Arithmetic Operations of Generalized Trapezoidal Picture Fuzzy Numbers by Vertex Method

In this article, we define the arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method which is assembled on a combination of the (α, γ, β)-cut concept and standard interval analysis. Various related properties are explored. Finally, some computations of picture fuzzy functions over generalized picture fuzzy variables are illustrated by using our proposed technique.

Aggregation operators on intuitionistic trapezoidal fuzzy number and its application to multi-criteria decision making problems

Intuitionistic trapezoidal fuzzy numbers and their operational laws are defined. Based on these operational laws, some aggregation operators, including intuitionistic trapezoidal fuzzy weighted arithmetic averaging operator and weighted geometric averaging operator are proposed. Expected values, score function, and accuracy function of intuitionitsic trapezoidal fuzzy numbers are defined. Based on these, a kind of intuitionistic trapezoidal fuzzy multi-criteria decision making method is proposed. By using these aggregation operators, criteria values are aggregated and integrated intuitionistic trapezoidal fuzzy numbers of alternatives are attained. By comparing score function and accuracy function values of integrated fuzzy numbers, a ranking of the whole alternative set can be attained. An example is given to show the feasibility and availability of the method.

Ranking Method for Complementary Judgment Matrixes with Fuzzy Numbers Based on Hausdorff Metric Distance

A method for ranking complementary judgment matrixes with traspezoidal fuzzy numbers based on Hausdorff metric distance and fuzzy compromise decision approach is proposed. With regard to fuzzy number complementary judgment matrixes given by a decider group whose members have various weights, the expert＇s information was aggregated first by means of simple weight average（SWA） method and Bonissone calculational method. Hence a matrix including all the experts＇ preference information was got. Then the matrix＇ column members were added up and the fuzzy evaluation values of the alternatives were got. Lastly, the Hausdorff metric distance and fuzzy compromise decision approach were used to rank the fuzzy evaluation values and then the ranking values of all the alternatives were got. Because exact numbers and triangular fuzzy numbers could all be transformed into trapezoidal fuzzy numbers, the method developed can rank complementary judgment matrixes with trapezoidal fuzzy numbers, triangular fuzzy numbers and exact numbers as well. An illustrative example is also given to verify the developed method and to demonstrate its feasibility and practicality.

Obtaining Crisp Priorities for Triangular and Trapezoidal Fuzzy Judgments

This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices.Crisp judgments cannot be given for real-life situations,as most of these include some level of fuzziness and com-plexity.In these situations,judgments are represented by the set of fuzzy numbers.Most of the fuzzy optimization models derive crisp priorities for judgments repre-sented with Triangular Fuzzy Numbers(TFNs)only.They do not work for other types of Triangular Shaped Fuzzy Numbers(TSFNs)and Trapezoidal Fuzzy Numbers(TrFNs).To overcome this problem,a sum of squared error(SSE)based optimization model is proposed.Unlike some other methods,the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments.A fuzzy number is simulated using the Monte Carlo method.A threshold-based constraint is also applied to minimize the deviation from the initial judgments.Genetic Algorithm(GA)is used to solve the optimization model.We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods.Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments.Thus,the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29%compared to the existing studies.

Optimal Solution of Fuzzy Transportation Problem Using Octagonal Fuzzy Numbers

In this paper a fuzzy transportation problem under a fuzzy environment is solved using octagonal fuzzy numbers.The transportation problem is significant and has been widely studied in the field of applied mathematics to solve a system of linear equations in many applications in science.Systems of concurrent linear equations play a vital major role in operational research.The main perspective of this research paper is to find out the minimum amount of transportation cost of some supplies through a capacitated network formerly the availability and the demand notes are octagonal fuzzy numbers.Octagonal fuzzy numbers are used and showed a membership function.To illustrate this method,a fuzzy transportation problem is solved by using octagonal fuzzy numbers using the ranking technique.It is shown that it is the best optimal solution and it is demonstrated with a numerical example.

Floyd-Warshall Algorithm Based on Picture Fuzzy Information

The Floyd-Warshall algorithm is frequently used to determine the shortest path between any pair of nodes.It works well for crisp weights,but the problem arises when weights are vague and uncertain.Let us take an example of computer networks,where the chosen path might no longer be appropriate due to rapid changes in network conditions.The optimal path from among all possible courses is chosen in computer networks based on a variety of parameters.In this paper,we design a new variant of the Floyd-Warshall algorithm that identifies an All-Pair Shortest Path(APSP)in an uncertain situation of a network.In the proposed methodology,multiple criteria and theirmutual associationmay involve the selection of any suitable path between any two node points,and the values of these criteria may change due to an uncertain environment.We use trapezoidal picture fuzzy addition,score,and accuracy functions to find APSP.We compute the time complexity of this algorithm and contrast it with the traditional Floyd-Warshall algorithm and fuzzy Floyd-Warshall algorithm.

Lectotype Optimization of Offshore Platforms by Use of Three-Scale Fuzzy Analytical Hierarchy Process

A Three-Scale Fuzzy Analytical Hierarchy Process (T-FAHP) is proposed by introducing the Three-Scale Analytical Hierarchy Process (T-AHP) and the trapezoid fuzzy number. A multi-objective optimization model based on the T-FAHP is presented subsequently, in which many factors influencing the lectotype of offshore platform are taken into account synthetically, such as the original investment, the maintenance, cost, the ability of resisting fatigue and corrosion, the construction period, the threat to the environment, and so on. With this method, the experts can give the relatively precise ranking weight of each index and at the same time the requirement of consistence checking can be met, The result of a calculation example shows that the T-FAHP is practical.

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.

Fuzzy multi-criteria decision-making approach with incomplete information based on evidential reasoning

The weights of criteria are incompletely known and the criteria values are incomplete and uncertain or even default in some fuzzy multi-criteria decision-making problems.For those problems,an approach based on evidential reasoning is proposed,in which the criteria values are integrated on the basis of analytical algorithm of evidential reasoning,and then nonlinear programming models of each alternative are developed with the incomplete information on weights.The genetic algorithm is employed to solve the models,producing the weights and the utility interval of each alternative,and the ranking of the whole set of alternatives can be attained.Finally,an example shows the effectiveness of the method.

Solving Fully Fuzzy Nonlinear Eigenvalue Problems of Damped Spring-Mass Structural Systems Using Novel Fuzzy-Affine Approach

The dynamic analysis of damped structural system by using finite element method leads to nonlinear eigenvalue problem(NEP)(particularly,quadratic eigenvalue problem).In general,the parameters of NEP are considered as exact values.But in actual practice because of different errors and incomplete information,the parameters may have uncertain or vague values and such uncertain values may be considered in terms of fuzzy numbers.This article proposes an efficient fuzzy-affine approach to solve fully fuzzy nonlinear eigenvalue problems(FNEPs)where involved parameters are fuzzy numbers viz.triangular and trapezoidal.Based on the parametric form,fuzzy numbers have been transformed into family of standard intervals.Further due to the presence of interval overestimation problem in standard interval arithmetic,affine arithmetic based approach has been implemented.In the proposed method,the FNEP has been linearized into a generalized eigenvalue problem and further solved by using the fuzzy-affine approach.Several application problems of structures and also general NEPs with fuzzy parameters are investigated based on the proposed procedure.Lastly,fuzzy eigenvalue bounds are illustrated with fuzzy plots with respect to its membership function.Few comparisons are also demonstrated to show the reliability and efficacy of the present approach.

Fuzzy-TOPSIS Evaluation of Power Product-Service System:A Framework Driven by Big Data

Power product-service system(power PSS),which combines industrial electric products with electric energy services,is an effective solution for power enterprises under the background of the rapid development of power systems.In the life cycle of power PSS,evaluation decision of power PSS alternatives is of great significance for subsequent implementation.To address the power PSS alternative evaluation problem,a power PSS evaluation framework is explored driven by the big data of stakeholder comments.Based on the multi-stakeholder comments of power PSS evaluation decision’s influence factors,the index system is constructed through analyzing and summarizing the co-occurrence matrix and semantic network diagram of high-frequency words.To determine the fuzzy index value of power PSS alternative,the stakeholders’vague opinions expressed by trapezoidal fuzzy number are integrated by group decision method.Fuzzy concept is introduced into the classical Technique for Order Preference by Similarity to an Ideal Solution(TOPSIS)method and fuzzy-TOPSIS method is put forward by using the fuzzy index value.The improved TOPSIS is adopted to sequence the power PSS alternatives.The case of power PSS evaluation of six alternatives for a power enterprise shows that the explored framework is effective and can provide a feasible solution for power PSS alternative evaluation.

EXTENSION OF THE TOPSIS METHOD BASED ON PROSPECT THEORY AND TRAPEZOIDAL INTUITIONISTIC FUZZY NUMBERS FOR GROUP DECISION MAKING

Considering the decision maker＇s risk psychological factors and information ambiguity under uncertainty, a novel TOPSIS based on prospect theory （PT） and trapezoidal intuitionistic fuzzy numbers （YrIFNs） for group decision making is investigated, in which the criteria values and the criteria weights take the form of TrIFNs, and weights of decision makers are unknown. Firstly, distance measures for TrIFNs are used to induce value function under trapezoidal intuitionistic fuzzy environment. Secondly, the concepts of distance measures and trapezoidal intuitionistie fuzzy weighted averaging operator are employed to induce the weights of decision makers and thus the decision makers＇ options can be aggregated. Then the PT-based separation measures and relative closeness coefficient are defined and an algorithm for ranking alternatives under trapezoidal intuitionistic fuzzy environment is proposed. Finally, a numerical example further illustrates the practicality and effectiveness of the proposed TOPSIS method.

A MULTI-ATTRIBUTE LARGE GROUP EMERGENCY DECISION MAKING METHOD BASED ON GROUP PREFERENCE CONSISTENCY OF GENERALIZED INTERVAL-VALUED TRAPEZOIDAL FUZZY NUMBERS

In this paper, a new decision making approach is proposed for the multi-attribute large group emergency decision-making problem that attribute weights are unknown and expert preference information is expressed by generalized interval-valued trapezoidal fuzzy numbers （GITFNs）. Firstly, a degree of similarity formula between GITFNs is presented. Secondly, expert preference information on different alternatives is clustered into several aggregations via the fuzzy clustering method. As the clustering proceeds, an index of group preference consistency is introduced to ensure the clustering effect, and then the group preference information on different alternatives is obtained. Thirdly, the TOPSIS method is used to rank the alternatives. Finally, an example is taken to show the feasibility and effectiveness of this approach. These method can ensure the consistency degree of group preference, thus decision efficiency of emergency response activities can be improved.

CONFLICT MEASURE MODEL FOR LARGE GROUP DECISION BASED ON INTERVAL INTUITIONISTIC TRAPEZOIDAL FUZZY NUMBER AND ITS APPLICATION

The problem of measuring conflict in large-group decision making is examined with every decision preference expressed by multiple interval intuitionistic trapezoidal fuzzy numbers （IITFNs）. First, a distance measurement between two IITFNs is given and a function of conflict between two members of the large group is proposed. Second, members of the large group are clustered. A measurement model of group conflict, which is applied to aggregating large-group preferences, is then proposed by employing the conflict measure of clusters. Finally, a simulation example is presented to validate the models. These models can deal with the preference analysis and coordination of a large-group decision, and are thus applicable to emergency group decision making.

Analysis of migraine in mutlicellular organism based on trapezoidal neutrosophic cubic hesitant fuzzy TOPSIS method

In this paper,we define a new idea of trapezoidal neutrosophic cubic hesitant fuzzy number based on migraine diseases.We define and the migraine diseases on trapezoidal neutrosophic cubic hesitant fuzzy number and operational laws of trapezoidal neutrosophic cubic hesitant fuzzy number and hamming distance of TrNCHFNs.The new concept of trapezoidal neutrosophic cubic hesitant fuzzy TOPSIS method is introduced.Furthermore,we extend MCDM method based on the trapezoidal neutrosophic cubic hesitant fuzzy TOPSIS method.Finally,an illustrative example is given to verify and demonstrate the practicality and effectiveness of the proposed method.

Trapezoidal fuzzy model to optimize transportation problem

The main aim of this paper is to develop an approach based on trapezoidal fuzzy numbers to optimize transportation problem in fuzzy environment.The present algorithm has representation of availability,demand and transportation cost as trapezoidal fuzzy numbers.This algorithm is found quicker in terms of runtime as comparison to fuzzy VAM discussed in[Kaur A.,Kumar A.,A new method for solving fuzzy transportation problem using ranking function,Appl.Math.Model.35:5652–5661,2011;Ismail Mohideen S.,Senthil Kumar P.,A comparative study on transportation problem in fuzzy environment,Int.J.Math.Res.2:151–158,2010].On the other hand this technique gives much better results than some classical methods like north-west corner and least cost method.Another benefit of this algorithm is that for certain transportation problems it directly gives optimal solution.It is one of the simplest methods to apply and perceive.Practical usefulness of the new method over other existing methods is demonstrated with two numerical examples.

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.

Multi-Objective Vendor Selection Problem of Supply Chain Management Under Fuzzy Environment

Survival of a company in today's competitive business environment depends mainly on its supply chain.An adequate supply chain gives a competitive edge to a com-pany.Sourcing,which is the initial stage of a supply chain,can be made efficient by making an appropriate selection of vendors.Appropriate vendor selection results not only in reduced purchasing costs,decreased production lead time,increased customer satisfaction but also in improved corporate competitiveness.In general,the vendor selection problem is a multi-objective decision-making problem that involves some quantitative and qualitative factors.So,we have considered a multi-objective ven-dor selection problem(MOV SP)with three multiple objective goals:minimization of net ordering price,minimization of rejected units and minimization of late delivered units.In most of the cases,information about the price of a unit,percentage of rejected units,percentage of late delivered units,vendor rating value and vendor quota flexibil-ity may not be known precisely due to some reasons.In this paper,imprecision in input information is handled by the concept of a simulation technique,where the parameter follows the uniform distribution.Deterministic,stochastic,a-cut and ranking function approaches are used to get the crisp value of the simulated data sets.The four differ-ent algorithms,namely-fuzzy programming,goal programming,lexicographic goal programming and D1-distance algorithm,have been used for solving the MOVSP.In last,three different types of simulated data sets have been used to illustrate the work.

Effective inventory management using postponement strategy with fuzzy cost

In this article, we consider a two level supply chain to evaluate the impact ofpostponement strategy on the retailer. Here the cost parameters are fuzzified.Signed distance method is used to defuzzify and to obtain the estimation of thetotal cost in the fuzzy sense. The common variable production cost, commonfixed cost and the common unit holding cost per unit time are assumed to befuzzy in nature. Inventory models are formulated for postponement system andindependent system such that the total average inventory cost function per unittime is minimized. Algorithms are given to derive the optimal solutions of theproposed model. Theoretical analysis and the computational procedure helps tostudy the impact of deterioration rate on the optimal inventory policies. Acomparative study between the postponement system and independent systemconsidering fuzzy costs is also made.