We study two-stage robust optimization problems with mixed discrete-continuous decisions in both stages. Despite their broad range of applications, these problems pose two fundamental challenges: (i) they constitute infinite-dimensional problems that require a finite-dimensional approximation, and (ii) the presence of discrete recourse decisions typically prohibits duality-based solution schemes. We address the first challenge by studying a K-adaptability formulation that selects K candidate recourse policies before observing the realization of the uncertain parameters and that implements the best of these policies after the realization is known. We address the second challenge through a branch-and-bound scheme that enjoys asymptotic convergence in general and finite convergence under specific conditions. We illustrate the performance of our algorithm in numerical experiments involving benchmark data from several application domains.

Highlights• We study multi-period routing problems with arrival-time consistency requirements.• We present the first-ever exact method for the Consistent Traveling Salesman Problem.• We propose novel valid inequalities and associated separation techniques.• Instances with up to 50 customers and 5 periods are solved to guaranteed optimality.• Consistency can be enforced with a modest (< 2% on average) increase in routing costs. AbstractWe develop an exact solution framework for the Consistent Traveling Salesman Problem. This problem calls for identifying the minimum-cost set of routes that a single vehicle should follow during the multiple time periods of a planning horizon, in order to provide consistent service to a given set of customers. Each customer may require service in one or multiple time periods and the requirement for consistent service applies at each customer location that requires service in more than one time period. This requirement corresponds to restricting the difference between the earliest and latest vehicle arrival-times, across the multiple periods, to not exceed some given allowable limit. We present three mixed-integer linear programming formulations for this problem and introduce a new class of valid inequalities to strengthen these formulations. The new inequalities are used in conjunction with traditional traveling salesman inequalities in a branch-and-cut framework. We test our framework on a comprehensive set of benchmark instances, which we compiled by extending traveling salesman instances from the well-known TSPLIB library into multiple periods, and show that instances with up to 50 customers, requiring service over a 5-period horizon, can be solved to guaranteed optimality. Our computational experience suggests that enforcing arrival-time consistency in a multi-period setting can be achieved with merely a small increase in total routing costs.

This paper studies robust variants of an extended model of the classical Heterogeneous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles with different capacities, availabilities, fixed costs and routing costs is used to serve customers with uncertain demand. This model includes, as special cases, all variants of the HVRP studied in the literature with fixed and unlimited fleet sizes, accessibility restrictions at customer locations, as well as multiple depots. Contrary to its deterministic counterpart, the goal of the robust HVRP is to determine a minimum-cost set of routes and fleet composition that remains feasible for all demand realizations from a pre-specified uncertainty set. To solve this problem, we develop robust versions of classical node-and edge-exchange neighborhoods that are commonly used in local search and establish that efficient evaluation of the local moves can be achieved for five popular classes of uncertainty sets. The proposed local search is then incorporated in a modular fashion within two metaheuristic algorithms to determine robust HVRP solutions. The quality of the metaheuristic solutions is quantified using an integer programming model that provides lower bounds on the optimal solution. An extensive computational study on literature benchmarks shows that the proposed methods allow us to obtain high quality robust solutions for different uncertainty sets and with minor additional effort compared to deterministic solutions.

Deliver today or deliver tomorrow?

The consistent traveling salesman problem aims to identify minimum-cost routes to be followed by a single vehicle so as to provide a set of customers with service that adheres to arrival-time consistency across the multiple time periods of a planning horizon. In this paper, we address this problem via a new, exact algorithm that decomposes the problem into a sequence of single-period traveling salesman problems with time windows within a branch-and-bound search procedure. The new algorithm is highly competitive, as it is able to solve to guaranteed optimality instances with up to 100 customers requiring service over a five-period horizon, effectively doubling the size of instances that were solvable by the previous state of the art. Furthermore, and in contrast to all previous approaches, the new algorithm accounts for route duration limits, whenever applicable, as well as incorporates the flexibility for the vehicle to idle before providing service to a customer. We in fact show that, for the benchmark instances we considered, the cost of implementing consistent routes can be reduced significantly if the vehicle is allowed to idle en route, further motivating the need for algorithmic schemes to incorporate this realistic option.