Fuzzy Decision Making

All you ever wanted to know about fuzzy decision making

but were afraid to ask!

Rita Almeida Ribeiro

CA3/CTS/UNINOVA

email: rar@uninova.pt

URL: www.uninova.pt/ca3/

22-10-2010

2 Out 2010

Topics

Basic Concepts of Fuzzy Set theory

Decision Making: Crisp vs Fuzzy

1. Fuzzy Optimization - Fuzzy Multiple Objective Decision

Making (FMODM)

Examples

2. Fuzzy Multiple Attribute Decision Making (FMADM)

Examples

3. Fuzzy Inference Systems

Examples

Conclusions

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Decision Making Contexts (or environments)

(1) Decision making under certainty (complete information).

E.g. If employee has children he gets child support

(3) Decision making under uncertainty (least information).

E.g. If cold take the car

(2) Decision making under risk (less than complete information).

E.g. Probability of finishing the project in 10 months?

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Basics on Fuzzy Set Theory [Zadeh, 65]

A fuzzy set A in X is a set of ordered pairs, A={xi, μ(xi)| xi E X}

where (xi) represents the grade of membership of xi in A.

Corresponding Discrete Fuzzy Sets:

Around3={2.1/0.1,..., 3/1,...,3.9/0.1} Cold={-10/1,...0/1,...,5/0}

Continuous Fuzzy Sets:

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Basics on Fuzzy Sets Theory

Linguistic Variables

They include a set of labels (values), each corresponding to a compatibility

function (i.e. fuzzy set). The objective is to enable expressing imprecise

semantic concepts using a consistent mathematical formulation and then

perform approximate reasoning.

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Basics on Fuzzy Sets Theory [Zadeh, 65]

Not Pleasant Very Hot

Negation (complement)

Modifiers (concentration): e.g. very, few, more or less, quite, many....

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OPERATORS [Klir & Folger 1988]

INTERSECTION (AND)

• T-norms (e.g. min, product, drastic product...)

• Compensatory (e.g. Hammaker, Dubois&Prade, Yager...)

UNION (OR)

• T-conorms (e.g. Max, sum, drastic sum...)

• Compensatory (e.g. Dubois & Prade...)

IMPLICATION (Inference)

• Kleene, Lukasiewicz, Mandani, Sugeno....

AGGREGATION:

• Intersection, weighted average, Yager, Sugeno.....

Note: Fuzzy logic and decision making use/need these operators

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FUZZY MULTICRITERIA DECISION MAKING

MCDM

MADM MODM

Objectives

Constraints

Select Optimal Decision

Maxmin Model (Bellman & Zadeh, 70) FMADM FMODM

Zimmermann, 76, 78,96

Alternatives

Attributes

Select Best Decision

FUZZY

9 Out 2010

Basics on MCDM

MODM - consists of a set of conflicting goals that usually are difficult to

achieve simultaneously. MODM deals with problems where the alternatives

are not pre-defined, so the decision-maker has to select the more

promising alternative facing the quantity of (limited) resources available.

MADM - the alternatives are pre-determined and known. The decision-

maker has to select/prioritize/rank a finite number of alternative actions.

The choice of alternatives is performed based on the attributes

classification.

“A decision process in which the goals and constraints, but not necessarily

the system under control, are fuzzy in nature” (maxmin model) [Bellman &

Zadeh 1970]

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Maxmin Model (Symmetric) [Bellman and Zadeh 1970]

Consider a fuzzy set A in X is a set of ordered pairs, where (xi)

represents the grade of membership of xi in A.

Assuming a set of goals and constraints, the fuzzy decision D is,

• Hence the “optimum” is:

“No difference from goals and constraints “SYMMETRIC MODEL”

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FUZZY OPTIMIZATION

Fuzzy Constraints

Fuzzy coefficients

Decision Variables Mathematical

relationships Result variables

Degrees of violation

Goal(s): Max Z

fuzzy coefficients

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Fuzzy Optimization Modes of Imprecision

Fuzzy Constraint boundaries

E.g. total time of painting should be less than around 50 hours

Fuzzy goals

E.g. profits should be above $100,000.

Fuzzy coefficients in the objective function variables

E.g. price of chair2 is around $20

Fuzzy coefficients in the constraints variables

E.g. cost per hour of producing product 1 is around $10

Formally,

NON-

LINEAR

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Example - Unfeasible problem

Crisp infeasible problem!

(Constraint 4)

40

10 15 20 25 30 35 40

10

15

20

25

30

35

(1)

(3)

(4)

(2)

(5)

satisfying (1) (2) (3) (5)

x 1

Fuzzy constraints Fuzzy coefficients Both

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Example 2 - Unfeasible problem

10

15

20

25

30

35

40

10 15 20 25 30 35 40

(1)

(3) (4)

(2)

(5)

satisfying (1) (2) (3) and (5)

x1

x2

(1) (2)

(3)

3 fuzzy results with different fuzzifications

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Optimization Resolution Algorithms

Search Techniques

SA

evolutionary

algorithms tabu search

GA Evolutionary

strategies SA/GA

Source: adapted from IEEE Computer, June 94

calculus-based guided random enumerative

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FUZZY MULTICRITERIA DECISION MAKING

MCDM

MADM MODM

Objectives

Constraints

Select Optimal Decision

Maxmin Model (Bellman & Zadeh, 70) FMADM FMODM

Zimmermann, 76, 78,96

Alternatives

Attributes

Select Best Decision

FUZZY

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Multiple Attribute Decision Making (FMADM)

Consequences

States of Nature

C1 C2 … Cn

Actions

A1 x

11 x

12 … x

1n

A2 x

21 x

22 … x

2n

.

.

.

.

.

.

.

.

.

…

…

…

.

.

.

Am x

m

1

xm

2

… xm

n

“Decision making is a process of choosing among alternative courses of

action for the purpose of attaining a goal or goals”

MADM - the alternatives are pre-determined and known. The decision-

maker has to select/prioritize/rank a finite number of alternative actions.

The choice of alternatives is performed based on the attributes

classification.

EXAMPLES :

Job selection (e.g. company prestige, location, salary, career prospects…)

Missile selection (e.g. velocity, range, reliability, potential... )

Evaluation of proposals (e.g.leasing of cars, project selection, statue to

choose)

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Types and Challenges of MADM Problems

Main types of problems:

1. Classification

2. Selection

Main Challenges:

1. Selection of operators

2. Definition of weights

Main Methods:

1. Dominance

2. Maxmin 8. ELECTRE

3. Maximax 9. TOPSIS

4. Conjunctive/disjunctive method

5. AHP

6. Lexicographic method

7. Weigthed product or average

[Chen & Hwang, 1992]

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FMADM Application: best site to land

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FMADM- Example 2

Prioritization of equipment repairs under battle conditions

A Fuzzy Decision Support System for Equipment Repair under Battle Conditions. M. S. Marques, R. A. Ribeiro, A. G. Marques. Fuzzy Sets and Systems, Nr 115 (2000), pp 141-157.

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Alternatives

Attributes

Select Best Decision

FMADM FMODM

Objectives

Constraints

Select Optimal Decision

Maxmin Model (Bellman & Zadeh, 70)

FIS/FKBS

Zimmermann, 76

Inputs,Outputs

Rules

Infer Best Decision

Mamdani, 74

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Fuzzy Inference Systems

Fuzzy Knowledge Based System (FBKS)/Fuzzy Inference Systems (FIS)/Fuzzy Expert System (FES)

FIS

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FIS: MODI

Alarm System

Input Telemetry

Nominal

Terrain Recognition

Terrain detected

No Terrain detected

Alarm

Unknown Situation

The System doesn’t

know the actual

situation.

Critical alarm

Terrain type, MPa and

Confidence-level.

Lack of information

to detect the terrain.

If alarm is not critical it

will infer about the

terrain

Alarm Reasoning

IF Set-Points are satisfied (terrain, translation, rotation) and the variables are nominal Then No-alarm.

IF Set-Points are satisfied (terrain, translation, rotation) and one or more variables have deviations Then alarm-level (critical, severe, moderate etc)

IF Set-Points are NOT satisfied (terrain, translation,rotation) Then unknown situation.

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CONCLUSION

Absolute certainty is difficult to find in the real world.

Natural language is an imprecise mode of expressing concepts.

Since the majority of problems faced by decision makers are ill- structured they require specific tools and methods.

“... they have to be capable of actually making or recommending

decisions, taking as their inputs the kinds of empirical data that are

available in the real world. ... idealised models of optimising

entrepreneurs, equipped with complete certainty about the world ....

are of little use”. H. Simon

25 Out 2010

Bibliography

Bellman, R. E., and Zadeh, L. A. (1970). “Decision-Making in a Fuzzy Environment.” Management Science, Vol.17(No.4), 141-164.

Chen, S.-J. and C.-L. Hwang (1992). Fuzzy Multiple Attribute Decision Making, Springer-Verlag.

Klir, G. J. and T. A. Folger (1988). Fuzzy Sets, Uncertainty, and Information, Prentice-Hall International Editions.

Lai, Y.-J., and Hwang, C.-L. (1992). Fuzzy Mathematical Programming: Methods and Applications, Springer-Verlag.

Lai, Y.-J., and Hwang, C.-L. (1994). Fuzzy Multiple Objective Decision Making, Springer-Verlag.

Ribeiro, R. A., and Pires, F. M. (1999) “Fuzzy Linear Programming via Simulated Annealing.”Kybernetika, Vol 35, Nr 1, 57:67.

Ribeiro, Rita A. (1996). “Fuzzy Multiple Attribute Decision Making: A Review and New Preference Elicitation Techniques”, Fuzzy Sets and Systems . 78, pp 155-181.

Zadeh, L. A. (1965). "Fuzzy Sets." Information and Control 8: 338-353.

Zadeh, L. A. (1987). The Concept of a Linguistic Variable and its Application to Approximate Reasoning - I. Fuzzy Sets and Applications: Selected Papers by L. A. Zadeh. R. R. Yager, S. Ovchinnikiv, R. Tong and H. T. Nguyen. New York, John Wiley & Sons. Reprinted from: Information Sciences, 8 (1975): 219-269.

Zimmermann, H.-J. (1996). Fuzzy Set Theory and its Applications, Kluwer Academic publisher, Boston.