You can find out the basics of Data Envelopment Analysis (DEA) in numerous books and papers.

The summary is as follows:

• Data envelopment analysis (DEA) is a technique that can be used to measure the multiple dimensions of performance (efficiency) of producing units.

•DEA allows multiple inputs and outputs to be used that develop a single efficiency score of those units.

•The basic idea of the DEA is finding the "best" producer among many other producers (comparative scores of the units).

•If one producer (unit1) is either making more output with the same input or making the same output with less input than producer 2 (unit 2) then producer 1 is more efficient than producer 2.

The score of a unit is defined as the ratio of its weighted outputs to the weighted inputs, i.e.

Scoring function=(weighted sum of outputs)/(weighted sum of inputs)

•The procedure of comparing the producers can be formulated as a linear optimization problem. Without writing the equations that can be easily found elsewhere, the idea is finding weights (decision variables) in the above scoring function that maximize this function given the input data (i.e. inputs for all units and the outputs for these units) and subject to the constraint that the scoring function must be

Dear Pawas,

In linear programming, weights are assigned to the sumproduct of the inputs and outputs columns. The model maximizes efficiency subject to the constraint that makes the sum of the weight equal to 1. Because we are using nonlinear as opposed to simplex programming, it does not matter what initial weights we start out with. Given any initial guess of the weight, the model will iterate on them to fulfill the maximum and constraint conditions.

I attach a small Excel program that illustrates the computation. Hope it helps.

Thank you Lall B. Ramrattan sir for the excel. It made things clearer but again but i am unable to comprehend how DEA is analyzing that particular weight is important so it should be given some weight and other input/output is not important and hence leaving it by assigning a zero weight to it. How this tool is deciding this weight distribution.

Thank you Alexander Kolker sir for the description. Sir, I understand that using the LPP its assigning the weights, but is there any basis for this assignment, can we know how LPP is deciding which factor should be given to a particular weight and why

Pawas Trivedi As I have indicated in my earlier reply, LPP is deciding the values to assign for the weights that give the maximal possible value of the scoring function (objective function) given the above mentioned constraints.

Again, I strongly suggest that you get any of the numerous books or papers with the detailed explanation of the DEA methodology.

Thank you Alexander Kolker sir. I will read them well.

Dear Pawas,

I will try to give an intuitive explanation for the weights for a one-input, one-output case.

The data are plotted as order pairs (input, output) in the x-y plane.

A frontier is fitted with the initial data. Then a distant metric correlated with the weight is used to fit the distance of each ordered pair from the initial frontier. This is called a projection of the points on the frontier to each for all ordered pairs. The LP program maximize efficiency subject to the constraints.

The weights are then iterated on to adjust the distance of the ordered pairs from the frontier to maximize the efficiency coefficient.

Thank you so much Lall B. Ramrattan sir. This clears my doubt.

Hi Pawas.

I could also explain a bit about the efficiency scores in different contexts. While LP models can indeed be used in DEA to maximize efficiencies of the producer units, it is more difficult to maximize the overall efficiency of all units simultaneously.

Assuming that one central decision-maker (or decision-makers) wishes to improve the efficiencies of all units simultaneously, it leads to a Multi-Objective LP (MOLP) with several non-dominated solutions. It is possible to compute single solutions from the efficient frontier, but it does not give you a good overall picture of the problem.

However, in such a case, it is possible to compute all non-dominated solutions to present all solutions to the decision-maker(s). Also, in this case, it is possible to observe that conventional efficiency scores are not reliable measures to use when deciding which producer units should receive more inputs (or resources) and which units should have less. If you are interested, here is a recent paper that studies the problem in this setting.

Article Efficient Allocation of Resources to a Portfolio of Decision...

The paper assumes that producers cannot increase their efficiency scores immediately. However, traditional DEA efficiency scores remain unreliable even without this constraint (at least in the problems studied in this paper).

The constraint not allowing efficiency scores to increase immediately may not be entirely realistic in most practical cases, but assuming that all producers can magically obtain maximum efficiencies is not realistic either. A more reasonable approach would be to limit possible efficiency increases based on, for example, past experience or expert opinions.

Best regards,

Juho

Thank you so much for the insights and the paper. The fact you mentioned is so true Juho Andelmin. Can we put these restrictions on the weights itself on the basis of past experience or expert opinions as you mentioned? Will it be reasonable?

Hi Pawas Trivedi

Imposing constraints on the weights is indeed one method to prevent unrealistic increases in efficiency scores. I'm not too familiar with this area of research, but intuitively it would seem that adding such constraints would be the most robust way to deal with unrealistic efficiency increases. So I believe your intuition is also correct in this regard.

I found some literature about the effects of some weight restriction approaches. Here is one example, although it seems that this is an acive research topic so there are likely several studies about different appoaches to impose weight constraints.

Article Data envelopment analysis with common weights: the weight re...

Thanks a lot, Juho Andelmin for the help.

Hi Pawas Trivedi! As Juho Andelmin said, it is possible to assign weights to the variables to deal with unrealistic efficiency. Maybe when you consider that among the variables exist a hierarchical order, this is not taken into account in DEA's weights. You can use the specific model Assurance region, implementing some constraints to represent contextualized weights for the variables.

I can recommend this paper in which I used the Assurance region.

Article Assessing the efficiency of science, technology and innovati...