In this presentation, you will be introduced to the concept of Integer Programming and its application in conference scheduling. We will delve into the fundamentals of Integer Programming and its practical utilization in optimizing the allocation of talks to specific time slots and rooms within a conference program. By the conclusion of the talk, attendees will gain a clearer comprehension of the potential of this powerful tool in creating a conference schedule that is both efficient and effective, ultimately maximizing attendee satisfaction. Whether you are involved in conference organization or simply curious about optimization algorithms, this presentation is tailored to meet your interests.
2. Mathematical Modelling
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OSS Contributor & Creator of PyScaffold
Dr. Florian Wilhelm
• HEAD OF DATA SCIENCE
FlorianWilhelm.info
florian.wilhelm@inovex.de
FlorianWilhelm
@FlorianWilhelm
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4. Definition:
The objective of Operations Research (OR) is the
development and use of mathematical methods to support
decision-making processes in domains such as
manufacturing or finance.
Operations Research
Introduction
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5. ● Assignment/Allocation Problem
How to schedule some talks in an agenda under some
constraints like room sizes to optimize the conference
experience?
● Transportation Problem
Which supplier should deliver to which factories given some
costs to satisfy each factory with minimal costs?
● Shortest Path Problem
Given some graph with a cost on each edge, what is the
shortest path from source to sink?
● Maximum Flow Problem
Given some pipe system with a source and sink, what is the
maximum flow through the system?
Common Problem Classes in Operations Research
Introduction
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6. Definition of Linear Programming (LP)
Introduction
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LPs can be solved with the Simplex algorithm in cubic on average
but exponential number of steps in worst case scenario or interior
point methods in about .
7. Definition of Mixed Integer Programming (MIP)
Introduction
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MIPs are NP-hard, i.e. complexity grows exponentially with n
Dropping the integrality constraints, i.e. "linear relaxation", leads to
an LP problem.
8. If all variables need to be integer, it is a (pure)
integer linear program (ILP, IP).
If all variables need to be 0 or 1 (binary, boolean), it is a
0 - 1 linear program.
Special Cases of Mixed Integer Programming
Introduction
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9. Given a set of items, each with a weight and a value,
determine which items to include in the collection so that the
total weight is less than or equal to a given limit and the
total value is as large as possible.
- Wikipedia
Knapsack problem
Introduction
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10. 10
Solving a MILP
Introduction
Dropping the integrality constraints, i.e. “linear relaxation”, leads to an LP problem.
Two major method classes for solving MILPs:
● Cutting plane methods
● Branch and bound methods
Often combined as Branch and Cut methods
Source: Introduction to Optimization by Laurent Lessard, Spring 2017–18
11. Cutting Planes
Cutting Planes Methods
Introduction
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Idea
● solve the LP relaxation problem
● while solution is not integral:
○ add constraint that excludes solution but no integer points
○ solve LP relaxation again
13. Cutting Planes Methods
Introduction
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0
1
2
3
1 2 3
0
Optimal solution is 4.5 at (2, 2.5)
● Relaxing the problem and solving the LP instead of MIP
● The LP solution is always an upper bound for the MIP
14. Cutting Planes Methods
Introduction
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0
1
2
3
1 2 3
0
Optimal solution is 4.16 at (2.16, 2)
● Add a cut to exclude the LP solution but no feasible
point
● Solve the LP again
● … repeat until LP solution is also an MIP solution
16. Branch & Bound
Branch & Bound Methods
Introduction
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Idea
1. solve the relaxed LP and for a fractional split into two
subproblems
○ with constraint
○ and with constraint
2. repeat step 1. and build a tree of subproblems
3. eliminate branches of the tree using
17. How can we use MILPs for a
Conference Scheduling?
Use-Case
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Monday Room 1 Room 2 …
Morning
Afternoon
A
B
C
D
A
B
C
D
18. ● each talk must be assigned exactly once,
● each room/timeslot combination can only be occupied by
one talk at most,
● the length of the timeslot must match the length of the talk
● some tutorials have part 1 & 2, thus need to be consecutive
What Constraints do we have?
Use-Case
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19. 1. the preferences for day and time of some speakers, e.g.
keynotes, need to be considered
2.popularity of a talk should be reflected in the room
capacity,
3.avoid parallel talks that attract the same audience,
4.have in the same session, i.e. consecutive block of talks, the
same main track, e.g. PyData vs. PyCon,
5.or even the same sub track, e.g. PyData: Data Handling,
What is our objective?
Use-Case
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Precedence: 1 > 2 > 3 > 4 > 5
20. 1. Framework to formulate the problem, e.g.
a.Pyomo (OSS)
b.PuLP (OSS)
c.AMPL (Commercial)
d.…
2.Solver to solve the canonical problem, e.g.
a.HiGHS (OSS)
b.CBC (OSS)
c.Gurobi (Commercial)
d.IBM CPlex (Commercial)
e.…
Solving MILPs in Python
Use-Case
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26. Solving the MILP Problem and Displaying model.vBScheduleSchedule
Use-Case
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27. Pyomo / HiGHS notebook:
https://github.com/FlorianWilhelm/pytanis/blob/main/notebooks/pyconde-pydata-berlin-2023/50_scheduling_v1.ipynb
Check out the Full Source Code
Use-Case
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Pytanis includes a Pretalx client and all
the tooling you need for conferences
using Pretalx, from handling the initial
call for papers to creating the final
program.
31. 1. MILPs are an important tool in the domain of
Operations Research
2.Many optimal decision use-cases can be formulated
mathematically as a MILP, e.g. Knapsack problem
3.Solving a (M)ILP is NP-hard but good heuristics methods
exists like Branch & Cut
4.Tools like Pyomo translate your problem in a standardized
form and solver like HiGHS solve it
5.Modelling a problem as MILP, especially the Linearity
requirement, is the hardest part
Main Take-Aways
Summary
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32. ● Introduction to Optimization by
Laurent Lessard
● Mixed Integer Programming for
time table scheduling by Francisco
Espiga
● Schedule Optimisation using Linear
Programming in Python by Lewis
Woolfson
● Some icons taken from
Flaticon.com
References
Summary
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