Bin Packing

Given a set of \( N \) items with sizes \( s_1, s_2, \dots, s_N \) and bins of capacity \( C \), assign each item to a bin so the total number of bins in use is minimised, subject to each bin's load not exceeding \( C \):

\[ \min_{b \in \{0, 1, \dots, N-1\}^N} \left| \{\, b_i : i = 1, \dots, N \,\} \right| \quad \text{subject to} \quad \sum_{i : b_i = k} s_i \le C,\ \forall k \]

Each gene \( b_i \) names the bin item \( i \) lands in. With \( N \) items the search space is \( [0, N-1]^N \); putting every item in its own bin is the trivial upper bound, so a packing always exists.

Fitness penalises any overload: \( f(x) = -\,|\{b_i\}| - \lambda \sum_k \max\!\left(0,\; \sum_{i : b_i = k} s_i - C\right) \). The penalty weight \( \lambda \) defaults to 1000, so any feasible packing beats any infeasible one regardless of bin count.