**ABSTRACT**

This study develops the context-dependent DEA by incorporating value judgment, inspired by Russell measure of technical efficiency to measurement technical efficiency. Next, a way of identifying progress or regress from a current period to next one is extended. This extension of DEA models, is illustrated by an empirical application to bank branches.

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**How to cite this article**

*Asian Journal of Applied Sciences, 1: 147-157.*

**DOI:**10.3923/ajaps.2008.147.157

**URL:**https://scialert.net/abstract/?doi=ajaps.2008.147.157

**INTRODUCTION**

Data Envelopment Analysis (DEA), developed by Charnes *et al*. (1978), provides a non-parametric methodology for evaluating the efficiency of each of a set of comparable Decision Making Units (DMUs), relative to one another. In the original model of Charnes *et al*. (1978) efficiency is represented by the ratio of weighted sum of outputs to the weighted sum of inputs in a specific time period. Many additional theoretical papers in the field have adapted models and applications (Tone, 2001; Sengopta, 2005; Sueyoshi and Sekitani, 2005).

In previous research efforts, Fare and Lovell (1978) approached the measurement of **technical efficiency** by suggesting some desirable properties that an ideal **technical efficiency** measure should satisfy and then, proposed a measure which satisfied them. This measure was called Russell measure of technical efficiency. Russell measure was extended to the multiple output case by Fare *et al*. (1983). Unfortunately, this approach has a difficulty in the efficiency measurement, because, the objective function is formulated as a non-linear programming problem. Sueyoshi and Sekitani (2007) proposed a re-formulation of the Russell measure by a second-order cone programming model and applied the primal-dual interior point algorithm to solve the Russell measure.

In the applications of DEA presented in the literature, the models presented are designed to obtain a measure of efficiency in a single period. In many instances, however, the DMUs involved may examine in several periods. In such situations, we are often interested to know if there is a progress or regress from period t to t+1. Tulkens and Eeckaut (1995) have presented a way to measure non-parametrically efficiency, progress and regress from panel data.

In this study, we consider the measurement of efficiency and progress or regress of DMUs from DEA perspective. We assume that the productive activities of DMUs are observed in T periods. Over the time periods, it is important to know that whether a specific DMU_{o} has progress or regress from period t to t+1. We extend the context-dependent DEA by incorporating value judgment to determine progress and regress measures. The objective here is two-fold: first, this study proposes a re-formulation of Russell measure by incorporating value judgment into the inputs and outputs. Next, we extend the context-dependent DEA by incorporating value judgment into the inputs and outputs to determine progress and regress measures of DMUs in two successive periods.

**A RE-FORMULATION OF RUSSELL MEASURE**

Assume we have n decision making units, each consumes m inputs to produce s outputs. We denote by y_{rj} the level of the r-th output, r = 1,..., s and by x_{ij} the level of the i-th input, i = 1,..., m to the j-th unit. The Russell graph measure of **technical efficiency** was defined as a combination of the input and output Russell measures of **technical efficiency** (Fare *et al*., 1983). For a given DMU_{o}: (x_{o},y_{o}), the value of this measure can be obtained from the following nonlinear programming problem:

(1) |

In this formulation, the constraints θ_{i}<=1 and φ_{r}≥1 are the requirements for dominance. As we can see, the dominance factors θ_{i} and φ_{r} appear in the objective function in an additive way. Instead of combining the input and output Russell measures in an additive way as in (1), Pastor *et al*. (1999) defined a measure as the ratio between them. They proposed the following model:

(2) |

These formulations are nonlinear and do not require a priori information on the importance of the attributes (inputs and outputs). However, different attributes play different roles in the evaluation of a DMU′s performance. In order to incorporate such a priori information, let ρ_{io} and γ_{ro} are weights related to the inputs and outputs of DMU_{o}, respectively, such that

We define an efficiency index μ_{o} as follows:

(3) |

The numerator in (3) is a weighted sum of the dominance factors θ_{i} and the denominator is a weighted sum of the dominance factors φ_{r}. The larger the γ_{ro} is the more importance the y_{ro} and the smaller the ρ_{io} is the more importance the x_{io}. In an effort to estimate the efficiency of DMU_{o}, we formulate the following fractional program:

(4) |

Minimizing μ_{o} in (4) means that the numerator is minimized and simultaneously, the denominator is maximized and hence (4) measures how far DMU_{o} is from the frontier. Hence, we are looking for a point on the frontier so that the weighted distance from x_{o} to frontier is minimized and simultaneously, the weighted distance from y_{o} to frontier is maximized. Let μ_{o}* be the optimal objective value to (4). Based on this value, we define an efficient DMU_{o} as follows:

**Definition 1 **A DMU

_{o}: (x

_{o}, y

_{o}) is efficient if and only if μ*

_{o}= 1.

The linear fractional model (4) can be converted to a non-fractional form in the usual manner of Charnes and Cooper (1962). Specifically, make the transformation

and let we have

(5) |

**MEASURING PROGRESS AND REGRESS**

**Context-Dependent DEA **It is assumed that there are n decision making units (DMU

_{j}: j = 1,...,n) and their productive activities are examined in T periods. In the t-th period, each DMU

_{o}uses x

_{o}

^{(t)}(an m-dimensional vector of inputs) in order to produce y

_{o}

^{(t)}(an s-dimensional vector of outputs). Based upon (4), we propose a stratification procedure for period t in the same manner to the original context-dependent DEA (Seiford and Zhu, 2003; Morita

*et al*., 2005) as J

_{l+1}

^{(t)}= J

_{l}

^{t}− E

_{l}

^{t}, where E

_{l}

^{(t)}= {DMU

_{o}∈ J

_{l}

^{(t)}: π

_{ol}

^{(t)}= 1} and π

_{ol}

^{(t)}is the optimal value to the following linear programming model for each

*l*:

(6) |

in which J_{1}^{(t)} = {(x_{j}^{(t)}, y_{j}^{(t)}): j = 1,...,n}. The DMUs in E_{1}^{(t)} define the first-level efficient frontier in period t. When *l* = 2, model (6) gives the second-level efficient frontier after the exclusion of the first-level efficient units and so on. It is easy to show that the sets J_{l}^{(t)} and E_{l}^{(t)} have the following properties:

• | |

• | The DMUs in E_{l}^{(t)} are dominated by the DMUs in E_{l′}^{(t)} for l > l′ |

**Progress and Regress Measures **Over the time periods, it is important to know that whether a specific DMU

_{o}has progress or regress from period t to t+1. In economics and management, the notion of progress has been associated with outward shifts of production frontiers and similarly, inward shifts refer to regress. Here, we provide suitable definitions and methods for measuring progress and regress in DEA context.

**Definition 2 **A specific DMU

_{o}

^{(t)}∈E

_{l}

^{(t)}is said to induce progress with respect to the context E

_{l}

^{(t)}from period t to t+1, if and only if DMU

_{o}

^{(t+1)}is un-dominated at period t+1 by units in the set J

_{l}

^{(t)}.

**Definition 3 **A specific DMU

_{o}

^{(t)}∈E

_{l}

^{(t)}is said to induce

*regress*with respect to the context E

_{l}

^{(t)}from period t to t+1, if and only if DMU

_{o}

^{(t+1)}is dominated by one or several units in the set J

_{l}

^{(t)}.

For any DMU_{o}^{(t)} at reference set J_{l}^{(t)} in period t, proceed according to the following three-steps:

**Step 1**

Compute the efficiency of DMU_{o}^{(t+1)} using the following model:

(7) |

**Step 2 **If π

_{ol}

^{(t+1)}< 1, then DMU

_{o}

^{(t+1)}is found inefficient with respect to the context E

_{l}

^{(t)}and it has a regress from t to t+1. The regress degree of DMU

_{o}is denoted by rd

_{ol}

^{(t+1)}= π

_{ol}

^{(t+1) }−1<0. If π

_{ol}

^{(t+1)}≥ 1, then DMU

_{o}

^{(t+1)}is found efficient. Go to step 3.

**Step 3 **Solve the following linear programming model:

(8) |

if π_{ol}^{(t+1)} = 1, there is no progress or regress. But, if π_{ol}^{(t+1)} > 1, then DMU_{o}^{(t+1)} has a progress from t to t+1. The progress degree of DMU_{o} is denoted by pd_{ol}^{(t+!)} = π_{ol}^{(t+1)} −1 > 0.

**AN EMPIRICAL STUDY**

We shall illustrate our general approach for progress and regress measurement with the analysis of bank branches activities. The data set consists of 50 bank branches located in 7 regions in Iran. The data for this analysis are derived from operations during 2004 and 2005. We use nine variables from the data set as inputs and outputs. Inputs include number of staff (x_{1}), number of computer terminals (x_{2}), operational costs(excluding staff costs) (x_{3}) and space (x_{4}) and outputs include deposits (y_{1}), loans (y_{2}), number of subscribers (y_{3}), charges (y_{4}) and profits (y_{5}). Table 1 and 2 contain a listing of the original data in two periods. Table 3 and 4 show the period measures. By using the DEA model (5), we obtain three levels of efficient frontiers in period one as follows:

Table 1: | Bank branches data in period 1 |

Table 2: | Bank branches data in period 2 |

Table 3: | Efficiency scores in period 1 |

We let ρ_{1} = 0.35, ρ_{2} = 0.15, ρ_{3} = 0.3, ρ_{4} = 0.2, γ_{1} = 0.5, γ_{2} = 0.15, γ_{3} = 0.1, γ_{4} = 0.2, γ_{5} = 0.05, |

Table 4: | Efficiency scores in period 2 |

Table 5: | Progress and regress measures |

To determine the measure of progress or regress from period 1 to 2, we have used the models (7) and (8). Table 5 displays the measure of progress and regress. As the Table 5 indicates, 32 companies have a progress and other 18 companies have regress from period 1 to period 2. Also, company 41 has the most progress and company 35 has the most regress.

**CONCLUSIONS**

This study is concerned with the measurement of efficiency and progress or regress of DMUs from DEA perspective. Context-dependent DEA by incorporating value judgment, inspired by the Russell measure of **technical efficiency** is developed to measurement the efficiency of DMUs with respect to a given evaluation context for T periods. Different strata of efficient frontier are used as evaluation context. Then, the movement of DMUs with respect to their stratum interpreted as progress or regress. The current study proposes a re-formulation of the Russell measure by incorporating value judgment into the inputs and outputs. An illustrative application of the methodology to a sample of bank branches is given.

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Direct Link