1 / 36

Workforce Scheduling

Workforce Scheduling. 1. Days-Off Scheduling 2. Shift Scheduling 3. Cyclic Staffing Problem (& extensions) 4. Crew Scheduling. Off-Days Scheduling: “Scheduling workers who fall asleep on the job is not easy.”. Topic 1. Days-Off Scheduling. Not. Days-Off Scheduling.

oleg
Download Presentation

Workforce Scheduling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


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

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

  3. 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. Constraints • Demand per day nj, j = 1,2,…,7 • k1 out of every k2 weekends (day 1 & 7) off • Work 5 out of 7 days • Work no more than 6 consecutive days

  5. Optimal Schedule • Algorithm for one week • Repeat for next week • Cyclic schedule when repeat

  6. Lower Bounds on Minimum Workforce W • Weekend constraint • Total demand constraint • Maximum daily demand constraint

  7. 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. Algorithm • Schedule weekends off • Determine additional off days (in pairs) • Categorize employees • Assign off-day pairs

  9. Example - analysis • Data • Bounds: • max(n1,...,n7) = 3, then W >= 3 • , so W >= 3 • n = max(n1, n7) = 2, k1 = 1 and k2 = 3, so

  10. 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 / on 1 2: on / off 1 3: on / on 2

  11. Topic 2 Shift Scheduling

  12. Shift Scheduling • Fixed cycle of length m periods • Have bi people assigned to ith period • Have n shift patterns: • Cost cj of assigning a person to shift j • Integer decision variable: xj = # people assigned to j

  13. 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. Topic 3 Cyclic Staffing (& extensions)

  15. 24 23 1 22 2 60 42 110 34 21 116 3 24 4 3 6 3 130 20 6 3 4 18 6 2 124 20 19 6 5 2 140 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. 24 7 3 6 18 6 130 4 50 6 17 4 7 110 58 6 5 16 102 5 8 80 5 5 5 100 5 5 90 15 9 96 98 72 96 14 10 13 11 12 Call center agents # of agents needed

  16. Cyclic Staffing Problem • An m-period cyclic schedule (e.g. 24 hours a day) • Minimize cost • Constraint bi for ith period • Each worker works for k consecutive periods and is free for the next m-k • Example: (5, 7)-cyclic staffing problem

  17. Integer Program Formulation • Shift patterns • (5, 7) example: 7 different patterns

  18. 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. Example (3,5)-cyclic staffing problem Step 1:

  20. Optimal Solution • Add together: • Step 2a: Add constraint: • No feasible solution • Step 2b: Add constraint: • Solution:

  21. 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. Example • Two days off in a week + no more than 6 consecutive workdays

  23. 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. Topic 44 Crew Scheduling

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

  26. Notation • Cost cj of round trip j • Define

  27. Integer Program Minimize Subject to

  28. 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. Row Prices • Say that is a set of feasible row prices if for • Cost of covering a job

  30. Change Partition • Let Z1 (Z2) denote the objective value of partition 1 (2) • Then • Potential savings of including column j is • If all negative then optimal

  31. 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. 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. Helpdesk model Groningen • bit: person i is available at day t (no holiday) • rit: person i has other duties at day t • xit: person i has desk duty at day t

  34. Schedules that satisfy all wishes

  35. 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. Helpdesk model Leidschendam • bit: person i is available at day t (no holiday) • rit: person i has other duties at day t • wij: person i prefers shift j • xijt: person i has desk duty at day t and shift j

More Related