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Two-Bar Charts Packing Problem

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Abstract

We consider a Bar Charts Packing Problem (BCPP), in which it is necessary to pack bar charts (BCs) in a strip of minimum length. The problem is, on the one hand, a generalization of the bin packing problem (BPP) and, on the other hand, a particular case of the project scheduling problem with multidisciplinary jobs and one limited non-accumulative resource. Earlier, we proposed a polynomial algorithm that constructs the optimal packing for a given order of non-increasing BCs. This result generalizes a similar result for BPP. We focus on the Two-Bar Charts Packing Problem (2-BCPP), where each BC consists of two bars. For this case, we show that the proposed algorithm A constructs a packing in polynomial time with a length of \(2OPT+1\), where OPT is the minimum possible packing length. As far as we know, this is the first non-trivial guaranteed estimate for 2-BCPP. We also conducted a numerical experiment to compare the solutions built by our approximate algorithms with the optimal solutions constructed by the CPLEX package. The experimental results confirmed the high efficiency of the developed algorithms.

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Abbreviations

A :

Approximation algorithm with guaranteed estimate

A1:

Algorithm A without the first stage

A_LO:

Algorithm A with lexicographic preordering of BCs

A1_LO:

Algorithm A1 with lexicographic preordering of BCs

BC:

Bar chart

2-BC:

Bar chart consisting of two bars

BCPP:

Bar Charts Packing Problem

2-BCPP:

Two Bar Charts Packing Problem

BF:

Algorithm Best Fit for bin packing

BLP:

Boolean Linear Programming

BPP:

Bin packing problem

2-DVPP:

Two-dimensional vector packing problem

CPLEX:

IBM ILOG CPLEX Optimization Studio

CPLEX1:

CPLEX with warm start

FF:

Algorithm First Fit for bin packing

FFD:

Algorithm First Fit Decreasing for bin packing

G :

Order-preserving greedy algorithm

GA :

Greedy algorithm

GA_LO:

Algorithm GA with lexicographic preordering of BCs

OPT :

Minimum packing length (the optimal value of the objective function)

SPP:

Strip packing problem

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Acknowledgements

The work is supported by Mathematical Center in Akademgorodok under Agreement No. 075-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.

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  1. Sobolev Institute of Mathematics, SB RAS, Novosibirsk, Russia, 630090

    Adil Erzin, Gregory Melidi, Stepan Nazarenko & Roman Plotnikov

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  1. Adil Erzin
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Erzin, A., Melidi, G., Nazarenko, S. et al. Two-Bar Charts Packing Problem. Optim Lett 15, 1955–1971 (2021). https://doi.org/10.1007/s11590-020-01657-1

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