# Part 11: Modeling¶

We ask you not to publish your solutions on a public repository. The instructors interested to get the source code of our solutions can contact us.

## The Logical Clause Constraint and Watched Literals¶

• Implement the constraint Or.java for modeling the logical-clause constraint: (x[0] or x[1] or x[2] or … or x[n-1]).
• Test your implementation in OrTest.java..
• The implementation should use the watched-literals technique.

A reminder about the watched-literals technique:

• The constraint should only listen to the changes of two unbound variables with propagateOnFix(this) and dynamically listen to other ones whenever one of these two becomes fixed. Keep in mind that
any call to x[i].propagateOnFix(this) has a stateful effect on backtrack.
• Why two? Because as long as there is one unfixed one, the constraint is still satisfiable and nothing needs to be propagated, and whenever it is detected that only one is unfixed and all the other ones are fixed to false, the last one must be fixed to true (this is called unit propagation in SAT solvers).
• The two unfixed variables should be at indices wL (watched left) and wR (watched right). As depicted below, wL (resp. wR) is the leftmost (resp. rightmost) unfixed variable.
• Those indices are stored in a StateInt so that they can only increase during search and thus help achieve incrementality.
• When propagate is called, it means that one of the two watched variables is fixed (x[wL] or x[wR]) and consequently the two pointers must be updated.
• If during the update a variable fixed to true is detected, then the constraint can be deactivated since it will always be satisfied.

## The Reified Logical-Clause Constraint¶

• Implement the constraint IsOr.java for modeling the reified logical-clause constraint: b iff (x[0] or x[1] or x[2] or … or x[n-1]).
• Test your implementation in IsOrTest.java..
• In case b is true, you can post your previous Or constraint

(create it once and for all and post it when needed to avoid creating objects during search that would trigger garbage collection).

## Steel Mill Slab Problem: Modeling, Redundant Constraints, and Symmetry Breaking¶

A number of TODO tasks must be completed in Steel.java so as to improve gradually the performance for solving this problem optimally:

1. Model the objective function denoting the total loss to be minimized. You should use Element constraints to denote the loss in each slab. The precomputed array loss gives for each load (index) the loss that would be incurred. It is precomputed as the difference between the smallest capacity that can accommodate the load and the load value. A Sum constraint constraint can then be used to denote the total loss.
2. Model a Boolean variable reflecting the presence or not of each color in each slab. The color is present if at least one order with this color is present. The IsOr constraint previously implemented can be used for that.
3. Restrict the number of colors present in slab j to be at most 2. Your model can now be run, although it will not be able to solve optimally even the easiest instance data/steel/bench_20_0 in reasonable time.
4. Add a redundant constraint for bin packing, stating that the sum of the loads is equal to the sum of the elements. Do you observe an improvement in the solving time?
5. Add symmetry-breaking constraints. Two possibilities: the loads of slabs must be decreasing or the losses must be decreasing. Do you observe an improvement in the solving time?
6. Implement a dynamic symmetry-breaking during search. Select an order x representing the slab where this order is placed. Assume that the maximum index of a slab containing an order is m. Then create m+1 branches with x=0 ,x=1, …, x=m, x=m+1 since all the decisions x=m+2, x=m+3, … would be subproblems symmetrical to x=m+1. You should now be able to solve quickly and optimally the instance ‘data/steel/bench_20_0’, by reaching a zero-loss solution.