A certainty equivalent MIP formulation of this problem, under non-stationary stochastic continuous demand assumption, is given by [26]. The same problem, but under discrete demand assumption, is tackled here using the Stochastic CP framework and the results for the scenario reduction algorithms are given in Fig. 7.

![0_image_0.png](0_image_0.png)

![0_image_1.png](0_image_1.png)

The performances of scenario reduction algorithms exhibit **a completely different pattern in the**
service-level version of the production/inventory example. In contrast to our previous observations, now the "mostlikely" method outperforms other two methods. **In only 1 case out of a total of 11 cases**
the "mostlikely" method is surpassed by LHS. It should also be noted that Dupacova et al. method gives an infeasible solution in one case and LHS in two cases, **whereas the "mostlikely" method**
consistently produces feasible solutions. An explanation **for this outcome lies in the very nature of the**
problem under consideration and the probability redistribution mechanism of the scenario reduction algorithms. In Dupacova et al., in each iteration two closest scenarios are chosen and reduced to one. The deleted scenario's probability is added to the preserved one's probability. Likewise, in LHS, probabilities are redistributed in accordance with the outcome of simulation experiments. These two methods modify the existing probability structure substantially, whereas the "mostlikely" method chooses the most probable scenarios and then only normalises their probabilities, which is less of a radical change compared to other two methods. Therefore, the better performance of the "mostlikely" method hinges on these aspects of scenario reduction algorithms in conjunction with those of chanceconstrained problems where probabilities are not only a factor that affects the expected value of the objective function but feasibility itself.

## 9 Robust Solutions

Inspired by robust optimization methods in operations research [13], we can also find robust solutions to stochastic constraint programs. That is, solutions in which similar decisions are made in different 19