New multi-commodity flow formulations for the pooling problem

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.

Relaxations and discretizations for the pooling problem

“…We skip the details of this hybrid formulation (HYB) since it can be easily obtained by combining the previous sections. Similar formulations can also be found in Boland et al (2015).…”

Pooling problems under perfect and imperfect competition

“…They highlight the "concentration," "proportion," and an augmented "proportion" formulation, which are also known under their more cryptic names as the P, Q, and PQ formulations, respectively. Alfaki and Haugland [2] and Boland et al [5] proposed several multi-commodity flow formulations for the continuous version of the pooling problem.…”

A multi-commodity flow formulation for the optimal design of wastewater treatment networks

“…In Alfaki and Haugland's work [28], Λ contains the compositions of flows entering the sources, which are known constants. In the work of Boland et al [29], Λ can also be the compositions of flows entering the terminals, which are variables in the optimization. In this paper, we allow Λ to have any values that, together with water, enable a valid and efficient…”

“…Note that in [28], the commodity flows are the total flows entering the sources, for which Λ may not be square and invertible. When using the demand commodity flows as in [29], Λ is not constant, so even if Λ is invertible, the strong nonlinear relationship between f o,v and f i,v will overly complicate the optimization problem.…”