1

FUZZY DECISIONMAKING

Decision Making

A – set of alternatives or possible actions.– set of states (various conditions) of the

environment in which decisions are taken.– set of consequences resulting from the choice of a

particular alternative.– is a mapping A specifying a consequence

for each element of the environment. The space Adefines the solution space.D – decision function D : . Reflects thepreference structure of the decision maker.

389

( , , , , )A D

2

Decision function

The decision function D incorporates the goals of thedecision maker. It induces a preference ordering onthe set of consequences such that

390

iff ( ) ( )i j i jD D

where , , and is the preference relation, i.e.,

consequence is preferred to consequencei j

i j

Example

A person is driving a car on a cold winter day down aroad. Suddenly, a dog jumps in front of the car. Thedriver can decide between two actions: he can breakhard applying full power to the brakes, or he can brakesoft knowing that the car cannot come to a stopbefore a collision with the animal.

391

{slippery road, not slippery road}

{brake soft, brake hard}A

3

Example

392

road is not slippery

brake soft

brake hard

brake soft

brake hard

hit dog slight ly

slip and hit tree

hit dog slight ly

do not hit anything

states alternatives solutionset

consequences preferenceorderingD

D1

D2

D3

D4A

Example

Multidimensional consequences in multicriteriadecision making

393

ConsequenceCar

Damage toanimal tree

hit dog slightly minor minor none

slip and hit tree major none minor

do not hitanything none none none

4

Fuzzy decisions

Consider that the states of the environment areknown to the decision maker. In this case, theelements of can be incorporated in the set A.Thus, is a mapping : A .Best decision alternative a* for n decision criteria:

394

* max ( ( ))a A

a D a

1( ) ( ), , ( )na a a

Decision problem

The set A cannot always be defined explicitly. This setcan be defined implicitly by the specification of anumber of constraints that need to be satisfied.Suppose that the alternatives are represented byvectors x A n.The optimization problem can then be formulated as

395

maximize ( )subject to ( ) 0, 1, ,i

Dg i l

xx

5

Fuzzy goals and fuzzy constraints

Let A be a given set of possible alternatives whichcontains a solution to a decision making problemunder consideration.A fuzzy goal G is a fuzzy set on A, characterized by G:A [0,1], represents the degree to which thealternatives satisfy the specified decision goal.A fuzzy constraint C is a fuzzy set on A characterizedby C: A [0,1], constrains the solution to a fuzzyregion within the set of possible solutions.

396

Fuzzy goal

Goal: “Product concentration should be about 80%”.

397

About 80 %

mem

bers

hip

grad

e

0

0.5

1

7570 80 85 9590

6

Fuzzy constraint

Constraint: “Product concentration should benot substantially higher than 75%”.

398

Not substantiallyhigher than 75 %

mem

bers

hip

grad

e

0

0.5

1

7570 80 85 9590

Bellman and Zadeh’s model

Fuzzy decision F is a confluence of (fuzzy) decisiongoals and (fuzzy) decision constraintsBoth the decision goals and the decision constraintsshould be satisfied:

Maximizing decision (optimal decision a*)

399

( ) ( ) ( ),D G CD G C a a a a A

* argmax( ( ) ( ))G Ca A

a a a

7

Optimal fuzzy decision

Maximizing decision using min:a* = arg max D

400

mem

bers

hip

grad

e

Fuzzy Decision D

xma*

BZ model : example

401

interferon dosage [mg]maximizing decision

Small dosage (fuzzy constraint) Large dosage (fuzzy goal)1

fuzzy decision

8

Several goals and constraints

Fuzzy goal Fj, j = 1,2,...,nFuzzy constraint Gi, i = 1,2,...,mMembership functions: Fj(x), Gi(x) : X [0,1]

Fuzzy decision (Bellman and Zadeh model):D(x) = F1(x) ... Fn(x) G1(x) ... Gm(x)

Optimal decision:

402

* arg max ( )x X

x D x

Example: form basket team

Criteria to form a good basketball team:Criterion 1: “much taller that 5 ft”Criterion 2: “pretty good shooting rate”Criterion 3: “salary is about $50k/year”Criterion 4: “able to get along with team-mates”

Each criterion i is described by a fuzzy set Ai.Decision: D = A1 A2 A3 A4

where “ ” is an aggregation operator.Important extension: criteria can be weighted.

403

9

Hierarchical aggregation

Example of hierarchical aggregation of goals andconstraints:

404

G1 G2 F1 F2

T1

T2

M

Final decision

e.g. product

e.g. arithmetic meane.g. minimum

Fuzzy goalsFuzzy constraints

Yager’s model

A special case of Bellman and Zadeh’s modelDiscrete set of alternativesMultiple decision criteriaEvaluation of alternatives for each criterion by using afuzzy set, leading to judgements (ratings, membershipvalues)Use of fuzzy aggregation operators for combining thejudgements (decision function)Decision criteria can be weightedAlternatives ordered by the decision function

405

10

Discrete choice problem

Set of alternatives A = {a1,…, an}Set of criteria C = {c1,…, cm}Judgements ij from evaluation of each alternative foreach criterion. Evaluation matrix:

Evaluations made: using membership functionsrepresenting fuzzy criteria, or by direct evaluation ofalternatives.

406

mnmm

n

n

c

caa

1

1111

1

Discrete choice problem

Weight factors denote importance of criteriaAn aggregation function (decision function) combinesweight factors and judgements for the criteria

Decision function orders the alternatives according topreferenceA higher aggregated value corresponds to a morepreferred alternative

407

1( , , ), {1, , }wj mjD j n

( ) ( )k l k lD a D a a a

11

Weighted aggregation

Weights represent relative importance of objectivefunctions and the constraintsThe problem is described by:

The solution is given by

For general fuzzy optimization with simultaneoussatisfaction of constraints, t-norms must be extendedto their weighted counterparts.

408

0 0 1 1( , ) ( , ( ), ( ), , ( ))T T Tm mD T G G Gx w w a x a x a x

*( , ) sup ( , )D Dx

x w x w

Weight factors

Weights represent the relative importance of variousconstraints and the goal within the preferencestructure of the decision makerThe higher the weight of a particular criteria, the largerits importance on the aggregation resultImportance of criteria can also be done directly in themembership functions.Normalization of weights for t-norms

409

0

1m

ii

w

12

Weighted conjunction

Minimum operator

Product operator

Hamacher t-norm

Yager t-norm

410

iwi

m

iGD )]([),(

0xwx

m

i

wi iGD

0)]([),( xwx

otherwise,1

10)(,if0

),(

0 )()(1m

i GG

i

i

iiw

Gi

D

xx

x

wx

mi ii GwD 0

2))(1(1,0max),( xwx

Application: logistic system

411

Order # 1

Order # 2

Order # 3 Order # 1Order # 2

Order # 3

Request the components

Suppliers

(Delay)

Component stock

Order stock

Scheduling

decision process

Order # 1

Poisson

Exponential

A

B

C

B

A

B

C

A

B

C

E

A

D

A

B

C

D

E

B

A

C

E

Order # 1

Order # 2

Order # 3 Order # 1Order # 2

Order # 3

Order # 1Order # 2

Order # 3

Request the components

Suppliers

(Delay)

Component stock

Order stock

Scheduling

decision process

Scheduling

decision process

Order # 1

PoissonPoisson

ExponentialExponential

A

B

C

B

A

B

C

A

B

C

A

B

C

E

A

D

A

B

C

D

E

B

A

C

E

13

Logistic system

Two criteria are considered:Priority u1 has three possible values: 0.25, 0.5and 1.Lateness u2

412

Logistic system

The aggregation of the criteria is given by

Cost function to be optimized (by meta-heuristics):

413

1( )

DO

fD x

1 2( ) (1 )D wu w ux

14

Optimization results

Optimization using ant colony optimization:

414

Priority L< 0 L = 0 L> 0 min(L) max(L)0.25 150 51 40 -15 11

fclassic 0.5 59 23 15 -12 101 75 52 21 -12 11Total 284 126 76 -15 110.25 121 86 33 -18 11

ffuzzy 0.5 54 38 7 -14 171 77 56 18 -14 15Total 252 180 58 -18 17

Fuzzy Linear Programming

Formulation of the optimization problem:fuzzy maximize cTx

x IRn

~subject to Ax b

x 0

Vectors b and c and matrix A have crisp elementsFuzzy goal: F(cTx) (call this G0(a0

Tx))Fuzzy constraints: Gi(ai

Tx), aiT is row i of A, i = 1,2,...,m

Optimal vector x* is found by

415

0

( ) ( )m

Ti i

i

D x G a x

15

Example: maximizing profit

Company makes two products:

P1 with profit $0.40 per unit.

P2 with profit $0.30 per unit.

P1 takes twice time to produce compared to P2.

Total labor time per day is 500 hours. It can beextended to 600 hours with overtime work.

Supply of material is sufficient for 400 units of bothproducts, but it can be extended to 500 units per day.

416

Maximizing profit

Objective: determine the number of units to produceper day of units P1 and P2 in order to maximize profit.Let x1 and x2 represent the number of units of theproducts P1 and P2, respectively:

fuzzy maximize z = 0.4 x1+0.3 x2x IRn

~subject to x1 + x2 400 material

~2x1 + x2 500 labor hoursx 0

417

16

Membership functions

Parameters: a0T = cT = [0.4 0.3]

a1T = [1.0 1.0]

a2T = [2.0 1.0]

418

450 500 550 600 6500

0.2

0.4

0.6

0.8

1

labor hours

mem

bers

hip

labor constraint

350 400 450 500 5500

0.2

0.4

0.6

0.8

1

amount

mem

bers

hip

materials constraint

Objective membership function

Solving two conventional linear programmingproblems, the MF for the objective has the followingparameters:

zl =130zu = 160

419

100 120 140 160 1800

0.2

0.4

0.6

0.8

1

profits

mem

bers

hip

objective

zl

zu

17

Optimization settings

The linear programming problem can be solved usingthe Simplex (Nelder-Mead) algorithmConsidering the lower limits for material and laborhours, the profit is $130 (classical LP)Four weighted aggregation operators are considered:minimum, product, Yager (s=2) and HamacherThree different weights are considered:

w0 = 1.0, w1 = 1.0, w2 = 1.0w0 = 1.0, w1 = 0.5, w2 = 1.0w0 = 1.0, w1 = 0.25, w2 = 0.5

420

Regions of optimal solutions

421

0 100 200100

200

300

400

500

x 2

minimum

0 100 200100

200

300

400

500

x 2

product

0 100 200100

200

300

400

500

x 2

Yager

0 100 200100

200

300

400

500

x1 x1

x 2

Hamacher

18

Optimal solutions

422

146.4532466466660.50.251.0Hamacher

150.0522489489330.50.251.0Yager

147.3515479479360.50.251.0Product

150.6552477477750.50.251.0Minimum

144.0520460460601.00.51.0Hamacher

145.1517467467501.00.51.0Yager

143.1499466466331.00.51.0Product

147.7543467467761.00.51.0Minimum

142.6528449449791.01.01.0Hamacher

140.8522443443791.01.01.0Yager

140.0500450450501.01.01.0Product

145.05504503501001.01.01.0Minimum

ProfitLaborMaterialx2x1w2w1w0Aggregation

Solutions with Yager operator

423

– non-weighted, – more weight on labor,* – more weight on profit.

0 50 100 150 200100

150

200

250

300

350

400

450

500

x1

x 2

Yager

19

Model-Based Predictive Control

424

reference rHc

Hp

predicted output y

control input u

past output y

k-1 k+1 ... k H+ c... k H+ p

Optimisation Issues

In general, non-convex optimization problemSearch space increases exponentially with the numberof decision stages consideredRough quantization of the input and the state spacedesiredSearch algorithms:

Dynamic programmingBranch-and-boundEvolutionary methods (e.g. genetic algorithms)

425

20

Search for optimal solution

426

...

...

...

. . .. . .

. . .

. . .

. . .

. . .

...

...

. . .

. . .

. . .

. . .

. . .

. . .

k+Hpk k+1 k+2 . . . . . .

x( )k

1

2

N

y k+( 2)y k+( 1) y k+H( )c. . . y k+H( )p. . .

...

...

...

...

Predictive control with fuzzy criteria

427

ProcessDecision makingalgorithm

Model Goals andconstraints

Humanknowledge

Controller

ru y

d

21

Fuzzy objective functions

Use fuzzy goals and fuzzy constraints (fuzzy criteria).

428

0

1

e11

0 0

y u

y k+i( )e k+i( ) u k+i( 1)Ky

+Ky-

Sy- Sy

+ Ku+Ku

-Hu

- Hu+OuKe

- Ke+0

Fuzzy objective functions

Policy generates discrete control actions:

= u(k),...,u(k + Hp – 1) .

Fuzzy criterion jl – denotes criterion l at time stepk + j, with l = 1,...,m and j = 1,..., Hp.

jl– membership value representing satisfaction of

decision criteria after applying control action u(k + j).Total number of decision criteria: M = m Hp.

429

22

Aggregation of fuzzy criteria

Membership value for the control sequence isobtained by using aggregation operators , g and cto combine the membership values

jl:

430

)()(

)()(

)()(

)1(1

2)1(2221

1)1(1111

mpHgnpHgnpHpH

mgngn

mgngn

ccgg

ccgg

ccgg

Optimal decision

Usually: same operator is used for all aggregations.Possible to use weights for each criterion.Optimal sequence of control actions * is found by themaximization of :

431

)1(,),(

* maxargpHkk uu

23

Example: container gantry crane

432

water side

l

Containerload

M

Trolley q

x

Containership

T2

T1

Container gantry crane

2 inputsT1 – torque of motor 1T2 – torque of motor 2

3 outputs to be controlled:x – position of the trolleyh – length of the rope

– swing of the load

Criteria: errors for the horizontal displacement ex, ropelength el and swing angle e .Weights can be considered for each criterion.

433

24

Container gantry crane

Control goals: position trolley at horizontal position xwhile reducing the swing angle to zero and reducetransport time.Two sets of d.c. motors: one for trolley motion, one forhoisting motion.System studied by a simulation model including themodel of motors, etc.Maximum trolley speed: 3.2 ms–1.Maximum trolley acceleration: 0.8 ms –2.

434

Control results without weights

435

0 10 20 30 40 50 60 70 80 900

20

40

60

Pos

ition

[m]

Container gantry crane controller

0 10 20 30 40 50 60 70 80 90

10

12

14

16

Rop

e le

nght

[m]

0 10 20 30 40 50 60 70 80 90-2

-1

0

1

2

Time [sec]

Sw

ing

[deg

]

25

Control results with weights

436

0 10 20 30 40 50 60 70 80 900

20

40

60P

ositi

on [m

]

Container gantry crane controller

0 10 20 30 40 50 60 70 80 90

10

12

14

16

Rop

e le

nght

[m]

0 10 20 30 40 50 60 70 80 90-2

-1

0

1

2

Time [sec]

Sw

ing

[deg

]

Application: fault isolation

437

y

F1 Fn

y

System

Model NormalOperation

Inputs Outputs

FaultDetection

ModelFault 1

ModelFault n

F1y

Fny

FaultIsolationFault

Information

y

FDI

-

-

- y

…

......

26

Fault isolation

At each time instant k, a residual ij is computed foreach fault i and for each output j:

438

ˆ( )ij ij ijk y y

i1 i2

i

i1= 0y1

y2

i2= 0

Fault isolation

Let di(k) be a fuzzy decision factor:

where t is a triangular norm (fuzzy intersection).A vector of fuzzy decision factors is given by:

Fault is isolated when for tk consecutive time instants:

di > T

439

1( ) ( , , )

i imi id k t

1 2( ) [ ( ) ( ) ( )]nD k d k d k d k

27

Example: pneumatic industrial valve

440

Possible faults

441

Faults Description

F1 Valve clogging

F2 Valve seat erosion

F3 Internal leakage

F4 Medium evaporation or critical flow

F5 Flow rate sensor fault

28

Fault detection and isolation

Detection and isolation times (in seconds)

442

Faults Abrupt faults Incipient faults

detection isolation detection isolation

F1 51 155 519 750

F2 51 114 114 449

F3 51 115 156 394

F4 51 52 51 183

F5 51 133 85 125