(189b) An Approximate Optimal Blending Method Under Uncertainties

Maximizing profit while meeting product quality specifications is always the primary goal of the refining industry in a highly competitive market with increasingly stringent environmental regulations. For a refinery, the liquid products, such as gasoline and diesel, are the most important profits generators, usually yielding 60%-70% of revenues [1]. Generally, these liquids are produced by mixing different feedstock according to certain recipes in a well-developed blender system. However, noting that the properties of feedstocks in fact are subject to fluctuations and cannot be measured in real time, blending recipes designed based on the nominal feedstock parameters can be infeasible in reality and result in a substantial loss.

In this study, we are concerned with the optimal design of blending recipes to keep the final product on specification with high probability under uncertainties as well as maximizing the profit for the refinery. In practice, the product can be off specification with a small probability, usually 5%, due to uncertainties. Thus, the well-known chance-constrained programming is naturally employed to solve the blending problem modeled by a linear program with uncertain coefficients. By assuming a Gaussian distribution and using the Boole inequality, the joint chance-constrained programming formulation can be approximated conservatively as an individual chance-constrained formulation and further converted to a second-order cone program [2, 3], which is convex and easy to solve by CPLEX or GUROBI. However, this solution can be too conservative to apply in practice since the optimal assignment of violation chance for each quality constraint is very difficult to calculate. To handle this challenge, we develop a simple global optimization method for the individual chance-constrained programming to specify the violation chance for each constraint and investigate the implementation issues in this framework. To speed up the algorithm, the optimality-based variable tightening procedure is also integrated within our global optimization framework. In addition, the start-of-the-art global optimization software [4, 5] are used to compare and show the superior performance of the proposed approach. 

Moreover, we also notice that the global optimal solution for the individual chance-constrained programming is still a conservative solution for the original joint chance-constrained formulation. To further improve the quality of the solution, we employ a simple bisection search method to relax the required 5% violation chance in the algorithm and still achieve a feasible but more profitable result. This algorithm is tested in a pure blending problem with 12 chance constraints and a crude oil procurement problem with 14 chance constraints.

 

Reference

[1] Mendez, C. A., Grossmann, I. E., Harjunkoski, I., Kabore, P., 2006. A simultaneous optimization approach for off-line blending and scheduling of oil-refinery operations. Computers and Chemical Engineering 30, 614-634.

[2] Prekopa, A., 1995. Stochastic programming. Kluwer Academic Publishers, Netherlands.

[3] Nemirovski, A., Shapiro, A., 2006. Convex approximations of chance constrained programs. SIAM Journal on Optimization 17, 969-996.

[4] Misener, R., Floudas, C. A., 2014. ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations, Journal of Global Optimization 59, 503-526.

[5] Achterberg, T., 2009. SCIP: solving constraint integer programs. Mathematical Programming Computation, 1, 1â??41.