Best Practice in Mathematical Modeling

Department of Informatics, University of Fribourg Best Practices in Mathematical Modelling : Lecture of T. Hürlimann 1
Tony Hürlimann
Department of Informatics
Bd. Perolles 90
1700 Freiburg
tony.huerlimann@unifr.ch
Best Practice in
Mathematical Modelling
Chap 1: Introduction
Master Course in Decision Support
at the
Department of Informatics, University of Fribourg, 2015

Department of Informatics, University of Fribourg Best Practices in Mathematical Modelling : Lecture f T. Hürlimann 2
Main Objectives of the Course
 Learn to read and write mathematical indexed
notation
 To be able to read and
write the code on the right
 Modelling: skills to translate real problems into
mathematical notation
 Given a problem in human language, write it in
mathematics.
 Implement the model in a modelling language
and solve it on a computer
 We shall use and extensively learn the modeling
language LPL to implement the models.

Overview of this First Lecture
 Goal: Relax and listen to the lecture
 No need to understand all the formalisms right now
 Motivation to use mathematics is given
 A game  Slitherlink  Route planning
 Mathematics – Reality: strange relation ?
 Coloring vertices versus scheduling exams
 „Real“ Examples
 Work schedule for bus drivers
 Cutting of paper rolls
 Location and logistics (supply-chain)
 Rostering: Work schedule of 800 persons at the
Zurich airport !

A 2-Persons Game
Problem : Two players play the following number game:
Each chooses (secretly) a positive number. The numbers are then
uncovered at the same time and compared. If the numbers are equal, neither of the
players will get a payoff. If the numbers differ by one, then the player who has chosen
the higher number obtains the sum of both, otherwise the player with the smaller
number obtains the smaller of both. The play is repeated endlessly. Which number
and how often should a player choose a number in each round?
1 1 4 1 2 3 2 1 2 2 1 3 1 4 1 2 5 2 4 2 3 4 4 2 1 2 3 1 4 2 3 1 5 2 4 5 3 5 4 1 1 1 3 1 3 1 4 3 4 4
3 1 3 3 5 3 5 1 3 2 1 4 3 5 3 3 5 3 2 1 4 5 4 4 1 1 5 4 3 5 3 2 1 3 2 5 1 2 3 5 1 1 2 1 3 3 2 1 3 4
4 1 2 1 1 4 4 2 3 3 4 3 2 4 4 1 1 1 1 1 3 1 3 2 2 4 2 2 3 3 5 2 3 2 4 4 3 4 2 3 3 5 3 1 3 2 1 1 3 3
1 5 4 3 5 2 3 4 3 3 1 1 5 2 1 5 2 1 1 5 3 3 1 3 1 2 5 3 2 3 5 2 2 3 1 4 4 1 5 3 1 5 2 1 4 3 1 3 5 3
2 3 1 1 3 5 3 3 3 1 4 1 1 5 4 3 5 1 1 1 1 2 2 2 1 3 3 3 3 4 3 2 2 3 5 3 2 3 3 5 2 2 3 4 3 1 3 4 4 5
1 4 2 2 3 3 5 1 1 1 5 2 5 4 5 2 2 4 1 1 1 3 2 1 1 4 2 2 3 2 1 2 1 4 2 4 3 3 5 1 2 5 3 1 1 1 2 2 1 1
3 4 3 1 5 1 3 1 2 3 1 4 3 2 1 5 1 1 1 1 4 5 1 1 4 5 3 1 3 4 5 3 3 5 4 4 5 5 4 4 1 2 2 1 4 2 4 5 1 5
Table of Numbers from which you can choose (optimally) [ above we choose from the
second row ] : (This table is hidden from the class)
4 1 12 1 4 4 32 3
How was this table built and why should this strategy be „optimal“? (see next page!)
Let us play in the class room (interactively I show the following number one by one)
The students guess one number and I discover one, and so on...

The 2-Persons Game:
A Mathematical Formulation
model GAME;
set i, j := 1..100;
parameter p{i,j} :=
if(i>j+1,-j , i=j+1,i+j ,
i=j-1,-i-j, i<j-1,i);
variable x{i} "Strategy";
constraint R: sum{i} x = 1;
maximize gain: min{j}(sum{i} p*x);
end
Choose number 1 with frequency 24.75%
Never choose another number !
(Both players can follow this strategy,
then in the long run nobody will win!)
The table on the previous page was built on
these frequencies !
The optimal strategy
of a player is:
Mathematical Formulation
Computer-executable Formulation
Coded in the mathematical
modeling language LPL
SEE: lpl.unifr.ch/puzzles/Solver.jsp?name=gameh

Another Game : Sudoku I
model sudoku;
set i,j,k;
set h,g;
parameter P{i,j};
parameter S;
binary variable x{i,j,k};
constraint
N{i,j}: sum{k} x = 1;
R{i,k}: sum{j} x = 1;
C{j,k}: sum{i} x = 1;
B{h,g,k}: sum{h1 in h,g1 in g}
x[(h-1)*S+h1,(g-1)*S+g1,k] = 1;
F{i,j,k|P[i,j]=k}: x = 1;
solve;
end
Model with:15’625 binary variables
2’787 linear constraints
SEE: lpl.unifr.ch/puzzles/Solver.jsp?name=sudoku
A 25x25 Sudoku:

And this is a complete mathematical description of the
Sudoku game :
Another Game : Sudoku II

Another Game like Sudoku :
Slitherlink
Find a simple loop that follows the borders of the cells in
a way that the number inside a cell represents how many
of its four sides are segments in the loop. If the cell is
empty, then the number does not matter.
A Slitherlink Puzzle…. .... and its solution

Slitherlink: Solution
 Try to solve it:
 Special configuration: 1 at a
corner, 3 and 0!
 The 2 in a bottom line
 The 3 in a corner
 etc.
In the Internet you can run various puzzles:
 Solution using a
mathematical model!
SEE: lpl.unifr.ch/puzzles/Solver.jsp?name=slitherlink

A Practical Small Example:
Manufacturing Cans
 Manufacturing cans with minimal usage of material
 How should the relation be between height and width of a can for a
given volume ?
h=height , r=radius
Surface
r=radius
With a radius of 0.8dm the height
will be only 0.5dm and the surface
then is 6.5dm2
, hence 18% higher
than in its minimum !
We suppose: The surface is directly
proportional to the amount of material used.
Exercise: Verify the solution!

Mathematics is everywhere
 A modern application is: Ciphering while transfering electronic money:
 Multipling two large number is easy !
 Factoring a large number into its (two) prime numbers is difficult !
 One can exploit this asymmetry (easy—difficult) in cryptography
Everybody can (with a little patience) do the following multiplication :
193'707'721 * 761'838'257'287 = 147'573'952'589'676'412'927 !
But: The mathematicien Cole (1861–1926) spent his weekends of 3
years to factor the number 267
-1 = 147'573'952'589'676'412'927 into
its two prim factors:193'707'721 und 761'838'257'287.
This assymetry in the complexity is used in real-life cryptography, to cipher a text in a
simple way. However, the text cannot be easily deciphered !
If you know the two prime factors, you can decipher a text !
If you know only the result of the multiplication, you can encipher !
Prime numbers have been considered „useless“ for centuries.
Our modern money economy would collapse without them !
example

Mathematics: How does it work?
An Abstract Problem
 „Abstract“ Problem: Vertex coloring
 Graph consists of vertices and edges
 Example: net with locations and routes
At least 4
colors are
necessary
An abstract
problem (just
baublery !?)
Is this useful?
Well ....

A Concrete Application
 Exams schedule
 A number of exams, a number of students
 How to find a Conflict-free Schedule!
Time Window 1 : Exams No.: 1 6
Time Window 2 : Exams No.: 3 4 8
Time Window 3 : Exams No.: 2 9
Time Window 4 : Exams No.: 5 7 10
Correspondence Mathematics – „Reality“ :
Vertex = Exam
Edge = Collision
Color = Time window
Number of colors = Length of schedule

Another Concrete Application
 Use memory for variable in a program
 10 variables used in a program at different life-time
 How much memory to use ?
Memory location 1 : var 1, 3, 10
Memory location 2 : var 2, 4, 6, 7, 9
Memory location 3 : var 5, 8
Vertex = program variable
Edge = Overlap life-time
Color = Memory location
Number of colors = Memory size

Still another Concrete Application
 Semaphores at a crossroads
 Number of traffic lanes, Number of crossing roads
 Collision-free plan of semaphore phases ?
Time window 1 : Lanes No.: AB BA DC EB EC ED
Time window 2 : Lanes No.: AD BD DB
Time window 3 : Lanes No.: BC DA EA
Time window 4 : Lanes No.: AC
Vertex = Traffic Lane
Edge = Collision
Color = Time Window
Number of colors = Length of phases

Alternative Solution
Time window 1 : Lanes No.: AB AC AD
Time window 2 : Lanes No.: BA BC BD ED
Time window 3 : Lanes No.: DA DB DC
Time window 4 : Lanes No.: EA EB EC
Vertex = Traffic Lane
Edge = Collision
Color = Time window
Number of colors = Length of phases
 Semaphores at a crossroads
 Number of traffic lanes, Number of crossing roads
 Collision-free plan of phases ?
A more „equilibrarted“ solution

For Larger Problems:
We need smart Mathematics and fast Computers
More difficult
to solve.
Needs
computer power
Needs
math. methods !
100 Exams
24 Time windows
Graph density 20%

Vertex Coloring:
A Mathematical Model
We shall go through this model later in the course !
This model is a first approach. It is no really „smart“ for larger problems

Scheduling Bus Drivers:
The Problem
 Number of excursions planed
 Monday: 14, Tuesday: 12, Wednesday: 18, Thursday:
16, Friday: 15, Saturday: 16, Sunday: 19
 Problem: How many drivers need to be hired?
 At least 19 (see Sunday)!
 Well! It depends on the working plan
 maximally 5-days working
 2 consecutive days off

7 working contracts
(5 days consecutive) : S1-S7
------------------------------------------------------
Mo Tu We Th Fr Sa Su
S1 * * * * *
S2 * * * * *
S3 * * * * *
S4 * * * * *
S5 * * * * *
S6 * * * * *
S7 * * * * *
Unknown quantities
(Number of drivers under the 7 different
working contracts)
------------------------------------------------------
S1 S2 S3 S4 S5 S6 S7
x1 x2 x3 x4 x5 x6 x7
Operation schedule
(Number of drivers per day)
------------------------------------------------------
Mo : x1 + x4 + x5 + x6 + x7 ≥ 14
Tu : x1 + x2 + x5 + x6 + x7 ≥ 12
. . .
model Plans "Schedule of Drivers";
integer variable x1; x2; x3; x4; x5; x6; x7;
constraint
mon: x1 + x4 + x5 + x6 + x7 >= 14;
tue: x1 + x2 + x5 + x6 + x7 >= 12;
wen: x1 + x2 + x3 + x6 + x7 >= 18;
thu: x1 + x2 + x3 + x4 + x7 >= 16;
fri: x1 + x2 + x3 + x4 + x5 >= 15;
sat: x2 + x3 + x4 + x5 + x6 >= 16;
son: x3 + x4 + x5 + x6 + x7 >= 19;
minimize
obj: x1 + x2 + x3 + x4 + x5 + x6 + x7;
end
Mathematical Model
Solution  on a computer
S1 = work from Monday to Friday, then 2 days off
x1 = number of drivers working under contract S1
The Formulation

2 Mo Tu We Th Fr Sa Su
1 14 14
2 19 19 19 19 19
33 drivers , 123 total salary
DEMAND 14 12 18 16 15 16 19
Actual 14 14 19 19 19 19 19
Overplus 0 2 1 3 4 3 0
------------------------------------------
3 Mo Di Mi Do Fr Sa So
1 3 3 3 3
2 11 11
3 16 16 16 16 16
DEMAND 14 12 18 16 15 16 19
Actual 14 14 19 16 16 16 19
Overplus 0 2 1 0 1 0 0
------------------------------------------
1 5 5 5 5 5
2 8 8 8 8
3 6 6 6 6 6
4 5 5 5 5 5
DEMAND 14 12 18 16 15 16 19
Actual 14 13 18 16 16 16 19
Overplus 0 1 0 0 1 0 0
1 3 3 3 3 3
2 7 7 7 7
3 2 2
4 9 9 9 9 9
5 6 6 6
DEMAND 14 12 18 16 15 16 19
Actual 14 12 18 16 15 16 19
Overplus 0 0 0 0 0 0 0
------------------------------------------
1 4 4 4 4 4
2 8 8 8 8 8
3 2 2 2 2 2
4 2 2 2 2 2
5 3 3 3 3 3
6 3 3 3 3 3
DEMAND 14 12 18 16 15 16 19
Actual 14 12 18 16 15 16 19
Overplus 0 0 0 0 0 0 0
5 Rostering systems with 2 to 6 different working contracts (schedules):
Various Solutions

Cutting Material:
The Problem
Given widths of the rolls: 152cm , 122cm , 102cm.
Demanded sizes (rectangles) :
20000 pieces of 24x 33 cm , 15000 pieces of 36x 80 cm , 5000 pieces of 29x100 cm
5000 pieces of 39x103 cm , 5000 pieces of 29x100 cm , 5000 pieces of 39x 93 cm
5000 pieces of 19x 75 cm , 15000 pieces of 29x 68 cm , 15000 pieces of 19x 29 cm
Conditions:
• 2-stage Guillotine-cuts
• First generate larger rectangles of length between 19 and 170cm (how?)
• Rotatation of rectangles of 90º is ok.
Problem: How many and in what length should the larger rectangles be cut into and how
should the final rectangles then be cut from them in order to minimize the waste ?

Case Study: Cutting Sheets
Solution Approach
1. First generate a number of „interesting“ larger rectangles (the sheet patterns).
Cut the demanded rectangles from these larger pieces of widths
(102,122,152) and lengths [19..170], using dynamic programming.
-- this is a well known and well studied problem: 2-stage unconstrained
guillotine cuts from rectangles given a number of smaller rectangles.
-- in the second step these rectangles are given as patterns.
2. Using these patterns, decide how many rectangles to cut from the rolls.
-- the demanded rectangles must be produced.
-- minimize the waste using linear optimization.
Decompose the problem into two sequential steps

Location & Transport Logistics
A Real Problem

Location & Transport Logistics
The Model Context
Holcim – a large company – uses mathematical optimization in its
strategic and operative logistics.
Problem: Decide where to produce and to pack ciment and in
what quantities and how to transport it to the clients in order to
minimize costs.
Solution: (1) The mathematical structure (the business logic) is
easily formulated in pure mathematics (as a linear programming
model). The data are read/written to/from databases. The whole
model consists of more than 10‘000 variables and constraints and
can be solved in two minutes on a current personal computer.
(2) Build an easy-to-use interface to database on top of the
model and the database interface.

Rostering (Airport of Zurich)
The Problem
 800 persons with different skills and contracts.
 250 work shifts: Check-in (Swiss), announcement,
ticket controls, etc.
 Find a monthly plan (30 days): Who works at which
day in which shift ?
 Demand must be fulfilled
 Working laws and collective contracts must be observed
and the skills must match.
 Maximize „Satisfaction“ (fullfill workers wishes)
This is a current project with FH Winterthur+Software company+client at the airport ZH

 Use a large mathematical optimization problem with
 500‘000 variables
 20‘000-40‘000 linear constraints
 Such problems can be treated and „solved“ with
current techniques of Operations Research.
To compare the size of the model above, look at this following small
trivial example with 2 variables and 2 linear constraints :
„My grandfather and grandmother are together 150 years old. Their ages
differ by 4 years. How old are they?“
The model is :
(x = Age of grandfather, y = Age of grandmother)
The solution is :
The Solution Approach

 Find an efficient way to „translate“ the problem into
the „right“ mathematical structure (Modelling).
 Use mathematical toolboxes to solve the problem:
Operations Research develops powerful methods
since 60 years (Solution Methods).
 Implement the approach together with user-
interfaces and data binding (Needs
Programming).
 Use fast computers (Hardware).
At the present time, 20 persons are in charge of planifying the schedules.
Decreasingimportance!
How to approach such a large problem

Day in month 
Persons
Overtime (in 1/4 hours)
Violates time constraint
Distance between shifts too short
violates working contracts
Unsatisfying solution: Working time is partially massively exceeded !
Extract of a (not-so-good) solution
Does not fullfill a workers wish
to get a free day on Tuesday
Worker 57 is on shift
no 79 on this Sunday

Better solution. However still violations of working laws.
Extract of a (better) solution

Summary
How to attack complex and real problems ?
 For standard problems use standard software:
 Text processing, accounting, etc.
 „Packages of various industries“ ...
 For relatively well structured problems use
specialized software packages
 Time-tabling in high school, tour planing.
 Machine control processes
 For complex decision problems no standard
solution exists
 Schedules of all sorts
 Layout-, Location-
 rostering-, processes-
Good Modeling skills are in high demand !
Good Solution methods needed !
Software skills: Rapid-Prototyping !
Hardware: the faster the better !

Best Practice in Mathematical Modeling

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Similar to Best Practice in Mathematical Modeling

Similar to Best Practice in Mathematical Modeling (20)

Recently uploaded

Recently uploaded (20)

Best Practice in Mathematical Modeling