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New multi-commodity flow formulations for the pooling problem

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Abstract

The pooling problem is a nonconvex nonlinear programming problem with numerous applications. The nonlinearities of the problem arise from bilinear constraints that capture the blending of raw materials. Bilinear constraints are well-studied and significant progress has been made in solving large instances of the pooling problem to global optimality. This is due in no small part to reformulations of the problem. Recently, Alfaki and Haugland proposed a multi-commodity flow formulation of the pooling problem based on input commodities. The authors proved that the new formulation has a stronger linear relaxation than previously known formulations. They also provided computational results which show that the new formulation outperforms previously known formulations when used in a global optimization solver. In this paper, we generalize their ideas and propose new multi-commodity flow formulations based on output, input and output and (input, output)-commodities. We prove the equivalence of formulations, and we study the partial order of formulations with respect to the strength of their LP relaxations. In an extensive computational study, we evaluate the performance of the new formulations. We study the trade-off between disaggregating commodities and therefore increasing the size of formulations versus strengthening the relaxed linear programs and improving the computational performance of the nonlinear programs. We provide computational results which show that output commodities often outperform input commodities, and that disaggregating commodities further only marginally strengthens the linear relaxations. In fact, smaller formulations often show a significantly better performance when used in a global optimization solver.

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Acknowledgments

This research was supported by the ARC Linkage Grant No. LP110200524, Hunter Valley Coal Chain Coordinator (hvccc.com.au) and Triple Point Technology (tpt.com). The authors would like to thank Dr Hamish Waterer for his contributions, both computationally and theoretically, to this research. The authors would also like to thank the two anonymous referees for their helpful comments which improved the quality of the paper.

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Authors and Affiliations

  1. H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Natashia Boland

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, NSW, 2308, Australia

    Thomas Kalinowski & Fabian Rigterink

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  1. Natashia Boland
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  3. Fabian Rigterink
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Correspondence to Fabian Rigterink.

Appendix: Detailed computational results

Appendix: Detailed computational results

Table 10 Relative LP objectives and their baselines for all formulations and instances
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Table 11 Total solve times of the NLP, relative NLP lower and upper bounds and their baselines for all formulations and instances
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Table 12 Average relative LP objectives and number of times a formulation found the maximum LP objective for a set of instances
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Table 13 Average relative NLP lower bounds and number of times a formulation found the maximum NLP lower bound for a set of instances
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Table 14 Average relative NLP upper bounds and number of times a formulation found the minimum NLP upper bound for a set of instances
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Boland, N., Kalinowski, T. & Rigterink, F. New multi-commodity flow formulations for the pooling problem. J Glob Optim 66, 669–710 (2016). https://doi.org/10.1007/s10898-016-0404-x

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