FLEA

1
FLEA

• Prof. dr. ir. J. Hellendoorn

2
Distance functions

• Assume a linear vector space, then a distance
  function should satisfy
• d(a,b) ? 0
• d(a,b) d(b,a)
• d(a,b) 0 if and only if a b
• d(a,b) ? d(a,c) d(c,b)
• In practice (traffic, chess, social life) these
  conditions often dont hold

3
Two N-dimensional points

• Two N-dimensional points fA(xi) and fB(xi), where
  for each dimension i 0 ? fA(xi) ? 1 and 0 ?
  fB(xi) ? 1

Y
B
A
X
0
4
Hamming distance

• The Hamming or taxi cab distance from A to B is
  defined by
• The relative Hamming distance is

5
Euclidean distance

• The shortest line from A to B
• The relative Euclidean distance

6
Example
7
General relative distance

• The general relative distance function is defined
  by
• p 1 gives Hammings distance function
• p 2 gives Euclidean distance function
• p ? 1 can be compared with - ? lt s lt ?

8
Decision making with distance functions

• An alternative is determined by the values ?ij
• The best alternative is the alternative that
• has the greatest distance from abad
• is closest to aideal
• or another a

9
Example
10
Order of alternatives w.r.t. aideal

• dH Hamming distance, dE Euclidean distance,
  dmax max-min distance

11
Order of alternatives w.r.t. agood
12
Order w.r.t. abad and a0
13
Order of alternatives w.r.t. a0
14
Order of alternatives w.r.t. abad
15
Weight factors

• Suppose,
• If we add weight factors, than according to
  Ronald Yager this can be performed by
• where wj are weight factors and

16
Example

• w1 w2 0.5Both alternativesare good
• w1 0.3, w2 0.7

wrong
intuitively right
17
Property of weight factors

• The decision function should keep the properties
  positive, continuous, monotonous, non decreasing
  in s, etc.
• In case of equal weights the function should be
  reduced to the original one
• In case of increasing weights the decision
  function should increase
• In case of increasing weight for an aspect, the
  influence of this aspect should increase

18
Two decision functions

• Two decision functions that satisfy these
  properties
• Di only satisfies criterion 4 when s ? 0

19
Decision with equal weights
20
Decision with unequal weights
21
Conclusions for the order

• Both examples
• Very pessimistic A5, A4, A3, A2, A1
• Very optimistic A1, A2, A3, A4, A5
• Equal weights
• Bit pessimistic A4, (A3, A5), A2, A1
• Bit optimistic (A4, A3), (A5, A2), A1
• Unequal weights
• Bit pessimistic A3, (A4, A5), A2, A1
• Bit optimistic A3, A4, A5, A2, A1

22
Geometrical interpretation of D

• Consider the function Di and the graphic
• The question is what isdistance? This
  dependson s

23
Distance with s 1

• Suppose a decision space with two criteria, hence
  two dimensions

s 1, Di(1) constant
24
Distance with s ? ?
s ? ?, Di(?) max(m1, m2) Only one of the
ms determines Di(?)(unless one of the ms is
larger than agood )
25
Distance with s ? -?
s ? -?, Di(-?) min(m1, m2) Only one of the
ms determines Di(-?)(unless one of the ms is
smaller than abad )
26
Distance with s 0
27
Distance with s 2
28
Properties of s

• s 1 straight lines
• s gt 1 convex lines
• s lt 1 concave lines
• Changing s influences the optimism or pessimism
  about one of the criteria

29
Introduction of weights

• The use of weights graphically changes the slope
  of the lines or ellipses, example

30
Other valuations

• Until now we assumed that the ms were
  independent, this is not always the case
• Three other kinds of comparing alternatives
• Criteria-criteria dependency
• Criteria-alternative dependency
• Alternative-alternative dependency

31
Criteria-criteria dependency

• Suppose you want to buy a car
• Aspects price, fuel consumption, comfort, space
• Usually more space implies more comfort, hence
  mspace? ? mcomfort?
• Often a higher price implies more fuel
  consumption, hence mprice? ? mconsumption ?
• Often higher price means more comfort, hence
  mprice? ? mcomfort?

32
Criteria-alternative dependency

• Some criteria only hold for some of the
  alternatives
• Example car
• small four wheel drive ? Subaru
• four wheel drive in Africa ? Toyota
• crosswise parking ? Smart

33
Alternative-alternative dependency

• A company with Dell computers will buy another
  Dell computer, server or even network and printer
  rather then another brand (thats why Dell
  introduced printers)
• Employees from Nokia or Siemens will buy their
  own telephones
• Some people will always buy Miele

34
Hierarchical decision making

• In complicated environments it is useful to
  decide in more than one step

Operator
Operator 1
Operator 2
Operator 3
Operator 4
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
Single step decision making
Multi step (hierarchical) decision making
35
Advantages

• In each decision step it is possible to apply
  different decision functions or operators
• Similar criteria or criteria belonging together
  can be grouped
• In some blocks criteria-criteria dependency can
  be used
• In blocks with groups of criteria weights get
  more sense

36
Example purchasing a car

• Aspects
• A single step decision takes these 15 aspects
  together in a 15 dimensional vector space

• Price
• Weight
• Fuel consumption
• Repair costs
• Max. velocity

• Acceleration
• Braking distance
• Comfort
• Front space
• Back space

• Boot space
• Drive
• Road holding
• Reliability brand
• Reliability dealer

37
Grouping the criteria
Final choice
Financial
Technique
Comfort
Safety
Price Weight Fuel consumption Repair costs
Max. velocity Acceleration Braking
distance Reliability brand
Comfort Front space Back space Boot
space Reliability dealer
Drive Road holding Weight Braking distance
38
Criteria-criteria dependency

• Higher weight ? lower acceleration
• Higher weight ? more safety (?)
• Higher weight ? longer braking distance
• Lower acceleration ? less safety (?)
• Higher price ? more comfort, space, weight, fuel
  consumption

39
Role of and and or

• Usually the reasoning is when the alternative
  satisfies criterion 1 and criterion 2 and
  criterion 3 and ...
• The prison-example shows the ambiguity of and

40
Voice example

• Speaking is accomplished by three functions air
  pressure, larynx and vocal cords
• Doctor 1 No speaking is possible when one of the
  functions is eliminated
• Doctor 2 Speaking is possible even when only one
  function works
• Suppose working is described by 0-1

41
Two doctors
42
In practice

• Never either 0 or 1
• Never either doctor 1 or doctor 2, rather
  somewhere between maximum and minimum
• Dependency of prosthesis

43
Front and rear wheel drive

• Normal road or function
• Heavy terrain and function
• Other situations g front (1-g) rear,
  compare Hurwicz-function

44
Other situations
45
Determining weights directly

• Direct method, e.g., by drawing a line between
  aspects and weights
• Changing one line changes all the weights

0
Aspect 1
2
Aspect 2
4
Aspect 3
6
Aspect 4
8
Aspect 5
10
46
Determining weights indirectly

• Pair wise comparison, e.g., when there are many
  aspects
• For each pair of aspects, determine which aspect
  is more important and to what degree
• Bring all the comparisons in one framework, e.g.,
  divide 100 points per pair and

47
Example
Six pairs ab 12 3367 ac 13
2575 ad 14 2080 bc 23 4060 bd
12 3367 cd 34 4357
In this case, normed (sum 1)
48
Non-numeric values

• Humans usually dont evaluate in numbers but
  linguistically, e.g.,
• as important as
• somewhat more important
• much more important
• less important
• etc.

49
Saatys linguistic terms
50
Other values

• Saaty uses intermediate values 2, 4, 6, 8 to
  express doubt between two linguistic terms
• wij gt 1, if ci is more important than cj
• wij 1/ wji, e.g., ½ or ¼
• In general, humans can distinguish 72 classes

51
Example

• Suppose the following objects are compared w.r.t.
  weight
• Radio, laptop, small suitcase, projector, large
  suitcase
• A1 as important as
• S3 somewhat more important
• M5 much more important
• V7 very much more important
• X9 absolutely more important
• D0 doubt between two classes

52
Comparison
53
Scales
same importance
1
2
3
4
5
6
7
8
9
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
?
1/?
1/9
1/8
1/7
1/6
1/5
1/4
1/3
1/2
Importance
increments positively
increments negatively