Examples of operational research

Simple introduction to operations research

1. 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

2. 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”.

3. 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.

4. 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.

5. Operations Research
everywhere.
I hope that from now on,
you will see plenty
of OR opportunities everywhere.

6. POLL
This teaching is adaptive.
Tell me what you want, and I'll try to adapt.
Email: [email protected]
(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.

7. 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

8. 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.

9. OP. RESEARCH: History
- Some precursors
- “Real” birth in world war II
- Then widely spread everywhere
- industry
- business

10. 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

11. 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.

12. WORLD WAR II
- Modern techniques
everywhere
- Supply chain
= critical
- Both tactical and
strategical
elements.

13. - attacks
= bombing
- defense =
anti-aircraft
artillery
+ radar
+ planes
UK is an island

14. VS

15. 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

16. An optimization problem: defending UK
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

17. An optimization problem: defending UK
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 ?

18. An optimization problem: defending UK
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)

19. An optimization problem: defending UK
Computing g(x,y) on a computer:
- simulate the time steps of a airplane
flight / bombing
- evaluate the damage
- return the damage
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 ?
Not so easy!

20. 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

21. Two optimization levels
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
Main optimization
problem

22. The second one is antagonist
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
Secondary optimization
problem: the German army
will optimize its behavior

23. Overview
1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
Sometimes OR = solved by just modelization.
submarine

24. Overview
1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
Example:
UK trying to destroy U-boats (which destroyed
UK convoys).
submarine

25. Overview
1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
Sometimes OR = solved by just modelization.


submarine
Approaches
U-boat: Escapes!

26. Overview
1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
Sometimes OR = solved by just modelization.


submarine
Approaches
Escapes!
Bomb!

27. 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.
submarine
Approaches
Escapes!
Bomb!

28. 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.
Just a relevant pen&paper model ==> big improv.
by changing the depth of explosion.
submarine
Approaches
Escapes!
Bomb!

29. Overview
1) Basic definitions
2) Illustrative example: UK in WWII
3) Categories of Op. Research (OR)
Modelization is crucial – nothing works
without a good model.
But often we need maths&algorithms:
optimization, statistics, Zermelo-like algos,...
a) based on optimization
b) based on statistics
c) based on time-decomposition
d) others
e) all-in-one

30. Optimizing plane is important;
turbulence matters
Simulations
Real-world

31. 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

32. 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

33. 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

34. Questions
Minimum of:
f(x) = x
2
f(x) = (x-1)
2
f(x)= ( (x-1)
2
-1)
2

f(x)= -x
2
f(x)= -x
2
for x >0
Final question: how to find the maximum of f
if you just have a program for finding minima ?

35. 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;
}

36. Minimization: the simplest
algorithm ever, random search
I sample
plenty of points.
I evaluate the value
for each point.
I keep the best one.
Here!

37. Minimization: the simplest
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;
}

38. 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.

39. 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.

40. 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;
}

41. 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;
}

42. The (1+1)-evolution strategy:
should I stay or should I go ?
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;
}
I start
here.

43. I should go!
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;
}
I test
here:
better!

44. ...
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;
}
So here
I go.

45. … we can not always succeed ..
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;
}
...and
here
I test.

46. Should I stay or should I go now ?
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;
}
...I stay
and I test
again

47. Should I stay or should I go now ?
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;
}
...
and I try
again
...

48. Should I stay or should I go now ?
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;
}
...
and
again
...

49. Should I stay or should I go now ?
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;
}
...
and
again
...

50. Should I stay or should I go now ?
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;
}
...
Found!

51. Should I stay or should I go now ?
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;
}

52. 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;
}
There is a
parameter s
for the size of
modifications

53. 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;
}
If success, then I
increase the size of
modifications!

54. 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;
}
Otherwise, I
decrease the size of
modifications!

55. 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;
}
There are plenty of variants of this algorithm.
These variants are evolution strategies.

56. 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 ?

57. 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 ?

58. Without warehouses
● Either many small
long-distance
transportations (huge cost)
● or one big
transportation (big delay,
big cost)

59. Warehouses
● Less travels with
big transportation
● Total price decreased
● Maximum delay reduced
(==> no breakdown)
● Less pollution

60. 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

61. 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)

62. 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 (…)

63. 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.

64. 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

65. 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:

66. 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...

67. Statistics crucial notion 1:
confidence interval
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.
Statistics crucial notion 2:
bias

68. Example of application:
industrial owens

69. Statistics
Steel quality
(for my factory)
Carbon quantity

70. Statistics crucial notion 3:
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.

71. Statistics crucial notion 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

72. Let's see a nice, counter-intuitive
example of application of
statistics:
==> protecting bombers

73. Let's come back to
anti-aircraft artillery.
UK bombers also attacked Germany.
But, many anti-bombers artillery.
Many planes destroyed.
Problem: how to protect planes ?
(Hamburg
Bombing with
incendiary bombs;
1000°C, ~190km/h
Wind – death toll
42600)

74. 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 ?

75. Using statistics for
protecting bombers
q(X)= P( part X damaged | plane came back)
= 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)

76. Using statistics for
protecting bombers
q(X)= P( part X damaged | plane came back)
= 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 ! ! !

77. Using statistics for
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.

78. Using statistics for
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.
m2=frequency of positive answers in category 2.
…
==> Who should receive advertisements ?

79. Using statistics for
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

80. Using statistics for
sending mails
m1=frequency of positive answers in category 1.
m2=frequency of positive answers in category 2.
…
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.

81. 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: Zermelo
(remember, irrigation, water, crops, etc)
d) others
e) all-in-one

82. 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.

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

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

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

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



Dirty
water
Seine

87. 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)
Problem: if big rainfalls reach dirty water,
then dirty water might pollute the Seine
(yes, in Taiwan it is more impressive,
sometimes it is 16.7 dm in 24 hours and gusts
can reach 250 km/h...)
Dirty
water
Seine

88. 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)
Problem: if big rainfalls reach dirty water,
then dirty water might pollute the Seine
(yes, in Taiwan it is more impressive,
sometimes it is 16.7 dm in 24 hours and gusts
can reach 250 km/h...)
Dirty
water
Seine

89. 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)
Problem: if big rainfalls reach dirty water,
then dirty water might pollute the Seine
(yes, in Taiwan it is more impressive,
sometimes it is 16.7 dm in 24 hours and gusts
can reach 250 km/h...)
Seine
Dirty
water
→ Seine!

90. Another beautiful application
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)

91. Big water network
Dirty
water
Dirty
water
Dirty
water
Dirty
water
Clean
water
Clean
water
Clean
water
Clean
water

92. 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

93. Water vs dirty 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.

94. Water vs dirty water
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
goes to stock 2 at rate 0.02m
3
/s)
- if (stock[2]>0.3) then dirty water ==> Seine,
0.1m
3
/s
==> Maximize your number of points at the end
of the storm

95. 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.

96. 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.

97. 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.

98. 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.

99. Shrinking horizon
moving window of
30 time steps

100. 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

101. 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