Hi everybody, I am working on a deterministic equivalent of a two stage stochastic problem with a plannning horizon of multiple periods. Stage 1 are all decisions for the next period, everything else is Stage 2. Uncertainty is modelled by a set of demand scenarios. Possible outcomes are defined for every period (independently) by a different histogram. For complexity reasons it is not possible to combine all outcomes of each period against all others. On the one hand I want to have a set of possible outcomes that represents the histograms (with their relative frequencies), on the other hand some “extreme scenarios” regarding the demand (not regarding the probability!) should be included. It has to be mentioned that these cases do not necessarily have a very low probability because the relative frequencies of the histograms do not have to be Gaussian distributed. My question is, if there is “a common” methodology, because having these two properties should be somehow desired in a lot of applications?! Subsequently some of my “ideas” are added, but I am not sure if I am “on the right track”. I considered using Monte Carlo sampling, but this method might fail when only few scenarios are generated and the relative frequencies are more or less equally distributed. In this case the extreme scenarios (regarding the amount) are usually missed out. I also thought about sorting all histograms by the amount (descending) and combining always the i-th entry. The weighting of a scenario results by multiplication of the corresponding frequencies. With this method I might miss the probable scenarios because in some histograms high frequencies belong to high amounts where in others high frequencies belong to low amounts. Finally I considered a combination of both, where each method provides 50% of the total weight of the scenario set. It would be nice to get some advice or remarks.
asked
stefan |

Your problem description is ambiguous and hard to understand and I've not understand it completely even after reading it for two times. Therefore, I strongly recommend you rewrite as people here are not completely familiar with your problem and they have limited time as well. Anyway, as far as I understand you have a problem with correct sampling of scenarios for your problem. There are various methods for sampling scenarios including 1) bound based construction, 2) Monte Carlo and Quasi Monte Carlo based methods, 3) EVPI-based sampling, 4) Moment matching principle, and 5) Probability metric based approximation. I recommend reading the following papers in order to solve your problems: Dupacova, J., Consigli, G., Wallace, S.W., 2000. Scenarios for multistage stochastic programs. Annals of Operations Research 100, 25-53. Heitsch, H., Romisch, W., 2003. Scenario reduction algorithms in stochastic programming. Computational Optimization and Applications 24, 187-206.
answered
Ehsan ♦ |

Maybe it helps to go a few steps backward. I consider the demand for different products to be uncertain. Hence, every demand is a random variable. As a result, the dimension of the scenario space grows with the number demands. Does this way of modelling make sense, or is it meaningless to consider (for example) 50 scenarios in the case of 15 demands? Is there any alternative?!