Pooling Problem: Alternate Formulations and Solution Methods

Abstract

The pooling problem, which is fundamental to the petroleum industry, describes a situation in which products possessing different attribute qualities are mixed in a series of pools in such a way that the attribute qualities of the blended products of the end pools must satisfy given requirements. It is well known that the pooling problem can be modeled through bilinear and nonconvex quadratic programming. In this paper, we investigate how best to apply a new branch-and-cut quadratic programming algorithm to solve the pooling problem. To this effect, we consider two standard models: One is based primarily on flow variables, and the other relies on the proportion of flows entering pools. A hybrid of these two models is proposed for general pooling problems. Comparison of the computational properties of flow and proportion models is made on several problem instances taken from the literature. Moreover, a simple alternating procedure and a variable neighborhood search heuristic are developed to solve large instances and compared with the well-known method of successive linear programming. Solution of difficult test problems from the literature is substantially accelerated, and larger ones are solved exactly or approximately.pooling problem, bilinear programming, branch-and-cut, heuristics, variable neighborhood search

GloMIQO: Global mixed-integer quadratic optimizer

“…In addition to relaxing bilinear terms using the convex hull, the BARON preprocessing routines detect connected multivariable terms within quadratic equations [26]. -Branch-and-cut for QCQP [18,19,20,21] Audet et al [18] discuss their implementation of a branch-and-cut global optimization algorithm for QCQP which made contributions to generating cutting planes and boundupdating strategies. -Couenne [27,88] Like αBB and BARON, Couenne addresses generic MINLP to ε-global optimality and relaxes bilinear terms using the convex hull.…”

“…, M} Major applications of MIQCQP include quality blending in process networks, separating objects in computational geometry, and portfolio optimization in finance. Specific instantiations of MIQCQP in process networks optimization problems include: pooling problems [5,12,20,28,53,65,66,67,79,93,97,98,100,101,107,137,139], distillation sequences [8,54,58], wastewater treatment and total water systems [9,13,22,29,50,62,71,73,108,109], hybrid energy systems [23,24,49], heat exchanger networks [39,56], reactorseparator-recycle systems [75,76], separation systems [119], data reconciliation [115], batch processes [86], crude oil scheduling [78,80,81,104,103], and natural gas production [82,…”

APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes

“…Nonconvex bilinear terms in the standard pooling problem arise from tracking the levels of linearlyblending fuel qualities about the pooling nodes to meet constraints on the composition of the final products [Visweswaran, 2009]. Among the many notable contributions towards solving the standard pooling problem [Adhya et al, 1999, Almutairi and Elhedhli, 2009, Audet et al, 2004, Ben-Tal et al, 1994, Chakraborty, 2009, Quesada and Grossmann, 1995, Foulds et al, 1992, Greenberg, 1995, Haverly, 1978, Lasdon et al, 1979, Lodwick, 1992, Pham et al, 2009, Tawarmalani and Sahinidis, 2002, the most directly relevant to the work presented in this paper and the computational tool APOGEE are those of: Floudas and Visweswaran , who were the first to rigorously solve the pooling problem to global optimality; Foulds et al [1992], who developed a linear relaxation of the QCQP by replacing each bilinear term with their convex and concave hulls [Al-Khayyal andFalk, 1983, McCormick, 1976]; Ben-Tal et al [1994], who introduced an alternative q-formulation of the pooling problem that often has fewer nonconvex bilinear terms than the original p-formulation [Audet et al, 2004]; and Tawarmalani and Sahinidis [2002], who showed that augmenting the q-formulation with reformulation-linearization technique cuts Adams, 1999, Sherali andAlameddine, 1992] proposed by Quesada and Grossmann [1995] produces a linear relaxation of the pooling problem that strictly dominates both the p-and q-formulations. Depending on the formulation, the standard pooling problem can be classified as a linear objective with quadratic constraints (p-formulation) or a quadratic objective with quadratic constraints (q-and pq-formulations).…”

“…The generalized pooling problem increases the complexity of the pooling problem by allowing flow between the intermediate storage nodes and transforming the network topology from a pre-determined structure into an optimally-chosen configuration [Audet et al, 2004, Meyer and Floudas, 2006. Choosing the interconnections between source, intermediate, and output nodes is combinatorially complex.…”

“…These bilinear terms give rise to a multiplicity of local optima which prevent linear, convex, and stochastic solvers from certifying global optimality of the quadratically-constrained quadratic program (QCQP) that mathematically represents the standard pooling problem [Floudas, 2000, Floudas and Pardalos, 1995. Adding further complexity to industrially-relevant pooling problems are the binary decisions associated with the activation or deactivation of specific process units and/or connecting pipelines and the variety of piecewise nonconvex equations needed to mathematically model complex product qualities such as those in the EPA definition of polluting emissions [Audet et al, 2004, Meyer and Floudas, 2006.…”