## Abstract

This paper studies optimal path problems integrated with the concept of second order stochastic dominance. These problems arise from applications where travelers are concerned with the trade off between the risks associated with random travel time and other travel costs. Risk-averse behavior is embedded by requiring the random travel times on the optimal paths to stochastically dominate that on a benchmark path in the second order. A general linear operating cost is introduced to combine link- and path-based costs. The latter, which is the focus of the paper, is employed to address schedule costs pertinent to late and early arrival. An equivalent integer program to the problem is constructed by transforming the stochastic dominance constraint into a finite number of linear constraints. The problem is solved using both off-the-shelf solvers and specialized algorithms based on dynamic programming (DP). Although neither approach ensures satisfactory performance for general large-scale problems, the numerical experiments indicate that the DP-based approach provides a computationally feasible option to solve medium-size instances (networks with several thousand links) when correlations among random link travel times can be ignored.

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## Notes

Absolute risk aversion implies that the amount of returns someone is willing to expose to risk decrease as the total returns increases (Arrow 1965).

Unless otherwise specified, path A dominate path B should always be interpreted as “the random travel time on path A dominate that on path B.

It may appear strange that we used the terms “first order” and “second order” in the above definitions. The reason for the terminology is the connection between the increasing (convex) order seen above and the classical notion of first (second) order stochastic dominance. According to these classical definitions, a random variable

*X*dominates another random variable*Y*in first order (denoted*X*≽_{(1)}*Y*) if the CDF of*X*never lies above the integral of the CDF of*Y*. Similarly, a random variable*X*dominates another random variable*Y*in second order (denoted*X*≽_{(2)}*Y*) if the integral of the CDF of*X*up to*t*never lies above the integral of the CDF of*Y*up to*t*for any*t*∈**R**(note the distinction between the ≽_{(k)}and ≽_{ k }notations,*k*= 1,2, which are associated with utility and disutility, respectively). It can be shown that \(X \preceq_{\mbox{icx}} Y\) if and only if −*X*≽_{(2)}−*Y*, and \(X \preceq_{\rm{i}} Y\) if and only if −*X*≽_{(1)}−*Y*; we refer again to Muller and Stoyan (2002) for details.Since \(\pi_{\hat{l}}^{is}\) and \(\pi_k^{is}\) are both random variables with known probability mass functions, we can calculate the value of \(E(\pi_{\hat{l}}^{is}-\eta)_{+}\) or \(E(\pi_k^{is}-\eta)_{+}\) for any given

*ϕ*.CPLEX is a linear programming solver, KNITRO is linear/nonlinear programming solver and MINTO is an integer programming solver. They are available through the NEOS server at http://www-neos.mcs.anl.gov/.

This property follows from the fact that, given two random variables

*X*and*Y*, if*X*≽_{2}*Y*then*E*(*X*) ≤*E*(*Y*). Therefore, if path 1^{1,18}were not SSD-admissible, there would exist at least one other LET path.

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## Acknowledgements

Dr. Richard A. Waltz kindly provided a free copy of KNITRO to be used with AMPL. The authors would like to thank Dr. Pattharin Sarutipand from Northwestern University for her assistance with building AMPL models, and Leilei Zhang from University of Illinois at Chicago for her help with numerical experiments and for valuable discussions on the topic of Section 3.4. We also thank three anonymous referees for their constructive comments.

This research was partially supported by National Science Foundation (CMMI-0928577 and CMMI-1033051) and Northwestern’s Center for Commercialization and Innovative Transportation Technology.

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Nie, Y.(., Wu, X. & Homem-de-Mello, T. Optimal Path Problems with Second-Order Stochastic Dominance Constraints.
*Netw Spat Econ* **12**, 561–587 (2012). https://doi.org/10.1007/s11067-011-9167-6

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DOI: https://doi.org/10.1007/s11067-011-9167-6