Unfortunately, the world is replete with examples of alternatives that violate such proportionality rules. Finally, ratio analysis does not lend itself to explicit treatments of risk and uncertainty. The production function approach to CEA can deal with variable returns to scale and with other nonlinearities in technology. Numerous estimates of costs and benefits for various alternatives at different scale levels are used to fit a nonlinear production function by regression. This technique can deal with several measures of input and can therefore overcome some of the difficulties of the cost benefit ratio.
Production functions can also incorporate risk described with random variables.
But, the multiple regression production function also has some drawbacks. First, the use of regression tends to measure efficiency relative to average performance instead of best performance. That is, all the observations are pooled to fit the production function, a measure of average efficiency, then each alternative is compared to that average measure. Also, multiple regression requires that a single indicator for output be used. Thus, multiple outputs must be combined into a single effectiveness indicator, similar to ratio analysis.