This paper studies the cyclic dynamic gaming case of the r-interdiction median problem with fortification (CDGC-RIMF), which is important for strengthening a facility’s reliability and invulnerability under various possible attacks. We formulated the CDGC-RIMF as a bi-objective mixed-integer linear programming (MILP) model with two opposing goals to minimize/maximize the loss from both the designer (leader) and attacker (follower) sides. The first goal was to identify the most cost-effective plan to build and fortify the facility considering minimum loss, whereas the attacker followed the designer to seek the most destructive way of attacking to cause maximum loss. We found that the two sides could not reach a static equilibrium with a single pair of confrontational plans in an ordinary case, but were able to reach a dynamically cyclic equilibrium when the plan involved multiple pairs. The proposed bi-objective model aimed to discover the optimal cyclic plans for both sides to reach a dynamic equilibrium. To solve this problem, we first started from the designer’s side with a design and fortification plan, and then the attacker was able to generate their worst attack plan based on that design. After that, the designer changed their plan again based on the attacker’s plan in order to minimize loss, and the attacker correspondingly modified their plan to achieve maximum loss. This game looped until, finally, a cyclic equilibrium was reached. This equilibrium was deemed to be optimal for both sides because there was always more loss for either side if they left the equilibrium first. This game falls into the subgame of a perfect Nash equilibrium—a kind of complete game. The proposed bi-objective model was directly solved by the CPLEX solver to achieve optimal solutions for small-sized problems and near-optimal feasible solutions for larger-sized problems. Furthermore, for large-scale problems, we developed a heuristic algorithm that implemented dynamic iterative partial optimization alongside MILP (DIPO-MILP), which showed better performance compared with the CPLEX solver when solving large-scale problems.
An a posteriori verification method for generalized real-symmetric eigenvalue problems in large-scale electronic state calculations
