If I have 5 kinds of shifts and I need between 2 and 4 people on each shift, and at most 8 people on the first 3 shift types, what form of global cardinality constraint should I use?

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BlackTom AI · Accepted Answer

We’re given a scheduling/assignment problem involving 5 shift types and constraints on how many people can be assigned to each type, plus an aggregate constraint across a subset of those types.
Option a: global_cardinality_low_up_closed. This form typically enforces that for each value (here, each shift type), the count of that value lies within a closed interval [lb, ub], and it is explicit about both the lower and upper bounds being enforced. The word 'closed' emphasizes inclusion of the bounds. In our case, we do want lower and upper bounds (between 2 and 4 people) for each of the 5 shift types, so this form could be applicable for that part. However, the extra constraint—at most 8 people on the first 3 shift types—does not come from the per-value bounds alone; it requires a cross-value constraint (a sum across several values). While GCC variants can express per-value bounds precisely, they don’t inherently capture the combined sum across a subset of values as part of the same per-value specification, so this option may be insufficient by itself if the solver expects the cross-sum to be included via another constraint.
Option b: global_cardinality_closed. This would enforce equal upper and lower bounds for each value, effectively fixing the count for each shift type to a single number (lb = ub). Our problem specifies a range (2 to 4) rather than an exact count, so this option does not fit the requirements for the per-type counts. It would not allow the 2–4 flexibility.
Option c: global_cardinality_low_up. This form expresses that each value (each shift type) must appear at least a lower bound and at most an upper bound, matching the 2–4 requirement per shift type. The description aligns well with the first constraint. The additional cross-type constraint (first 3 types totaling at most 8) would typically be handled by a separate linear (or arithmetic) constraint, not by changing the basic GCC form. Therefore, this option is a good fit for the per-type bounds and leaves room for the aggregate constraint to be added separately.
Option d: global_cardinality. This is the general form of the GCC constraint, but without specifying whether it’s low_up, or closed, or the precise semantics of the bounds. Depending on the solver, this could be ambiguous or default to exact counts or to another default. Since our problem clearly requires a lower and upper bound (2 and 4) for each type, a more explicit variant like low_up is preferable to avoid misinterpretation.
Option e: can’t use global cardinality. Given that GCC can express per-type counts with a range and we can typically express the cross-part constraint separately, there is no fundamental reason to reject GCC here. So this option is incorrect on the basis that GCC is indeed usable for at least the per-type bounds, and the problem’s cross-sum can be added as an additional constraint.
In summary, the most fitting form among the choices for capturing that each shift type should have between 2 and 4 people is the one that explicitly supports a lower and upper bound per value. That aligns with option c (global_cardinality_low_up), which provides the 2–4 range for each shift type, while the extra at-most-8 constraint across the first three types would be handled via an extra constraint rather than through altering the GCC form itself.

FIT5216 - MUM S1 2025 Week 5 participation quiz

If I have 5 kinds of shifts and I need between 2 and 4 people on each shift, and at most 8 people on the first 3 shift types, what form of global cardinality constraint should I use?

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