196 M.
A
NDY
R
UDHITO
E
T
A
L
Conversely, for every weighted directed graph G = V, A, can be defined a matrix A
n n
max
R
, which is called the weighting matrix of graph G, where A
ij
=
.
, if
, if
, A
A i
j i
j i
j w
. The mean weight of a path is defined as the sum of the weights of the individual arcs of this path, divided by the length of this path. If such a path is
a circuit one talks about the mean weight of the circuit, or simply the cycle mean. It follow that a formula for maximum mean cycle mean
max
A in GA is
max
A =
n
k 1
k 1
ii k
n
A
1 i
.
. The matrix A
n n
max
R
is said to be irreducible if its precedence graph G = V, A is strongly connected, that is for every i, j
V, i j, there is a path from i to j. We can show
that matrix A
n n
max
R
is irreducible if and only if A
2
A
...
1
n
A
ij
for every i, j where i
j Schutter, 1996.
Given A
n n
max
R
. Scalar
R
max
is called the max-plus eigenvalue of matrix A if there
exists a vector v
n max
R
with v
n 1
such that A
v =
v. Vector v is called max-plus
eigenvector of matrix A associated with
. We can show that
max
A is a max-plus eigenvalue of matrix A. For matrix B =
max
A A, if
ii
B
= 0, then i-th column of matrix
B
is an eigenvector corresponding with eigenvalue
max
A. The eigenvector is called fundamental max-plus eigenvector
associated with eigenvalues
max
A Bacelli, et al., 2001. A linear combination of fundamental max-plus eigenvector of matrix A is also an eigenvector
assosiated with
max
A. We can show that if matrix A
n n
max
R
is irreducible, then
max
A is the unique max-plus egenvalue of A and the max-plus eigenvector associated with
max
A is
v
, where v
i
for every i {1, 2, ..., n} Bacelli, et al., 2001.
3. INTERVAL MAX-PLUS ALGEBRA
In this section we will review some concepts and results of interval max-plus algebra, matrix over interval max-plus algebra and interval max-plus eigenvalue. Further details can
be found in Rudhito, et al. [7] and Rudhito [8].
The closed max-plus interval x in R
max
is a subset of R
max
of the form
x = [
x
,
x
] = {x
R
max
x
m
x
m
x
},
which is shortly called interval. The interval x y if and only if
y
m
x
m
x
m
y
.
Especially x = y if and only if
x
=
y
and
x
=
y
. The number x
R
max
can be represented
as interval [x, x]. Define IR
:= {x = [
x
,
x
]
x
,
x
R,
m
x
m
x
} { }, where
:= [
,
]. Define x
y = [
x
y
,
x
y
] and x
y = [
x
y
,
x
y
] for every x, y IR
. We can show that IR
,
,
is a commutative idempotent semiring with neutral element
= [
,
] and unity element 0 = [0, 0]. This commutative idempotent semiring
Ap p li c a t i on of Fu z z y Nu mb er M a x -Plu s Alg eb ra to Closed Serial Queuing Network… 197
IR
,
,
is called interval max-plus algebra, and is written as IR
max
. Relation “
Im
”defined on IR
max
as x
Im
y x
y = y is a partial order on IR
max
. Notice that x
y = y
x
m
y
and
x
m
y
.
Define IR
n m
max
:= {A = A
ij
A
ij
IR
max
, i = 1, 2, ..., m, j = 1, 2, ..., n}. The
elements of IR
n m
max
are called matrices over interval max-plus algebra or shortly interval matrices
. The operations on interval matrices can be defined in the same way with the operations on matrices over max-plus algebra. For any matrix A
IR
n m
max
, Define the matrix
A
=
ij
A
R
n m
max
and
A
=
ij
A
R
n m
max
, which are called lower bound matrix and upper bound matrix of A, respectively. Define a matrix interval of A, that is [
A
,
A
] = {A
R
n m
max
A
m
A
m
A
} and I
n m
max
R
b
= { [
A
,
A
] A R
n n
max
}. The matrix interval [
A
,
A
] and [
B
,
B
] IR
n m
max
b
are equal if
A
=
B
and
A
=
B
. We can show that for every matrix interval A
IR
n m
max
we can determine matrix interval [
A
,
A
] I
n m
max
R
b
and conversely. The matrix interval [
A
,
A
] is called matrix interval associated with the interval
matrix A, and is written as A [
A
,
A
]. Moreover, we have A [
A
,
A
], A
B [
A
B
,
A
B
] and A
B [
A
B
,
A
B
].
Define IR
n max
:= { x = [x
1
, x
2
, ... , x
n
]
T
| x
i
IR
max
, i = 1, 2, ... , n }. Note that
I R
n max
can be viewed as IR
1 max
n
. The elements of IR
n max
are called interval vectors over
I R
max
or shortly interval vectors. An interval vector x IR
n max
is said to be not equal to interval vector
, and is written as x , if there exists i {1, 2, ..., n} such that x
i
. Interval matrix A
IR
n n
max
, where A [
A
,
A
], is said to be irreducible if every matrix A
[
A
,
A
] is irreducible. We can show that interval matrix A
IR
n n
max
, where A
[
A
,
A
] is irreducible if and only if
A
n n
max
R
is irreducible Rudhito, et al. [7].
4. FUZZY NUMBER MAX-PLUS ALGEBRA
