1. Rostering Problem Liverpool Hope University

2. Covering problem Computational problem that ask whether a certain combinatorial structure covers another Minimization problem and usually linear programs The dual problem are packing problems

3. Staff Rostering Problem Definition Personnel scheduling, or rostering, is the process of constructing work timetables for its staff so that an organisation can satisfy the demand for its goods or services The first part of this process involves determining the number of staff, with particular skills, needed to meet the service demand. Individual staff members are allocated to shifts so as to meet the required staffing levels at different times, and duties are then assigned to individuals for each shift.

4. Staff Rostering Problem Definition All industrial regulations associated with the relevant workplace agreements must be observed during the process It is extremely difficult to find good solutions to these highly constrained and complex problems and even more difficult to determine optimal solutions that minimise costs, meet employee preferences, distribute shifts equitably among employees and satisfy all the workplace constraint

5. Staff rostering problem Different algorithms solve the problem in different ways but they all need to be able to differentiate between good rosters and bad rosters

6. Motivation Cost savings. Employee salaries are a significant proportion of expenditure for most organisations. Better scheduling can reduce this expense: – Through minimising over coverage (not assigning more employees than are required at any time). – Via cutting the reliance on expensive, short notice workers to fill gaps in schedules when it may appear to be the only solution. – Through increased work performance due to reduced fatigue and stress amongst workers caused by poor scheduling (e.g. overwork, insufficient rest, bad shift combinations etc).

7. Motivation Higher staff retention and a recruiting aid. In the healthcare industry for example, a number of countries have experienced a reduction in the number of people training to become nurses and/or an increase in the number of nurses leaving the profession. As the populations of these countries age, the demand for healthcare will increase and these nurse shortage problems will become more acute. In order to encourage more people to become nurses and to reduce the number of people leaving the nursing profession, various initiatives have been proposed. One of these is to allow more part time contracts and to provide the nurses with more flexibility and input on when they work. This allows, for example, more parents with young children to remain in nursing.

8. Motivation Reduction in absenteeism and tardiness. Many organisations incur a reduction in productivity due to staff absenteeism and tardiness. The reasons for personnel arriving late or taking days off are various. This can partly be attributed, though, to dissatisfaction with their schedules or fatigue due to poor scheduling. This can be reduced through better rostering and giving the workers more say in their work patterns. For example, an employee is less likely to be absent for a shift which they actually requested.

9. Motivation Personal preferences. Increasing the employees' satisfaction with their schedules by providing them with more choice and allowing them to better plan and use their leisure time can also increase general morale levels. This, in turn, can lead to benefits such as higher productivity and lower staff turnover with its associated costs. Increased quality of service. As another example from the healthcare industry, nurses are able to spend more time with patients if they are not overworked or the ward/department is not understaffed as a result of poor scheduling. In the worst case, fatigue and stress can result in medical error endangering the patient's health and safety and damaging the hospital's reputation.

10. Mathematical Model A shift is a work duty characterized by a starting time and a duration. The set of shifts S is an input of the problem. It contains also a particular shift, called rest shift, representing days off. The shifts in S cover different parts of the day and may be disjoint in time, or overlapping. A staff member can work any shift in S and can be assigned only one shift per day. The solution to this problem by a roster design that can be visualized by a table (called roster table) with a row for each staff member and seven columns indexed by the days of the week

11. Mathematical Model assume that a week starts on Monday and ends on Sunday. Each position in the roster table is called a cell and corresponds to a pair (employee, day) The set of cells C is given by all the pairs of employees p in P and days d in D

12. Integer Programming Formulation P the set of staff members, by C the set of cells of the roster table, and by S the set of shifts, given by the work shifts plus the rest shift. We associate with the seven days of the week the integers 1, 2,..., 7, where 1 represents Monday, 2 represents Tuesday, etc., and D = {1, 2,..., 7) is the set of days in the week.

13. Integer Programming Formulation we define C(d) as the set of cells corresponding to day d, and d(c) as the day of the week associated with cell c in the roster table. Let denote a 0-1 variable for cell and shift, such that If shifts s is assigned to cell c otherwise

14. Constraints Four types of contraints – Assignment constraints – Work shifts covering constraints – Forbidden sequence constraints – Weekly rest shifts constraints

15. Assignment constraints The assignment constraints require that one shift must be assigned for each staff member and each day, then For all

16. Work shifts covering constraints The work shifts covering constraints state that the number of staff members assigned to a given work shift s on a day d must be greater than or equal to the given demand Where is the rest shift. The purpose of these constraints is to ensure the minimum level of services to be provided. Forall

17. Forbidden sequence constraints The forbidden sequence constraints are used to represent the agreement restrictions on consecutive shifts (for example, two consecutive "night" shifts or two consecutive shifts that do not allow for a sufficiently long break may be forbidden): Where such that if shift t cannot follow shift s Forall

18. Weekly rest shifts constraints The weekly rest shifts constraints express the number of rest shifts that are to be assigned every week. Let assume a fixed number of rest shifts per week equal to two Where W(p) is the set of cell indices corresponding to row p of the roster table. Forall

19. Objective Function The objective function expresses an approximated measure of staff satisfaction by the proximity of certain shifts. When certain pairs of shifts are assigned to consecutive cells (i.e., in consecutive days) the objective function is increased accordingly Let, be the benefit of assigning a shift s immediately before a shift t; then, the staff satisfaction can be expressed by the following quadratic function:

20. Objective Function The main component of the objective function is the sequencing of rest shifts: Rosters where the two weekly rest shifts are assigned to consecutive days are typically preferred. This is reflected in the weights w with the property that is some orders of magnitude larger than, if or. According to this weight structure, each optimal solution will assign the maximum number of consecutive rest shifts. A second important component of the objective function is the assignment of the same work shift in consecutive days. For the same argument expressed above, this is reflected in the weights, which are larger than if.

21. Problem Complexity The staff rostering problem is classified according to mathematical complexity theory as NP-Hard. For problems within this complexity group there is no known algorithm for finding provably optimal solutions within practical time limits. One of the reasons why these problems are so difficult to solve is due to their exponentially large search spaces

22. Automatic Solver http://www.staffrostersolutio ns.com/index.html http://www.staffrostersolutio ns.com/index.html The engine interfaces with existing workforce management solutions via an xml-based modelling lanaguage.

23. Simple Example Staff = {E1, E2, E3, E4, E5} Contracts = {C1, C2, C3} Skills = {A, B, C} Shifts = {Morning (00.00-08.00), Afternoon (08.00-16.00), Night (16.00-00-00)} Scheduling period = 1 month i.e. E1 has contract C1 and skill A, E2 has contract C2 and skills B&C…

24. Simple Example Constraints Monday: min 3 staffs in morning shift Tuesday: min 1 staff with skill B in morning or night shift Wednesday: max 3 staffs with skill B in morning shift Thursday: min 2 staffs from 00.00 to 08.00 Friday: min 2 staffs with skill C in night shift Saturday: min 1 staff with skill C in afternoon shift Sunday: min 3 staffs in morning shift

25. Solution