Calculate All Expected Value Statistics |
Contents
The Calculate All Expected Value Statistics option on the SP Solver tab:
controls whether LINGO displays information regarding the expected values for a number of statistics when solving stochastic programming (SP) models. To illustrate, when solving the SPGAS.LG4 model when this option is enabled, you will see the following expected values at the top of the solution report:
Expected value of:
Objective (EV): 1400.000
Wait-and-see model's objective (WS): 1326.667
Perfect information (EVPI = |EV - WS|): 73.33333
Policy based on mean outcome (EM): 1479.444
Modeling uncertainty (EVMU = |EM - EV|): 79.44444
These values are a guide as to how the stochastic nature of the model is impacting the objective value. The following is a brief description of these expected values:
Expected Value of Objective (EV) - is the expected value for the model's objective over all the scenarios, and is the same as the reported objective value for the model.
Expected Value of Wait-and-See Model's Objective (WS) - reports the expected value of the objective if we could wait and see the outcomes of all the random variables before making our decisions. Such a policy would allow us to always make the best decision regardless of the outcomes for the random variables, and, of course, is not possible in practice. For a minimization, it's true that WS <= EV, with the converse holding for a maximization. Technically speaking, WS is a relaxation of the true SP model, obtained by dropping the nonanticipativity constraints.
Expected Value of Perfect Information (EVPI) - is the absolute value of the difference between EV and WS. This corresponds to the expected improvement to the objective were we to obtain perfect information about the random outcomes. As such, this is a expected measure of how much we should be willing to pay to obtain perfect information regarding the outcomes of the random variables.
Expected Value of Policy Based On Mean Outcome (EM) - is the expected true objective value if we (mistakenly) assume that all random variables will always take on exactly their mean values. EM is computed using a two-step process. First, the values of all random variables are fixed at their means, and the resulting deterministic model is solved to yield the optimal values for the stage 0 decision variables. Next, a) the stage 0 variables are fixed at their optimal values from the previous step, b) the random variables are freed up, c) the nonanticipativity constraints are dropped, and d) this wait-and-see is solved. EM is the objective value from this WS model.
Expected Value of Modeling Uncertainty (EVMU) - is the absolute value of the difference EV - EM. It is a measure of what we can expect to gain by taking into account uncertainty in our modeling analysis, as opposed to mistakenly assuming that random variables always take on their mean outcomes.
| Note: | The above approach for computing EM and EVMU makes unambiguous sense only for models with a stage 0 and a stage 1. If there are later random variables in stages 2, 3, etc., then there are complications. For example, for decisions in later stages, we have seen the outcomes from the random variables in earlier stages, so considering these random variables to take on their mean value does not make sense. For models with a stage 0 and stage 1, EVMU will be an accurate measure of the expected value of modeling uncertainty. For models with additional stages beyond 0 and 1, EVMU will merely be an approximation of the true expected value of modeling uncertainty. |
| Note: | Computing these expected value statistics can be very time consuming for large models. If speed is an issue, you may wish to disable this feature on the Solver|Options|SP Solver tab. |
The next component of the solution lists various statistics regarding the class and size of the model: