Data Envelopment Analysis

Charnes, Cooper and Rhodes in 1978, published data envelopment analysis method caused a big shock in management science and economics (Chiang Gao, Nan-Xu Huang, Toshiyuki, 2003). This method can handle multiple input and output at the same time, because the efficient frontier of data envelopment analysis, is by each assessment unit and the combined line is the most favourable conditions. Therefore, in this line as target of other units which have a compare function, and the result of analysis also is to make each unit be accepted (Chiao-Pin Bao, 2007).

Basic theory of Data Envelopment Analysis (DEA) derived from Farrell Envelope Curve in economics (Envelope Curve) principle put forward by the Production front (Production Frontier), as the basis of Production efficiency measure. Farrell's (1957) research laid the DEA the default measure efficiency on the basis of production function approach, but its research is limited to handle the case of a single output, because it needs to solve the multiple input and output of the problem. .

After the efficiency evaluation of Farrell (1957), followed by Charnes et al. (1978), they expanded the previous evaluation model with multiple inputs and developed the multiple output efficiency evaluation model of the modified CCR model. CCR model for input and output items as fixed scale reward relations is under the overall technical efficiency of evaluation. But if the production process is not a fixed size consideration, the CCR is not applicable. So Banker et al. (1984) proposed the BCC model assuming that the size is the limit of fixed remuneration. The BCC model discusses the relationship between input and output items to change the size of the compensation, the pure technical efficiency and scale efficiency rate of the evaluation. In the field of DEA, both CCR model and BCC model are universally recognized as the most influential models, therefore this study will significantly focus on these two models.

2.3.1 CCR Model

The CCR model is the origin of data envelopment analysis, and the first step in the field into the DEA. Performance evaluation model (Charnes, Cooper and Rhodes, 1978, 1879 and 1978) proposed by hereinafter referred to as the CCR model, emphasize the fixed size hypothesis which means that each additional input will increase output. This mode is generally also called technical efficiency (Technical Efficiency, TE). Content of the discussion is described as : with n decision making unit of a similar nature (Decision Making Unit, DMU), each decision unit have m as input item and s as output items with the purpose evaluate the k^th DMUk performance and that considering the position by the critics, emphasized by the critics of the output relative to the fixed inputs great situation, this is a CCR model of investment orientation (Chiao-Pin Bao, 2007 ), for the DMUk establish evaluation model are as follows:

ε is a smallest positive number (Charnes called the Archimedes number, Non-Archimedean small number, standing on the practical application 10-4 or 10-6 ).

Model (1) the efficiency value is under the same output level, compare the efficiency of the resources input, so called investment efficiency (input-based efficiency). This mode will limit the ratio of output divided by input of less than 1, in order to meet the definition of efficiency. The weight u_r and v_i as unknown, when calculating the target decision-making k unit, the weight will be selected for a specific values, so that the efficiency value hk is maximum. When the efficiency of the decision-making unit is 1, compared to other units known as an efficient decision-making, less than 1 called relatively inefficient (Chiang Gao, Nan-Xu Huang, Toshiyuki, 2003).

Since the mode (1) as a fraction of the linear programming objective function (fractional linear programming) type, in addition to the operation is not easy and there are infinitely many solutions, so will the model is transformed into linear programming model that also the denominator is set to 1 (Chiang Gao, Nan-Xu Huang, Toshiyuki, 2003). Model as performance of each DMU, but in (2) formula there is m+s variables, but limit type have m + n + s + 1, in terms of solving the dual proposition, reducing the number of constraints, making the model more efficient computing (Boussofiane, Dyson, and Thanassoulis, 1991). (2) formula of antitheses as below:

restricted

: The r^th output items of surplus variables

(3) formula in addition to calculate more efficient, but also can be see DMU inefficient, is terms of reference unit efficiency, through CCR efficiency evaluation models, can produce the following three conditions:

1. θk*<1，the DMUk input, is greater than the weighted average of all DMU input amounts, operating performance of optimization, the DMU_k inficiency with CCR.

2. θk*=1，however si-

or sr+

not equal 0, has a divergent efficiency (Radical Efficiency), also known as the weak efficiency (weak efficiency).

3. θk*>1，with si-

and sr+

equal 0, then has the efficiency of called Pareto-Koopmans’

efficiency, refers to an efficient DMU.

2.3.2 BCC Model

Banker, Charnes and Cooper (1984) proposed can measure the Pure Technical Efficiency (PTE) and the Scale Efficiency (SE) of BCC model, this mode is an extension of the concept and scope of CCR developed derived. BCC remove the CCR model, with the assumption of constant returns to scale changed to variable return to Scale (Return to Scale, VRS), in order to evaluate each decision-making unit of pure technical efficiency. Also the efficiency of the CCR model efficiency value is divided by the value of the BCC model,

namely the scale efficiency of decision making units. Its input-oriented programming style

The (4) model also is not easy to solve, but through the fixed value of the denominator shall be converted into a linear programming model to form the following model：

free

2.3.3 The Characteristics of Data Envelopment Analysis and Limits

According to Lewin, A. Y. and Minton, J. W.(1989) research, it indicated that DEA method has the following features in the evaluation which if with comprehensive index, it can learn about the actual use of resources, and the management for decision analysis.

1. Dealing with multiple inputs and multiple output efficiency evaluation at the same time.

2. Don't need to preset weights and is not affected by subjective factors. Evaluation is more fair and reasonable in the process.

3. Not because of different units and evaluation.

4. Easy to evaluate relatively efficient or relatively inefficient DMU.

According to Sun (2004) research which indicated that DEA theory limits are as follows：

1. The purpose is to measure the relative efficiency of input and output, rather than absolute efficiency.

2. According to the rule of thumb, the number of DMU for input and output of the project are at least two times.

It is because the input and output value is easily affected by extreme value, we should then choose input and output values in advance.