A column generation and Combinatorial Benders Decomposition algorithm for the Selective Dial-A-Ride-Problem

In this paper, we address the electric vehicle routing problem with time windows and propose two branch-and-price-and-cut methods based on a column generation algorithm. One is an exact algorithm whereas the other is a heuristic method. The pricing sub-problem of the column generation method is solved using a label correcting algorithm. The algorithms are strengthened with the state-of-the-art acceleration techniques and a set of valid inequalities. The acceleration techniques include: (i) an intermediate column pool to prevent solving the pricing sub-problem at each iteration, (ii) a label correcting method employing the ng-route algorithm adopted to our problem, (iii) a bidirectional search mechanism in which both forward and backward labels are created, (iv) a procedure for dynamically eliminating arcs that connect customers to remote stations from the network during the path generation, (v) a bounding procedure providing early elimination of sub-optimal routes, and (vi) an integer programming model that generates upper bounds. Numerical experiments are conducted using a benchmark data set to compare the performances of the algorithms. The results favour the heuristic algorithm in terms of both the computational time and the number of instances solved. Moreover, the heuristic algorithm is shown to be specifically effective for larger instances. Both algorithms introduce a number of new solutions to the literature.

In this paper, we introduce the Electric Vehicle Routing Problem with Flexible Time Windows (EVRPFTW) in which vehicles are allowed to serve customers before and after the earliest and latest time window bounds, respectively. The objective of this problem is to assign electric vehicles to feasible routes and make schedules with minimum total cost that includes the traveling costs, the costs of using electric vehicles and the penalty costs incurred for earliness and lateness. The proposed mathematical model is solved by a column generation procedure. To generate an integer solution, we solve an integer programming problem using the routes constructed by the column generation algorithm. We further develop a linear programming model to compute the optimal times to start service at each customer for the selected routes. A number of well-known benchmark instances is solved by our solution procedure to evaluate the operational gains obtained by employing flexible time windows.

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.

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms.

The generalized lock scheduling problem (GLSP) is a mixed integer optimization problem which consists of a ship placement (SP) and a lockage operation scheduling (LOS) sub-problem. In previous research, the GLSP is solved by different exact and heuristic methods, which are confirmed inferior with respect to computation time and solution quality. Consequently, none of those methods is efficient for handling practical large-scale GLSP. For the first time, we show that high-quality solutions of GLSP can be efficiently obtained by using an innovative approach proposed in this paper. Specifically, an ingenious solution structure of GLSP is designed, by which the GLSP is converted to a combinatorial optimization problem. Furthermore, an adaptive large neighborhood search (ALNS) heuristic based on the principle of destruction and reconstruction of solutions is proposed for solving the GLSP. Test results using a large number of instances reported in the literature are compared with those obtained by two exact methods, the mixed integer linear programming (MILP) and combinatorial Benders' decomposition (CBD) method. The results show that our ALNS achieves optimal solutions within less time in terms of most of the small-scale instances. Much better solutions are obtained by the ALNS within a few minutes for those large-scale instances that cannot be solved to optimality by exact methods within 2 h. Especially, the advantage of the proposed method is more remarkable when there is no specific chronological rules forced, which indicates that the proposed method is capable of handling the GLSP in a broader scope of situations.

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.

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.

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.

Column generation is an increasingly popular basic tool for the solution of large-scale mathematical programming problems. As problems being solved grow bigger, column generation may however become less efficient in its present form, where columns typically are not optimizing, and finding an optimal solution instead entails finding an optimal convex combination of a huge number of them. We present a class of column generation algorithms in which the columns defining the restricted master problem may be chosen to be optimizing in the limit, thereby reducing the total number of columns needed. This first article is devoted to the convergence properties of the algorithm class, and includes global (asymptotic) convergence results for differentiable minimization, finite convergence results with respect to the optimal face and the optimal solution, and extensions of these results to variational inequality problems. An illustration of its possibilities is made on a nonlinear network flow model, contrasting its convergence characteristics to that of the restricted simplicial decomposition (RSD) algorithm.

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

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.

Considered is a carrier that requires decision support to organize an efficient transport of loaded and empty containers in the hinterland of a sea port. Loaded containers are handled as pickup-and-delivery requests, however, requests for empty containers are incomplete because either the pickup location of a container or the delivery location of a container is a priori unknown. The problem is modelled as a generalization of the pickup-and-delivery problem (PDP) with less-than-truckload (LTL) requests. Practically speaking, by using LTL request we are able to consider 20-foot and 40-foot containers simultaneously. This is closer to reality than most previous models discussed in the literature which use full truckload requests, i.e., only containers of homogeneous size are possible. Three types of decisions are involved in the proposed model: a determination of pickup or delivery locations for the incomplete requests, a routing of vehicles, and a routing of empty containers. The presented model is validated by means of a numerical example computed by a MIP solver.

Designing networks with resiliency to edge failures using two-stage robust optimization

Dantzig–Wolfe decomposition of the daily course pattern formulation for curriculum-based course timetabling

During the past few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges. One way to describe choice behavior is to assume that each customer belongs to a segment, which is characterized by a consideration set, i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature. In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. (2004) [Gallego, G., G. Iyengar, R. Phillips, A. Dubey. 2004. Managing flexible products on a network. Technical Report CORC TR-2004-01, Department of Industrial Engineering and Operations Research, Columbia University, New York], and the follow-up dynamic programming decomposition heuristic of van Ryzin and Liu (2008) [van Ryzin, G. J., Q. Liu. 2008. On the choice-based linear programming model for network revenue management. Manufacturing Service Oper. Management 10(2) 288–310]. We focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-hard and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective and that the overall approach leads to high-quality, practical solutions.