DEA is a linear programming based technique for measuring the relative performance of organizational units where the presence of multiple inputs and out- puts makes comparisons difficult. The DEA mathe-
matical model is as follows: M a x h = r i
u r y v i x r j0 ij 0
r u r y r j i v i x i j
j = 1, Λ, n( for all j)
performance of organizational units where the pres- ence of multiple inputs and outputs makes compar- isons difficult. The DEA mathematical model is as follows:
The u’s and v’s are variables of the problem and are constrained to be greater than or equal to some small positive quantity in order to avoid any input or output being ignored in computing the efficiency.
ur, vi ≥
The solution to the above model gives a value h, the efficiency of the unit being evaluated. If h = 1 then this unit is efficient relative to the others. But if it is less than l then some other units are more efficient than this unit, which determines the most favorable set of weights. This flexibility can be a weakness because the judicious choice of weights by a unit possibly un- related to the value of any input or output may allow a unit to appear efficient.
To solve the model, we need to convert it into linear programming formulation:
Max h =
r u r y r y
subject to dual variable
i v i x i j 0 = 1 0 0 ( % )
r u r y r j −
i v i x i j
−vi ≤ − i = 1, 2, Λ, m −ur ≤ − r = 1, 2, Λ, t
j = 1, Λ, n
Z0 λ s s − r + i j
We call this formulation CCR (Charnes, Cooper, and Rhodes, 1978) model. The dual model can be con- structed by assigning a dual variable to each constraint
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in the primal model. This is shown below.
s + i −
− r s r
λ j x i j = x i j 0 Z 0 − s + i , i = 1 , Λ , m j
j λ j y r j = y r j 0 + s − r , r = 1 , Λ , t λ j , s + i , s − r ≥ 0
The dual variables λ’s are the shadow prices related to the constraints limiting the efficiency of each unit to be no greater than 1. Binding constraint implies that the corresponding unit has an efficiency of 1 and there will be a positive shadow price or dual variable. Hence positive shadow prices in the primal, or positive values for the λ’s in the dual, correspond to and identify the peer group for any inefficient unit.
The above models assume constant return to scale. If we add a variable to the model, we can construct a DEA model with variable return to scale. Variable re- turns means that we might get different levels of out- put due to reduced performance or economics of scale. This version of the model is popularly known as BCC (Banker, Charnes, and Cooper 1984)
The concern with the DEA model is that by a judi- cious choice of weights a high proportion of units will turn out to be efficient and DEA will thus have little discriminatory power. The first thing to note is that a unit which has the highest ratio of one of the outputs to one of the inputs will be efficient, or have an effi- ciency very close to one by putting as much weight as possible on that ratio and the minimum weight zero on the other inputs and outputs. Further empirical studies justify that the number of decision making units evalu- ated should be greater than two times the total number of variables.
Application To Efficiency Measurement of State Transport Undertakings
Since DEA was first introduced by Charnes, Cooper, and Rhodes (1978), this methodology has been widely applied to the efficiency measurement of many orga- nizations. Sherman and Gold (1985) used DEA model for evaluating bank branch operating efficiency. Shang and Sueyoshi (1995) applied the model to the selection