Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by a⊕b = max{a, b} and a⊗b = a+b, respectively. Each system of linear equation in max-plus algebra we can write in the matrix form A⊗ x = b, where A and b are matrix and vector of suitable size. If we replace the matrix elements with matrix interval A = [A,A] and vector elements by vector int...

A variety of problems in non-linear time-evolution systems such as manufacturing plants, operations research, computer networks, etc., can be modelled as min-max-plus systems in which operations of min, max and addition appear simultaneously. It is well know that systems with only maximum (or minimum) constraints can be modelled as max-plus system and handled by max-plus algebra which changes t...

This paper summarizes results on some topics in the max-plus convex geometry, mainly concerning the role of multiorder, Kleene stars and cyclic projectors, and relates them to some topics in max algebra. The multiorder principle leads to max-plus analogues of some statements in the finite-dimensional convex geometry and is related to the set covering conditions in max algebra. Kleene stars are ...

. The High-Variety, Low-Volume (HVLV) scheduling problem is one of the most arduous and combinatorial optimization problems. This paper presents an analytical scheduling model using a tropical algebra called (max,+) algebra. The aim is to find an allocation for each operation and to define the sequence of operations on each machine, so that the resulting schedule has a minimal completion time a...

Let A = (aij ) ∈ Rn×n,N = {1, . . . , n} and DA be the digraph (N, {(i, j); aij > −∞}). The matrix A is called irreducible if DA is strongly connected, and strongly irreducible if every maxalgebraic power of A is irreducible. A is called robust if for every x with at least one finite component, A(k) ⊗ x is an eigenvector of A for some natural number k. We study the eigenvalue–eigenvector proble...

In this paper, we focus on the cyclic job-shop problem. This problem consists in determining the order of a set of generic tasks on machines in order to minimize the cycle time of the sequence. We propose an exact method to solve this problem. For each solution, a linear max-plus model (possibly non causal) is obtained. To evaluate the performance of a considered schedule, we build the causal m...

Exotic semirings such as the “(max;+) semiring” (R[f 1g;max;+), or the “tropical semiring” (N[f+1g;min;+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (lo...

Abstract: Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively. We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known re...

The eigenvalue problem for an irreducible nonnegative matrix A = a ij ] in the max algebra system is A x = x, where (A x) i = max j (a ij x j) and turns out to be the maximum circuit geometric mean, (A). A power method algorithm is given to compute (A) and eigenvector x. This method generalizes and simpliies an algorithm due to Braker and Olsder. The algorithm is developed by using results on t...