Abstract
Process design procedures under uncertainty result in stochastic optimization problems whose resolution is complex due to the large uncertainty space, which hinders the application of optimization approaches, as well as the establishment of relationships between input and output variables. On the other hand, supervised machine learning (SML) offers tools with which to develop surrogate models, which are computationally inexpensive and efficient. This paper proposes a procedure based on modern design of experiments, deterministic optimization, SML tools, and global sensitivity analysis (GSA) to reduce the size of the uncertainty space for stochastic optimization problems. The proposal is illustrated with a case study based on the stochastic design of flotation plants. The results reveal that surrogate models of stochastic formulation enable the prediction of the structure, profitability parameters, and metallurgical parameters of designed flotation plants, as well as reducing the size of the uncertainty space via GSA and, consequently, establishing relationships between the input and output variables of the stochastic formulation.
Highlights
The procedures proposed in previous research for process design are commonly based on mathematical programming, which exhibits the following characteristics: first, it implements a superstructure to represent design alternatives from which a set of optimal alternatives can be selected; second, it uses mathematical expressions to model design alternatives, constraints, and goals
This approach assumes that the parameters are known; in practice, uncertainties are prevalent in industrial processes due to inaccurate measurement, forecast error, or lack of information, and these effects on process output variables could be critical
The integration of supervised machine learning (SML) tools and mathematical optimization has been reported in previous research to replace rigorous models in mixed-integer nonlinear programming (MINLP)/mixed-integer linear programming (MILP) problems [15,16,17,18] raised from design procedures, optimization under uncertainty [19], surrogate modeling of stochastic simulators to perform sensitivity analysis [20], and to overcome MINLP problems in the dynamic optimization of sequential batch processes [21]; its application in metallurgical process design has not been reported
Summary
The procedures proposed in previous research for process design are commonly based on mathematical programming, which exhibits the following characteristics: first, it implements a superstructure to represent design alternatives from which a set of optimal alternatives can be selected; second, it uses mathematical expressions to model design alternatives, constraints, and goals. The integration of SML tools and mathematical optimization has been reported in previous research to replace rigorous models in MINLP/MILP problems [15,16,17,18] raised from design procedures, optimization under uncertainty [19], surrogate modeling of stochastic simulators to perform sensitivity analysis [20], and to overcome MINLP problems in the dynamic optimization of sequential batch processes [21]; its application in metallurgical process design has not been reported This manuscript presents a methodology to reduce the uncertainty space of stochastic programming problems, which emerge from design procedures. The codes utilized in this study were developed in JupyterLab, using the python kernel, and they are attached as Supplementary Materials
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