1. Data Envelope Analysis (DEA)
DEA is used to study the efficiency of each of many Decision Making Units (DMU)
Examples of applications: Hospital units, schools, bank branches, i.e. many similar operating units

2. DEA Example 1

3. DEA is solved by LP
In the DEA methodology, formally developed by Charnes, Cooper and Rhodes (1978), efficiency is defined as a weighted sum of outputs to a weighted sum of inputs, where the weights structure is calculated by means of linear programming.

4. DEA main idea
Every unit is allowed to enjoy the pricing of inputs and outputs that puts it into the best possible light. If it still cannot compete with the best units, then this unit is not efficient.

5. Example 2
Unit 1 produces 100 pieces of items per day, and the inputs are 10 dollars of materials and 2 labour-hours
Unit 2 produces 80 pieces of items per day, and the inputs are 8 dollars of materials and 4 labour-hours
Unit 3 produces 120 pieces of items per day, and the inputs are 12 dollars of materials and 1.5 labour-hours

6. Efficiency of Unit 1
maximize Efficiency = (u1 * 100) / (v1 * 10 + v2 * 2)
Efficiency cannot be larger than 1:
(u1 * 100) / (v1 * 10 + v2 * 2) <= 1
(u1 * 80) / (v1 * 8 + v2 * 4) <= 1
(u1 * 120) / (v1 * 12 + v2 * 1.5) <= 1
all u and v >= 0.

7. Transformed to LP
But since linear programming cannot handle fraction, we need to transform the formulation, such that we limit the denominator of the objective function to be = 1, and only allow the linear programming to maximize the numerator.

8. Final LP model: Max Efficiency = u1 * 100
(u1 * 100) - (v1 * 10 + v2 * 2) <= 0
(u1 * 80) - ( v1 * 8 + v2 * 4) <= 0
(u1 * 120) - (v1 * 12 + v2 * 1.5) <= 0
v1 * 10 + v2 * 2 = 1
all u and v >= 0.