Ş. İlker Birbil

Solving Stochastic Mathematical Programs with Complementarity Constraints Using Simulation

Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we report some of the recent progress we had in broadening the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. We first describe the method and the class of stochastic mathematical programs with complementarity constraints that we are interested in solving and then outline a set of sufficient conditions for its almost-sure convergence. We also illustrate an application of the method to solving a toll pricing problem in transportation networks. These developments also make it possible to solve certain stochastic bilevel optimization problems and Stackelberg games, involving expectations or steady-state functions, using simulation.

Solving a robust airline crew pairing problem with column generation

In this study, we solve a robust version of the airline crew pairing problem. Our concept of robustness was partially shaped during our discussions with small local airlines in Turkey which may have to add a set of extra flights into their schedule at short notice during operation. Thus, robustness in this case is related to the ability of accommodating these extra flights at the time of operation by disrupting the original plans as minimally as possible. We focus on the crew pairing aspect of robustness and prescribe that the planned crew pairings incorporate a number of predefined recovery solutions for each potential extra flight. These solutions are implemented only if necessary for recovery purposes and involve either inserting an extra flight into an existing pairing or partially swapping the flights in two existing pairings in order to cover an extra flight. The resulting mathematical programming model follows the conventional set covering formulation of the airline crew pairing problem typically solved by column generation with an additional complication. The model includes constraints that depend on the columns due to the robustness consideration and grows not only column-wise but also row-wise as new columns are generated. To solve this difficult model, we propose a row and column generation approach. This approach requires a set of modifications to the multi-label shortest path problem for pricing out new columns (pairings) and various mechanisms to handle the simultaneous increase in the number of rows and columns in the restricted master problem during column generation. We conduct computational experiments on a set of real instances compiled from local airlines in Turkey.

Combination of Metaheuristic and Exact Algorithms for Solving Set Covering-Type Optimization Problems

We propose a new generic framework for solving combinatorial optimization problems that can be modeled as a set covering problem. The proposed algorithmic framework combines metaheuristics with exact algorithms through a guiding mechanism based on diversification and intensification decisions. After presenting this generic framework, we extensively demonstrate its application to the vehicle routing problem with time windows. We then conduct a thorough computational study on a set of well-known test problems, where we show that the proposed approach not only finds solutions that are very close to the best-known solutions reported in the literature, but also improves them. We finally set up an experimental design to analyze the effects of different parameters used in the proposed algorithm.

Single-Leg Airline Revenue Management with Overbooking

Airline revenue management is concerned with identifying the maximum revenue seat allocation policies. Because a major loss in revenue results from cancellations and no-shows, overbooking has received significant attention in the literature over the years. In this study, we propose new static and dynamic single-leg overbooking models. In the static case we introduce two models: the first one aims to determine the overbooking limit and the second one is about finding the overbooking limit and the booking limits to allocate the virtual capacity among multiple fare classes. Because the second static model is hard to solve, we also introduce computationally tractable models that give upper and lower bounds on its optimal expected net revenue. In the dynamic case, we propose a dynamic programming model, which is based on two streams of events. The first stream corresponds to the arrival of booking requests and the second one corresponds to the cancellations. We conduct simulation experiments to illustrate the effectiveness of the proposed models.

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate our approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems.

An entropic regularization approach for mathematical programs with equilibrium constraints

A new smoothing approach based on entropic regularization is proposed for solving a mathematical program with equilibrium constraints (MPEC). With some known smoothing properties of the entropy function and keeping real practice in mind, we reformulate an MPEC problem as a smooth nonlinear programming problem. In this way, a difficult MPEC problem becomes solvable by using available nonlinear optimization software. To support our claims, we use an online solver and test the performance of the proposed approach on a set of well-known test problems.

Robust Crew Pairing for Managing Extra Flights

This paper discusses a modeling approach to robust crew pairing when a set of extra flights is likely to be added to the regular flight schedule. The set of these possible extra flights is known at the planning stage. We demonstrate that these extra flights may be incorporated into the schedule if necessary by modifying the planned crew pairings appropriately and without delaying or canceling existing flights. To this end, we either identify a pair of crews whose schedules may be (partially) swapped while adding an extra flight into the schedule or show that an extra flight may be inserted into the schedule of a crew without affecting others. We note that deadheading may be necessary in either case. For these two types of solutions, we define the appropriate feasibility rules with respect to the common airline regulations. We then propose two robust mathematical programming models that consider incorporating such solutions into the set of selected pairings while keeping the increase in the crew cost at an acceptable level. The baseline solution for comparison is found by a conventional crew pairing model in the literature which ignores robustness at the planning stage and relies on recovery procedures at the time of operation. We also propose the variations of the two models, where the double counting of the possible solutions across extra flights is prevented. Finally, we conduct computational experiments on a set of data generated from the actual data of an airline company. We solve the crew pairing problem both with the proposed robust models and the conventional model. Our results demonstrate the benefits of the proposed modeling approach and indicate that the proposed robust models provide natural options to recovery without disrupting the existing flights at a relatively small incremental cost, which is visible at the planning stage.

On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach.

A global optimization method for solving fuzzy relation equations

A system of fuzzy relation equations can be reformulated as a global optimization problem. The optimum solution of this new model corresponds to a solution of the system of fuzzy relation equations whenever the solution set of the system is nonempty. Moreover, even if the solution set of the fuzzy relation equations is empty, a solution to the global optimization problem provides a point such that the difference between the right and the left hand side of the fuzzy relation equations is minimized. The new global optimization problem has a nonconvex and nondifferentiable objective function. Therefore, a recent stochastic search approach is applied to solve this new model. The performance of the approach is tested on a set of problems with different dimensions.

Multiagent cooperation for solving global optimization problems: an extendible framework with example cooperation strategies

This paper proposes the use of multiagent cooperation for solving global optimization problems through the introduction of a new multiagent environment, MANGO. The strength of the environment lays in its flexible structure based on communicating software agents that attempt to solve a problem cooperatively. This structure allows the execution of a wide range of global optimization algorithms described as a set of interacting operations. At one extreme, MANGO welcomes an individual non-cooperating agent, which is basically the traditional way of solving a global optimization problem. At the other extreme, autonomous agents existing in the environment cooperate as they see fit during run time. We explain the development and communication tools provided in the environment as well as examples of agent realizations and cooperation scenarios. We also show how the multiagent structure is more effective than having a single nonlinear optimization algorithm with randomly selected initial points.