FLEA

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FLEA

• Prof. dr. ir. J. Hellendoorn

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Decision making

• Decision making takes place in all areas of life
• Sometimes, results of decision making are visible
  sooner or later, sometimes the results cannot be
  judged
• Many different decision problems
• Number of deciders
• Number of decision steps
• Uncertainty

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Number of decision makers

• Individual decisions, more person or group
  decisions
• One common target or different targets (game
  theory)
• Separation in central and decentral decisions and
  decisions on different levels

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Number of decision steps

• One step decisions
• Multi-step decisions
• Series of consecutive and mutually dependent
  decision steps.
• Time can play a role, like in dynamic programming
  (Bellman).
• The final decision is taken in several steps

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Dynamic programming automaton
x t
z t
System
One time storage
z t1
Minimize (or maximize) f (x1,,xN)
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Optimal policy

• An optimal policy has the property that whatever
  the state the system is in at a particular stage,
  and whatever the decision taken in that state,
  then the resulting decisions are optimal for the
  subsequent state.
• An optimal policy is made up of optimal
  subpolicies.
• An optimal policy from any state is independent
  of how that state was achieved and comprises
  optimal subpolicies from then on.

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Routing example
16
B
E
16
15
18
22
H
23
22
18
19
16
A
C
F
J
11
22
29
I
15
19
15
13
22
D
G
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Recurrence relation
If f n(in) the cost of an optimal policy when
there are n stages remaining and the decision is
in state in then f n(in) min p(in , in-1)
f n-1(in-1) in-1 where p(a ,
b) is the premium associated with the single
stagecoach journey from a to b
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The example in numbers
f 1(H) 23 f 0(J) 23 0 23 f 1(I)
29 f 0(J) 29 0 29
f 2(E) min(16 f 1(H), 18 f 1(I))
min(16 23, 18 29) 39 f 2(F)
min(18 f 1(H), 11 f 1(I))
min(18 23, 11 29) 40 f 2(G) min(15 f
1(H), 13 f 1(I)) min(15 23,
13 29) 38
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The example II
f 3(B) min(16 f 2(E), 15 f 2(F))
min(16 39, 15 40) 55 f 3(C)
min(22 f 2(E), 16 f 2(F) , 22 f 2(G))
56 f 3(D) min(19 f 2(F), 22
f 2(I)) min(19 40, 22 38) 59
f 4(A) min(22 f 3(B), 19 f 3(C) , 15 f
3(D)) 74
Route A - D - F - I - J
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Extensions

• Negative premiums
• Stop when ?i. f n(i) f n-1(i), with p(i,i)
  0
• Approximation

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Knapsack problem
Vehicle with capacity of 10 tons and four products
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Other notation

• Maximize the sum 8x1 12x2 15x3 19x4,
  subject to two kinds of restriction, that 2x1
  3x2 4x3 6x4 ? 10, and x1, x2, x3, x4 should
  be equal to 0 or 1.
• As long as the problem is small enough it is
  possible to examine all the ways of deciding.
• Now we focus on the principle of dynamic
  programming

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In formula
f n(T) max(f n-1(T), Vn f n-1(T -
Wn)), if T ? Wn f n(T) f n-1(T) , else f
0(T) 0, ?T
In words the best that we can do in stage n and
with state T is the better of not loading any of
product n (f n-1(T)) and loading product n and
facing the next decision with a state reduced by
its weight (f n-1(T - Wn)) provided that T is
large enough to allow this to happen.
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The recurrent process
f 1(T) 19, if T ? 6 f 1(T) 0, if T lt
6 f 2(10) max(f 1(10) ,15 f 1(6)) 34 f
2(9) max(f 1(9) ,15 f 1(5)) 19 f
2(1) f 1(1) 0 f 2(0) 0
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The recurrent process II
f 3(10) max(f 2(10) ,12 f 2(7)) 34 f
3(9) max(f 2(9) ,15 f 2(5)) 31 f
3(1) f 2(1) 0 f 3(0) 0
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The recurrent process III
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Extensions

• Nonlinearity
• Adding more dimensions volume, price,
• Sensitivity analysis

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Uncertainty

• Uncertainty makes decision making interesting and
  difficult
• Decision processes
• Processes with precise information
• Processes with uncertain, imprecise, or
  incomplete information

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Level of knowledge

• Processes with precise information
• Target either known or uncertain
• Processes with uncertain, imprecise, or
  incomplete information
• Probabilistic uncertainty with known distribution
  function
• Statistical or stochastic methods
• Interval of uncertainty
• Vague, fuzzy knowledge

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This course

• Single step decisions
• Single person decisions
• Usage of uncertain information
• Not of complete fuzzy sets
• Usage of fuzzy decision matrices

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Example of fuzzy sets in DA I
m
A
B
1
B is better then A
0
Perf
m
A
B
1
Intuitively, B is better then A
0
Perf
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Example of fuzzy sets in DA II
m
B
1
What is better, A or B ?
A
0
Perf
m
A
B
1
Can A and B be compared?
0
Perf
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The decision problem

• Number of alternatives a1, a2, , an
• Number of criteria c1, c2, , cm
• Number of weight factors w1, w2, , wm , each
  weight factor belongs to the corresponding
  criterion
• Decision function D, D ?(wi , ci )

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How to determine ?

• Uncertainty in available alternatives
• Relative importance of each aspect
• Evaluation of each alternative with regard to
  each aspect
• Role of traditional arguments
• Value of prize money

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Examples from sport tennis

• Number of won sets is decisive
• Not the number of won games
• Not the number of won rally's
• The playing field is crisp, the ball is either in
  or out
• A net service may be regarded as neither right
  nor wrong

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Examples from sport decathlon

• Each of the numbers (100m, 400m, steeple chase,
  jump, javelin) delivers points
• The sum of the points determines the winner
• Participants can determine their own personal
  playing and training scheme to achieve an optimal
  result

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Skating

• Four distances 500m, 1500m, 5000m, and 10000m.
• Measured times t500, t1500, t5000 , t10000.
  These times are calculated back to 500m.
• The evaluation criterion
• The person with the least K is winner

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Skating alternatives

• Alternative decision functions to determine a
  winner
• Ranking on each distance, mean or sum of the
  rankings
• Similar, but 10 points for the winner, 6 or 8
  points for number two, etc. Compare Formula 1
• Take the mean of the whole field, and compare
  results with the mean, deviation from the mean
  brings points. Compare bridge. Disadvantage
  complex for the audience.

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Decision problem dice and coin

• Decide between two alternatives
• Throwing a coin
• Throwing a dice
• When one throws a coin, there are two
  possibilities
• Heads 4 points
• Tails 5 points

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Throwing a dice

• Number of points is equal to the number of dots,
  but 1 means throw again and 6 delivers 11 points.

Heads 4
Tails 5
Coin
1 throw again
2 2
DA
3 3
Dice
4 4
5 5
6 11
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Chances
H4
½
C5
Coin
½
1
1
1/6
22
DA
1
1/6
33
1
1/6
Dice
44
1/6
55
1/6
1/6
611
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Choices 1 and 2

• Maximum of the mathematical expectation
• Coin
• Dice
• Choice is for dice, optimistic.
• Maximum of minimal profit
• Coin minimum 4
• Dice minimum 2
• Choice is for coin, careful, pessimistic.

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Choices 3 and 4

• Maximum of the maximal profit
• Coin maximum 5
• Dice maximum 11
• Choice is for dice, opportunistic.
• Minimum of maximal spread
• Coin largest spread 1
• Dice largest spread 9
• Choice is for coin, careful, sure.

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Lottery

• Three different lotteries
• 10 lottery tickets, 1/ticket, 1st price 8.
• 10 lottery tickets, 1/ticket, 1st price 12.
• 1,000,000 lottery tickets, 1,000,000/ticket,1st
  price 1,200,000,000,000.
• A. seems unattractive with a mathematical
  expectation of 0.8. This is, however, extremely
  high for a lottery.

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Mathematical expectation risk

• Lottery B has a mathematical expectation of 1.2,
  so the lottery loses money. Very attractive for
  gamblers.
• Lottery C also has a mathematical expectation of
  1.2, however, who buys a ticket?
• Repeat the coin-dice example for higher amounts
  of money.

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Decision operator

• The decision to play or buy something (like in a
  lottery, gambling, betting) is dependent of the
  mathematical expectation and
• the character (careful, optimistic, reckless)
• the circumstances (rich, poor, lonesome, tension,
  charity)

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Savings

• Suppose you saved 100,000.
• Offer you put everything on one number of the
  roulette. If you win, you get 10,000,000.
• Expectation
• This is excellent, why not do it?

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Personal utility

• According to Bernouilli personal utility is not
  dependent of the amount of money but the value of
  money for the person in question.
• Bernouilli assumes that the value (use) of money
  is proportional to the logarithm of the amount of
  money.
• Does not take into account personal circumstances.

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Personal utility example

• A possesses 100,000 and gambles x.
• In case of loss 100,000 x.
• In case of profit 100,000 100x.
• Utility before playing
• Utility after playing

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Decision to play

• If Ubefore lt Uafter then the personal
  expectation value is positive.
• If x 100000, Ubefore5, Uafter ? dont play.
• If x 500, Ubefore5, Uafter 5.0026 play.
• We did not take into account personal, subjective
  factors.

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General decision problem

• Alternatives A a1, a2, , an
• Criteria C c1, c2, , cm
• Weights W w1, w2, , wm
• Target D

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Decision table

• The decision problem can be described by a table
  or matrix

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Three different approaches I, II

• For each alternative ai the function Dis
  calculated. The best value (the maximum value) of
  D determines the best alternative.
• For each criterion the alternatives are mutually
  compared. The best alternative is closest to an
  optimum or far from the worst alternative.

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Three different approaches III

• Build up a knowledge base that compares
  subjectively the alternatives and delivers a
  (vague) conclusion.

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The max-min criterion

• Max-min is the oldest method
• The best choice is

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Example

• Consider two alternatives, what is the best
  alternative when throwing heads?
• Clearly a1, but this seems too careful and
  pessimistic.

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Alternative from Savage

• Calculate the amount of money that is missed in
  case of a wrong choice.
• With a1 and tails one missed a2 and tails, i.e.,
  10000.
• So one regrets (use again max-min)

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General decision operator

• Best alternative is i, where
• D is a monotonous, non-decreasing function in s

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Several instantiations

• Intersection
• Harmonic mean
• Geometric mean
• Algebraic mean

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Several instantiations II

• Quadratic mean
• Union
• Generally if sltt then Di(s)lt Di(t), so

harm. mean
geom. mean
algebr. mean
quad. mean
min
max
s
-?
?
-1
0
1
2
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What is s?

• s can be regarded as a measure of optimism or
  carefulness.
• When s?-? the decision is based only on the
  minimum of the membership degrees very
  pessimistic!
• When s?? the decision is based only on the
  maximum of the membership degrees very
  optimistic!

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What is s? II

• For s lt 1 negative values (weak properties) have
  relatively high influence
• For s 1 the decision operator D is sensible for
  absolute changes in ?
• For s 0 the decision operator D is sensible for
  relative changes in ?

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Hurwicz-operator

• Another method to describe the optimism index is
  by
• Disadvantage of this method is that it regards
  only extreme values of ?ij

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Example of the influence of s
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Several values of s

• s ? -?, minimumhence alternative a1
• s ? 0, geometric mean, hence alternative a2
• s ? 1, algebraic mean, hence alternative a3

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Several values of s II

• s 2, quadratic meanhence alternative a4
• s ? ?, maximum,hence alternative a5

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In figure as a function from s
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Zooming in

• Important mean values lie around s ? 0
• We therefore map -? lt s lt ? on 0 ? r ? 1

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Functions r1, r2, and r3
pessimistic
optimistic
r
0
0.5
1
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Zooming in, in figure
well distinguishing functions