(14b) A Decomposition Algorithm for the Optimal Design of Integrated Sites Under Uncertainty

In this work we present a novel implementation of Benders decomposition for the optimal design of large-scale chemical integrated sites subject to uncertainty. An integrated site is a process network where several manufacturing plants producing different chemicals, final products and intermediates, are closely coupled (Wassick, 2009; Terrazas-Moreno et al., 2010). The objective of this work is to design an integrated site with maximum "effective production capacity", given a capital investment constraint. A set of failure modes can occur at random times, and with random durations, in any of the plants in the network, decreasing its production capacity. Therefore, the effective production represents a fraction of the production if no failures were to happen. To maximize the effective production capacity (minimize the effect of failures) a superstructure of the integrated site is postulated that includes parallel production units and intermediate storage tanks. The production capacity of the plants is also a degree of freedom, allowing excess capacity for building up inventories in the storage tanks. This problem can be modeled as a two-stage mixed-integer stochastic programming problem with endogenous uncertainties. Random failures can occur in a process only if it is selected from the superstructure. Since the optimal flowsheet is not known a priori, the failures (endogenous uncertainties) that affect the flowsheet and the structure of the scenario tree are decision-dependent. The design variables of the integrated site are first-stage decisions, while the second-stage decisions involve operational variables such as internal flows within the integrated site, and the rates of accumulation or depletion of storage tank levels. Since the number of scenarios grows exponentially in the number of failure modes, the solution to large-scale instances of the problem becomes intractable. To overcome this challenge we propose a decomposition technique based on a novel implementation of Benders decomposition. The master problem includes the design variables and the operational variables of some of the scenarios with highest probability. The sub-problem includes the operational variables of the rest of the scenarios. The contribution of our method is that we solve the sub-problem in a reduced space that includes only the scenarios relevant to the flowsheet defined in the master problem. Furthermore, since the structure of the scenario tree in the sub-problem is known, we do not need to enforce non-anticipativity constraints. The result is a sub-problem with significantly fewer variables and constraints. In addition to the fact that the master problem can be solved for a small number of scenarios, this technique makes our approach computationally efficient. The main theoretical challenge of the method is to prove that the algorithm is equivalent to Benders decomposition even when the master problem and the sub-problem are solved for scenarios that are defined on different spaces. The method was tested with the design of an industrial integrated site consisting of nine potential production plants and about one hundred possible failure modes. The Pareto-optimal curve for maximizing the effective production capacity and minimizing the capital investment was obtained using the ?'-constraint method. Reductions in computational time of up to several orders of magnitude were obtained with the proposed Benders decomposition scheme when compared to the full-space solution of this example.

References

Wassick, J. M., (2009). Enterprise-wide optimization in an integrated chemical complex. Computers and Chemical Engineering, 33(12), 1950 - 1963. Terrazas-Moreno S., I. E. Grossmann, J. M. Wassick and S. J. Bury (2010). Optimal design of reliable integrated chemical production sites. Submitted for publication