Routing with time windows (VRPTW) has been an area of research that have attracted many researchers within the last 10 { 15 years. In this period a number of papers and technical reports have been published on the exact solution of the VRPTW. The VRPTW is a generalization of the well-known capacitated routing problem (VRP or CVRP). In the VRP a eet of vehicles must visit (service) a number of customers. All vehicles start and end at the depot. For each pair of customers or customer and depot there is a cost. The cost denotes how much is costs a vehicle to drive from one customer to another. Every customer must be visited exactly ones. Additionally each customer demands a certain quantity of goods delivered (know as the customer demand). For the vehicles we have an upper limit on the amount of goods that can be carried (known as the capacity). In the most basic case all vehicles are of the same type and hence have the same capacity. The problem is now for a given scenario to plan routes for the vehicles in accordance with the mentioned constraints such that the cost accumulated on the routes, the xed costs (how much does it cost to maintain a vehicle) or a combination hereof is minimized. In the more general VRPTW each customer has a time window, and between all pairs of customers or a customer and the depot we have a travel time. The vehicles now have to comply with the additional constraint that servicing of the customers can only be started within the time windows of the customers. It is legal to arrive before a time window \opens" but the vehicle must wait and service will not start until the time window of the customer actually opens. For solving the problem exactly 4 general types of solution methods have evolved in the literature: dynamic programming, Dantzig-Wolfe (column generation), Lagrange decomposition and solving the classical model formulation directly. Presently the algorithms that uses Dantzig-Wolfe given the best results (Desrochers, Desrosiers and Solomon, and Kohl), but the Ph.D. thesis of Kontoravdis shows promising results for using the classical model formulation directly. In this Ph.D. project we have used the Dantzig-Wolfe method. In the Dantzig-Wolfe method the problem is split into two problems: a \master problem" and a \subproblem". The master problem is a relaxed set partitioning v vi problem that guarantees that each customer is visited exactly ones, while the subproblem is a shortest path problem with additional constraints (capacity and time window). Using the master problem the reduced costs are computed for each arc, and these costs are then used in the subproblem in order to generate routes from the depot and back to the depot again. The best (improving) routes are then returned to the master problem and entered into the relaxed set partitioning problem. As the set partitioning problem is relaxed by removing the integer constraints the solution is seldomly integral therefore the Dantzig-Wolfe method is embedded in a separation-based solution-technique. In this Ph.D. project we have been trying to exploit structural properties in order to speed up execution times, and we have been using parallel computers to be able to solve problems faster or solve larger problems. The thesis starts with a review of previous work within the eld of VRPTW both with respect to heuristic solution methods and exact (optimal) methods. Through a series of experimental tests we seek to dene and examine a number of structural characteristics. The rst series of tests examine the use of dividing time windows as the branching principle in the separation-based solution-technique. Instead of using the methods previously described in the literature for dividing a problem into smaller problems we use a methods developed for a variant of the VRPTW. The results are unfortunately not positive. Instead of dividing a problem into two smaller problems and try to solve these we can try to get an integer solution without having to branch. A cut is an inequality that separates the (non-integral) optimal solution from all the integer solutions. By nding and inserting cuts we can try to avoid branching. For the VRPTW Kohl has developed the 2-path cuts. In the separationalgorithm for detecting 2-path cuts a number of test are made. By structuring the order in which we try to generate cuts we achieved very positive results. In the Dantzig-Wolfe process a large number of columns may be generated, but a signicant fraction of the columns introduced will not be interesting with respect to the master problem. It is a priori not possible to determine which columns are attractive and which are not, but if a column does not become part of the basis of the relaxed set partitioning problem we consider it to be of no benet for the solution process. These columns are subsequently removed from the master problem. Experiments demonstrate a signicant cut of the running time. Positive results were also achieved by stopping the route-generation process prematurely in the case of time-consuming shortest path computations. Often this leads to stopping the shortest path subroutine in cases where the information (from the dual variables) leads to \bad" routes. The premature exit from the shortest path subroutine restricts the generation of \bad" routes signi cantly. This produces very good results and has made it possible to solve problem instances not solved to optimality before. The parallel algorithm is based upon the sequential Dantzig-Wolfe based algorithm developed earlier in the project. In an initial (sequential) phase unsolved problems are generated and when there are unsolved problems enough vii to start work on every processor the parallel solution phase is initiated. In the parallel phase each processor runs the sequential algorithm. To get a good workload a strategy based on balancing the load between neighbouring processors is implemented. The resulting algorithm is eÆcient and capable of attaining good speedup values. The loadbalancing strategy shows an even distribution of work among the processors. Due to the large demand for using the IBM SP2 parallel computer at UNIC it has unfortunately not be possible to run as many tests as we would have liked. We have although managed to solve one problem not solved before using our parallel algorithm.

We present a new technique (RIIPS) for solving rostering problems in the presence of service uncertainty. RIIPS stands for "Rostering by Iterating Integer Programming and Simulation". RIIPS allows great complexity of the stochastic system being rostered. This modelling freedom comes at a price, as the approach can be extremely computationally intensive. Therefore any reduction in computational effort using, for example, efficiency improvement techniques, is of great interest. We specify several ways in which these may be applied. 1 INTRODUCTION In rostering applications, one wishes to minimise the costs of staffing, subject to the constraint of maintaining reasonable customer service. The prototypical example is the call centre. In these systems, customers dial a number, possibly join a queue of waiting customers, and then receive service from a service agent. The quality of a customer's service is usually defined in terms of a measure called "customer grade of service" (CGOS), which...

In this paper, we investigate the possibility of integrating Artificial Intelligence (AI) and Operations Research (OR) techniques for solving the Crew Rostering Problem (CRP). CRP calls for the optimal sequencing of a given set of duties into rosters satisfying a set of constraints. The optimality criterion requires the minimization of the number of crews needed to cover the duties. This kind of problem has been traditionally solved with OR techniques. In recent years, a new programming paradigm based on Logic Programming, named Constraint Logic Programming (CLP), has been successfully used for solving hard combinatorial optimization problems. CLP maintains all the advantages of Logic Programming such as declarativeness, non-determinism and an incremental style of programming, while overcoming its limitations, mainly due to the inefficiency in exploring the search space. CLP achieves good results on hard combinatorial optimization problems which, however, are not comparable with those ...

This paper details a new simulation and optimisation based system for personnel scheduling (rostering) of Customs staff at the Auckland International Airport, New Zealand. An integrated approach using simulation, heuristic descent and integer programming techniques has been developed to determine near-optimal staffing levels. The system begins by using a new simulation system embedded within a heuristic search to determine minimum staffing levels for arrival and departure work areas. These staffing requirements are then used as the input to an integer programming model which optimally allocates full and part-time staff to each period of the working day. These shifts are then assigned to daily work schedules having a six-day-on, three-day-off structure. The application of these techniques has resulted in significantly lower staffing levels, while at the same time creating both high quality rosters and ensuring that all passenger processing targets are met. This paper charts the development of this system, outlines failures where they have occurred, and summarises the on-going impacts of this work on the organisation.

The Airline Crew Assignment Problem (ACA) consists of assigning lines of work to a set of crew members such that a set of activities is partitioned and the costs for that assignment are minimized. Especially for European airline companies, complex constraints defining the feasibility of a line of work have to be respected. We developed two different algorithms to tackle the large scale optimization problem of Airline Crew Assignment. The first is an application of the Constraint Programming (CP) based Column Generation Framework. The second approach performs a CP based heuristic tree search. We present how both algorithms can be coupled to overcome their inherent weaknesses by integrating methods from Constraint Programming and Operations Research. Numerical results show the superiority of the hybrid algorithm in comparison to CP based tree search and column generation alone.

This paper presents the design and implementation of an air-crew assignment system for producing and refining a solution to this problem based on the Artificial Intelligence principles and techniques of abductive reasoning as captured by the framework of Abductive Logic Programming (ALP). The system offers a high-level of flexibility in addressing both the tasks of crew scheduling and rescheduling. It can be used to generate a valid and good quality initial solution and then to help the human operators adjust and refine further this solution in order to meet extra requirements of the problem. These additional needs can arise either due to new foreseen requirements that the company wants to have or experiment with for a particular period in time or due to unexpected events that have occurred while the solution (crew-roster) is in operation. Our work shows the ability and flexibility of abduction, and more specifically of ALP, in tackling problems of this type with complex and changing r...

Deciding what to branch on at each node is a key element of search algorithms. We present four families of methods for selecting what question to branch on. They are all information-theoretically motivated to reduce uncertainty in remaining subproblems. In the first family, a good variable to branch on is selected based on lookahead. In real-world procurement optimization, this entropic branching method outperforms default CPLEX and strong branching. The second family combines this idea with strong branching. The third family does not use lookahead, but instead exploits features of the underlying structure of the problem. Experiments show that this family significantly outperforms the state-of-the-art branching strategy when the problem includes indicator variables as the key driver of complexity. The fourth family is about branching using carefully constructed linear inequality constraints over sets of variables. 1

The crew pairing optimization problem is faced by airline companies as an intensive part of the crew scheduling process. Crew scheduling is the assignment of cockpit and cabin crews to the flight legs that a company has to carry out during a predefined period of time. Due to the significant contribution of the crew cost to the overall operating cost of an airline company, the automation of the crew scheduling procedure is highly desirable. However, the crew pairing optimization subproblem of crew scheduling is extremely difficult and combinatorial in nature due to the large number and complexity of the involved constraints. The requirement for optimality makes it even more difficult. Many attempts have been made in the past 40 years to tackle the crew pairing optimization problem using methods from Operations Research. In this paper, an approach based on pure Constraint Logic Programming is presented, which leads to an elegant and flexible modeling of the problem. The whole process is ...

. We introduce a hybrid approach integrating CP-based Column Generation and pure CP methods. It is being applied to the large scale optimization problem of Airline Crew Assignment. The combination of methods from CP and OR results in an algorithm that overcomes the inherent weaknesses of each approach. First numerical results show the superiority of the hybrid algorithm in comparison to direct CP and CP-based Column Generation alone. Keywords: Airline Crew Assignment, hybrid OR-CP method, Constrained Based Column Generation 1 Introduction Scheduling ying crews of airline companies is a hard combinatorial problem, given the complexity of the constraints that have to be satised and the huge search space that has to be explored. The problem is often tackled by breaking it down into the crew pairing subproblem which consists of constructing pairings from ight legs, and the crew rostering (or assignment) subproblem which consists of assigning the constructed pairings to crew me...

The daily operations of the Danish railway operator DSB S-tog suffer from disruptions of various magnitude almost every day. Disruptions initiated by e.g. signalling problems or rolling stock failures cause train delays and cancellations. Changes in the train schedule affect the train driver duties. If a train is cancelled or delayed, the driver assigned to the train task might not be able to reach the station of his next train departure in time. In practice, if a driver is not available for the train departure, another driver, for instance, a reserve, is assigned to the task. If there are no drivers available to cover the task on time, the train is delayed or cancelled, causing further disruptions. The train driver recovery is performed by dispatchers, who often work under tremendous pressure. The size of the schedule (more than 2 000 train tasks, which are covered by approximately 270 drivers on a weekday) and a high frequency of the train departures with headways down to 2 minutes at certain network segments makes it difficult to find a good recovery solution fast.