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.
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 DMUo 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 yrj the level of the r-th output, r = 1,..., s and by xij 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 DMUo: (xo,yo), the value of this measure can be obtained from the following nonlinear programming problem:
(1) |
In this formulation, the constraints θi<=1 and φr≥1 are the requirements for dominance. As we can see, the dominance factors θi and φr 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 ρio and γro are weights related to the inputs and outputs of DMUo, respectively, such that
We define an efficiency index μo as follows:
(3) |
The numerator in (3) is a weighted sum of the dominance factors θi and the denominator is a weighted sum of the dominance factors φr. The larger the γro is the more importance the yro and the smaller the ρio is the more importance the xio. In an effort to estimate the efficiency of DMUo, we formulate the following fractional program:
(4) |
Minimizing μo in (4) means that the numerator is minimized and simultaneously, the denominator is maximized and hence (4) measures how far DMUo is from the frontier. Hence, we are looking for a point on the frontier so that the weighted distance from xo to frontier is minimized and simultaneously, the weighted distance from yo to frontier is maximized. Let μo* be the optimal objective value to (4). Based on this value, we define an efficient DMUo as follows:
Definition 1
A DMUo: (xo, yo) 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
(5) |
MEASURING PROGRESS AND REGRESS
Context-Dependent DEA
It is assumed that there are n decision making units (DMUj: j
= 1,...,n) and their productive activities are examined in T periods. In the
t-th period, each DMUo uses xo(t) (an m-dimensional
vector of inputs) in order to produce yo(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 Jl+1(t)
= Jlt − Elt,
where El(t) = {DMUo ∈ Jl(t):
πol(t) = 1} and πol(t)
is the optimal value to the following linear programming model for each l:
(6) |
in which J1(t) = {(xj(t), yj(t)): j = 1,...,n}. The DMUs in E1(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 Jl(t) and El(t) have the following properties:
| • | |
| • | The DMUs in El(t) are dominated by the DMUs in El′(t) for l > l′ |
Progress and Regress Measures
Over the time periods, it is important to know that whether a specific
DMUo 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 DMUo(t) ∈El(t)
is said to induce progress with respect to the context El(t)
from period t to t+1, if and only if DMUo(t+1) is un-dominated
at period t+1 by units in the set Jl(t) .
Definition 3
A specific DMUo(t) ∈El(t)
is said to induce regress with respect to the context El(t)
from period t to t+1, if and only if DMUo(t+1) is dominated
by one or several units in the set Jl(t).
For any DMUo(t) at reference set Jl(t) in period t, proceed according to the following three-steps:
Step 1
Compute the efficiency of DMUo(t+1) using the following
model:
(7) |
Step 2
If πol(t+1) < 1, then DMUo(t+1)
is found inefficient with respect to the context El(t)
and it has a regress from t to t+1. The regress degree of DMUo is
denoted by rdol(t+1) = πol(t+1)
−1<0. If πol(t+1) ≥ 1, then DMUo(t+1)
is found efficient. Go to step 3.
Step 3
Solve the following linear programming model:
(8) |
if πol(t+1) = 1, there is no progress or regress. But, if πol(t+1) > 1, then DMUo(t+1) has a progress from t to t+1. The progress degree of DMUo is denoted by pdol(t+!) = πol(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 (x1), number of computer terminals (x2), operational costs(excluding staff costs) (x3) and space (x4) and outputs include deposits (y1), loans (y2), number of subscribers (y3), charges (y4) and profits (y5). 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 ρ1 = 0.35, ρ2 = 0.15, ρ3 = 0.3, ρ4 = 0.2, γ1 = 0.5, γ2 = 0.15, γ3 = 0.1, γ4 = 0.2, γ5 = 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.