AE solutions to two-sided interval linear systems over max-plus algebra

This paper introduces a concept of AE solutions to two-sided interval max-plus linear systems, a rather general concept which includes many known concepts of solutions to interval systems, in particular, weak, strong, tolerance and control solutions as its special cases. We state full characterizations of AE solutions for the two-sided interval max-plus systems, including both linear inequalities and linear equations. Moreover, we provide a specific example to illustrate an efficient method of finding the AE solution set.


Introduction
The max-plus algebra (R max , ⊕, ⊗) has appeared under the name extremal algebra for many years [1,2]. In this algebra, several types of solvability of interval linear systems, including both interval linear equations and interval linear inequalities, have been studied in the literature, see, e.g., weak and strong solvability in [3,4], tolerance solvability in [5][6][7], and control solvability in [8]. A detailed discussion of interval solutions can be found in Chap. 6 of [9] and a brief review of the max-plus linear systems in [10].
Two-sided systems of max-plus linear systems containing a vector x ∈ R n max on both sides had been studied by many authors. The theories of these systems in the classical algebra can be found in, e.g., [11][12][13][14] and the references therein.
In classical linear algebra, interval linear systems are often used for modeling information systems and engineering problems. Over the past decades, these uncertain systems have been discussed widely, see, e.g., [9,[15][16][17][18]. One of the main difficulties when dealing with interval uncertainty is how to understand the concepts of solutions. Shary first proposed the concept of AE solutions for interval linear equations in 2002, a rather general concept which includes many traditional concepts of solutions to interval systems as its special cases [19]. Sharaya considered the interval linear system with both equations and inequalities [20]. Recently, Hladík [21] extended the AE solutions to the general interval linear systems of equations, inequalities, or both. Some new results on AE solutions can be found in, e.g., [22][23][24]. Similar methods have been utilized for interval inequalities and interval linear programming for some special cases [25][26][27][28][29][30].
In this paper, we first extend the concept of AE solutions to max-plus algebra in twosided interval systems. A practical motivation or studying AE solutions of two-sided interval systems in max-plus algebra is presented in Example 1. Necessary and sufficient conditions for checking the AE solvability of two-sided interval systems in max-plus algebra are formulated in Sects. 4 and 5. Then, in Sect. 6, we present an efficient method to find the AE solution set of two-sided interval linear systems. In Sect. 7, we show that each particular result of weak, strong, tolerance and control solutions established for interval linear systems in existing literature is a special case of the main results of this paper. The conclusion and future application are drawn in Sect. 8.

Preliminaries
In this section, let us introduce some notations first (see [9]). By a max-plus algebra we understand a triple (R max , ⊕, ⊗), where R max = R ∪ {ε}, R is the set of all real numbers and ⊕, ⊗ are binary operations defined as where ε = -∞. Let E be a matrix consisting entirely of ε. The operations ⊕, ⊗ are extended to matrices and vectors in the same way as in conventional linear algebra. If A ∈ R m×n max , B ∈ R n×p max , we define the product A ⊗ B ∈ R m×p max with entries (A ⊗ B) ij defined for i = 1, . . . , m, j = 1, . . . , p as follows: If A ∈ R m×n max , B ∈ R m×n max , we define the sum A ⊕ B ∈ R m×n max with entries (A ⊕ B) ij defined for i = 1, . . . , m, j = 1, . . . , n as follows:

An interval matrix is defined as
b} is understood as one-column interval matrix. In the maxplus algebra, the set of all m × n interval matrices will be denoted by IR m×n max , and the set of all m-dimensional interval vectors by IR m max . Given A ∈ IR m×n max and B ∈ IR m×n max , the corresponding two-sided interval linear system of inequalities is the family of systems Similarly, the corresponding interval system of equations of the form represents the set of all systems of linear max-plus systems of the form The following lemmas will be used in the proofs of our results.
Proof Note that a ij ≤ b ij , and for i = 1, . . . , m, we have

AE solutions to interval max-plus systems
Let us recall the concept of AE solutions of interval inequalities in classical algebra [19,21,22]. The interval matrix can be split as A = A ∀ + A ∃ , where A ∀ is the interval matrix comprising universally quantified coefficients, and A ∃ concerns existentially quantified coeffi- Analogously, in the max-plus algebra, by using forall-exists quantification of interval parameters, we decompose the interval matrix as A = A ∀ ⊕ A ∃ , where the components in the matrix A ∀ at the positions associated with the existential quantifier are intervals [ε, ε] and components in the matrix A ∃ at the positions associated with the universal quantifier are intervals [ε, ε]. Now we extend the concept of AE solutions of an interval linear system in classical algebra to the max-plus algebra.
The next example demonstrates how this type of solution can be applied to an application problem.
Example 1 A sportswear company produces m types of sport suit, including top and bottom. All the clothes are made from three kinds of material: cotton, polyester fibre and chemical fiber. Due to the varieties of product and material, the production time of each type of suit corresponding to three different kinds of material is different. Each clothing item is finished only after all the material is completed.
Suppose that the production time data of each type of top and bottom, corresponding to three different kinds of material are interval times [a ij , a ij ] and [b ij , b ij ]. Before processing, preparing time x j for each material is required. If the working durations a ij and b ij are fixed, the time at which each type of top and bottom are completed is The packaging should be completed only after the suits, including both tops and the bottoms, are finished. To optimize production, the company needs to set the preparation time x j for each material such that each type of top and bottom is completed at the same time. This task is equivalent to solving the system of equations for each i ∈ {1, 2, . . . , m}, which can be simplified to the matrix form In production, the company may not be able to improve parts of its interval production times [a ij , a ij ] and [b ij , b ij ]. In this situation, an AE solution (x 1 , x 2 , . . . , x n ) T of system (3.3), when n = 3, may exist if the company is able to fix the other parts of production times a ij and b ij .

Characterization of AE solutions for two-sided interval linear inequalities
Consider a two-sided interval system of equations of the following form: where A ∈ IR m×n max , B ∈ IR m×n max . A sufficient and necessary characterization of AE solutions to interval system of maxplus linear inequalities (2.1) is described in the following theorem.

Theorem 4.1 A vector x ∈ R n max is an AE solution of two-sided interval linear max-plus inequalities (2.1) if and only if
Due to the isotone properties presented in Lemmas 2.1 and 2.2, we have Particularly, letting A ∀ = A ∀ and B ∀ = B ∀ in (4.2), we have Therefore, inequalities (4.1) hold. Conversely, assume that vector x ∈ R n max satisfies inequalities (4.1). According to Lemma 2.1, for all A ∀ ∈ A ∀ , B ∀ ∈ B ∀ , we have This completes the proof.

Characterization of AE solutions for two-sided interval linear equations
Consider a two-sided interval system of equations of the following form: where A ∈ IR m×n max , B ∈ IR m×n max . A sufficient and necessary characterization of AE solutions to interval system of maxplus linear equations (2.3) is described in the following theorem.

Theorem 5.1 A vector x ∈ R n max is an AE solution of two-sided interval linear max-plus equations (2.3) if and only if
Proof Assume that x is an AE solution of system A ⊗ x = B ⊗ x, then it is an AE solution of A ⊗ x ≤ B ⊗ x and an AE solution of B ⊗ x ≤ A ⊗ x. Therefore, inequalities (5.1) and (5.2) hold due to Theorem 4.1.
Conversely, assume that vector x ∈ R n max satisfies inequalities (5.1) and (5.2). For the opposite implication, suppose that vector x is not an AE solution of Therefore, there exists an i 0 ∈ {1, . . . , m} such that If inequality (5.3) is satisfied, due to the isotone property, we have which leads to a contradiction of (5.1).

If inequality (5.4) is satisfied, due to the isotone property, we have
which leads to a contradiction of (5.2).
Thus vector x is an AE solution of A ⊗ x = B ⊗ x. This completes the proof.

Deriving the solution algorithm
From Theorems 4.1 and 5.1, we observe that the equivalent systems of AE solutions are both in a general form of a two-sided systems of max-plus linear inequalities In this section, we present Example 2 in order to show how to find the AE solution set of a two-sided interval linear system of inequalities A ⊗ x ≤ B ⊗ x, by using the following theorems proposed in [10]. And this method is also suitable for finding the AE solution . . , j i |M | ), i 1 < · · · < i |M | and C ∈ R n×n max has the following components: . . . where By Theorem 4.1, we can obtain the equivalent system of AE solutions, By Theorem 6.1, we know that the trivial solution x = (ε, ε, . . . , ε) T is not a unique solution of system (6.2), because {j ∈ N = {1, 2, 3} : Next, by Theorem 6.2, we have By Theorems 6.3 and 6.4, we derive that the solution set is because c 21 + c 12 = 4 > 0, and the solution set Thus, a vector x is an AE solution of system (6.1) if and only if x satisfies for instance, vector x = (0, -3, 1) T is an AE solution of system (6.1).

Special cases of AE solutions of two-sided interval linear systems
As we know, mathematical definitions of various traditional solution types (weak, strong, tolerance, control) of the two-sided interval linear systems of equations A ⊗ x = B ⊗ x were presented in [10] as follows: In this section, we first extend the analogous concepts of solutions for two-sided interval linear systems of inequalities A ⊗ x ≤ B ⊗ x. From the definition of AE solutions, it is easy to obtain that the weak, strong, tolerance and control solutions are all special cases of AE solutions.
Then we propose the full characterizations of four different types of solution of system A ⊗ x ≤ B ⊗ x.   Moreover, we find that the equivalent conditions for checking such solvability types of two-sided interval systems of equations in the max-plus algebra formulated in [10] are also special cases in Theorem 5.1.