Interval Type-2 Fuzzy Numbers Research Articles

The solution for type-2 fuzzy linear programming model based on the nearest interval approximation

Linear programming is an important branch of operations research. The model is mature in theory and widely used in real life. However, various complex realistic scenarios involve fuzzy information. In this paper, we consider a fuzzy linear programming (FLP) model in which all parameters are trapezoidal interval type-2 fuzzy numbers (IT2FNs) and propose a solution method based on the nearest interval approximation and the best-worst cases (BWC) method. We prove the nearest interval approximation interval of trapezoidal IT2FNs, then the trapezoidal IT2FNs in the model are transformed into interval numbers which both upper and lower limits are interval numbers. With the help of best-worst cases (BWC) method, the sub-models of the transformed interval linear programming model are proposed, and four sub-solutions with different specific meanings can be obtained by solving them respectively. Finally, an application example is presented to show the rationality and practical significance of the method.

Introducing a trapezoidal interval type-2 fuzzy regression model

The uncertainty is an important attribute about data that can arise from different sources including randomness and fuzziness, therefore in uncertain environments, especially, in modeling, planning, decision-making, and control under uncertainty, most data available contain some degree of fuzziness, randomness, or both, and at the same time, some of this data may be anomalous (outliers). In this regard, the new fuzzy regression approaches by creating a functional relationship between response and explanatory variables can provide efficient tools to explanation, prediction and possibly control of randomness, fuzziness, and outliers in the data obtained from uncertain environments. In the present study, we propose a new two-stage fuzzy linear regression model based on a new interval type-2 (IT2) fuzzy least absolute deviation (FLAD) method so that regression coefficients and dependent variables are trapezoidal IT2 fuzzy numbers and independent variables are crisp. In the first stage, to estimate the IT2 fuzzy regression coefficients and provide an initial model (by original dataset), we introduce two new distance measures for comparison of IT2 fuzzy numbers and propose a novel framework for solving fuzzy mathematical programming problems. In the second stage, we introduce a new procedure to determine the mild and extreme fuzzy outlier cutoffs and apply them to remove the outliers, and then provide the final model based on a clean dataset. Furthermore, to evaluate the performance of the proposed methodology, we introduce and employ suitable goodness of fit indices. Finally, to illustrate the theoretical results of the proposed method and explain how it can be used to derive the regression model with IT2 trapezoidal fuzzy data, as well as compare the performance of the proposed model with some well-known models using training data designed by Tanaka et al. [55], we provide two numerical examples.

An extended gained and lost dominance score method based risk prioritization for Fine-Kinney model with interval type-2 fuzzy information

The Fine-Kinney model is one of the most extensively adopted classical occupational risk analysis techniques which has been regarded as a helpful means for addressing the risk identification and evaluation problems in various fields. However, current extended Fine-Kinney models have limited capacity to cope with occupational risk evaluation and prioritization problems considering interactive relationships among multi-risk parameters under inter-uncertain and intra-uncertain environments. To overcome the limitations, this paper proposes a hybrid risk prioritization approach for the Fine-Kinney model by integrating GLDS (gained and lost dominance score) method, interval type-2 fuzzy sets (IT2FSs), and Maclaurin symmetric mean (MSM) operator. First, the trapezoidal interval type-2 fuzzy numbers (TrIT2FNs) are introduced to depict the inter-uncertain and intra-uncertain risk evaluation information. Then, an ordered weighted averaging (OWA) operator is constructed to aggregate the complex and uncertain risk evaluation information from heterogeneous decision-makers in which the similarity-measure-based weighting method is incorporated. Next, the GLDS method combined with the weighted MSM operator is developed to address the risk prioritization problem in the Fine-Kinney model by considering the inter-dependencies among multi-risk parameters. Finally, a numerical example of occupational risk evaluation for potential hazards in construction excavation is presented within IT2FSs to demonstrate the application and availability of the extended Fine-Kinney model. After that, the comparison and sensitivity analyses are conducted to further illustrate the superiority and rationality of the proposed approach.

A novel risk analysis approach for occupational safety using Bayesian Network and interval type-2 fuzzy sets: the case of underground mining

Occupational safety problems are no longer acceptable for any industrial environment. Lack of comprehensive and reliable evaluations for occupational safety causes many undesired events and harm to employees during the industrial process. In this study, it is aimed to develop an applicable risk analysis methodology for evaluating the undesired occupational events that occurred in the multi-process system where no historical accident records. The difficulty in obtaining and analyzing the data required for the determination of the occupational safety risks especially in the manually executed processes has been overcome with the Bayesian Network and interval type-2 fuzzy sets by using the expert judgments. While BN enables to development of a comprehensive reasoning approach about the occurrence of the events, interval type-2 fuzzy sets better represent the ambiguity in the judgments by covering the uncertainty in a wider mathematical range with less computational effort according to other fuzzy sets. During multi-processes in industrial activity, various occupational undesired events may occur, including rare events with very serious consequences or frequent events with very low severity consequences. To able to consider all kinds of events occurring in an industrial environment from a holistic risk perspective, a novel fuzzy scale for specifying the probability and consequence of the events are proposed by the interval type-2 fuzzy numbers. Therefore, all undesired events regardless the probability and consequence which may occur during the multi-processes in a system and the main root causes of these events can be observed within the proposed methodology. A case study is used to emphasize the effect of the proposed methodology for risk analysis of occupational safety in underground mining. The results have indicated that occupational safety education is the most contributing factor to occurring the undesired occupational events in underground mining. We believe that this study could help evaluate the safety risk of the multi-process systems comprehensively and holistically and proposing strategic planning for mitigating the occupational safety risks.

Designing a single-vendor and multiple-buyers’ integrated production inventory model for interval type-2 fuzzy demand and fuzzy rule based deterioration

In this paper, a single-vendor and multiple-buyers’ integrated production inventory model is investigated where demand of the item at the buyers’ location is considered as interval type-2 fuzzy number (IT2FN). Deterioration rate of the item is assumed to change in accordance with the weather conditions of a particular region. It relies upon the values of certain attributes that have a direct influence on the extent of deterioration. These parameter values are easily forecasted and thereby can be utilized to determine the item depletion rate, which is executed here using Mamdani fuzzy inference scheme. Besides, a nearest interval approximation formula for the defuzzification of IT2FN is developed and applied in the proposed integrated production inventory model. The model optimizes the total number of shipments to be made to the buyers within a complete cycle so as to minimize the overall integrated cost incurred. A detailed illustration of the theoretical results is further demonstrated with the help of numerical example, followed by sensitivity analysis which provides insights into better decision making.

A Restricted Multi-Objective Solid Transportation Problem with Budget Constraint Involving Stochastic Variable and Interval Type-2 Fuzzy Number

This paper proposes a new concept in transportation problem in which if the negligible amount quantity is transported through a distribution center, then the decision maker (DM) cannot deliver that negligible amount quantity through the particular distribution center and the DM puts some restriction on transportation problem. Here, we develop six transportation models with budget at each destination, of which three models are without restriction and another three models are with restriction. Also, apart from source, demand and capacity constraints, an extra constraint on the total budget at each destination is also imposed. Here, all the parameters in Models 1 and 2 are crisp in nature and are solved in crisp environment, whereas all the parameters in Models 3–6 are interval type-2 fuzzy and random in nature, respectively, and are solved in uncertain environment. To reduce Models 3–6 into its crisp equivalent, we use the expected value of fuzzy number and chance constraint programming technique, respectively. Weighted sum method is also used to give the preference of the objective function and a gradient-based optimization technique-generalized reduced gradient (GRG) method are applied and using LINGO-13 software to get the optimal solutions. A numerical example is provided to illustrate the models and programming. Finally, a sensitivity analysis is presented for Models 1 and 2 with respect to the weight function.

Interval Type-2 Fuzzy Standardized Cumulative Sum Control Charts in Production of Fertilizers

Statistical process control is a method used for controlling processes in which causes of variations and correction actions can be observed. Control chart is one of the powerful tools of statistical process control that are used to control nonconforming products. Previous literature suggests that fuzzy charts are more sensitive than conventional control charts, and hence, they provide better quality and conformance of products. Nevertheless, some of the data used are more suitable to be presented in interval type-2 fuzzy numbers compared to type-1 fuzzy numbers as interval type-2 fuzzy numbers have more ability to capture uncertain and vague information. In this paper, we develop an interval type-2 fuzzy standardized cumulative sum (IT2F-SCUSUM) control chart and apply it to data of fertilizer production. This new approach combines the advantages of interval type-2 fuzzy numbers and standardized sample means which can control the variability. Twenty samples with a sample size of six were examined for testing the conformance. The proposed IT2F-SCUSUM control chart unveils that 15 samples are “out of control.” The results are also compared to the conventional CUSUM chart and type-1 fuzzy CUSUM chart. The conventional chart shows that 13 samples are “out of control.” In contrast, the type-1 fuzzy CUSUM chart shows that the process is “out of control” for 14 samples. In the analysis of average run length, the proposed IT2F-SCUSUM chart outperforms the other two CUSUM charts. Thus, we can conclude that the IT2F-SCUSUM chart is more sensitive and takes lesser number of observations to identify the shift in the process. The analyses suggest that the IT2F-SCUSUM chart is a promising tool in examining conformance of the quality of the fertilizer production.

Type 2 Fully Fuzzy Linear Programming

Since its inception, fuzzy linear programming (FLP) has proved to be a more powerful tool than classical linear programming to optimize real-life problems dealing with uncertainty. However, the proposed models are partially fuzzy; in other words, they suppose that only some aspects can be uncertain, while others have to be crisp. Furthermore, the few methods that deal with fully fuzzy problems use Type 1 fuzzy membership function, while Type 2 fuzzy logic captures the uncertainty in a more suitable way. This work presents a fully fuzzy linear programming approach in which all parameters are represented by unrestricted Interval Type 2 fuzzy numbers (IT2FN) and variables by positive IT2FN. The treated comparative results show that the proposed achieves a better optimized function while permitting consideration of both equality and inequality constraints.

An advanced TOPSIS method with new fuzzy metric based on interval type-2 fuzzy sets

Decision-making techniques are among important topics applied in operations research. Multi-Criteria Decision-Making (MCDM) is one of the very commonly used subjects of this issue. Technique for Order Preference by Similarity to Ideal Solutions (TOPSIS) is among the most utilized MCDM approaches in terms of both convenience and efficiency. To provide the reality and ease of application of these methods as much as possible Interval Type-2 Fuzzy Numbers (IT2FN)s, a special kind of type-2 fuzzy sets (T2FS)s, are frequently used together with MCDM methods. However, before the final stage of the solution to the problem, defuzzification techniques are applied for obtaining the optimum solution. Switching to crisp numbers before the last step of the procedure reduces the effectiveness of using the IT2FNs. The aim of this paper is to contribute to the literature with a new idea of fuzzy metric function whose result is again a fuzzy number. The result of this study will be a serious contribution to the literature with its metric function structure that is a fuzzy number. A metric function whose result will be a fuzzy set can also be made for different fuzzy structures. Also, a new partial order relation will be given for IT2FNs to define this fuzzy metric function. Moreover, this new fuzzy metric will be adapted to the TOPSIS method. Hence, it will be assured that the defuzzification operations are used only in the final sorting stage and the advantage of using fuzzy numbers will be able to continue in further steps. The advanced TOPSIS with a new fuzzy metric will be applied to a video chat program selection problem as an example.

Green supplier selection for textile industry: a case study using BWM-TODIM integration under interval type-2 fuzzy sets.

Although environmental awareness has reached a high level, enterprises-regardless of their working domains-follow the concept of greenness for their practices. This awareness among the stakeholders and supply chain experts has a positive impact on the purchasing departments of enterprises in various sectors to consider greenness in their procurement processes. The critical decision that must be made in green supply chain management (GSCM) is supplier selection. In the textile industry, a highly competitive market in recent years, suppliers for this industry have crucial roles in business activities considering environmental issues. Therefore, green supplier selection (GSS) in the textile industry is considered a must-be process for the stakeholders. In this study, a GSS problem is tackled as a multi-criteria decision process. Best worst method (BWM) and TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) methods are merged under an improved fuzzy concept of interval type-2 fuzzy sets (IT2FSs). In determining green suppliers' evaluation criteria, BWM with interval type-2 fuzzy numbers (IT2F-BWM) is used. In selecting green suppliers, an interval type-2 fuzzy TODIM (IT2F-TODIM) is applied. Considering the characteristics of IT2FSs, BWM, and TODIM methods either individually and in integrated style, the proposed approach can handle uncertainty in the decision-making of GSS. To demonstrate the applicability of the approach, a case study in the Turkish textile industry is performed. Three green supplier alternatives (S1, S2, and S3) are assessed under forty-two sub-criteria. The study shows the most significant sub-criteria are recognized as dye and print quality, product design and pattern suitability, profit on the product, variation in price, and purchase cost. S2 green supplier has been selected as the most appropriate one. A sensitivity analysis is also fulfilled to check variation in the ranking of green suppliers.

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.

Regional energy internet project investment decision making framework through interval type-2 fuzzy number based Choquet integral fuzzy synthetic model

With the rapid development of traditional energy system, the essential obstacles to the coordination between different energy sources is increasingly obvious, which greatly hinders the further improvement of energy efficiency. As a promising energy supply and consumption system, Regional Energy Internet (REI) has significant practical values in improving the coordination of traditional energy and promoting large-scale renewable energy penetration. Due to the large scale, high ambiguity and numerous key interactive criteria, the investment decision making of REI project is a key problem in application management, but it is challenging and has not been solved. Therefore, this paper explores a targeted and scientific REI investment decision making framework through the interval type-2 fuzzy number, Choquet integral and fuzzy synthetic evaluation model according to different preferences of investors to ensure the successful implementation and effective resources integration of potential REI projects. To verify the effectiveness of the framework, a case study with three typical REI projects simulated from a two-stage stochastic optimization model is studied, and corresponding investment countermeasures are proposed as a reference to allocate resources and prevent risk. Besides, since REI investment decision-making has not been deeply studied, this paper contributes to literature and expands knowledge.

An Extended Prospect Theory for Robot Evaluation and Selection considering Risk Preferences and Interactive Criteria

The evaluation and selection process can be regarded as a complex multiple criteria decision analysis (MCDA) problem which involves various interaction relationships among criteria under high uncertain environment. In addition, the decision-makers are always bounded rational in the risk decision-making process. However, the current robot evaluation and selection approach seldom considers the decision-maker’s risk preference and interactive criteria under high uncertain environment. Thus, the purpose of this paper is to develop a hybrid MCDA approach for solving the robot evaluation and selection problem. In the proposed framework, the interval type-2 fuzzy set is used to express the uncertain evaluation information provided by decision-makers. Next, the distance measure of interval type-2 fuzzy numbers is developed to determine the fuzzy measure of each criterion. Then, the extended prospect theory based on developed Choquet integral is proposed to evaluate and prioritize the robot by considering the decision-maker’s risk preference and interactive criteria. Finally, a case study of robot evaluation and selection in the auto industry is selected to exemplify the application of the proposed framework. After that, comparison and sensitivity studies are conducted to further demonstrate the robustness, effectiveness, and reasonableness of the developed approach.

Multi-objective vendor managed inventory system with interval type-2 fuzzy demand and order quantities

PurposeThe proposed IT2FMOVMI model intends to concurrently minimize total cost and warehouse space for the single vendor-retailer, multi-item and a consolidated vendor store. Regarding demand and order quantities with the deterministic and type-1 fuzzy numbers, we have also formulated the classic/crisp MOVMI model and type-1 fuzzy MOVMI (T1FMOVMI) model. The suggested solution technique can solve both crisp MOVMI and T1FMOVMI problems. By finding the optimal ordered quantities and backorder levels, the Pareto-fronts are constructed to form the solution sets for the three models.Design/methodology/approachA multi-objective vendor managed inventory (MOVMI) is the most recognized marketing and delivery technique for the service provider and the retail in the supply chain in Industry 4.0. Due to the evolving market conditions, the characteristics of the individual product, the delivery period and the manufacturing costs, the demand rate and order quantity of the MOVMI device are highly unpredictable. In such a scenario, a MOVMI system with a deterministic demand rate and order quantity cannot be designed to estimate the highly unforeseen cost of the problem. This paper introduces a novel interval type-2 fuzzy multi-objective vendor managed inventory (IT2FMOVMI) system, which uses interval type-2 fuzzy numbers (IT2FNs) to represent demand rate and order quantities. As the model is an NP-hard, the well-known meta-heuristic algorithm named NSGA-II (Non-dominated sorted genetic algorithm-II) with EKM (Enhanced Karnink-Mendel) algorithm based solution method has been established.FindingsThe experimental simulations for the five test problems that demonstrated distinct conditions are considered from the real-datasets of SAPCO company. Experimental study concludes that T1FMOVMI and crisp MOVMI schemes are outclassed by IT2FMOVMI model, offering more accurate Pareto-Fronts and efficiency measurement values.Originality/valueUsing fuzzy sets theory, a significant amount of work has been already done in past decades from various points of views to model the MOVMI. However, this is the very first attempt to introduce type-2 fuzzy modelling for the problem to address the realistic implementation of the imprecise parameters.

Development of a multi-period multi-attribute group decision-making method using type- 2 fuzzy set of linguistic variables

Recently, fuzzy linguistic variables have gained a great deal of attention from researchers in many decision-making problems. In this problems, type-2 fuzzy sets have been used to better cover linguistic data uncertainty. However, many of research works in this regard have been performed in type-2 fuzzy domain and in static mode. Since the decision making problems in the real world usually fluctuate over time, so it need to use decision making models in multi period of time. In the present research, a Multi-Period (Dynamic) Multi-Attribute Group Decision-Making (MPMADM) method is presented based on type-2 fuzzy sets where decision-making attributes are first expressed in linguistic terms and then incorporated, as interval type-2 fuzzy numbers, into problem solving where a new integrating operator called Multi-Period Trapezoidal Interval Type-2 Fuzzy Number Weighted Arithmetic averaging (MPTIT2FNWA) is defined on type-2 interval fuzzy numbers to integrate decision-making information in multiple periods of time. Once finished with explaining the proposed method, a numerical example is given to evaluate the proposed method in terms of effectiveness and applicability, with the results compared to those of other methods.

Selection of solar tracking system using extended TOPSIS technique with interval type-2 pythagorean fuzzy numbers

The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is considered among the most frequently used techniques to deal with multi-criteria group decision-making (MCGDM) conflicts. In this article, we have presented an extended TOPSIS technique in the framework of interval type-2 trapezoidal Pythagorean fuzzy numbers (IT2TrPFN). We first projected a novel approach to evaluate the distance between them using ordered weighted averaging operator and (alpha ,beta )-cut. Subsequently, we widen the concept of TOPSIS method formed on the distance method with IT2TrPFNs and applied it on MCGDM dilemma by considering the attitudes and perspectives of the decision-makers. Lastly, an application of solar tracking system and numerous contrasts with the other existing techniques are presented to express the practicality and feasibility of our projected approach.

Sustainability evaluation of photovoltaic poverty alleviation projects using an integrated MCDM method: A case study in Guangxi, China

Photovoltaic poverty alleviation project (PPAP) is initiated to reduce rural poverty and pursue cleaner production, but the lack of reasonable sustainability evaluation has become a challenge. This study evaluates the sustainability of PPAP in order to improve the poverty alleviation performance and efficiency considering an integrated multi-criteria decision-making (MCDM) method combining benefits, opportunities, costs, risks (BOCR) theory, analytic hierarchy process (AHP), interval type-2 fuzzy numbers (IT2FNs), and technique for order preference by similarity to an ideal solution (TOPSIS). First of all, this paper identifies 16 criteria based on BOCR theory from the perspectives of benefits, opportunities, costs, risks and constructs the criteria system. Second, an integrated BOCR-AHP-IT2FTOPSIS method is proposed and applied in processing ambiguous information and comprehensive sustainability evaluation. Third, an empirical case in Guangxi Zhuang Autonomous Region of China, a comparative analysis and a sensitivity analysis are used to rank the alternative PPAPs and verify the effectiveness and robustness of proposed method. The results show that the integrated MCDM method is capable to evaluate and rank the sustainability levels of PPAPs with the contribution of BOCR theory-based criteria system. The work offers a novel sustainability evaluation angle for PPAPs and has potential applications in providing understandings and making decisions.

Ranking interval type-2 fuzzy number based on a novel value-ambiguity ranking index and its application in risk analysis

Consequence of decision making problems under fuzzy domain mostly necessitate choosing the best alternative among the sets of alternatives. This requirement is generally fulfilled through the methodology called the ranking of fuzzy numbers. In this paper, an attempt under fuzzy domain to develop a coherent methodology of ordering fuzzy numbers is thrived. The ill-defined quantity ‘value’ and the amount of uncertainty in the ill-defined quantity ‘ambiguity’ of a fuzzy number constituted the important tools in ranking fuzzy numbers. As such, an effort has been made to employ these quantities in ranking interval type-2 fuzzy numbers constructing a novel value-ambiguity ranking index. A through comparative study with the existing method and the present method is executed to exhibit the essence of the proposed ranking method. The comparative study exhibit the superiority of the present method of ranking interval type-2 fuzzy numbers over the existing method. Further, the method is not just limited to interval type-2 fuzzy numbers, it can be broadly applied to arbitrary fuzzy numbers. Finally, a risk analysis problem has been discussed and the proposed novel value-ambiguity ranking index of ranking interval type-2 fuzzy numbers has been applied successfully.

An Extended Interval Type-2 Fuzzy ORESTE Method for Risk Analysis in FMEA

The failure mode and effect analysis (FMEA) is a prospective engineering technique which can prevent risk in advance by identifying, assessing potential failure modes and their effects. However, the traditional FMEA approach possesses the limitations in improperly identifying risk factor weights, inefficiently calculating risk priority number and improperly evaluating failure modes in practice. This paper proposes a hybrid risk analysis framework for FMEA based on interval type-2 fuzzy numbers and ORESTE method. First, the interval type-2 fuzzy numbers are utilized to express the uncertain risk evaluation information provided by experts. Second, the power average (PA) operator for interval type-2 fuzzy numbers is constructed to fuse individual risk assessment information and build the group risk assessment matrix. Then, the extended ORESTE method is introduced to determine the risk priority of each failure mode, in which distance measure of interval type-2 fuzzy numbers is incorporated. Finally, a case of ocean-going fishing vessel in a marine industry is selected to illustrate the application and feasibility of the proposed FMEA approach. A comparative analysis is performed to verify the effectiveness of the extended ORESTE method.

Interval type-2 fuzzy aggregation operator in decision making and its application

Type-2 fuzzy sets (T2FSs) can deal with higher modeling and uncertainties which exist in the real-world application, specifically in the control systems. Particularly the climate changes are always uncertain and thus, the type-2 fuzzy controller is an effective system to handle those situations. Polyhouse is a methodology used to cultivate the plants. It breaks the seasonal hurdle of the formulation and it is also suitable for the conflictive climate conditions. Controlling and directing internal parameters of the polyhouse play an essential role in the growth of the plant. Among those, humidity is an important element when one deals with the growth of the plant in polyhouse. It affects the weather, as well as the global change of the climate and hence, the inner climate of the polyhouse will be disturbed. In this paper, operational laws for triangular interval type-2 fuzzy numbers and derived triangular interval type-2 weighted geometric (TIT2WG) operator with their desired mathematical properties using Dombi triangular norms. Also, humidity control is analyzed using interval type-2 fuzzy controller (IT2FC) with the use of derived aggregation operator which is the aim of the paper. Further stability of the system has been analyzed by applying four different defuzzification methods and the method is recommended which gives a better response.