Examples of operational research

HOW A FRENCH GUY CAN TEACH
IN A MANDARIN-SPEAKING COUNTRY
Problem: I speak French, you speak Mandarin.

Solutions:
+ English
+ Images
+ Stupid questions (I do not ask stupid
questions because of stupidity,
it's just for checking that there's
no misunderstanding)
+ Interrupt me + send emails
+ Feel free of discussing in Mandarin

I don't want so speak Chinese to you

In France, we say

“this is Chinese to me”

for “I can't catch a word”.

==> Please tell me if I “speak Chinese to you”.

DISCLAIMER

Fact 1: Many applications in these lessons are
about military applications.
Reason: Easy to understand + many precursor
operation research works are military.

Fact 2: My main application fields are ecological
and energetical.
Implication: We will see such applications later,
with much more details.

WHAT IS THE POINT
IN THESE 3 HOURS ?
This is about Operations Research (OR).
Key points:
- OR can help a company/country to save up
plenty of $.

- new OR works usually do not originate at the
head of companies / countries; originate in
young people who want to do new things.

- you can be very useful if you understand the
goal of OR.

Operations Research
everywhere.

I hope that from now on,

you will see plenty

of OR opportunities everywhere.

POLL

This teaching is adaptive.
Tell me what you want, and I'll try to adapt.
Email: olivier.teytaud@inria.fr
(emails will be considered as private, no forward, no archive;
only an anonymous agregation is public)

More mathematics ?
More illustrative examples ?
More C / C++ / Java / Matlab examples ?
More examples in domain X or Y ?
More algorithmic elements ?
Today, illustrative examples mainly ==> simple.

Overview

1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
a) based on optimization
b) based on statistics
c) based on time-decomposition
d) others
e) all-in-one

OP. RESEARCH: Definitions

Goal = near optimal decisions to
complex decision-making problems.

Similar to:
- Decision sciences
- Management sciences (part of)

Close to artificial intelligence
==> we will see plenty of vocabulary,
because vocabulary = crucial for finding
relevant literature.

OP. RESEARCH: History

- Some precursors

- “Real” birth in world war II

- Then widely spread everywhere
- industry
- business

WORLD WAR II viewed from Europe

- France + others quickly defeated
==> Hitler close to complete victory
- But UK resists, so Germany must fight both
on East and West
- Also UK territory can be used by US
for bombing germany
- The surprising resistance of UK is a key
element of World War II
- Op. Research = key element of WW II.

WORLD WAR II

- Modern techniques
everywhere

- Supply chain
= critical

- Both tactical and
strategical
elements.

UK is an island

- attacks
= bombing

- defense =
anti-aircraft
artillery
+ radar
+ planes

An optimization problem: defending UK

We know that Q German bombers might
take off at locations A, B, C and D.
We know that their flights have maximal length L.
We can build N radar antennas.
We can set up M anti-aircraft artillery.
We can distribute P air-fighters on airports E, F, G and H.
Choose:
- the positionning of the N radar antenna
- the anti-aircraft artillery
- the positionning of the air-fighters


Step 1: defining the variables & constraints.

- Two “Float” variables for each positionning (antenna, artillery).
==> 2(N+M) variables
- One “int” variables for each airport (number of airplanes),
==> e=nb of planes in E,
==> f=nb of planes in F,
==> g=nb of planes in G,
==> h=nb of planes in H.
Constraints:
- Number of planes: e+f+g+h=P
- Antennas and artillery not in the sea


Keeping 2(N+M)+4 variables take too much room.
Let's say x = vector ( e,f,g,h,x1,x2,x3,...,x(2(N+M)+4) ).

Step 2: defining the objective function.

= something which quantifies to which extent a proposed
solution is good or not.

f(x) = very big number if x is a bad strategy.
= very small number if x is a very good strategy.

Then, we will look for x* such that f(x*) = minimum.
Definition: x* is the optimum of f.

Can you define f for our radars / antenna / planes ?


Reminder:
We know that German bombers might
take off at locations A, B, C and D.
We know that their flights have maximal length L.
We don't know which starting point they will choose.

f(x) = maximum expected damage of the possible
German attacks

New variables:
a = nb of planes which take off at A,
b,c,d = nb of planes which take off at B,C,D.
z = vector defining German trajectories
==> y=(a,b,c,d,z)


Computing g(x,y) on a computer:
- simulate the time steps of a airplane
flight / bombing
- evaluate the damage
- return the damage Not so easy!
So we can write g(x,y) = damage if strategy y.

But what is f(x) ?

f(x) = max g(x,y) (maximum on y)

Is it clear for you ?

Defending UK: the complete picture

Main function:
Find x* such f(x*) is minimum. // x* = optimal
// UK strategy

Function f(x):
Find y such that g(x,y) is maximum.
Return g(x,y)

Function g(x,y)
Return damage if
- UK has strategy x
- Germany has strategy y

Two optimization levels

Main function:
// UK strategy

Function f(x): Main optimization
problem
Return g(x,y)

Function g(x,y)
Return damage if
- UK has strategy x

The second one is antagonist

Main function:
// UK strategy

Function f(x):
Return g(x,y)
Secondary optimization
Function g(x,y) problem: the German army
will optimize its behavior
Return damage if
- UK has strategy x

Overview

Sometimes OR = solved by just modelization.

submarine

Overview

Example:

submarine

UK trying to destroy U-boats (which destroyed
UK convoys).

Overview


Approaches

U-boat: Escapes!

submarine

Overview


Approaches

Bomb! Escapes!

submarine

Overview

Simulation: at which depth should the bomb
explode in order to maximize the probability of
destroying the U-boat
A relevant simulator helped a lot UK.

Approaches

Escapes!

Bomb!
submarine

Overview

Simulation: based the plane speed, the detection
time (by the U-boat), the U-boat behavior, the
U-boat speed, the bomb's damage radius...
Draw a figure with this and you have the solution.

Approaches

Escapes!

Bomb!
submarine

Just a relevant pen&paper model ==> big improv.
by changing the depth of explosion.

Overview
Modelization is crucial – nothing works
without a good model.
But often we need maths&algorithms:
optimization, statistics, Zermelo-like algos,...
c) based on time-decomposition
d) others
e) all-in-one

Optimizing plane is important;
turbulence matters

Real-world

Simulations

Optimizing aerodynamics is important

Better
aerodynamics
==> less fuel.

Turbulences
generated by
planes
at takeoff/landing
can be dangerous.
(or even en-route)

Feedback from
experience:
often crucial

Optimization-based OR
Plenty of parameters in a plane:
length of wings, width of wings,
position of gas turbines, etc.
Let's say x1, x2, … , xk.

If you have a flight simulator, you can define a function
f(x1,x2,.;.,xk) = efficiency (aerodynamics)
Or for short: f(x) = efficiency
Then, it is important to find x* such that f(x*) is minimum.
Also important for
- trains
- hard drives
- automobiles

Optimization: definitions

It is important to find x* such that
f(x*) is minimum.
Definition: such a x* is termed a “minimum” of f.
It is an “optimum” (the best solution).
Minimization: optimum = minimum
Maximization: optimum = maximum

Questions

Minimum of:
f(x) = x2
2
f(x) = (x-1)
2 2
f(x)= ( (x-1) -1)
f(x)= -x2
f(x)= -x2 for x >0

Final question: how to find the maximum of f
if you just have a program for finding minima ?

Minimization: the simplest
algorithm ever

double * RandomSearch( double (*f)(double* x) )
{ bestValue=-MAXDOUBLE;
for ( 10000 times )
{
x=randomVector()
value = f(x)
If (value<bestValue)
{ bestValue=value; xstar=x;}
}
return xstar;
}

algorithm ever, random search

I sample
plenty of points. Here!

I evaluate the value
for each point.

I keep the best one.

algorithm ever, random search

RandomSearch( f )
{ bestValue=-MAXDOUBLE;
for ( 10000 times )
{
x=randomVector()
value = f(x)
If (value<bestValue)
{ bestValue=value; xstar=x;}
}
return xstar;
}

Sorry, we must stop, we need a tool:
Gaussian random variables.

Just a brief overview:
standard Gaussian
vectors (normalized)
= random vectors
- distributed
around 0,
- with
density decreasing
with the distance,
- equally in all
directions.

Minimization: almost as simple,
the (1+1)-evolution strategy
OnePlusOne( f )
{ x=randomVector;s=1;
for ( 10000 times )
{
xp = x+s*randomGaussianVector()
value = f(x)
if (value<bestValue)
{ bestValue=value; x=xp;s=2*s}
else s=0.84*s;
}
return x;
}

The (1+1)-evolution strategy:
should I stay or should I go ?
OnePlusOne( f )
{I start
x=randomVector;s=1;
here.( 10000 times )
for
{
Xp = x+s*randomGaussianVector()
value = f(x)
else s=0.84*s;
}
return x;
}

I should go!

OnePlusOne( f )
for (I10000 times )
test
{ here:
better!
value = f(x)
else s=0.84*s;
}
return x;
}

...

OnePlusOne( f )
forSo here times )
( 10000
{ I go.
value = f(x)
else s=0.84*s;
}
return x;
}

… we can not always succeed ..

...and
OnePlusOne( f )
here
fortest.
I ( 10000 times )
{
value = f(x)
else s=0.84*s;
}
return x;
}

Should I stay or should I go now ?

OnePlusOne( f )
for ( 10000 times )
{
...I stay
and I test f(x)
value =
again
else s=0.84*s;
}
return x;
}


OnePlusOne( f )
... for ( 10000 times )
{
and I try
again Xp = x+s*randomGaussianVector()
... value = f(x)
else s=0.84*s;
}
return x;
}


OnePlusOne( f )
for ( 10000 times )
{
value = f(x)
...
and(value<bestValue)
if
again { bestValue=value; x=xp;s=2*s}
else s=0.84*s;
...
}
return x;
}


OnePlusOne( f )
...
and
for ( 10000 times )
{ again
...
value = f(x)
else s=0.84*s;
}
return x;
}


OnePlusOne( f )
for ( 10000 times )
{
...
value = f(x) Found!
else s=0.84*s;
}
return x;
}


OnePlusOne( f )
for ( 10000 times )
{
value = f(x)
else s=0.84*s;
}
return x;
}

OnePlusOne( f ) There is a
{ x=randomVector;s=1; for the size s
parameter
of
for ( 10000 times ) modifications
{
value = f(x)
else s=0.84*s;
}
return x;
}

OnePlusOne( f )
for ( 10000 times )
{
value = f(x)
else s=0.84*s;
}
return x; If success, then I
increase the size of
} modifications!

OnePlusOne( f )
for ( 10000 times )
{
value = f(x)
else s=0.84*s;
} Otherwise, I
return x; decrease the size of
} modifications!

OnePlusOne( f )
for ( 10000 times )
{
value = f(x)
else s=0.84*s;
}
return x; There are plenty of variants of this algorithm.
}
These variants are evolution strategies.

Other application

I have many shops.
They sell products from my factory.
Bringing products factory → shops is hard.

I want to install 7 warehouses:
- reduced costs for transportation
- less breakdowns

Let's simplify: there's no breakdown.
Where should be the 7 warehouses ?

Without warehouses

● Either many small
long-distance
transportations (huge cost)
● or one big

transportation (big delay,
big cost)

Warehouses

● Less travels with
big transportation

● Total price decreased

● Maximum delay reduced
(==> no breakdown)
● Less pollution

Formalization

x=position of the warehouses.

f(x) = benefit when using the 7 warehouses

= cost of transportation
with no warehouse

- cost of transportation with
the warehouses

==> with more sophisticated models, a very
important case of operation research

Formalization

For computing f(x), I need:
(1) cost of transportation with no warehouse
(2) cost of transportation with the warehouses

==> it is a secondary optimization problem:
optimizing the tactical use.
Cost = minimum cost_if_policy(y)
==> very usual:
- one strategical level: positionning
the infrastructure
- one tactical level: optimal use of the
Infrastructure (policy)

Sometimes many levels

Level 1: size of the infrastructure
- number/size of factories
- number/size of trucks
- number/size of warehouses
- number/size of trains, boats (…)

Level 2: positionning
- where is factory A
- where is factory B
- where is warehouse C (…)

Level 3: scheduling of supply chain
- when/where does truck X start
- when/where does train Y start (…)

Logistics: combining all this for
fast supply chain.

Not only positionning, for a fixed
number of trucks, men, boats.

You can also see which number of boats,
trucks, warehouses, is best.

Statistics in a (small) nutshell

Sample = some examples
= e.g. x1, x2, x3,...., xn.

- You are a sample of Taiwanese students;
- I am a (small) sample of French researchers.

Sample average m = (x1+...+xn)/n

Standard deviation:

Statistics crucial notion 1:
confidence interval

With probability 95%, the true mean is
between m - 2/sqrt(n) and m + 2/sqrt(n).

Under some conditions:
- no bias in the sample
- no correlation in the sample

Sorry for statisticians, I know
I simplify so much...

2:
confidence interval
bias
American election in 1936
(during great depression).
Who will be elected, Landon or Roosevelt ?

Literary Digest's poll: sample = 2.3 millions
==> predicts Landon.
Sample=people motivated for answering
the poll = minority which hates
Roosevelt.
==> they all vote for Landon. Big bias.

Gallup poll: sample = 50 000 persons (more properly
sampled). Predicted the right answer.

Example of application:
industrial owens

Statistics

Steel quality
(for my factory)

Carbon quantity

conditional probabilities

Probability for a dice ?

P(3) = 1/6

P(odd) = P(even) = 1/2

P(1 or 2) = 1/3

P(1 | 1,2,3,or 4 ) = ¼ <== this is conditioning
= probability of 1, given that we get 1, 2, 3 or 4.

Bayes theorem
Bayes published a theological book:
Divine Benevolence,
or an Attempt to Prove That the Principal
End of the Divine Providence and
Government is the Happiness of
His Creatures (1731)

But he also developped a special case of
Bayes theorem:
P(A | B) = P(A and B) / P(B)
P(3 | 1,2 or 3) = (1/6) / (½) = 1/3

Let's see a nice, counter-intuitive
example of application of
statistics:

==> protecting bombers

Let's come back to
anti-aircraft artillery.

UK bombers also attacked Germany. Bombing with
(Hamburg

But, many anti-bombers artillery. incendiary bombs;
1000°C, ~190km/h
Many planes destroyed. Wind – death toll
42600)
Problem: how to protect planes ?

Using statistics for
protecting bombers
1) For each part X of the plane, estimate
q(X)= P( part X damaged | plane came back)
E.g.
q(part1) = 0.1
q(part2) = 0.0
q(part3) = 0.02

2) Assume that P(part X damaged) = constant
(does not depend on X)

3) Then, which part should we reinforce ?

protecting bombers
= P ( PXD | PCB)
E.g. q(part1) = 0.1
q(part2) = 0.0
q(part3) = 0.02
Define c = P(PCB)/P(PXD) = constant.

P(PXD | PCB) = P(PXD and PCB) / P( PCB)
P(PCB | PXD) = P(PCD and PXD) / P( PXD)
= P(PXD | PCB) * P(PCB)/P(PXD)
= c * q(X)

protecting bombers
= P ( PXD | PCB)
E.g. q(part1) = 0.1
q(part2) = 0.0
q(part3) = 0.02
Define c = P(PCB)/P(PXD) = constant.

P(PCB | PXD) = c * q(X)
==> Probability that plane survives to “part X
is damaged” increases if q(x) increase.
==> Conclusion: reinforce part2 ! ! !

sending mails

Divide customers in 5 categories
- Women ( < 25 y.o.)
- Men (<25 y.o.)
- Women (25-45)
- Men (25-45)
- >45

You have money for sending Q advertisements.
You have P1, P2, P3, P4, P5 customers
in categories above.

sending mails

You have P1, P2, P3, P4, P5 customers
in categories above.

Data:
Sample of past advertisements:
m1=frequency of positive answers in category 1.
…
==> Who should receive advertisements ?

sending mails

m1=frequency of positive answers in category 1
= number of positive answers / number of ads

e.g. if 17 ads sent,
m1=mean(0,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0)=5/17

sending mails
…
1 = std deviation for category 1
2 = std deviation for category 2
…
b=benefit per customer with positive answer.

If ( (m1+21/sqrt(n1)) x b < price of one mailing),
then discard category 1.

Send ads to non-discarded categories.

Overview

c) based on time-decomposition: Zermelo
(remember, irrigation, water, crops, etc)
d) others
e) all-in-one

Our river in Paris
is the “Seine”.

A French
politician said
he would soon
swim across it.

After all, he never
did it.

For your health,
don't do it.

Nevertheless,
we try
to keep it
as clean
as possible.

Another beautiful application

This is Paris.
Beautiful town.
With plenty of people
(10 millions in IDF).


This is Paris.
Beautiful town.
With plenty of people
(10 millions in IDF).
Producing plenty of fecal
matter ==> dirty water.

Dirty water should be separated from the Seine.
And usually it is.
Something like this:

Seine

Dirty
water

Problem: if big rainfalls reach dirty water,
then dirty water might pollute the Seine

Seine

Dirty
water

No typhoon in France.
But we can have heavy rains/winds in Paris:
- 0.96 dm in 24 hours happened in 1987.
- gusts at 169 km/h in 1999 (very unusual in France)


Seine

Dirty
water
(yes, in Taiwan it is more impressive,
sometimes it is 16.7 dm in 24 hours and gusts
can reach 250 km/h...)

No typhoon in France.
But we can have heavy rains/winds in Paris:
- 0.96 dm in 24 hours happened in 1987.
- gusts at 169 km/h in 1999 (very unusual in France)


Seine
Dirty
water
→ Seine! (yes, in Taiwan it is more impressive,
sometimes it is 16.7 dm in 24 hours and gusts
can reach 250 km/h...)


Three water networks:

- dirty water: should go to cleaning stations

- clean water: can go to the Seine, but can't be drunk

- drinkable water (France: tap water = drinkable)

Big water network

Dirty Dirty Dirty Dirty
water water water water

Clean Clean Clean Clean
water water water water

Water vs dirty water

Challenge:
Summer storms.
Not comparable to a Taiwanese typhoon.
But a lot of water.
Can make dirty water become very big.
Can invade clean water.

Your mission:
- Get read of dirty water
- Protect clean water


State: level in each stock,
valves' status
(open or closed)

At each time step,
i(x) liters of water reach stock x.
you can open or close valves
==> get a new state.

Your mission:
- Get read of dirty water
- Protect clean water ===> Zermelo.


Typically:
(0, 1, 0, 0, 0, 1, 0, 1, 0.42, 0.2, 0.0, 0.8, 0.3)
(valves) (stock levels)

Plenty of rules:
- if (valve 4 opens, then water from stock 1
3
goes to stock 2 at rate 0.02m /s)
- if (stock[2]>0.3) then dirty water ==> Seine,
3
0.1m /s

==> Maximize your number of points at the end
of the storm

Shrinking horizon

Too many time steps!

At each time step, make a decision
using only 30 time steps.

Example of heuristic function:

zermeloValue(state)=the margin before any stock
can go to the Seine.

Shrinking horizon

moving window of
30 time steps

Examples of operational research

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