Integer programming model extensions for a multi-stage nurse rostering problem

Annals of Operations Research - In the variant of the well studied nurse rostering problem proposed in the Second International Nurse Rostering Competition, multiple stages have to be solved...     

A scatter search methodology for the nurse rostering problem

The benefits of automating the nurse scheduling process in hospitals include reducing the planning workload and associated costs and being able to create higher quality and more flexible schedules. This has become more important recently in order to retain nurses and to attract more people into the profession. Better quality rosters also reduce fatigue and stress due to overwork and poor scheduling and help to maximise the use of leisure time by satisfying more requests. A more contented workforce will lead to higher productivity, increased quality of patient service and a better level of healthcare. This paper presents a scatter search approach for the problem of automatically creating nurse rosters. Scatter search is an evolutionary algorithm, which has been successfully applied across a number of problem domains. To adapt and apply scatter search to nurse rostering, it was necessary to develop novel implementations of some of scatter search's subroutines. The algorithm was then tested on publicly available real-world benchmark instances and compared against previously published approaches. The results show the proposed algorithm is a robust and effective method on a wide variety of real-world instances.     

A systematic two phase approach for the nurse rostering problem

Nurse rostering is an NP-hard combinatorial problem which makes it extremely difficult to efficiently solve real life problems due to their size and complexity. Usually real problem instances have complicated work rules related to safety and quality of service issues in addition to rules about quality of life of the personnel. For the aforementioned reasons computer supported scheduling and rescheduling for the particular problem is indispensable. The specifications of the problem addressed were defined by the First International Nurse Rostering Competition (INRC2010) sponsored by the leading conference in the Automated Timetabling domain, PATAT-2010. Since the competition imposed quality and time constraint requirements, the problem instances were partitioned into sub-problems of manageable computational size and were then solved sequentially using Integer Mathematical Programming. A two phase strategy was implemented where in the first phase the workload for each nurse and for each day of the week was decided while in the second phase the specific daily shifts were assigned. In addition, local optimization techniques for searching across combinations of nurses’ partial schedules were also applied. This sequence is repeated several times depending on the available computational time. The results of our approach and the submitted software produced excellent solutions for both the known and the hidden problem instances, which in respect gave our team the first position in all tracks of the INRC-2010 competition.     

Roster evaluation based on classifiers for the nurse rostering problem

The personnel scheduling problem is a well-known NP-hard combinatorial problem. Due to the complexity of this problem and the size of the real-world instances, it is not possible to use exact methods, and thus heuristics, meta-heuristics, or hyper-heuristics must be employed. The majority of heuristic approaches are based on iterative search, where the quality of intermediate solutions must be calculated. Unfortunately, this is computationally highly expensive because these problems have many constraints and some are very complex. In this study, we propose a machine learning technique as a tool to accelerate the evaluation phase in heuristic approaches. The solution is based on a simple classifier, which is able to determine whether the changed solution (more precisely, the changed part of the solution) is better than the original or not. This decision is made much faster than a standard cost-oriented evaluation process. However, the classification process cannot guarantee 100 % correctness. Therefore, our approach, which is illustrated using a tabu search algorithm in this study, includes a filtering mechanism, where the classifier rejects the majority of the potentially bad solutions and the remaining solutions are then evaluated in a standard manner. We also show how the boosting algorithms can improve the quality of the final solution compared with a simple classifier. We verified our proposed approach and premises, based on standard and real-world benchmark instances, to demonstrate the significant speedup obtained with comparable solution quality.     

The nurse rostering problem: A critical appraisal of the problem structure

This paper is concerned with the problem of nurse rostering within hospitals. We analyse a class of four benchmark instances from the nurse rostering literature to provide insight into the nature of the problem. By highlighting the structure of the problem we are able to reduce the relevant solution space. A mixed integer linear programme is then able to find optimal solutions to all four instances of this class of benchmark problems, each within half an hour. Our second contribution is to extend current mathematical approaches to nurse rostering to take better account of the practical considerations. We provide a methodology for handling rostering constraints and preferences arising from the continuity from one scheduling period to the next.     

Integer programming models for the q-mode problem

The q-mode problem is a combinatorial optimization problem that requires partitioning of objects into clusters. We discuss theoretical properties of an existing mixed integer programming (MIP) model for this problem and offer alternative models and enhancements. Through a comprehensive experiment we investigate computational properties of these MIP models. This experiment reveals that, in practice, the MIP approach is more effective for instances containing strong natural clusters and it is not as effective for instances containing weak natural clusters. The experiment also reveals that one of the MIP models that we propose is more effective than the other models for solving larger instances of the problem.     

A hybrid heuristic ordering and variable neighbourhood search for the nurse rostering problem

This paper is concerned with the development of intelligent decision support methodologies for nurse rostering problems in large modern hospital environments. We present an approach which hybridises heuristic ordering with variable neighbourhood search. We show that the search can be extended and the solution quality can be significantly improved by the careful combination and repeated use of heuristic ordering, variable neighbourhood search and back-tracking. The amount of computational time that is allowed plays a significant role and we analyse and discuss this. The algorithms are evaluated against a commercial Genetic Algorithm on commercial data. We demonstrate that this methodology can significantly outperform the commercial algorithm. This paper is one of the few in the scientific nurse rostering literature which deal with commercial data and which compare against a commercially implemented algorithm.     

Integer programming formulations for the minimum weighted maximal matching problem

Given an undirected graph, the problem of finding a maximal matching that has minimum total weight is NP-hard. This problem has been studied extensively from a graph theoretical point of view. Most of the existing literature considers the problem in some restricted classes of graphs and give polynomial time exact or approximation algorithms. On the contrary, we consider the problem on general graphs and approach it from an optimization point of view. In this paper, we develop integer programming formulations for the minimum weighted maximal matching problem and analyze their efficacy on randomly generated graphs. We also compare solutions found by a greedy approximation algorithm, which is based on the literature, against optimal solutions. Our results show that our integer programming formulations are able to solve medium size instances to optimality and suggest further research for improvement.     

Integer programming approaches to the travelling salesman problem

The availability of an LP routine where we can add constraints and reoptimize, makes it possible to adopt an integer programming approach to the travelling-salesman problem.Starting with some of the constraints that define the problem we use either a branching process or a cutting planes routine to eliminate fractional solutions. We then test the resulting integer solution against feasibility and if necessary we generate the violated constraints and reoptimize until a genuine feasible solution is achieved.Usually only a small number of the omitted constraints is generated.The generality of the method and the modest solution times achieved leads us to believe that such an LP approach to other combinatorial problems deserves further consideration.     

Integer programming models for the routing and spectrum allocation problem

One of the most promising solutions to deal with huge data traffic demands in large communication networks is given by flexible optical networking, in particular the flexible grid (flexgrid) technology specified in the ITU-T standard G.694.1. In this specification, the frequency spectrum of an optical fiber link is divided into narrow frequency slots. Any sequence of consecutive slots can be used as a simple channel, and such a channel can be switched in the network nodes to create a lightpath. In this kind of networks, the problem of establishing lightpaths for a set of end-to-end demands that compete for spectrum resources is called the routing and spectrum allocation problem (RSA). Due to its relevance, this problem has been intensively studied in the last couple of years. It has been shown to be NP-hard (Christodoulopoulos et al. in IEEE J Lightw Technol 29(9):1354–1366, 2011; Wang et al. in IEEE J Opt Commun Netw 4(11):906–917, 2012) and several models and formulations have been proposed, leading to different solution approaches. In this work, we explore integer programming models for RSA, analyzing their effectiveness over known instances. We resort to several modeling techniques, to find natural formulations of this problem. Since integer programming techniques are known to provide successful practical approaches for several combinatorial optimization problems, the aim of this work is to explore a similar approach for RSA.     

Adaptive neighborhood search for nurse rostering

This paper presents an adaptive neighborhood search method (ANS) for solving the nurse rostering problem proposed for the First International Nurse Rostering Competition (INRC-2010). ANS uses jointly two distinct neighborhood moves and adaptively switches among three intensification and diversification search strategies according to the search history. Computational results assessed on the three sets of 60 competition instances show that ANS improves the best known results for 12 instances while matching the best bounds for 39 other instances. An analysis of some key elements of ANS sheds light on the understanding of the behavior of the proposed algorithm.     

Integer linear programming models for a cement delivery problem

We consider a cement delivery problem with an heterogeneous fleet of vehicles and several depots. The demands of the customers are typically larger than the capacity of the vehicles which means that most customers are visited several times. This is a split delivery vehicle routing problem with additional constraints. We first propose a two phase solution method that assigns deliveries to the vehicles, and then builds vehicle routes. Both subproblems are formulated as integer linear programming problems. We then show how to combine the two phases in a single integer linear program. Experiments on real life instances are performed to compare the performance of the two solution methods.     

Parametric linear programming techniques for the indefnite quadratic programming problem

In this paper, we discuss a multiparametric technique for findinga global minimum for an indefinite quadratic programming problembased on the spectral decomposition of the matrix of the quadraticform. Special attention is given to the case where this matrixhas only rank 1, where the multiparametric linear program turnsout to be a single-parameter linear program. An extension ofthe traditional linear parametric procedure is introduced whichin general solves this problem efficiently. However, an exampleis presented showing that this technique may take an exponentialnumber of steps.     

Integer programming models and linearizations for the traveling car renter problem

The traveling car renter problem (CaRS) is an extension of the classical traveling salesman problem (TSP) where different cars are available for use during the salesman’s tour. In this study we present three integer programming formulations for CaRS, of which two have quadratic objective functions and the other has quadratic constraints. The first model with a quadratic objective function is grounded on the TSP interpreted as a special case of the quadratic assignment problem in which the assignment variables refer to visitation orders. The second model with a quadratic objective function is based on the Gavish and Grave’s formulation for the TSP. The model with quadratic constraints is based on the Dantzig–Fulkerson–Johnson’s formulation for the TSP. The formulations are linearized and implemented in two solvers. An experiment with 50 instances is reported.     

Integer programming models for the multidimensional assignment problem with star costs

We consider a variant of the multidimensional assignment problem (MAP) with decomposable costs in which the resulting optimal assignment is described as a set of disjoint stars. This problem arises in the context of multi-sensor multi-target tracking problems, where a set of measurements, obtained from a collection of sensors, must be associated to a set of different targets. To solve this problem we study two different formulations. First, we introduce a continuous nonlinear program and its linearization, along with additional valid inequalities that improve the lower bounds. Second, we state the standard MAP formulation as a set partitioning problem, and solve it via branch and price. These approaches were put to test by solving instances ranging from tripartite to 20-partite graphs of 4 to 30 nodes per partition. Computational results show that our approaches are a viable option to solve this problem. A comparative study is presented.     

A hybrid model of integer programming and variable neighbourhood search for highly-constrained nurse rostering problems

This paper presents a hybrid multi-objective model that combines integer programming (IP) and variable neighbourhood search (VNS) to deal with highly-constrained nurse rostering problems in modern hospital environments. An IP is first used to solve the subproblem which includes the full set of hard constraints and a subset of soft constrains. A basic VNS then follows as a postprocessing procedure to further improve the IP’s resulting solutions. The satisfaction of the excluded constraints from the preceding IP model is the major focus of the VNS. Very promising results are reported compared with a commercial genetic algorithm and a hybrid VNS approach on real instances arising in a Dutch hospital. The comparison results demonstrate that our hybrid approach combines the advantages of both the IP and the VNS to beat other approaches in solving this type of problems. We also believe that the proposed methodology can be applied to other resource allocation problems with a large number of constraints.     

Integer linear programming models for grid-based light post location problem

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.     

Integer linear programming formulation of the material requirements planning problem

Lot sizing procedures for discrete and dynamic demand form a distinct class of inventory control problems, usually referred to asmaterial requirements planning. A general integer programming formulation is presented, covering an extensive range of problems: single-item, multi-item, and multi-level optimization; conditions on lot sizes and time phasing; conditions on storage and production capacities; and changes in production and storage costs per unit. The formulation serves as a uniform framework for presenting a problem and a starting point for developing and evaluating heuristic and tailor-made optimum-seeking techniques.     

Progress control in iterated local search for nurse rostering

This paper describes an approach in which a local search technique is alternated with a process which ‘jumps’ to another point in the search space. After each ‘jump’ a (time-intensive) local search is used to obtain a new local optimum. The focus of the paper is in monitoring the progress of this technique on a set of real world nurse rostering problems. We propose a model for estimating the quality of this new local optimum. We can then decide whether to end the local search based on the predicted quality. The fact that we avoid searching these bad neighbourhoods enables us to reach better solutions in the same amount of time. We evaluate the approach on five highly constrained problems in nurse rostering. These problems represent complex and challenging real world rostering situations and the approach described here has been developed during a commercial implementation project by ORTEC bv.