We formulate the NP-hard n-dimensional knapsack feasibility problem as an equivalent absolute value equation (AVE) in an n-dimensional noninteger real variable space and propose a finite succession of linear programs for solving the AVE. Exact solutions are obtained for 1880 out of 2000 randomly generated consecutive knapsack feasibility problems with dimensions between 500 and one million. For...

2 Knapsack Problem (continued) 2.1 Algorithm Recap Definition: We define states (sub-problems) as the following: the maximum value for a knapsack with capacity j and we are given the first i items. Then, we have ∀i, j, the corresponding state, a[i, j] is max(a[i−1, j−wi] + vi, a[i−1, j]), where the first option is the value if item i is in the knapsack and the second option is the value if item...

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose ...

New variants of greedy algorithms, called advanced greedy algorithms, are identified for knapsack and covering problems with linear and quadratic objective functions. Beginning with single-constraint problems, we provide extensions for multiple knapsack and covering problems, in which objects must be allocated to different knapsacks and covers, and also for multi-constraint (multi-dimensional) ...

The 0/1 knapsack problem is a well known problem occurring in many real world problems. The problem is NP-Complete. The multiobjective 0/1 knapsack problem is a generalization of the 0/1 knapsack problem in which multiple knapsacks are considered. A new evolutionary algorithm for solving multiobjective 0/1 knapsack problem is proposed in this paper. This algorithm used a ε-dominance relation fo...

In this paper we propose an approach for solving problems of optimal resource capacity allocation to a collection of stochastic dynamic competitors. In particular, we introduce the knapsack problem for perishable items, which concerns the optimal dynamic allocation of a limited knapsack to a collection of perishable or non-perishable items. We formulate the problem in the framework of Markov de...

The quadratic multiple knapsack problem (QMKP) is a generalization of the quadratic knapsack problem from a single knapsack to k knapsacks. In this study, the mathematical model of the QMKP is generalized as covering the constraints that we may face in real life problems. A new hybrid algorithm which combines F-MSG algorithm and genetic algorithm is proposed to solve the Generalized QMKP (GQMKP...

A necessary and sufficient condition for the identification of dominated terms in a general knapsack problem is derived. By general we mean a knapsack problem which is unbounded, equality constrained and which has a parametric right hand side. The given condition yields published results in the literature. A report on computational experiments for large scale knapsack problems, demonstrating th...

To enhance the performance of harmony search (HS) algorithm on solving the discrete optimization problems, this paper proposes a novel harmony search algorithm based on teaching-learning (HSTL) strategies to solve 0-1 knapsack problems. In the HSTL algorithm, firstly, a method is presented to adjust dimension dynamically for selected harmony vector in optimization procedure. In addition, four s...