Finding the $$\mathrm{K}$$ Mean-Standard Deviation Shortest Paths Under Travel Time Uncertainty

Abstract

The mean-standard deviation shortest path problem (MSDSPP) incorporates the travel time variability into the routing optimization. The idea is that the decision-maker wants to minimize the travel time not only on average, but also to keep their variability as small as possible. Its objective is a linear combination of mean and standard deviation of travel times. This study focuses on the problem of finding the best-\(K\) optimal paths for the MSDSPP. We denote this problem as the KMSDSPP. When the travel time variability is neglected, the KMSDSPP reduces to a \(K\)-shortest path problem with expected routing costs. This paper develops two methods to solve the KMSDSPP, including a basic method and a deviation path-based method. To find the \(k+1\)th optimal path, the basic method adds \(k\) constraints to exclude the first-\(k\) optimal paths. Additionally, we introduce the deviation path concept and propose a deviation path-based method. To find the \(k+1\)th optimal path, the solution space that contains the \(k\)th optimal path is decomposed into several subspaces. We just need to search these subspaces to generate additional candidate paths and find the \(k+1\)th optimal path in the set of candidate paths. Numerical experiments are implemented in several transportation networks, showing that the deviation path-based method has superior performance than the basic method, especially for a large value of \(K\). Compared with the basic method, the deviation path-based method can save 90.1% CPU running time to find the best \(1000\) optimal paths in the Anaheim network.