The stochastic knapsack problem is a natural generalization of classical knapsack problem. In this problem, we are given a set of items each associated with a probability distribution on sizes and a profit, and a knapsack of unit capacity. The size of an item is unknown before inserting it into the knapsack, and it is revealed as soon as the item inserted into the knapsack. The objective is to ...

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under wreath product. On the other hand, the class of knapsack-semilinear groups, where solutions sets of knapsack equations are effectively semilinear, is closed under...

Problem Definition This problem deals with packing a maximum reward set of items into a knapsack of given capacity, when the item-sizes are random. The input is a collection of n items, where each item i ∈ [n] := {1, · · · , n} has reward r i ≥ 0 and size S i ≥ 0, and a knapsack capacity B ≥ 0. In the stochastic knapsack problem, all rewards are deterministic but the sizes are random. The rando...

This paper develops an algorithm based on the Modular Approach to solve singly constrained separable discrete optimization problems (Nonlinear Knapsack Problems). The Modular Approach uses fathoming and integration techniques repeatedly. The fathoming reduces the decision space of variables. The integration reduces the number of variables in the problem by combining several variables into one v...

The Lasserre/Sum-of-Squares hierarchy is a systematic procedure to strengthen LP relaxations by constructing a sequence of increasingly tight formulations. For a wide variety of optimization problems, this approach captures the convex relaxations used in the best available approximation algorithms. The capacitated covering IP is an integer program of the form min{cx : Ux ≥ d, 0 ≤ x ≤ b, x ∈ Z+}...

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig-Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursi...

In this paper, we propose the assignment problem of resources humans’ methodology by using the multiple knapsack approach to formulate the studied problem, and solve it by using the genetic algorithm for obtaining the optimal assignment of the new qualified employees that ensures the maximization the productivity by introducing constraints corresponding to the cost of each posts, and capacity o...

In this paper, we study the robustness over the cardinality variation for the knapsack problem. For the knapsack problem and a positive number α ≤ 1, we say that a feasible solution is α-robust if, for any positive integer k, it includes an α-approximation of the maximum k-knapsack solution, where a k-knapsack solution is a feasible solution that consists of at most k items. In this paper, we s...

The facet of Knapsack ploytope, i.e. convex hull of 0-1 points satisfying a given linear inequality has been presented in this current paper. Such type of facets plays an important role in set covering set partitioning, matroidal-intersection vertex- packing, generalized assignment and other combinatorial problems. Strong covers for facets of Knapsack ploytope has been developed in the first pa...