Problemas de asignación de recursos humanos a través del problema de asignación multidimensional

149 páginas. Doctorado en Optimización.El problema de asignación de personal aparece en diversas industrias. La asignación eficiente de personal a trabajos, proyectos, herramientas, horarios, entre otros, tiene un impacto directo en términos monetarios para el negocio. El problema de asignación multidimensional (PAM) es la extensión natural del problema de asignación y puede ser utilizado en aplicaciones donde se requiere la asignación de personal. El caso más estudiado de PAM es el problema de asignación en tres dimensiones, sin embargo en años recientes han sido propuestas algunas heurísticas de búsqueda local y algoritmos meméticos para el caso general. En este trabajo de tesis se realiza un estudio profundo de PAM comenzando con un resumen del estado del arte de algoritmos, heurísticas y metaheurísticas para su resolución. Se describen algunos algoritmos y se propone uno nuevo que resuelve instancias de tamaño medio para PAM. Se propone la generalización de las conocidas heurísticas de variación de dimensión como una búsqueda local generalizada que proporciona un nuevo estado del arte de búsquedas locales para PAM. Adicionalmente, se propone un algoritmo memético con una estructura sencilla pero efectiva y que es competitivo con el mejor algoritmo memético conocido para PAM. Finalmente, se presenta un caso particular de problema de asignación de personal: el Problema de Asignación de Horarios (PAH). El PAH considera la asignación de personal a uno, dos o más conjuntos de objetos, por ejemplo puede ser requerida la asignación de profesores a cursos a periodos de tiempo a salones, para determinados grupos de estudiantes. Primero, se presenta el PAH así como una breve descripción de su estado del arte. Luego, se propone una nueva forma de modelar este problema a través de la resolución de PAM y se aplica sobre el PAH en la Universidad Autónoma Metropolitana, unidad Azcapotzalco (UAM-A). Se describen las consideraciones particulares del PAH en la UAM-A y proponemos una nueva solución para éste. Nuestra solución se basa en la resolución de múltiples PA3 a través de los algoritmos y heurísticas propuestos.Personnel assignment problems appear in several industries. The e cient assignment of personnel to jobs, projects, tools, time slots, etcetera, has a direct impact in terms monetary for the business. The Multidimensional Assignment Problem (MAP) is a natural extension of the well-known assignment problem and can be used on applications where the assignment of personnel is required. The most studied case of the MAP is the three dimensional assignment problem, though in recent years some local search heuristics and memetic algorithms have been proposed for the general case. Let X1; : : : ;Xs be a collection of s 3 disjoint sets, consider all combinations that belong to the Cartesian product X = X1 Xs such that each vector x 2 X, where x = (x1; : : : ; xs) with xi 2 Xi 8 1 i s, has associated a weight w(x). A feasible assignment is a collection A = (x1; : : : ; xn) of n vectors if xi k 6= xj k for each i 6= j and 1 k s. The weight of an assignment A is given by w(A) = Pn i=1 w(xi). A MAP in s dimensions is denoted as sAP. The objective of sAP is to nd an assignment of minimal weight. In this thesis we make an in depth study of MAP beginning with the state-ofthe- art algorithms, heuristics, and metaheuristics for solving it. We describe some algorithms and we propose a new one for solving optimally medium size instances of MAP. We propose the generalization of the called dimensionwise variation heuristics for MAP and a new generalized local search heuristic that provides new state-of-theart local searches for MAP. We also propose a new simple memetic algorithm that is competitive against the state-of-the-art memetic algorithm for MAP. In the last part of this thesis, we study a particular case of personnel assignment problem: the School Timetabling Problem (STP). The STP considers the assignment of personnel to other two or more sets, for example the assignment of professors to courses to time slots to rooms can be required. First, we provide a brief description of the state-of-the-art for STP. Then, we introduce a new approach for modeling this problem through the resolution of several MAP and we apply our solution on a real life case of study: STP at the Universidad Autonoma Metropolitana campus Azcapotzalco (UAM-A). We provide the particular aspects for STP at UAM-A and we provide a new solution for this problem. Our approach is based on solving several 3AP considering the introduced model and our proposed techniques.Consejo Mexiquense de Ciencia y Tecnología (Comecyt).Consejo Nacional de Ciencia y Tecnología (México

Solution Approaches to the Three-index Assignment Problem

This thesis explores the axial Three-Index Assignment Problem (3IAP), also called the Multidimensional Assignment Problem. The problem consists in allocating n jobs to n machines in n factories, such that exactly one job is executed by one machine in one factory at a minimum total cost. The 3IAP is an extension of the classical two-dimensional assignment problem. This combinatorial optimisation problem has been the subject of numerous research endeavours, and proven NP-hard due to its inextricable nature. The study adopts an algorithmic approach to develop swift and e ective methods for solving the problem, focusing on balancing computational e ciency and solution accuracy. The Greedy-Style Procedure (GSP) is a novel heuristic algorithm for solving the 3IAP, guaranteeing feasible solutions in polynomial time. Speci c arrangements of cost matrices can lead to the generation of higher-quality feasible solutions. In addressing the 3IAP, analysing the tie-cases and the matrix ordering led to new variants. Further exploration of cost matrix characteristics has allowed two new heuristic classes to be devised for solving 3IAP. The approach focuses on selecting the best solution within each class, resulting in an optimal or a high-quality approximate solution. Numerical experiments con rm the e ciency of these heuristics, consistently delivering quality feasible solutions in competitive computational times. Moreover, by employing diverse optimisation solvers, we propose and implement two e ective methods to achieve optimal solutions for 3IAP in good CPU times. The study introduces two local search methods based on evolutionary algorithms to solve 3IAP. These approaches explore the solution space through random permutations and the Hungarian method. Building on this, a hybrid genetic algorithm that integrates these local search strategies has been proposed for solving the 3IAP. Implementing the Hybrid Genetic Algorithm (HGA) produces high-quality solutions with reduced computational time, surpassing traditional deterministic approaches. The e ciency of the HGA is demonstrated through experimental results and comparative analyses. On medium to large 3IAP instances, our method delivers comparable or better solutions within a competitive computational time frame. Two potential future developments and expected applications are proposed at the end of this project. The rst extension will examine the correlation between cost matrices and the optimal total cost of the assignment and will investigate the dependence structure of matrices and its inuence on optimal solutions. Copula theory and Sklar's theorem can help with this analysis. The focus will be on understanding the stochastic dependence of cost matrices and their multivariate properties. Furthermore, the impact of variations in cost distributions, is often modelled based on economic sectors. The second extension involves integrating variable costs de ned by speci c probability distributions, enhancing the comprehensive analysis of economic scenarios and their impact on the assignment problem. The study considers various well-de ned probability distributions and highlights more practical applications of the assignment problem in real-world economics. The project's original contribution lies in its algorithmic approach to investigating the 3IAP, which has led to the development of new, fast, and e cient heuristic methods that strategically balance computational speed and the accuracy of the solutions achieved

Gauge field marginal of an Abelian Higgs model

We study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon--Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the H{ö}lder--Besov-type norms introduced in arXiv:1808.09196 are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables

Graded-index optical fiber emulator of an interacting three-atom system: illumination control of particle statistics and classical non-separability

We show that a system of three trapped ultracold and strongly interacting atoms in one-dimension can be emulated using an optical fiber with a graded-index profile and thin metallic slabs. While the wave-nature of single quantum particles leads to direct and well known analogies with classical optics, for interacting many-particle systems with unrestricted statistics such analoga are not straightforward. Here we study the symmetries present in the fiber eigenstates by using discrete group theory and show that, by spatially modulating the incident field, one can select the atomic statistics, i.e., emulate a system of three bosons, fermions or two bosons or fermions plus an additional distinguishable particle. We also show that the optical system is able to produce classical non-separability resembling that found in the analogous atomic system.Comment: 14 pages, 5 figure

From correlation functions to scattering amplitudes

We study the correlators of half-BPS protected operators in N=4 super-Yang-Mills theory, in the limit where the positions of the adjacent operators become light-like separated. We compute the loop corrections by means of Lagrangian insertions. The divergences resulting from the light-cone limit are regularized by changing the dimension of the integration measure over the insertion points. Switching from coordinates to dual momenta, we show that the logarithm of the correlator is identical with twice the logarithm of the matching MHV gluon scattering amplitude. We present a number of examples of this new relation, at one and two loops.Comment: typos corrected, references adde

Correlation functions of the chiral stress-tensor multiplet in N=4 SYM

We give a new method for computing the correlation functions of the chiral part of the stress-tensor supermultiplet that relies on the reformulation of N=4 SYM in twistor space. It yields the correlation functions in the Born approximation as a sum of Feynman diagrams on twistor space that involve only propagators and no integration vertices. We use this unusual feature of the twistor Feynman rules to compute the correlation functions in terms of simple building blocks which we identify as a new class of N=4 off-shell superconformal invariants. Making use of the duality between correlation functions and planar scattering amplitudes, we demonstrate that these invariants represent an off-shell generalisation of the on-shell invariants defining tree-level scattering amplitudes in N=4 SYM

Calculating Scattering Amplitudes Efficiently

We review techniques for more efficient computation of perturbative scattering amplitudes in gauge theory, in particular tree and one-loop multi-parton amplitudes in QCD. We emphasize the advantages of (1) using color and helicity information to decompose amplitudes into smaller gauge-invariant pieces, and (2) exploiting the analytic properties of these pieces, namely their cuts and poles. Other useful tools include recursion relations, special gauges and supersymmetric rearrangements.Comment: LaTex, 45 pages with 11 encapsulated figures. Presented at the Theoretical Advanced Study Institute (TASI 95), Boulder, CO, June 4-30, 1995. Eqs. (29) and (55), and postscript bug fixe