Estimating the Benefits of Considering Travel Time Variability in Urban Distribution

Urban distribution systems typically require carriers to deliver goods to receivers within specified time windows. This paper presents procedures that allow the variable nature of travel times to be incorporated within optimization methods for the vehicle routing problem with time windows. Expressions are presented for determining the penalty costs associated with truck arrivals at customer locations with time windows when the travel times between customers are normally distributed. Formulations of this problem using stochastic programming and robust optimization (RO) are presented. Stochastic programming procedures require numerical integration techniques to be used and thus are computationally demanding. However, RO allows solutions to be generated with only a limited number of travel time scenarios between customers and, thus provides a practical means of incorporating travel time variability. A procedure was developed for estimating the cost savings of implementing the RO solution. The benefits of explicitly considering the variability of travel times between customers were estimated using a case study based on a distribution problem in Melbourne, Australia.