Systems of Fuzzy Number Max-plus Linear Equations

This paper discusses the solution of systems of fuzzy number max-plus linear equations through the greatest fuzzy number subsolution of the system. We show that if entries of each column of the coecient matrix are not equal to infinite, the system has the greatest fuzzy number subsolution. The greatest fuzzy number subsolution of the system could be determined by first finding the greatest interval subsolution of the alpha-cuts of the system and then modifying it if needed, such that each its components is a family of alpha-cut of a fuzzy number. Then, based on the Decomposition Theorem on Fuzzy Set, we can determine the membership function of the elements of greatest subsolution of the system. If the greatest subsolution satisfies the system then it is a solution of the system.DOI : http://dx.doi.org/10.22342/jims.17.1.10.17-28


Introduction
The max-plus algebra can be used to model and analyze networks, like the project scheduling, production system, queuing networks, etc [1], [3], [4]. The networks modeling with max-plus algebra approach is usually a system of maxplus linear equations and it can be written as a matrix equation A ⊗ x = b where x and b are input vector and output vector respectively.
Recently, the fuzzy networks modeling have been developed. In this paper, the fuzzy networks refer to networks whose activity times are fuzzy numbers. The fuzzy scheduling are as in [2], [10] and the fuzzy queueing networks as in [7]. When we follow the notions of modeling and analyzing fuzzy networks with max-plus algebra approach, for the input-output fuzzy system we will use systems of fuzzy number max-plus linear equations. For this reason, this paper will discuss existence and computation of solutions of systemÃ⊗x =b of fuzzy number max-plus linear equations.
The solution of the fuzzy relational equation has been investigated by many researchers. Among their results, are the minimal solution of a fuzzy relational equation with the supinf composite operation [8], the complete set of minimal solutions for fuzzy relational equations with max-product composition [11], the solutions for fuzzy relational equations with supmin composition [12], and the solutions for fuzzy relational equations with supinf composition [9]. The solutions for fuzzy relational equations with max-plus composition will be discussed in this paper.
We first review some basic concepts of max-plus algebra, matrices over maxplus algebra, and the solution of system of max-plus linear equations A ⊗ x = b. Further details can be found in [1].
Let R ε := R ∪ {ε} be the set of all real numbers and ε := −∞. Defined two operations on R ε such that for every a, b ∈ R, a ⊕ b := max(a, b), a ⊗ b := a + b.
Then (R, ⊕, ⊗) is a commutative idempotent semiring whose neutral element ε = −∞ and unity element e = 0. Moreover, (R, ⊕, ⊗) is a semifield, that is (R, ⊕, ⊗) is a commutative semiring, where for every a ∈ R there exist −a such that a⊗(−a) = 0. Then, (R, ⊕, ⊗) is called the max-plus algebra, and is written as R max . The algebra R max has no zero divisors, that is for every x, y ∈ R, if x ⊗ y = ε , then x = ε or y = ε. The relation " m " on R max defined by x m y if x ⊕ y = y, is a partial order on R max . The operations on R max are consistent with respect to the order m , that is for every a, b, c ∈ R max , if a m b, then a ⊕ c m b ⊕ c and a ⊗ c m b ⊗ c. We define x 0 := 0 , x k := x ⊗ x k−1 and ε k := ε, for k = 1, 2, ... .
The operations ⊕ and ⊗ on R max can be extended to the set in R m×n max , where R m×n max := {A = (A ij )|A ij ∈ R max , for i = 1, 2, ..., m and j = 1, 2, ..., n}. Specifically, for A, B ∈ R n×n max we define We also define matrix Υ ∈ R m×n max , with (Υ) ij := ε for every i and j, and We can show that (R m×n max , ⊕) is an idempotent commutative semigroup, (R n×n max , ⊕, ⊗) is an idempotent semiring whose neutral element is the matrix Υ and unity element is the matrix E, and R m×n max is a semimodule over R max . For any matrix A ∈ R n×n max , define A 0 = E n and A k : , is a partial order on R m×n max . In (R n×n max , ⊕, ⊗), operations ⊕ and ⊗ are consistent with respect to the order m , that is for every Define R n max := {[x 1 , x 2 , ..., x n ] T |x i ∈ R max , i = 1, 2, ..., n}. Note that R n max can be viewed as R n×1 max . The elements of R n max are called vectors over R max or shortly vectors. Definition 1.1. Given A ∈ R m×n max and b ∈ R m max , a vectorx is called subsolution of the system of max-plus linear equations x for every subsolutionx of the system A ⊗ x = b.
max with the entries of each column are not all equal to ε and b ∈ R m max , then the greatest subsolution of system A ⊗ x = b exists and is given byx = −(A T ⊗ (−b)).

Main Results
We begin the discussion by developing some basic concepts of interval maxplus algebra, matrices over interval max-plus algebra, and solutions of systems of interval max-plus linear equations. The concepts are developed based on [6].
A (closed) interval x in R max is a subset of R max of the form Then (I(R) m×n max ,⊕,⊗) is an idempotent semiring whose neutral element is the matrix [Υ] , with ([Υ]) ij := [ε] for every i and j, and unity element is the matrix We can also show that I(R) m×n max is a semimodule over I(R) max .
For any matrix and is written as Define .., n}. The Elements of I(R) m×n max are called interval vectors over I(R) max or shortly interval vectors.
is the greatest interval subsolution of the interval system x andx is a subsolution of A⊗x = b and A ⊗ x = b, respectively. Sincex and x are the greatest subsolution of ). Since the operation ⊗ for matrices is consistent with Hence,x andx are a subsolution of A ⊗x = b and A ⊗x = b, respectively. Since isx the greatest subsolution of A ⊗x = b,x mx . Then, we can show thatx mx . Suppose thatx mx . Since the relation m in R max is a total relation, there is an index i such thatx Thus, If the greatest interval subsolution was satisfies the system, then it is an interval solution of the system. In the further discussion, we assume that the reader know about some basic concepts in fuzzy set and fuzzy number. Further details can be found in [5] and [13].
Such matrices will be called fuzzy number matrix. The operations⊕ and⊗ in F (R) max can be extended to the operations of fuzzy number matrices in F (R) n×n max . Specifically, for the matricesÃ,B ∈ F (R) n×n max , we define For everyÃ ∈ F (R) n×n max and number α ∈ [0, 1] define α-cut matrix of matrixÃ, that is the interval matrix A α := (A α ij ), with A α ij is the α-cut ofÃ ij for every i and j. Note that matrix A α = (A α ij ) ∈ R m×n max and A α = (A α ij ) ∈ R m×n max are lower bound and upper bound of matrix A α , respectively. We can conclude that the matrices A,B ∈ F (R) m×n max , are equal if and only if A α = B α , that is A α ij = B α ij for every α ∈ [0, 1] and for every i and j. For every fuzzy number matrixÃ, .., n}. The elements in F (R) n max are called fuzzy number vectors over F (R) max , or shortly fuzzy number vectors.
Definition 2.7. GivenÃ ∈ F (R) m×n max andb ∈ F (R) m max . A fuzzy number vector x * ∈ F (R) n max is called fuzzy number solution of systemÃ⊗x =b ifx * satisfies the system. A fuzzy number vectorx ∈ F (R) n max is called fuzzy number subsolution of the system ifÃ⊗x F mb .
A fuzzy number vector x ∈ F (R) n max is called greatest fuzzy number subsolution ofÃ⊗x =b ifx F mx for every fuzzy number subsolutionx of the system. Definition 2.9. GivenÃ ∈ F (R) m×n max with the entries of each column are not all equal toε andb ∈ F (R) m max . Define a fuzzy number vectorx whose components arex i , that is a fuzzy number with the α- The bounds ofx α i are defined recursively as bellow.
Notice that the α-cut family of the components of the fuzzy number vectoȓ x as in Definition 2.9 is really an α-cut family of a fuzzy number. That is because (i ) according to the Theorem 2.3,x α i is an interval, thenx α i is also an interval, Definition 2.9, it is clear thatx α i is a nested α-cut family, (iv )Ã ∈ F (R) m×n max and b ∈ F (R) m max , then A 0 ij and b 1 ij are bounded, respectively. Meanwhile from the Definition 2.9, we havex 0 hencex 0 i is also bounded. Further, use to Decomposition Theorem in fuzzy set, we can get the components of fuzzy number vectorx , that isx i = α∈[0,1]c α i wherec α i is a fuzzy set in R with its membership function µ(x) = αχ c α i , where χ c α i is the characteristic function ofx α i . With the above definedx , we have thatx is the greatest fuzzy number subsolution, wherex α i Imx α i for every α ∈ [0, 1] and for every i = 1, 2, ..., n. The theorem below gives a condition for the existence of the greatest fuzzy number subsolution the systemÃ⊗x =b. We show that the fuzzy number vector as defined in the Definition 2.9 is the greatest fuzzy number subsolution of the systemÃ⊗x =b.
Theorem 2.10. GivenÃ ∈ F (R) m×n max with the entries of each column are not all equal toε andb ∈ F (R) m max , then the fuzzy number vectorx which whose components are defined as in Definition 2.9 is the greatest fuzzy number subsolution ofÃ⊗x =b.
Proof. According to the Theorem 2.3, the interval vectorx α whose components , and ) i for every i = 1, 2, ..., n, is the greatest subsolution of system A α⊗ [x ] = b α for every α ∈ [0, 1]. Letx be a fuzzy number vector whose components are fuzzy numberx i , wherex α i = [x α i ,x α i ] whose bounds are defined as in Definition 2.9. From the definition ofx α i ,x α i Imx α i . Thus,x α Imx α for every α ∈ [0, 1]. Sincex α is the greatest subsolution of A α⊗ [x ] = b α for every α ∈ [0, 1], A α⊗x α Im b α for every α ∈ [0, 1]. Sincex α Imx α for every α ∈ [0, 1] and the operation⊗ on interval matrix is consistent with respect to the order " Im ", A α⊗x α Im A α⊗x α Im b α for every α ∈ [0, 1]. Hence,Ã⊗x F mb sox is a subsolution of the systemÃ⊗x =b. Letx ∈ F (R) n max be a fuzzy number subsolution ofÃ⊗x =b, thenÃ⊗x F mb or A α⊗x α Im b α , for every α ∈ [0, 1]. Sincex α is the greatest This contradicts the fact thatx is a fuzzy number subsolution of the systemÃ⊗x =b. the greatest subsolution of the system A αj ⊗ x = b αj , A αj ⊗x αj j b αj . This conclusion contradicts the fact thatx is a fuzzy number subsolution of the system A⊗x =b. (ii ) For casex α k ≻ mx α k there are two possibilities: (a) Supposex α k ≺ m x α k ≺ mx α k , thenx α k mx α k . Sincex α k is the greatest subsolution of the system This result contradicts the fact thatx is a fuzzy number subsolution of the systemÃ⊗x =b.