Mathematical modeling and global optimization of large-scale extended pooling problems with the (EPA) complex emissions constraints

Abstract

Environmental Protection Agency (EPA) Title 40 Code of Federal Regulations Part 80.45: Complex Emissions Model [40CFR80.45, 2007] codifies a mathematical model of gasoline emissions for reformulated gasoline (RFG) as a function of eleven fuel properties. In this paper we propose an extended pooling problem to maximize the profit of blending reformulated gasoline on a predetermined network structure of feed stocks, intermediate storage tanks, and gasoline products subject to applicable environmental standards. A mixed-integer nonlinear programming (MINLP) model is introduced which is nonconvex due to the presence of bilinear, polynomial, and fractional power terms. A mixed-integer linear programming (MILP) relaxation of the extended pooling problem is proposed and we introduce several test cases from small to medium to large scale and solve them to global optimality. The large-scale test case involves 14 feed stocks, 5 pools, and 10 products and consists of 1104 continuous variables, 150 binary variables, and 640 nonlinear terms. The nonconvexities in the large-scale case study include 410 bilinear terms, 40 polynomial terms, and 10 terms raised to a fractional power.