Linear programming and the DEA approach Anders Ringgaard Kristensen - - PowerPoint PPT Presentation

Linear programming and the DEA approach

Anders Ringgaard Kristensen

Absolute effectiveness

Let

• Y be a vector of products
• x be a vector of factors
• U(x, Y) be the utility function of a farmer
• T be the set of technically possible combinations of

factors and products for the farm in question. How efficient is the farmer if his current situation is described by the product-factor combination (xi, Yi)? His absolute effectiveness is defined as If U is unknown the absolute effectiveness may not be calculated

Absolute efficiency

If the utility function U is unknown, we may instead calculate the absolute efficiency, Fi, (where the ideal is to produce as much as possible with as little as possible) as An absolute efficiency of Fi = 1.0 indicates that the current input xi could not result in higher output than Yi. A value of Fi > 1.0 means that more output could have been produced with the same input. Calculation of the absolute efficiency requires that T is known.

Relative efficiency

If the set T is unknown, it may be estimated as T* based on data from other farms. If i = 1, … , 100, any observation of (xi, Yi) is a valid member

• f T

If we have enough observations (farms) we may estimate T as T* We then define the relative efficiency as

Relative efficiency, sow herd, one factor, one product

Graphical illustration: Constant returns to scale

Solved by linear programming

Graphical illustration: Variable returns to scale

Solved by linear programming

One factor, two products

How relative efficient is farmer i?

His input-output combination is (xi, Yi) We have J other farms with input-output combinations (xj, Yj), j = 1, … , J. Let λ1, … , λJ be weighting factors so that

• ∑j λj = 1
• λ1, … , λJ ≥ 0

Then ∑j xjλj is a weighted average of inputs And ∑j Yjλj is a weighted average of outputs using the same weighting factors. Assume that xi ≥ ∑j xjλj We will then determine F in such a way that FYi ≥ ∑j Yjλj If it is necessary to multiply Yi by a factor F > 1 it means that

• ther farms produce more efficient than farm i.

We want the weighting λ1, … , λJ that maximizes F

A linear programming problem

The variables of the problem: F, λ1, … , λJ The coefficients: x1, … , xJ , Y1, … , YJ The objective function:

• F = Max!

The constraints:

• xi ≥ ∑j xjλj
• FYi ≤ ∑j Yjλj
• ∑j λj = 1
• λ1, … , λJ ≥ 0