
| Backtracking (BT)   | Forward Checking (FC)   | Scenario-Based (SB)   |           |         |         |               |         |
|---------------------|-------------------------|-----------------------|-----------|---------|---------|---------------|---------|
| No. Stages          | Nodes                   | CPU/sec               | Nodes     | CPU/sec | Failure | Choice Points | CPU/sec |
| 1                   | 28                      | 0.01                  | 10        | 0.01    | 4       | 5             | 0.00    |
| 2                   | 650                     | 0.09                  | 148       | 0.03    | 4       | 8             | 0.02    |
| 3                   | 17,190                  | 2.72                  | 3,604     | 0.76    | 8       | 24            | 0.16    |
| 4                   | 510,356                 | 83.81                 | 95,570    | 19.07   | 42      | 125           | 1.53    |
| 5                   | 15,994,856              | 3,245.99              | 2,616,858 | 509.95  | 218     | 690           | 18.52   |
| 6                   | -                       | -                     | -         | -       | 1260    | 4035          | 474.47  |

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

will be sold at a low price. Crop yields are uncertain, depending upon weather conditions during the growing season.

This problem has 4 stages and 10,000 scenarios. In Fig. 2, we again see that Dupacova et al's algorithm and Latin hypercube sampling are very effective, and both are far ahead of the most likely scenario method (which requires approximately one third the scenarios before the first decision is made correctly).

## 8.4 Production/Inventory Control

Uncertainty plays a major role in production and inventory planning. In this simplified production/inventory planning example, there is a single product, a single stocking point,

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

![0_image_2.png](0_image_2.png)

production capacity constraints and stochastic demand. The objective is to find the minimum expected cost policy. The cost components take into account holding costs, backlogging costs, fixed replenishment (or setup) costs and unit production costs. The optimal policy gives the timing of the replenishments as well as the order-up-to-levels. Hence, the exact order quantity can be known only after the realization of the demand, using the scenario dependent order-up-to-level decisions. This problem has 5 stages and 1,024 scenarios. In Fig.

3, we again see that Dupacova et al's algorithm and Latin hypercube sampling are very effective, but both are now only a small distance ahead of the most likely scenario method.

9 Robust solutions Inspired by robust optimization methods in operations research [Kouvelis and Yu, 1996], stochastic OPL also allows us to find robust solutions to stochastic constraint programs. That is, solutions in which similar decisions are made in the different scenarios. It will often be impossible or undesirable for all decision variables to be robust. We therefore identify those decision variables whose values we wish to be identical across scenarios using commands of the form:

## Robust <Var>;

For example, in production/inventory problem of Sec.8.4 the decision variables "order-up-to-levels" and "replenishment periods" can be declared as robust variables. The values of these two sets of decision variables are then fixed at the beginning of the planning horizon. A robust solution dampens the nervousness of the solution, an area of very active re-