The constrained compartmentalised knapsack problem

The constrained compartmentalised knapsack problem is an extension of the classical integer constrained knapsack problem which can be stated as the following hypothetical situation: a climber must load his/her knapsack with a number of items. For each item a weight, a utility value and an upper bound are given. However, the items are of different classes (food, medicine, utensils, etc.) and they have to be loaded in separate compartments inside the knapsack (each compartment is itself a knapsack to be loaded by items from the same class). The compartments have flexible capacities which are lower and upper bounded. Each compartment has a fixed cost to be included inside the knapsack that depends on the class of items chosen to load it and, in addition, each new compartment introduces a fixed loss of capacity of the original knapsack. The constrained compartmentalised knapsack problem consists of determining suitable capacities of each compartment and how these compartments should be loaded, such that the total items inside all compartments does not exceed the upper bound given. The objective is to maximise the total utility value minus the cost of the compartments. This kind of problem arises in practice, such as in the cutting of steel or paper reels. The problem is modeled as an integer non-linear optimisation problem and for which some heuristic methods are designed. Finally, computational experiments are given to analyse the methods.