FUZZY DECISION
MAKING
Decision Making
( A, , , , D )
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 A
defines the solution space.
D – decision function D :
. Reflects the
preference structure of the decision maker.
389
1
Decision function
The decision function D incorporates the goals of the
decision maker. It induces a preference ordering on
the set of consequences such that
i
where
i
,
, and
j
consequence
j
i
iff D ( i )
D( j )
is the preference relation, i.e.,
is preferred to consequence
j
390
Example
A person is driving a car on a cold winter day down a
road. Suddenly, a dog jumps in front of the car. The
driver can decide between two actions: he can break
hard applying full power to the brakes, or he can brake
soft knowing that the car cannot come to a stop
before a collision with the animal.
{slippery road, not slippery road}
A {brake soft, brake hard}
391
2
Example
hit dog slight ly
D
har d
slip and hit tree
D
of t
b rak e s
hit dog slight ly
D
do not hit anything
D
s of t
b rak e
brak
e
ro a
d
is n
ot s
lipp
e ry
b rak
states
e h ar
d
alternatives A
solution
set
consequences
D
1
2
3
4
preference
ordering
392
Example
Multidimensional consequences in multicriteria
decision making
Car
Damage to
animal
tree
hit dog slightly
minor
minor
none
slip and hit tree
major
none
minor
do not hit
anything
none
none
none
Consequence
393
3
Fuzzy decisions
Consider that the states of the environment are
known to the decision maker. In this case, the
elements of can be incorporated in the set A.
Thus, is a mapping : A
.
Best decision alternative a* for n decision criteria:
a*
max D ( (a))
a A
(a)
1
(a ),
,
n
(a)
394
Decision problem
The set A cannot always be defined explicitly. This set
can be defined implicitly by the specification of a
number of constraints that need to be satisfied.
Suppose that the alternatives are represented by
n.
vectors x A
The optimization problem can then be formulated as
maximize D ( x)
subject to g i (x) 0, i 1,
,l
395
4
Fuzzy goals and fuzzy constraints
Let A be a given set of possible alternatives which
contains a solution to a decision making problem
under consideration.
A fuzzy goal G is a fuzzy set on A, characterized by G:
A
[0,1], represents the degree to which the
alternatives satisfy the specified decision goal.
A fuzzy constraint C is a fuzzy set on A characterized
by C: A
[0,1], constrains the solution to a fuzzy
region within the set of possible solutions.
396
Fuzzy goal
Goal: “Product concentration should be about 80%”.
membership grade
1
About 80 %
0.5
0
70
75
80
85
90
95
397
5
Fuzzy constraint
Constraint: “Product concentration should be
not substantially higher than 75%”.
membership grade
1
Not substantially
higher than 75 %
0.5
0
75
70
80
85
90
95
398
Bellman and Zadeh’s model
Fuzzy decision F is a confluence of (fuzzy) decision
goals and (fuzzy) decision constraints
Both the decision goals and the decision constraints
should be satisfied:
D G
C
D
(a)
G
(a)
C
(a), a
A
Maximizing decision (optimal decision a*)
a*
arg max(
G
(a)
C
( a ))
a A
399
6
Optimal fuzzy decision
membership grade
Maximizing decision using min:
a* = arg max
D
Fuzzy Decision
D
m
a*
x
400
BZ model : example
Small dosage (fuzzy constraint)
Large dosage (fuzzy goal)
1
fuzzy decision
maximizing decision
interferon dosage [mg]
401
7
Several goals and constraints
Fuzzy goal Fj, j = 1,2,...,n
Fuzzy constraint Gi, i = 1,2,...,m
Membership 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:
x*
arg max D ( x)
x X
402
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
8
Hierarchical aggregation
Example of hierarchical aggregation of goals and
constraints:
Final decision
e.g. product
T2
e.g. arithmetic mean
e.g. minimum
T1
G1
M
G2
Fuzzy constraints
F1
F2
Fuzzy goals
404
Yager’s model
A special case of Bellman and Zadeh’s model
Discrete set of alternatives
Multiple decision criteria
Evaluation of alternatives for each criterion by using a
fuzzy set, leading to judgements (ratings, membership
values)
Use of fuzzy aggregation operators for combining the
judgements (decision function)
Decision criteria can be weighted
Alternatives ordered by the decision function
405
9
Discrete choice problem
Set of alternatives A = {a1,…, an}
Set of criteria C = {c1,…, cm}
Judgements ij from evaluation of each alternative for
each criterion. Evaluation matrix:
a
a
1
n
c
1
11
1n
c
m
m1
mn
Evaluations made: using membership functions
representing fuzzy criteria, or by direct evaluation of
alternatives.
406
Discrete choice problem
Weight factors denote importance of criteria
An aggregation function (decision function) combines
weight factors and judgements for the criteria
Dw (
1j
,
,
mj
),
j {1,
, n}
Decision function orders the alternatives according to
preference
A higher aggregated value corresponds to a more
preferred alternative
D(ak )
D(al )
ak
al
407
10
Weighted aggregation
Weights represent relative importance of objective
functions and the constraints
The problem is described by:
D (x, w ) T ( w, G0 (aT0 x), G1 (a1T x),
, Gm (aTm x))
The solution is given by
D (x* , w ) sup D(x, w )
x
For general fuzzy optimization with simultaneous
satisfaction of constraints, t-norms must be extended
to their weighted counterparts.
408
Weight factors
Weights represent the relative importance of various
constraints and the goal within the preference
structure of the decision maker
The higher the weight of a particular criteria, the larger
its importance on the aggregation result
Importance of criteria can also be done directly in the
membership functions.
Normalization of weights for t-norms
m
wi 1
i 0
409
11
Weighted conjunction
m
Minimum operator
D ( x, w )
Product operator
D ( x, w )
[Gi (x)]wi
i 0
m
[Gi (x)]wi
i 0
0 if i, Gi (x) 0
Hamacher t-norm
1
D(x, w )
1
Yager t-norm
D (x, w )
1 Gi (x)
m
i 0 wi Gi ( x)
max 0,1
, otherwise
Gi (x)) 2
m
i 0 wi (1
410
Application: logistic system
Poisson
Suppliers
(Delay)
Order # 1
A
B
Request the components
Component stock
A
C
A
B
B
Order # 2
E
Exponential
C
A
B
C
E
Order # 3
A
C
D
E
Order # 1
Order stock
Order # 3
Order # 2
Order # 1
A
Scheduling
B
decision process
C
D
411
12
Logistic system
Two criteria are considered:
Priority u1 has three possible values: 0.25, 0.5
and 1.
Lateness u2
412
Logistic system
The aggregation of the criteria is given by
D ( x)
wu1 (1 w) u2
Cost function to be optimized (by meta-heuristics):
f
1
D ( x)
O
D
413
13
Optimization results
Optimization using ant colony optimization:
fclassic
ffuzzy
Priority
L< 0
L=0
L> 0
min(L)
max(L)
0.25
150
51
40
-15
11
0.5
59
23
15
-12
10
1
75
52
21
-12
11
Total
284
126
76
-15
11
0.25
121
86
33
-18
11
0.5
54
38
7
-14
17
1
77
56
18
-14
15
Total
252
180
58
-18
17
414
Fuzzy Linear Programming
Formulation of the optimization problem:
cTx
fuzzy maximize
x
IRn
subject to
~
Ax b
x 0
Vectors b and c and matrix A have crisp elements
Fuzzy goal: F(cTx) (call this G0(a0Tx))
Fuzzy constraints: Gi(aiTx), aiT is row i of A, i = 1,2,...,m
Optimal vector x* is found by
m
Gi (aiT x)
D ( x)
i 0
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14
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 be
extended to 600 hours with overtime work.
Supply of material is sufficient for 400 units of both
products, but it can be extended to 500 units per day.
416
Maximizing profit
Objective: determine the number of units to produce
per day of units P1 and P2 in order to maximize profit.
Let x1 and x2 represent the number of units of the
products P1 and P2, respectively:
fuzzy maximize z = 0.4 x1+0.3 x2
x IRn
~
subject to
x1 + x 2
400
material
~
2x1 + x2
x 0
500
labor hours
417
15
Membership functions
Parameters:
a0T = cT = [0.4 0.3]
a1T = [1.0 1.0]
a2T = [2.0 1.0]
labor constraint
1
1
materials constraint
0.8
membership
membership
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
450
500
550
600
0
350
650
labor hours
400
450
500
550
amount
418
Objective membership function
Solving two conventional linear programming
problems, the MF for the objective has the following
parameters:
objective
zl =130
zu = 160
zu
1
membership
0.8
0.6
0.4
0.2
0
100
120
zl
140
160
180
profits
419
16
Optimization settings
The linear programming problem can be solved using
the Simplex (Nelder-Mead) algorithm
Considering the lower limits for material and labor
hours, the profit is $130 (classical LP)
Four weighted aggregation operators are considered:
minimum, product, Yager (s=2) and Hamacher
Three different weights are considered:
w0 = 1.0, w1 = 1.0, w2 = 1.0
w0 = 1.0, w1 = 0.5, w2 = 1.0
w0 = 1.0, w1 = 0.25, w2 = 0.5
420
Regions of optimal solutions
500
minimum
500
400
x2
x2
400
300
300
200
100
0
200
100
200
100
0
100
200
Hamacher
500
500
400
400
x2
x2
Yager
300
300
200
100
0
product
200
100
x1
200
100
0
100
x1
200
421
17
Optimal solutions
Aggregation w0
w1
w2
x1
x2
Material
Labor
Profit
Minimum
1.0
1.0
1.0
100
350
450
550
145.0
Product
1.0
1.0
1.0
50
450
450
500
140.0
Yager
1.0
1.0
1.0
79
443
443
522
140.8
Hamacher
1.0
1.0
1.0
79
449
449
528
142.6
Minimum
1.0
0.5
1.0
76
467
467
543
147.7
Product
1.0
0.5
1.0
33
466
466
499
143.1
Yager
1.0
0.5
1.0
50
467
467
517
145.1
Hamacher
1.0
0.5
1.0
60
460
460
520
144.0
Minimum
1.0
0.25
0.5
75
477
477
552
150.6
Product
1.0
0.25
0.5
36
479
479
515
147.3
Yager
1.0
0.25
0.5
33
489
489
522
150.0
Hamacher
1.0
0.25
0.5
66
466
466
532
146.4
422
Solutions with Yager operator
– non-weighted, – more weight on labor,
* – more weight on profit.
Yager
500
450
400
x
2
350
300
250
200
150
100
0
50
x1 100
150
200
423
18
Model-Based Predictive Control
Hp
Hc
reference r
past output y
predicted output ^y
control input u
k-1
k+1 ...
k+Hc
...
k+Hp
424
Optimisation Issues
In general, non-convex optimization problem
Search space increases exponentially with the number
of decision stages considered
Rough quantization of the input and the state space
desired
Search algorithms:
Dynamic programming
Branch-and-bound
Evolutionary methods (e.g. genetic algorithms)
425
19
Search for optimal solution
...
y( k+2)
y(k+1)
y(k+Hp)
.. .
y( k+Hc)
...
...
.
.
.
...
.. .
.
.
.
.
.
.
. ..
.
.
.
1
...
x(k)
.
.
.
2 .
.
.
...
...
N
. ..
...
.
.
.
. ..
.
.
.
.
.
.
...
k
k+1
k+2
...
k+Hp
...
426
Predictive control with fuzzy criteria
Controller
Model
r
Human
knowledge
Goals and
constraints
Decision making
algorithm
u
Process
d
y
427
20
Fuzzy objective functions
Use fuzzy goals and fuzzy constraints (fuzzy criteria).
1
e
0
-
1
1
y
u
0
+
Ke 0 Ke
-
Ky
e(k+i )
-
Sy
+
Sy
+
Ky
0
-
Ku
y(k+i)
+
Hu Ou Hu
+
Ku
u(k+i 1)
428
Fuzzy objective functions
Policy
generates discrete control actions:
= u(k),...,u(k + Hp – 1)
.
Fuzzy criterion jl – denotes criterion l at time step
k + j, with l = 1,...,m and j = 1,..., Hp.
– membership value representing satisfaction of
jl
decision criteria after applying control action u(k + j).
Total number of decision criteria: M = m Hp.
429
21
Aggregation of fuzzy criteria
Membership value for the control sequence is
obtained by using aggregation operators , g and
to combine the membership values jl:
(
(
(
11
21
H p1
g
g
g
1ng
g
g
)
2 ng
g
(
)
H p ng
1( ng 1)
(
)
2 ( ng 1)
(
c
c
c
H p ( ng 1)
1m
c
c
)
2m
c
c
)
H pm
)
430
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 the
maximization of :
*
arg max
u ( k ), ,u ( k H p 1)
431
22
Example: container gantry crane
T2
x
Trolley
T1
q
Container
load
l
M
Container
ship
water side
432
Container gantry crane
2 inputs
T1 – torque of motor 1
T2 – torque of motor 2
3 outputs to be controlled:
x – position of the trolley
h – length of the rope
– swing of the load
Criteria: errors for the horizontal displacement ex, rope
length el and swing angle e .
Weights can be considered for each criterion.
433
23
Container gantry crane
Control goals: position trolley at horizontal position x
while reducing the swing angle to zero and reduce
transport time.
Two sets of d.c. motors: one for trolley motion, one for
hoisting motion.
System studied by a simulation model including the
model of motors, etc.
Maximum trolley speed: 3.2 ms–1.
Maximum trolley acceleration: 0.8 ms –2.
434
Control results without weights
Container gantry crane controller
Position [m]
60
40
20
0
0
10
20
30
40
50
60
70
80
90
10
20
30
40
50
60
70
80
90
10
20
30
40
50
Time [sec]
60
70
80
90
Rope lenght [m]
16
14
12
10
0
Swing [deg]
2
1
0
-1
-2
0
435
24
Control results with weights
Container gantry crane controller
Position [m]
60
40
20
0
0
10
20
30
40
50
60
70
80
90
10
20
30
40
50
60
70
80
90
10
20
30
40
50
Time [sec]
60
70
80
90
Rope lenght [m]
16
14
12
10
0
Swing [deg]
2
1
0
-1
-2
0
436
Application: fault isolation
FDI
Fault
Isolation
Fault
Information
Model
Fault n
..
.
Model
Fault 1
Model Normal
Operation
Inputs
System
y
-
y
F1
ŷFn
ŷF1
.
..
-
…
Fn
-y
y
Fault
Detection
Outputs
437
25
Fault isolation
At each time instant k, a residual ij is computed for
each fault i and for each output j:
ij
(k )
yˆ ij
yij
i
i1
i2
y2
y1
=0
i1
i2
=0
438
Fault isolation
Let di(k) be a fuzzy decision factor:
i
d i (k ) t (
i1
,
,
im
)
where t is a triangular norm (fuzzy intersection).
A vector of fuzzy decision factors is given by:
D( k ) [d1 ( k ) d 2 ( k )
d n ( k )]
Fault is isolated when for tk consecutive time instants:
di > T
439
26
Example: pneumatic industrial valve
440
Possible faults
Faults
Description
F1
Valve clogging
F2
Valve seat erosion
F3
Internal leakage
F4
Medium evaporation or critical flow
F5
Flow rate sensor fault
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27
Fault detection and isolation
Detection and isolation times (in seconds)
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
442
28