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Quay crane scheduling is a key aspect of container terminal operation, which can be regarded as a decision-making process with uncertainty. Each task involves stochastic loading and unloading operation times owing to the existence of uncertainty. In this study, we investigate the unidirectional quay crane scheduling problem for a stochastic processing time, which requires that all the quay cranes move in the same direction either from bow to stern, or vice versa, throughout the planning horizon. The problem is formulated as a two-stage stochastic mixed-integer programming model, where the binary first-stage decision variables correspond to the assignment of tasks to quay cranes, and the mixed-integer second-stage decision variables are related to the generation of detailed schedules. To make the model solvable, we develop an alternative equivalent reformulation with a special structure that involves binary variables in the first stage and continuous variables in the second stage. To solve this reformulated model, an integer L-shaped method is presented for small-size instances, and a simulated annealing algorithm is presented for large-size instances to obtain near-optimal solutions. Numerical experiments show that the integer L-shaped method and simulated annealing algorithm could efficiently solve the unidirectional quay crane scheduling problem with uncertainty. The results also indicate that the stochastic model has distinct advantages in terms of shortening the completion time of vessels and improving the service level of container terminals compared with the expected value problem solutions. [ABSTRACT FROM AUTHOR]

Path flow identification is of particular interest for a number of traffic applications, such as OD demand estimation, link flow inference, and toll freeway revenue management. Optimal positioning of active sensors can help to identify path flows. Due to the stochastic nature of transportation systems, we propose a scenario based two stage stochastic programming framework which considers the uncertainty of the link-path matrix. The first stage model aims to minimize the total traffic sensor installation cost and the expected penalty for uncovered and undifferentiated paths. The second stage model attempts to minimize uncovered and undifferentiated paths for a given sensor location pattern and a specific scenario. In addition, a mean risk measure is also incorporated into the two stage stochastic programming framework, and consequently a mean risk two stage stochastic programming model is proposed. Both models have the same structure, where the first stage and second stage decision variables are binary. The second stage decision variable can be relaxed to a continuous variable without changing the nature of the model. To solve the two stochastic programming models, a branch and bound based integer L-shaped algorithm is presented. Finite steps convergence is guaranteed for the algorithm. To handle the problem with a large number of scenarios, a sampling technique is introduced, and the confidence bound is analyzed with respect to the scenario size. Extensive numerical experiments are conducted to verify the effectiveness of the proposed models and algorithm. The most important numerical results are as follows: (i) the stochastic programming framework is capable of capturing the reality more efficiently and accurately, (ii) the path differentiation factor is more critical than the path coverage factor in determining the sensor placement pattern, and (iii) in the partial parameter setting case, the mean risk based stochastic programming model results in a significantly different sensor placement pattern compared to the normal stochastic programming model. The study contributes to practical sensor placement design. [ABSTRACT FROM AUTHOR]