Methods and Models for Decision Making

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Methods and Models for Decision Making

Alberto Colorni – Dipartimento INDACO, Politecnico di Milano Alessandro Lué – Consorzio Poliedra, Politecnico di Milano

5°

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© Alberto Colorni

MMDM – Lesson 5

• (1) Introduction (2) Tools & frame

• (3) Mental models (4) Design & decision

• (5) Classification (6) Ranking-1, risk analysis

• (7) Ranking-2, multicriteria (8) A tentative case

• (9) Rating problems (10) Seminar M. Henig

• (11) Group decision (12) Genetic alg. + …

• (13) Research topics (14) Case results (if any …)

• (15) Conclusions

Index:

• Variable space (decision) & objective space (results)

• Three phases in MCDM (or MODM) problems

• Phase 1  from indicators to utilities

• Phase 2  elimination of dominated solutions

• Phase 3  DM preferences & final choice

• The second is the only “objective” phase

• Utility functions & preferences of this DM

• Sensitivity analysis with respect to the weight vector

2

God in 7 steps:

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© Alberto Colorni 3

A tentative case:

the “Colorni award”

(the best italian newspaper on the web)

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The best italian newspaper on the web

(you are in the jury …)

• What are the alternatives ?

• What are the attributes ?

• What are the utility functions ?

• What are the weights ?

• What is the ranking ? (the personal one and the collective one)

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Alternatives

• Repubblica 

• Corriere Sera 

• Sole24Ore 

• Ansa online 

• RaiNews24 

• Foglio 

• Gazzetta d. Sport ? NO, because it is too specific (only sports)

• Novella 2000 ? NO, because it is a magazine (weekly)

What are the rules ? (if there are rules …)

The alternatives must be “similar” (but the concept of similarity is subjective)

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Attributes

• A lot of (tentative) attributes; note that the attributes must generate a set of indicators that could be measurables

• Four main indicators:

 items in the homepage

 upgrades during the day

 daily visitors (declaration)

 quality, in a scale [1, 10]

• A search (made by the students) to obtain the data set

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Performance matrix

7

A1 A2 A3 A4 A5 A6

I1 29 24 23 16 6 8 items

I2 120 240 288 288 288 1 upgrades

I3 3.6 3.3 0.03 0.02 0.23 0.01 visitors

I4 8 8 7 8 8 2 quality

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Utility functions

• One utility function for each attribute/indicator

• Three steps for each utility function (easy version)

• The results 

• The evaluation matrix (in a common [1, 10] scale)

• Phase 2  dominated alternatives ?

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Utility functions

9

5

ind.1

10

0

10 20 30

ind.2

10

0

100 300 .5

ind.3

10

0

… 3.3 5

0 0

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Evaluation matrix

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A1 A2 A3 A4 A5 A6

C1 1 6 7 10 2 6

C2 5.5 8.5 9.75 9.75 9.75 0

C3 10 10 0.5 0.4 4.6 0

A6 (Il Foglio) is dominated, but …

C4 8 8 7 8 8 2

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Pairwise comparison (to obtain the weights)

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C1 C2 C3 C4

C1 1 *

C2 * 1 *

C3 1

C4 * * * 1

C4 > C2 > C1 > C3

weights?

(numerical values)

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Determination of weights / 1

C1 C2 C3 C4

C1 1 1/3 2 1/4

C2 3 1 3 1/2

C3 1/2 1/3 1 1/5

C4 4 2 5 1

(reciprocal matrix) ^coherent?

C1 C2

C4 C3

3

5 4 2

2

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Determination of weights / 2

1 1/2 2 1/4

2 1 4 1/2

1/2 1/4 1 1/8

4 2 8 1

(consistent matrix) ^{coherent ?}

C1 C2

C4 C3

2

4

8 4 2

2

2/15 4/15 4/15 8/15 collective weights

w1

w2

w3

w4

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Weights (the preference structure)

• We are 30 (approximately) DMs

• What is the way to obtain (shared) weights ?

• Discussion …

• A common vector of weights ?

• Total utility of each candidate  final ranking

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General utilities

1 6 7 10 2 6

5.5 8.5 9.75 9.75 9.75 0

10 10 0.5 0.4 4.6 0

8 8 7 8 8 2

98 120 109.⁵ 123.⁴ 111.⁶ 28 … /15

2

4 1 8

… /15

Ranking: A4 – A2 – A5 – A3 – A1 – A6

1. RAI news 2. Corriere 3. ANSA 4. Sole24 5. Repubblica 6. Foglio

weighted sum

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A lot of rankings (the personal paths)

• A set of (∼ 30) individual rankings

• What can we do ?

• It is a group decision  see lesson 6

• The final result is the “less conflicting” (or more shared) ranking

• Two main procedures  Distillation or Maastricht

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© Alberto Colorni 17

DI – Deeper Investigations

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DI-1  Utility & dominance

cost

alt 1 alt 2 alt 3 0.3 0.8 0.7

25 47 12

1 6 5

250 710 940

7 6 4

53 71 98

409 524 912 air quality

water quality flora

fauna

landscape employment

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Utility functions

alt 1 alt 2 alt 3

80 60 70

70 50 80

40 80 70

50 70 90

70 60 40

50 70 80

90 70 40

air quality water quality flora

fauna landscape employment

cost cost

u₇ 100

0

u₂

Water quality

100

0

u₄

fauna

100

0 alt 1 alt 2 alt 3

0.3 0.8 0.7

25 47 12

1 6 5

250 710 940

7 6 4

53 71 98

409 524 912 air quality

flora

landscape employment

cost

water quality fauna

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Pareto dominance

The Pareto criteria has to be verified only after the application of the utility

functions

alt 1 alt 2 alt 3

80 60 70

70 50 80

40 70 70

50 70 90

70 30 40

50 70 90

90 40 40

fauna landscape employment cost

cost

alt 1 alt 2 alt 3 0.3 0.7 0.5

25 47 12

1 5 5

250 940 710

7 3 4

53 71 83

409 912 912

fauna landscape employment

(attributi)

(criteria)

utility

alt2 dominated by alt3 Performance

matrix

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The preferential rank (and the structure) between 2 values of an ttribute doesn’t depend on the value of the other attribute

Example the purchase of a radio

2 attributes • price ↓

• ratio signal/noise ↑

Counterexample1 choice in a menu

2 attributes • food (fish, meat)

• beverage (white wine, red wine)

Counterexample2 the chemical reaction

2 attributi • reagent A

• reagent B (4,5) > (3,5) but (4,2) < (3,2)

complementary goods, synergic effects keep the ratio 1:1

DI-2  Mutual preferential independence

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separabilità U(x,y)= f(u_x(x) , u_y(y))

additività U(x,y)= f₁ [u_x(x)] + f₂ [u_y(y)]

Mutual preferential indipendence

2 steps:

• determine u_x(x) e u_y(y)

• determine the functions f₁ e f₂

More formally

The mutual preferential indipendence is a necessary condition (but not sufficient) for the additivity

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DI-3  Tasso di sostituzione

Il decisore è sempre disposto a compensare il peggioramento della prestazione di un attributo con un opportuno miglioramento della prestazione di uno qualsiasi degli altri.

Tasso marginale di sostituzione

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Tasso marginale

1 1 1

λ

_A

λ

_B

λ

_C

λ

_E

λ

_D

A B

C

D E

λ

_B

<λ

_A

< λ

_C

λ

_E

<λ

_A

< λ

_D

x

y

A’

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DI-4  Weight assignment

y

x₀ x y₀ y₁

x₁

2 domande:

1. è preferibile (x₀, y₁) o (x₁,y₀)?

se la risposta è (x₁,y₀)

⇒

w₁>w₂

2. supponiamo di essere in (x₀, y₁) , qual è il valore di x per cui (x₀, y₁) è indifferente a (x, y₀)?

w₁ u_x(x₀) + w₂ u_y(y₁) = w₁ u_x(x) + w₂ u_y(y₀) w₁ + w₂ = 1

x

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Weight assignment: pair comparison

verde pubblico (m² /ab) x₁

x2 % occupazione

•^A

•^B principio di compensazione

tasso di sostituzione

1

w2

w

n pesi ⇒ n-1 confronti a coppie

⇒ n-1 rapporti

j

wi

w

∑

⁼

i

i 1

& w

80 ? 82

7.0 6.7

è il rapporto tra le differenze di utilità

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DI-5  The quality of life

• La qualità della vita nelle città italiane

• Indagine del Sole24ore (29 dicembre 2008)

• Analizzati 6 settori, con 36 indicatori

• Funzioni di utilità, pesi, classifiche

• Confronto tra classifiche basate sugli indicatori e

percezione di un campione (circa 700 ab. per provincia)

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Test examples

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 Describe how the Sudoku can be seen as a decision (or operational research) problem

or

 What is the Pareto frontier ? Describe it and show an example or

 Define the concept of lottery. Elaborate with two numerical examples or

 Describe the main ideas behind the “C-K theory”

or

 … ( a concept to be explained) …

Example-1: free text concerning an argument

(no more than 300 words or 2000 characters)

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In a problem characterized by two attributes (x ^andy), you know the utility functions of such attributes:

Moreover, you know that the decision-maker is indifferent to two following situation (A and B)

A B

Please discuss if the situation K is preferable to H Example-2

x x

u_x( ) =1− ) 2

(y y u_y =







=

2 1 1 y x







=

= 0 2 1 y x









=

4 1 2 1

y x









=

2 1 4 3

y x

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A multicriteria decision problem (6 alternatives, 3 criteria = utilities) is showed in this matrix, with its weight vector.

1. Are there dominated alternatives ?

2. What is the ranking and the final choice ?

3. Is the result changing if w₂ increase ? Is there a rank reversal ? Explain (briefly) all the answers.

Example-3

a₁ a₂ a₃ a₄ a₅ a₆

c₁ 60 40 20 70 100 80 w₁ 0.20

c₂ 40 40 35 35 35 40 w₂ 0.40

c₃ 20 30 60 40 50 50 w₃ 0.40