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

New Auction Algorithms for the Assignment Problem and Extensions

We consider the classical linear assignment problem, and we introduce new auction algorithms for its optimal and suboptimal solution. The algorithms are founded on duality theory, and are related to ideas of competitive bidding by persons for objects and the attendant market equilibrium, which underlie real-life auction processes. We distinguish between two fundamentally different types of bidding mechanisms: aggressive and cooperative. Mathematically, aggressive bidding relies on a notion of approximate coordinate descent in dual space, an epsilon-complementary slackness condition to regulate the amount of descent approximation, and the idea of epsilon-scaling to resolve efficiently the price wars that occur naturally as multiple bidders compete for a smaller number of valuable objects. Cooperative bidding avoids price wars through detection and cooperative resolution of any competitive impasse that involves a group of persons. We discuss the relations between the aggressive and the cooperative bidding approaches, we derive new algorithms and variations that combine ideas from both of them, and we also make connections with other primal-dual methods, including the Hungarian method. Furthermore, our discussion points the way to algorithmic extensions that apply more broadly to network optimization, including shortest path, max-flow, transportation, and minimum cost flow problems with both linear and convex cost functions

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D