A computational study of exact knapsack separation for the generalized assignment problem

Abstract

The Generalized Assignment Problem is a well-known NP-hard combinatorial optimization problem which consists of minimizing the assignment costs of a set of jobs to a set of machines satisfying capacity constraints. Most of the existing algorithms are of a Branch-and-Price type, with lower bounds computed through Dantzig-Wolfe reformulation and column generation.In this paper we propose a cutting plane algorithm working in the space of the variables of the basic formulation, whose core is an exact separation procedure for the knapsack polytopes induced by the capacity constraints. We show that an efficient implementation of the exact separation procedure allows to deal with large-scale instances and to solve to optimality several previously unsolved instances.

Separation Heuristic for the Rank-1 Chvatal-Gomory Inequalities for the Binary Knapsack Problem

“…We choose GAP as our benchmark test set since it is a well-known difficult problem which has been studied extensively, for example see (Avella et al, 2010). In the computational test, we generate cuts from each knapsack constraint of GAP for a given fractional solution.…”

“…Note that the two optimization softwares automatically add lifted cover inequalities. The exact knapsack facet generation results can be found in (Avella et al, 2010), where the row generation procedure was used to solve the problem. To implement heuristic separation procedure, we used Mosel language and LP solver provided by Xpress (2007).…”

“…where   is the value of initial LP relaxation, UB is the optimal IP value (for those instances where the optimal values are unknown, it refers to the best known value), and LB is the value of LP relaxation strengthened by the corresponding knapsack cuts (KF : exact knapsack facet (Avella et al, 2010), HCG : heuristically separated rank-1 C-G cuts, XLC : lifted cover cuts by Xpress (2007), CLC : lifted cover cuts by CPLEX (2007)). …”

“…To make comparisons between the strength of the rank-1 C-G cuts separated by the proposed heuristic procedure and those of the exact rank-1 C-G cuts and the knapsack facets, we implemented the exact rank-1 C-G cut separation procedure (Park and Lee, 2011) and the exact knapsack facet generation procedure (Avella et al, 2010) by using the Mosel language (Xpress, 2007). We tested the procedures on small instances of GAP in OR-Library (Beasley, 1990) since the exact rank-1 C-G separation procedure and the exact knapsack facet generation procedure require relatively much computation time.…”

“…This method is based on the observation that BKP can be solved in pseudo-polynomial time by a dynamic programming algorithm, so that we can formulate the facet generation problem for BKP as an optimization problem on the polar (Nemhauser and Wolsey, 1988) of the knapsack polytope, and the problem can be solved by a row generation method. For example, Avella et al (2010) applied the facet generation method to solve the generalized assignment problem. There are two main difficulties in applying this method in practice.…”