Thomas Riis Stidsen

A Branch and Bound Algorithm for a Class of Biobjective Mixed Integer Programs

Most real-world optimization problems are multiobjective by nature, involving noncomparable objectives. Many of these problems can be formulated in terms of a set of linear objective functions that should be simultaneously optimized over a class of linear constraints. Often there is the complicating factor that some of the variables are required to be integral. The resulting class of problems is named multiobjective mixed integer programming (MOMIP) problems. Solving these kinds of optimization problems exactly requires a method that can generate the whole set of nondominated points (the Pareto-optimal front). In this paper, we first give a survey of the newly developed branch and bound methods for solving MOMIP problems. After that, we propose a new branch and bound method for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables. The proposed method is able to find the full set of nondominat...

Integer programming for the generalized high school timetabling problem

Recently, the XHSTT format for high school timetabling was introduced. It provides a uniform way of modeling problem instances and corresponding solutions. The format supports a wide variety of constraints, and currently 38 real-life instances from 11 different countries are available. Thereby, the XHSTT format serves as a common ground for researchers within this area. This paper describes the first exact method capable of handling an arbitrary instance of the XHSTT format. The method is based on a mixed-integer linear programming (MIP) model, which is solved in two steps with a commercial general-purpose MIP solver. Computational results show that our approach is able to find previously unknown optimal solutions for 2 instances of XHSTT and proves optimality of 4 known solutions. For the instances not solved to optimality, new non-trivial lower bounds were found in 11 cases, and new best known solutions were found in 9 cases. Furthermore, the approach is compared with the finalists of Round 2 of the International Timetabling Competition 2011 and is shown to be competitive with one of the finalists.

A multilevel variable neighborhood search heuristic for a practical vehicle routing and driver scheduling problem

The worlds second largest producer of pork, Danish Crown, also provides a fresh meat supply logistics system within Denmark. This is used by the majority of supermarkets in Denmark. This article addresses an integrated vehicle routing and driver scheduling problem arising at Danish Crown in their fresh meat supply logistics system. The problem consists of a 1-week planning horizon, heterogeneous vehicles, and drivers with predefined work regulations. These regulations include, among other things, predefined workdays, fixed starting time, maximum weekly working duration, and a break rule. The objective is to minimize the total delivery cost that is a weighted sum of two kinds of delivery costs. A multilevel variable neighborhood search heuristic is proposed for the problem. In a preprocessing step, the problem size is reduced through an aggregation procedure. Thereafter, the aggregated weekly planning problem is decomposed into daily planning problems, each of which is solved by a variable neighborhood search. Finally, the solution of the aggregated problem is expanded to that of the original problem. The method is implemented and tested on real-life data consisting of up to 2,000 orders per week. Computational results show that the aggregation procedure and the decomposition strategy are very effective in solving this large scale problem, and our solutions are superior to the industrial solutions given the constraints considered in this work.

Heuristic methods for shared backup path protection planning

Protecting communication networks against failures is becoming increasingly important as they have become an integrated part of our society. Cable failures are fairly common, but it is unacceptable for a single cable failure to disconnect communication for more than a few seconds - hence protection schemes are employed. In contrast to manual intervention, automatic protection schemes such as Shared Backup Path Protection (SBPP) can recover from failure quickly and efficiently. SBPP is a simple but efficient protection scheme that can be implemented in backbone networks with technology available today. In SBPP backup paths are planned in advance for every failure scenario in order to recover from failures quickly and efficiently. The SBPP problem is an NP-hard optimization problem, and previous work confirms that it is time-consuming to solve the problem in practice using exact methods. We present heuristic algorithms and lower bound methods for the SBPP planning problem. Experimental results show that the heuristic algorithms are able to find good quality solutions in minutes. A solution gap of less than 3.5% was achieved for more than half of the benchmark instances (and a gap of less than 12% for the remaining instances.)

Enabling optimization in LCA: from “ad hoc” to “structural” LCA approach—based on a biodiesel well-to-wheel case study

PurposeApplied life cycle assessment (LCA) studies often lead to a comparison of rather few alternatives; we call this the “ad hoc LCA approach.” This can seem surprising since applied LCAs normally cover countless options for variations and derived potentials for improvements in a product life cycle. In this paper, we will suggest an alternative approach to the ad hoc approach, which more systematically addresses the many possible variations to identify the most promising. We call it the “structural LCA approach.” The goals of this paper are (1) to provide basic guidelines for the structural approach, including an easy expansion of the LCA space; (2) to show that the structural LCA approach can be used for different types of optimization in LCA; and (3) to improve the transparency of the LCA work.MethodsThe structural approach is based on the methodology “design of experiments” (Montgomery 2005). Through a biodiesel well-to-wheel study, we demonstrate a generic approach of applying explanatory variables and corresponding impact categories within the LCA methodology. Explanatory variables are product system variables that can influence the environmental impacts from the system. Furthermore, using the structural approach enables two different possibilities for optimization: (1) single-objective optimization (SO) based on response surface methodology (Montgomery 2005) and (2) multiobjective optimization (MO) by the hypervolume estimation taboo search (HETS) method. HETS enables MO for more than two or three objectives.Results and discussionUsing SO, the explanatory variable “use of residual straw from fields” is, by far, the explanatory variable that can contribute with the highest decrease of climate change potential. For the respiratory inorganics impact category, the most influencing explanatory variable is found to be the use of different alcohol types (bioethanol or petrochemical methanol) in biodiesel production. Using MO, we found the Pareto front based on 5 different life cycle pathways which are nondominated solutions out of 66 different analyzed solutions. Given that there is a fixed amount of resources available for the LCA practitioner, it becomes a prioritizing problem whether to apply the structural LCA approach or not. If the decision maker only has power to change a single explanatory variable, it might not be beneficial to apply the structural LCA approach. However, if the decision maker (such as decision makers at the societal level) has power to change more explanatory variables, then the structural LCA approach seems beneficial for quantifying and comparing the potentials for environmental improvement between the different explanatory variables in an LCA system and identifying the overall most promising product system configurations among the chosen PWs.ConclusionsThe implementation of the structural LCA approach and the derived use of SO and MO have been successfully achieved and demonstrated in the present paper. In addition, it is demonstrated that the structural LCA approach can lead to more transparent LCAs since the potentially most important explanatory variables which are used to model the LCAs are explicitly presented through the structural LCA approach. The suggested structural approach is a new approach to LCA and it seems to be a promising approach for searching or screening product systems for environmental optimization potentials. In the presented case, the design has been a rather simple full factorial design. More complicated problems or designs, such as fractional designs, nested designs, split plot designs, and/or unbalanced data, in the context of LCA could be investigated further using the structural approach.

Integer programming techniques for educational timetabling

Educational timetabling problems require the assignment of times and resources to events, while sets of required and desirable constraints must be considered. The XHSTT format was adopted in this work because it models the main features of educational timetabling and it is the most used format in recent studies in the field. This work presents new cuts and reformulations for the existing integer programming model for XHSTT. The proposed cuts improved hugely the linear relaxation of the formulation, leading to an average gap reduction of 32%. Applied to XHSTT-2014 instance set, the alternative formulation provided four new best known lower bounds and, used in a matheuristic framework, improved eleven best known solutions. The computational experiments also show that the resulting integer programming models from the proposed formulation are more effectively solved for most of the instances.

Network Survivability: End-to-End Recovery Using Local Failure Information

This chapter presents an advanced shared protection approach called Failure Dependent Path Protection (FDPP). Under this approach, several protection paths can be assigned to connections in the context of a shared protection framework. After formalizing the survivable online routing problem, two possible implementations are compared, one based on heuristics and the other on ILP. Building upon the concepts of routing already exposed, the chapter then presents two case studies. The first one employs Shortcut Span Protection to examine how different protection strategies affect resource provisioning, while the second is a thorough analysis of the performance of path protection in terms of connection availability, both for dedicated and shared path protection in heterogeneous network topologies.

Staff optimization for time-dependent acute patient flow

Abstract The emergency department is a key element of acute patient flow, but due to high demand and an alternating rate of arriving patients, the department is often challenged by insufficient capacity. Proper allocation of resources to match demand is, therefore, a vital task for many emergency departments. Constrained by targets on patient waiting time, we consider the problem of minimizing the total amount of staff-resources allocated to an emergency department. We test a matheuristic approach to this problem, accounting for both patient flow and staff scheduling restrictions. Using a continuous-time Markov chain, patient flow is modeled as a time-dependent queueing network where inhomogeneous behavior is evaluated using the uniformization method. Based on this modeling approach, we recursively evaluate and allocate staff to the system using integer linear programming until the waiting time targets are respected in all queues of the network. By comparing to discrete-event simulations of the associated system, we show that this approach is adequate for both modeling and optimizing the patient flow. In addition, we demonstrate robustness to the service time distribution and the associated system with multiple classes of patients.

A fix-and-optimize matheuristic for university timetabling

University course timetabling covers the task of assigning rooms and time periods to courses while ensuring a minimum violation of soft constraints that define the quality of the timetable. These soft constraints can have attributes that make it difficult for mixed-integer programming solvers to find good solutions fast enough to be used in a practical setting. Therefore, metaheuristics have dominated this area despite the fact that mixed-integer programming solvers have improved tremendously over the last decade. This paper presents a matheuristic where the MIP-solver is guided to find good feasible solutions faster. This makes the matheuristic applicable in practical settings, where mixed-integer programming solvers do not perform well. To the best of our knowledge this is the first matheuristic presented for the University Course Timetabling problem. The matheuristic works as a large neighborhood search where the MIP solver is used to explore a part of the solution space in each iteration. The matheuristic uses problem specific knowledge to fix a number of variables and create smaller problems for the solver to work on, and thereby iteratively improves the solution. Thus we are able to solve very large instances and retrieve good solutions within reasonable time limits. The presented framework is easily extendable due to the flexibility of modeling with MIPs; new constraints and objectives can be added without the need to alter the algorithm itself. At the same time, the matheuristic will benefit from future improvements of MIP solvers. The matheuristic is benchmarked on instances from the literature and the 2nd International Timetabling Competition (ITC2007). Our algorithm gives better solutions than running a state-of-the-art MIP solver directly on the model, especially on larger and more constrained instances. Compared to the winner of ITC2007, the matheuristic performs better. However, the most recent state-of-the-art metaheuristics outperform the matheuristic.

A Strategic View of University Timetabling

Abstract University timetabling has traditionally been studied as an operational problem where the goal is to assign lectures to rooms and timeslots and create timetables of high quality for students and teachers. Two other important decision problems arise before this can be solved: what rooms are necessary, and in which teaching periods? These decisions may have a large impact on the resulting timetables and are rarely changed or even discussed. This paper focuses on solving these two strategic problems and investigates the impact of these decisions on the quality of the resulting timetables. The relationship and differences between operational, tactical and strategic timetabling problems are reviewed. Based on the formulation of curriculum-based course timetabling and data from the Second International Timetabling Competition (ITC 2007), three new bi-objective mixed-integer models are formulated. We propose an algorithm based on the ϵ-constraint method to solve them. The algorithm can be used to analyze the impact of having different resources available on most timetabling problems. Finally, we report results on how the three objectives – rooms, teaching periods, and quality – influence one another.