Algorithms for Restoration Planning in a Telecommunications Network

Column Generation is a very powerful class of combinatorial optimization algorithms that has been used successfully to solve a variety of large scale optimization problems. Its application has helped many companies in various industries increase revenue and reduce costs significantly, particularly in transportation, energy, manufacturing, and telecommunication companies. In this chapter, we will first discuss the motivations for column generation, then we will provide an intuitive but rigorous treatment of the mechanisms of column generation – how it works, why it works. We will then give descriptions on the branch and price algorithm and several examples of column generation’s successful applications in one of the world’s largest airlines. We will discuss monthly airline crew schedule optimization for bidlines, crew pairing optimization, and integrated modeling of fleet and routing in the optimization of aircraft scheduling. Part of the focus is on business requirements and priorities in these areas and how the column generation models are built to effectively meet these challenges. Some airline industry domain-specific details are provided to allow the readers to better appreciate the scheduling problems’ complexities that made the master-subproblem approach in column generation essential. We will also discuss the significant run-time speedups for these large scale scheduling problems due to various practical model enhancements, as well as progress in the large scale optimization space made possible by technologies such as parallel processing, big data, and better chips. At last, we will briefly discuss several example variants of column generation and their applications in various industries. We will also review recent applications of optimization techniques to machine learning as well as the future potentials of large scale optimization in this field. This chapter can be used as a primer on the fundamentals of column generation techniques since it clearly addresses essential theoretical concepts that are sometimes elusive to researchers and graduate students who are new to this area. The chapter should also be helpful to practitioners who would like to gain insights into how to build effective column generation models to solve real world large scale optimization problems.

We consider a ground set E and a submodular function acting on it. We first propose a “set multicovering” problem in which the value (price) of any is . We show that the linear program (LP) of this problem can be directly solved by applying a submodular function minimization (SFM) algorithm on the dual LP. However, the main results of this study concern prize‐collecting multicovering with submodular pricing, that is, a more general and more difficult “multicovering” problem in which elements can be left uncovered by paying a penalty. We formulate it as a large‐scale LP (with variables representing all subsets of E) that could be tackled by column generation (CG; for a CG algorithm for “set‐covering” problems with submodular pricing). However, we do not solve this large‐scale LP by CG, but we solve it in polynomial time by exploiting its integrality properties. More exactly, after appropriate restructuring, the dual LP can be transformed into an LP that has been thoroughly studied (as a primal) in the SFM theory. Solving this LP reduces to optimizing a strong map of submodular functions. For this, we use the Fleischer–Iwata framework that optimizes all these functions within the same asymptotic running time as solving a single SFM, that is, in , where and γ is the complexity of one submodular evaluation. Besides solving the problem, the proposed approach can be useful to (a) simultaneously find the best solution of up to functions under strong map relations in time, (b) perform sensitivity analysis in very short time (even at no extra cost), and (c) reveal combinatorial insight into the primal–dual optimal solutions. We present several potential applications throughout the paper, from production planning to combinatorial auctions.

Current beam orientation optimization algorithms for radiotherapy, such as column generation (CG), are typically heuristic or greedy in nature because of the size of the combinatorial problem, which leads to suboptimal solutions. We propose a reinforcement learning strategy using a Monte Carlo Tree Search (MCTS) that can find a better beam orientation set in less time than CG. We utilize a reinforcement learning structure involving a supervised learning network to guide the MCTS and to explore the decision space of beam orientation selection problems. We previously trained a deep neural network (DNN) that takes in the patient anatomy, organ weights, and current beams, then approximates beam fitness values to indicate the next best beam to add. Here, we use this DNN to probabilistically guide the traversal of the branches of the Monte Carlo decision tree to add a new beam to the plan. To assess the feasibility of the algorithm, we used a test set of 13 prostate cancer patients, distinct from the 57 patients originally used to train and validate the DNN, to solve five-beam plans. To show the strength of the guided MCTS (GTS) compared to other search methods, we also provided the performances of Guided Search, Uniform Tree Search and Random Search algorithms. On average, GTS outperformed all the other methods. It found a better solution than CG in 237 s on average, compared to 360 s for CG, and outperformed all other methods in finding a solution with a lower objective function value in less than 1000 s. Using our GTS method, we could maintain planning target volume (PTV) coverage within 1% error similar to CG, while reducing the organ-at-risk mean dose for body, rectum, left and right femoral heads; the mean dose to bladder was 1% higher with GTS than with CG.

Solution method for the location planning problem of logistics park with variable capacity

We apply column generation with branch‐and‐price optimization to a multi‐target, multi‐task assignment problem, with precedence constraints. Column generation transforms the nonlinear program with separable costs and constraints into a linear program. This reformulation divides the original problem into a number of smaller problems, where one of these smaller problems accounts for the coupling constraints between agents and must be known by every agent. All other divisions consider local constraints affecting only one agent; these smaller problems are known by only one corresponding agent. Because of this reformulation, the assignment problem can be solved in a distributed manner. A theorem is proven which details the central analytical result of the paper, allowing a nonlinear program to be reformulated as a linear program. Simulation results for a multi‐target, single‐task assignment problem, as well as a multi‐target, multi‐task assignment problem with precedence constraints are presented. Published 2011. This article is a US Government work and is in the public domain in the USA.

An effective resource allocation technique is required for each Ready Mixed Concrete (RMC). Finding the optimum solution for large scale RMC dispatching problems with available computing facilities is intractable. Two kinds of techniques have been implemented to deal with this problem: (i) evolutionary techniques and (ii) numerical techniques. For the purposes of this paper we selected a technique from each category and compared them under the same conditions. Robust Genetic Algorithm (Robust-GA) and Column Generation (CG) were selected and tested with different sizes of real RMC problems. The results show that on average CG solutions are obtained with a 20% reduced cost. However, Robust-GA converges 40% faster than CG, while the number of unassigned customers for both the techniques is almost the same.

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

We propose a column generation algorithm for solving sequence dependent flowshop scheduling problems (SDST flowshop) to minimize the total weighted tardiness. The continuous relaxation of the set partitioning formulation of the Dantzig-Wolfe decomposition for SDST flowshop is solved by the column generation. The pricing subproblem for the column generation is decomposed into each job-level subproblem. In order to strengthen the lower bound, the additional constraints are imposed to the pricing problems in the proposed algorithm. The cut generation algorithm is developed for the column generation. Computational experiments show that the proposed method can derive the solutions with a smaller duality gap compared with those of the conventional algorithm of ordinary column generation and Lagrangian relaxation with cuts.

PurposeIt is worth mentioning that in supply chain management (SCM), managerial decisions are often based on optimization of resources. Till the early 2000s, supply chain optimization problems were being addressed by conventional programming approaches such as Linear Programming, Mixed-Integer Linear Programming and Branch-and-Bound methods. However, the solution convergence in such approaches was slow. But with the advent of Swarm Intelligence (SI)-based algorithms like particle swarm optimization and ant colony optimization, a significant improvement in solution of these problems has been observed. The purpose of this paper is to present and analyze the application of SI algorithms in SCM. The analysis will eventually lead to development of a generalized SI implementation framework for optimization problems in SCM.Design/methodology/approachA structured state-of-the-art literature review is presented, which explores the applications of SI algorithms in SCM. It reviews 56 articles published in peer-reviewed journals since 1999 and uses several classification schemes which are critical in designing and solving a supply chain optimization problem using SI algorithms.FindingsThe paper revels growth of swarm-based algorithms and seems to be dominant among all nature-inspired algorithms. The SI algorithms have been used extensively in most of the realms of supply chain network design because of the flexibility in their design and rapid convergence. Large size problems, difficult to manage using exact algorithms could be efficiently handled using SI algorithms. A generalized framework for SI implementation in SCM is proposed which is beneficial to industry practitioners and researchers.Originality/valueThe paper proposes a generic formulation of optimization problems in distribution network design, vehicle routing, resource allocation, inventory management and supplier management areas of SCM which could be solved using SI algorithms. This review also provides a generic framework for SI implementation in supply chain network design and identifies promising emerging issues for further study in this area.

New Heuristics and Adaptive Memory Procedures for Boolean Optimization Problems, Lars M. Hvattum, Arne Lokketangen, and Fred Glover Convergent Lagrangian Methods for Separable Nonlinear Integer Programming: Objective Level-Cut and Domain-Cut Methods, Duan Li, Xiaoling Sun, and Jun Wang The Generalized Assignment Problem, Robert M. Nauss Decomposition in Integer Linear Programming, Ted K. Ralphs and Matthew V. Galati Airline Scheduling Models and Solution Algorithms for the Temporary Closure of Airports, Shangyao Yan and Chung-Gee Lin Determining an Optimal Fleet Mix and Schedules: Part I - Single Source and Destination, Hanif D. Sherali and Salem M. Al-Yakoob Determining an Optimal Fleet Mix and Schedules: Part II - Multiple Sources and Destinations, and the Option of Leasing Transshipment Depots, Hanif D. Sherali and Salem M. Al-Yakoob An Integer Programming Model for the Optimization of Data Cycle Maps, David Panton, Maria John, and Andrew Mason Application of Column-Generation Techniques to Retail Assortment Planning, Govind P. Daruka and Udatta S. Palekar Noncommercial Software for Mixed-Integer Linear Programming, Jeff T. Linderoth and Ted K. Ralphs

We survey the main results obtained by the author in his PhD dissertation supervised by Anass Nagih and Lucas Letocart. It was defended in December 2008 at The Computer Science lab of the Paris-Nord University (L.I.P.N.), Villetaneuse, France. The thesis is written in French and is available from http://www-lipn.univ-paris13.fr/~touati/TM_Thesis.pdf. Column generation algorithms are instrumental in many areas of applied optimization, where linear programs with an enormous number of variables need to be solved. Although successfully used in many applications, this method suffers from well-known “instability” issues that somewhat limit its efficiency. This work focuses on accelerating strategies in a column generation scheme; on the first iterations using diversification, on the last iterations using reoptimization and on the overall GC scheme using an improving technique to solve the pricing problems (dynamic programming with blocs). The effectiveness of these approaches is validated on the Vehicle Routing Problem with Time Windows.

A column generation based heuristic for sensor placement, activity scheduling and data routing in wireless sensor networks

The wireless relay mesh network (WRMN) is designed to provide robust and fault tolerant communications between relay and user nodes in broadband wireless networks. In this paper, we investigate the benefits of decoupled optimization of routing and scheduling in WRMNs using the physical interference model and spatial reuse to maximize overall throughput. We model the routing optimization as a linear program using multicommodity flows (MCF). We refer to this problem as multicommodity flow routing optimization (MCF-ROPT). Using the flow per link determined from MCF-ROPT, we develop an optimization formulation to schedule the link traffic such that interference is minimized and time slots are reused appropriately based on spatial TDMA (STDMA). Furthermore, our scheduling approach incorporates the effect of reuse of multiple carriers on the transmission schedule. We refer to this problem as SM-TSS (STDMA multicarrier traffic sensitive scheduling). The SM-TSS is NP-hard and thus is solved using column generation. We compare our formulations with decoupled optimizations that use the protocol interference model and/or single carrier systems and show that our approach guarantees higher throughput by mitigating interference effectively.

Service type assignment and container routing with transit time constraints and empty container repositioning for liner shipping service networks

Column generation is one of the most successful approaches for solving large-scale linear programming problems. However, degeneracy difficulties and long-tail effects are known to occur as problems become larger. In recent years, several stabilization techniques of the dual variables have proven to be effective. We study the use of two types of dual-optimal inequalities to accelerate and stabilize the whole convergence process. Added to the dual formulation, these constraints are satisfied by all or a subset of the dual-optimal solutions. Therefore, the optimal objective function value of the augmented dual problem is identical to the original one. Adding constraints to the dual problem leads to adding columns to the primal problem, and feasibility of the solution may be lost. We propose two methods for recovering primal feasibility and optimality, depending on the type of inequalities that are used. Our computational experiments on the binary and the classical cutting-stock problems, and more specifically on the so-called triplet instances, show that the use of relevant dual information has a tremendous effect on the reduction of the number of column generation iterations.