Interval-valued Triangular Fuzzy Numbers Research Articles

A behavior three-way decision approach under interval-valued triangular fuzzy numbers with application to the selection of additive manufacturing composites

Additive manufacturing composites, also recognized as three-dimensional (3D) printing composites, are highly anticipated for their potential to replace industrial materials due to the availability of multiple printing processes and optional materials. However, research gaps exist in cognitive deficiencies and psychological behaviors of decision-makers, as well as experimental error effects caused by material testing, resulting in material selection as a challenging issue. Therefore, this study proposes a novel behavior three-way decision model under the interval-valued triangular fuzzy number (IVTFN) to settle the selection issue of 3D printing composites. The research contributions are summarized as follows. First, the IVTFN is presented to account for the impacts of cognitive deficiency and experimental errors, based on which the concepts of information entropy and fuzzy measure are further developed to conduct the criterion weights. In addition, by integrating the prospect theory and regret theory, a framework for constructing the behavioral decision matrix is presented. Moreover, a novel behavior three-way decision model with the perspectives of objective and preference is proposed to classify the decision region. This study presents a comprehensive methodology integrating the three-way decision model and multi-criteria decision-making method to achieve both alternative ranking and alternative classifying. Finally, a research case of 3D printing composites reinforced by continuous hybrid fibers is adopted to illustrate the validity of the methodology. Comparative analysis and sensitivity analysis are also performed. This study offers valuable insights and tools for systematically tackling the 3D printing composite material selection issues.

Evaluation and Selection of a Green and Sustainable Supplier by Using a Fuzzy ARAS Mathematical Modeling

In the present scenario, the main aim of each and every organization is to be competitive, eco-friendly, and sustainable in order to run effectively and efficiently. Therefore, to meet and fulfill the above objectives, each organization must identify and choose the appropriate and best green and sustainable supplier, as a green supplier considers green technology, green products, green packaging, etc., while a sustainable supplier considers all three characteristics of sustainability, namely social, economic, and environmental. The problem of the evaluation and selection of green and sustainable suppliers is an example of a multi-criteria decision-making (MCDM) problem. In this research paper, to handle the uncertainty hidden in the green supply chain, the extended version of a MCDM technique named fuzzy additive ratio assessment (F-ARAS) is implemented for the evaluation of the relative weights of the green and sustainable supplier selection criteria, namely social, economic, and environmental, and also to determine the best green and sustainable supplier through fuzzy theory by using linguistic terms, which includes the application of interval-valued triangular fuzzy numbers (IVTFNs) in an organization. Finally, the technique, F-ARAS, has performed well and enables the organization’s managers to attain their required solution and final decision.

Providing a Framework for Evaluating the Quality of Health Care Services Using the HealthQual Model and Multi-Attribute Decision-Making Under Imperfect Knowledge of Data

Due to the increasing importance of evaluating the quality of health care services using the patient-centred approach, this study aimed to propose a novel framework by combining the SERVQUAL model and multi-attribute decision-making (MADM) methods using interval-valued triangular fuzzy numbers (IVTFN). In this study, after an initial overview of related work and expert opinions, a list of the most important dimensions and indicators for measuring the quality of health care services was extracted and localized. Then, to determine the importance of each of the identified factors, one of MADM’s acceptable methods called step-wise weight assessment ratio analysis (SWARA) was used. Then, in order to use the developed framework for comparing different health centres and ranking them, after collecting evaluation data in the form of linguistic variables, another practical method in the field of MADM has been used, namely, Additive Ratio Assessment (ARAS) method. The dimensions and sub-dimensions identified are, on the one hand, appropriate to the conditions of the case study and, on the other hand, the findings from the implementation show that among the dimensions of health service quality, responsiveness and then reliability has the highest rank in this case. Also, the use of IVTFN, on the one hand, eliminates the problems related to the use of Likert scale in other quality assessment methods and, on the other hand, reduces the possibility of facing imperfect knowledge of data which is a common problem in the field of qualitative evaluations. Utilizing the results of this study can significantly help decision makers in their choice of strategies to improve service quality. Furthermore, improving the quality of services can play an important role in promoting the competitiveness and performance of health care providers by increasing patient satisfaction with the services received. Also, as a side effect, the developed framework can be used to compare the performance of different hospitals and health centres, as well as their ranking.

On modeling regression in full interval-valued fuzzy environment

We apply the general extension-principle-based approach to make predictions based on a regression model in a full interval-valued fuzzy environment. We use triangular interval-valued fuzzy numbers that model the uncertainty of the observed inputs and outputs to derive the predicted outputs in full accordance with Zadeh's extension principle. On one side, we enhance the Monte Carlo based algorithm introduced in the literature for simulating the output predictions of a fuzzy regression model by reducing the universe of random selections still keeping the accuracy of the empirical results; and on the other side, we solve quadratic models to derive the left endpoints of the α-cut intervals of the exact results. We use one real-life problem from hydrology engineering with data recalled from the literature to carry out numerical experiments and illustrate our proposed methodology.

Fuzzy Reasoning Numerical Spiking Neural P Systems for Induction Motor Fault Diagnosis.

The fuzzy reasoning numerical spiking neural P systems (FRNSN P systems) are proposed by introducing the interval-valued triangular fuzzy numbers into the numerical spiking neural P systems (NSN P systems). The NSN P systems were applied to the SAT problem and the FRNSN P systems were applied to induction motor fault diagnosis. The FRNSN P system can easily model fuzzy production rules for motor faults and perform fuzzy reasoning. To perform the inference process, a FRNSN P reasoning algorithm was designed. During inference, the interval-valued triangular fuzzy numbers were used to characterize the incomplete and uncertain motor fault information. The relative preference relationship was used to estimate the severity of various faults, so as to warn and repair the motors in time when minor faults occur. The results of the case studies showed that the FRNSN P reasoning algorithm can successfully diagnose single and multiple induction motor faults and has certain advantages over other existing methods.

A Bi-Objective Model for Scheduling Construction Projects Using Critical Chain Method and Interval-Valued Fuzzy Sets

Numerous constraints affect construction projects, and lack of management may lead to schedule deviation. In the execution phase of the project, due to the lack of timely access to the required resources and the existence of uncertainty, the project activities do not progress following the schedule, and as a result, schedule deviation occurs. The scheduling addresses resource constraints by the critical chain method and deals with delays in activities by placing buffers that have emerged as a method for scheduling construction projects. This paper presents a new bi-objective mathematical model which aims to reduce delay and increase quality, based on the critical chain method and resource constraint for scheduling construction projects. In the proposed model, the activities have been considered multi-mode ones. Moreover, this paper has assumed each activity to be executed in a normal or crashing way. Due to the uncertainty in real-world problems, the duration of the activity is expressed using triangular interval-valued fuzzy numbers. A new interval-valued fuzzy solution process is presented in this paper using a two-step approach. First, the equivalent crisp model is given; then in the second step, a goal programming approach is applied for transforming the bi-objective model into the single-objective one. Finally, the mathematical model is implemented on a case study adapted from the literature, and sensitivity analysis of the results is conducted.

A novel advertising media selection framework for online games in an intuitionistic fuzzy environment

Research background: The critical role of online games in e-commerce and the great competition among providers to enhance market share has significantly increased the need to use effective advertising patterns, techniques, and tools to attract users. There are two significant challenges to planning online media game selection. The first challenge is that there is no agreement on media selection criteria for online game advertising. The second challenge relates to the complexity of choosing advertising media. Purpose of the article: Given the multidimensionality and uncertainty in evaluating and selecting advertising media, especially in the case of online games, the need to provide a systematic framework for evaluating and selecting media is critical. Methods: The present study aims to provide a systematic framework based on multi-attribute decision-making (MADM) methods to evaluate and select the appropriate media for online game advertising. For this purpose, first, by reviewing the literature, a relatively comprehensive list of media selection criteria for online game advertising was extracted and then provided to experts in online game marketing and advertising in the fuzzy Delphi questionnaire. Then, based on their opinions, a localized decision model was obtained. Also, the Step-wise Weight Assessment Ratio Analysis (SWARA) method helped to determine the criteria? importance. In the next step, a preliminary list of online game advertising media was prepared and evaluated by experts based on the criteria obtained in the previous step. Finally, the media was ranked using the Additive Ratio ASsessment (ARAS) method. Findings & value added: Awareness of the criteria affecting the selection of online game advertising media and having a systematic framework for applying these criteria in advertising media selection decisions play a vital role in practical decisions. This research addresses one of the main gaps in the field of study by proposing a quantitative methodology for integrating information based on the knowledge of experts in the decision-making processes select advertising media for online games. Most traditional media selection processes are based solely on experience and estimation, and in practice, they are unable to systematically prioritize the alternatives due to the multiplicity of media available and the complexity of the decision-making process Interval-valued triangular fuzzy numbers (IVTFNs) can address the shortcomings of previous research while considering the uncertainties in this decision-making process. The findings of this framework can be good support for e-commerce managers and online game advertising practitioners.

A new approach for material ordering and multi-mode resource constraint project scheduling problem in a multi-site context under interval-valued fuzzy uncertainty

Taking both material procurement and project scheduling problem simultaneously has been proved the most noteworthy areas in project management. The ordering amount is a key factor due to the fact that through keeping a full stock of required materials, the probability of material shortage and corresponding inefficiency decline but the costs of inventory holding increase significantly. Besides, the delay in individual activities and consequently in project completion increase if a small amount of needed material is purchased. The multi-site context as one of the new resource constraint project scheduling problem (RCPSP) extensions has been widely utilized in some applications in real-world cases, especially in construction projects. In this paper, a new mathematical model is proposed for the combination of ordering problems considering all-unit discount policy and multi-mode RCPSP (MRCPSP) in the multi-site context. The developed model aims to minimize the project total cost as well as the completion time of the project. In the novel proposed mathematical model, the transportation time is crucial and depends on the site that activities are assigned. With respect to the inherent uncertainties in project management, the duration of activities, transportation time between sites, and lead time are regarded as triangular interval-valued fuzzy numbers. In addition, a new hybrid multi-objective uncertainty approach is presented to handle interval-valued fuzzy mathematical model. The results obtained by solving the presented model employing an application example from MPSPLIB data set and a real case study about seawater intake systems are appraised considering key parameters to validate the proposed model and provide some managerial insights.

Ishita approach to construct an interval-valued triangular fuzzy regression model using a novel least-absolute based discrepancy

Regression models in the presence of fuzziness imposed researchers to construct regression models in three cases based on the fuzziness of the model’s components — independent, dependent variables and the parameters. The purpose of this study is to construct full interval-valued triangular fuzzy regression model (IVTFRM) (when the regression components are represented as interval-valued triangular fuzzy numbers (IVTFNs). This paper proposes an approach (named as Ishita approach) using a mathematical linear optimization problem to find IVTFRM’s parameters of the minimum discrepancy between the given patterns and predicted values of the dependent variable. For the same and behind the fact that the more uncertainty of an IVTFN is reduced, the more accuracy is gotten; the paper proposed a novel least absolute discrepancy between two IVTFNs which take into the consideration the distance between the maximum points of the piecewise uncertainty function of the two IVTFNs. Moreover, compared to the existing approach, the proposed model is general to fit any type of datasets whereas the existing model can only fit positive interval-valued triangular fuzzy data. The proposed approach is demonstrated with a real-life application. The model of proposed Ishita approach for the real-life application shows its superiority over the model of the only one existing approach.

Application of interval-valued triangular fuzzy numbers and their functional to the healthcare problems

Aim. In healthcare field there exist different types of uncertainty due to medical error generated by human and technologies. In general the crisp value generate loss of precision and inaccuracy about result and therefore the available data is not sufficient to assessed clinical process up to desired degree of accuracy. Therefore fuzzy set theory play as an important and advance role in accuracy of results in healthcare related problems. Methods. Here for more accuracy of result, we use functional fuzzy numbers in this paper. This study uses a new fuzzy fault tree analysis for patient safety risk modelling in healthcare. In this paper we will use level (λ, ρ) interval-valued triangular fuzzy number, their functional, t-norm operation and centre of gravity defuzzification method to evaluate fuzzy failure probability and estimate reliability of system. The effectiveness of these methods is illustrated by an example related to healthcare problems and then we analyse the result obtained with the other existing techniques. Tanaka et al.’s approach has been used to give the rank of basic events of the considered problems. Also, we use functional of fuzzy numbers to analyse the change in fuzzy failure probability. Results. The paper examines the application of the failure tree, t-norm and functional fuzzy numbers in the context of interval-valued triangular fuzzy numbers. The research examined two types of healthcare-specific problems and the corresponding defuzzification techniques for the purpose of reliability analysis using the existing methods. The authors concluded that t-norm is not associated with significant accumulation and identified how a functional fuzzy number affects reliability. Similarly, using the V index method, the least critical events were found for each system.

A New Multi-Attribute Decision-Making Framework for Policy-Makers by Using Interval-Valued Triangular Fuzzy Numbers

Policy-makers are often hesitant to invest in unproven solutions because of a lack of the decision-making framework for managing innovations as a portfolio of investments that balances risk and return, especially in the field of developing new technologies. This study provides a new portfolio matrix for decision making of policy-makers to identify IoT applications in the agriculture sector for future investment based on two dimensions of sustainable development as a return and IoT challenge as a risk using a novel MADM approach. To this end, the identified applications of IoT in the agriculture sector fall into eight areas using the meta-synthesis method. The authors extracted a set of criteria from the literature. Later, the fuzzy Delphi method helped finalise it. The authors extended the SWARA method with interval-valued triangular fuzzy numbers (IVTFN SWARA) and used it to the weighting of the characteristics. Then, the alternatives were rated using the Additive Ratio Assessment (ARAS) method based on interval-valued triangular fuzzy numbers (IVTFN ARAS). Finally, decision-makers evaluated the results of ratings based on two dimensions of sustainability and IoT challenge by developing a framework for decision-making. Results of this paper show that policy-makers can manage IOT innovations in a disciplined way that balances risk and return by a portfolio approach, simultaneously the proposed framework can be used to determine and prioritise the areas of IoT application in the agriculture sector.

A Distinctive Symmetric Analyzation of Improving Air Quality Using Multi-Criteria Decision Making Method under Uncertainty Conditions

This world has a wide range of technologies and possibilities that are available to control air pollution. Still, finding the best solution to control the contamination of the air without having any impact on humans is a complicated task. This proposal helps to improve the air quality using the multi-criteria decision making method. The decision to improve air quality is a challenging problem with today’s technology and environmental development level. The multi-criteria decision making method is quite often faced with conditions of uncertainty, which can be tackled by employing fuzzy set theory. In this paper, based on an objective weighting method (CCSD), we explore the improved fuzzy MULTIMOORA approach. We use the classical Interval-Valued Triangular Fuzzy Numbers (IVTFNs), viz. the symmetric lower and upper triangular numbers, as the basis. The triangular fuzzy number is identified by the triplets; the lowest, the most promising, and the highest possible values, symmetric with respect to the most promising value. When the lower and upper membership functions are equated to one, we get the normalized interval-valued triangular fuzzy numbers, which consist of symmetric intervals. We evaluate five alternatives among the four criteria using an improved MULTIMOORA method and select the best method for improving air quality in Tamil Nadu, India. Finally, a numerical example is illustrated to show the efficiency of the proposed method.

A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique

Multi-criteria decision-making (MCDM) plays a vibrant role in decision-making, and the characteristic object method (COMET) acts as a powerful tool for decision-making of complex problems. COMET technique allows using both symmetrical and asymmetrical triangular fuzzy numbers. The COMET technique is immune to the pivotal challenge of rank reversal paradox and is proficient at handling vagueness and hesitancy. Classical COMET is not designed for handling uncertainty data when the expert has a problem with the identification of the membership function. In this paper, symmetrical and asymmetrical normalized interval-valued triangular fuzzy numbers (NIVTFNs) are used for decision-making as the solution of the identified challenge. A new MCDM method based on the COMET method is developed by using the concept of NIVTFNs. A simple problem of MCDM in the form of an illustrative example is given to demonstrate the calculation procedure and accuracy of the proposed approach. Furthermore, we compare the solution of the proposed method, as interval preference, with the results obtained in the Technique for Order of Preference by Similarity to Ideal solution (TOPSIS) method (a certain preference number).

Type-2 Duality Trapezoidal Fuzzy Fractional Transportation Problem using Fuzzy Optimization Techniques

The authors focused goal programming technique for solving type-2 duality fuzzy fractional transportation problem by using interval-valued triangular intuitionistic fuzzy numbers. The proposed method solves three types of models in which variables are taken as type-2 fuzzy variables which have spring up to the fuzzy fractional transportation problem. In these models, duality is applied and a particular type of non-linear membership functions are used to resolve duality fractional transportation problem including fuzzy parameters. A numerical example for examining the performance of the proposed model is envisaged here.

Collaborative filtering recommendation algorithm based on interval-valued fuzzy numbers

Most collaborative filtering recommendation algorithms use crisp ratings to represent the users’ preferences. However, users’ preferences are subjective and changeable, crisp ratings can’t measure the uncertainty of users’ preferences effectively. In order to solve this problem, this paper proposes the interval-valued triangular fuzzy rating model. This model replaces crisp ratings with interval-valued triangular fuzzy numbers on the basis of users’ rating statistics information, which can measure the users’ preferences in a more reasonable way. Based on this model, the collaborative filtering recommendation algorithm based on interval-valued fuzzy numbers is designed. The algorithm calculates the users’ similarity by the interval-valued triangular fuzzy numbers, and takes the ambiguity of ratings into consideration in the prediction stage. Our experiments prove that, compared with other fuzzy and traditional algorithms, our algorithm can increase the prediction precision and rank accuracy effectively with a little time cost, and has an obvious advantage when implemented in a sparse dataset which has more users than items. Thus our method has strong effectiveness and practicability.

ASSESSING CONSTRUCTION LABOURS’ SAFETY LEVEL: A FUZZY MCDM APPROACH

Risk decision matrix has widely been favoured by the researchers in the area of construction safety risk assessment. Although it provides the construction safety professionals with the final illustration of the risks magnitude, it suffers from major shortcomings, including inability to considering the importance of probability and severity, impaired analysis resulting from the use of raw numbers for ratings, and the limited range of classifications for assessing the risks. All these shortages give an impaired insight to the concerned parties, deteriorating the involved workers’ safety. As such, this paper aims to develop a novel Risk Assessment Model (RAM) through the integration of the Fuzzy Best Worst Method (FBWM) with the Interval-Valued Fuzzy Technique for Order of Preference by Similarity to Ideal Solution (IVFTOPSIS). Based on the application of RAM to a real-life case study, it was observed that the developed RAM contributes to the body of construction safety risk assessment in five unique ways: (1) computing the importance of the two risk parameters (i.e. probability and severity) using fuzzy-reference-based comparisons, (2) obviating the needs for having statistical data, (3) prioritizing the identified risks using the combination of interval-valued triangular fuzzy numbers with TOPSIS, (4) providing the safety analysts with wider ranges of classifications for conducting risk assessment, and (5) providing the safety professionals with appropriate evaluation strategies for controlling the analysed risks. The developed model in the study can be applied to any projects, giving a conclusive plan to the concerned safety professionals for adopting the further prudent mitigation measurements.

A multi-criteria decision-making method based on triangular interval-valued fuzzy numbers and the VIKOR method

In many cases, complex problems cannot be accurately described by precise numerical values. Fuzzy theory provides a suitable tool for solving these problems. However, if decision makers cannot reach an agreement on the method for defining linguistic variables based on fuzzy sets, TIVFNs (triangular interval-valued fuzzy numbers) can provide more accurate modeling. Therefore, solving fuzzy MCGDM (multiple criteria group decision-making) problem with an unknown expert weight and criterion weight in TIVFNs has become an important research direction. In this paper, TIVF-VIKOR (triangular interval-valued fuzzy VIKOR) method, which is suitable for the environment of TIVFNs, is proposed to solve the problem of fuzzy MCGDM. To achieve this goal, the TIVF-VIKOR method is innovatively adopted similarity and coefficient of variation are combined to calculate expert weight, and deviation maximization method based on divergence matrix is used to calculate criterion weight. VIKOR method is used to find the compromise solutions, which are converted into the form of binary connection number, and the optimal compromise solution is obtained after ranking. The proposed method is applied to the problem of machine fault detection, and the validity and feasibility of the method are illustrated. Compared with the TOPSIS∖ELECTRE method, the ranking results of the three methods are equivalent, and the fluctuation of the TIVF-VIKOR method is more distinct.

Ranking triangular interval-valued fuzzy numbers based on the relative preference relation

In this paper, we first use a fuzzy preference relation with a membership function representing preference degree forcomparing two interval-valued fuzzy numbers and then utilize a relative preference relation improved from the fuzzypreference relation to rank a set of interval-valued fuzzy numbers. Since the fuzzy preference relation is a total orderingrelation that satisfies reciprocal and transitive laws on interval-valued fuzzy numbers, the relative preference relation isalso a total ordering relation. Practically, the fuzzy preference relation is more reasonable on ranking interval-valuedfuzzy numbers than defuzzification because defuzzification does not present preference degree between fuzzy numbersand loses messages. However, fuzzy pair-wise comparison for the fuzzy preference relation is more complex and difficultthan defuzzification. To resolve fuzzy pair-wise comparison tie, the relative preference relation takes the strengths ofdefuzzification and the fuzzy preference relation into consideration. The relative preference relation expresses preferencedegrees of interval-valued fuzzy numbers over average as the fuzzy preference relation does, and ranks fuzzy numbersby relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown intraditional fuzzy numbers, such as triangular and trapezoidal fuzzy numbers, for previous approaches. In this paper, weextend and utilize the relative preference relation on interval-valued fuzzy numbers, especially for triangular intervalvalued fuzzy numbers. Obviously, interval-valued fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve fuzzy information.

Effective aggregation of expert opinions to inform environmental management: An integrated fuzzy group decision-making framework with application to cadmium-contaminated water treatment alternatives evaluation

Application of group decision making to practical environmental management problems indicates that effectively avoiding the loss of decision information and reasonably aggregating expert views are two main challenges during the decision-making process. Hence, this paper focuses on the utilization of interval-valued triangular fuzzy numbers for expressing expert opinions and the introduction of Group-G1 and Hamming-based weighted polymerization operator to incorporate the fuzzy information provided by individual experts, into group consensus opinion. The developed integrated fuzzy group decision-making framework was then applied to the evaluation and selection of the most desirable technology for removing low-concentration cadmium ions from water. It has been determined that hydroxide precipitation was the most preferred option at the current stage because of its relative simplicity and high maturity. However, the rankings generated based on expert opinions suggested that adsorption technologies, especially biosorption, could also be preferred because of its low-cost and environment-friendly characteristics. Finally, the reliability of the resulting priority-rankings was investigated by sensitivity analysis. The result showed this fuzzy group decision-making framework is potentially of great value for developing robust environmental recommendations and thus enables experts to make informed decisions and achieve optimal results.

Comment on “Least-squares approach to regression modeling in full interval-valued fuzzy environment”

To the best of our knowledge, only the existing approach (Rabiei et al. in Soft Comput 18(10), 2043–2059, 2014) has been proposed to construct a full interval-valued fuzzy linear regression model [a regression model when the observation of the response, independent variables as well as the regression coefficients are triangular interval-valued fuzzy numbers (TIVFNs)]. However, after a deep study, it is observed that a mathematical incorrect assumption has been considered in this approach. Furthermore, it is observed that to resolve this mathematical incorrect assumption, there is a need to propose the multiplication of an unrestricted TIVFN (regression coefficient) with a restricted TIVFN [observed values of independent variable(s)]. Keeping the same in mind, in this paper, the same type of multiplication is proposed, and with the help of proposed multiplication, a modified approach is proposed to construct a full interval-valued fuzzy regression model. Also, the modified results of some existing real-life problems are obtained with the help of the modified approach.