1. Lecture 23: Workforce Scheduling 1
© J. Christopher Beck 2005

2. Outline Bank Personnel Scheduling Demo Days-off Scheduling
Example
Lower bounds
Algorithm
Example
© J. Christopher Beck 2005

3. Workforce Timetabling
Arrange shifts and assign people to them
Constraints:
Number of people per shift
Minimum Days off (x/k days must be off)
Weekends
Nurses, call centres, hotels, restaurants, plane crew, …
Bank Demo
© J. Christopher Beck 2005

4. Days Off Scheduling (Example 12.2.2)
Find minimum workforce size: W
nj people required on day j
Assume Sun to Sat are numbered 1 to 7
Each employee given 1 out of every 3 weekends off
Each employee works exactly 5 days in 1 week (Sun – Sat)
Each employee works no more than 6 consecutive days
© J. Christopher Beck 2005

5. Days Off Scheduling (Example 12.2.2)
Find lower bounds on W
There are 3 that are interesting
1 of 3 weekends off
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req
© J. Christopher Beck 2005

6. Step 1: Weekends off W = 3, number the employees Let n = max(n1,n7)
Assign 1st weekend off to first W – n employees, 2nd weekend off to the next W – n and so on
Employee 1 comes after employee W
Cycle until you assign employee 1 to the same weekend off again
© J. Christopher Beck 2005

7. Step 1: Weekends off W = 3 n = 2 So 1 employee is off each weekend day4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 2 3
© J. Christopher Beck 2005

8. Step 1: Weekends off W = 3 n = 2 So 1 employee is off each weekend day4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

9. Step 1.5: Construct Off-Day Pairs
Let uj = W – nj, j = 2, … = n – nj for j = 1, 7
uj is the “over-capacity” on day j
Let uk = max(uj)
Construct a list of day-pairs with over-capacity
© J. Christopher Beck 2005

10. Step 1.5: Construct Off-Day Pairs
Pick uk
Pick another day um (m ≠ k), um > 0
If all um = 0, m ≠ k, choose m = k
Add (k, m) to list and decrease uk and um by 1
Repeat n times
Pairs (k,k) are called “non-distinct” pairs
© J. Christopher Beck 2005

11. Step 1.5: Construct Off-Day Pairs
W = 3
n = 2
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat nj uj
(2, 1)
uk = u2 = 3
um = u1 = 1
© J. Christopher Beck 2005

12. Step 1.5: Construct Off-Day Pairs
W = 3
n = 2
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat nj uj
(2, 1)
(2, 2)
uk = u2 = 2
um = u2 = 2
© J. Christopher Beck 2005

13. Step 1.5: Construct Off-Day Pairs
W = 3
n = 2
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat nj uj
(2, 1)
(2, 2)
© J. Christopher Beck 2005

14. Step 2: Categorize Workers
Week 1 employees fall into 4 groups:
T1: WE1 off, 0 off days, WE2 off
T2: WE1 off, 1 off day, WE2 on
T3: WE1 on, 1 off day, WE2 off
T4: WE1 on, 2 off days, WE2 on
n people working each weekend, so
|T3|+|T4| = n
|T2|+|T4| = n
So |T3| = |T2|
© J. Christopher Beck 2005

15. Step 2: Categorize Workers
n = 2
(2, 1)
(2, 2)
T1 = {}
T2 = {1}
T3 = {2}
T4 = {3}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

16. Step 2: Categorize Workers
Assign off-day pairs to workers
First to T4 – they get both days off
Second to (T2,T3) pairs
T3 gets earlier day
T2 gets later day
© J. Christopher Beck 2005

17. Step 3: Assign Off-Day Pairs
W = 3
n = 2
(2, 1)
(2, 2)
T1 = {}
T2 = {1}
T3 = {2}
T4 = {3}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

18. Step 3: Assign Off-Day Pairs
W = 3
n = 2
(2, 1)
(2, 2)
T1 = {}
T2 = {1}
T3 = {2}
T4 = {3}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

19. Step 4: Assign Off-Day Pairs for Week i
Categorize workers for week i
Case 1: All off-days are distinct
T4(i) = T4(i-1), T3(i) = T3(i-1)
T4 gets both days off, T3 gets earlier day, T2 gets later day
Case 2: Not all off-days are distinct
Week i schedule is identical to week 1
© J. Christopher Beck 2005

20. Step 4: Assign Off-Day Pairs for Week 2
(2, 1)
(2, 2)
T1 = {}
T2 = {2}
T3 = {3}
T4 = {1}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

21. Step 4: Assign Off-Day Pairs for Week 2
(2, 1)
(2, 2)
T1 = {}
T2 = {2}
T3 = {3}
T4 = {1}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

22. Step 4: Assign Off-Day Pairs for Week 3
(2, 1)
(2, 2)
T1 = {}
T2 = {3}
T3 = {1}
T4 = {2}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

23. Step 4: Assign Off-Day Pairs for Week 3
(2, 1)
(2, 2)
T1 = {}
T2 = {3}
T3 = {1}
T4 = {2}
day 1 2 3 4 5 6 7
Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req S M T W F 1 X 2 3
© J. Christopher Beck 2005

24. Alg 12.2.1 Overview Step 0: Calculate W Step 1: Assign Weekends Off
Step 1.5: Construct Off-Day Pairs
Step 2: Categorize Week 1 Workers
Step 3: Assign Off Day Pairs for Week 1
Step 4: Assign Off Day Pairs for Week i
© J. Christopher Beck 2005

25. Example 12.2.3 Find a schedule using Alg 12.2.1 day 1 2 3 4 5 6 7 Sun
Mon
Tue
Wed
Thu
Fri
Sat
Req
© J. Christopher Beck 2005