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This paper proposes a dynamic vehicle routing problem (DVRP) model with nonstationary stochastic travel times under traffic congestion. Depending on the traffic conditions, the travel time between two nodes, particularly in a city, may not be proportional to distance and changes both dynamically and stochastically over time. Considering this environment, we propose a Markov decision process model to solve this problem and adopt a rollout-based approach to the solution, using approximate dynamic programming to avoid the curse of dimensionality. We also investigate how to estimate the probability distribution of travel times of arcs which, reflecting reality, are considered to consist of multiple road segments. Experiments are conducted using a real-world problem faced by Singapore logistics/delivery company and authentic road traffic information.Index Terms-Dynamic vehicle routing problem, approximate dynamic programming, uncertain travel times, rollout algorithm.

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Abstract:In this paper, we examine a sustainable economic order quantity (S-EOQ) problem with a stochastic lead-time and multi-modal transportation options. With the S-EOQ, decisions of order quantities, as well as a reorder point could be influenced by various factors, including unit price, stock-out cost, lead-time variability and emission costs. For a better understanding, we present a mathematical model of the concomitant S-EOQ problem and, using numerical experiments, explore various scenarios to determine the effects of incorporating sustainability considerations into the traditional inventory model on operational decisions including the choice of transportation modal combination and the sourcing decisions