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The pooling problem is a classical NP-hard problem in the chemical processand petroleum industries. This problem is modeled as a nonlinear, nonconvexnetwork flow problem in which raw materials with different specifications areblended in some intermediate tanks, and mixed again to obtain the finalproducts with desired specifications. The analysis of the pooling problem isquite an active research area, and different exact formulations, relaxationsand restrictions are proposed. In this paper, we focus on a recently proposedrank-one-based formulation of the pooling problem. In particular, we study arecurring substructure in this formulation defined by the set of nonnegative,rank-one matrices with bounded row sums, column sums, and the overall sum. Weshow that the convex hull of this set is second-order cone representable. Inaddition, we propose an improved compact-size polyhedral outer-approximationand families of valid inequalities for this set. We further strengthen theseconvexification approaches with the help of various bound tightening techniquesspecialized to the instances of the pooling problem. Our computationalexperiments show that the newly proposed polyhedral outer-approximation canimprove upon the traditional linear programming relaxations of the poolingproblem in terms of the dual bound. Furthermore, bound tightening techniquesreduce the computational time spent on both the exact, linear programming andmixed-integer linear programming relaxations.

Improved Rank-One-Based Relaxations and Bound Tightening Techniques for the Pooling Problem