An Application of Non-additive Measures and Corresponding Integrals in Tourism Management

القياسات الغير جمعية والتكاملات المقابلة لها قدمت في الاصل (أولا) من قبل شوكيت في عام 1953 وتم تعريفها بشكل مستقل من قبل سوجينو في عام 1974 من أجل تمديد القياس التقليدي عن طريق استبدال خاصية الإضافة إلى خاصية غير مضافة. ومن السمات(الميزات) الهامة للقياسات الغير جمعية والتكاملات الضبابية أنها يمكن أن تمثل أهمية مصادر المعلومات الفردية والتفاعلات فيما بينها. هناك العديد من التطبيقات حول القياسات الغير جمعية والتكاملات الضبابية مثل، معالجة الصور، صنع القرار متعدد المعايير(المقاييس)، اندماج المعلومات، التصنيف، والتعرف على الأنماط. في هذا البحث قدمنا نموذجًا رياضيًا لمناقشة تطبيق القياسات الغير جمعية والتكاملات المقابلة لها في ادارة السياحة. أولا وصفنا مسألة ادارة السياحة لآحدى شركات السياحة في العراق ثم طبقنا التكاملات الضبابية ( سوجينو, ﭽوكيت, شالكريت) المتعلقة بالقياسات الغير جمعية لتقييم درجة اشباع (رضا, قبول) السائح بالبقاء في مدينة معينة وبالتالي تحديد أفضل تقييم للمدن المقصودة.Non-additive measures and corresponding integrals originally have been introduced by Choquet in 1953 (1) and independently defined by Sugeno in 1974 (2) in order to extend the classical measure by replacing the additivity property to non-additive property. An important feature of non –additive measures and fuzzy integrals is that they can represent the importance of individual information sources and interactions among them. There are many applications of non-additive measures and fuzzy integrals such as image processing, multi-criteria decision making, information fusion, classification, and pattern recognition. This paper presents a mathematical model for discussing an application of non-additive measures and corresponding integrals in tourism management. First, the problem of tourism management is described for one of the tourism companies in Iraq. Then, fuzzy integrals (Sugeno integral, Choquet integral, and Shilkret integral) are applied with respect to non-additive measures to evaluate the grade of the gratification of the tourist of staying in a particular town for determining the best evaluation

Stochastic dominance with respect to a capacity and risk measures

Pursuing our previous work in which the classical notion of increasing convex stochastic dominance relation with respect to a probability has been extended to the case of a normalised monotone (but not necessarily additive) set function also called a capacity, the present paper gives a generalization to the case of a capacity of the classical notion of increasing stochastic dominance relation. This relation is characterized by using the notions of distribution function and quantile function with respect to the given capacity. Characterizations, involving Choquet integrals with respect to a distorted capacity, are established for the classes of monetary risk measures (defined on the space of bounded real-valued measurable functions) satisfying the properties of comonotonic additivity and consistency with respect to a given generalized stochastic dominance relation. Moreover, under suitable assumptions, a "Kusuoka-type" characterization is proved for the class of monetary risk measures having the properties of comonotonic additivity and consistency with respect to the generalized increasing convex stochastic dominance relation. Generalizations to the case of a capacity of some well-known risk measures (such as the Value at Risk or the Tail Value at Risk) are provided as examples. It is also established that some well-known results about Choquet integrals with respect to a distorted probability do not necessarily hold true in the more general case of a distorted capacity.Choquet integral ; stochastic orderings with respect to a capacity ; distortion risk measure ; quantile function with respect to a capacity ; distorted capacity ; Choquet expected utility ; ambiguity ; non-additive probability ; Value at Risk ; Rank-dependent expected utility ; behavioural finance ; maximal correlation risk measure ; quantile-based risk measure ; Kusuoka's characterization theorem

On the Extension of Pseudo-Boolean Functions for the Aggregation of Interacting Criteria

The paper presents an analysis on the use of integrals defined for non-additive measures (or capacities) as the Choquet and the \Sipos{} integral, and the multilinear model, all seen as extensions of pseudo-Boolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a ``neutral level'', allowing to introduce the absolute notion of attractive or repulsive act. It is shown that in this case, the Sipos integral is a suitable solution, although not unique. Properties of the Sipos integral as a new way of aggregating criteria are shown, with emphasis on the interaction among criteria.

On the Integrability of Tonelli Hamiltonians

In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped and still interesting information on the dynamics of the system can be deduced. Moreover, we prove that on the n-dimensional torus this weaker condition implies classical integrability in the sense of Liouville. The main idea of the proof consists in relating the existence of independent integrals of motion of a Tonelli Hamiltonian to the size of its Mather and Aubry sets. As a byproduct we point out the existence of non-trivial common invariant sets for all Hamiltonians that Poisson-commute with a Tonelli one.Comment: 19 pages. Version accepted by Trans. Amer. Math. So

Convergence of the Fourth Moment and Infinite Divisibility

In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly.Comment: 10 page

Bernstein's problem on weighted polynomial approximation

We formulate and discuss a necessary and sufficient condition for polynomials to be dense in a space of continuous functions on the real line, with respect to Bernstein's weighted uniform norm. Equivalently, for a positive finite measure $\mu$ on the real line we give a criterion for density of polynomials in $L^p(\mu)$