Topic 41
Workforce Scheduling
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Workforce Scheduling

Overview:





Days-Off Scheduling
Shift Scheduling
Cyclic Staff Problem
Crew Scheduling
Simultaneous Worker and Job Scheduling
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
2
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
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
3
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
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
4
Optimal Schedule

Algorithm for one week

Repeat for next week

Cyclic schedule when repeat
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
5
Lower Bounds on Minimum
Workforce W

Weekend constraint

 k2 max n1 , n7 
W 
  B1
k2  k1


Total demand constraint

1 7

W    n j   B2
 5 j 1 
Maximum daily demand constraint
W  max n1 , n2 ,..., n7   B3
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
6
Optimal Schedule

Define W  max B1 , B2 , B3 
n  max n1 , n7 
W  n j
uj  
n

n
j


Have
7
u
j 1
July 11, 2017
j
j  2,...,6
j  1,7
 2n
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
7
Algorithm

Schedule weekends off

Determine additional off days

Categorize employees

Assign off-day pairs
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
8
Optimality

Can be shown that schedule is feasible
 Schedule is optimal

Why?

A cyclic optimal schedule exists
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
9
Topic 42
Shift Scheduling
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Shift Scheduling



Fixed cycle of length m periods
Have bi people assigned to ith period
Have n shift patterns:
(a1 j , a2 j ,..., amj )


Cost cj of assigning a person to shift j
Decision variable: xj = people assigned
to j
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
11
Integer Program
Minimize
Subject to
c1 x1  c2 x2  ...  cn xn
a11x1  a12 x2  ...  a1n xn  b1
a21x1  a22 x2  ...  a2 n xn  b2

am1 x1  am 2 x2  ...  amn xn  bm
xj  0
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
12
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
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
13
Topic 43
Cyclic Staffing
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Cyclic Staffing Problem

An m-period cyclic schedule

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
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
15
Integer Program Formulation

Shift patterns:
1
1

1

A  1
1

0
0

July 11, 2017
0 0 1 1 1 1

1 0 0 1 1 1
1 1 0 0 1 1

1 1 1 0 0 1
1 1 1 1 0 0

1 1 1 1 1 0
0 1 1 1 1 1
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
16
Solution



Solution to LP relaxation ‘almost right’
'
'
'
STEP 1: Solve LP relaxation to get x1 , x 2 ,..., xn
if integer STOP; otherwise continue
STEP 2: Formulate two new LPs with
x1  x2  ...  xn  x1  x 2  ...  x
'
n
x1  x2  ...  xn  x1'  x '2
'
n
'
'

 ...  x 
The best integer solution is optimal
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
17
Example
(3,5)-cyclic staffing problem
1

1
A  1

0
0

July 11, 2017
0 0 1 1

1 0 0 1
1 1 0 0

1 1 1 0

0 1 1 1
 3
 
 4
b   6
 
 4
7
 
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
18
Solution

Solve the LP relaxation
min cx
x
s.t.
Ax  b
x0
Non-integer
x  (1.5,0,4.5,0,2.5)
'
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
19
Solution


Add together:
1.5  0  4.5  0  2.5  8.5
Add constraint
x1  x2  x3  x4  x5  8

No feasible solution
Add constraint

Solution: x  (2,0,4,1,2)

x1  x2  x3  x4  x5  8
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Optimal
20
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: unknown cycle length

Difficulty: many patterns  larger problem
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
21
Example
Two days off in
week
+ no more than 6
consecutive
workdays

July 11, 2017
0

a
0
1

1
1

1

1
A  
1

0
1

1
1

1

0
0
0

1
0
0

1
1
1

1
1
1

1
1
1

1
1
1

0
1
1

0
0
1

1
1
0

1
1
1

1
1
1

1
1
1

0
1
0

0
0
1

1
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson






















22
Cyclic Staffing with Overtime


24-hour operation
8-hour shifts with up to 8 hour overtime
0 1 1 1

0 0 1 1
0 0 0 1

 1 0 U
0 0 0 0


U
A  U 1 0 
0 U 1
0 0 0 0


0 0 0 0

0 0 0 0
0 0 0 0
Lecture Notes for Planning
 and Scheduling
July 11, 2017
Prepared by Siggi Olafsson
1 1 1 1 1

1 1 1 1 1
1 1 1 1 1

1 1 1 1 1
0 1 1 1 1
0 0 1 1 1

0 0 0 1 1
0 0 0 0 1
23
Penalties for
Under/Overstaffing

Demand not fixed




'
i
''
i
Linear penalty c for understaffing
Linear penalty c for overstaffing
'
Let xi denote the level of understaffing

IP min c x  c ' x '  c '' b  Ax  x '

Ax  Ix  b
'
x, x  0
'
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
24
Topic 44
Crew Scheduling
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
Crew Scheduling


Have m jobs, say flight legs
Have n feasible combination of jobs a
crew is permitted to do
2
1
m6
5
3
4
6
July 11, 2017
Set partitioning
problem
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
26
Notation


Cost cj of round trip j
Define
1 if leg i is part of round trip j
aij  
otherwise
0
1 if round trip j is selected
xj  
otherwise
0
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
27
Integer Program
Minimize
Subject to
c1 x1  c2 x2  ...  cn xn
a11 x1  a12 x2  ...  a1n xn  1
a21 x1  a22 x2  ...  a2 n xn  1

am1 x1  am 2 x2  ...  amn xn  1
x j  0,1
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
28
Set Partitioning




Constraints called partitioning equations
The positive variables in a feasible
solution called a partition l
l
l
J  { j : x j  1}
NP-Hard
Well studied like TSP, graph-coloring,
bin-packing, etc.
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
29
Row Prices

Say that
  ( ,  ,...,  )
l
l
1
l
2
l
m
is a set of feasible row prices if for j  J
m
 a
i 1

l
i
ij
 cj
Cost of covering a job
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
30
l
Change Partition


Let Z1 (Z2) denote the objective value of
partition 1 (2)
Then
m
1
jJ 2 i 1


Z  Z    a  c j
2
1
i ij

Potential savings of including column j is
 j    a  c j 
m
1
i ij

1
If all negative theni optimal
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
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
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
32
Connections

Here



Timetabling with resource constraints




Production requirements per shift fixed
Optimize number of people per shift
Number of people fixed
Optimize job schedule
Determines production requirements per
shift
Optimize simultaneously
July 11, 2017
Lecture Notes for Planning and Scheduling
Prepared by Siggi Olafsson
33