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Most nurse rostering problems have a number of objectives. However, as many of these objectives conflict with each other, a feasible solution which simultaneously satisfies all of them rarely exists. Instead, the objectives are often treated as goals or soft constraints with user specified priorities or weights. The objectives are then often combined into a single (often weighted) sum.

An alternative approach is Pareto optimisation, which aims to return the Pareto optimal front for a multiobjective problem. The Pareto optimal front consists of all the non-dominated solutions (a solution is non-dominated if there is no other solution which is better than it for all objectives). The user can then select a solution from this front which best represents their trade-off preference for the objectives. A good introduction to Pareto optimisation approaches to scheduling and timetabling can be found in [87, 156].

Arthur and Ravindran [20] solve a nurse rostering problem using goal programming followed by heuristic assignment. In the first phase, a goal programming model is used to assign days on/off to nurses over weekly periods. One of three shift types is then assigned to each nurse for the on days using heuristics based on minimising under-cover. Although the approach does not permit the substitution of different grade/class nurses for each other or consider part time workers, feasible extensions are suggested to accommodate these requirements.

Musa and Saxena [191] use a zero-one integer goal programming method to solve a very basic nurse rostering problem. Although the problem includes full and part time nurses of different grades, the only instance tested had eleven nurses, a two week scheduling horizon and one shift type. Seven goals are defined relating to cover requirements, weekends and consecutive days off and minimum/maximum number of days worked.

Franz et al. [105] use integer goal programming to solve a slightly different health personnel scheduling problem. In this scenario, nurses can be assigned to a number of different clinics with different geographical locations. This complicates the problem as travel costs and nurse preferences for working in different locations have to be considered. The problem is simpler in another respect though as cover only has to be provided at each clinic Monday-Friday, 8:00am-9:00pm. Each clinic has a varying skill mix and staff number requirements to provide a satisfactory service for the predicted patient numbers. The objectives or goals of the problem are to maximise staff to patient ratios in

order to reduce waiting times, minimise travel costs for the staff and maximise staff preferences for working in specific clinics at certain times. Although integer solutions to the problem could not be produced in acceptable computation times, a number of modifications to the problem are considered, one of which enables the fast production of solutions which are comparable to the manually created ones.

Berrada et al. [39] test three techniques for solving a nurse scheduling problem with multiple objectives. Although the problem is simplified by not considering shift rotations, a number of common soft constraints or objectives are included. For example, no isolated working days, a maximum length of consecutive working days, grouping days off together and personal shift and day off requests. The objectives are assigned a priority ordering to reflect scheduling preferences. Two mathematical programming techniques are tested to produce (loosely) non- dominated solutions with respect to the objectives used. A tabu search with a neighbourhood based on swapping a working and non-working day for an individual nurse is also applied (this swap is possible as cover requirements for a specific day do not represent a strict hard constraint). All three techniques produced schedules of a similar satisfactory quality although the tabu search required more computation time. Further experiments with tabu search on a very similar problem formulation can be found in [102].

Jaszkiewicz [141] uses a metaheuristic approach to solve a multiple objective nurse scheduling problem for a surgery unit in a Polish hospital. The five objectives defined are similar to those discussed in other nurse rostering problems

in the form of constraints or objectives. For example, preferred lengths of consecutive working days, non-working days, shift rotations, balance of shift types worked and equal assignments of surplus nurses over the week. A population of initial solutions is created using a constraint programming method and then a simulated annealing approach is employed to identify the Pareto optimal front, or at least a good approximation to it. The algorithm uses dynamically altered weights for each objective to guide the search over the trade off surface. A randomly selected move (from three) may be applied to a solution and the move accepted probabilistically. The approach was able to fairly quickly produce solutions that dominated those produced manually.

Gascon et al. [109] developed a goal programming model to solve a problem requiring the scheduling of flying squad nurses. Rather than always working in the same care unit, a flying squad nurse can be assigned to one of a number of units in order to meet cover demand. Working in different care units helps the nurses maintain the skills required to operate in that unit but frequent movement between units is undesirable as it lessens the quality of service provided. Solutions to the problem must allocate days on and off to the nurses as well as specifying which unit they are stationed at on their working days. Although the model assumes that the nurses work the same shift type, there are a number of objectives and constraints to satisfy. A combined priority ordering and weighted method for the objectives is used in solving the problem.

Through surveys, feedback from head nurses, hospital regulations and analysing published recommended work practices for nurses, Azaiez and Al Sharif [21]

formulate a goal programming model for a nurse scheduling problem in a Saudi Arabian hospital. Initially, the problem was too large to solve, so a heuristic was introduced to decompose the problem. The nurses were split into groups (ensuring a balance of skills), schedules for these groups were found separately and then the overall schedule was formed by recombining the individual group schedules. The authors found that for the majority of instances, this method was able to produce optimal schedules (all goals completely satisfied). After the system was tried and the diverse workforce typical in Saudi Arabian hospitals was surveyed for a second time, an improvement in the rosters was generally noticed. Cost savings through reduced overtime (one of the goals) was also introduced but a few of the nurses did not appreciate this achievement as it prevented them from having the opportunity of earning more money.