I would like to use a Data Envelopment analysis (DEA) model in which one input is a probability value and one output is an expected value (whose value depends on the probability value used as input). In such a situation, should be the case to use the chance-constrained DEA? I know that it can use when inputs and outputs are random variable with a known probability distribution. However, in my case, I have as a input a probability value.
Thank you very much
If your DEA data set is only the two variables (the probabilistic input and the expected output value) it should not be a problem using DEA model, however if you have another non-probabilistic input, beside the probabilistic one here you may have a problem in your results. To avoid the problem I would suggest scaling up the probabilistic input values by multiplying cross board by 10, or 100 depending on the size of the other variables. This will not change the efficiency ranking of DMUs but makes data more comparable.
Have a go at Stochastic DEA
http://astro.temple.edu/~banker/DEA/18.pdf
ISSUE: How to deal with expected value of output and probability of input in DEA model?
BASIC MODEL: Data Envelopment Analysis (DEA) is a nonparametric tool tool for efficiency measurement. Productive unit efficiency is obtained by the weighted sum of outputs divided by the weighted sum of inputs, thus:
(1) E = (ΣUiYiq) / (ΣVjXjq)
... where V = weight assigned to j-th input; and U = weights assigned to output. For short (1) may be written as E = A / B. You question asks how to deal with this situation:
(2) E* = A^ / Pr(B)
... where A^ = a-hat is the expected value of A; and Pr(B) is the probability of B. Output depends on input; therefore, the expected output depends on the input factor is not a problem. What to make of the "probability of input?" It appears that Pr(B) where B is already weight-times-input matrix is an application of Kahneman-Tversky's Prospect Theory where:
(3) U = Σw(Pi)v(Xi)
... where w = eight; v = function that gives the possible outcome; and Xi = observed outcome; and Pi = respective probability of each observation. Recall that B = (ΣVjXjq) or B = sum of weight-times-input. This is equivalent to w and Xi in Kahneman-Tversky and Pr(B) is Kahneman-Tvertsky's U.
Why is you output written as "expected value of output"? It must be written as "expected value" because you do not have a "real input" value, but a probability of that value.
Therefore, you do not have a problem and you do not need to do any conversion or scaling. This is a matter of the use of formal language, i.e. formal statement, of condition in mathematics. Continue with your analysis as if you have to deal with a straight forward equation (1). The logic for the expected output and probable input is found in (3).
REFERENCES: I attached here 2 articles that might be helpful: (i) DEA modeling, and (ii) Propspect Theory by Kahneman-Tvertsky. I hope this has been helpful. Cheers.
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