A strong P-formulation for global optimization of industrial water-using and treatment networks

The problem of finding the optimal flow allocation within an industrial water-using and treatment network can be formulated into nonconvex nonlinear program or nonconvex mixed-integer nonlinear program. The efficiency of global optimization of the nonconvex program relies heavily on the strength of the problem formulation. In this paper, we propose a variant of the commonly used P-formulation, called the P \(^*\) ${}^{\ast }$ -formulation, for the water treatment network (WTN) and the total water network (TWN) that includes water-using and water treatment units. For either type of networks, we prove that the P \(^*\) ${}^{\ast }$ -formulation is at least as strong as the P-formulation under mild bound consistency conditions. We also prove for either type of networks that the P \(^*\) ${}^{\ast }$ -formulation is at least as strong as the split-fraction based formulation (called SF-formulation) under certain bound consistency conditions. The computational study shows that the P \(^*\) ${}^{\ast }$ -formulation significantly outperforms the P- and the SF-formulations. For some problem instances, the P \(^*\) ${}^{\ast }$ -formulation is faster than the other two formulations by several orders of magnitudes.