(371i) An Efficient Approach to Solving Stochastic Optimal Control Problems with Application to Sustainability Assessment of a Complex Integrated Model

Sustainability has emerged as a new discipline for analysis and evaluation of all ecosystems components and human development in an integrated way. One of the main goals of sustainability analysis is the creation and maintenance of the conditions under which humans and nature can coexist in productive harmony, and support present and future generations. It has been recognized that this topic is, operationally and conceptually, one of the most difficult that modern science has faced. Literature reports systematic mathematical programming approaches to model integrated systems and the different dimensions of sustainability. Some models are deterministic, but a complex ecosystem could be affected by time dependent uncertainties that can change its behavior significantly. As a general approach, the economic, ecological and social components of an ecosystem can be represented as a compartmental dynamic model and then posed as an optimal control problem by maximizing/minimizing alternative sustainability indicators. The task is to provide optimal policy guidelines which drive the system towards a more sustainable behavior. Further, if time dependent uncertainties are included in the formulation, the resulting problem is a stochastic optimal control problem.

This paper intends to contribute in this area in two ways. First, we have developed a novel approach to address stochastic optimal control problems based on the Better Optimization of Nonlinear Uncertain Systems (BONUS) algorithm (Diwekar, 2015). This method avoids the need for excessive model evaluations through a reweighting scheme using kernel density estimation and relies on sampling to estimate the probabilistic objective function. A batch reactor for biodiesel production is used as a case study to illustrate the proposed approach. Results for a maximum profit problem indicate that the optimal objective function and the optimal profiles are similar to those obtained by the Stochastic Maximum Principle (SMP) technique. Also, by searching on the whole decision space through sampling, the BONUS approach avoids local optima commonly found by the SMP technique.

As a second contribution, we have reformulated an integrated compartmental model for sustainability analysis (Kotecha et al., 2013) as a stochastic optimal control problem by considering the uncertain behavior of some model parameters (mortality and birth rate; wages). The resulting problem has then been solved through our efficient approach based on BONUS. The aim is to maximize the sustainability of the system, analyzing it in a scenario of instability and using ecological indicators as a measure of sustainability. The results show the complexity of ecosystem behavior and the big impacts that human activities have over the ecological components, as well as the difficulty to mitigate them. Results can also help decision makers to identify critical policy guidelines that allow such integrated systems to be driven into a more sustainable framework.

Keywords: Sustainability theory, stochastic optimal control, BONUS

References

Diwekar, U.; David, A., BONUS Algorithm for Large Scale Stochastic Nonlinear Programming Problems, Springer Briefs in Optimization, 2015.

Kotecha, P.; Diwekar, U.; Cabezas, H., Model-based approach to study the impact of biofuels on the sustainability of an ecological system, Clean Technologies and Environmental Policy, 15, 21–33, 2013.