S3. Change the probability of the scenario ω
(¯s)**, that is the nearest to** ω
(s
∗
):

$$\pi^{(\bar{s})}:=\pi^{(\bar{s})}+\pi^{(s^{*})}$$
$$\pi^{(s^{*})}:=0$$

All node probabilities are adjusted, as the sum of the probabilities of possible realizations at each node equals 1.

S4. If ns > N then go to step 1, where N **is the desired number of scenarios remain.**
This algorithm is incorporated into Stochastic OPL and can be called by scenario DGR <Num>;
Dupacova et al. report power production planning problems on which this method offers 90%
accuracy sampling 50% of the scenarios and 50% accuracy sampling just 2% of the scenarios.

## 8 Some Examples

To illustrate the potential of this framework for decision making under uncertainty, we now describe a wide range of problems that we have modelled. In each problem, we illustrate the effectiveness of different scenario reduction techniques.

## 8.1 Portfolio Diversification

The portfolio diversification problem of [2] can be modelled **as a stochastic COP. Suppose we have $**P to invest in any of I investments and we wish to exceed a wealth of $G after t **investment periods.**
To calculate the utility, we suppose that exceeding $G is equivalent to an income of q**% of the excess**
while not meeting the goal is equivalent to borrowing at a cost r**% of the amount short. This defines**
a concave utility function for r > q**. The uncertainty in this problem is the rate of return, which**
is a random variable, on each investment in each period. The objective is to determine the optimal investment strategy, which maximizes the investor's expected utility. A mathematical formulation of the problem, and corresponding stochastic and certainty-equivalent OPL representations are given in the appendix.

The test problem has 8 stages, in which the number of states are [5,4,4,3,3,2,2,2], and 5760 scenarios.

The CP model has 27 decision variables and 18 constraints for **one scenario, and 33,438 decision** variables and 22,292 constraints for 5,760 scenarios. To compare the effectiveness of different scenario reduction algorithms, we adopt a two step procedure. In the first step, the scenario reduced problem is solved and the first period's decision is observed. We then **solve the full-size (non scenario reduced)** problem to optimality with this first decision fixed. The difference between the objective values of these two solutions is normalized by the range [optimal solution, observed worst solution] to give a normalized error for committing to the scenario reduced first decision. In Fig. 2, we see that Dupacova et al's algorithm is very effective, that Latin hypercube sampling is a small distance behind, and both are far ahead of the most likely scenario method (which requires approximately half the scenarios before the first decision is made correctly).

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