The multiple vehicle balancing problem

This paper deals with the multiple vehicle balancing problem (MVBP). Given a fleet of vehicles of limited capacity, a set of vertices with initial and target inventory levels and a distribution network, the MVBP requires to design a set of routes along with pickup and delivery operations such that inventory is redistributed among the vertices without exceeding capacities, and routing costs are minimized. The MVBP is NP\u2010hard, generalizing several problems in transportation, and arising in bike\u2010sharing systems. Using theoretical properties of the problem, we propose an integer linear programming formulation and introduce strengthening valid inequalities. Lower bounds are computed by column generation embedding an ad\u2010hoc pricing algorithm, while upper bounds are obtained by a memetic algorithm that separate routing from pickup and delivery operations. We combine these bounding routines in both exact and matheuristic algorithms, obtaining proven optimal solutions for MVBP instances with up to 25 stations

Network Utility Maximization With Nonconcave Utilities Using Sum-of-Squares Method

The Network Utility Maximization problem has recently been used extensively to analyze and design distributed rate allocation in networks such as the Internet. A major limitation in the state-of-the-art is that user utility functions are assumed to be strictly concave functions, modeling elastic flows. Many applications require inelastic flow models where nonconcave utility functions need to be maximized. It has been an open problem to find the globally optimal rate allocation that solves nonconcave network utility maximization, which is a difficult nonconvex optimization problem. We provide a centralized algorithm for off-line analysis and establishment of a performance benchmark for nonconcave utility maximization. Based on the semialgebraic approach to polynomial optimization, we employ convex sum-of-squares relaxations solved by a sequence of semidefinite programs, to obtain increasingly tighter upper bounds on total achievable utility for polynomial utilities. Surprisingly, in all our experiments, a very low order and often a minimal order relaxation yields not just a bound on attainable network utility, but the globally maximized network utility. When the bound is exact, which can be proved using a sufficient test, we can also recover a globally optimal rate allocation. In addition to polynomial utilities, sigmoidal utilities can be transformed into polynomials and are handled. Furthermore, using two alternative representation theorems for positive polynomials, we present price interpretations in economics terms for these relaxations, extending the classical interpretation of independent congestion pricing on each link to pricing for the simultaneous usage of multiple links

Robust Toll Pricing

We study a robust toll pricing problem where toll setters and users have different level of information when taking their decisions. Toll setters do not have full information on the costs of the network and rely on historical information when determining toll rates, whereas users decide on the path to use from origin to destination knowing toll rates and having, in addition, more accurate traffic data. In this work, we first consider a single origin-destination parallel network and formulate the robust toll pricing problem as a distributionally robust optimization problem, for which we develop an exact algorithm based on a mixed-integer programming formulation and a heuristic based on two-point support distribution. We further extend our formulations to more general networks and show how our algorithms can be adapted for the general networks. Finally, we illustrate the usefulness of our approach by means of numerical experiments both on randomly generated networks and on the road network of the city of Chicago

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 dicult 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 a local airline in Turkey