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Institute for Advanced Management Systems ResearchDepartment of Information Technologies
Faculty of Technology, Abo Akademi University
The Past is Crisp, but the Future isFuzzy - Tutorial
Robert Fuller
Directory
Table of Contents Begin Article
c 2010 [email protected] 25, 2010
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Table of Contents
1. Triangular and trapezoidal fuzzy numbers
2. Material implication
3. Fuzzy implications
4. The theory of approximate reasoning
5. Crisp and fuzzy relations
6. Simplified fuzzy reasoning schemes
7. Tsukamotos and Sugenos fuzzy reasoning scheme
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Table of Contents (cont.) 3
8. Fuzzy programming versus goal programming
9. Multiple Objective Programs
10. Application functions for MOP problems
11. The efficiency of compromise solutions
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Section 1: Triangular and trapezoidal fuzzy numbers 4
1. Triangular and trapezoidal fuzzy numbers
A fuzzy set A of the real line R is defined by its membership function (de-noted also by A) A : R [0, 1]. If x R then A(x) is interpreted as thedegree of membership ofx in A.
Figure 1: Possible membership functions for monthly small salary and
big salary.
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Section 1: Triangular and trapezoidal fuzzy numbers 5
Ifx0 is the amount of the salary then x0 belongs to fuzzy set
A1 = small
with degree of membership
A1(x0) =
1
x0 2000
4000if2000 x0 6000
0 otherwise
and to
A2 = big
with degree of membership
A2(x0) =
1 ifx0 6000
1 6000 x0
4000if2000 x0 6000
0 otherwise
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Section 1: Triangular and trapezoidal fuzzy numbers 6
Definition 1.1. A fuzzy set A is called triangular fuzzy number with peak(or center) a, left width > 0 and right width > 0 if its membership
function has the following form
A(t) =
1 a t
ifa t a
1 t a
ifa t a +
0 otherwise
and we use the notation A = (a,,). The support ofA is (a , b + ).A triangular fuzzy number with centera may be seen as a fuzzy quantity
x is close to a orx is approximately equal to a.
Definition 1.2. A fuzzy setA is called trapezoidal fuzzy number with toler-ance interval [a, b], left width and right width if its membership function
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Section 1: Triangular and trapezoidal fuzzy numbers 7
1
aa-! a+"
Figure 2: A triangular fuzzy number.
has the following form
A(t) =
1 a t
ifa t a
1 ifa t b
1 t b
ifa t b +
0 otherwise
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Section 1: Triangular and trapezoidal fuzzy numbers 8
and we use the notation A = (a,b,,).
A trapezoidal fuzzy number may be seen as a fuzzy quantity
x is approximately in the interval [a, b].
1
aa-!
b+"
b
Figure 3: Trapezoidal fuzzy number.
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Section 2: Material implication 9
2. Material implication
Let p =
x is in A
and q =
y is in B
are crisp propositions, where A andB are crisp sets for the moment.
The full interpretation of the material implication p q is that: the degreeof truth ofp q quantifies to what extend q is at least as true as p, i.e.
(p q) =
1 if(p) (q)0 otherwise
(p) (q) (p q)
1 1 10 1 1
0 0 1
1 0 0
Table 1: Truth table for the material implication.
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Section 3: Fuzzy implications 10
3. Fuzzy implications
Consider the implication statement
if pressure is high then volume is small
Figure 4: Membership function for big pressure.
The membership function of the fuzzy set A, big pressure, can be interpreted
as
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Section 3: Fuzzy implications 11
1 is in the fuzzy set big pressure with grade of membership 0
4 is in the fuzzy set big pressure with grade of membership 0.75
x is in the fuzzy set big pressure with grade of membership 1, x 5
A(u) =
1 ifu 5
1 5 u4
if1 u 5
0 otherwise
The membership function of the fuzzy set B, small volume, can be inter-
preted as
B(v) =
1 ifv 1
1 v 1
4if1 v 5
0 otherwise
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Section 3: Fuzzy implications 12
Figure 5: Membership function for small volume.
5 is in the fuzzy set small volume with grade of membership 0
2 is in the fuzzy set small volume with grade of membership 0.75
x is in the fuzzy set small volume with grade of membership 1, x 1
Ifp is a proposition of the form x is A where A is a fuzzy set, for example,big pressure and q is a proposition of the form y is B for example, smallvolume then we define the implication p q as
A(x) B(y)
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Section 3: Fuzzy implications 13
For example,
x is big pressure y is small volume A(x) B(y)
Remembering the full interpretation of the material implication
p q =
1 if(p) (q)0 otherwise
We can use the definition
A(x) B(y) = 1 ifA(x) B(y)0 otherwise
4 is big pressure 1 is small volume = 0.75 1 = 1
The most often used fuzzy implication operators are listed in the following
table.
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Section 3: Fuzzy implications 14
Name Definition
Early Zadeh x y = max{1 x, min(x, y)}ukasiewicz x y = min{1, 1 x + y}
Mamdani x y = min{x, y}
Larsen x y = xy
Standard Strict x y = 1 ifx y0 otherwise
Godel x y =
1 ifx yy otherwise
Gaines x y =
1 ifx yy/x otherwise
Kleene-Dienes x y = max{1 x, y}Kleene-Dienes-ukasiewicz x y = 1 x + xy
Yager x y = yx
Table 2: Fuzzy implication operators.
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4. The theory of approximate reasoning
In 1979 Zadeh introduced the theory of approximate reasoning. This theoryprovides a powerful framework for reasoning in the face of imprecise and
uncertain information.
Entailment rule:
Mary is very young
very young young
Mary is young
Conjuction rule:
pressure is not very high
pressure is not very low
pressure is not very high and not very low
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Section 4: The theory of approximate reasoning 16
Disjunction rule:
pressure is not very high
or pressure is not very low
pressure is not very high or not very low
Projection rule:
(x, y) is close to (3, 2)
x is close to 3
(x, y) is close to (3, 2)
y is close to 2
How to make inferences in fuzzy environment?
1: if pressure is BIG then volume is SMALL
observation pressure is 4
conclusion volume is ?
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Section 4: The theory of approximate reasoning 17
Figure 6: BIG(4) = SMALL(2) = 0.75.
1: if pressure is BIG then volume is SMALL
observation pressure is 4
conclusion volume is 2
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5. Crisp and fuzzy relations
A classical relation can be considered as a set of tuples, where a tuple is anordered pair. A binary tuple is denoted by (u, v), an example of a ternarytuple is (u,v,w) and an example ofn-ary tuple is (x1, . . . , xn).
Definition 5.1. LetX andY be nonempty sets. A fuzzy relation R is a fuzzysubset ofX Y. IfX = Y then we say thatR is a binary fuzzy relation inX.
Let R be a binary fuzzy relation on R. Then R(u, v) is interpreted as thedegree of membership of(u, v) in R.
Example 5.1. A simple example of a binary fuzzy relation on U = {1, 2, 3},called approximately equal can be defined as
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S ti 5 C i d f l ti 19
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Section 5: Crisp and fuzzy relations 19
R(1, 1) = R(2, 2) = R(3, 3) = 1,
R(1, 2) = R(2, 1) = R(2, 3) = R(3, 2) = 0.8,R(1, 3) = R(3, 1) = 0.3.
The membership function ofR is given by
R(u, v) =
1 ifu = v0.8 if|u v| = 1
0.3 if|u v| = 2
In matrix notation it can be represented as
R =
1 0.8 0.30.8 1 0.8
0.3 0.8 1
Fuzzy relations are very important because they can describe interactionsbetween variables.
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6. Simplified fuzzy reasoning schemes
Suppose that we have the following rule base
1: if x is A1 then y is z1also
2: if x is A2 then y is z2. . . . . . . . . . . .
n: if x is An then y is zn
fact: x is x0
action: y is z0
where A1, . . . , An are fuzzy sets.
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Section 6: Simplified fuzzy reasoning schemes 21
Suppose further that our data base consists of a single fact x0. The problemis to derive z0 from the initial content of the data base, x0, and from thefuzzy rule base = {1, . . . , n}.
1: if salary is small then loan is z1
also
2: if salary is big then loan is z2fact: salary is x0
action: loan is z0
A deterministic rule base can be formed as follows
1: if 2000 s 6000 then loan is max 1000
3: if s 6000 then loan is max 2000
4: if s 2000 then no loan at all
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Section 6: Simplified fuzzy reasoning schemes 22
Figure 7: Discrete causal link between salary and loan.
The data base contains the actual salary, and then one of the rules is applied
to obtain the maximal loan can be obtained by the applicant.
In fuzzy logic everything is a matter of degree.
Ifx is the amount of the salary then x belongs to fuzzy set
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Section 6: Simplified fuzzy reasoning schemes 23
A1 = small with degree of membership 0 A1(x) 1
A2 = big with degree of membership 0 A2(x) 1
Figure 8: Membership functions for small and big.
In fuzzy rule-based systems each rule fires.
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Section 6: Simplified fuzzy reasoning schemes 24
The degree of match of the input to a rule (wich is the firing strength) is
the membership degree of the input in the fuzzy set characterizing the an-
tecedent part of the rule.
The overall system output is the weighted average of the individual rule
outputs, where the weight of a rule is its firing strength with respect to the
input.
To illustrate this principle we consider a very simple example mentionedabove
1: if salary is small then loan is z1
also2: if salary is big then loan is z2
fact: salary is x0
action: loan is z0
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Section 6: Simplified fuzzy reasoning schemes 25
Then our reasoning system is the following
input to the system is x0
the firing level of the first rule is 1 = A1(x0)
the firing level of the second rule is 2 = A2(x0)
the overall system output is computed as the weghted average of theindividual rule outputs
z0 =1z1 + 2z2
1 + 2
that isz0 =
A1(x0)z1 + A2(x0)z2A1(x0) + A2(x0)
A1(x0) = 1 (x0 2000)/4000 if2000 x0 6000
0 otherwise
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p y g
Figure 9: Example of simplified fuzzy reasoning.
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p y g
A2(x0) =
1 ifx0 60001 (6000 x0)/4000 if2000 x0 6000
0 otherwise
It is easy to see that the relationship
A1(x0) + A2(x0) = 1
holds for all x0 2000.
It means that our system output can be written in the form.
z0 = 1z1 + 2z2 = A1(x0)z1 + A2(x0)z2
that is,z0 =
1
x0 2000
4000
z1 +
1
6000 x04000
z2
if2000 x0 6000.
And z0 = 1 ifx0 6000. And z0 = 0 ifx0 2000.
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p y g
Figure 10: Input/output function derived from fuzzy rules.
The (linear) input/oputput relationship is illustrated in figure 10.
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Section 7: Tsukamotos and Sugenos fuzzy reasoning scheme 29
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7. Tsukamotos and Sugenos fuzzy reasoning scheme
Tsukamotos reasoning scheme
1 : ifx is A1 and y is B1 then z is C1
also
2 : ifx is A2 and y is B2 then z is C2
fact : x is x0 and y is y0
cons. : z is z0
Sugeno and Takagi use the following architecture
1 : ifx is A1 and y is B1 then z1 = a1x + b1y
also
2 : ifx is A2 and y is B2 then z2 = a2x + b2y
fact : x is x0 and y is y0
cons.: z
0
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Figure 11: An illustration of Tsukamotos inference mechanism. The firing
level of the first rule: 1 = min{A1(x0), B1(y0)} = min{0.7, 0.3} =0.3, The firing level of the second rule: 2 = min{A2(x0), B2(y0)} =min{0.6, 0.8} = 0.6, The crisp inference: z0 = (8 0.3 + 4 0.6)/(0.3 +0.6) = 6.
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Figure 12: Example of Sugenos inference mechanism. The overall system
output is computed as the firing-level-weighted average of the individual
rule outputs: z0 = (5 0.2 + 4 0.6)/(0.2 + 0.6) = 4.25.
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8. Fuzzy programming versus goal programming
Consider the following simple linear program
x min,
subject to x 1, x R
What if the decision makers aspiration level (or goal) is
b0 = 0.5?
The goal is set outside of the conceivable values of the objective function
under given constraint.
The linear inequality system
x 0.5
x 1
does not have any solution.
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In goal programming we are searching for a solution from the decision
set, which minimizes the distance between the goal and the decision set.
That is,
|x 0.5| min, subject to x 1, x R
The unique solution is x = 1.
In fuzzy programming we are searching for a solution that might not even
belong to the decision set, and which simultaneously minimizes the (fuzzy)
distance between the decision set and the goal.
We want to be as close as possible to the goal and to the constraints.
Depending on the definition of closeness, the fuzzy version can have the
form
max min{|x 0.5|, |x 1|}, subject to 1/2 x 1.
The unique solution is x = 0.75.
The fuzzy problem can be stated as: Find an x R such that
{ x is as close as poss. to 0.5 and x is as close as poss. to 1}
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Figure 13: Illustration of the optimal solution. By using the minimum oper-
ator to aggregate the fuzzy statements x is close to 0.5 and x is close to 1
we get that the optimal solution is x = 0.75.
In our case (see Figure 13),
max
1
|x 0.5|
1/2, 1
|x 1|
1/2
, subject to 1/2 x 1.
If we use the minimum operator to aggregate the objective functions then
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we have the following single-objective problem
max min1 |x 0.5|
1/2
, 1 |x 1|
1/2, subject to 1/2 x 1.
Which - in this very special and simple case - can be written in the form,
max min{|x 0.5|, |x 1|}, subject to 1/2 x 1,
which has a unique optimal solution x = 0.75.
As an example consider the following two-variable linear program
x1 + x2 max
subject to
x1 1x2 1
x1, x2 R,
What if the decision makers aspiration level is b0 = 3?
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The aspiration level can not be reached since the maximal value of the ob-
jective function is equal to two?
Figure 14: A simple LP. The unique optimal solution is (1, 1) and the opti-mal value of the objective function is 2.
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Figure 15: The desired value of the objective function is set to 3. This value,
however, is unattainable on the decision set [0, 1] [0, 1].
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Figure 16: An illustration of the fuzzy LP. The optimal solution is outside
of the decision space [0, 1] [0, 1]. It is - with certain fuzzy coefficients - asclose as possible to the goal and to the constraints.
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9. Multiple Objective Programs
Consider a multiple objective program (MOP)
maxxX
f1(x), . . . , f k(x)
where fi : Rn R, i = 1, . . . , k are objective functions, Rk is the criterion
space, x Rn is the decision variable, Rn is the decision space, X Rn
is called the set of feasible alternatives. The image ofX inRk
, denoted byZX , i.e. the set offeasible outcomes is defined as
ZX = {z Rk|zi = fi(x), i = 1, . . . , k , x X}.
MOP problems may be interpreted as a synthetical notation of a conjuction
statement maximize jointly all objectives: maximize the first objective andmaximize the second objective. A good compromise solution to MOP is
defined as an x [0, 1] [0, 1] being as good as possible for the wholeset of objectives.
Definition 9.1. An x X is said to be efficient (or nondominated or
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Pareto-optimal) for the MOP iff there exists no y X such that
fi(y) fi(x)
for all i with strict inequality for at least one i. The set of all Pareto-optimalsolutions will be denoted by X.
Consider the following Multiple Objective Linear Program (MLP)
{x1 + x2, x1 x2} maxsubject to
x X = {x R2 | 0 x1, x2 1}
The Pareto optimal solutions (the north-east boundary of the image of the
decision space) to this problem are X = {(1, x2) | x2 [0, 1]}.
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Figure 17: The decision space is [0, 1] [0, 1].
10. Application functions for MOP problems
An application function hi for the MOP
maxxXf1(x), . . . , f k(x),
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Figure 18: The image of the decision space. Sometimes called the criterion
space.
is defined as hi : R [0, 1], where hi(t) measures the degree of fulfillmentof the decision makers requirements about the i-th objective by the valuet. Suppose that the decision maker has some preference parameters, forexample
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Figure 19: Explanation of the image of the decision space.
reference points which represents desirable leveles on each criterion
reservation levels which represent minimal requirements on each cri-
terion
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Figure 20: The problem: {x1 + x2, x1 x2} max; subject to x1, x2
[0, 1]. The set of its Pareto optimal solutions is X = {(1, x2), x2 [0, 1]}.
If the value of an objective function (at the current point) exceeds his desir-
able level on this objective then he is totally satisfied with this alternative.
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Section 10: Application functions for MOP problems 45
If h h l f bj ti f ti ( t th t i t) i b l
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If, however, he value of an objective function (at the current point) is below
of his reservation level on this objective then he is absolutely not satisfied
with this alternative.
Let mi denote the value
min{fi(x)|x X}
i.e. mi is the worst possible value for the i-th objective and let Mi denote
the valuemax{fi(x)|x X}
i.e. Mi is the largest possible value for the i-th objective on X. It is clearthat the inequalities
mi fi(x) Mi,
hold for each x in X. The most commonly used linear application functionfor the i-th objective can be defined as
hi(fi(x)) = 1 Mi fi(x)
Mi mi.
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Section 10: Application functions for MOP problems 46
It is clear that h (f (x)) 0 h (f (x)) min{f (x)|x X} m
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It is clear that hi(fi(x)) = 0 hi(fi(x)) = min{fi(x)|x X} = mi,and hi(fi(x)) = 1 hi(fi(x)) = max{fi(x)|x X} = Mi.
Letr1 = m1 = min{f1(x) = x1 + x2 | 0 x1, x2 1} = 0
and
r2 = m2 = min{f2(x) = x1 x2 | 0 x1, x2 1} = 1
be the reservation levels and letR1 = M1 = max{f1(x) = x1 + x2 | 0 x1, x2 1} = 2
and
R2 = M2 = max{f2(x) = x1 x2 | 0 x1, x2 1} = 1
be the reference points for the firts and the second objectives, respectivelyin the MLP problem,
{x1 + x2, x1 x2} max
subject to
0 x1, x2 1.
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Section 10: Application functions for MOP problems 47
Then we can build the following application functions
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Then we can build the following application functions
h1(f1(x)) = h1(x1 + x2) = 1 2 (x1 + x2)
2
=x1 + x2
2
,
h2(f2(x)) = h2(x1 x2) = 1 1 (x1 x2)
2=
1 + x1 x2
2.
Consider now the MLP problem with k objective functions
maxxX
f1(x), f2(x), . . . , f k(x)
.
With the notation of
Hi(x) = hi(fi(x)),
Hi(x) may be considered as the degree of membership ofx in the fuzzy setgood solutions for the i-th objective.
Then a good compromise solution to MLP may be defined as an x Xbeing as good as possible for the whole set of objectives.
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Section 10: Application functions for MOP problems 48
Taking into consideration the nature of Hi it is quite reasonable to look for
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Taking into consideration the nature ofHi, it is quite reasonable to look forsuch a kind of solution by means of the following auxiliary problem
maxxX
H1(x), . . . , H k(x).For max
H1(x), . . . , H k(x)
may be interpreted as a synthetical notation
of a conjuction statement maximize jointly all objectives, and Hi(x) [0, 1], it is reasonable to use a t-norm T to represent the connective and. Inthis way
maxxX
H1(x), . . . , H k(x)
turns into the single-objective problem
maxxX
T(H1(x), . . . , H k(x)).
Let us suppose the decision maker chooses the minimum operator to rep-
resent his evaluation of the connective and in the problem of: maximize the
first objective and maximize the second objective.
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Section 10: Application functions for MOP problems 49
Then the original biobjective problem turns into the single-objetive LP
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Then the original biobjective problem turns into the single-objetive LP,
max minx1 + x2
2
,1 + x1 x2
2
subject to
0 x1, x2 1.
That is,
max x1 + x2
2
1 + x1 x2
2
subject to
0 x1, x2 1,
Then an optimal solution is x1
= 1, and x2
= 1 and (f1(1, 1), f2(1, 1)) =
(2, 0) is a Pareto-optimal solution to the original biobjective problem.
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Section 10: Application functions for MOP problems 50
Consider the following linear biobjective programming problem
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Consider the following linear biobjective programming problem
max{2x1 + x2, x1 2x2}
subject tox1 + x2 4,
3x1 + x2 6,
x1, x2 0.
The first objective
2x1 + x2,
attains its maximum at point (4, 0), whereas the second one
x1 2x2,
has its maximum at point (2, 0).
The Pareto optimal solutions are {(x1, 0), x1 [2, 4]}.
Let r1 = 4, and r2 = 5 be the reservation levels and let R1 = 7, R2 = 3be the reference points for the firts and the second objectives, respectively
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Section 10: Application functions for MOP problems 51
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3
2 4
(4, 0)(2, 0)
o_1
o_2
x_1
Figure 21: Illustration of the decision space and the objectives of the biob-
jective problem. Pareto optimal solutions are: {(x1, 0), x1 [2, 4]}.
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Section 10: Application functions for MOP problems 52
in the MLP problem,
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in the MLP problem,
max{2x1 + x2, x1 2x2}
subject tox1 + x2 4,
3x1 + x2 6,
x1, x2 0.
Then we can build the following application functions
h1(f1(x)) =
1 iff1(x) 7
1 7 f1(x)
3 if4 f1(x) 7
0 iff1(x) 4
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Section 10: Application functions for MOP problems 53
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h2(f2(x)) =
1 iff2(x) 3
1 3 f2(x)
2if5 f2(x) 3
0 iff2(x) 5
Let us suppose the decision maker chooses the minimum operator to rep-
resent his evaluation of the connective and in the problem of: maximize the
first objective and maximize the second objective.
Then the original biobjective problem turns into the single-objetive LP,
max min{h1(f1(x1, x2)), h2(f2(x1, x2))}
subject to
x1 + x2 4,
3x1 + x2 6,
x1, x2 0.
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Section 10: Application functions for MOP problems 54
That is,
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,
max min{h1(2x1 + x2), h2(x1 2x2)}
subject tox1 + x2 4,
3x1 + x2 6,
x1, x2 0.
Which can be written in the form,
max
subject to
h1(2x1 + x2) h2(x1 2x2)
x1 + x2 4,
3x1 + x2 6,
x1, x2 0.
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Section 10: Application functions for MOP problems 55
That is
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max
subject to
1 7 (2x1 + x2)
3
1 3 (x1 2x2)
2
x1 + x2 4,
3x1 + x2 6,
x1, x2 0.
Its optimal solution
x = (23/7, 0)
is also an efficient solution for the original biobjective problem since it lies
in the segment
{(x1, 0), x1 [2, 4]}.
The optimal values of the objective functions are 46/7 and 23/7.
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56
11. The efficiency of compromise solutions
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One of the most important questions is the efficiency of the obtained com-
promise solutions.
Theorem 11.1. Letx be an optimal solution to
maxxX
T(H1(x), . . . , H k(x))
where T is a t-norm, Hi(x) = hi(fi(x)), hi is an increasing applicationfunction, i = 1, . . . , k. Ifhi is strictly increasing on the interval [ri, Ri] fori = 1, . . . , k. Then x is efficient for the problem
maxxX
f1(x), . . . , f k(x)
if either
(i) x is unique;
(ii) T is strict andHi
(x
) = hi
(fi
(x
)) (0, 1)for
i = 1, . . . , k.
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Section 11: The efficiency of compromise solutions 57
Proof. (i) Suppose that x is not efficient. Ifx were dominated, then x
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X such thatfi(x
) fi(x)
for all i and with a strict inequality for at least one i.
Consequently, from the monotonicity ofT and hi we get
T(H1(x), . . . , H k(x
)) T(H1(x), . . . , H k(x
))
which means that x is also an optimal solution to the auxiliary problem.So x is not unique.
(ii) Suppose that x is not efficient. Ifx were dominated, then x X
such thatfi(x
) fi(x)
for all i and with a strict inequality for at least one i. Taking into considera-tion that
Hi(x) = hi(fi(x
)) (0, 1)
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Section 11: The efficiency of compromise solutions 58
for all i and T is strict, and hi is monoton increasing we get
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T(H1(x), . . . , H k(x
)) < T(H1(x), . . . , H k(x
))
which means that x
is not an optimal solution to the auxiliary problem. Sox is not efficient.
If we use linear application functions then they are strictly increasing on
[ri, Ri], and, therefore any optimal solution x to the auxiliary problem is
an efficient solution to the original MOP problem if either
(i) x is unique;
(ii) T is strict and Hi(x) (0, 1), i = 1, . . . k.
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