Crisp and Fuzzy

8/2/2019 Crisp and Fuzzy

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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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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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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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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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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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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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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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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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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


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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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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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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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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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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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Section 8: Fuzzy programming versus goal programming 33


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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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Section 9: Multiple Objective Programs 40


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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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Section 10: Application functions for MOP problems 42


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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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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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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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Then we can build the following application functions


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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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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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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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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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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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in the MLP problem,


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in the MLP problem,

max{2x1 + x2, x1 2x2}


3x1 + x2 6,

x1, x2 0.


h1(f1(x)) =

1 iff1(x) 7

1 7 f1(x)

3 if4 f1(x) 7

0 iff1(x) 4

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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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That is,


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,

max min{h1(2x1 + x2), h2(x1 2x2)}


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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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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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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Crisp and Fuzzy

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