Mathematical models and decomposition methods for the multiple knapsack problem

Abstract

We consider the multiple knapsack problem, that calls for the optimal assignment of a set of items, each having a profit and a weight, to a set of knapsacks, each having a maximum capacity. The problem has relevant managerial implications and is known to be very difficult to solve in practice for instances of realistic size. We review the main results from the literature, including a classical mathematical model and a number of improvement techniques. We then present two new pseudo-polynomial formulations, together with specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. Extensive computational experiments show the effectiveness of the proposed approaches.

Faster Algorithms for $k$-Subset Sum and Variations

“…The Multiple Knapsack problem has been more intensively studied in recent years as applications for it arise naturally (in fields such as transportation, industry, and finance, to name a few). Some notable studies on variations of the problem are given by Lahyani et al [19] and Dell'Amico et al [13]. Special cases and variants of Multiple Subset Sum, such as the k-Subset Sum problem, have been studied in [10,9] where simple pseudopolynomial algorithms were proposed.…”

A branch-and-price algorithm for the temporal bin packing problem

“…Very recently, Hessler et al [30] proposed efficient stabilized branch-andprice algorithms. Their column-generation sub-problem is a multidimensional knapsack problem (see, e.g., Dell'Amico et al [19]) either binary, bounded, or unbounded, that they solved as a shortest path problem with resource constraints.…”

“…Otherwise let P + denote the subset of columns of P with a positive ξ * value. In the remaining case, r ∈ P for all P ∈ P + , and for any s ∈ N \ {r} equation (19) defines an integer γ value (otherwise we have found the r, s couple to be used for branching). But the latter case is not possible because it implies that any item is packed within r in all columns of P + , which results to be identical.…”

Popularity‐aware back‐tracing partition cooperative cache distribution for space–terrestrial integrated networks

“…The optimisation problem can be model as a multiple knapsack problem [25] based on (5), where the cache nodes are knapsacks with capacity , and file set with size are items. The total cost is the average distribution delay time.…”