An Approximation-Based Approach for Chance-Constrained Vehicle Routing and Air Traffic Control Problems

We proposed a polynomial approximation-based approach to solve a specific type of chance-constrained optimization problem that can be equivalently transformed into a convex program. This type of chance-constrained optimization is in great needs of many applications, and most solution techniques are problem-specific. Our essential contribution is to provide an all-purpose solution approach through Monte Carlo and establish the linkage between our obtained optimal solution with the true optimal solution. Thanks to fast-advancing computer hardware, our method would be increasingly appealing to businesses, including small businesses. We present the numerical results including the air traffic flow management (ATFM) and the capacitated routing problem (CVRP) with stochastic demand to show that our approach with Monte Carlo will yield high-quality, timely, and stable solutions. We apply the approach to the ATFM problem to efficiently solve the weather-affected traffic flow management problem. Since there are massive independent approximation processes in the polynomial approximation-based approach, a distributed computing framework is designed to carry out the computation. For the CVRP problem, we conclude that our chance-constrained method has some strategic advantages to serve a logistics company well when resource costs and service guarantees are of concern.