The Doctor Rostering Problem - Asser Fahrenholz

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Chapter 4. Solving the DRP 17 this, only a subproblem of the DRP is tried and tested in an optimal solver (see chapter 5). To overcome the problem of exponential search time, approximations or heuristics are used. 4.3 Heuristics for the DRP Heuristics are trial-and-error methods, using some evaluation or objective function to evaluate solutions. It is important to note that optimality of the results obtained from heuristics can not be guaranteed and a measure of how good the heuristic is, can be hard to find. The main goal of a heuristic is to find high quality or near-optimal approximations to solutions of a computationally difficult problem, in a relatively short amount of time. A key part of heuristics is the fact that they are tailored specifically for the problem to which they are applied, using problem specific knowledge to guide the search for better solutions. Heuristics can be either deterministic, meaning, given the same input, then the heuristic will produce the same result every time, or non-deterministic and are non-exhaustive, since if they were exhaustive, they could be classified as an exact method, and not an approximation. Different types of heuristics exist to reach a solution. First, there are constructive- or construction heuristics, that builds a solution from scratch. These are considered fast as they are often only used once or in a one-pass manner. The second type of heuristics are optimisation heuristics, such as the class of Local Search (LS) heuristics, exploring the solution space around a given solution, using a well defined function. The last notion of heuristics that will be addressed here, are Metaheuristics. These are a class of more general, in some sense advanced or sophisticated, heuristics, using some algorithm- dependent mechanism to guide the search for a better solution. In this project, three heuristics will be addressed; a greedy construction heuristic, the Greedy Randomized Adaptive Search Procedure metaheuristic and the Simulated Annealing metaheuristic. Due to the structure of this problem, I (similar to Dowsland [8]) allow infeasible solutions in the heuristics, thus not only seeking global optimum of objective function value, but also seeking minimisation of violations of hard constraints. Minimising these violations is of higher priority than the objective function value.
Chapter 4. Solving the DRP 18 4.3.1 A construction heuristic The initial solution is the basis for optimisation and as such, is an important part of any heuristic modelled to solve this problem. One goal of a construction heuristic is often to be fast as to allow time to be focused on other parts of the process. In the context of this DRP, several construction heuristics can be identified: Zero solution is a straightforward and easy solution to construct and consists of noth- ing but an empty schedule. This is a very fast (O(1)) solution to generate but is infeasible (over all shifts) due to constraint H4 on page 10. Random solution consists of randomly assigning doctors to shifts that are unassigned, until all shifts are assigned a doctor. This can be done by assigning each shift a random doctor, or, for each doctor, assigning that doctor to approximately n m random shifts. The random solution does not guarantee feasibility and is worse, in running time (O(n)), than the zero solution. Greedy solution is based on the concept of choosing what looks best, given a candidate set and a selection function, here and now. That is, it bases its choice upon knowledge of the present and past shifts, but not the future ones as they have not yet been assigned a doctor. It makes no reconsideration of past choices, contrary to e.g. dynamic programming algorithms, that bases decisions on all previous decisions and as such, may reconsider the past. Several notions of the candidate set are used in the literature, but in the context of this project, I define the Local Candidate List as a list unique to each shift consisting of all doctors, ordered by their heuristic value when assigning them to the given shift. The greedy algorithm picks the best doctor from the candidate list and assigns it to the shift at hand. The basic version of the greedy is deterministic, i.e. given the same input, greedy always returns the same output. In an adaptive greedy algorithm (Resende and Ribeiro [20]), the doctor is randomly chosen from the local candidate list. To rule out the possibility of choosing the worst doctor for the shift, I introduce the Re- stricted Candidate List as the n best, measured by either a heuristic value or by a percentage, elements in the candidate list. Figure 4.2 on the next page illustrates various kinds of RCL parameters for a minimisation problem. Another option for the RCL list, is to define it as a list of ordered pairs, consisting of all unassigned shifts coupled with the doctor that has the highest heuristic value when assigned to the specific shift. This option was discarded due to the time required to construct the list being O(n · m) and the fact that the list would have to be reconstructed for each shift. The greedy solution does not guarantee feasible solutions, though the odds has improved vastly, from that of the random solution.
Page 1 and 2: TECHNICAL UNIVERSITY OF DENMARK The
Page 3 and 4: TECHNICAL UNIVERSITY OF DENMARK Abs
Page 5 and 6: Acknowledgements Foremost, I would
Page 7 and 8: Contents vi 4.3.1 A construction he
Page 9 and 10: List of Figures 4.1 The transformat
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Page 13 and 14: Chapter 1. Introduction and problem
Page 15 and 16: Chapter 2. The Doctor Rostering Pro
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Appendix C. Tests 88 C.1.1 Instance
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Appendix C. Tests 94 JST PZH HP SDA
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Appendix C. Tests 100 Cons 12 Imps
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Appendix C. Tests 102 Test 3 RCL 1
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Bibliography 118 [10] Michael R. Ga
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