Discrete Applied Mathematics 111 (2001) 231–262
Lower bounds and algorithms for the 2-dimensional vector
Alberto Caprara, Paolo Toth
University of Bologna, DEIS, Viale Risorgimento 2, 40136 Bologna, Italy
Received 1 December 1998; revised 4 November 1999; accepted 10 April 2000
Given n items, each having, say, a weight and a length, and n identical bins with a weight
and a length capacity, the 2-Dimensional Vector Packing Problem (2-DVPP) calls for packing
all the items into the minimum number of bins. The problem is NP-hard, and has applications
in loading, scheduling and layout design. As for the closely related Bin Packing Problem (BPP),
there are two main possible approaches for the practical solution of 2-DVPP. The ÿrst approach
is based on lower bounds and heuristics based on combinatorial considerations, which are fast
but in some cases not eective enough to provide optimal solutions when embedded within a
branch-and-bound scheme. The second approach is based on an integer programming formulation
with a huge number of variables, whose linear programming relaxation can be solved by column
generation, typically requiring a considerable time, but obtaining extensive information about
the optimal solution of the problem. In this paper we ÿrst analyze several lower bounds for
2-DVPP. In particular, we determine an upper bound on the worst-case performance of a class
of lower bounding procedures derived from BPP. We also prove that the lower bound associated
with the huge linear programming relaxation dominates all the other lower bounds we consider.
We then introduce heuristic and exact algorithms, and report extensive computational results on
several instance classes, showing that in some cases the combinatorial approach allows for a
fast solution of the problem, while in other cases one has to resort to the huge formulation for
ÿnding optimal solutions. Our results compare favorably with previous approaches to the problem.
? 2001 Elsevier Science B.V. All rights reserved.
Keywords: Two-dimensional vector packing problem; Lower bounds; Branch-and-bound;
Given n items, the jth having a weight w
¿ 0(j =1;:::;n), and n identical bins
of capacity c¿0, the Bin Packing Problem (BPP) calls for packing all the items
into the minimum number of bins, subject to the capacity constraint. This problem is
Corresponding author. Tel.: +39-051-2093028; fax: +39-051-2093073.
E-mail address: firstname.lastname@example.org (P. Toth).
0166-218X/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.