Institute for Advanced Management Systems Research
Department of Information Technologies
Åbo Akademi University
An Introduction to Fuzzy Linear
Programs - Tutorial
Robert Fullér
Directory
• Table of Contents
• Begin Article
c 2010
October 2, 2010
rfuller@abo.fi
Table of Contents
1. Fuzzy programming versus goal programming
2. Fuzzy numbers
3. Measures of possibility
4. Zimmermann’s approach
5. Fuzzy linear programming with fuzzy number coefficients
3
1. Fuzzy programming versus goal programming
Consider the following simple linear program
x → min, subject to x ≥ 1, x ∈ R
What if the decision maker’s 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
does not have any solution.
Toc
JJ
II
x≥1
J
I
Back
J Doc
Doc I
Section 1: Fuzzy programming versus goal programming
4
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 turn into the following singleobjective problem
max min{|x − 0.5|, |x − 1|}, subject to 1/2 ≤ x ≤ 1.
The unique solution is x∗ = 0.75.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 1: Fuzzy programming versus goal programming
5
Figure 1: Illustration of the optimal solution. By using the minimum operator 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.
The fuzzy problem is to find an x ∈ R such that,
’x is as close as pos. to 0.5’ and ’x is as close as pos. to 1’
If we use the minimum operator to aggregate the objective functions then we have the following single-objective problem (FigToc
JJ
II
J
I
Back
J Doc
Doc I
Section 1: Fuzzy programming versus goal programming
6
ure 1)
|x − 0.5|
|x − 1|
, subject to 1/2 ≤ x ≤ 1.
max min 1−
, 1−
1/2
1/2
Which turns into,
max min{|x − 0.5|, |x − 1|}, subject to 1/2 ≤ x ≤ 1
which has a unique optimal solution x∗ = 0.75.
We will consider the following two-variable linear program
x1 + x2 → max
subject to
x1 , x2 ≤ 1
x1 , x2 ∈ R,
What if the decision maker’s aspiration level is b0 = 3?
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 1: Fuzzy programming versus goal programming
7
The aspiration level can not be reached since the maximal value
of the objective function is equal to two?
Figure 2: A simple LP. The unique optimal solution is (1, 1).
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 1: Fuzzy programming versus goal programming
8
Figure 3: The desired value of the objective function is set to 3.
Toc
JJ
II
J
I
Back
J Doc
Doc I
9
2. Fuzzy numbers
A fuzzy set A of the real line R is defined by its membership
function (denoted also by A)
A : R → [0, 1].
If x ∈ R then A(x) is interpreted as the degree of membership
of x in A.
A fuzzy set in R is called normal if there exists an x ∈ R such
that A(x) = 1. A fuzzy set in R is said to be convex if A
is unimodal (as a function). A fuzzy number A is a fuzzy set
of the real line with a normal, (fuzzy) convex and continuous
membership function of bounded support.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 2: Fuzzy numbers
10
Definition 2.1. A fuzzy set A is called triangular fuzzy number
with peak (or center) a, left width α > 0 and right width β > 0
if its membership function has the following form

a−t



1−
if a − α ≤ t ≤ a


α

t−a
A(t) =
1
−
if a ≤ t ≤ a + β



β



0
otherwise
and we use the notation A = (a, α, β).
The support of A is (a − α, b + β). A triangular fuzzy number
with center a may be seen as a fuzzy quantity
”x is close to a”or”x is approximately equal to a”.
Toc
JJ
II
J
I
Back
J Doc
Doc I
If A is not a fuzzy number then there exists an γ ∈ [0, 1] such that [A]
is not
a convex
Section
2: Fuzzysubset
numbersof R.
11
1
a-!
a
a+"
Fig. 1.2. Triangular fuzzy number.
Figure 4: A triangular fuzzy number.
Definition
1.1.42.2.
A fuzzy
set A
called
fuzzy number
peak
Definition
A fuzzy
setis of
the triangular
real line given
by the with
mem(or center) a, left width α > 0 and right width β > 0 if its membership
bership function
function has the following
form
|a − t|

a−t

1
−
if |a − t| ≤ α,

A(t) = 
(1)
 1 − αα if a − α ≤ t ≤ a


otherwise,
t−a
A(t) = 0
1−
if a ≤ t ≤ a + β



β a symmetrical triangular fuzzy
Where α > 0 will be
 called


otherwise
JJ
II 0 J
I
J Doc Doc I
Toc
Back
Section 2: Fuzzy numbers
12
number with center a ∈ R and width 2α and we shall refer to
it by the pair (a, α).
Let A = (a, α) and B = (b, β) be two fuzzy numbers of the
form (1), λ ∈ R. Then we have
A + B = (a + b, α + β), λA = (λa, |λ|α).
(2)
Which can be interpreted as
”x is approximately equal to a”
+
”y is approximately equal to b”
=
”x + y is approximately equal to a + b”
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 2: Fuzzy numbers
13
Figure 5: A = (a, 1), B = (b, 2), A + B = (a + b, 3)
Definition 2.3. A fuzzy set A is called trapezoidal fuzzy number
with tolerance interval [a, b], left width α and right width β if
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 2: Fuzzy numbers
14
its membership function has the following form

a−t


1−
if a − α ≤ t ≤ a



α


 1
if a ≤ t ≤ b
A(t) =
t−b


 1−
if a ≤ t ≤ b + β


β



0
otherwise
and we use the notation
A = (a, b, α, β).
(3)
The support of A is (a − α, b + β). A trapezoidal fuzzy number
may be seen as a fuzzy quantity
”x is approximately in the interval [a, b]”.
Toc
JJ
II
J
I
Back
J Doc
Doc I
The support of A is (a − α, b + β).
15
1
a-!
a
b
b+"
Fig. 1.3. Trapezoidal fuzzy number.
Figure 6: Trapezoidal fuzzy number.
A trapezoidal fuzzy number may be seen as a fuzzy quantity
3. Measures of possibility
”x is approximately in the interval [a, b]”.
Definition
1.1.6 Any
fuzzy
number
A ∈ F can be
as distribuFuzzy numbers
can
also
be considered
asdescribed
possibility

% number
&
tions. If A ∈ F is afuzzy
and x ∈ R a real number
a−t


if tdegree
∈ [a − α,ofa]possibility of the
L
then A(x) can be interpreted
as the


α


statement ”x is A”. 

1
if t ∈ [a, b]
%
&
A(t) =

tJ− b)
JJ
II
I
J Doc Doc I
Toc
Back

Section 3: Measures of possibility
16
Let A, B ∈ F be fuzzy numbers. The degree of possibility that
the proposition
”A is less than or equal to B”
is true denoted by Pos[A ≤ B] and defined by the extension
principle as
Pos[A ≤ B] = sup min{A(x), B(y)},
(4)
x≤y
In a similar way, the degree of possibility that the proposition
”A is greater than or equal to B”
is true, denoted by Pos[A ≥ B], is defined by
Toc
JJ
II
J
I
Back
J Doc
Doc I
Finally, the degree of possibility that the proposition is true ”A is equal to
Section 3: Measures of possibility
A
17
Pos[A!B]=1
a
B
b
Fig. 1.13. Pos[A ≤ B] = 1, because a ≤ b.
Figure 7: Pos[A ≤ B] = 1, because a ≤ b.
B” and denoted by Pos[A = B], is defined by
Pos[A = B] = sup min{A(x), B(x)} = (A − B)(0),
Pos[A ≥ B]x = sup min{A(x), B(y)}.
(1.21)
(5)
x≥y
Let A = (a, α) and B = (b, β) fuzzy numbers of symmetric triangular form.
It is easy to compute that,
Let A = (a, α) and B = (b, β) fuzzy numbers of symmetric
if athat,
≤b
to1 compute
triangular form. It is easy


a−b
Pos[A ≤ B] = 1 −
(1.22)
otherwise

α+β



0
if a ≥ b + α + β
II
I ”A Back
J Doc
Doc I
The Toc
degree ofJJ
necessity
that theJproposition
is less than
or equal
to B”
The degree of necessity that the proposition ”A is less than or equal to B”
Section 3: Measures of possibility
18
B
A
Pos[A!B]
b
a
Fig. 1.14. Pos[A ≤ B] < 1, because a > b.
Figure 8: Pos[A ≤ B] < 1, because a > b.
is true, denoted by Pos[A ≤ B], is defined as
Nes[A
≤ B] = 1 − Pos[A ≥ B].
1
if a ≤ b




If A = (a, α) and B = (b, β) are fuzzyanumbers
− b of symmetric triangular form
(6)
Pos[A ≤ B] =
1−
otherwise

α
+
β


 0
if a ≥ b + α + β
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
19
Pos[A = (a, α) ≤ B = (b, β)] = 1 ⇐⇒ a ≤ b.
Figure 9: Pos[A = (a, α) ≤ B = (b, β)] = 1 ⇐⇒ a ≤ b
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
20
Figure 10: The left-hand side of B does not play any role in computing
Pos[A ≤ B].
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
21
Figure 11: The right-hand side of B does matter as to the value of Pos[A ≤
B] if a > b.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
22

1
if a ≥ b




b−a
Pos[A ≥ B] =
1−
otherwise

α
+
β


 0
if a ≤ b − (α + β)
(7)
The degree of possibility that the proposition
”A is equal to B”
is true, denoted by Pos[A = B], is defined by
Pos[A = B] = sup min{A(x), B(x)},
x
Let x ∈ R and let A be a fuzzy number. The degree of possibility of the statement that ’A takes value x’ is defined by A(x).
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
23
Figure 12: Pos[A = (a, α) ≥ B = (b, β)] = 1 ⇐⇒ a ≥ b
Let A = (a, α) be a fuzzy number of symmetric triangular form
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
24
Figure 13: The right-hand side of B does not matter as to the value of
Pos[A ≥ B].
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 3: Measures of possibility
25
and let x ∈ R. Then,
Pos[A ≤ x] =
Pos[A ≥ x] =
Toc
JJ

1



if a ≤ x
A(x) = 1 −



0

1



A(x) = 1 −



0
II
J
I
a−x
otherwise
α
if a ≥ x + α
if a ≥ x
x−a
otherwise
α
if x ≤ a − α
Back
J Doc
Doc I
Section 3: Measures of possibility
26
Figure 14: Pos[A ≥ x].
Toc
JJ
II
J
I
Back
J Doc
Doc I
27
4. Zimmermann’s approach
The conventional model of linear programming (LP) can be
stated as
ha0 , xi → min; subject to Ax ≤ b.
In many real-world problems instead of minimization of the
objective function ha0 , xi it may be sufficient to determine an x
such that
a01 x1 + · · · + a0n xn ≤ b0 ; subject to Ax ≤ b.
(8)
where b0 is a predetermined aspiration level.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
28
Suppose that the decision maker’s aspiration level is set below
the optimal value of the objective function. In this case linear
inequality system (8) does not have a solution.
Then we use flexible constraints. We state the fuzzy linear programming problem as follows (Zimmermann [1])
Find an x∗ ∈ Rn such that it satisfies the following inequalities
as much as possible (in the sense we defined above)
a01 x1 + · · · + a0n xn ≤ (b0 , d0 )
a11 x1 + · · · + a1n xn ≤ (b1 , d1 )
..
.
am1 x1 + · · · + amn xn ≤ (bm , dm )
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
29
Figure 15: Illustration of constraint satisfacion hai , xi ≤ (bi , di ).
Let us µi (x) denote the degree of satsifaction of the i-th con-
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
30
straint by x ∈ Rn . Then we have,

1
if hai , xi ≤ bi ,



hai , xi − bi
µi (x) =
1−
if bi < hai , xi ≤ bi + di ,

di


0
if hai , xi > bi + di ,
for i = 0, 1, . . . , m.
• if for an x ∈ Rn the value of hai , xi is less or equal than bi
then x satisfies the i-th constraint with the maximal conceivable degree one;
• if bi < hai , xi < bi + di then x is not feasible in classical sense, but the decision maker can still tolerate the
violation of the crisp constraint, and accept x as a solution
with a positive degree, however, the bigger the violation
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
31
the less is the degree of acceptance;
• if hai , xi > bi + di then the violation of the i-th costraint
is untolerable by the decision maker, that is, µi (x) = 0.
Then the (fuzzy) solution of the FLP problem is defined as a
fuzzy set on Rn whose membership function is given by
µ(x) = min{µ0 (x), µ1 (x), . . . , µm (x)},
In this setup µ(x) denotes the degree to which all inequalities
are satisfied at point x ∈ Rn .
The maximizing solution x∗ of the FLP problem satisfies the
equation
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
32
λ∗ = µ(x∗ ) = max µ(x).
x
That is, λ∗ denotes the maximal degree to which both the aspiration level and the constraints can be satisfied.
max λ
µ0 (x) ≥ λ
µ1 (x) ≥ λ
...
µm (x) ≥ λ
0 ≤ λ ≤ 1, x ∈ Rn .
In this case we get the following LP problem
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
33
max λ
λd0 + ha0 , xi ≤ b0 + d0 ,
λd1 + ha1 , xi ≤ b1 + d1 ,
······
λdm + ham , xi ≤ bm + dm ,
0 ≤ λ ≤ 1, x ∈ Rn .
For example, let the optimal value of LP problem
ha0 , xi → min; subject to Ax ≤ b
be equal to 2, that is, ha0 , x∗ i = 2, and let the aspiration level
be 1 and let all the tolerance leveles be equal to one. Then our
FLP problem turns into the following LP
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
34
max λ
λ + ha0 , xi ≤ 1 + 1,
λ + ha1 , xi ≤ b1 + 1,
······
λ + ham , xi ≤ bm + 1
0 ≤ λ ≤ 1, x ∈ Rn .
It is clear that the solution of this LP is different from the solution of the original LP since for x = x∗ we get λ = 0.
Really, from λ + 2 ≤ 2 and 0 ≤ λ ≤ 1 it follows that λ = 0.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
35
Let us consider the problem
x1 + x2 → min
subject to
x1 ≥ 1
x2 ≥ 1
x1 , x2 ∈ R,
Let b0 = 1 and let d0 = d1 = d2 = 1. The aspiration level
is less than 2, the value we can reach under given constraints.
Then we get the following fuzzy linear programming problem:
Find an x∗ ∈ R such that
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
36
Figure 16: The optimization problem: unreachable goal (we are in the red).
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
37
x1 + x2 ≤ (1, 1)
subject to
x1 ≥ (1, 1)
x2 ≥ (1, 1)
x1 , x2 ∈ R
Then we have
µ0 (x) =
Toc
JJ

1



1−



0
II
if x1 + x2 ≤ 1,
x1 + x2 − 1
otherwise
1
if x1 + x2 > 2,
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
38
Figure 17: Degrees of satisfactions of the goal x1 + x2 ≤ (1, 1) by the
values of the objective function x1 + x2 .

if x1 + x2 ≤ 1,
 1
2 − (x1 + x2 ) otherwise
µ0 (x) =

0
if x1 + x2 > 2,
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
39
Figure 18: Degrees of satisfactions of the first constraint x1 ≥ (1, 1) by the
values of x1 .
µ1 (x1 , x2 ) =
Toc
JJ
II

1



1−



0
J
if x1 ≥ 1,
1 − x1
otherwise
1
if x1 < 0,
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
40

 1 if x1 ≥ 1,
x1 otherwise
µ1 (x1 , x2 ) =

0 if x1 < 0,

 1 if x2 ≥ 1,
x2 otherwise
µ2 (x1 , x2 ) =

0 if x2 < 0,
The fuzzy solution is defined by
µ(x1 , x2 ) = min{µ0 (x1 , x2 ), µ1 (x1 , x2 ), µ2 (x1 , x2 )}
and the maximizing solution is obtained from
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
41
λ∗ = max min{µ0 (x1 , x2 ), µ1 (x1 , x2 ), µ2 (x1 , x2 )}.
x1 ,x2
Figure 19: Degrees of satisfactions of the goal x1 + x2 ≤ (1, 1) by the
values of the objective function x1 + x2 .
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
42
Figure 20: Degrees of satisfactions of the first constraint x1 ≥ (1, 1) by the
values of x1 .
That is,
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
43
max λ
2 − (x1 + x2 ) ≥ λ
x1 ≥ λ
x2 ≥ λ
0 ≤ λ ≤ 1, x1 , x2 ∈ R.
The optimal solutions are x∗1 = x∗2 = λ∗ = 2/3.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 4: Zimmermann’s approach
44
Figure 21: Illustration of the solution.
Toc
JJ
II
J
I
Back
J Doc
Doc I
45
5. Fuzzy linear programming with fuzzy number coefficients
Assume that all parameters in (8) are fuzzy numbers of symmetric triangular form. Then the following flexible (or fuzzy)
linear programming (FLP) problem can be obtained by replacing crisp parameters aij , bi with symmetric triangular fuzzy
numbers ãij = (aij , α) and b̃i = (bi , di ) respectively,
(a01 , α)x1 + · · · + (a0n , α)xn ≤ (b0 , d0 )
(a11 , α)x1 + · · · + (a1n , α)xn ≤ (b1 , d1 )
..
.
(am1 , α)x1 + · · · + (amn , α)xn ≤ (bm , dm )
(9)
Here d0 and di are interpreted as the tolerance levels for the
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
46
objective function and the i-th constraint, respectively.
We denote by µi (x) the degree of satisfaction of the i-th restriction at the point x ∈ Rn in (9), i.e.
µi (x) = Pos(ãi1 x1 + · · · + ãin xn ≤ b̃i ).
Especially, µ0 (x) denotes the degree of satisfaction of the aspiration level of the decision maker at point x ∈ Rn .
µ0 (x) = Pos(ã01 x1 + · · · + ã0n xn ≤ b̃0 ).
Then the (fuzzy) solution of the FLP problem (9) is defined as
a fuzzy set on Rn whose membership function is given by
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
47
µ(x) = min{µ0 (x), µ1 (x), . . . , µm (x)},
In this setup µ(x) denotes the degree to which all inequalities
are satisfied at point x ∈ Rn .
The maximizing solution x∗ of the FLP problem (9) satisfies
the equation
λ∗ = µ(x∗ ) = max µ(x).
x
That is, λ∗ denotes the maximal degree to which both the aspiration level and the constraints can be satisfied.
From (6) it follows that the degree of satisfaction of the i-th
restriction at x in (9) is the following:
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
48

1
if hai , xi ≤ bi ,




hai , xi − bi
µi (x) =
1−
otherwise,

α|x|1 + di



0
if hai , xi > bi + α|x|1 + di ,
where
|x|1 = |x1 | + · · · + |xn |
and
hai , xi = ai1 x1 + · · · + ain xn ,
for i = 0, 1, . . . , m.
To find a maximizing solution to FLP problem (9) we have to
solve the following nonlinear programming problem
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
49
λ + h0, xi → max
ha0 , xi − b0
1−
≥λ
α|x|1 + d0
ha1 , xi − b1
≥λ
α|x|1 + d1
······
ham , xi − bm
1−
≥λ
α|x|1 + dm
1−
0 ≤ λ ≤ 1, x ∈ Rn .
That is,
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
50
max λ
λ(α|x|1 + d0 ) − α|x|1 + ha0 , xi ≤ b0 + d0 ,
λ(α|x|1 + d1 ) − α|x|1 + ha1 , xi ≤ b1 + d1 ,
······
λ(α|x|1 + dm ) − α|x|1 + ham , xi ≤ bm + dm ,
0 ≤ λ ≤ 1, x ∈ Rn .
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
51
Example 5.1. As a very simple example consider the following
LP
x → min, subject to x ≥ 1, x ∈ R
that is,
1 × x → min
subject to 1 × x ≥ 1, x ∈ R,
with a unique solution x∗ = 1.
(10)
Let us set the aspiration level to 0.5 and derive from it the following FLP: Find an x∗ ∈ R from the possibilistic inequality
system
Toc
JJ
II
(1, α)x ≤ (0.5, d0 )
(1, α)x ≥ (1, d1 )
J
I
Back
(11)
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
52
which satisfies both inequalities as much as possible.
If α = 0.5, d0 = 0.5 and d1 = 0.5 we get the following statement: Find an x∗ ∈ R from the possibilistic inequality system
(1, 0.5)x ≤ (0.5, 0.5)
(1, 0.5)x ≥ (1, 0.5)
which satisfies both inequalities as much as possible.
After the multiplication by scalar we get the problem: Find an
x∗ ∈ R from the possibilistic inequality system
Inequality 1. (x, 0.5 × |x|) ≤ (0.5, 0.5)
Inequality 2. (x, 0.5 × |x|) ≥ (1, 0.5)
which satisfies both inequalities as much as possible.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
53
Figure 22: Explanation of the fuzzified aspiration level ’the value of the
objective should be less then 0.5’
Let us consider the case when x = 1. Then we get
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
54
Figure 23: Explanation of the fuzzified constraint,
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
55
Figure 24: The goal and the constraint are in a trade-off relation.
Inequality 1. (1, 0.5) ≤ (0.5, 0.5)
Inequality 2. (1, 0.5) ≥ (1, 0.5)
Inequality 1. is satisfied with degree 0.5 since µ0 (1) = 0.5 and
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
56
the second one is satisfied with degree one since µ1 (2) = 1.
That is, x = 1 can be considered as a solution to the FLP
problem with degree min{0.5, 1} = 0.5.
Figure 25: The situation for x = 1.
Let us consider the case when x = 2. Then we get
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
57
Figure 26: The situation for x = 2.
Inequality 1. (2, 1) ≤ (0.5, 0.5)
Inequality 2. (2, 1) ≥ (1, 0.5)
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
58
Inequality 1. is satisfied with degree 0 since
µ0 (2) = 0
and the second one is satisfied with degree one since µ1 (1) = 1.
Figure 27: The situation for x = 0.75.
Using (6) we get that the degree of satisfaction of the first inToc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
59
equality at point x is equal to

1
if x ≤ 0.5




x − 0.5
µ0 (x) =
1−
if 0.5 ≤ x ≤ 1

0.5|x| + 0.5



0
if x > 1
Using (6) we get that the degree of satisfaction of the second
inequality at point x is equal to
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
60

1
if −x ≤ −1




−x+1
µ1 (x) =
1−
if −1 < −x ≤ −0.5

0.5|x| + 0.5



0
if −x > −0.5
The fuzzy solution is
µ(x) = min{µ0 (x), µ1 (x)}
and the maximizing solution is obtained from
λ∗ = max µ(x) = max min{µ0 (x), µ1 (x)}
x
Toc
JJ
x
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
61
That is,
λ → max
x − 0.5
1−
≥λ
0.5|x| + 0.5
−x+1
1−
≥λ
0.5|x| + 0.5
0 ≤ λ ≤ 1, x ∈ R.
That is, the uniqe maximizing solution of (11) is then obtained
from the equation
1−
−x+1
x − 0.5
=1−
0.5|x| + 0.5
0.5|x| + 0.5
where x∗ = 0.75 and the degree of consistency is
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
62
Figure 28: µ0 and µ1 .
λ∗ = 1 −
Toc
JJ
II
0.75 − 0.5
= 5/7
0.5 × 0.75 + 0.5
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
63
The degree of consistency is smaller than one, because the aspiration level, b0 = 0.5, is set below 1, the minimal value of the
crisp goal function, 1 × x under the crisp constraint x ≥ 1.
Figure 29: Illustration of the optimal solution.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
64
Example 5.2. Consider the following LP
x1 + x2 → max
subject to
x1 ≤ 1
x2 ≤ 1
x1 , x2 ∈ R,
that is,
x1 + x2 → max
subject to
x1 + 0 × x2 ≤ 1
0 × x1 + x2 ≤ 1
x1 , x2 ∈ R,
with a unique solution x∗1 = x∗2 = 1 and x∗1 + x∗2 = 2.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
65
Figure 30: An illustration of example 2.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
66
Suppose that the decision maker sets the desired value of the
objective function to 3. Then we get the following fuzzy linear
program
(1, α)x1 + (1, α)x2 ≥ (3, d0 )
(1, α)x1 + (0, α)x2 ≤ (1, d1 )
(0, α)x1 + (1, α)x2 ≤ (1, d2 )
x1 , x2 ∈ R,
with α = 0.5, d0 = d1 = d2 = 1.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
67
That is,
(x1 , 0.5|x1 |) + (x2 , 0.5|x2 |) ≥ (3, 1)
(x1 , 0.5|x1 |) + (0, 0.5|x2 |) ≤ (1, 1)
(0, 0.5|x1 |) + (x2 , 0.5|x2 |) ≤ (1, 1)
x1 , x2 ∈ R,
After the addtions and multiplications by scalar we get
(x1 + x2 , 0.5(|x1 | + |x2 |)) ≥ (3, 1)
(x1 , 0.5(|x1 | + |x2 |)) ≤ (1, 1)
(x2 , 0.5(|x1 | + |x2 |)) ≤ (1, 1)
x1 , x2 ∈ R,
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
68
Figure 31: The fuzzified goal: if x1 + x2 ≥ 3 then the goal is satisfied with
degree 1.
Let us consider the situation when x1 = x2 = 1.
(2, 1) ≥ (3, 1)
(1, 1) ≤ (1, 1)
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
69
Figure 32: The fuzzified constraint: if x1 ≤ 1 then the first constraint is
satisfied with degree 1.
Using (6-7) we get
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
70
Figure 33: The value of the objective function for x1 = x2 = 1.
µ0 (x1 , x2 ) =

1
if x1 + x2 ≥ 3




3 − (x1 + x2 )
otherwise
1−

1 + 0.5(|x1 | + |x2 |)



0
if x1 + x2 ≤ 3 − (1 + 0.5(|x1 | + |x2 |))
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
71
µ1 (x1 , x2 ) =

1
if x1 ≤ 1




x1 − 1
otherwise
1−

1 + 0.5(|x1 | + |x2 |)



0
if x1 > 1 + (1 + 0.5(|x1 | + |x2 |))
µ2 (x1 , x2 ) =

1
if x2 ≤ 1




x2 − 1
otherwise
1−

1
+
0.5(|x
1 | + |x2 |)



0
if x2 > 1 + (1 + 0.5(|x1 | + |x2 |))
The fuzzy solution is defined by
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
72
µ(x1 , x2 ) = min{µ0 (x1 , x2 ), µ1 (x1 , x2 ), µ2 (x1 , x2 )}
and the maximizing solution is obtained from
λ∗ = max min{µ0 (x1 , x2 ), µ1 (x1 , x2 ), µ2 (x1 , x2 )}.
x1 ,x2
That is,
max λ
µ0 (x) ≥ λ
µ1 (x) ≥ λ
µ2 (x) ≥ λ
0 ≤ λ ≤ 1, x ∈ R2 .
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
73
To find a maximizing solution to FLP problem we have to solve
the following nonlinear programming problem
max λ
3 − (x1 + x2 )
1−
≥λ
1 + 0.5(|x1 | + |x2 |)
1−
x1 − 1
≥λ
1 + 0.5(|x1 | + |x2 |)
1−
x2 − 1
≥λ
1 + 0.5(|x1 | + |x2 |)
0 ≤ λ ≤ 1, x1 , x2 ∈ R.
The unique solution is x∗ = (4/3, 4/3) and λ∗ = 6/7.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
74
Figure 34: The optimal value of the fuzzy objective function.
(x1 + x2 , 0.5(|x1 | + |x2 |)) ≥ (3, 1)
(x1 , 0.5(|x1 | + |x2 |)) ≤ (1, 1)
(x2 , 0.5(|x1 | + |x2 |)) ≤ (1, 1)
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
75
Figure 35: The optimal value of fuzzy objective and its relation to the fuzzified goal.
(4/3 + 4/3, 0.5(4/3 + 4/3)) ≥ (3, 1)
(4/3, 0.5(4/3 + 4/3)) ≤ (1, 1)
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
76
Figure 36: Constraint satisfaction.
References
[1] H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets
and Systems, 1(1978) 45-55. 28
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
77
Figure 37: An illustration of example 2.
Toc
JJ
II
J
I
Back
J Doc
Doc I
Section 5: Fuzzy linear programming with fuzzy number coefficients
78
Figure 38: An illustration of the functions µ0 , µ1 and µ2 .
Toc
JJ
II
J
I
Back
J Doc
Doc I