1. Operational Research & ManagementOperations Scheduling Workforce Scheduling 1.Days-Off Scheduling 2.Shift Scheduling 3. Cyclic Staffing Problem (& extensions) 4.Crew Scheduling

2. Operational Research & ManagementOperations Scheduling Topic 1 Days-Off Scheduling Off-Days Scheduling: “Scheduling workers who fall asleep on the job is not easy.” Not

3. Operational Research & ManagementOperations Scheduling3 Days-Off Scheduling  Number of workers assigned to each day  Fixed size of workforce  Problem: find minimum number of employees to cover a weeks operation

4. Operational Research & ManagementOperations Scheduling4 Constraints  Demand per day n j, j = 1,2,…,7  k 1 out of every k 2 weekends (day 1 & 7) off  Work 5 out of 7 days  Work no more than 6 consecutive days

5. Operational Research & ManagementOperations Scheduling5 Optimal Schedule  Algorithm for one week  Repeat for next week  Cyclic schedule when repeat

6. Operational Research & ManagementOperations Scheduling6 Lower Bounds on Minimum Workforce W  Weekend constraint  Total demand constraint  Maximum daily demand constraint

7. Operational Research & ManagementOperations Scheduling7 Optimal Schedule  Define  First schedule weekends off (cyclic)  Furthermore,  Idea: Give W workers 2n days off during the week Work both days! Surplus when all workers present

8. Operational Research & ManagementOperations Scheduling8 Algorithm  Schedule weekends off  Determine additional off days (in pairs)  Categorize employees  Assign off-day pairs

9. Operational Research & ManagementOperations Scheduling9 Example - analysis  Data  Bounds: – max(n 1,...,n 7 ) = 3, then W >= 3 –, so W >= 3 – n = max(n 1, n 7 ) = 2, k 1 = 1 and k 2 = 3, so

10. Operational Research & ManagementOperations Scheduling10 Example - solution  Weekends off (one worker per weekend)  Calculate 2n surplus days (in pairs)  (Sun, Mon) and (Mon, Mon)  Weekly: assign pairs to worker (or to pair of workers) Week 1 1: off / on1 2: on / off1 3: on / on2

11. Operational Research & ManagementOperations Scheduling Topic 2 Shift Scheduling

12. Operational Research & ManagementOperations Scheduling12 Shift Scheduling  Fixed cycle of length m periods  Have b i people assigned to ith period  Have n shift patterns:  Cost c j of assigning a person to shift j  Integer decision variable: x j = # people assigned to j

13. Operational Research & ManagementOperations Scheduling13 Solution  NP-hard in general  Special structure in shift pattern matrix  Solve LP relaxation – Solution always integer when each column contains a contiguous set of ones

14. Operational Research & ManagementOperations Scheduling Topic 3 Cyclic Staffing (& extensions)

15. Operational Research & ManagementOperations Scheduling15 The outer ring shows the average arriving intensity at that hour. The inner ring shows the number of centralists necessary for that particular arriving intensity. 4 3 3 4 6 6 6 4 5 5 5 55 5 5 6 6 6 7 3 6 3 2 2 60 24 50 110 116 130 124 140 130 110 102 100 96 72 96 98 90 80 58 20 18 24 34 42 24 18 12 6 1 2 3 4 5 7 8 9 10 1113 14 15 16 17 19 20 21 22 23 # of agents needed Call center agents

16. Operational Research & ManagementOperations Scheduling16 Cyclic Staffing Problem  An m-period cyclic schedule (e.g. 24 hours a day)  Minimize cost  Constraint b i for i th period  Each worker works for k consecutive periods and is free for the next m-k  Example: (5, 7)-cyclic staffing problem

17. Operational Research & ManagementOperations Scheduling17 Integer Program Formulation  Shift patterns  (5, 7) example: 7 different patterns

18. Operational Research & ManagementOperations Scheduling18 Solution  Solution to LP relaxation ‘almost right’  STEP 1: Solve LP relaxation to get if integer STOP; otherwise continue  STEP 2: Formulate two new LPs with  The best integer solution is optimal

19. Operational Research & ManagementOperations Scheduling19 Example (3,5)-cyclic staffing problem Step 1:

20. Operational Research & ManagementOperations Scheduling20 Solution  Add together:  Step 2a: Add constraint: – No feasible solution  Step 2b: Add constraint:  Solution: Optimal

21. Operational Research & ManagementOperations Scheduling21 Extension 1: Days-Off Scheduling  We can represent our days-off scheduling problem as a cyclic staffing problem as long as we can determine all the shift patterns  Difficulty 1: unknown cycle length  Difficulty 2: many patterns  larger problem

22. Operational Research & ManagementOperations Scheduling22 Example  Two days off in a week + no more than 6 consecutive workdays

23. Operational Research & ManagementOperations Scheduling23 Extension 2: Cyclic Staffing with Overtime  24-hour operation  8-hour shifts with up to 8 hour overtime  3 shifts without overtime + 8 shifts with overtime

24. Operational Research & ManagementOperations Scheduling Topic 44 Crew Scheduling

25. Operational Research & ManagementOperations Scheduling25 Crew Scheduling  Have m jobs, say flight legs  Have n feasible combination of jobs a crew is permitted to do 1 2 3 4 5 6 Set partitioning problem

26. Operational Research & ManagementOperations Scheduling26 Notation  Cost c j of round trip j  Define

27. Operational Research & ManagementOperations Scheduling27 Integer Program Minimize Subject to

28. Operational Research & ManagementOperations Scheduling28 Set Partitioning  Constraints called partitioning equations  The positive variables in a feasible solution called a partition  NP-Hard  Well studied like TSP, graph-coloring, bin-packing, etc.

29. Operational Research & ManagementOperations Scheduling29 Row Prices  Say that is a set of feasible row prices if for  Cost of covering a job

30. Operational Research & ManagementOperations Scheduling30 Change Partition  Let Z 1 (Z 2 ) denote the objective value of partition 1 (2)  Then  Potential savings of including column j is  If all negative then optimal

31. Operational Research & ManagementOperations Scheduling31 Heuristic  Start with some partition  Construct a new partition as follows: – Find the column with highest potential savings – Include this column in new partition – If all jobs covered stop; otherwise repeat

32. Operational Research & ManagementOperations Scheduling32 Helpdesk (KPN)  18 employees (= 15 in DH + 3 in G)  6 required at desk (= 5 in DH + 1 in G)  5 in DH (= 2 early + 3 late shift)  Wishes (soft constraints) – holiday – other duties – preference for early shif – preference for late shift  determine schedule for the next 8 weeks: – that is fair – satisfies all wishes as much as possible

33. Operational Research & ManagementOperations Scheduling33 Helpdesk model Groningen  b it :person i is available at day t (no holiday)  r it :person i has other duties at day t  x it :person i has desk duty at day t

34. Operational Research & ManagementOperations Scheduling34 Schedules that satisfy all wishes

35. Operational Research & ManagementOperations Scheduling35 More wishes, more constraints  All have same number of desk duties  May conflict with other wishes, e.g. request for duty free days  Holidays may not lead to relatively more desk duties  Desk duties evenly spaced in time

36. Operational Research & ManagementOperations Scheduling36 Helpdesk model Leidschendam  b it :person i is available at day t (no holiday)  r it :person i has other duties at day t  w ij :person i prefers shift j  x ijt :person i has desk duty at day t and shift j