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Relaxations and discretizations for the pooling problem

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Abstract

The pooling problem is a folklore NP-hard global optimization problem that finds applications in industries such as petrochemical refining, wastewater treatment and mining. This paper assimilates the vast literature on this problem that is dispersed over different areas and gives new insights on prevalent techniques. We also present new ideas for computing dual bounds on the global optimum by solving high-dimensional linear programs. Finally, we propose discretization methods for inner approximating the feasible region and obtaining good primal bounds. Valid inequalities are derived for the discretized models, which are formulated as mixed integer linear programs. The strength of our relaxations and usefulness of our discretizations is empirically validated on random test instances. We report best known primal bounds on some of the large-scale instances.

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Notes

  1. For the p-formulation, specifications serve the role of commodities and (4a) is a commodity balance constraint.

  2. For convenience, an equation that is satisfied by all feasible points in a set is also referred to as a valid inequality.

  3. Some generalized instances can also be found in Alfaki and Haugland [4] but in our experience the pq-formulations of these instances were solved by \(\texttt {BARON}\) in less than 15 min and hence seem to be relatively ease.

  4. Amongst the various choices for discretizing \(\mathbb {P}\), flow discretization was by far the best choice but the solutions from solving \({{\mathrm{\mathcal {B}}}}(\mathbb {F}\mathbb {P})\) were still very poor in comparison to discretizing \(\mathbb {PQ}\).

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Acknowledgments

We thank one referee whose elaborate comments helped us significantly improve the readibility of this paper. Megan Ryan, a graduate student at Clemson University, assisted the first author with the computational experiments in Sect. 5.1.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, SC, USA

    Akshay Gupte

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Shabbir Ahmed & Santanu S. Dey

  3. ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Myun Seok Cheon

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Gupte, A., Ahmed, S., Dey, S.S. et al. Relaxations and discretizations for the pooling problem. J Glob Optim 67, 631–669 (2017). https://doi.org/10.1007/s10898-016-0434-4

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