The robust vehicle routing problem with time windows

Abstract

This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.

Routing Optimization Under Uncertainty

“…In these robust optimization models, uncertain parameters are characterized by uncertainty sets without any information on their probability distributions. The budget of uncertainty robust optimization approach introduced by Bertsimas andSim (2003, 2004) has been adopted to address routing optimization problems (see Sungur et al, 2008;Souyris et al, 2007;Agra et al, 2013;and Lee et al, 2012). Although these robust optimization models are more computationally amiable than their stochastic counterparts, they may represent uncertainty inadequately and result in possibly conservative solutions.…”

Models and Algorithms for Stochastic and Robust Vehicle Routing with Deadlines

“…(Please, provide the mansucript number!) Lee et al (2012) and Agra et al (2013) considered the VRP with time windows where one seeks to find a feasible solution to any realization where the travel time uncertainty is defined using the notion of budget of uncertainty as introduced in Bertsimas and Sim (2004). Jaillet et al (2014) proposed a mathematical framework for a routing problem with soft time windows when exact probability distributions of travel times are not known for a single uncapacitated vehicle.…”

A maritime inventory routing problem with stochastic sailing and port times

“…Agra et al [3] solved a full-load ship routing and scheduling problem with uncertain travel times using robust optimization. Weather conditions affect both sailing speeds and the loading and unloading operations for supply vessels servicing offshore installations, and various heuristic strategies to achieve robust weekly voyages and schedules were analyzed by Halvorsen-Weare and Fagerholt [9].…”

“…For instance, We can see that t 11 = 1, t E 11 = 2, t 21 = 1, t E 21 = 2, t 31 = 7, t E 31 = 8, t 12 = 14, t E 12 = 15, t 32 = 8.5, t E 32 = 9.5. Suppose there is backlog at nodes (1, 2) and (3,2). In addition to the inequalities (37) for (1, 2) and (3, 2) defining the lower bound on the backlog, the following inequalities, corresponding to the critical paths to nodes (1, 2) and (3, 2) are added to limit the time variables:…”