Path finding problems have many real-world applications in various fields, such as operations research, computer science, telecommunication, transportation, etc. In this paper, we examine three definitions of optimality for finding the optimal path under an uncertain environment. These three stochastic path finding models are formulated as the expected value model, dependent-chance model, and chanceconstrained model using different criteria to hedge against the travel time uncertainty. A simulation-based genetic algorithm procedure is developed to solve these path finding models under uncertainties. Numerical results are also presented to demonstrate the features of these stochastic path finding models.

In recent empirical studies on values of time and reliability, many have suggested that travelers are interested not only in travel time saving but also in reduction in travel time variability. Variability introduces uncertainty for travelers such that they do not know exactly when they will arrive at their destination. Thus, it is considered as a risk (or an added cost) to a traveler making a trip. Continuing research is reported on route choice models and the effect on travel time reliability in an uncertain environment caused by demand and supply variations. The goal is to examine what the aggregate impact of changes in variability might be on network assignment and how travelers with different risk-taking behaviors respond to these changes.

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A mean-variance model was developed for determining the optimal toll and capacity in a build-operate-transfer (BOT) roadway project subject to traffic demand uncertainty. This mean-variance model involves two objectives: maximizing mean profit and minimizing the variance (or standard deviation) of profit. The variance associated with profit is considered as a risk. Because maximizing expected profit and minimizing risk are often conflicting, there may not be a single best solution that can simultaneously optimize both objectives. Hence, it is necessary to explicitly consider this as a multiobjective problem so that a set of nondominated solutions can be generated. In this study, the optimal toll and capacity selection for the BOT problem under demand uncertainty is formulated as a special case of the stochastic network design problem. A simulation-based multiobjective genetic algorithm was developed to solve this stochastic bilevel mathematical programming formulation. Numerical results are also presented as a case study.

Solving optimization problems with multiple objectives under uncertainty is generally a very difficult task. Evolutionary algorithms, particularly genetic algorithms, have shown to be effective in solving this type of complex problems. In this paper, we develop a simulation-based multi-objective genetic algorithm (SMOGA) procedure to solve the build-operate-transfer (BOT) network design problem with multiple objectives under demand uncertainty. The SMOGA procedure integrates stochastic simulation, a traffic assignment algorithm, a distance-based method, and a genetic algorithm (GA) to solve a multi-objective BOT network design problem formulated as a stochastic bi-level mathematical program. To demonstrate the feasibility of SMOGA procedure, we solve two mean-variance models for determining the optimal toll and capacity in a BOT roadway project subject to demand uncertainty. Using the inter-city expressway in the Pearl River Delta Region of South China as a case study, numerical results show that the SMOGA procedure is robust in generating 'good' non-dominated solutions with respect to a number of parameters used in the GA, and performs better than the weighted-sum method in terms of the quality of non-dominated solutions.