The concept of k-additivecapacity seems to be of particular interest, since the value of k is directly related to the complexity of the model (the number of subsets of at most k elements) and it has a clear interpretation in many domains of decision making. In social welfare, the generalized Gini index pro- posed by Weymark  corresponds in fact to a Choquetintegral w.r.t. a symmetric k-additivecapacity, and has a very natural interpretation in terms of the weight the decision maker puts on envy in the society (see Gajdos , Miranda et al. ). For the 2-additive case, it corresponds to a deci- sion maker (DM) who is inequality averse in the sense that any Pigou-Dalton transfer increases his measure of welfare, wherever this transfer is applied on the income distribution. In multicriteria decision making, any interaction be- tween two criteria can be represented and interpreted by a Choquetintegral w.r.t. a 2-additivecapacity, but not more complex interaction. The Choquetintegral w.r.t. a 2-additivecapacity is very used in many applications such that the evaluation of discomfort in sitting position (see Grabisch et al. ), the construction of performance measurement systems model in a supply chain context (see Berrah and Clivill´e , Clivill´e et al. ) and complex system design (see Pignon and Labreuche ).
In the CPT model the GLS implies the separation of the domain of the gains from that of the losses, with respect to a subjective reference point. This sepa- ration, technically, depends on a characteristic S-shaped utility function, steeper for losses than for gains, and on two different weighting functions, which distort, in different way, probabilities relative to gains and losses. We aim to generalize CPT, maintaining the S-shaped utility function, but replacing the two weighting functions with a bi-weighting function. This is a function with two arguments, the first corresponding to the probability of a gain and the second correspond- ing to the probability of a loss of the same magnitude. We call this model the bipolar Cumulative Prospect Theory (bCPT). The bCPT will allow gains and losses within a mixed prospect to be evaluated conjointly. In the next we dis- cuss our motivations. The basic one, stems from the data in Wu and Markle (2008) and Birnbaum and Bahra (2007). Both of these papers, following a rig- orous statistical procedure, reported systematic violations of GLS. Moreover, if we look through the Wu-Markle paradox showed above, we understand that the involved probabilities are very clear, since they are the three quartiles 25%, 50% and 75%. Similarly, the involved outcomes have the “right” size: neither so small to give rise to indifference nor so great to generate unrealism. Now suppose to look at the experiment in the other sense, from non mixed prospects to mixed ones. The two preferences L + ≻ H + and L − ≻ H − , under the hypothesis of GLS, should suggest that L should be strongly preferred to H. Surprisingly enough, H ≻ L . What happened? Clearly, the two preferences L + ≻ H + and L − ≻ H − did not interact positively and, on the contrary, the trade-off between
To use the Choquetintegral preliminary we have to identify the fuzzy measure on the basis of expert knowledge. This identification is complicated by exponential increasing complexity in the sense that it is necessary to set a value of fuzzy measure for each subset of criteria. Setting the values of all coefficients of the fuzzy measure is very difficult or even impossible for the expert. Note that even in case of three criteria for determining the fuzzy measure it is necessary to obtain 2 3 8 coefficients. Despite this complexity Choquetintegral still can be applied in practice. For this Grabisch proposed the concept of - order fuzzy measure or - additive fuzzy measure . This order can be less than the number of aggregated criteria, . Essence of the - additivity concept consists in simplification of the task of fuzzy measures determining by excluding from consideration the dependencies between more than criteria. According to the - additivity concept in most practical cases it is possible to use the Choquetintegral with respect to 2-order fuzzy measure or, equivalently, the 2-order Choquetintegral because it allows to model the interaction between the criteria while remaining relatively simple . The paper  is entirely devoted to the question under what conditions such a simplification (using of the 2-order Choquetintegral) is correct. This paper presents necessary conditions that should satisfy the expert preferences in order that they can be formalized using the 2-order Choquetintegral.
 introduced the so called k-additive capacities, see also Grabisch , and Miranda and Grabisch . The 2-additive case in particular (see Miranda, Grabisch, and Gil, ; Mayag, Grabisch, and Labreuche, [43, 44]) is a good trade-off between the range of the model and its complexity (only n(n+1)/2 real coefficients are required to define a 2-additivecapacity). The Choquetintegral with respect to a 2-additivecapacity is an interesting and effective modelling tool, see for instance Marques Pereira and Bortot [41, 42], Berrah and Clivill´e , Clivill´e, Berrah, and Maurice , Berrah, Maurice, and Montmain . In this paper we propose an aggregation model based on Choquet integration with respect to a 2-additivecapacity. The groups considered are formed by “experts” (people with specific technical competence) and “non-experts” (peo- ple with less specific technical competence, usually experts in related fields), because the typically complementary bias of the two classes contribute to a more balanced estimate. In this paper we exploit further the synergies between experts and non-experts in an MCDM framework, aggregating the individual estimates by means of non-additiveChoquet integration, and representing the complementary bias by the multiagent interaction structure underlying the ca- pacity.
proved by  in his experiment of dictionaries read by some people, the at- tributes without reference level are less taken into account in the final decision. The set X ′ we use in this paper is the set of binary alternatives or binary actions denoted by B. A binary action is a fictitious alternative which takes ei- ther the neutral value (neither satisfactory nor unsatisfactory) 0 for all criteria, or the neutral value 0 for all criteria except for one or two criteria for which it takes the satisfactory value 1. The binary actions are used in many applications through the MACBETH methodology [3, 1, 11]. Since these alternatives have a very simple structure and make sense for the DM, he should have no difficulty to express preference on them. In , a characterization of the representation of an ordinal information (a preferential information containing only a strict pref- erence and an indifference relations) on binary actions by a 2-additiveChoquetintegral have been proposed. This Characterization is based on two axioms, the classic cycles and the MOPI (MOnotonicity of Preferential Information) prop- erty which is related to the special kind of monotonicity induced by a 2-additivecapacity. Our aim is to solve the following fundamental problem: Is the cardinal information of the decision maker representable by a Choquetintegral w.r.t. a 2-additivecapacity? If the answer is positive, one can extend the preference relation over the whole set X .
Multiattribute utility theory (MAUT)  is a commonly used framework for dealing with decision with multiple criteria, of which the additive utility model is one of its best-known representatives. In the additive utility model, however, the contributions of criteria to the overall utility are added independently, so that it is not possible to represent any interaction effect between the criteria. So far in the MAUT literature, two main models have been proposed that are able to deal with interactive criteria, namely the Choquetintegral model [2, 3], and the generalized additive independence (GAI) model . The Choquetintegral model is a particular instance of the decomposable model, where marginal utility functions defined on each attribute are aggregated by some aggregation function. It is based on the Choquetintegral w.r.t. a capacity (a.k.a. fuzzy measure, nonadditive measure, etc.). Decomposable models are characterized by weak separability, which allows to induce from the preference relation of the decision maker on alternatives a preference relation on the values taken by each attribute. In particular, weak separability entails that a preference between two values of an attribute is unconditional of the value taken by the other attributes. By contrast, the GAI model does not necessarily satisfy this condition, so that the well-known menu example, where white wine is preferred to red wine if the main dish is fish, and the converse preference holds in case of meat, can be easily dealt with.
In order to take into account some appropriate measure of inconsistency be- tween criteria which may be present in the main matrix A in modulating the weighted averaging scheme of the AHP, it is natural to extend the standard weighted mean aggregation to the more general framework of Choquet integra- tion. Comprehensive reviews of Choquet integration con be found in Grabisch and Labreuche [22, 23, 24], Grabisch, Kojadinovich, and Mayer , plus also Wang and Klir , Grabisch, Nguyen and Walker , Grabisch, Murofushi and Sugeno . The Choquetintegral is defined with respect to a non-additivecapacity and corresponds to a large class of aggregation functions, including the classical weighted mean - the additivecapacity case - and the ordered weighted means (OWA) - the symmetric capacity case. General reviews of aggregation functions can be found in Calvo, Mayor, and Mesiar , Beliakov, Pradera, and Calvo , Grabisch, Marichal, Mesiar, and Pap .
integral equation (4). The proof of the latter in the special case where C is a multiple of the Lebesgue measure can be found in [7, Lemma 1] and in [2, §3.5.3]. A more general case is dealt with in [6, Theorem 1]. We give a different proof in Proposition 1 below. The lemmas below are straightforward and well-known but we give proofs for completeness. As before, X is a locally finite Borel measure without atoms and X = A − C is a decomposition as the difference of two nonnegative locally finite Borel measures without atoms. We set
Kendall’s tau, a rank correlation parameter estimated by pairwise comparisons, to give the joint probability from inputs of marginal probabilities. The most valuable information from copula is the changing direction and range in probability when one wants to control one risk by manipulating another. Copula can connect risks via joint probability distributions and thus the decision maker can know if the conditions set to individual risks conflict to each other or a broader target. Besides, the information from copula is free to human judgment so intuitively it is a very useful tool to determine human irrationality of over weighting some risk and set the boundary of safe alternatives. On the other hand, Choquet fuzzy integral model gives weights to risks and aggregates risks to a weighted average. Human preferences can play the key role to determine those weights and the structure of Choquet fuzzy integral also gives a way to deal with the property of sub-additivity. The most valuable information of Choquet fuzzy integral model is the priority sequence of risks and their relative weights which are obtained from the decision maker’s preference. This information is useful to point out high-priority risks which the decision maker may want to emphasize and put more resources on. Therefore, Choquet fuzzy integral model can provide a preference-wise realistic and efficient way to manage risks. However, the water that bears the boat is the same that swallows it. Human may be irrational or contradict to selves, so the weights obtained from extra biased human preference may result in a decision which deviated from the preset goal of management.
In the present paper, IntegralRepresentation Method (IRM) and Generalized IntegralRepresentation Method (GIRM) are explained from very basic level to advanced level, and the relationships with other numerical methods such as Finite Difference Method (FDM) and Collocation Method (CM) etc. are clarified.
ABSTRACT. Technological advances have increased the diversity of payment instruments and transaction channels, heightening consumers’ expectations for services in this regard. Coupled with an increasing competitiveness of the banking industry, this has emphasized the great importance of understanding consumers’ choices of payment instruments. In order to meet their customers’ expectations, banks have to understand what determines their choices of payment instruments. This study aims to uncover these determinants of payment instrument choice, through the use of cognitive mapping to structure the decision problem, and its combination with the Choquetintegral to identify the overall preferred payment instrument from the user perspective. The results show that direct debits and electronic cards constitute the preferred payment instruments, and automated teller machines (ATMs) and point-of- sale (POS) the overall preferred transaction channels. Understanding consumers’ choices of payment instrument, the factors underlying them and their interactions can contribute to better planning by banks at the distribution channel level. Strengths, limitations and managerial implications of our proposal are also discussed.
In the initialization step, after EEG signals acquisition, preprocessing and artifact removal, the following classifiers are trained and then given input vectors to classify: Bayesian Li- near Discriminant Analysis (BLDA), Artificial Neural Network (ANN), Fisher Linear Discriminant Analysis (FLDA), Support Vector Machine with Linear kernel (SVM-LIN), Support Vec- tor Machine with Radial-Basis kernel (SVM-RBF), Shrunken Regularized Linear Discriminant Analysis (SRLDA) and Stepwise Linear Discriminant Analysis (SWLDA), which are seven of the most popular classifiers in the BCI community suitable for the P300 paradigm , . From the output of the classifiers, given the correct result for each trial, n fuzzy measures, i.e. one for each class, are learned by means of the heuristic algorithm proposed by Grabisch . We have cho- sen a suboptimal but quick approach since, at this stage, only an estimate about the contribution of each classifier is needed. After that, the ensemble of classifiers to be used in the next step is identified by means of a strategy based on the algorithm for feature selection using the Shapley value and the interaction index proposed by Mikenina and Zimmermann . In order to avoid excessive complexity we have considered an ensemble of four classifiers.
are considered together. As example of aggregation functions, one can find the weighted sum, the Choquetintegral or the Sugeno integral. When F is a Choquetintegral, several methods for the determination of the fuzzy measure are available. For instance, linear methods , quadratic methods [4, 6] and heuristic-based methods  are available in the literature. For the Sugeno integral, heuristics  and methods based on fuzzy relations [16, 17] can be found. Unfortunately, these methods address only the problem of constructing the aggregation function and refer more to learning procedures than to true decision making approaches. The way the first step is dealt with is generally not explained, especially with elaborate aggregators such as fuzzy integrals.
The case study of the intranet of a Brazilian university illustrates the application of that methodology. After collecting the data and consolidating users’ opinions a fuzzy triangular number was determined that resulted from the frequencies of users’ opinions for the set of constructs making up the metrics being assessed. That Brazilian university’s name is Ibmec and this case study is conducted in their Rio de Janeiro campus, here denoted by .
• The number of parameters of the model, i.e. the number of coefficients of the M¨obius transform, is too small to have all the constraints satisfied. In this case, in order to increase the number of free parameters, and therefore to be more likely to be able to model the DM’s initial preferences, the approach usually consists in incrementing the order of k-additivity. It may happen however that even with a general (n-additive) capacity, the constraints imposed by the DM, still being in accordance with the pre- viously mentioned natural axioms, cannot be satisfied. In such a case, some more specific axioms underlying the Choquetintegral model are violated (see e.g. ) and the Choquetintegral cannot be considered as sufficiently flexible for modeling the initial preferences of the DM.
The multi-criteria decision between alternatives is a problem including both quantitative and qual- itative criteria. The conventional approaches to multi-criteria decision problem tend to be less e¤ective in dealing with the vague or imprecise nature of the linguistic assessment. Under many situations, the values of the qualitative criteria are often imprecisely de…ned for the decision-makers. Choquetintegral has been used for the solution of multiple criteria decision-making problems in the literature. Marichal et al.  analyzed an ordinal sorting procedure (TOMASO) for the assignment of alternatives to graded classes and present a freeware constructed from this procedure. Meyer et al.  presented a multiple criteria decision support approach in order to build a ranking and suggest a best choice on a set of alternatives. The aggregation is performed through the use of a fuzzy extension of the Choquetintegral. Demirel et al.  used it for warehouse location selection. This paper proposes a multi-criteria decision-making method using fuzzy Choquetintegral for the selection of bike model. We …rst determine the main and sub-criteria and the hierarchy for the bike selection problem, then make a multi-criteria evaluation of the bike models to illustrate how the generalized Choquetintegral is used to do this. The Choquetintegral is a ‡exible aggregation op- erator being introduced by Sugeno  and it is the generalization of the weighted average method, the Ordered Weighted Average (OWA) operator, and the max–min operator .
Our version of transitivity is stated in a way that appears to be a natural extension of the transitivity axiom for exact preferences. Because of the symmetry axiom, we do not need an elaborate formula to define our version of transitivity, and our notion of tran- sitivity appears to be less restrictive than other existing notions of transitivity. In par- ticular, for our notation of transitivity, the sizes of R x z ( ) , and R z y ( ) , have noth- ing to do with the size of R x y ( ) , This avoids the above mentioned problems. In addi- tion, our transitivity axiom implies the usual transitivity axiom for exact orders as a special case.
Due to the importance of the relationship between selected criteria, experts usually prescribe appropriate chemotherapy drugs after examining factors such as disease-related area, the patient’s weight, the patient's age, rate of progression of the disease, etc. These factors or criteria are interdependent and experts should choose the best treatment given the interplay between them. This interaction and dependency are replied by Choquetintegral method. Also incomplete AHP by taking incomplete data on criteria and alternatives into a hierarchical structure with complete data, transforms a complex issue into a hierarchical structure and it's application on scrutinizing the issue of chemotherapy, would be a very efficient and flexible way to choose the best alternatives.
A subset of an abelian semigroup is called an asymptotic basis for the semigroup if every element of the semigroup with at most ﬁnitely many exceptions can be represented as the sum of two distinct elements of the basis. The representation function of the basis counts the number of representations of an element of the semigroup as the sum of two distinct elements of the basis. Suppose there is given function from the semigroup into the set of nonnegative integers together with inﬁnity such that this function has only ﬁnitely many zeros. It is proved that for a large class of countably inﬁnite abelian semigroups, there exists a basis whose representation function is exactly equal to the given function for every element in the semigroup.