The fuzzy p-median problem
In many location models, the strong crisp assumptions, like known demands and distances, are not realistic in most cases. The fuzzy p-median problem relaxes this hypothesis giving to the decision maker a necessary degree of freedom to solve real-world problems. It allows a decision maker to improve an optimal covering of a location problem by considering partially feasible solutions in which some demand is left uncovered. Here we revise the main facts and results about this problem emphasising different specific algorithms of resolution. Finally we show that this fuzzy version can be used to analyse the global structure of a given instance of the crisp problem.
A fuzzy method to repair infeasibility in linearly constrained problems
Abstract In this paper we introduce a fuzzy method to deal with infeasibility in linearly constrained programs. Given an infeasible instance, we determine how much we should perturb the right-hand side coefficients in order to attain feasibility and propose a ‘feasible reformulation’ of the problem. Although we prove that our algorithm always finds such a reformulation the convenience of using it can be decided by the analyst. By this, we mean that the method also provides a simple way to compute lower bounds on the changes on every right-hand side coefficient, and if the decision maker considers that some of the magnitudes are unacceptable, he or she simply stops at this step. We think tha…
Fuzzy Linear Programming in Practice: An Application to the Spanish Football League
FLP problems are perhaps one of the most and best studied topics of Soft Computing, and are among the most fruitful in applications and in theoretical and practical results. Areas of application of FLP problems are many and varied and in fact suppose an extraordinary example of technology transfer in action. In this paper, Fuzzy Linear Programming models are applied to the European football game in which the inherent uncertainty of the parameters relating to the football teams in the Spanish Football League serve to establish these models and so optimize the returns of the investments made to maintain a high quality competition. In this context, fuzzy DEA models are established which provid…
Controlling risk through diversification in portfolio selection with non-historical information
We deal with the portfolio selection problem for investors having information on the expected returns of the assets based not only on historical data. In the absence of a way of measuring the risk of non-historical information, the investor may try to adjust it through the consideration of a suitable set of diversification constraints. With this aim, we relate the concept of value of information (recently introduced by Kao and Steuer) to a qualitative subjective measure of the investor’s level of confidence in his/her non-historical information. As an illustration, we analyze the behavior of the proposed indicator in the Spanish IBEX35 index for risk, upper bound, semicontinuous variable an…
The fuzzy p-median problem: A global analysis of the solutions
Abstract We apply fuzzy techniques to incorporate external data into p-median problems. So we can detect certain solutions that would be discarded by usual crisp and fuzzy algorithms but that contrasted with this additional information can be advantageous. This usually reveals a pathology of the model and hence our methods provide some fuzzy validation criteria for p-median models.
Building Composite Indicators With Unweighted-TOPSIS
Composite indicators have been widely used in a large number of fields, including innovation and entrepreneurship as a useful tool for conveying summary information about overall performance in a relatively simple way. The construction of composite indicators implies several stages concerning collection of data, selection of criteria and individual indicators, normalization and weighting of criteria and indicators, aggregation, and comparison of overall performance of the alternatives or options. This article aims at contributing to the construction of synthetic indicators by showing with a real example, how the proposed methodology can overcome the problem of the establisment of the decisi…
Fuzzy Mathematical Programming for Portfolio Management
The classical portfolio selection problem was formulated by Markowitz in the 1950s as a quadratic programming problem in which the risk variance is minimized. Since then, many other models have been considered and their associated mathematical programming formulations can be viewed as dynamic, stochastic or static decision problems. In our opinion, the model formulation depends essentially on two factors: the data nature and the treatment given to the risk and return goals. In this communication, we consider several approaches to deal with the data uncertainty for different classical formulations of the portfolio problem. We make use of duality theory and fuzzy programming techniques to ana…
Unweighted TOPSIS: a new multi-criteria tool for sustainability analysis
In this work, we show the usefulness of a new Multi-Criteria Decision Analysis (MCDA) method, called Unweighted TOPSIS (uwTOPSIS), in helping decision making, aiming at achieving the objectives of ...
A fuzzy mathematical programming approach to the assessment of efficiency with DEA models
In many real applications, the data of production processes cannot be precisely measured. This is particularly worrying when assessing efficiency with frontier-type models, such as data envelopment analysis (DEA) models, since they are very sensitive to possible data errors. For this reason, the possibility of having available a methodology that allows the analyst to deal with imprecise data becomes an issue of great interest in these contexts. To that end, we develop some fuzzy versions of the classical DEA models (in particular, the BCC model) by using some ranking methods based on the comparison of α-cuts. The resulting auxiliary crisp problems can be solved by the usual DEA software. We…
Fuzzy Degree of Geographic Appropriateness for Social Impact Investing
Impact investing is an investment practice that is characterized by the explicit intentionality of attaining a social impact and the requisite of report and measure this impact in a transparent way. The investment decision making process has two main stages. In the first stage, filters are applied regarding four critical issues: target geography, impact theme, asset class and target return category. In this phase, the set of possible investment alternatives are determined based on their appropriateness for impact investment in terms of those four essential aspects. In a second stage, efficient portfolios are obtained taking into account financial criteria (maximizing expected return, minimi…
Finding socially responsible portfolios close to conventional ones
Abstract An increasing number of investors are interested in sustainable, responsible and impact investment (SRI). However, there is a concern about the possible financial sacrifice associated to this kind of investments. The design of Decision Support Systems assisting socially responsible investors in their investment decisions can contribute to stimulate SRI. In this paper the financial content of a portfolio selection model is discussed in order to justify that it can be integrated into a Decision Support System designed for investors interested in socially responsible investment but initially reluctant to pay a financial cost in exchange for increasing the social responsibility of thei…
Soft computing-based aggregation methods for human resource management
Abstract We are interested in the personnel selection problem. We have developed a flexible decision support system to help managers in their decision-making functions. This DSS simulates experts’ evaluations using ordered weighted average (OWA) aggregation operators, which assign different weights to different selection criteria. Moreover, we show an aggregation model based on efficiency analysis to put the candidates into an order.
RANKING DECISION MAKING UNITS BY MEANS OF SOFT COMPUTING DEA MODELS
This paper presents a method for ranking a set of decision making units according to their level of efficiency and which takes into account uncertainty in the data. Efficiency is analysed using fuzzy DEA techniques and the ranking is based on the statistical analysis of cases that include representative situations. The method enables the removal of the sometimes unrealistic hypothesis of a perfect trade-off between increased inputs and outputs. This model is compared with other DEA models that work with imprecise or fuzzy data. As an illustration, we apply our ranking method to the evaluation of a group of Spanish seaports, as well as teams playing in the Spanish football league. We compar…
Presymplectic manifolds and conservation laws
In this paper we make use of a new structure called seeded fibre bundle. This allows us to combine the symplectic formalism and general relativity. A theorem of existence is obtained and some examples and properties are studied.
Viability of infeasible portfolio selection problems: A fuzzy approach
Abstract This paper deals with fuzzy optimization schemes for managing a portfolio in the framework of risk–return trade-off. Different models coexist to select the best portfolio according to their respective objective functions and many of them are linearly constrained. We are concerned with the infeasible instances of such models. This infeasibility, usually provoked by the conflict between the desired return and the diversification requirements proposed by the investor, can be satisfactorily avoided by using fuzzy linear programming techniques. We propose an algorithm to repair infeasibility and we illustrate its performance on a numerical example.
Grading investment diversification options in presence of non-historical financial information
Modern portfolio theory deals with the problem of selecting a portfolio of financial assets such that the expected return is maximized for a given level of risk. The forecast of the expected individual assets’ returns and risk is usually based on their historical returns. In this work, we consider a situation in which the investor has non-historical additional information that is used for the forecast of the expected returns. This implies that there is no obvious statistical risk measure any more, and it poses the problem of selecting an adequate set of diversification constraints to mitigate the risk of the selected portfolio without losing the value of the non-statistical information owne…
Finding Satisfactory Near-Optimal Solutions in Location Problems
We develope and analyze a heuristic procedure to solve a fuzzy version of the p-median problem in which we allow part of the demand not to be covered in order to reduce the transport cost. This can be used to improve a given solution of the crisp p-median problem as well as to give to the decision-maker a range of alternative locations that can be adequate according to his or her own criteria.
On the Computation of the Efficient Frontier of the Portfolio Selection Problem
An easy-to-use procedure is presented for improving theε-constraint method for computing the efficient frontier of the portfolio selection problem endowed with additional cardinality and semicontinuous variable constraints. The proposed method provides not only a numerical plotting of the frontier but also an analytical description of it, including the explicit equations of the arcs of parabola it comprises and the change points between them. This information is useful for performing a sensitivity analysis as well as for providing additional criteria to the investor in order to select an efficient portfolio. Computational results are provided to test the efficiency of the algorithm and to i…
A Fuzzy-Clustering Based Approach for Measuring Similarity Between Melodies
Symbolic melodic similarity aims to evaluate the degree of likeness of two or more sequences of notes. In this work, we propose the use of fuzzy c-means clustering as a tool for the measurement of the similarity between two melodies with a different number of notes. Moreover, we present an algorithm, FOCM, implemented in a computer program written in C\(\sharp \) able to read two melodies from files with MusicXML format and to perform the clustering to calculate the dissimilarity between any two melodies. In addition, for each iteration step in the convergence process of the algorithm, a family of intermediate states (transition melodies) are obtained that can be used as new thematic materi…
Fuzzy tuning systems: the mathematics of musicians
We present some mathematical properties which determine tuning methods. We introduce the concept of fuzzy tuning systems and we analyze four of the systems coexisting within the current orchestras: Pythagorean, Just Intonation, Holder's and Equal Temperament systems. We show that the theoretical and practical tuning methods are the same. We introduce the idea of compatibility between tuning systems and we give some sufficient conditions to determine an appropriate number of notes into which the octave must be divided.
Fuzzy Techniques for Improving Satisfaction in Economic Decisions
The authors use fuzzy set theory to improve classical decision-making problems by incorporating the inherent vagueness in decision-makers’ preferences into the model. They specifically study two representative models: the p-median problem and the portfolio selection problem. The first one is a location problem, which on the one hand fits many real world management situations and on the other hand is suitable for a theoretical analysis of the techniques. The version of the portfolio selection problem presented here is a harder problem, which allows the authors to show the scope of their methods. Some numerical examples are provided to illustrate how fuzzy optimal solutions improve classical …
Compatibility of the Different Tuning Systems in an Orchestra
Focusing on the daily practice of musicians, we give flexibility to the mathematical treatment of musical notes, tuning systems and the relations between them. This allows us to connect the theory and the practice of music. Using the techniques of fuzzy logic, we describe the concepts with fuzzy sets and introduce the α-compatibility as a degree of interchangeability between tuning systems. To show how our proposal works, we use a fragment of Haydn and analyze the compatibility of the notes taken from 48 recordings for the tuning systems of Pythagoras, Zarlino and Equal Temperament of 12 notes.
Fuzzy portfolio selection based on the analysis of efficient frontiers
We present an algorithm for analyzing the geometry of the efficient frontier of the portfolio selection problem with semicontinuous variable and cardinality constraints, and use it as a basis to solve a fuzzy version of the problem, designed to obtain efficient portfolios, in the Markowitz's sense, for which the trade-off between expected return and assumed risk fits better the investor's subjective criteria. We illustrate our proposal with an example solved with LINGO and Mathematica.
Mathematics and Soft Computing in Music
Mathematics is the fundamental tool for dealing with the physical processes that explain music but it is also in the very essence of this art. Musical notes, the first elements which music works with, are defined for each tuning system as very specific frequencies; however, instrumentalists know that small changes in these values do not have serious consequences. In fact, sometimes consensus is only reached if the entire orchestra alters the theoretical pitches. The explanation for this contradiction is that musicians implicitly handle very complex mathematical processes involving some uncertainty in the concepts and this is better explained in terms of fuzzy logic. Modelling the notes as f…
Ranking corporate sustainability: a flexible multidimensional approach based on linguistic variables
Corporate sustainability implies a compromise between the present environmental, social, and economic needs of a firm's stakeholders and their future needs. Corporate sustainability is therefore a multidimensional concept. Nowadays, several independent rating agencies rate firms in terms of environmental, social, and governance (ESG) criteria. These ratings are usually used by main sustainability indices such as the Dow Jones Sustainability Index, FTSE4 Good, Stoxx Sustainability Index, or Euronext Vigeo Family to select companies to invest in. Only those firms performing better than the average of their sector are selected. However, although providing linguistic ratings about the performan…
Obtaining the Compatibility between Musicians Using Soft Computing
Modeling the musical notes as fuzzy sets provides a flexible framework which better explains musicians’ daily practices. Taking into account one of the characteristics of the sound: the pitch (the frequency of a sound as perceived by human ear), a similarity relation between two notes can be defined. We call this relation compatibility. In the present work, we propose a method to asses the compatibility between musicians based on the compatibility of their interpretations of a given composition. In order to aggregate the compatibilities between the notes offered and then obtain the compatibility between musicians, we make use of an OWA operator. We illustrate our approach with a numerical e…
Soft-computing based heuristics for location on networks: The p-median problem
We propose a genetic algorithm for the fuzzy p-median problem in which the optimal transport cost of the associated crisp problem is unknown. Our algorithm works with two populations: in one, the solutions with a better crisp transport cost are favored by the selection criterion, whereas in the second one, solutions with a better fuzzy satisfaction level are preferred. These populations are not independent. On the contrary, the first one periodically invades the second one, thus providing new starting points for finding fuzzy improvements. Our computational results also reveal the importance of choosing adequate functions for selecting the parents. Our best results are obtained with functio…
The Spanish Football Crisis
Abstract This article analyses the economic results of Spanish professional football teams using data collected over the past few years. These data differ substantially from the information presented in the annual reports of the “Liga de Futbol Profesional” (Spanish professional football league), which are obtained from the initial budgets elaborated each season. Special attention is paid to the two most important teams in the Spanish league: Real Madrid and FC Barcelona. The results of this research show that Spanish football has the same problems as those evidenced by the Italian and English football leagues and, therefore, requires similar solutions in order to be able to survive in fort…
Companies’ Selection Methods for Inclusion in Sustainable Indices: A Fuzzy Approach
Sustainability indices handle concepts which are both, of numerical and non-numerical nature. In this context, the use of Fuzzy Logic is highly useful as allows a more faithful representation of reality. Usually these indices follow a three-step process to define sustainable investment universes. First step consists of sustainability assessment. In the second step, assets are rated based on the previously assessed sustainability scores and finally, best assets are selected. This last step relies on the construction of a global score reflecting the performance of the assets in main sustainability dimensions. In this Chapter we are concerned with the third step of the selection process. We re…
Project portfolio selection and planning with fuzzy constraints
Abstract Selecting a project portfolio is a complex process involving many factors and considerations from the time it is proposed to the time the project portfolio is finally selected. Given that making a good selection is of crucial importance, it is essential to develop well-founded mathematical models to lead the organization to its final goal. To achieve this, such models have to reflect as closely as possible both the real situation of the organization as well as its targets and preferences. However, since the process of selecting and implementing project portfolios occurs in real environments and not in laboratories, uncertainty and a lack of knowledge regarding some data is always a…
Modos de enseñanza en los videotutoriales de matemáticas: equilibrio entre eficacia puntual y utilidad formativa
Resumo Presentamos una propuesta de análisis de videotutoriales de internet basado en los estilos cognitivos Holístico-analítico y Verbal-imaginativo. Para ello, se distinguen entre dos modos de enseñanza: la eficacia puntual (resolver un problema aquí y ahora) y la utilidad formativa (resolver un problema ahora y para problemáticas afines). La utilidad docente de un videotutorial puede depender de un equilibrio adecuado entre estos modos de enseñanza. Se trata de lograr resolver un problema ahora, pero con sustentos generales y a más largo plazo. La caracterización de los videotutoriales, en este sentido, podría ayudar a los profesores en la elección de videos, ya que según sus necesidades…
Estudio de fracciones en contextos sonoros
En este ensayo resaltamos los vínculos cognitivos entre las matemáticas y la música, que pueden favorecer procesos de enseñanza y aprendizaje de las fracciones en la matemática escolar. Las actividades exhibidas provienen de una investigación que promueve experiencias sensoriales en el campo rítmico para favorecer la construcción de nociones matemáticas. Dichas actividades posibilitan un engranaje armónico entre las matemáticas y la música convergentes en el tiempo y el sonido como los objetos de estudio común entre éstas disciplinas. El análisis de la investigación que aquí reportamos es producto de la aplicación de las actividades en diferentes programas de formación de profesorado de Méx…
Construction of quality indicators based on pre-established goals: application to a colombian public university
This study creates indicators of adequacy and excellence based on multiple-criteria decision-making (MCDM) methods and fuzzy logic. The calculation of indicators presents two main difficulties: The nature of the data (numerical, interval, and linguistic values are mixed) and the objective of each criterion (which does not have to reach either the maximum or the minimum). A method is proposed, based on similarity measures with predetermined ideals, that is capable of overcoming these difficulties to provide easy-to-interpret information about the quality of the alternatives. To illustrate the usefulness of this proposed method, it has been applied to data collected from students across nine …
Marginal analysis for the fuzzy p-median problem
The solutions to the fuzzy p-median problem make it possible to leave part of the demand uncovered in order to obtain significant reductions in costs. Moreover, the fuzzy formulation provides the decision-maker with many flexible solutions that he or she may prefer to the classical crisp solution. We introduce some marginal analysis techniques to study how solutions depend on membership functions. Taking into account the internal structure of the problem, we propose a practical criterion to fix the tolerances for the uncovered demand, which happens to be the most sensitive aspect of the fuzzy p-median.
Fuzzy logic helps to integrate music theory and practice
Among the many practices of composers, instrumentalists and singers which clearly correspond with fuzzy logic our focus here is on tuning. Different criteria have been used to select the sounds that music uses. A set containing these sounds (musical notes) is called a tuning system. Several tuning systems coexist in a classical orchestra. The pitches of the notes are different and very precisely defined for each system; however the consequences of small deviations from these theoretical frequencies are not serious. Actually, the orchestra members are aware of the necessity of reaching a consensus and adjust their instruments to tune well. Because of this, many musicians feel that the mathem…
Fuzzy Portfolio Selection Models for Dealing with Investor’s Preferences
This chapter provides an overview of the authors’ previous work about dealing with investor’s preferences in the portfolio selection problem. We propose a fuzzy model for dealing with the vagueness of investor preferences on the expected return and the assumed risk, and then we consider several modifications to include additional constraints and goals.
Increasing offensive or defensive efficiency? An analysis of Italian and Spanish football☆
This paper analyses technical efficiency of Italian and Spanish football during three recent seasons, to shed light on the sport performance of professional football clubs. To achieve this we have used mathematical optimisation methods, particularly DEA models, which enable the calculation of the frontiers of efficient production. Some of the most interesting results are the following. Firstly, the Spanish league is clearly more homogeneous and competitive than the Italian league. Secondly, to obtain a better classification in the Italian league, it is much more important to improve defensive, rather than offensive, efficiency. The popular maxim holds in Italy: the best attack begins with a…
Soft Computing Methods for Personnel Selection Based on the Valuation of Competences
Personnel selection based on candidates' competences is a difficult task due to the imprecise description of the applicants' competences and to the existence of several experts simultaneously evaluating those attributes. In this context, fuzzy sets theory provides suitable tools for the attainment of the maximum possible information from imprecise data. In this work, personnel selection methods are proposed that rely on the definition of an ideal candidate. Aggregated fuzzy valuations of each candidate are obtained taking into account the individual valuations provided by the experts. Then, candidates are ranked based on their similarity with the ideal candidate. Three different scenarios a…
On the importance of perspective and flexibility for efficiency measurement: effects on the ranking of decision-making units
[EN] The efficiency of a firm can be assessed from several perspectives and using a variety of methodologies. Data envelopment analysis (DEA) is one of the most commonly used methodologies. Conventional DEA analyses or models allow one to classify decision-making units (DMUs) into efficient and inefficient ones based on their efficiency scores, which could also be used for ranking DMUs; however, such rankings generally show many ties. Super-efficiency DEA analyses have been proposed to address the tie issue. On the other hand, conventional DEA analyses only take account of a single perspective in estimating efficiency scores. Cross-efficiency DEA analyses provide an alternative that takes a…
On the construction, comparison, and exchangeability of tuning systems
The aim of this article is to describe mathematically different tuning systems, to study their mathematical properties, and to propose a construction allowing their comparison. In order to reach these goals, we introduce a concept of similarity between tuning systems and then we provide two sufficient conditions for the particular case in which a tuning system generated by an interval and a circulating temperament are compared. Finally, we show by means of an example that, for two tuning systems to be exchangeable, some well-known results determining the suitable number of notes per octave are not enough.
Soft Computing Techniques for Portfolio Selection: Combining SRI with Mean-Variance Goals
A fuzzy portfolio selection model is presented incorporating a socially responsible goal without discarding a priori financially good portfolios or weakening a priori the financial goals. Hence, the optimal portfolios it provides could be either efficient from the strictly financial point of view or non-efficient if leaving the efficient frontier substantially improves the degree of social responsibility. The model can be used to direct heuristic procedures in order to select a reduced number of various alternatives from which the investor can directly make a final decision.
Measuring Social Responsibility: A Multicriteria Approach
In this chapter we present a portfolio selection model for Socially Responsible Investment. The model, following the spirit of Socially Responsible Investment, consists of two different steps. Firstly, a social screening is applied in order to obtain the feasible set of assets accomplishing the socially responsible investment policy of the assets’ manager. In this step, an indicator is obtained for the measurement of the social responsibility degree of an asset. Assets are then ranked using this indicator from the most socially responsible to the less socially responsible. In a second step, once the feasible set is obtained, composed of those socially responsible assets verifying the screen…
A fuzzy framework to explain musical tuning in practice
A theoretical tuning system is a set of pitches that can be used to play music. It is a fact that the human ear perceives notes with very close frequencies as if they were the same note. Therefore, in our approach a musical note and its pitch sensation are modeled as L-R fuzzy numbers with a modal interval and a bounded support. We pay particular attention to the 12-tone equal temperament (12-TET) for being the most widely used tuning system and we define the fuzzy 12-TET composed of 12 fuzzy notes. A similarity relation between a fuzzy note and a theoretical note can be defined, and subsequently a similarity class associated to each one of the fuzzy notes in the fuzzy 12-TET arises. Finall…
Mercury$$^\mathrm{\textregistered }$$: A Software Based on Fuzzy Clustering for Computer-Assisted Composition
We present Mercury, a new software for computer-assisted composition based on fuzzy clustering algorithms. This software is able to generate a big number of transitions between any two different melodies, harmonic progressions or rhythmical patterns. Mercury works with symbolic music notation. The software is, therefore, able to read music and to export the generated musical production into MusicXML format. This paper focusses on some theoretical aspects of the CFT algorithm implemented in the software in order to create those complete transitions, overviewing not only the structure of the program but the user’s interface and its music notation module. Finally, the wide variety of compositi…
An exact algorithm for the fuzzy p-median problem
In this paper we propose a fuzzy version of the classical p-median problem. We consider a fuzzy set of constraints so that the decision-maker will be able to take into account solutions which provide significantly lower costs by leaving a part of the demand uncovered. We propose an algorithm for solving the problem which is based on Hakimi's works and we compare the crisp and the fuzzy approach by means of an example.
Measuring Sustainability with Unweighted TOPSIS: An Application to Sustainable Tourism in Spain
The measurement of sustainability is complex due to its multiple dimensions of different relative importance and different natures. From the perspective of sustainability, three types of tourism can be established: traditional tourism, sustainable tourism, and sustainable impact tourism. In the context of multiple-criteria decision analysis, this paper presents a flexible method for assess and rank decision alternatives based on their sustainability. The proposal does not require the relative importance of each criterion to be precisely assigned beforehand, which reduces the subjectivity of the decision making and yields results that can be of interest for decision makers. To show the diffe…
Teaching modes in mathematical instructional videos: balance between specific effectiveness and educational utility
Abstract We present a proposal for the analysis of Internet video tutorials based on the Holistic-analytical and Verbal-imaginative cognitive styles. We will distinguish between aspects of teaching modes: Specific effectiveness (solving a problem here and now) and a Educational utility (solving a problem now and for related problems). From those aspects, we suggest the idea of the balance degree between them, that is, to solve a problem now, but with general and longer-term support. The characterization of video tutorials in different teaching modes could help the teachers’ choice of videos, according to their needs, decisions should be different. Besides we must not only consider the teach…