A novel algorithm for the global solution of mixed-integer bi-level multi-follower problems and its application to Planning & Scheduling integration

Optimization problems involving a leader decision maker with multiple follower decision makers are referred to as bi-level multi-follower programming problems (BMF-P). In this work, we present novel algorithms for the exact and global solution of two classes of bi-level programming problems, namely (i) bi-level multi-follower mixed-integer linear programming problems (BMF-MILP) and (ii) bi-level multi-follower mixed-integer convex quadratic programming problems (BMF-MIQP) containing both integer and continuous variables at all optimization levels. Based on multi-parametric programming theory, the main idea is to recast the lower level, follower, problems as multi-parametric programming problems, in which the optimization variables of the upper level, leader, problem are considered as parameters for the lower level problems. The resulting exact multi-parametric mixed-integer linear or quadratic solutions are then substituted into the upper level problem, which can be solved as a set of single-level, independent, deterministic mixed-integer optimization problems. The proposed algorithm is applied for the solution of the challenging problem of planning and scheduling integration.