Editing Data envelopment analysis (section)

==Example==
Assume that we have the following data:

* Unit 1 produces 100 items per day, and the inputs per item are 10 dollars for materials and 2 labour-hours
* Unit 2 produces 80 items per day, and the inputs are 8 dollars for materials and 4 labour-hours
* Unit 3 produces 120 items per day, and the inputs are 12 dollars for materials and 1.5 labour-hours

To calculate the efficiency of unit 1, we define the objective function (OF) as

*<math>Max Efficiency :(100u_1)/(10v_1+2v_2)</math>

which is subject to (ST) all efficiency of other units (efficiency cannot be larger than 1):

*Efficiency of unit 1: <math>(100u_1)/(10v_1+2v_2)\leq 1</math>     
*Efficiency of unit 2: <math>(80u_1)/(8v_1+4v_2)\leq 1</math>        
*Efficiency of unit 3: <math>(120u_1)/(12v_1+1.5v_2)\leq 1</math>

and non-negativity:

*<math>u,v \geq 0</math>

A fraction with decision variables in the numerator and denominator is nonlinear. Since we are using a linear programming technique, we need to linearize the formulation, such that the denominator of the objective function is constant (in this case 1), then maximize the numerator.

The new formulation would be:

* OF
**<math>Max Efficiency :100u_1</math>
*ST
** Efficiency of unit 1: <math>100u_1-(10v_1+2v_2)\leq 0</math>
** Efficiency of unit 2: <math display="inline">80u_1-(8v_1+4v_2)\leq 0</math>
** Efficiency of unit 3: <math>120u_1-(12v_1+1.5v_2)\leq 0</math>
**Denominator of nonlinear OF'':''   <math>10v_1+2v_2=1</math>
** Non-negativity: <math>u,v \geq 0</math>