The wastewater treatment network (WWTN) optimization problem can be viewed as an extension of the generalized pooling problem (GPP), where not only the different streams are mixed and split, but also some chemical components are removed from the system. It is well known that a strong formulation is key to efficient global solution of GPP, and multi-commodity flow formulations have been recognized as such strong formulations. The paper presents the first multi-commodity flow formulation for the WWTN optimization problem. We discuss the challenges in adapting the classical multi-commodity flow approaches for a WWTN, and propose the conditions under which the selected commodities are adequate for modeling a WWTN. We prove that the proposed formulation is at least as strong as a classical component based formulation. We show the significant computational advantage of the proposed formulation over the component based formulation through 16 problem instances from the literature.

废水处理网络（WWTN）优化问题可以看作是广义汇集问题（GPP）的扩展，在该问题中，不仅将不同的流进行混合和分配，而且还从系统中除去了一些化学成分。众所周知，强大的配方是GPP有效的全球解决方案的关键，并且多商品流程配方已被认为是这种强大的配方。本文介绍了第一个针对WWTN优化问题的多商品流程公式。我们讨论了将经典的多商品流方法应用于WWTN时遇到的挑战，并提出了所选商品足以用于WWTN建模的条件。我们证明了所提出的配方至少与基于经典成分的配方一样坚固。