Search

We focus on optimization models involving individual chance constraints, in which only the right-hand side vector is random with a finite distribution. A recently introduced class of such models treats the reliability levels / risk tolerances associated with the chance constraints as decision variables and trades off the actual cost / return against the cost of the selected reliability levels in the objective function. Leveraging recent methodological advances for modeling and solving chance-constrained linear programs with fixed reliability levels, we develop strong mixed-integer programming formulations for this new variant with variable reliability levels. In addition, we introduce an alternate cost function type associated with the risk tolerances which requires capturing the value-at-risk (VaR) associated with a variable reliability level. We accomplish this task via a new integer linear programming representation of VaR. Our computational study illustrates the effectiveness of our mathematical programming formulations. We also apply the proposed modeling approach to a new stochastic last mile relief network design problem and provide numerical results for a case study based on the real-world data from the 2011 Van earthquake in Turkey. (C) 2018 Elsevier Ltd. All rights reserved.

The uncertainties inherent in the airport flight arrival and departure traffic may lead to the unavailability of gates for accommodating the scheduled flights. Mechanical failures, severe weather conditions, heavy traffic volume at the airport are some typical causes of the uncertainties in the input data. Incorporating such random disruptions is crucial in constructing effective flight-gate assignment plans. We consider the gate assignment problem in the presence of uncertainty in arrival and departure times of the flights and represent the randomness associated with these uncertain parameters by a finite set of scenarios. Using the scenario-based approach, we develop new stochastic programming models incorporating alternate robustness measures to obtain assignments that would perform well under potential random disruptions. In particular, we focus on the number of conflicting flights, the idle and buffer times as robustness measures. Minimizing the expected variance of idle times or the expected total semi-deviation of idle times from a specified buffer time value are some examples of the objectives that we incorporate in our models to appropriately distribute the idle times among gates, and by this way, to decrease the number of potential flight conflicts. The proposed stochastic optimization models are formulated as computationally expensive large-scale mixed-integer programming problems, which are hard to solve. In order to find good feasible solutions in reasonably short CPU times, we employ tabu search algorithms. We conduct an extensive computational study to analyze the proposed alternate formulations and show the effectiveness of the proposed solution methods.

P-efficient points (pLEPs), introduced by Prekopa (1990), are utilized to solve probabilistically constrained problems which have been receiving significant attention in a wide range of application fields. Due to their non-convex feasible regions such problems are hard to solve. In a special case, the probabilistic constraint is enforced jointly on constraints involving randomness only in the right hand side and the vector of random variables has a finite distribution. For this special class of problems, the concept of the p-efficient point has been widely used to develop associated ecient solution methods. Those methods require the generation of pLEPs. We consider a finite random vector that is characterized by a set of scenarios and propose a mathematical programming based approach to generate pLEPs. Using a mixed-integer formulation, we obtain a method to generate pLEPs based on a new preprocessing technique and iterative outer approximations. Moreover, some valid inequalities are derived to strengthen the formulations of the outer approximations, which result in improved computational performance of the proposed generation method. We also develop a heuristic algorithm to generate “quasi pLEPs” efficiently. To the best of our knowledge, developing an optimizationbased approach for generating multiple pLEPs and developing a method for generating pLEPs for random variables characterized by a set of scenarios are novel. We also present some numerical results to show the computational efficiency and effectiveness of the proposed methods.