We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

/pdf/branch-and-price-column-generation-for-solving-huge-integer-4jvlny6blr.pdf

Branch-And-Price: Column Generation for Solving Huge Integer Programs

https://www.mansci.ovgu.de/mansci_media/publikationen/2007/typology-EGOTEC-5t0pvr6fjifln4r4oav60tt612.pdf

An improved typology of cutting and packing problems

C4.5: Programs for Machine Learning (書評)

In network design, the gap between theory and practice is woefully broad. This book narrows it, comprehensively and critically examining current network design models and methods. You will learn where mathematical modeling and algorithmic optimization have been under-utilized. At the opposite extreme, you will learn where they tend to fail to contribute to the twin goals of network efficiency and cost-savings. Most of all, you will learn precisely how to tailor theoretical models to make them as useful as possible in practice. Throughout, the authors focus on the traffic demands encountered in the real world of network design. Their generic approach, however, allows problem formulations and solutions to be applied across the board to virtually any type of backbone communication or computer network. For beginners, this book is an excellent introduction. For seasoned professionals, it provides immediate solutions and a strong foundation for further advances in the use of mathematical modeling for network design. (Less)

Routing, Flow, And Capacity Design In Communication And Computer Networks

A survey on problems and methods in generalized assembly line balancing

http://solab.kaist.ac.kr/files/papers/intl_journal/2003_1.pdf

Algorithms for the variable sized bin packing problem

In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.

/pdf/robust-vehicle-routing-problem-with-deadlines-and-travel-49ns6cambf.pdf

Robust vehicle routing problem with deadlines and travel time/demand uncertainty

A bilevel programming approach is used to tackle an optimization problem for area traffic control and equilibrium flows. The signal timing plan is defined by common cycle time, and by starts and durations of greens. The system performance index is defined as the sum of a weighted linear combination of rate of delay and number of stops per unit time for all traffic streams, which is evaluated by the traffic model from TRANSYT. User equilibrium traffic assignment is formulated as a variational inequality problem. Approximate mathematical expressions for various components of the performance index and the average delay to a vehicle at the downstream junction in the TRANSYT model for both undersaturated and oversaturated links have been derived. For a locally optimal search, the gradient projection method is used in deciding whether feasible descent directions leading to a Karush-Kuhn-Tucker point, which is potentially a local optimum, can be identified. A global search heuristic is proposed in this paper by which successively better Karush-Kuhn-Tucker points can be found with reasonable computation effort. The mixed search procedure, including the locally optimal search and global search heuristic, is proposed. Encouraging results for Allsop and Charlesworth's network have confirmed that the approximately optimizing mixed search procedure in solving the area traffic control optimization problem can achieve substantially better results than does the nonoptimizing calculations of mutually consistent signal timings and link flows.

Optimization of Area Traffic Control for Equilibrium Network Flows

We consider the routing and wavelength assignment (RWA) problem on WDM ring networks without wavelength conversion. When the physical network and required connections are given, RWA is the problem to select a suitable path and wavelength among the many possible choices for each connection such that no two paths using the same wavelength pass through the same link. We give an integer programming formulation of the problem and propose an algorithm to solve it. Although the formulation has exponentially many variables, we solve the linear programming relaxation of it by using the column generation technique. We solve the column generation problem efficiently by decomposing the problem into several subproblems. After solving the linear programming relaxation, we apply the branch-and-price procedure to get an optimal solution. We test the proposed algorithm on some randomly generated data. Test results show that the algorithm gives optimal solutions to almost all instances under the given node limit of the branch-and-bound tree.

http://solab.kaist.ac.kr/files/papers/intl_journal/2000_5.pdf

Sungsoo Park

Papers

Algorithms for the variable sized bin packing problem

Robust vehicle routing problem with deadlines and travel time/demand uncertainty

Optimization of Area Traffic Control for Equilibrium Network Flows

Optimal routing and wavelength assignment in WDM ring networks

Efficient solution procedure and reduced size formulations for p-hub location problems