A recipe is a set of ingredients. The well-ordering problem: given a set of recipes \(r_1\ldots{}r_m\) comprised of ingredients \(y_1\ldots{}y_n\), find an ordering \(\sigma\) of the ingredients that maximizes \[ \large\sum_{i=1}^{n}\sum_{j=1}^{m} p_\sigma(i, j) \] where \[ \large p_\sigma(i, j) = \begin{cases} 1 & r_j \subseteq \{y_{\sigma^{-1}(k)} \mid 1 \leq k \leq i\} \\ 0 & \mathrm{otherwise.} \end{cases} \]

In other words, we assign one point to each recipe in each round it is satisfiable by the already-collected ingredients. We reduce the well-ordering problem to integer linear programming:

\[ \large \begin{equation} \begin{split} \text{maximize } & \sum_{k=1}^n\sum_{j=1}^m v_{k,j} \\ \text{subject to } & \begin{aligned}[t] \sum_{i=0}^{n}c_{k, i} &= k & k \in [1, n] \\ c_{k,i} &\le c_{k+1, i} & i \in [1, n], k \in [1, n-1] \\ v_{k,j} &\ge -\lvert{}r_j\rvert + 1 + \sum_{i \in r_j}c_{k,i} & k \in [1, n], j \in [1, m] \\ v_{k,j} &\le c_{k,i} & k \in [1, n], j \in [1, m], i \in r_j \\ \end{aligned} \\ \text{binary } & \begin{aligned}[t] v_{k,j} & & k \in [1, n], j \in [1, m] \\ c_{k,i} & & k \in [1, n], i \in [1, n] \\ \end{aligned} \\ \end{split} \end{equation} \]

\(c_{k,i}\) is set when \(y_i\) is collected in rounds 1 through \(k\). \(v_{k,j}\) is set if \(r_j\) is satisfiable in round \(k\). The first and second constraints ensure exactly one ingredient is added in each round. The third and fourth constraints model that \(v_{k,j} = \bigwedge\limits_{i \in r_j} c_{k,i}\).

The system is solved in-browser using jvail's port of GLPK to WebAssembly. Alternatively, COIN-OR's cbc can find the optimal solution in under thirty seconds on a desktop computer. Recipes adapted from teijo/iba-cocktails.