An empirical analysis of exact algorithms for the unbounded knapsack problem

Abstract

This work presents an empirical analysis of exact algorithms for the unbounded knapsack problem, which includes seven algorithms from the literature, two commercial solvers, and more than ten thousand instances. The terminating stepoff, a dynamic programming algorithm from 1966, presented the lowest mean time to solve the most recent benchmark from the literature. The threshold and collective dominances are properties of the unbounded knapsack problem first discussed in 1998, and exploited by the current state-of-the-art algorithms. The terminating step-off algorithm did not exploit such dominances, but has an alternative mechanism for dealing with dominances which does not explicitly exploits collective and threshold dominances. Also, the pricing subproblems found when solving hard cutting stock problems with column generation can cause branch-and-bound algorithms to display worst-case times. The authors present a new class of instances which favors the branch-and-bound approach over the dynamic programming approach but do not have high amounts of simple, multiple and collective dominated items. This behaviour illustrates how the definition of hard instances changes among algorithm approachs. The codes used for solving the unbounded knapsack problem and for instance generation are all available online.

Enhanced Moth Search Algorithm for the Set-Union Knapsack Problems

“…Second, it would be interesting to investigate other swarm intelligence, such as earthworm optimization algorithm (EWA) [38], fruit fly optimization algorithm (FOA) [39], invasive weed optimization algorithm (IWO) [40], cuckoo search (CS) [41], krill herd (KH) [42], for solving SUKP. Finally, it is certainly worth extending MS to other more complex combinatorial optimization problems including the knapsack problem with setup (KPS) [43], the 0-1 multidimensional knapsack problem (MKP) [44], unbounded knapsack problem (UKP) [45], constrained knapsack problems in dynamic environments (DKPs) [46].…”

Community Participation Strategy for Sustainable Urban Regeneration in Xiamen, China

“…Our method first obtains the break type b and residual capacity r using the common greedy algorithm [2,5]. Then, the types in the set N are divided into two disjoint subsets, N 1 and N 2 , where the profit density of types in N 1 and N 2 are equal to and less than the break type, respectively.…”

“…It is worth noting that for the UKP, even state-of-the-art dynamic programming algorithms such as EDUK have a time complexity of O(nC) [2,5,49]. Furthermore, the most special case of the UKP, where all types have the same profit density, is known as the Unbounded Subset Sum Problem (USSP).…”

“…1 Introduction Definition 1. [1,2,3,4,5] Given a multiset N consisting of n ′ item types and a total of n items, where all items of the j-th type have same profit of p j and weight of w j , and p j , w j ∈ N + . Each type can be selected unlimited times.…”

A nonlinear multidimensional knapsack problem in the optimal design of mixture experiments

“…Several extensions of the knapsack problem have been described in the literature. In one extension, the decision variables z c can take integer values other than 0 or 1 (Becker and Buriol, 2019). If there is an upper bound for z c , as is the case in the mixture design problem, the problem is called bounded.…”