Crowder et al. (1983) suggested using strong inequalities derived from the individual linear constraints of the problem (knapsack relaxations).
What makes a vector solution x infeasible is that the total weight of the individual x has exceeded the knapsack capacity.
It closely corresponds to the knapsack problem.
Thus, for a large knapsack of 20,000 elements, the adaptive penalty function used in this research computes the minimum and the maximum penalties as approximately 1 and 10, respectively.
Pmax represents the profit of the solution and cap is the knapsack size.
However, we can simplify the problem further, so that finding such inequalities is easier, and then build from these inequalities to obtain facets of the knapsack polytopes.
The general problem of ordering categories to minimize wasted gaps in allocation is somewhat similar to a knapsack problem.
Finding the shortest path corresponds to minimising the cost of placing items in the knapsack, and therefore maximising the profit.
