A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid

The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.Choquet integral, Sugeno integral, capacity, bipolarity, preferences

Axiomatizations of the Choquet integral on general decision spaces

PhDWe propose an axiomatization of the Choquet integral model for the general case of a heterogeneous product set X = X1 Xn. Previous characterizations of the Choquet integral have been given for the particular cases X = Y n and X = Rn. However, this makes the results inapplicable to problems in many fields of decision theory, such as multicriteria decision analysis (MCDA), state-dependent utility (SD-DUU), and social choice. For example, in multicriteria decision analysis the elements of X are interpreted as alternatives, characterized by criteria taking values from the sets Xi. Obviously, the identicalness or even commensurateness of criteria cannot be assumed a priori. Despite this theoretical gap, the Choquet integral model is quite popular in the MCDA community and is widely used in applied and theoretical works. In fact, the absence of a sufficiently general axiomatic treatment of the Choquet integral has been recognized several times in the decision-theoretic literature. In our work we aim to provide missing results { we construct the axiomatization based on a novel axiomatic system and study its uniqueness properties. Also, we extend our construction to various particular cases of the Choquet integral and analyse the constraints of the earlier characterizations. Finally, we discuss in detail the implications of our results for the applications of the Choquet integral as a model of decision making

The posterity of Zadeh's 50-year-old paper: A retrospective in 101 Easy Pieces – and a Few More

International audienceThis article was commissioned by the 22nd IEEE International Conference of Fuzzy Systems (FUZZ-IEEE) to celebrate the 50th Anniversary of Lotfi Zadeh's seminal 1965 paper on fuzzy sets. In addition to Lotfi's original paper, this note itemizes 100 citations of books and papers deemed “important (significant, seminal, etc.)” by 20 of the 21 living IEEE CIS Fuzzy Systems pioneers. Each of the 20 contributors supplied 5 citations, and Lotfi's paper makes the overall list a tidy 101, as in “Fuzzy Sets 101”. This note is not a survey in any real sense of the word, but the contributors did offer short remarks to indicate the reason for inclusion (e.g., historical, topical, seminal, etc.) of each citation. Citation statistics are easy to find and notoriously erroneous, so we refrain from reporting them - almost. The exception is that according to Google scholar on April 9, 2015, Lotfi's 1965 paper has been cited 55,479 times

Feature and Decision Level Fusion Using Multiple Kernel Learning and Fuzzy Integrals

The work collected in this dissertation addresses the problem of data fusion. In other words, this is the problem of making decisions (also known as the problem of classification in the machine learning and statistics communities) when data from multiple sources are available, or when decisions/confidence levels from a panel of decision-makers are accessible. This problem has become increasingly important in recent years, especially with the ever-increasing popularity of autonomous systems outfitted with suites of sensors and the dawn of the ``age of big data.\u27\u27 While data fusion is a very broad topic, the work in this dissertation considers two very specific techniques: feature-level fusion and decision-level fusion. In general, the fusion methods proposed throughout this dissertation rely on kernel methods and fuzzy integrals. Both are very powerful tools, however, they also come with challenges, some of which are summarized below. I address these challenges in this dissertation. Kernel methods for classification is a well-studied area in which data are implicitly mapped from a lower-dimensional space to a higher-dimensional space to improve classification accuracy. However, for most kernel methods, one must still choose a kernel to use for the problem. Since there is, in general, no way of knowing which kernel is the best, multiple kernel learning (MKL) is a technique used to learn the aggregation of a set of valid kernels into a single (ideally) superior kernel. The aggregation can be done using weighted sums of the pre-computed kernels, but determining the summation weights is not a trivial task. Furthermore, MKL does not work well with large datasets because of limited storage space and prediction speed. These challenges are tackled by the introduction of many new algorithms in the following chapters. I also address MKL\u27s storage and speed drawbacks, allowing MKL-based techniques to be applied to big data efficiently. Some algorithms in this work are based on the Choquet fuzzy integral, a powerful nonlinear aggregation operator parameterized by the fuzzy measure (FM). These decision-level fusion algorithms learn a fuzzy measure by minimizing a sum of squared error (SSE) criterion based on a set of training data. The flexibility of the Choquet integral comes with a cost, however---given a set of N decision makers, the size of the FM the algorithm must learn is 2N. This means that the training data must be diverse enough to include 2N independent observations, though this is rarely encountered in practice. I address this in the following chapters via many different regularization functions, a popular technique in machine learning and statistics used to prevent overfitting and increase model generalization. Finally, it is worth noting that the aggregation behavior of the Choquet integral is not intuitive. I tackle this by proposing a quantitative visualization strategy allowing the FM and Choquet integral behavior to be shown simultaneously

Use of aggregation functions in decision making

A key component of many decision making processes is the aggregation step, whereby a set of numbers is summarised with a single representative value. This research showed that aggregation functions can provide a mathematical formalism to deal with issues like vagueness and uncertainty, which arise naturally in various decision contexts

Concurrent, Integrated and Multicriteria Design Support for Mechatronic Systems

RÉSUMÉ Les systèmes mécatroniques sont une combinaison coopérative de composantes mécaniques, électroniques, de contrôle et logiciels. Dans les dernières décennies, Ils ont trouvé diverses applications dans l'industrie et la vie quotidienne. En raison de leur aspect multi-physique, du nombre élevé de leurs composantes et des interconnexions dynamiques entre les différents domaines impliqués dans leur fonctionnement, les dispositifs mécatroniques sont souvent considérés comme hautement complexes ce qui rend la tâche de les concevoir très difficile pour les ingénieurs. Cette complexité inhérente a attiré l’attention de la communauté de recherche en conception, en particulier dans le but d’atteindre une conception optimale des systèmes multi-domaines. Ainsi, cette thèse, représente une recherche originale sur le développement d'un paradigme de conception systématique, intégrée et multi-objectifs pour remplacer l'approche de conception séquentielle traditionnelle qui tend à traiter les différents domaines de la mécatronique séparément. Dans le but d'augmenter l'efficacité, la fiabilité, la facilité de contrôle et sa flexibilité, tout en réduisant la complexité et le coût effectif, ainsi que l'intégration systèmes, cette thèse présente de nouvelles approches pour la conception concurrente et optimale des systèmes mécatroniques aux stades de design conceptuel et détaillé. Les modèles mathématiques et les fondements qui soutiennent cette pensée sont présentés dans cette thèse. Les contributions des travaux de recherche de ce doctorat ont commencé par l'introduction d'un vecteur d'indices appelé le profile mécatronique multicritère (PMM) utilisé pour l'évaluation des concepts lors de la conception des systèmes mécatroniques. Les intégrales floues non linéaires de la théorie de décisions multicritères sont utilisées pour agréger les critères de conception et pour gérer les interactions possibles entre elles. Ensuite, une méthodologie de conception conceptuelle systématique est proposée et formulée. Le soutien à l'intégration d'outils d’aide à la décision multicritère dans le processus de conception est un autre objectif de cette thèse où un certain nombre de cadres de travail sont proposés pour aider les ingénieurs concepteurs à évaluer l’importance de certains critères et des paramètres d'interaction. Ces cadres de travail ne s'appliquent pas uniquement l'évaluation de la conception et de la conception optimales, mais aussi à la détermination des possibles façons d'améliorer les concepts développés. Des méthodes basées sur l’exploitation de données ainsi que des algorithmes d'optimisation sémantique sont utilisées pour identifier les paramètres flous avec le peu d’information disponibles sur les différents choix de concepts et les préférences des concepteurs.----------ABSTRACT Mechatronic systems are a combination of cooperative mechanical, electronics, control and software components. They have found vast applications in industry and everyday life during past decades. Due to their multi-physical aspect, the high number of their components, and the dynamic inter-connections between the different domains involved, mechatronic devices are often considered to be highly complex which makes the design task very tedious and non-trivial. This inherent complexity, has attracted a great deal of attention in the research community, particularly in the context of optimal design of multi-domain systems. To this end, the present thesis represents an original investigation into the development of a systematic, integrated and multi-objective design paradigm to replace the traditional sequential design approach that tends to deal with the different domains separately. With the aim of increasing efficiency, reliability, controllability and flexibility, while reducing complexity and effective cost, and finally facilitating system integration, this thesis presents new approaches towards concurrent and optimal design of mechatronic systems in conceptual and detailed design stages. The mathematical models and foundations which support this thinking are presented in the thesis. The contributions of our research work start with introducing an index vector called Mechatronic Multi-criteria Profile (MMP) used for concept evaluation in design of mechatronic systems. Nonlinear fuzzy integrals from multicriteria decision theory are utilized to aggregate design criteria and for handling possible interactions among them. Then, a systematic conceptual design methodology is proposed and formulated. Supporting the incorporation of multicriteria decision making tools into the design process, is another focus of this work where a number of frameworks are proposed to help the designers with assessment of criteria importance and interaction parameters. These frameworks are not only applicable in optimal design and design evaluation procedures, but also for determining possible ways for design improvements. Both data-driven methods as well as semantic-based optimization algorithms are used to identify the fuzzy parameters with limited available information about the design alternatives and designer preferences. Moreover, a fuzzy-based multi-objective approach has been undertaken for proposing and formulating a detailed design methodology. A unified performance evaluation index is introduced by the means of Choquet integrals and then optimized using a constrained particle swarm optimization (PSO) algorithm

Indicators for the characterization of discrete Choquet integrals

Ordered weighted averaging (OWA) operators and their extensions are powerful tools used in numerous decision-making problems. This class of operator belongs to a more general family of aggregation operators, understood as discrete Choquet integrals. Aggregation operators are usually characterized by indicators. In this article four indicators usually associated with the OWA operator are extended to discrete Choquet integrals: namely, the degree of balance, the divergence, the variance indicator and Renyi entropies. All of these indicators are considered from a local and a global perspective. Linearity of indicators for linear combinations of capacities is investigated and, to illustrate the application of results, indicators of the probabilistic ordered weighted averaging -POWA- operator are derived. Finally, an example is provided to show the application to a specific context

Enhancing strategic management using a quantified VRIO: Adding value with the MCDA approach

The field of strategic management has been popularized since the 1960s, as an aid for the search of success factors amongst the internal and external surroundings of an organization. Strategic management has observed and created strategies that are considered as pillars in the present way of applying contemporary management operations. Even though strategic management relies on managers’ capability to comprehend the current economic trends, this area has left a variety of questions unanswered, especially regarding the analyses of the combination of quantitative and qualitative decision criteria. This dissertation aims to enhance strategic management by developing a quantified valuable, rare, inimitable and organized (VRIO) framework, with the aid of the multiple criteria decision analysis (MCDA) approach. To accomplish this objective, the VRIO framework is combined with the Choquet integral (CI) and a real-life application is carried out to support strategic management. The dual methodology used in this dissertation offers an innovative process for business improvement. The benefits and limitations are also presented and discussed