Lecture 18:
Service Scheduling &
Timetabling
© J. Christopher Beck 2008
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Outline
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Introduction to Service Scheduling
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Characteristics of Service Scheduling
Differences with Manufacturing
UofT APSC Timetabling
Reservations without Slack
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Example 9.2.1
IP formulation
Algorithm 9.2.2
© J. Christopher Beck 2008
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Readings
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P Ch 3
P Ch 9.6, 9.2
Questions I Like:
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9.1, 9.2, 9.4, 9.6
© J. Christopher Beck 2008
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Examples of Service
Scheduling
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We will look at:
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Timetabling
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Sports Scheduling
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Classrooms
ACC Basketball
Workforce scheduling
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Workers in a call centre
© J. Christopher Beck 2008
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Example Classroom
Scheduling
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Assign classes to
rooms such that
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Rooms are big enough
No two classes as in the same
room at the same time
No prof has to teach two
classes at one time
No students have to take two
classes at one time
© J. Christopher Beck 2008
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Service Scheduling
Characteristics: Resources
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Classroom, hotel, rental car, stadium,
operating room, plane, ship, airport
gate, dock, railroad track, person
(nurse/pilot)
Synchronization of resources may be
important
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Need a plane and a pilot
Classroom, AV equipment, prof, students
© J. Christopher Beck 2008
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Service Scheduling
Characteristics: Resources
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Each resource may have its own
characteristics
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Classroom: capacity, equipment, cost,
accessibility
Truck: capacity, refrigeration, speed
Person: specialist (surgeon, nurse) with
skills (languages)
© J. Christopher Beck 2008
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Service Scheduling
Characteristics
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Activities may have
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Time windows
Capacity requirements/constraints
Resources may have
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Setup/transition time – runways!
Operator/tooling requirements
Workforce scheduling constraints
Shift patterns, break requirements
 Union and safety rules
© J. Christopher Beck 2008
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Differences from
Manufacturing
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Impossible to “store” goods
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Resource availability often varies
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If you don’t fill a hotel room you can’t “get
back” the lost time
May even be part of the objective function
Saying “no” to a customer is common
“No available seats on that flight” (even if
there are some!)
 Beck
Try2008to book a restaurant for 8 PM
© J. Christopher
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Classroom Timetabling
APSC, U of T ’06-’07
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4000 students, 1200 first year
7 departments, 9 degree programs
79 programs of study
(programs/options and year)
219 faculty members, 12 buildings, 80
lab rooms
75% of the timetables are delivered to
students conflict-free
© J. Christopher Beck 2008
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APSC Timetabling
Process
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Data acquisition
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departments send data on courses,
staffing, resources, enrollment limits, …
time frame: months
Decide roll-over strategy
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what can be kept from previous year?
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curriculum changes, staff changes, …
were there problems with last year’s
schedule?
© J. Christopher Beck 2008
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APSC Timetabling
Process
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Schedule 1st year
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Separately for each department
many shared courses but each course is
“owned” by one department
Usually leaves some
Automated “first” schedule classes unscheduled
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Very iterative and mostly manual changes
May be changed later (e.g., to fix a big
problem in later year schedule)
© J. Christopher Beck 2008
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APSC Timetabling
Process
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Schedule 2nd year
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automated schedule (with 1st year fixed)
manual scheduling, rescheduling
may make changed to 1st year
Schedule 3rd & 4th year is similar
Assign rooms
Upload to ROSI
© J. Christopher Beck 2008
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APSC Timetabling
Student Goals
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Conflict-free for 1 to 3 and for core 4th
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next best: minimize conflict, limit to
tutorials and labs
4th year: minimize conflicts among popular
courses
Deliver required courses
Meet student criteria
© J. Christopher Beck 2008
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APSC Timetabling
Student Goals
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9-5 with lunch between 12 and 2
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Minimize number of gaps in a day
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next best: 9-6, then 9-7, lunch between 11
and 2
Existing gaps should be short
Provide some study time
All student groups should have equal
access to resources
© J. Christopher Beck 2008
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APSC Timetabling
Faculty Goals
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Meet staff criteria
Professors should have one day per
week with no teaching
Conflict-free
© J. Christopher Beck 2008
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APSC Timetabling
Process
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Data collection starts in January
Schedule available in August
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But not completely finished until the course
drop dates (i.e., mid-Sept)
About 25 people involved
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About 3 full-time
© J. Christopher Beck 2008
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Classroom
Timetabling
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UC Berkeley
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30,000 students, 80 depts, 4000 classes,
250 rooms
3 schedulers and 1 supervisor
Read about it in Ch 9.6
© J. Christopher Beck 2008
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Reservation
Systems
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Hotel rooms, car rentals,
airline tickets (and
classroom scheduling)
You want to have the use of a resource
for a given period of time
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With slack: pj < dj – rj
Without slack pj = dj – rj
May not be able to schedule all requests
© J. Christopher Beck 2008
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Objectives
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Maximize $$
Maximize resource usage
Minimize number of rejected requests
Minimize $$ of rejected requests
© J. Christopher Beck 2008
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Reservations without Slack
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n activities, m resources
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Weight is
often
equivalent
to profit
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All activities and resources are independent
pj = d j – rj
Activities have weight wj or wij (see next
slide)
May have resource subsets
You don’t want to rent any car, you want to
rent an SUV
 Some substitutability of resources
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© J. Christopher Beck 2008
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Weights Can Get Complicated
(Ex. 9.2.1)
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Car rental agency with 4 car types
Customer j wants either a subcompact
or midsize
Customer k wants a subcompact but
there are none left
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wij = (qj – ci) * pj
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Yield management
qj is the price charged per day to customer j
ci is the cost (to the rental agency) per day of a
car in class i
© J. Christopher Beck 2008
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IP Formulation
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H slots
xij: binary variable that is 1 if activity j is
assigned to resource i
Jt: set of activities that need a resource
in slot t
© J. Christopher Beck 2008
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IP Formulation
m
maximize
Does not represented
resource subsets!
n
 w x
i 1 j 1
ij ij
m
 xij  1, j  1,..., n
i 1
x
jJ t
ij
Every activity is assigned to
at most one resource
 1, i  1,..., n, t  1,..., H
© J. Christopher Beck 2008
Each resource has
only one activity
per time slot
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IP Formulation
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General problem is hard
Special cases are easy
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All activities have duration of 1 –
independent problem for each time slot
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See p. 209
All weights are 1, all resources in a single
set, durations are arbitrary
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Maximize the number of scheduled activities
© J. Christopher Beck 2008
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Alg 9.2.2: Maximize # of
Scheduled Activities
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All weights are 1, all resources in a
single set, durations are arbitrary
Order activities in ascending order of
release date
Let J be the set of already scheduled
activities
Step 1: J = {}, j = 1
© J. Christopher Beck 2008
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Alg 9.2.2
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Step 2: If a resource is available at rj,
assign it to activity j, add activity j to J,
and goto 4.
Step 3: Let j* in J with max. completion
time. If Cj > Cj*, then don’t schedule j.
Else replace j* with j in J
Step 4: If j = n, STOP. Else j = j +1 and
goto Step 2.
© J. Christopher Beck 2008
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Exercise 9.1:
3 resources, 0 slack
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Use Alg 9.2.2 to find max # activities
Find schedule that maximizes sum of
durations of activities scheduled
What is the minimum number of
resources needed to do all activities?
activities
rj
dj
© J. Christopher Beck 2008
1
2
8
2
7
8
3
5
9
4
2
4
5
1
4
6 7 8
0 4 8
3 10 10
9 10
0 0
1 3
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