Edouard Wagneur
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Tropical algebra is the algebra constructed over the tropical semifield \(R_{max}= (R\cup \{-\infty\},\max, +)\)
. We show here that every \(m\)
-dimensi...
Tropical algebra is the algebra constructed over the tropical semifield \(\mathbb{R}_{max}\)
. After revisiting the classification of 2-dimensional `(\math...
We show here that every \(m\)
-dimensional semiring module \(M\)
over an idempotent semiring \(S\)
with strongly independent basis can be embedded
in...
Tropical algebra is the study of algebraic structures (tropical maps, idempotent semimodules, etc), constructed over the semifield `(\underline{\mathbb{R}} ...
BibTeX referenceThe Injectivity Modules of a Tropical Map
In <font size=2>[Wagneur, E., Linear Algebra and its Application, 435, 1786-1795, 2011]</font>, we show that any \(m\)
-dimensional tropical torsion module...
We prove here a tropical version of the well-known Whitney embedding theorem (1944) stating that a smooth <i>m</i>-dimensional compact manifold can be e...
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We study here a flow-line which covers a wide range of real-world situations, from plastic molding to steel milling and the service industries, where the dur...
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We give an explicit description of the basic solutions of max-linear systems <img src="/cgi-bin/mimetex.cgi?A\otimes x\leq B\otimes x"> with two inequalities.
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We consider an inequality of the type <img src="/cgi-bin/mimetex.cgi?A\cdot x\le B\cdot x"> over the idempotent semifield <img src="/cgi-bin/mimetex.cgi?R_{...
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We study conflict and cooperation issues in a two-stage production system. The objective of the first stage is to minimize the sum of the completion times o...
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We investigate here the "anatomy" of idempotent semimodules, i.e. we look for the equivalent of the classical decomposition of a module over a principal ide...
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We consider the problem of minimizing makespan in a no-wait flow-shop with three machines. Lot streaming (lot sizing) is the process of creating sublots t...
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In classical module theory, a module over a principal ideal domain may be split into the direct sum of a free module and a torsion module. This decompositio...
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A module over a principal ideal domain splits into a direct sum of a free module and a torsion module. This decomposition does not hold in general for semim...
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We are interested here in the reachability and controllability problems for DEDS in the max-algebra. We show that these problems lead to an eigenvector prob...
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We consider the problem of minimizing makespan in two-machine no-wait flowshops with multiple products requiring lot streaming. A "product" (or lot) consist...
BibTeX referenceShort Exact Sequences of Pseudomodules: a Canonical Approach to the Decomposition of Pseudomodules
We show here how to define short exact sequences in pseudomodules. More precisely, for a given subpseudomodule <i>M</i> of a pseudomodule <i>N</i>, we defin...
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A semimodule <i>M</i> over an idempotent semiring <i>P</i> is also idempotent. When <i>P</i> is linearly ordered and conditionnally complete, we call it a p...
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We consider a permutation flow-shop with <i>n</i> jobs and <i>m</i> machines, where the jobs processing times are given by a monotone nondecreasing function...
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It is an old observation that for the Generalized Assignment Problem (GAP) a Linear Programming (LP) relaxation introduces only few fractions in the solutio...
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If the processing time of each job in a flow-shop also depends on waiting time (or on the time spent prior to processing), then the choice of a sequence inf...
BibTeX referenceThe Preference for Quality
We investigate here the problems raised by the demand for quality. In particular, we show that the (inverse) demand, which yields the price the consumers ar...
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We investigate the complexity status of the preemptive openshop scheduling problem. After reviewing recent studies of the shop with various objective functi...
BibTeX referenceOpenshops with Jobs Overlap
We consider the complexity status of scheduling <i>n</i> jobs in an openshop with <i>m</i> machines, when overlapping of jobs is permitted, for some classic...
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The (<b>R</b>,max,+) algebra gives raise, to a "linear algebra" very usefull in the context of DEDS which can be represented by envent graphs. Some propert...
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L'algèbre (max, +) est un outil important en Recherche Opérationnelle et en Automatique dans le contexte de l'analyse des systèmes à événements discrets. L'...
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Consider a flow-shop with m processors and <i>n</i> jobs, whose processing times are state dependent. Since the state often also depend on the sequence, a h...
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The concept of moduloïd over a dioïd has been introduced in M. Gondran and M. Minoux [8] for the algebraic structure left invariant under the action of a ma...
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The problem of two processors in series and <i>n</i> parts is considered, when processing times depend on some state variable. It is first shown that the mi...
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We determine here sufficient conditions for finite dimensional moduloïds and pseudomodules to be lattices. As could be expected, completeness of the scalar ...
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Consider a flow-shop with <i>n</i> parts, whose processing times are state dependent. Since the state often depends on the sequence, a hierarchical approach...
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Consider the optimal control problem for the two processor flow-shop when processing time is a (linear) function of the state. Since the latter also depends...
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Consider the optimal control problem for the two processor flow-shop when processing times for <i>n</i> tasks are (linear) functions of the state. Since th...
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We study here the classification problem for a particular class of moduloïds over a dioïd, called simple pseudomodules. After introducing the concept of can...
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In the analysis of discrete event systems, we need to consider sequences and/or matrices of elements in a given numerical set <i>D</i> (ℕ, Z, Q or &#x...
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In this paper, I give a counterexample to the closedness property of the set of possible current production vectors <i>Y<sub>f</sub></i> which is used by J....
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Nous présentons un aperçu des principales idées permettant la formulation du phénomène de négociation collective dans le langage de la théorie des jeux. Apr...
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This research deals with the Planning of Operation on a two-machine cell consisting of: a) a (NC) turret-mounted multi-tool punch, b) a (NC) shear. The cell...
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A process planning problem typically encountered in flexible manufacturing systems treating laminated material (e.g. metal sheets) is modelled and solved. I...
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A Travelling Salesman Problem in a three-dimensional spatial domain is associated with a scheduling problem encountered in a flexible manufacturing cell incl...
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In consumer theory, each good and its price may be considered conjugate. However, it is not clear which good is conjugate to the budget, or which dual varia...
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The problem of exogenity in economics is a highly relative one. For example in microeconomics (production theory or consumer theory) quantities are exogeneo...
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A regular consumption unit is defined by (Ω, ⩽, W<sub>S</sub>), where Ω is an open connected subset of ℝ<sup>n</sup> and ⩽ i...
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