Fuzzy Interval Research Articles

Triangular approximation of intuitionistic fuzzy numbers on multi-criteria decision making problem

Most of the engineering applications depend on incomplete and imprecise information which are represented by nonlinear mathematical functions for defining membership and nonmembership functions in intuitionistic fuzzy setup. Shuyang Li and Hongxing Li have studied the approximation of conventional intuitionistic fuzzy numbers in which the membership function is bounded and nonmembership function is unbounded in nature. The concepts of intuitionistic fuzzy sets (IFSs) and interval valued intuitionistic fuzzy sets (IVIFSs) have bounded membership function and bounded nonmembership function. As the generalization of both IFSs and IVIFSs, the notion of intuitionistic fuzzy numbers of new type with bounded membership function and bounded nonmembership function has been introduced by Lakshmana et al. In this paper, the procedure for weighted triangular approximation of intuitionistic fuzzy number of new type is provided with some suitable illustrations. Moreover, some useful properties of the triangular approximation on intuitionistic fuzzy numbers have also been discussed.

Enhanced fuzzy time series forecasting model based on hesitant differential fuzzy sets and error learning

Most of existing fuzzy time series forecasting models lead to unsatisfactory forecasting performance due to deficiencies in constructing fuzzy intervals. This paper adopts the Fuzzy Silhouette criterion to determine the optimal number and length of fuzzy intervals effectively and objectively. The universe of discourse with equal and unequal intervals are merged by aggregating hesitant information. Second, multiple relevant factors are considered to reveal the complex attributes of the real-world data, and differential fuzzy sets are included to reconnoiter deep trends. Finally, an optimal error learning mechanism is applied to enhance the forecasting performance. Key strengths of the developed model are verifying the possibility of increasing the performance of conventional fuzzy time series models by integrating them with new components and processes. Three typical stock index datasets are selected to evaluate the performance of the proposed model and experimental results prove that it outperforms compared models in forecasting accuracy and profitable ability.

The depth and the attracting centre for a continuous map on a fuzzy metric interval

<p>Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω- limit points of x under f, respectively. Write ω(f) = ∪x∈Iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and Ωn+1(f) = Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2.</p>

Fuzzy Alexandrov Topologies Associated to Fuzzy Interval Orders

We characterize the fuzzy T0 - Alexandrov topologies on a crisp set X, which are associated to fuzzy interval orders R on X. In this way, we generalize a well known result by Rabinovitch (1978), according to which a crisp partial order is a crisp interval order if and only if the family of all the strict upper sections of the partial order is nested.

M-polar neutrosophic soft mapping with application to multiple personality disorder and its associated mental disorders

Multiple personality disorder (MPD) or dissociative identity disorder is the mental disease in which one can observe the existence of two or more than two personalities in a single person. We define the controversies nearby the diagnosis of MPD with its associated mental disorders. We discuss the various symptoms of MPD, dissociative amnesia, depersonalization or derealization disorder, and major depression disorder. After this exploration, we perceive that these disorders enclose parallel symptoms and it is difficult to identify the accurate type of disorder with its severeness. Since in experimental diagnosis the indeterminacy and falsity parts are often neglected. Due to this problem, we cannot see the accuracy in the patient’s improvement record and cannot predict the duration of treatment. To eradicate these boundaries, we present the m-polar neutrosophic soft set (MPNSS) and m-polar neutrosophic soft mapping (MPNS-mapping) with its inverse mapping. These notions are proficient and valuable to diagnose the disorder appropriately by connecting it with the mathematical modeling. The connection of m-polar neutrosophic set (MPNS) with the soft set characterizes a relation among patients, symptoms, and treatments which decreases the complexity of the case study. We build a chart based on a fuzzy interval [0, 1] to range the types of disorders. We establish an algorithm based on MPNS-mapping to identify the disease appropriately and to select the finest treatment for the corresponding disease of every patient. At last, we introduce the generalized MPNS-mapping which will helps a doctor to save the patient’s improvement record and to predict the period of treatment until the disease is cured.

Aircraft selection by applying AHP and TOPSIS in interval type-2 fuzzy sets

The selection of the appropriate aircraft can bring competitive advantages to airlines, however, there are a number of factors which introduce a degree of uncertainty to the selection process. By removing this uncertainty, airlines can increase their chances of achieving their long-term goals. New Multi-Criteria Decision Making (MCDM) methods provide decision-makers with a satisfactory solution for choosing suitable aircraft. Therefore, we focused on the multi-dimensional evaluation and selection of the most suitable commercial aircraft alternatives by using new MCDM method. This article provides decision support to airline planners on the selection of commercial aircraft under uncertainty. In the study, unlike other studies in the literature on aircraft selection, the model presented here uses an Interval Type-2 Fuzzy Analytical Hierarch Process (IT2FAHP) and Interval Type-2 Fuzzy Technique for Order Preference by Similarity to an Ideal Solution (IT2FTOPSIS) hybrid methods. The proposed model for aircraft selection allows commercial airlines to evaluate the aircraft in terms of specific criteria: economic performance, technical performance, and environmental impact, and, as a result, it helps decision makers select appropriate aircraft in an uncertain environment. In addition to use by commercial airlines, the methods in the study can also be applied to the selection of training aircraft, cargo aircraft and military aircraft. Our findings show that the Airbus A321neo is the most suitable commercial aircraft in terms of technical aspects, economic aspects and environmental aspects for airlines.

An Interval Fuzzy, Double-Sided, Chance-Constrained Stochastic Programming Model for Planning the Ecological Service Value of Interconnected River Systems

The western region of Jilin Province is an ecologically fragile area with scarce water resources. The effective allocation of the limited water resources in order to obtain a higher ecological service value is an urgent requirement. In this paper, an interval fuzzy, double-sided chance-constrained, stochastic programming (IFDCP) model was used based on the interconnected river system network project in the western Jilin Province. With the objective of maximizing the value of regional ecological services, the water diversion and supplement schemes were optimized and adjusted. The model results showed that the restored water surface area of all lakes and ponds in the western region of Jilin Province was higher than the initially planned scheme in the high flow year. The water surface area fulfilled the minimum constraints, but did not fulfill the initial scheme in the normal flow year. In the low flow year, the lower limit of some of the regions had to be decreased in order to meet the allocation of the limited water resources. The proportion of floodwater resource utilization gradually increased with an increase in the flood amount. The ecological service value produced in the normal and high flow years was found to be higher than the initial scheme. The marsh wetland can produce higher ecological service value. Therefore, the core of the model optimization was introducing more water to the marsh wetland after fulfilling the basic consumption of ponds and the reed wetland. In addition, the IFDCP model was more flexible in water diversion and supplement as compared to other models that had been developed previously.

Measuring the production performance indicators for food processing industry

Production performance measurement is the most important activity to achieve the higher productivity in food industries. The Overall Equipment Effectiveness (OEE) is a standard for measuring production productivity. However, the uncertainty of OEE components could mask the true production performance estimation in food processing lines. In this study, the production speed and stoppage duration measurements as major uncertainties to estimate the OEE and its components were investigated. In order to handle the uncertainties, two methods based on fuzzy arithmetic and interval arithmetic were conducted to supplement the classical-OEE. To survey the classical-OEE, the operational data field of edible oil purification process over a period of two years were examined. The comparison results of filed data revealed that the new OEE index has been increased around 4% by maintenance improvements. In such index, the total availability, performance efficiency and quality rate were improved by 3%, 2% and 5%, respectively. The research results showed that the spread values of Fuzzy-OEE are more explicit to show the worst and best values, while the interval arithmetic of OEE reflects the results by tighter limits. The proposed methods are more convenient to quantify uncertainty with limits to understand the OEE variations helping to make better decisions on uncertainties in food production lines.

Arithmetics of Vectors of Fuzzy Sets

The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.

Closed Forms of the Interval Type 2 Fuzzy Sets Additions Based on Archimedean T-norms with Application in Decision Making Aggregation

Fuzzy addition is one of the operations widely used in the application of fuzzy systems. The analytical and closed form expressions of the fuzzy additions of the generalized fuzzy numbers and the interval type 2 fuzzy numbers (IT2FS) are developed. The generalized fuzzy numbers is a non-flat and subnormal fuzzy number with different left and right shape functions. Classical fuzzy operations based on the minimum t-norm could lead to uncontrollable fuzziness. Fuzzy addition based on generalized t-norm can better control ambiguity. In order to control the ambiguity of fuzzy addition, this paper is to develop a general type 1 fuzzy set and interval type 2 fuzzy set (IT2FS) additions based on t-norm. The closed forms of the generalized IT2FS additions based on some popular t-norms are formulated. The specific analytical and closed form expressions based on Archimedean t-norms: e.g. drastic, Łukasiewicz, product and the families of Schweizer-Sklar and Yager t-norms are derived. The IT2FS are widely used to characterize the natural words, e.g. the decision making linguistic judgments, the fuzzy real time control systems with linguistic rules. Furthermore, based on the analytical T1FS additions results, the application to the linguistic multiple criteria decision making aggregation of the IT2FS additions based on t-norms is introduced. A comparative MCDM numerical example is demonstrated for the verification of the developed closed form expressions.

Portfolio selection of renewable energy-powered desalination systems with sustainability perspective: A novel MADM-based framework under data uncertainties

Harnessing renewable energy (RE) sources to power desalination systems can migrate the pressures of freshwater scarcity and fossil fuel depletion. In this paper, a novel mathematical framework was developed to assess the sustainability of different RE-powered desalination systems, which is very essential for their portfolio selection, by resorting to the fuzzy multi-attribute decision making (MAMD) methods. In the framework, an evaluation system including ten attributes from four dimensions is introduced, while fuzzy triangular numbers and interval values are employed to capture the epistemic and aleatory uncertainties of decision information, respectively. More importantly, three fuzzy MADM methods were proposed in the framework to make the following methodological contributions: (1) fuzzy decision-making trial and evaluation laboratory-based analytic network process (DANP) offers a reliable weighting result by addressing the interrelationships among the attributes; (2) fuzzy full consistency method (FUCOM) can easily but rigorously quantify the qualitative attributes by using the simplified pair-wise comparisons; (3) interval vector-aided technique for order of preference by similarity to ideal solution (VATOPSIS) generates a rational ranking sequence by incorporating the balance among multi-attributes. To illustrate the proposed framework, six RE-powered desalination systems were studied, showing that the solar thermal-powered multi-effect distillation would be the best option under current conditions. Finally, the effectiveness and advantages of the proposed framework were verified by results comparison and discussion.

Interval analysis of the HIV dynamics model solution using type-2 fuzzy sets

The objective of this study is to analyze the numerical intervals of the Human Immunodeficiency Virus (HIV) using the solution of a dynamics model at each time iteration, considering the infection rate of CD4+ T lymphocytes and the production rate of the virus as outputs of an interval type-2 Fuzzy Rule-Based System (FRBS). The mathematical model consists of a system of ordinary differential equations in the case of the HIV-seropositive individuals who are under antiretroviral treatment. The input variables of FRBS are adherence to treatment and medication potency, which are important factors in the treatment of HIV-seropositive individuals. If the infection rate of CD4+ T lymphocytes and the production rate of the virus are output variables of a type-1 FRBS, then we obtain curves for the solution of the HIV dynamic model. When the rates are outputs obtained through the interval type-2 FRBS, we can obtain ranges. This means that a type-2 FRBS provides more information about the behavior of the uninfected and infected CD4+ T lymphocytes, free virus particles, and virus-specific cytotoxic T lymphocytes (CTL) that attack infected cells in the bloodstream of a HIV-seropositive individual. The theory of the single level constraint interval arithmetic allows us to obtain a single interval that containing the intervals of all iterations for uninfected and infected CD4+ T lymphocytes, free virus particles, and CTL. Thus, the novel contribution of this research is twofold. Firstly, the use of interval type-2 fuzzy rule-base analysis to obtain more informative and robust results not possible with standard approaches is new. Secondly, the use of the fuzzy constraint interval representation for fuzzy type-2 intervals in the context of the type-2 fuzzy rule base is new. Moreover, constraint intervals unify the theory and its use in single-level constraint representations simplifies the calculations, though this aspect is not pursued in detail here.

Developing a Labeled Affective Magnitude scale and Fuzzy Linguistic scale for tactile feeling

AbstractAffective design is the inclusion or representation of human emotions and subjective impressions in product design processes. In affective design, a number of different scales are commonly used to reveal and measure subjective emotions related to the design features of products. Osgood's Semantic Differential Scale (SDS) is one of the scales that has often been used for this purpose. However, there are some drawbacks in the SDS due to the ordinal nature of the scale that leads to losses or distortions of a significant amount of information and this makes it difficult to justify parametric statistical analysis. In this study, two scales, namely a Labeled Affective Magnitude (LAM) scale and a Fuzzy Linguistic scale, are developed. The LAM scale is an alternative scale based on magnitude estimation and has ratio properties. The Fuzzy Linguistic scale is an interval scale for which responses are linguistic descriptors that are identified with fuzzy numbers or intervals. The scales were developed for tactile feelings because they are an important factor in product evaluation. Statistical analysis was conducted to compare the scales. There was no significant difference between the newly constructed fuzzy scale and 11 point SDS, whereas there was a significant difference between the newly constructed LAM scale and 11 point SDS.

Use of fuzzy areas of preference in tasks of multi – criteria evaluation of freight air transport

This paper shows how, based on the theory of fuzzy sets, the problem of multi-criteria selection of cargo planes is solved. It is proposed to determine fuzzy preference intervals on the scales of aircraft efficiency criteria. At the same time it is allowed to intersect such intervals taking into account the limitations on the maximum and minimum degree of belonging of criterion values. For combinations of fuzzy intervals, Decision Maker makes its judgments on a fuzzy preference scale. The resulting aircraft efficiency model is checked for coverage of all points of the criterion space with a level of preference not lower than the specified one. This model can be used to estimate any number of cargo aircraft in automatic mode. Both clear and fuzzy criterion values can be submitted to the model input, and then the method allows to build the functions of aircraft belonging to all areas of preference. The software implementation of this model can be used to make decisions about the purchase, manufacture, repair of certain types of aircraft and to determine which types of aircraft should be decommissioned.

Method of estimating the Pareto-optimal solutions based on the usefulness

This article briefly discusses methods for finding Pareto-optimal solutions for multi-criteria problems in the conditions of uncertainty. It is noted that the quality of management depends on the adequacy of the rule base for the selected fuzzy controller set by experts. The conditions for the final decision of the choice of the Pareto-optimal fuzzy solution are determined. An algorithm for the method of estimating the Pareto-optimal solution taking into account utility is developed and presented. Multicriteria optimization of the functioning of control systems is considered when setting local criteria in the form of fuzzy intervals. Scalarization methods that are used in ranking criteria are considered. Quantitative determination of the criterion is applicable if its numerical comparison is possible. The objective function is determined by local criteria. Optimization is associated with the search for control actions that provide optimal values of the multidimensional objective function. For problems with threshold optimization, a similar estimate can also be obtained. Given that, when analyzing solution options, complexity scales are constructed, an estimate of the time to build the complexity scale is also introduced. The application of the threshold optimization method to search for the Pareto optimal solution is considered. An approach to evaluating Pareto-optimal solutions using the criteria for evaluating utility is proposed. An algorithm has been developed to evaluate the Pareto-optimal solution taking into account utility.

Application of Artificial Intelligence in College Dance Teaching and Its Performance Analysis

Artificial intelligence (AI) has not been widely and deeply implemented in dance teaching of colleges. To solve the problem, this paper identifies the problems with the AI in college education and dance teaching, theoretically discusses the application strategies of the AI in dance teaching of colleges, and demonstrates the promotive effect of AI in college dance teaching. Next, a performance analysis model was established based on fuzzy interval num-ber, grey system theory, and analytic hierarchy process (AHP), and adopted to evaluate the performance of the AI application in college dance teaching. The research results shed a new light on the application of cutting-edge tech-niques in college education.

Strategy selection by using interval type-2 fuzzy mcdm and an application

Competitive market conditions today are becoming more challenging in every sector. Companies must determine their road maps in terms of short, mid and long-term strategies. The selection of the most suitable strategy by a group of decision makers is a very complicated process. As the same linguistic expression can represent different meanings to different people the fuzzy decision-making process is a good tool to use in group decisions. Various methods and different linguistic terms are used to define fuzzy sets. In recent years, the interval type-2 fuzzy AHP and interval type-2 TOPSIS methods have been introduced in the literature, as well as the type-1 fuzzy AHP and type-1 fuzzy TOPSIS methods. The aim of this study is to present a new model for group decision-making with the Balanced Score Card, fuzzy AHP and fuzzy TOPSIS methods by using interval type-2 fuzzy sets. The proposed model was applied in the strategy selection process of a large-scale public transportation company that operates in Istanbul/Turkey.

N-Dimensional (S,N)-implications

The n-dimensional fuzzy logic (n-DFL) has been contributed to overcome the insufficiency of traditional fuzzy logic in modeling imperfect and imprecise information, coming from different opinions of many experts by considering the possibility to model not only ordered but also repeated membership degrees. Thus, n-DFL provides a consolidated logical strategy for applied technologies since the ordered evaluations provided by decision makers impact not only by selecting the best solutions for a decision making problem, but also by enabling their comparisons. In such context, this paper studies the n-dimensional fuzzy implications (n-DI) following distinct approaches: (i) analytical studies, presenting the most desirable properties as neutrality, ordering, (contra-)symmetry, exchange and identity principles, discussing their interrelations and exemplifications; (ii) algebraic aspects mainly related to left- and right-continuity of representable n-dimensional fuzzy t-conorms; and (iii) generating n-DI from existing fuzzy implications. As the most relevant contribution, the prospective studies in the class of n-dimensional interval (S,N)-implications include results obtained from t-representable n-dimensional conorms and involutive n-dimensional fuzzy negations. And, these theoretical results are applied to model approximate reasoning of inference schemes, dealing with based-rule in n-dimensional interval fuzzy systems. A synthetic case-study illustrates the solution for a decision-making problem in medical diagnoses.

Influence of impreciseness in designing tritrophic level complex food chain modeling in interval environment

In this paper, we construct a tritrophic level food chain model considering the model parameters as fuzzy interval numbers. We check the positivity and boundedness of solutions of the model system and find out all the equilibrium points of the model system along with its existence criteria. We perform stability analysis at all equilibrium points of the model system and discuss in the imprecise environment. We also perform meticulous numerical simulations to study the dynamical behavior of the model system in detail. Finally, we incorporate different harvesting scenarios in the model system and deploy maximum sustainable yield (MSY) policies to determine optimum level of harvesting in the imprecise environment without putting any unnecessary extra risk on the species toward its possible extinction.

Methods of Using Fuzzy Interval Logic During Processing of Space States of Complex Biophysical Objects

Methods of Using Fuzzy Interval Logic During Processing of Space States of Complex Biophysical Objects