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  • The minimum sum representation as an index of voting power

     Freixas Bosch, Josep; Kaniovski, Serguei
    European journal of operational research
    Date of publication: 2014-03-16
    Journal article

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    We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley-Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

  • Voting games with abstention: linking completeness and weightedness

     Freixas Bosch, Josep; Tchantcho, Bertrand; Tedjeugang, Narcisse
    Decision support systems
    Date of publication: 2014-01-01
    Journal article

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    Weighted games for several levels of approval in input and output were introduced in [9]. An extension of the desirability relation for simple games, called the influence relation, was introduced for games with several levels of approval in input in [24] (see also [18]). However, there are weighted games not being complete for the influence relation, something different to what occurs for simple games. In this paper we introduce several extensions of the desirability relation for simple games and from the completeness of them it follows the consistent link with weighted games, which solves the existing gap. Moreover, we prove that the influence relation is consistent with a known subclass of weighted games: strongly weighted games. (C) 2013 Elsevier B.V. All rights reserved.

  • On minimum integer representations of weighted games

     Freixas Bosch, Josep; Kurz, Sascha
    Mathematical social sciences
    Date of publication: 2014-01-01
    Journal article

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    We study minimum integer representations of weighted games, i.e. representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if they exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for t >= 4 types of voters without a minimum integer representation preserving types, i.e. where we additionally require that the weights are equal within equivalence classes of voters. (C) 2013 Elsevier B.V. All rights reserved.

  • The golden number and Fibonacci sequences in the design of voting systems

     Freixas Bosch, Josep; Kurz, Sascha
    European journal of operational research
    Date of publication: 2013
    Journal article

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    Some distinguished types of voters, as vetoes, passers or nulls, as well as some others, play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power. In this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters. With this purpose in mind we enumerate these special classes of games and find out that its number always follows a Fibonacci sequence with smooth polynomial variations. As a consequence we find several families of games with the same asymptotic exponential behavior except for a multiplicative factor which is the golden number or its square. From a more general point of view, our studies are related with the design of voting structures with a predetermined importance ranking.

  • Circumstantial power: some hints for finding optimal persuadable or bribable voters

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Date of publication: 2013
    Book chapter

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    The paper proposes new tools for measuring power for individulas in binary decision schemes. Two optimization problems are formulated for finding optimal persuadable voters and optimal bribable voters. By using three preorderings we find sufficient conditions to select a list of voters to be persuaded or bribed, given any particular ranking of their predictions.

  • Some open problems in simple games

     Bertini, Cesarino; Freixas Bosch, Josep; Gambarelli, Gianfranco; Stach, Izabella
    International game theory review
    Date of publication: 2013
    Journal article

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    This paper presents a review of literature on simple games and highlights various open problems concerning such games; in particular, weighted games and power indices.

  • Comparing power indices

     Bertini, Cesarino; Freixas Bosch, Josep; Gambarelli, Gianfranco; Stach, Izabella
    International game theory review
    Date of publication: 2013
    Journal article

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    This paper aims to give a global vision concerning the state of the art of studies on 13 power indices and to establish which of them are more suitable for describing the real situations which are, from time to time, taken into consideration. In such contexts, different comparisons have been developed in terms of properties, axiomatic grounds and so on. This analysis points out various open problems.

  • Egalitarian properties for power indices

     Freixas Bosch, Josep; Marciniak, Dorota
    Social choice and welfare
    Date of publication: 2013
    Journal article

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    In this study, we introduce and examine the Egalitarian property for some power indices on the class of simple games. This property means that after intersecting a game with a symmetric or anonymous game the difference between the values of two comparable players does not increase. We prove that the Shapley¿Shubik index, the absolute Banzhaf index, and the Johnston score satisfy this property. We also give counterexamples for Holler, Deegan¿Packel, normalized Banzhaf and Johnston indices. We prove that the Egalitarian property is a stronger condition for efficient power indices than the Lorentz domination.

  • Teoría de juegos: fundamentos matemáticos y aplicaciones

     Amer Ramon, Rafael; Carreras Escobar, Francisco; Freixas Bosch, Josep; Giménez Pradales, José Miguel; Llongueras Arola, Maria Dolors; Magaña Nieto, Antonio; Molinero Albareda, Xavier; Pons Vallès, Montserrat; Puente Del Campo, Mª Albina; Sales Ingles, Vicente
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  • Probabilistic power indices for voting rules with abstention

     Freixas Bosch, Josep
    Mathematical social sciences
    Date of publication: 2012-07
    Journal article

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    In this paper, we introduce eight power indices that admit a probabilistic interpretation for voting rules with abstention or with three levels of approval in the input, briefly (3, 2) games.Weanalyze the analogies and discrepancies between standard known indices for simple games and the proposed extensions for this more general context. A remarkable difference is that for (3, 2) games the proposed extensions of the Banzhaf index, Coleman index to prevent action and Coleman index to initiate action become non-proportional notions, contrarily to what succeeds for simple games. We conclude the work by providing procedures based on generating functions for weighted (3, 2) games, and extensible to (j,k) games, to efficiently compute them.

  • Complete voting systems with two types of voters: weightedness and counting

     Freixas Bosch, Josep; Molinero Albareda, Xavier; Roura Ferret, Salvador
    Annals of operations research
    Date of publication: 2012
    Journal article

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    We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.

    We investigate voting systems with two classes of voters, for which there is a hierarchy giving each member of the stronger class more influence or important than each member of the weaker class. We deduce for voting systems one important counting fact that allows determining how many of them are for a given number of voters. In fact, the number of these systems follows a Fibonacci sequence with a smooth polynomial variation on the number of voters. On the other hand, we classify by means of some parameters which of these systems are weighted. This result allows us to state an asymptotic conjecture which is opposed to what occurs for symmetric games.

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    Computation of several power indices by generating functions  Open access

     Alonso Meijide, José María; Freixas Bosch, Josep; Molinero Albareda, Xavier
    Applied mathematics and computation
    Date of publication: 2012
    Journal article

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    In this paper we propose methods to compute the Deegan-Packel, the Public Good, and the Shift power indices by generating functions for the particular case of weighted voting games. Furthermore, we define a new power index which combines the ideas of the Shift and the Deegan-Packel power indices and also propose a method to compute it with generating functions. We conclude by some comments about the complexity to compute these power indices.

    Postprint (author’s final draft)

  • On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices

     Freixas Bosch, Josep; Marciniak, Dorota; Pons Vallès, Montserrat
    European journal of operational research
    Date of publication: 2011
    Journal article

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  • A note on decisive symmetric games

     Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Decision support systems
    Date of publication: 2011
    Journal article

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    Binary voting systems, usually represented by simple games, constitute a main DSS topic. A crucial feature of such a system is the easiness with which a proposal can be collectively accepted, which is measured by the “decisiveness index” of the corresponding game. We study here several functions related to the decisiveness of any simple game. The analysis, including the asymptotic behavior as the number n of players increases, is restricted to decisive symmetric games and their compositions, and it is assumed that all players have a common probability p to vote for the proposal. We show that, for n large enough, a small variation, either positive or negative, in p when p=1/2 takes the decisiveness to quickly approach, respectively, 1 or 0. Moreover, we analyze the speed of the decisiveness convergence.

  • On the complexity of problems on simple games

     Freixas Bosch, Josep; Molinero Albareda, Xavier; Olsen, Martin; Serna Iglesias, Maria Jose
    RAIRO. Operations research
    Date of publication: 2011-10
    Journal article

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  • Decisiveness of decisive symmetric games

     Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Date: 2011-04-08
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  • On minimal integer representations for weighted games

     Freixas Bosch, Josep; Kurz, Sascha
    Date: 2011-03-04
    Report

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  • On $\alpha$-roughly weighted games

     Freixas Bosch, Josep; Kurz, Sascha
    Date: 2011-12-13
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  • On the generalized decisiveness of decisive symmetric games

     Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Congreso Nacional de Estadística e Investigación Operativa
    Presentation's date: 2010-09-17
    Presentation of work at congresses

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  • A new power index based on minimal winning coalitions without any surplus

     Alonso-Meijide, José María; Freixas Bosch, Josep
    Decision support systems
    Date of publication: 2010-04
    Journal article

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    In this paper we propose a new power index useful for the evaluation of each member in a committee, or democratic institution, and the degree of influence over the voting decision making system. The proposed solution is based on the observation that democratic organizations not only tend to form coalitions which can by themselves guarantee the control of the organization, but that they also do it in an extremely efficient way that avoids the inclusion of powerful members if they can be replaced by weaker ones. The mathematical foundation of the new measure is based on two different axiomatizations. A comparison with other well-known measures is also done.

  • Weighted games without a unique minimal representation in integers

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Optimization methods software
    Date of publication: 2010-04
    Journal article

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  • Hierarchies achievable in simple games

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Theory and decision
    Date of publication: 2010-04
    Journal article

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    A previous work by Friedman et al. (Theory and Decision, 61:305–318, 2006) introduces the concept of a hierarchy of a simple voting game and characterizes which hierarchies, induced by the desirability relation, are achievable in linear games. In this paper, we consider the problem of determining all hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, achievable in simple games. It is proved that only four hierarchies are non-achievable in simple games.Moreover, it is also proved that all achievable hierarchies are already obtainable in the class of weakly linear games. Our results prove that given an arbitrary complete pre-ordering defined on a finite set with more than five elements, it is possible to construct a simple game such that the pre-ordering induced by the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices coincides with the given pre-ordering.

  • On ordinal equivalence of the Shapley and Banzhaf values for cooperative games

     Freixas Bosch, Josep
    International journal of game theory
    Date of publication: 2010-09
    Journal article

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    In this paper I consider the ordinal equivalence of the Shapley and Banzhaf values for TU cooperative games, i.e., cooperative games for which the preorderings on the set of players induced by these two values coincide. To this end I consider several solution concepts within semivalues and introduce three subclasses of games which are called respectively: weakly complete, semicoherent and coherent cooperative games. A characterization theorem in terms of the ordinal equivalence of some semivalues is given for each of these three classes of cooperative games. In particular, the Shapley and Banzhaf values as well as the segment of semivalues they limit are ordinally equivalent for weakly complete, semicoherent and coherent cooperative games.

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    Detection of paradoxes of power indices for simple games  Open access

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Date of publication: 2010-06
    Book chapter

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  • Maximum tolerance and maximum greatest tolerance

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Date of publication: 2010
    Book chapter

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    On the notion of dimension and codimension of simple games  Open access

     Freixas Bosch, Josep; Marciniak, Dorota
    Date of publication: 2010-06
    Book chapter

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  • Some hints on finding the most important components in a system

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Date of publication: 2010
    Book chapter

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  • Teoría de juejos y sistemas de decisión colectiva con aplicaciones tecnológicas, económicas, políticas y sociológicas

     Freixas Bosch, Josep; Amer Ramon, Rafael; Carreras Escobar, Francisco; Giménez Pradales, José Miguel; Llongueras Arola, Maria Dolors; Molinero Albareda, Xavier; Pons Vallès, Montserrat; Puente Del Campo, Mª Albina; Sales Ingles, Vicente; Magaña Nieto, Antonio
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  • Simple games and weighted games: a theoretical and computational viewpoint

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Discrete applied mathematics
    Date of publication: 2009-04
    Journal article

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    It is a well-known result in the theory of simple games that a game is weighted if and only if it is trade robust. In this paper we propose a variant of trade robustness, that we call invariant-trade robustness, which is enough to determine whether a simple game is weighted. To test whether a simple game is invariant-trade robust we do not need to consider all winning coalitions; a reduced subset of minimal winning coalitions is enough. We make a comparison between the two methods (trade robustness and invariant-trade robustness) to check whether a simple game is weighted. We also provide by means of algorithms a full classification using both methods, for simple games with less than 8 voters according to the maximum level of (invariant-)trade robustness they achieve.

  • On the existence of a minimum integer representation for weighted voting systems

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Annals of operations research
    Date of publication: 2009-02
    Journal article

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    A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming. For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters

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    Anonymous yes-no voting with abstention and multiple levels of approval  Open access

     Freixas Bosch, Josep; Zwicker, William S.
    Games and economic behavior
    Date of publication: 2009-11
    Journal article

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    Els jocs simples simètrics (3, 2) serveixen de models per a sistemes de votació anònims en els quals cada votant pot votar "sí", abstenir-se, o votar "no", el resultat és "sí" o "no", i els votants juguen papers intercanviables. L'extensió a jocs simples simètrics (j,2), en els quals cada votant tria entre j nivells d'aprovació ordenats, també n'imita algunes regles naturals de decisió, com sistemes classificats passa/falla. Cada un dels jocs està determinat pel conjunt de perfils (anònims) mínims vencedors. Això fa possible comptar els sistemes possibles, i els recomptes suggereixen alguns patrons interessants. En el cas (3, 2), l'aproximació sosté una versió del Teorema de May, classificant tota les normes de votació anònima possibles amb abstenció en termes de quote functions. A diferència de la situació per jocs ordinaris simples aquests resultats revelen que la classe de jocs simples amb 3 o més nivells d'aprovació es manté ampli i variat, fins i tot després de la imposició de la simetria.

  • A minimum dimensional class of simple games

     Freixas Bosch, Josep; Marciniak, Dorota
    TOP
    Date of publication: 2009-12
    Journal article

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  • Hierarchies in voting simple games

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Date of publication: 2009-06
    Book chapter

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  • Minimum dimension and codimension for simple games

     Freixas Bosch, Josep; Marciniak, Dorota
    Date of publication: 2009-02
    Book chapter

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  • Weightedness for simple games with less than 9 voters

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Date of publication: 2009-06
    Book chapter

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  • Grup de recerca en teoria de jocs

     Amer Ramon, Rafael; Freixas Bosch, Josep; Giménez Pradales, José Miguel; Llongueras Arola, Maria Dolors; Magaña Nieto, Antonio; Pons Vallès, Montserrat; Puente Del Campo, Mª Albina; Sales Ingles, Vicente; Carreras Escobar, Francisco
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  • The allowable hierarchies in simple games

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Date: 2009
    Report

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    Estudi de les jerarquies que són possibles per jocs simples

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    Maximum tolerance and maximum greatest tolerance  Open access

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Date: 2009-09-23
    Report

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    An important consideration when applying neural networks is the sensitivity to weights and threshold in strict separating systems representing a linearly separable function. Two parameters have been introduced to measure the relative errors in weights and threshold of strict separating systems: the tolerance and the greatest tolerance. Given an arbitrary separating system we study which is the equivalent separating system that provides maximum tolerance or/and maximum greatest tolerance.

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    On the complexity of problems on simple games  Open access

     Freixas Bosch, Josep; Molinero Albareda, Xavier; Olsen, Martin; Serna Iglesias, Maria Jose
    Date: 2009-10-08
    Report

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    Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes–no voting system is a set of rules that specifies exactly which collections of “yea” votes yield passage of the issue at hand, each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game. We finalize with some considerations on the possibility of representing a game in a more succinct representation showing a natural representation in which the recognition problem is hard.

  • Problemes resolts de matemàtiques per a titulacions de grau

     Freixas Bosch, Josep
    Date of publication: 2009-11
    Book

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  • Ordinal equivalence of the Shapley and Banzhaf values

     Freixas Bosch, Josep
    Date: 2009
    Report

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    Estudi de la coincidència de les ordenacions donades per aquests valors en jocs TU cooperatius

  • A new power index based on minimal winning coalitions

     Alonso-Meijide, José María; Freixas Bosch, Josep
    Date: 2009
    Report

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    Nou índex de poder basat en coalicions shift-minimals

  • Consecutive expansions of k-out-of-n structures

     Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Date: 2009
    Report

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    Estudi dels sistemes obtinguts per expansions succesives

  • The influence of the node criticality relation on some measures of component importance

     Freixas Bosch, Josep; Pons Vallès, Montserrat
    Date: 2009
    Report

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    Estudi de la relació entre diverses mesures d'importància probabilístiques i la relació crítica associada al sistema

  • Weighted games without a unique minimal representation in integers

     Freixas Bosch, Josep; Molinero Albareda, Xavier
    Date: 2009
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  • Minimal dimensional classes of simple games

     Freixas Bosch, Josep; Marciniak, Dorota
    Date: 2009
    Report

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    Estudi de classes mínimes de jocs simples amb dimensió i codimensió així com de les relacions entre elles

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    A Fibonacci sequence for linear structures with two types of components  Open access

     Freixas Bosch, Josep; Molinero Albareda, Xavier; Roura Ferret, Salvador
    Date: 2009-07-23
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    We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters.

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    Consecutive expansions of k-out-of-n systems  Open access

     Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Operations research letters
    Date of publication: 2009-11
    Journal article

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    S'introdueixen expansions de sistemes k-out-of-n. Es proposen algunes fórmules per fer més fàcil de comptabilitzar aquestes mesures i estudiar-ne les jerarquies.

  • On the reliability of the self-dual k-out-of-n systems

     Carreras Escobar, Francisco; Freixas Bosch, Josep; Puente Del Campo, Mª Albina
    Technometrics
    Date of publication: 2008-02
    Journal article

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    Here we consider a class of k-out-of-n systems and study the variation of their reliability under different assumptions. We show that a way to achieve highly reliable self-dual k-out-of-n systems is to maintain the reliability of the components at a level of performance above 1/2. We extend our asymptotic results to systems with multiple self-dual k-out-of-n subsystems.