1. © J. Christopher Beck 20051 Lecture 24: Workforce Scheduling 2

2. © J. Christopher Beck 2005 2 Outline Shift Scheduling Example 12.3.1 Cyclic Staffing Algorithm 12.4.1 Example 12.4.2 Call Centre Scheduling

3. © J. Christopher Beck 2005 3 In Yesterday’s Episode … We created shift patterns and assigned one worker to each pattern All patterns cost the same, so we just wanted to minimize the number of workers SSMTWTFSSMTWTFSSMTWTFS 1XXXXXXX 2XXXXXX 3XXXXXX

4. © J. Christopher Beck 2005 4 Shift Scheduling Fixed cycle Month, week, day Predefined set of shift patterns Each worker is assigned to exactly one pattern Each pattern has its own cost Assign workers to patterns such that Staffing requirements are met and cost is minimized

5. © J. Christopher Beck 2005 5 Shift Scheduling Definition m periods: lengths may be different b i : # people required in period i n shift patterns n j = (a 1j, a 2j, …, a mj ) = (1, 0, 0, …, 0) a ij = 1: workers on pattern j work period i a ij = 0: workers on pattern j don’t work period i c j : cost of assigning a worker to pattern j x j : # of people assigned to pattern j

6. © J. Christopher Beck 2005 6 IP Formulation (see Example 12.3.1) minimize subject to Each period has enough workers Number workers assigned to a pattern is an positive integer or 0 Such an IP is hard unless there is special structure such as having no split shifts

7. © J. Christopher Beck 2005 7 LS Formulation Develop a local search formulation that doesn’t depend on having no split shifts

8. © J. Christopher Beck 2005 8 Apply Your LS to Example 12.3.1 PatternHoursTotal Hours Cost 110AM – 6PM850 21PM – 9PM860 312PM – 6PM630 410AM – 1PM315 56PM – 9PM316 HourStaff 10AM – 11AM3 11AM – 12PM4 12PM – 1PM6 1PM – 2PM4 2PM – 3PM7 3PM – 4PM8 4PM – 5PM7 5PM – 6PM6 6PM – 7PM4 7PM – 8PM7 8PM – 9PM8

9. © J. Christopher Beck 2005 9 A Variation of Example 12.3.1 PatternHoursTotal Hours Cost 110AM – 6PM850 21PM – 9PM860 312PM – 6PM630 410AM – 1PM315 56PM – 9PM316 62PM – 5PM, 7PM – 9PM 535 HourStaff 10AM – 11AM3 11AM – 12PM4 12PM – 1PM6 1PM – 2PM4 2PM – 3PM7 3PM – 4PM8 4PM – 5PM7 5PM – 6PM6 6PM – 7PM4 7PM – 8PM7 8PM – 9PM8 Would you solve this problem with IP or LS?

10. © J. Christopher Beck 2005 10 Cyclic Staffing m period cycle Each period i has requirement b i Each person works k consecutive periods and has the other m – k off c j is the cost of putting a worker on pattern j Find minimum cost schedule

11. © J. Christopher Beck 2005 11 (5,7)-Cyclic Staffing Problem 1001111 1100111 1110011 1111001 1111100 0111110 0011111 Periods Patterns 80701001057590100 3 8 9 11 4 Cost Requirements There are split shifts!

12. © J. Christopher Beck 2005 12 Different Special Structure Makes The Problem Easy Solve LP1 (x’ 1,x’ 2, …, x’ n ) If integral, you have optimal Otherwise LP2: LP1 + If integral, you have optimal Otherwise LP3: LP1 + Always integral, so you have optimal

13. © J. Christopher Beck 2005 13 Call Centre Scheduling: Demand time 6AM126PM demand

14. © J. Christopher Beck 2005 14 Call Centre Scheduling: Shift Patterns time 6AM126PM Breaks

15. © J. Christopher Beck 2005 15 Call Centre Scheduling Assign people to shifts to meet the demand and minimize costs It gets worse: workers with different skills! {English}, {English, French}, {French}

16. © J. Christopher Beck 2005 16 Solution Framework (Fig 12.4) Select solid tours Place breaksCompare fits Close enough? Modify target demand DONE Y N

17. © J. Christopher Beck 2005 17 Call Centre Scheduling Assign people to shifts to meet the demand and minimize costs It gets worse: workers with different skills! {English}, {English, French}, {French}

18. © J. Christopher Beck 2005 18 Call Centre Scheduling: Demand time 6AM126PM demand English French