Shortest path problem
Shortest path in stochastic time-dependent networks

In real-life, a transportation network is usually stochastic and time-dependent. The travel duration on a road segment depends on many factors such as the amount of traffic (origin-destination matrix), road work, weather, accidents and vehicle breakdowns. A more realistic model of such a road network is a stochastic time-dependent (STD) network.

There is no accepted definition of optimal path under uncertainty (that is, in stochastic road networks). It is a controversial subject, despite considerable progress during the past decade. One common definition is a path with the minimum expected travel time. The main advantage of this approach is that it can make use of efficient shortest path algorithms for deterministic networks. However, the resulting optimal path may not be reliable, because this approach fails to address travel time variability.

To tackle this issue, some researchers use travel duration distribution instead of its expected value. So, they find the probability distribution of total travel duration using different optimization methods such as dynamic programming and Dijkstra's algorithm. These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length. The terms travel time reliability and travel time variability are used as opposites in the transportation research literature: the higher the variability, the lower the reliability of predictions.

To account for variability, researchers have suggested two alternative definitions for an optimal path under uncertainty. The most reliable path is one that maximizes the probability of arriving on time given a travel time budget. An α-reliable path is one that minimizes the travel time budget required to arrive on time with a given probability.
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