The pooling problem,also called the blending problem,is fundamental in production planning of petroleum.It can be formulated as an optimization problem similar with the minimum-cost flow problem.However,Alfaki and Haugland(J Glob Optim 56:897–916,2013)proved the strong NP-hardness of the pooling problem in general case.They also pointed out that it was an open problem to determine the computational complexity of the pooling problem with a fixed number of qualities.In this paper,we prove that the pooling problem is still strongly NP-hard even with only one quality.This means the quality is an essential difference between minimum-cost flow problem and the pooling problem.For solving large-scale pooling problems in real applications,we adopt the non-monotone strategy to improve the traditional successive linear programming method.Global convergence of the algorithm is established.The numerical experiments show that the non-monotone strategy is effective to push the algorithm to explore the global minimizer or provide a good local minimizer.Our results for real problems from factories show that the proposed algorithm is competitive to the one embedded in the famous commercial software Aspen PIMS.
