Vehicle Routing with Space- and Time-Correlated Stochastic Travel Times: Evaluating the Objective Function

Abstract

We study how to model and handle correlated travel times in two-stage stochastic vehicle routing problems. We allow these travel times to be correlated in time and space, that is, the travel time on one link in one period can be correlated to travel times on the same link in the next and previous periods, as well as travel times on neighboring links (links sharing a node) in both the same and the following periods. Hence, we are handling a very high-dimensional dependent random vector. We shall discuss how such vehicle routing problems should be modeled in time and space, how the random vector can be represented, and how scenarios (discretizations) can meaningfully be generated to be used in a stochastic program. We assume that the stochastic vehicle routing problem is being solved by a search heuristic, and focus on the objective function evaluation for any given solution. Numerical procedures are given and tested. As an example, our largest case has 142 nodes, 418 road links and 60 time periods, leading to 25,080 dependent random variables. To achieve an objective-function evaluation stability of 1%, we need only fifteen scenarios for problem instances with 64 customer nodes and 9 vehicles.

On scenario construction for stochastic shortest path problems in real road networks

“…Following Guo, Wallace and Kaut (2019) and Huang and Gao (2018), this paper investigates various SSP problems with spatially and temporally correlated speeds based on real travel speeds from the Caltrans Performance Measurement System (PeMS) at http://pems.dot.ca.gov/. We establish that there are statistically significant correlations in time and space in this data set.…”

“…. , S + m − 1, S + m, where m is an integer and set as 4 in our paper, based on tests in Guo, Wallace and Kaut (2019). For each scenario set, we take the problem at hand (in our case the SSP problem) and use a solution method from the literature to find an optimal (or near-optimal) solution.…”

Understanding the marginal distributions and correlations of link travel speeds in road networks

“…There is an increased use of stochastic travel speeds (times) in various real-world decision-making problems such as vehicle routing 8 - 12 , shortest or reliable paths 13 - 17 , ridesharing 18 , 19 , traffic assignment 20 , 21 and traffic or travel time (speed) predictions 22 - 24 . As two important indicators to describe stochastic travel speeds, marginal distributions 8 - 13 , 15 - 17 , 21 , 23 and correlations 9 , 13 - 17 , 22 - 24 have been widely used in these studies.…”

“…For reliable path problems, the inaccuracy in the reliability objective is 15% on average without the consideration of correlations in link travel time 16 . However, these studies usually assume that the stochastic travel speeds obey certain continuous 10 - 12 , 15 - 17 , 21 or discrete 9 , 14 , 24 distributions and have some simple 9 , 14 - 16 or even no correlations 10 , 21 .…”