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

A column generation approach for solving the examination-timetabling problem

The Curriculum-Based Course Timetabling (CB-CTT) problem involves the task of scheduling lectures of courses to classrooms, considering teacher availability, a specified curricula, and a set of constraints. In the Polytechnic University of Victoria (PUV), Mexico, the CB-CTT problem occurs with the additional constraints that the courses must be assigned to teachers and that a teacher cannot teach more than one course in the same curriculum. This document proposes a methodology to solve this special case CB-CTT problem. The main contributions derived from the research are: (a) the solution of the special case of the CB-CTT problem; (b) the use of a Simulated Annealing based methodology, which relaxes the problem into three simpler subproblems (the assignment, distribution, and scheduling of courses), to reduce the search space and speed up the construction of a solution; and (c) the solution of a real world instance using the proposed methodology. The results show that the approach can construct solutions for the problem in the PUV in a few minutes, and of a similar quality to those manually constructed in a couple of weeks.

Curriculum-based course timetabling problem (CB-CTP) is a classical problem that is still being researched on nowadays. There are two phases in solving CB-CTP. The first phase is the construction phase where a population of initial solutions is generated. The second phase is the optimization phase where the final optimal solution is produced. While most papers concentrate on the optimization phase, this paper focuses on the construction phase due to the importance of initial solutions to the optimization phase. The initial solutions are generated by implementing graph coloring heuristic to the CB-CTP. The graph coloring heuristics used in this paper are largest degree and largest weighted degree. Result from computational experiment shows that both methods are able to produce population of initial solutions for almost every data instance.

This work applies a hybrid approach in solving the university curriculum-based course timetabling problem as presented as part of the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nd</sup> International Timetabling Competition 2007 (ITC2007). The core of the hybrid approach is based on an artificial bee colony algorithm. Past methods have applied artificial bee colony algorithms to university timetabling problems with high degrees of success. Nevertheless, there exist inefficiencies in the associated search abilities in term of exploration and exploitation. To improve the search abilities, this work introduces a hybrid approach entitled nelder-mead great deluge artificial bee colony algorithm (NMGD-ABC) where it combined additional positive elements of particle swarm optimization and great deluge algorithm. In addition, nelder-mead local search is incorporated into the great deluge algorithm to further enhance the performance of the resulting method. The proposed method is tested on curriculum-based course timetabling as presented in the ITC2007. Experimental results reveal that the proposed method is capable of producing competitive results as compared with the other approaches described in literature.

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.

This paper proposes an abstraction of emerging vehicle routing problems, the Vehicle Routing Problem with Black Box Feasibility. In this problem the routes of a basic VRP need to satisfy an unknown set of constraints. A black box function to test the feasibility of a route is provided. This function is considered of non-linear complexity (in the length of the route). Practical examples of such problems are combinations of VRP with Loading problems or VRP with Scheduling problems. The difficulty in addressing the VRP with Black Box Feasibility lies in the unknown problem structure and the costly feasibility check. We propose a column generation-based approach to locally optimize this problem. Columns are heuristically generated by so-called Collector ants, executing a construction heuristic while guided by pheromones. To find an integer solution we solve an integer Set Partitioning Problem defined on the set of columns generated by the ants. We test the proposed approach on two applications from the literature, the Three-Dimensional Loading Capacitated Vehicle Routing Problem and the Multi-Pile Vehicle Routing Problem, showing the applicability of our approach and its good behavior compared to dedicated approaches.

In this paper, we present an in-depth analysis of neighborhood relations for local search algorithms. Using a curriculum-based course timetabling problem as a case study, we investigate the search capability of four neighborhoods based on three evaluation criteria: percentage of improving neighbors, improvement strength and search steps. This analysis shows clear correlations of the search performance of a neighborhood with these criteria and provides useful insights on the very nature of the neighborhood. This study helps understand why a neighborhood performs better than another one and why and how some neighborhoods can be favorably combined to increase their search power. This study reduces the existing gap between reporting experimental assessments of local search-based algorithms and understanding their behaviors.

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].

The course timetabling problem is a well known constraint optimization problem which has been of interest to researchers as well as practitioners. Due to the NP-hard nature of the problem, the traditional exact approaches might fail to find a solution even for a given instance. Hyper-heuristics which search the space of heuristics for high quality solutions are alternative methods that have been increasingly used in solving such problems. In this study, a curriculum based course timetabling problem at Yeditepe University is described. An improvement oriented heuristic selection strategy combined with a simulated annealing move acceptance as a hyper-heuristic utilizing a set of low level constraint oriented neighbourhood heuristics is investigated for solving this problem. The proposed hyper-heuristic was initially developed to handle a variety of problems in a particular domain with different properties considering the nature of the low level heuristics. On the other hand, a goal of hyper-heuristic development is to build methods which are general. Hence, the proposed hyper-heuristic is applied to six other problem domains and its performance is compared to different state-of-the-art hyper-heuristics to test its level of generality. The empirical results show that the proposed method is sufficiently general and powerful.

We study the stochastic transportation-inventory network design problem involving one supplier and multiple retailers. Each retailer faces some uncertain demand, and safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. Shen et al. (2003) formulated this problem as a set-covering integer-programming model. The pricing problem that arises from the column generation algorithm gives rise to a new class of the submodular function minimization problem. In this paper, we show that by exploiting certain special structures, we can solve the general pricing problem in Shen et al. efficiently. Our approach utilizes the fact that the set of all lines in a two-dimension plane has low VC-dimension. We present computational results on several instances of sizes ranging from 40 to 500 retailers. Our solution technique can be applied to a wide range of other concave cost-minimization problems.

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

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.

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 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.

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.