Trapezoidal Fuzzy Numbers Research Articles

Profit analysis of skimmed milk powder producing system of milk plant using trapezoidal fuzzy numbers with different left and right heights

Profit analysis of skimmed milk powder producing system of milk plant using trapezoidal fuzzy numbers with different left and right heights

DECISION MAKING UNDER LINGUISTIC UNCERTAINTY CONDITIONS ON BASE OF GENERALIZED FUZZY NUMBERS

This article is devoted to the problem of decision making under linguistic uncertainty. The effective method for modelling linguistic uncertainty is the fuzzy set theory. There are several types of fuzzy number types proposed by L. Zadeh: fuzzy type-1, fuzzy type-2, Z-numbers. Chen proposed concept of generalized fuzzy numbers. Generalized trapezoidal fuzzy numbers (GTFN) one of effective approach which can be used for modeling linguistic uncertainty. GFTN very convenient model which allow take in account second order uncertainty. GFTN are formalized and major operations are described as practical problem is considered group decision making for supplier selection. In this case the criteria assessments are expressed by experts in linguistic form. Group decision making model is presented as 2 step aggregation procedure, in first step is aggregated value of alternative by expert, in second step by criteria. Numerical example with four criteria and three alternatives are presented and solved.

Application of intuitionistic fuzzy critical path method on airfreight ground operation systems

The main objective of this paper is to express how intuitionistic fuzzy critical path method helps and 11 improves the working process of the airfreight ground operation system. In this way, an example is discussed based 12 on the application of intuitionistic fuzzy critical path method for airfreight ground operation systems. Instead of 13 following the traditional method to find critical path and total completion duration of intuitionistic fuzzy critical 14 path analysis, we have proposed a new approach that is easier and simple to understand when compared with 15 existing methods. Here activity durations are represented as trapezoidal intuitionistic fuzzy numbers. Also, a 16 new centroid based ranking grade of modal value, left fuzziness and right fuzziness index of membership and 17 nonmembership functions of trapezoidal intuitionistic fuzzy numbers has been applied.

Preference implication-based approach to ranking fuzzy numbers

Dombi and Baczy'{n}ski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing fuzzy number ranking methods.

A Two-Dimensional Simulation Approach for Ranking Fuzzy Numbers by the Monte Carlo Technique

The ranking of fuzzy numbers has an important position in fuzzy application models and systems. In this paper, a two-dimensional approach is proposed by the Monte Carlo simulation technique, and the simulation formula of the possibility degree is established. The algorithms for ranking interval numbers, triangular fuzzy numbers and trapezoidal fuzzy numbers are elaborated on. Moreover, the axiomatic properties of the possibility degree formulae are investigated. It is found that the simulation formula of the possibility degree is reasonable for the satisfaction of the considered axiomatic properties. Finally, by considering a set of fuzzy numbers and the effect of a viewpoint, some comparisons are reported by carrying out numerical examples. The observations reveal that the proposed simulation method is effective to reflect the uncertainty of fuzzy numbers and coherent with the human intuition.

Fuzzy Representation of Principal's Preferences in Inspire Negotiation Support System.

We consider the negotiation problem, in which an agent negotiates on behalf of a principal. Our considerations are focused on the Inspire negotiation support system in which the principal’s preferences are visualised by circles. In this way, the principal describes the importance of each negotiation issue and the relative utility of each considered option. The paper proposes how this preference information may be implemented by the agent for determining a scoring function used to support decisions throughout the negotiation process. The starting point of our considerations is a discussion regarding the visualisation of the principal’s preferences. We assume here that the importance of each issue and the utility of each option increases with the size of the circle representing them. The imprecise meaning of the notion of “circle size” implies that in a considered case, the utility of an option should be evaluated by a fuzzy number. The proposed utility fuzzification is justified by a simple analysis of results obtained from the empirical prenegotiation experiment. A novel method is proposed to determine trapezoidal fuzzy numbers, which evaluates an option’s utility using a series of answers given by the participants of the experiment. The utilities obtained this way are applied to determine the fuzzy scoring function for an agent. By determining such a common generalised fuzzy scoring system, our approach helps agents handle the differences in human cognitive processes associated with understanding the principal’s preferences. This work is the first approach to fuzzification of the preferences in the Inspire negotiation support system.

Bi-objective trapezoidal fuzzy mixed integer linear program-based distribution center location decision for large-scale emergencies

This paper aims at solving three important and challenging issues: (i) How to build a 0-1 mixed integer linear program-based distribution center location model for large-scale emergencies (we call emergency distribution center (EDC) location model for short). (ii) How to develop an effective method to convert the trapezoidal fuzzy EDC location model into a crisp one. (iii) How to demonstrate the validity, flexibility and superiorities of the constructed model and developed method. Considering the acceptance degree of fuzzy constraints to be violated, this paper defines a novel flexible ranking relation for trapezoidal fuzzy numbers (TrFNs) by using the α-cut set and analyzes its desirable properties. A bi-objective trapezoidal fuzzy EDC location model (BTFELM) is built for reflecting the urgency and uncertainty of large-scale emergencies. Some essential definitions and theorems are proposed for equivalently converting the fuzzy constraints of the BTFELM into crisp ones. By employing the graded mean integration representation of TrFN, the BTFELM is ultimately transformed into a single objective crisp EDC location linear program. Wenchuan earthquake case is studied to illustrate the practicality and validity of the proposed method. Furthermore, the sensitivity and comparative analyses are provided to justify the flexibility and superiorities of the proposed method. Finally, the accuracy analyses are conducted to illustrate the reliability of the proposed method.

The integrated PFMEA approach with interval type-2 fuzzy sets and FBWM: A case study in the automotive industry

The change of market’s demand could be predictable to a certain degree at stable conditions but it may vary due to disruptive events. This research contributes by establishing the improvement of PFMEA (Process Failure Mode and Effect Analysis) analysis in the domain of assessment and determining severity risk factor, as well as identifying of failure priority. According to the researchers’ and practitioners’ suggestions, severity needs to be considered from the multiple aspects. The risk factor severity is considered from the aspects of product importance, quality, and cost. These aspects have different relative importance, which is determined in an exact way. The relative importance of the aspects, as well as the values of the risk factors, was described by linguistic expressions that are modeled by using the Interval type-2 trapezoidal fuzzy numbers (IT2TrFNs). IT2FBWM was used to determine weight vectors of risk factors. The priority of failures is determined according to the Action priority model which proposed by AIAG & VDA (Automotive Industry Action Group and German Association of the Automotive Industry). The proposed methodology is tested in a Case study where the real-life data originated from a company from the Republic of Serbia that operates as a part of an automotive supply chain.

Rational Supplier Selection Based on Two-Phase Deep Analysis considering Fuzzy QFD and Game Theory

Viewing from the perspective of a traditional manufacturing enterprise, the paper proposes a fuzzy comprehensive evaluation system model, which selects the most suitable supplier or suppliers to undertake the supply tasks and to achieve the purposes of reducing procurement costs and improving the supply quality and reliability, so as to fully meet customer demands. Based on decision-makers’ preferences and suppliers’ supply capabilities, subjective and objective weight of criteria are both developed for supplier selection with the corresponding weighting method, namely, quality function deployment (QFD) and entropy weight. Specially, the two-phase fuzzy QFD method combined with the trapezoidal fuzzy number (TFN) is creatively applied to realize the complicated conversion process. Furthermore, the game theory is employed to combine the advantages of the subjective and objective weighting method, by which the comprehensive weight can be determined. A numerical example is given to demonstrate the proposed modeling process. Through programming drawing images for comparative analysis and sensitivity analysis, study image trends, and draw conclusions, the proposed model effectively alleviates the weight deviation and has higher rationality and extensibility in supplier selection.

A Hybrid Fuzzy Goal Programming for Smart Phones and Rate Plan Selection

Smartphones are essential personal belongings today. However, smartphone buyers (SBs) are often hesitant in deciding what the best smartphone and rate plan to choose offered by telecom providers. Among many new smartphones, it is often difficult for users to choose the most suitable one in consideration of reaching the salient success aspiration level (SSAL) as close as possible, e.g., large battery capacity, and avoiding falling below the survival aspiration level (SAL) as far as possible, e.g., budget limitation, simultaneously. A novel hybrid weighted SSAL-SAL fuzzy goal programming (FGP) is proposed to assist SBs in finding satisfactory smartphones of their preferences with suitable rate plans. To meet different preferences, SBs can easily set different weights for each smartphone selection goal with linguistic terms, such as high, average, and low. In addition, these linguistic terms can easily be transformed into trapezoidal fuzzy numbers. The weights are then attached to goals in the objective function in the weighted SSAL-SAL-FGP model for choosing the most suitable smartphone. Moreover, this study also allows SBs to set the goal satisfaction levels as a preemptive priority for each goal in the FGP to find the most suitable rate plan option for the selected smartphone. The proposed approach allows SBs to select the best smartphones with suitable rate plans considering their priority preferences. In addition, SBs can save costs and precious time in finding their ideal smartphones and rate plan options. Finally, various examples are used to demonstrate the usefulness of the proposed weighted SSAL-SAL-FGP model, which allows SBs to set different weights for each goal with the intuitive method, such as linguistic terms or preemptive priority to select appropriate smartphones with suitable rate plans to match their habits and budgets. Besides, this study distributed an online questionnaire to investigate smartphone owners and found that female smartphone owners consider the battery capacity and the weight of the smartphone more crucial than the other criteria, while male smartphone owners prefer a smartphone with bigger screen size.

A framework for group decision making with multiplicative trapezoidal fuzzy preference relations

Trapezoidal fuzzy numbers are efficient to represent the quantitative vagueness and ambiguity of decision makers (DMs). Preference relations are powerful to express pairwise judgments of DMs with respect to alternatives. Combining their advantages, this paper focuses on group decision making (GDM) with multiplicative trapezoidal fuzzy preference relations (MTrFPRs) and proposes a new consistency-consensus based GDM method. To achieve this goal, a new consistency concept for MTrFPRs is proposed by analyzing the drawbacks of the previous ones. By utilizing this concept, optimal models for judging the consistency of MTrFPRs are built. Meanwhile, a consistency index is proposed to measure the consistency level of any given MTrFPR. After that, an algorithm for ranking alternatives from acceptable consistent MTrFPRs is proposed. For GDM, a central-derivation model for determining the fuzzy measure on a set of DMs is proposed. Furthermore, an optimal model for reaching the consensus threshold is constructed. Finally, a GDM method with MTrFPRs is proposed and an application example is utilized to show the efficiency of the proposed GDM method and to compare with the existing GDM methods. The proposed GDM method outperforms the existing GDM methods. It provides us with a very useful way for GDM based on MTrFPRs.

A Hybrid Methodology for Prioritizing of Store Plan Alternatives Produced with Rule-Based Design

With the widespread use of computers, in the design world, rule-based design methods, which can work in harmony with computer logic and can be easily adapted to the computer environment, have come to the fore. It is possible to analyze the logic of a design made with the rule-based design method. It is even possible to change the design which is placed on the rule base by removing the rules, in the computer environment, or creating new designs with the same design logic. In this paper, a new hybrid model is proposed based on Interval Type-2 Fuzzy Analytic Network Process (IT2 FANP) and Interval Type-2 Fuzzy TOPSIS (IT2 Fuzzy TOPSIS) for the evaluation of store plan alternatives produced with rule-based design method. The IT2 FANP method is used to determine the weight of criteria determined by experts in the selection of store plan alternatives. Then, the IT2 Fuzzy TOPSIS method is used to obtain the ranking of alternatives with Interval Type-2 trapezoidal fuzzy numbers. The originality of the paper comes from the first-time usage of a hybrid approach based on IT2 FANP and IT2 Fuzzy TOPSIS methodology in the prioritizing of store plan alternatives produced with rule-based design method.

A single-consignor multi-consignee multi-item model with permissible payment delay, delayed shipment and variable lead time under consignment stock policy

This article proposes a two-level fuzzy supply chain inventory model, in which a single consignor delivers multiple items to the multiple consignees with the consignment stock agreement. The lead time is incorporated into the model and is considered a variable for obtaining optimal replenishment decisions. In addition, crashing cost is employed to reduce the lead time duration. This article investigates four different cases under controllable lead time to analyze the best strategy, focusing on two delays such as delay-in-payments and delay-in-shipment. In all four cases, all associated inventory costs are treated as a trapezoidal fuzzy number, and a signed distance method is employed to defuzzify the fuzzy inventory cost. An efficient optimization technique is adopted to find the optimal solution for the supply chain. Four numerical experiments are conducted to illustrate the four cases. Any one of these experimental results will provide the best solution for the ideal performance of the business under controllable lead time in the consignment stock policy. Finally, the managerial insights, conclusion and future direction of this model are provided.

A fuzzy behavioral portfolio decision model with trapezoidal fuzzy return and aspiration

This paper deals with the stock portfolio selection problem involved with trapezoidal fuzzy number returns and multiple mental accounts. A fuzzy behavioral portfolio decision model is proposed to maximize the possibilistic mean value of portfolio return and ensure the portfolio return of each mental account exceeding the given minimum fuzzy aspiration level with a given probability. Then, some programming models are designed to solve the optimal portfolio strategy. Finally, one numerical example is given to illustrate the effectiveness of the proposed fuzzy behavioral portfolio decision approach.

Extension of particle swarm optimization algorithm for solving transportation problem in fuzzy environment

Transportation problem is one of the most common real life optimization problems seeking to minimize the total transportation cost while delivering a product from a number of sources to a number of destinations. In this paper, the transportation costs have been considered as generalized trapezoidal fuzzy numbers representing the uncertainty therein whereas the supply and the demand levels are crisp numbers. Particle Swarm Optimization has been extended to solve this fuzzy transportation problem. To illustrate the proposed extension of PSO, two numerical examples from existing literature have been solved and the results have been compared with those of the existing approaches. A few randomly generated problems of different dimensions have also been solved and the convergence rate of the proposed PSO has also been studied with respect to the variants of inertia weight, population size and the number of iterations. It has been observed that the proposed PSO works efficiently to obtain the optimal solutions and also removes the barricades of the traditional solution techniques.

Regional integrated energy system schemes selection based on risk expectation and Mahalanobis-Taguchi system

Regional integrated energy system (RIES) provides a platform for coupling utilization of multi-energy and makes various energy demand from client possible. The suitable RIES composition scheme will upgrade energy structure and improve integrated energy utilization efficiency. Based on a RIES construction project in Jiangsu province, this paper proposes a new multi criteria decision-making (MCDM) method for the selection of RIES schemes. Because that subjective evaluation on RIES schemes benefit under criteria has uncertainty and hesitancy, intuitionistic trapezoidal fuzzy number (ITFN) which has the better capability to model ill-known quantities is presented. In consideration of risk attitude and interdependency of criteria, a new decision model with risk coefficients, Mahalanobis-Taguchi system and Choquet integral is proposed. Firstly, the decision matrices given by experts are normalized, and then are transformed to minimum expectation matrices according to different risk coefficients. Secondly, the weights of criteria from different experts are calculated by Mahalanobis-Taguchi system. Mobius transformation coefficients based on interaction degree are to calculate 2-order additive fuzzy measures, and then the comprehensive weights of criteria are obtained by fuzzy measures and Choquet integral. Thirdly, based on group decision consensus requirement, the weights of experts are obtained by the maximum entropy and grey correlation. Fourthly, the minimum expectation matrices are aggregated by the intuitionistic trapezoidal fuzzy Bonferroni mean operator. Thus, the ranking result according to the comparison rules using the minimum expectation and the maximum expectation is obtained. Finally, an illustrative example is taken in the present study to make the proposed method comprehensible.

A new algorithm for fuzzy transportation problems with trapezoidal fuzzy numbers under fuzzy circumstances

Transportation problem is an important network structured linear programming problem that arises in several contexts and has deservedly received a great deal of attention in the literature. The central concept in this problem is to find the least total tra nsportation cost of a commodity in order to satisfy demands atdestinations using available supplies at origins in a crisp environment. In real life situations, the decision maker may not be sure about the precise values of the coefficients belonging to th e transportation problem. The aim of this paper is to introduce a formulation of Fully Fuzzy Transportation Problem involvingTrapezoidal fuzzy numbers for the transportation costs and values of supplies and demands. We propose a two step method for solvi ng fuzzy transportation problem where all of the parameters are represented by triangular fuzzy numbers i.e two Interval Transportation problems. Since the proposed approach is basedon classical approach it is very easy to understand and to apply on real life transportation problems for the decision makers. To illustrate the proposed approach f our application examples are solved. The results show that the proposed method is simpler and computationally more efficient than existing methods in the literature.

Multicriteria decision making approach using an efficient novel similarity measure for generalized trapezoidal fuzzy numbers.

Multicriteria Decision Making (MCDM) has a huge role to play while ruling out one suitable alternative among a pool of alternatives governed by predefined multiple criteria. Some of the factors like imprecision, lack of information/data, etc., which are present in traditional MCDM processes have showcased their lack of efficiency and hence eventually it has paved the ways for the development of Fuzzy multicriteria decision making (FMCDM). In FMCDM processes, the decision makers can model most of the real-life phenomena by fuzzy information-based preferences. The availability of a wide literature on similarity measure (SM) emphasizes the vital role of SM of generalized fuzzy numbers (GFNs) to conduct accurate and precise decision making in FMCDM problems. Despite having few advantages, most of the existing approaches possessed a certain degree of counter intuitiveness and discrepancies. Thus, we have attempted to propose a novel SM for generalized trapezoidal fuzzy numbers (GTrFNs) which could deliberately overcome the impediments associated with the earlier existing approaches. Moreover, a meticulous comparative study with the existing approaches is also presented. This paper provides us with an improved method to obtain the similarity values between GTrFNs and the proposed SM consists of calculating the prominent features of fuzzy numbers such as expected value and variance. We use fourteen different sets of GTrFNs, to compare the fruition of the present approach with the existing SM approaches. Furthermore, to show the utility and applicability of our proposed measure, we illustrate few practical scenarios such as the launching of an electronic gadget by a company, a problem of medical diagnosis and finally, a proper anti-virus mask selection in light of the recent COVID-19 pandemic. The obtained results with our proposed SM, for the mentioned FMCDM problems, are analytically correct and they depict the efficiency and novelty of the present article.

Critical path method for the analysis of bituminous road transport network under fuzzy environment of various fuzzy quantities

Determining the critical path in bituminous road transport network in the past period depended only on time and cost, whereas in current period the identifying the critical path has become complicated because of multi-criteria: time, distance, security, etc. We have proposed Integrated FAHP – FTOPSIS Methodology for determining the optimal critical path based on trapezoidal, hexagonal and octagonal fuzzy numbers. The aim of this research work is for transport department and society should have through the effective methodology to avoid critical path and for safe journey. Moreover, this methodology will carry for future course of action through Ministry of Road Transport to make safe roadways which is helpful and useful for effective transportations. We are exhibiting the proposed methodology by giving a numerical example and comparing trapezoidal, hexagonal and octagonal fuzzy numbers.

Evaluation of construction and demolition waste utilization schemes under uncertain environment: A fuzzy heterogeneous multi-criteria decision-making approach

The issue of sustainability has been a prime concern as a continued surging amount of construction and demolition waste (CDW) is being generated while natural resources are consumed and contaminated at an alarming rate. However, CDW utilization involves a range of options based on combinations of different crushing options and manufacture of various recycled products, which is confusing for practitioners to decide the optimal CDW utilization scheme from a comprehensive perspective. To assist decision making among CDW utilization schemes, a heterogeneous multi-criteria evaluation framework is proposed where interval numbers, triangular fuzzy numbers (TFNs) and trapezoidal fuzzy numbers (TrFNs) are used for comprehensive evaluation, enabling experts to present assessment information flexibly based on their preferences, knowledge and nature of different data forms. Firstly, a comprehensive index system is constructed from social, environmental, financial and technical dimensions for the evaluation of CDW utilization schemes. Afterwards, a novel method is proposed to unify heterogeneous evaluation values into rough numbers. Thirdly, to fully reflect both inner and outer dependencies among criteria, the analytic network process (ANP) method is utilized to derive weights of criteria and sub-criteria, before the fuzzy multi-attributive border approximation area comparison (MABAC) method is applied to obtain the final ranking. Finally, a case study of CDW utilization schemes selection problem is illustrated where the scheme of on-site crushing combined with production of recycled bricks and aggregates is selected as the best option with the sum of the distance Sk between alternatives and the border approximation areas obtained as [0.0037, 0.0544], and a comparative analysis is conducted to verify its effectiveness and validity of the proposed framework.