Online LIB problems: Heuristics for Bin Covering and lower bounds for Bin Packing

¹ Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, Mawson Lakes, SA 5095, Australia; luke.finlay@unisa.edu.au
² Centre for Informatics and Applied Optimisation (CIAO), University of Ballarat, Mount Helen, VIC 3350, Australia; p.manyem@ballarat.edu.au

Received: 25 January 2004
Accepted: 7 September 2005

Abstract

We consider the NP Hard problems of online Bin Covering and Packing while requiring that larger (or longer, in the one dimensional case) items be placed at the bottom of the bins, below smaller (or shorter) items — we call such a version, the LIB version of problems. Bin sizes can be uniform or variable. We look at computational studies for both the Best Fit and Harmonic Fit algorithms for uniform sized bin covering. The Best Fit heuristic for this version of the problem is introduced here. The approximation ratios obtained were well within the theoretical upper bounds. For variable sized bin covering, a more thorough analysis revealed definite trends in the maximum and average approximation ratios. Finally, we prove that for online LIB bin packing with uniform size bins, no heuristic can guarantee an approximation ratio better than 1.76 under the online model considered.