Application of Choquet Integral-Importance-Performance Analysis and TOPSIS Methods in Approaching the Preference Factors of Calligraphy and Seal Engraving Imagery

Classical Chinese characters,presented through calligraphy,seal engraving,or painting,can exhibit different aesthetics and essences of Chinese characters,making them the most important asset of the Chinese people.Calligraphy and seal engraving,as two closely related systems in traditional Chinese art,have developed through the ages.Due to changes in lifestyle and advancements in modern technology,their original functions of daily writing and verification have gradually diminished.Instead,they have increasingly played a significant role in commercial art.This study utilizes the Evaluation Grid Method(EGM)and the Analytic Hierarchy Process(AHP)to research the key preference factors in the application of calligraphy and seal engraving imagery.Different from the traditional 5-point equal interval semantic questionnaire,this study employs a non-equal interval semantic questionnaire with a golden ratio scale,distinguishing the importance ratio of adjacent semantic meanings and highlighting the weighted emphasis on visual aesthetics.Additionally,the study uses Importance-Performance Analysis(IPA)and Technique for Order Preference by Similarity to Ideal Solution(TOPSIS)to obtain the key preference sequence of calligraphy and seal engraving culture.Plus,the Choquet integral comprehensive evaluation is used as a reference for IPA comparison.It is hoped that this study can provide cultural imagery references and research methods,injecting further creativity into industrial design.

Research on Normal Pythagorean Neutrosophic Set Choquet Integral Operator and Its Application

We first propose the normal Pythagorean neutrosophic set(NPNS)in this paper,which synthesizes the distribution of the incompleteness,indeterminacy,and inconsistency of the Pythagorean neutrosophic set(PNS)and normal fuzzy number.We also define some properties of NPNS.For solving the decision-making problem of the nonstrictly independent and interacting attributes,two kinds of NPNS Choquet integral operators are proposed.First,the NPNS Choquet integral average(NPNSCIA)operator and the NPNS Choquet integral geometric(NPNSCIG)operator are proposed.Then,their calculating formulas are derived,their properties are discussed,and an approach for solving the interacting multi-attribute decision making based on the NPNS is studied.Finally,the two kinds of operators are applied to validate the stability of the new method.

Monotone Set-Valued Function Defined by Set-Valued Choquet Integral

Structural characteristics and absolute continuities of monotone set-valued function defined by set- valued Choquet integral are discussed. Similar to the single-valued monotone set function, several important structural characteristics of set-valued function are defined and have been proven the same as those in the original set functions, such as null-additivity, weakly null-additivity, order continuity, strong order continuity and property（S）. A counterexample shows that order continuity and strong order continuity of the original set functions are no longer kept in a monotone set-valued function when Choquet integrably bounded assumption is abandoned. Four kinds of absolute continuities are defined for set-valued function, and all been proven valid with respect to the original set functions.

Supplier Evaluation Model Based on Fuzzy Measures and Choquet Integral

To identify interactions among evaluation criteria and describe their importance,a new identification method making use of a fuzzy measure is presented.The relative weight and interaction degree of every evaluation criteria pair are obtained by using the diamond pairwise comparison method.Based on comparison results,the maximum eigenvector method of analytic hierarchy process （AHP）,the hierarchical clustering method,and the phi（s） transformation are utilized to generate values of the fuzzy measure for each subset of the evaluation criterion set.Overall evaluation on each supplier is aggregated by Choquet integral with respect to the fuzzy measure.Finally,an illustrative example demonstrates the practical feasibility and validity of the proposed method.

Some General Inequalities for Choquet Integral

With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f1, f2, …, fn are comonotone, we also obtain the Hölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.

A Choquet integral-based TODIM method for q-rung trapezoidal fuzzy numbers and its application in group decision-making

Purpose–The purpose of this paper is to develop a multi-attribute group decision-making(MAGDM)method under the q-rung orthopair trapezoidal fuzzy environment,which calculates the interaction between the criteria depending on the proposed q-rung orthopair trapezoidal fuzzy aggregation Choquet integral(q-ROTrFACI)and employ TODIM(an acronym in Portuguese of Interactive and Multi-criteria Decision Making)to consider the risk psychology of decision-makers,to determine the optimal ranking of alternatives.Design/methodology/approach–In MAGDM,q-rung orthopair trapezoidal fuzzy numbers(q-ROTrFNs)are efficient to indicate the quantitative vagueness of decision-makers.The q-ROTrFACI operator is defined and some properties are proved.Then,a novel similarity measure is developed by fusing the area and coordinates of the q-rung orthopair trapezoidal fuzzy function.Based on the above,a Choquet integral-based TODIM(CI-TODIM)method to consider the risk psychology of decision-makers is proposed and two cases are provided to prove superiority of the method.Findings–The paper investigates q-ROTrFACI operator to productively solve problems with interdependent criteria.Then,an approach is proposed to determine the center point of q–ROTrFNs and a q-rung orthopair trapezoidal fuzzy similarity is constructed.Furthermore,CI-TODIM method is devised based on the proposed q-ROTrFACI operator and similarity in q-rung orthopair trapezoidal fuzzy context.The illustration example of business models’solutions and hypertension health management are given to demonstrate the effectiveness and superiority of proposed method.Originality/value–Thepaperdevelops a novelCI-TODIMmethodthat effectivelysolves the MAGDM problems under the premise of fully considering the priority of criteria and the risk preference of decision-makers,which provides guiding advantages for practical decision-making and enriches the application of decision-making theory.

Multi-attribute group decision-making method with linguistic q-rung orthopair fuzzy information based on bi-direction Choquet integral

Purpose-The purpose of this paper is to develop the linguistic q-rung orthopair fuzzy set(LqROFS)information VIKOR method based on the bi-direction Choquet integral(BDCI),taking into account the correlation between information.The method can enrich the existing studies related to LqROFS information and better solve the problem of MAGDM problem.Design/methodology/approach-Since applying Choquet integral(CI)depict information interaction is a common operation in MAGDM.However,the traditional CI has some limitations.The unidirectional alignment may affect the MAGDM results.Therefore,a LqROFS-VIKOR method based on BDCI is proposed,where BDCI is used to aggregate the decision matrix.Furthermore,it is not reasonable to apply exact numbers to express the similarity between two qualitative data.Then a new method of defining similarity using linguistics is proposed.The similarity is used to calculate attribute weights.Findings-The validity and potential application of MAGMD method with linguistic q-rung orthopair fuzzy information based on BDCI are demonstrated in a numerical examples study.Originality/value-According to the study of available literature,the current research on LqROFS is incomplete.The existing studies of both similarity and aggregate operators have certain shortcomings.The definition of similarity proposed in this paper is more in line with reality.And compared with the existing methods,the BDCI-based aggregate operator can describe the interaction between information more reasonably.Based on this VIKOR method based on BDCI under the LqROFS environment can better select the alternative.

Diagnosis of lumbar degenerative disc disease by using L^(p)-spaces related to generalized interval-valued m-polar neutrosophic choquet integral Operator

Innovative and astonishing developments in the field of spine analysis can commence with this manuscript.The lumbar disks(L1−L2 to L5−S1)are most commonly impaired by degeneration due to their long-standing degeneration and associated strain.We investigate the indications,purposes,risk factors,and therapies of lumbar degenerated disc disease(L-DDD).We assume that the degeneration of five discs creates many effects,making it difficult to differentiate between the different types of degenerated discs and their seriousness.Since the indeterminacy and falsity portions of science or clinical diagnosis are often ignored.Due to this complexity,the reliability of the patient’s progress report cannot be calculated,nor can the period of therapy be measured.The revolutionary concept of interval-valued m-polar neutrosophic Choquet integral aggregation operator(IVmPNCIAO)is proposed to eliminate these problems.We associate generalized interval-valued m-polar neutrosophic Choquet integral aggregation operator(GIVmPNCIAO)with the statistical formulation of L^(p)-spaces and use it to identify the actual kind of degenerative disc in the lumbar spine.For the classification of interval-valued m-polar neutrosophic numbers(IVMPNNs),we set the ranking index and score function.These concepts are appropriate and necessary in order to better diagnose degeneration by associating it with mathematical modeling.We construct a pre-diagnosis map based on the fuzzy interval[0,1]to classify the types of degenerative discs.We develop an algorithm by using GIVmPNCIAO based on interval-valued m-polar neutrosophic sets(IVMPNNs)to identify the degenerative disc appropriately and to choose the most exquisite treatment for the corresponding degeneration of every patient.Furthermore,we discuss the sensitivity analysis with parameter p in GIVmPNCIAO to investigate the patient’s improvement record.

Probabilistic uncertain linguistic TODIM method based on the generalized Choquet integral and its application

Purpose-The purpose of this paper is to develop a probabilistic uncertain linguistic(PUL)TODIM method based on the generalized Choquet integral,with respect to the interdependencies between criteria,for the selection of the best alternate in the context of multiple criteria group decision-making(MCGDM).Design/methodology/approach-Owing to decision makers(DMs)do not always show completely rational and may have the preference of bounded rational behavior,this may affect the result of the MCGDM.At the same time,criteria interaction is a focused issue in MCGDM.Hence,a novel TODIM method based on the generalized Choquet integral selects the best alternate using PUL evaluation,where the generalized Choquet integral is used to calculate the weight of criterion.The generalized PUL distance measure between two probabilistic uncertain linguistic elements(PULEs)is calculated and the perceived dominance degree matrices for each alternate relative to other alternates are obtained.Furthermore,the comprehensive perceived dominance degree of each alternate can be calculated to get the ranking.Findings-Potential application of the PUL-TODIM method is demonstrated through an evaluation example with sensitivity and comparative analysis.Originality/value-As per author’s concern,there are no TODIM methods with probabilistic uncertain linguistic sets(PULTSs)to solve MCGDM problems under uncertainty.Compared with the result of existing methods,the final judgment value of alternates using the extended TODIM methodology is highly corroborated,which proves its potential in solving MCGDM problems under qualitative and quantitative environments.

Trapezoidal Intuitionistic Fuzzy Aggregation Operator Based on Choquet Integral and Its Application to Multi-Criteria Decision-Making Problems

The Choquet integral can serve as a useful tool to aggregate interacting criteria in an uncertain environment. In this paper, a trapezoidal intuitionistic fuzzy aggregation operator based on the Choquet integral is proposed for multi-criteria decision-making problems. The decision information takes the form of trapezoidal intuitionistic fuzzy numbers and both the importance and the interaction information among decision-making criteria are considered. On the basis of the introduction of trapezoidal intuitionistic fuzzy numbers, its operational laws and expected value are defined. A trapezoidal intuitionistic fuzzy aggregation operator based on the Choquet integral is then defined and some of its properties are investigated. A new multi-criteria decision-making method based on a trapezoidal intuitionistic fuzzy Choquet integral operator is proposed. Finally, an illustrative example is used to show the feasibility and availability of the proposed method.

A Novel Green Supplier Selection Method Based on the Interval Type-2 Fuzzy Prioritized Choquet Bonferroni Means

In view of the environment competencies,selecting the optimal green supplier is one of the crucial issues for enterprises,and multi-criteria decision-making(MCDM)methodologies can more easily solve this green supplier selection(GSS)problem.In addition,prioritized aggregation(PA)operator can focus on the prioritization relationship over the criteria,Choquet integral(CI)operator can fully take account of the importance of criteria and the interactions among them,and Bonferroni mean(BM)operator can capture the interrelationships of criteria.However,most existing researches cannot simultaneously consider the interactions,interrelationships and prioritizations over the criteria,which are involved in the GSS process.Moreover,the interval type-2 fuzzy set(IT2FS)is a more effective tool to represent the fuzziness.Therefore,based on the advantages of PA,CI,BM and IT2FS,in this paper,the interval type-2 fuzzy prioritized Choquet normalized weighted BM operators with fuzzy measure and generalized prioritized measure are proposed,and some properties are discussed.Then,a novel MCDM approach for GSS based upon the presented operators is developed,and detailed decision steps are given.Finally,the applicability and practicability of the proposed methodology are demonstrated by its application in the shared-bike GSS and by comparisons with other methods.The advantages of the proposed method are that it can consider interactions,interrelationships and prioritizations over the criteria simultaneously.

Profit Allocation Scheme Among Players in Supply-Chain Based on Shapley Value of Fuzzy Bi-cooperative Games

The Shapley value of fuzzy bi-eooperative game is developed based on the conventional Shapley value of bi-cooperative game. From the viewpoint that the players can participate in the coalitions to a certain extent and there are at least two independent cooperative projects for every player to choose, Shapley value which is introduced by Grabisch is extended to the case of fuzzy bi-cooperative game by Choquet integral. Moreover, the explicit fuzzy Shapley value is given. The explicit fuzzy Shapley function can be used to allocate the profits among players in supply-chain under the competitive and uncertain environment.

Shapley Value for Cooperative Games with Fuzzy Coalition

Fuzzy Shapley values are developed based on conventional Shapley value. This kind of fuzzy cooperative games admit the representation of rates of players＇ participation to each coalition. And they can be applicable to both supperadditive and subadditvie cooperative games while other kinds of fuzzy cooperative games can only be superadditive. An explicit form of the Shapley function on fuzzy games with λ-fuzzy measure was also proposed.

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan （2006）. Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integrals. Links of these newly introduced risk measures to multi-period comonotonic risk measures are represented. Finally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.

Fuzzy-TOPSIS Method with Multi-goal

To develop the technique for order preference by similarity to an ideal solution, namely, TOPSIS method with multi-goal in fuzzy decision environment. Firstly, a new approach to constructing fuzzy decision matrix by Choquet integral was proposed in multi-goal decision system. Secondly, the concepts of fuzzy positive-ideal solution and fuzzy negative-ideal solution related to the fuzzy decision matrix were given. Finally, the credibility measure was adopted to calculate the distances to fuzzy positive-ideal solution and fuzzy negative-ideal solution. The presented fuzzy-TOPSIS method embodies well both the predetermined preferences and the weights of goals.

“AniFair”:A GUI Based Software Tool for Multi-Criteria Decision Analysis--An Example of Assessing Animal Welfare

Multi-criteria decision analysis deals with decision problems in which multiple criteria need to be considered. The criteria might be measured on different scales so that comparability is difficult. One approach to help the user to organize the problem and to reflect on his or her assessment on the decision is Measuring Attractiveness by a Categorical Based Evaluation TecHnique (MACBETH). Here the user needs to provide qualitative judgment about differences of attractiveness regarding pairs of options. MACBETH was implemented in the M-MACBETH software using the additive aggregation model. The present article introduces the software tool “AniFair” which combines the MACBETH approach with the Choquet integral as an aggregation function, because the Choquet integral enables the modeling of interaction between criteria. With the Choquet integral, the user can define constraints on the relative importance of criteria (Shapley value) and the interaction between criteria. In contrast to M-MACBETH, with every instance of “AniFair” the user is made available at least two aggregation level. “AniFair” provides Graphical User Interfaces for the entering of information. The software tool is introduced via an example from the Welfare Quality Assessment protocol for pigs. With this, “AniFair” is applied to real data that were collected from thirteen farms in Northern Germany by an animal welfare expert. The “AniFair” results enabled a division of the farms into five groups of comparable performance concerning the welfare principle “Good feeding”. Hereby, the results differed in how much the interaction between criteria contributed to the Choquet integral values. The shares varied from 5% to 55%. With this, the vulnerability of aggregation results towards relative importance of and interaction between criteria was stressed, as changes in the ranking due to the definition of constraints could be shown. All results were exported to human readable txt or csv files for further analyses, and advice could be given to the farmers on how to improve their welfare situation.

GENERALIZED INTUITIONISTIC FUZZY HYBRID CHOQUET AVERAGING OPERATORS

In this study a new hybrid aggregation operator named as the generalized intuitionistic fuzzy hybrid Choquet averaging（GIFHCA） operator is defined.Meantime,some desirable properties are studied, and several important cases are examined.Furthermore,we define the generalized Shapley GIFHCA （GS-GIFHCA） operator,which does not only overall consider the importance of elements and their ordered positions,but also globally reflect the correlations among them and their ordered positions.In order to simplify the complexity of solving a fuzzy measure,we further define the generalizedλ-Shapley GIFHCA（GλS-GIFHCA） operator.

Boundedness of the Multilinear Maximal Operator with the Hausdorff Content

In this paper,we establish the strong and weak boundedness of the multilinear maximal operator in the setting of the Choquet integral with respect to theαdimensional Hausdorff content.Our results cover Orobitg and Verdera’s results in[8].

Behavioral Portfolio Selection with Loss Control

In this paper we formulate a continuous-time behavioral （4 la cumulative prospect theory） portfolio selection model where the losses are constrained by a pre-specified upper bound. Economically the model is motivated by the previously proved fact that the losses Occurring in a bad state of the world can be catastrophic for an unconstrained model. Mathematically solving the model boils down to solving a concave Choquet minimization problem with an additional upper bound. We derive the optimal solution explicitly for such a loss control model. The optimal terminal wealth profile is in general characterized by three pieces： the agent has gains in the good states of the world, gets a moderate, endogenously constant loss in the intermediate states, and suffers the maximal loss （which is the given bound for losses） in the bad states. Examples are given to illustrate the general results.

The Symmetric Banzhaf Value for Fuzzy Games with a Coalition Structure

In this paper,a generalized form of the symmetric Banzhaf value for cooperative fuzzy games with a coalition structure is proposed.Three axiomatic systems of the symmetric Banzhaf value are given by extending crisp case.Furthermore,we study the symmetric Banzhaf values for two special kinds of fuzzy games,which are called fuzzy games with multilinear extension form and a coalition structure,and fuzzy games with Choquet integral form and a coalition structure,respectively.