17

I-SBM-SE Rank 1 17 16 18 10 14 4 19 9 15

I-Additive-SE Rank 1 15 15 15 10 12 5 15 9 15

Approach		Bank1	Bank2	Bank3	Bank4	Bank5	Bank6	Bank7	Bank8	Bank9	Bank10
I-CCR	Rank	1	13	11	1	15	9	10	1	1	1
I-SBM-SE	Rank	7	13	11	3	20	12	8	6	2	5
I-additive-SE	Rank	7	14	11	3	15	13	8	6	2	4
		Bank11	Bank12	Bank13	Bank14	Bank15	Bank16	Bank17	Bank18	Bank19	Bank20
I-CCR	Rank	1	20	18	19	12	14	1	16	8	17
I-SBM-SE	Rank	1	17	16	18	10	14	4	19	9	15
I-Additive-SE	Rank	1	15	15	15	10	12	5	15	9	15

The non-zero slacks indicate the maximal increasable value of inputs or the maximal reducible value of outputs that can hold the efficient state for specific bank. For the case of lower bound of bank 1 with only one non-zero slacks “the total sum of four main deposits”, if the total sum of four main deposits is decreased more than 590714.7, then the bank 1 would turn to be inefficient bank. For the case of upper bound of bank 4, the bank 4 can increase some of inputs “payable interest” or “non-performing loans” and reduce some of outputs “the total sum of four main deposits”, “other deposits”, “loans granted” or “received interest” to still keep itself to be an efficient bank.

4.3 Rank

An acceptability index is adopted to compare and rank interval efficiencies for 20 bank branches.

Under interval SBM super-efficiency model:

For fully-efficient banks 1, 4, 8, 9, 10, 11 and 17, we have $Bank 1 \binom{≺}{0.122} Bank 8 \binom{≺}{0.066} Bank 10 \binom{≺}{0.023} Bank 17 \binom{≺}{0.1740} Bank 4 \binom{≺}{0.1603} Bank 9 \binom{≺}{0.9698} Bank 11$ .

For weakly efficient banks 2, 3, 6, 7, 15, 16 and 19, we have $Bank 16 \binom{≺}{0.0107} Bank 2 \binom{≺}{0.0421} Bank 6 \binom{≺}{0.06} Bank 3 \binom{≺}{0.0767} Bank 15 \binom{≺}{0.1396} Bank 19 \binom{≺}{0.045} Bank 7$ .

For inefficient banks 5, 12, 13, 14, 18, 20, we have $Bank 5 \binom{≺}{0.025} Bank 18 \binom{≺}{0.1114} Bank 14 \binom{≺}{0.5315} Bank 12 \binom{≺}{0.1714} Bank 13 \binom{≺}{0.2674} Bank 20$ .

Under interval additive super-efficiency DEA model:

For fully-efficient banks 1, 4, 8, 9, 10, 11 and 17, we have $Bank 1 \binom{≺}{0.185} Bank 17 \binom{≺}{0.0009} Bank 10 \binom{≺}{0.0675} Bank 8 \binom{≺}{0.2849} Bank 4 \binom{≺}{0.0168} Bank 9 \binom{≺}{2.0255} Bank 11$ .

For weakly-efficient banks 2, 3, 6, 7, 15, 16 and 19, we have $Bank 2 \binom{≺}{0.7056} Bank 6 \binom{≺}{0.0585} Bank 16 \binom{≺}{0.5526} Bank 3 \binom{≺}{0.2708} Bank 15 \binom{≺}{0.1327} Bank 19 \binom{≺}{0.2481} Bank 7$ .

The results are shown in the following Table 7.

From what has been discussed above, using interval CCR model, we cannot rank the banks whose efficiency scores are 1. interval SBM Super-efficiency model not only shows the ability of interval CCR, it also determine the banks entire ranking. As for the case of Bank 5, interval CCR has determined its ranking at 15, as compared to interval SBM Super-efficiency model’s ranking at 20. This may have been caused by differences in their discriminating power. In other words, for interval CCR to consider the “proportion” inputs, interval SBM Super-efficiency model and interval additive super-efficiency model both consider the input and output slacks between the specific banks and frontiers. Moreover, we could confirm that interval SBM super-efficiency score is no larger than interval CCR efficiency score. Furthermore, interval super SBM model has the similar rankings with the interval additive super-efficiency model with the exclusion of banks 10, 17, 2, 6, 16. This is because they are two different ranking mechanisms. Last but not least, these two approaches are both unit invariant.

5 Conclusion

So far, the DEA method with uncertainty is of great importance to evaluate enterprise performance. In reality, there are many variables that cannot easily and effectively identify “exact” values, especially with qualitative and environment factors. In this occasion, we describe the data with interval data, but the traditional DEA approach cannot fully resolve it. In this paper, we developed SBM super-efficiency model and additive super-efficiency model with data uncertainty to demonstrate their characteristics theoretically. Moreover, we compare interval CCR model, interval SBM of super-efficiency model and interval additive super-efficiency model through an illustrative example. The additive (slacks-based) super-efficiency DEA method is not only unit invariant, it also is always feasible under the condition of CRS or VRS assumption. Finally, it is hoped that this study makes a small contribution in interval DEA.

Footnotes

Acknowledgments

We appreciate the support from the Projects No. 71272160, No. 71472104 and No. 71673022 of the National Natural Science Foundation of China, the Project No. 043204001 of the China Postdoctoral Science Foundation, the Project No. 2015KJW02 of the Ministry of Education of China and Tsinghua University Initiative Scientific Research Program (Grant No. 20151080390). This paper is finished as expected.

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Two non-radial measures of super-efficiency in DEA with data uncertainty

Abstract

Keywords

1 Introduction

2 Preliminaries

Nomenclature

3 Non-radial DEA models with data uncertainty

3.1 Interval slacks-based super-efficiency DEA model

4 An example application

Table 1 Inputs and Outputs Inputs Outputs x1 Payable interest y1 The total sum of four main deposits x2 Personnel y2 Other deposits x3 Non-performing loans y3 Loans granted y4 Received interest y5 Fee

4.2 Interval additive super-efficiency DEA model

5 Conclusion

Footnotes

Acknowledgments

References

Table 1
Inputs and Outputs

Inputs Outputs

x1 Payable interest y1 The total sum of four main deposits

x2 Personnel y2 Other deposits

x3 Non-performing loans y3 Loans granted

y4 Received interest

y5 Fee