Lecture 24:
Workforce Scheduling
© J. Christopher Beck 2008
1
Outline


Bank Personnel Scheduling Demo
Shift Scheduling


Cyclic Staffing



Example 12.3.1
Algorithm 12.4.1
Example 12.4.2
Call Centre Scheduling
© J. Christopher Beck 2008
2
Readings


P Ch 12.1,
12.3, 12.4,
12.5, 12.7
Questions I
Like

12.5a
© J. Christopher Beck 2008
3
Workforce Timetabling


Arrange shifts and assign people to
them
Bank Demo
Constraints:




Number of people per shift
Minimum Days off (x/k days must be off)
Weekends
Nurses, call centres, hotels, restaurants,
plane crew, …
© J. Christopher Beck 2008
4
Shift
Scheduling

Fixed cycle


Predefined set of shift patterns



Month, week, day
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
© J. Christopher Beck
2008

5
Example 12.3.1
Pattern
Hours
Total Cost
Hours
1
10AM – 6PM
8
50
2
1PM – 9PM
8
60
3
12PM – 6PM
6
30
4
10AM – 1PM
3
15
5
6PM – 9PM
3
16
© J. Christopher Beck 2008
Hour
Staff
10AM – 11AM
3
11AM – 12PM
4
12PM – 1PM
6
1PM – 2PM
4
2PM – 3PM
7
3PM – 4PM
8
4PM – 5PM
7
5PM – 6PM
6
6PM – 7PM
4
7PM – 8PM
7
8PM – 9PM
8
6
Shift Scheduling Definition



Length of each period
may be different
m periods
bi: # people required in period i
n shift patterns

nj = (a1j, a2j, …, amj) = (1, 0, 0, …, 0)




aij = 1: workers on pattern j work period i
aij = 0: workers on pattern j don’t work period i
cj: cost of assigning a worker to pattern j
xj: # of people assigned to pattern j
© J. Christopher Beck 2008
7
IP Formulation
(see Example 12.3.1)
Such an IP is hard unless there is
special structure such as having
no split shifts
n
minimize
c x
j
a x
j
j 1
n
subject to
j 1
j
ij
 bi
i  1,..., m
Each period has enough workers
xj  0
© J. Christopher Beck 2008
j  1,..., n ; x j  int
Number workers assigned to a
pattern is an positive integer or 0
8
LS Formulation

Develop a local
search
formulation that
doesn’t depend
on having no
split shifts
© J. Christopher Beck 2008
9
Apply Your LS
to Example 12.3.1
Pattern
Hours
Total Cost
Hours
1
10AM – 6PM
8
50
2
1PM – 9PM
8
60
3
12PM – 6PM
6
30
4
10AM – 1PM
3
15
5
6PM – 9PM
3
16
© J. Christopher Beck 2008
Hour
Staff
10AM – 11AM
3
11AM – 12PM
4
12PM – 1PM
6
1PM – 2PM
4
2PM – 3PM
7
3PM – 4PM
8
4PM – 5PM
7
5PM – 6PM
6
6PM – 7PM
4
7PM – 8PM
7
8PM – 9PM
8
10
A Variation of
Example 12.3.1
Pattern
Hours
Total
Hours
Cost
1
10AM – 6PM
8
50
2
1PM – 9PM
8
60
3
12PM – 6PM
6
30
4
10AM – 1PM
3
15
5
6PM – 9PM
3
16
6
2PM – 5PM,
7PM – 9PM
5
35
Would you solve this problem with IP or LS?
© J. Christopher Beck 2008
Hour
Staff
10AM – 11AM
3
11AM – 12PM
4
12PM – 1PM
6
1PM – 2PM
4
2PM – 3PM
7
3PM – 4PM
8
4PM – 5PM
7
5PM – 6PM
6
6PM – 7PM
4
7PM – 8PM
7
8PM – 9PM
8
11
Cyclic Staffing





m period cycle
Each period i has requirement bi
Each person works k consecutive
periods and has the other m – k off
cj is the cost of putting a worker on
pattern j
Find minimum cost schedule
© J. Christopher Beck 2008
12
(5,7)-Cyclic Staffing Problem
Requirements
Patterns
Periods
Cost
1
0
0
1
1
1
1
3
1
1
0
0
1
1
1
8
1
1
1
0
0
1
1
9
1
1
1
1
0
0
1
11
1
1
1
1
1
0
0
11
0
1
1
1
1
1
0
11
0
0
1
1
1
1
1
4
80
70
75
90
100
© J. Christopher Beck 2008
100 105
There are split shifts!
13
(Different) Special Structure
Makes The Problem Easy



Solve LP1 (x’1,x’2, …, x’n)
If integral, you have optimal
Otherwise LP2:



LP1 + x1  x2   xn  x1  x2    xn 
If integral, you have optimal
Otherwise LP3:


LP1 + x1  x2   xn  x1  x2    xn 
Always integral, so you have optimal
© J. Christopher Beck 2008
14
Call Centre Scheduling:
Demand
demand
6AM
© J. Christopher Beck 2008
12
time
6PM
15
Call Centre Scheduling:
Shift Patterns
Breaks (moveable)
6AM
© J. Christopher Beck 2008
12
time
6PM
16
Solution Framework (Fig 12.4)
demand
Select solid
tours
Place breaks
Modify target
demand
Compare fits
N
Close
enough?
Y
DONE
© J. Christopher Beck 2008
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}
© J. Christopher Beck 2008
18
Call Centre Scheduling:
Demand
What do you do with
bilingual people?
demand
English
French
6AM
© J. Christopher Beck 2008
12
time
6PM
19