A new geometric approach to multiobjective linear programming problems

In this paper, we present a novel method for solving multiobjective linear programming problems (MOLPP) that overcomes the need to calculate the optimal value of each objective function. This method is a follow-up to our previous work on sensitivity analysis, where we developed a new geometric approach. The first step of our approach is to divide the space of linear forms into a finite number of sets based on a fixed convex polygonal subset of $\mathbb{R}^{2}$. This is done using an equivalence relationship, which ensures that all the elements from a given equivalence class have the same optimal solution. We then characterize the equivalence classes of the quotient set using a geometric approach to sensitivity analysis. This step is crucial in identifying the ideal solution to the MOLPP. By using this approach, we can determine whether a given MOLPP has an ideal solution without the need to calculate the optimal value of each objective function. This is a significant improvement over existing methods, as it significantly reduces the computational complexity and time required to solve MOLPP. To illustrate our method, we provide a numerical example that demonstrates its effectiveness. Our method is simple, yet powerful, and can be easily applied to a wide range of MOLPP. This paper contributes to the field of optimization by presenting a new approach to solving MOLPP that is efficient, effective, and easy to implement

Estimation des modèles probit polytomiques : un survol des techniques

Parce qu’il admet des structures très générales d’interdépendance entre les modalités, le probit polytomique (MNP) fournit une des formes les plus intéressantes pour modéliser les choix discrets qui découlent d’une maximisation d’utilité aléatoire. L’obstacle majeur et bien connu dans l’estimation de ce type de modèle tient à la complexité que prennent les calculs lorsque le nombre de modalités considérées est élevé. Cette situation est due essentiellement à la présence d’intégrales normales multidimensionnelles qui définissent les probabilités de sélection. Au cours des deux dernières décennies, de nombreux efforts ont été effectués visant à produire des méthodes qui permettent de contourner les difficultés de calcul liées à l’estimation des modèles probit polytomiques. L’objectif de ce texte consiste à produire un survol critique des principales méthodes mises de l’avant jusqu’à maintenant pour rendre opérationnel le cadre MNP. Nous espérons qu’il éclairera les praticiens de ces modèles quant au choix de technique d’estimation à favoriser au cours des prochaines années.The Multinomial Probit (MNP) model provides the most general framework to allow for interdependent alternatives in discrete choice analysis. The primary impediment to this methodology is related to the dimensionality of the response probabilities which are multifold normal integrals of about the size of the choice set. During the last two decades, numerous researches have been devoted to develop practical methodologies to replace these hard to compute choice probabilities in the estimation process. The main objective of this paper is to survey the major and the most important of these techniques

An adaptive method to solve multilevel multiobjective linear programming problems

This paper is a follow-up to a previous work where we defined and generated the set of all the possible compromises of multilevel multiobjective linear programming problems (ML-MOLPP). In this paper, we introduce a new algorithm to solve ML-MOLPP in which the adaptive method of linear programming is nested. First, we start by generating the set of all the possible compromises (set of all non-dominated solutions). After that, an algorithm based on the adaptive method of linear programming is developed to select the best compromise among all the possible compromises achieved. Finally, all the construction stages are carefully checked and illustrated with a numerical example

A new geometric approach for sensitivity analysis in linear programming

In this paper, we present a new geometric approach for sensitivity analysis in linear programming that is computationally practical for a decision-maker to study the behavior of the optimal solution of the linear programming problem under changes in program data. First, we fix the feasible domain (fix the linear constraints). Then, we geometrically formulate a linear programming problem. Next, we give a new equivalent geometric formulation of the sensitivity analysis problem using notions of affine geometry which consists write the coefficient vector of the objective function in polar coordinate and determining all the angles for which the solution remains unchanged. Finally, the approach is presented in detail and illustrated with a numerical example

Stiffening mechanisms in vermiculite-amorphous polyamide bio-nanocomposites

Sub-micron thick flakes were obtained by sonication of vermiculite that was first exfoliated by either thermal shock or chemical treatment with hydrogen peroxide. Dimer fatty acid polyamide nanocomposites with a mixed morphology were prepared via a solution-dispersion technique. The large (in the micrometre range) vermiculite flakes assumed random orientations in the matrix. BET surface area measurements indicated flake thickness below 100 nm but SEM showed that thicker flakes were also present. Filler content was varied up to 30 wt.%. At this loading, the tensile strength doubled, the modulus increased five-fold but the elongation-at-break decreased by a factor of ten. Dynamic mechanical analysis suggests that three stiffening mechanisms were operating. The reinforcing effect of the high stiffness inorganic flakes is the primary contributor. Together with the chain confinement effect, that expresses itself in an apparent increase in the glass transition temperature, this provided an adequate rationalisation of the stiffness variation below Tg. However, an additional stiffening effect is indicated at temperatures above Tg. The mechanism may involve dynamic network formation based on fluctuating hydrogen bonding interactions between the matrix polymer chains and the filler particles.National Research Foundation (NRF) via the South African/Algeria research partnership programme under Grant 87453.http://www.elsevier.com/locate/europolj2017-01-31hb2016Chemical Engineerin

Flame retarding polyamide 11 with exfoliated vermiculite nanoflakes

Polyamide 11‐based bionanocomposites were prepared by melt compounding with 10 wt% clays of different chemistry and morphology. This included vermiculite nanoflakes obtained by consecutive thermal and ultrasonic exfoliation in both neat and organo‐modified form. The mechanical reinforcement‐ and flame‐retardant performance of the vermiculite clays were compared to organo‐modified montmorillonite (Cloisite 30B) and needle‐shaped sepiolite (Pangel S9). Electron microscope investigations revealed different structures and dispersion levels of the clay nanoparticles in the polymer matrix. Tensile tests showed that the addition of clays led to considerable improvements in Young's modulus without compromising the elongation at break. Compared to the neat polymer, all clays reduced the peak heat release rate and the smoke production rate in cone calorimeter testing. Surprisingly, the needle‐shaped sepiolite clay and the two vermiculites outperformed the montmorillonite organoclay in the fire testing even though it featured the highest degree of exfoliation in the polymer matrix.The Deutsche Forschungsgemeinschaft (DFG) through Grant AN 212/18-1 and from the National Research Foundation (NRF) via the South African/Algeria research partnership program under Grant 87453.https://onlinelibrary.wiley.com/journal/154826342019-10-01hj2019Chemical Engineerin

Accelerated photo-oxidation of Polyamide 11 nanocomposites under various clays nanofillers

DATA AVAILABILITY STATEMENT : Data available on request from the authors.The effect of various clays nanofillers on the photo-oxidation of polyamide 11 (PA11) has been investigated by accelerated ultraviolet (UV) test up to 780 h. Organo-modified montmorillonite, halloysite nanotubes, and sepiolite were selected and incorporated separately to PA11 at 5 wt.%. The samples were prepared by melt compounding. Fourier transform infrared (FTIR) data showed a linear increase of carbonyl index (CI) in the first 360 h of exposure indicating a rapid oxidation of all samples without any induction period. Further, the nanocomposite samples exhibit faster oxidation kinetics than PA11, being however less pronounced for PA11/sepiolite. This is consistent with both the yellowing index (YI) evolution determined by UV–Vis spectroscopy and also the onset oxidation temperature (OOT) determined by differential scanning calorimetry (DSC).The DGRSDT (Direction Générale de la Recherche Scientifique et du développement Technologique) from Algeria.https://onlinelibrary.wiley.com/journal/15213900hj2023Chemical EngineeringChemistr

Tribological and mechanical properties of polyamide-11/halloysite nanotube nanocomposites

This article reports some morphological, tribological, and mechanical data on polyamide-11(PA11)/ halloysite nanotube (HNT) nanocomposites prepared by melt-compounding. HNTs extracted from the Djebel Deb- bagh deposit in Algeria were incorporated into the polymer at 1, 3, and 5 wt%. For comparison, commercial HNTs were also used under the same processing conditions. Scanning electron microscopy showed that both HNTs were homogeneously dispersed in the PA11 matrix, despite the presence of few aggregates, in particular at higher filler contents. The tribological properties were significantly improved, resulting in a decrease in the friction coefficient and the wear rate characteristics due to the lubricating effect of HNTs. This is consistent with optical profilometry data, which evidenced the impact of both types of HNTs on the surface topography of the nanocomposite samples, in which the main wear process was plastic deformation. Furthermore, Young’s modulus and tensile strength were observed to increase with the filler content, but to the detriment of elongation at break and impact strength. Regarding the whole data, the raw Algerian halloysite led to interesting results in PA11 nanocomposites, thus reveal- ing its potential in polymer engineering nanotechnology

La Méthode Adaptée et l’Optimisation Multicritère à Plusieurs Niveaux

This thesis primarily focuses on the development of methods for solving multi-level linear programming problems based on the principle of the adaptive method. Efficiently solving multi-level problems remains a complex challenge, which is also evident in the majority of the problems addressed in this work. The goal of this thesis is to generalize the adaptive method for solving multi-level multi-criteria linear programming problems. Chapter 3 presents a geometric study of solution stability in single-objective linear programming problems. Chapters 5 and 6 are dedicated to constructing procedures based on the principle of the adaptive method to solve multi-level linear programming problems. In Chapter 5, an advanced technical construction of an application is established to define a new sub-optimality estimate. These constructions are then utilized to develop two versions of an algorithm that solves single-objective multi-level linear programming problems. Lastly, in Chapter 6, we generalize the concept of non-dominated solutions in the context of multi-level linear problems and introduce a method to generate them. This new concept is employed to describe an algorithm that applies the adaptive method to achieve a unique compromise. The obtained results demonstrate the effectiveness and relevance of the developed methods in solving multi-level linear programming problems. This thesis makes significant contributions to the resolution of complex problems and opens up new prospects for future research in this field.La présente thèse se concentre principalement sur le développement de méthodes de résolution de problèmes de programmation linéaire à plusieurs niveaux, en se basant sur le principe de la méthode adaptée. La résolution efficace de problèmes multi-niveaux reste un défi complexe. Cette difficulté se retrouve également dans la plupart des problèmes abordés dans ce travail. L'objectif de cette thèse est de généraliser la méthode adaptée pour résoudre les problèmes de programmation linéaire multi-critère à plusieurs niveaux. Dans le chapitre 3, une étude géométrique de la stabilité des solutions d'un problème de programmation linéaire mono-critère est présentée. Les chapitres 5 et 6 sont consacrés à la construction de procédures basées sur le principe de la méthode adaptée pour résoudre les problèmes de programmation linéaire à plusieurs niveaux. Dans le chapitre 5, une construction technique avancée d'une application est établie pour définir une nouvelle estimation de sous-optimalité. Ces constructions sont ensuite utilisées pour développer deux versions d'un algorithme résolvant les problèmes de programmation linéaire multi-niveaux à objectif unique. Enfin, dans le chapitre 6, nous généralisons le concept de solutions non-dominées dans le contexte des problèmes linéaires multi-niveaux, et nous mettons en place une méthode pour les générer. Ce nouveau concept est utilisé pour décrire un algorithme qui applique la méthode adaptée afin d'atteindre un compromis unique. Les résultats obtenus démontrent l'efficacité et la pertinence des méthodes développées pour résoudre les problèmes de programmation linéaire à plusieurs niveaux. Cette thèse apporte ainsi des contributions significatives à la résolution de problèmes complexes et ouvre de nouvelles perspectives pour la recherche future dans ce domaine