An exact branch‐and‐price algorithm for multitasking scheduling on unrelated parallel machines

Models and hybrid methods for the onshore wells maintenance problem

Scheduling in a two-stage flowshop with parallel unrelated machines at each stage and shared resources

A column generation approach to capacitated p-median problems

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.

In this paper we formulate an inverse shortest path problem as a special linear programming problem. A column generation scheme is developed that is able to keep most columns of the LP model implicit and to generate necessary columns by shortest path algorithms. This method can get an optimal solution in finitely many steps. Some numerical results are reported to show that the algorithm has a good performance.

A column generation approach for location-routing problems with pickup and delivery

A shift scheduling problem for a call centre is solved by applying a column generation method. The objective is to find an employee schedule within the planning horizon that meets staffing requirements provided by the customer and that minimises the costs of excess-staffing and under-staffing. To solve the problem, two iterative models were used: the first being a mixed integer-linear programming model, the master problem, and the second being a column generation model. In a first approach, the master problem was solved considering a fixed number of shift patterns. The second approach was to use both models iteratively to generate shift patterns to be used in the master problem. Both approaches met all constraints relatively fast, but the column generation model decreased the cost considerably when cheaper shift patterns were generated. The models contribute to the current literature by providing flexibility to consider under-coverage and over-coverage at two levels.

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

This article considers the order batching problem in steelmaking and continuous-casting production. The problem is to jointly specify the slabs needed to satisfy each customer order and group all the slabs of different customer orders into production batches. A novel mixed integer programming model is formulated for the problem. Through relaxing the order assignment con- straints, a Lagrangian relaxation model is then obtained. By exploiting the relationship between Lagrangian relaxation and column generation, we develop a combined algorithm that contains nested double loops. At the inner loop, the subgradient method is applied for approximating the Lagrangian dual problem and pricing out columns of the master problem corresponding to the linear dual form of the Lagrangian dual problem. At the outer loop, column generation is employed to solve the master problem exactly and adjust Lagrangian multipliers. Computational experiments are carried out using real data collected from a large steel company, as well as on large-scaled problem instances randomly generated. The results demonstrate that the combined algorithm can obtain tighter lower bound and higher quality solution within an acceptable computation time as compared to the conventional Lagrangian relaxation algorithm. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 58: 370-388, 2011

We present an optimisation-based solution approach for a real ship planning problem,which is a combination of a variant of the multi-vehicle pickup and delivery problemwith time windows (m-PDPTW), and a multi-inventory model. This problem involves thedesign of a set of minimum cost routes for a fleet of heterogeneous ships servicing a set ofproduction and consumption harbours with a single product (ammonia). The production andinventory information at each harbour, together with the ship capacities and the location ofthe harbours, determine the number of possible arrivals at each harbour during the planningperiod, the time windows for start of service and the load quantity intervals at each arrival.We call this problem the inventory pickup and delivery problem with time windows -IPDPTW. In the mathematical programming model, we duplicate some of the variables anduse a Dantzig - Wolfe decomposition approach. Then the IPDPTW decomposes into a sub-problemfor each harbour and each ship. By synchronising the solutions from both types ofsubproblems, we get extra constraints in the master problem as compared to the masterproblem for the m-PDPTW discussed in the literature. The LP-relaxation of the masterproblem is solved by column generation, where the columns represent ship routes or harbourvisit sequences. Finally, this iterative solution process is embedded in a branch-and-boundsearch to make the solution integer optimal. Our computational results indicate that theproposed method works for the real planning problem.

A branch and price algorithm for EOS constellation imaging and downloading integrated scheduling problem

We present a new model and a column generation algorithm for the linear minimum cost multicommodity flow problem. The new model has two differences when compared to the classic formulation based on flows on paths: it has an additional set of variables and an additional set of inequalities. The extra variables are associated with flows on circuits where some arcs are reversed. The extra inequalities assure that an optimal solution to the extended model can be converted to an optimal solution to the path formulation. In order to obtain an optimal solution to the extended model, we use column generation. The extra variables are explicitly considered in the restricted master problem, from the beginning of the column generation process. The subproblem remains a set of shortest path problems. Our present work is related to the use of dual–optimal inequalities in the sense that we extend a model with additional primal variables/dual inequalities. However, our extended model also has additional primal inequalities/dual variables. We present computational results of our method in planar instances and compare them with standard column generation, a bundle method, and a general‐purpose solver. For the tested instances, there is an effective improvement in computational time of the column generation method when the extended model is used. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(1), 3–16 2007

The minimum sum-of-squares clustering problem (MSSC), also known as k-means clustering, refers to the problem of partitioning n data points into k clusters, with the objective of minimizing the total sum of squared Euclidean distances between each point and the center of its assigned cluster. We propose an efficient algorithm for solving large-scale MSSC instances, which combines column generation (CG) with dynamic constraint aggregation (DCA) to effectively reduce the number of constraints considered in the CG master problem. DCA was originally conceived to reduce degeneracy in set partitioning problems by utilizing an aggregated restricted master problem obtained from a partition of the set partitioning constraints into disjoint clusters. In this work, we explore the use of DCA within a CG algorithm for MSSC exact solution. Our method is fine-tuned by a series of ablation studies on DCA design choices, and is demonstrated to significantly outperform existing state-of-the-art exact approaches available in the literature. History: Accepted by Andrea Lodi, Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grant 2023-04466]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0938 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0938 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Nowadays, large-scale failures in elastic optical inter-data center networks (EO-DCNs) are critical issues for network survivability. Once the original connection or the primary DC fails, an alternative data center (DC) can serve the cloud services with replicated content. In this paper, we study disaster-resilient cloud services provisioning with distance-adaptive modulation formats, considering content placement, routing with modulation format adaption, protection of path and content, and spectrum allocation simultaneously. Both dedicated end-to-content backup path protection (DEBPP) and shared end-to-content backup path protection (SEBPP) are studied. First, integer linear program (ILP) models are formulated with the objective of minimum spectrum usage. For the sake of scalability, a column generation (CG) approach is developed to decompose the ILP model into a master problem and a pricing problem. The CG approach is shown to achieve comparable results to ILP models. Numerical simulations are conducted to investigate the spectrum usage of DEBPP and SEBPP with different traffic parameters and different network topologies.

Job shop scheduling, as one of the classical scheduling problems, has been widely studied in literatures, and proved to be mostly NP-hard. Although it is extremely difficult to solve job shop scheduling with no-wait constraint to optimality, the two-machine no-wait job shop scheduling to minimise makespan could be solvable in polynomial time when each job has exactly two equal length operations (proportionate job shop). In the present paper, an extension is attempted by considering a proportionate flexible two-stage no-wait job shop scheduling problem with minimum makespan, and a set-covering formulation is put forward which contains a master problem and a pricing problem. To solve this problem, a column generation (CG)-based approach is implemented. In comparison, a mixed integer programming model is constructed and optimised by Cplex. A series of randomly generated numerical instances are calculated. And the testing result shows that the mixed integer model handled by Cplex can only solve small scale cases, while the proposed CG-based method can conquer larger size problems in acceptable time.