The recently introduced ‘Improved Probabilistic Intuitionistic Fuzzy c-Means algorithm (IPIFCM)’ provides a Probabilistic Euclidean Distance Measure (PEDM) based computationally efficient clustering technique. Since IPIFCM is defined on Type-1 Atanassov Intuitionistic Fuzzy Sets (AIFS), it is unable to capture the uncertainty of the membership and non-membership values of a given datapoint induced by the hesitancy factor of the AIFS. Interval Type-2 Fuzzy Sets (IT2 FSs) deals with the uncertainty in the membership values. In this paper, we incorporate IT2 FSs in the IPIFCM algorithm by introducing upper bound and lower bound of the membership (and non-membership) values of each datapoint to model the change caused by the hesitancy factor. Accordingly, this paper proposes the ‘Probabilistic Interval Type-2 Intuitionistic clustering algorithm’ (P-IT2IFCM), which uses the interval probabilistic weights for PEDM to propose ‘Interval Type-2 Probabilistic Euclidean Distance Measure’ (IT2PEDM). The proposed algorithm provides superior results to existing Fuzzy c-Means (FCM) algorithms such as the basic FCM algorithm, IFCM algorithm, Kernelized-IFCM algorithm and IPIFCM algorithm when executed over various benchmark UCI datasets.

The recently reported ‘Improved Probabilistic Intuitionistic Fuzzy c-Means (IPIFCM) algorithm’ is a computationally efficient algorithm that does fuzzy clustering based on Probabilistic Euclidean Distance measure (PEDM). A significant issue with the IPIFCM algorithm is that it does not consider the hesitation factor's effect while updating the membership of a datapoint for a given cluster. Therefore, the convergence of the algorithm is not optimal. In this paper, we modify the membership function by adding the hesitation component to the IPIFCM clustering algorithm's objective function to propose 'Modified Improved Probabilistic Intuitionistic Fuzzy c-Means' (MPIFCM) clustering algorithm. The proposed MPIFCM algorithm helps in achieving a realistic clustering of the datapoints. This modification leads to the improvement in the accuracy as well as the convergence rate of the algorithm. Experiments over various benchmark UCI datasets confirm that our proposed algorithm provides better performance over its existing counterparts. Popular performance metrices such as accuracy, convergence rate, partition coefficients and cluster entropy are considered for comparative analysis of the performances of the studied algorithms.

Intuitionistic fuzzy soft sets (IFSSs) are employed to model real-life problems. In IFSS, a parameter reflecting the expert opinion regarding the purity of the information is added to generalize the classical concept of IFSS. The reliability of this generalized notion of IFSS in dealing with decision-making problems more accurately is high than the classical IFSS. The similarity between two generalized intuitionistic fuzzy soft sets (GIFSSs) is calculated by similarity measure. Sugeno integral gives an operation that is similar to the expected value, so it would be an effective tool to determine the expected total similarity degree between two GIFSSs. Hence, we have proposed a novel similarity measure based on the Sugeno integral of GIFSS. We have also studied some of the mathematical properties of our novel similarity measure. Along with the effect of the generalization parameter in the proposed similarity measure, its performance analysis over a decision-making problem discusses the superiority of the proposed similarity measure.

The current ongoing pandemic COVID-19 situation severely impacts the tourism sector. The principal economic component in some of the countries is their tourism sector. So, a proper strategy to recommence the tourism sector needs to be formulated. We chalk out a plan of action while solving multiple criteria sorting (MCS) problem. The dealing of COVID-19 involves hesitancy and uncertainty thus, Atannasov’s intuitionistic fuzzy set is used to model this situation. The paper introduces an intuitionistic fuzzy grey relational analysis sort (IFGRA-sort) technique to strategize the reopening of the tourism industry. The proposed technique successfully solves the tourism industry problem given in Current Issues in Tourism (2021): 1-11, Taylor and Francis.

Atanassov came up with an idea of the intuitionistic fuzzy set (AIFS), primarily a generalized extension of Zadeh's fuzzy set. AIFS is more supple in tackling the real world's unpredictability rather than Zadeh's fuzzy set. The researchers from various disciplines are attracted to the AIFS because of its proximity to real life. AIFS is very rich in literature in both aspects, i.e., theoretical and practical. At present, the intuitionistic fuzzy integral is a scorching issue of the AIFS. Furthermore, to aggregate and process the data, the existence of integral is compulsory. It is reasonably necessary to look that whether integral exists or not for the given data. Thus, our primary aim is to confirm the existence of a given additive definite integral (ADI). We have provided a necessary and sufficient condition to prove the existence of the given ADI. Finally, an example is fabricated for application grounds. It is shown that to aggregate a vast amount of data aggregation operator prompted from ADI is an outstanding option.

A Fuzzy Data Envelopment Analysis (FDEA) is a popular technique to measure the relative efficiency of a Decision Making Unit (DMU) with respect to other DMUs under uncertain/imprecise information represented in form of fuzzy input and fuzzy output. However, in a real life application, due to higher order of uncertainty, the fuzzy set may not be a suitable choice, as the membership value alone cannot represent the input/output information precisely. Therefore in the paper, we extend the FDEA model to Intuitionistic Fuzzy Data Envelopment Analysis (IFDEA) model, namely, Parabolic Intuitionistic Fuzzy based Data Envelopment Analysis, where the input and output are demonstrated by Parabolic Intuitionistic Fuzzy Numbers (PIFNs). Further the α-cut and ß-cut approach are used to convert the parabolic intuitionistic fuzzy inputs and outputs into their corresponding intervals and to compute the parametric efficiencies of the given DMUs. Additionally, we have used the section formula to defuzzify the parabolic intuitionistic fuzzy numbers into their crisp form to execute the optimization problem. Finally, the cross-efficiency technique is used to rank the DMUs.

Atanassov's intuitionistic fuzzy sets (A-IFSs) are used to deal with that information, which is incomplete as well as imprecise. In this paper, we defined a similarity measure by using Sugeno integral and technique of (α, β)-cut. Hwang et al. [Hwang, Chao-Ming, et al. "A similarity measure of intuitionistic fuzzy sets based on the Sugeno integral with its application to pattern recognition." Information Sciences 189 (2012): 93-109.] defined Sugeno integral based similarity measure for the first time. But, in Hwang et al.'s similarity measure, only α-cut is utilized that neglected the contribution of non-membership function. The non-membership function plays an equal role in the A-IFS theory. Therefore, we proposed the Sugeno integral based similarity measure concerning both the (α, β)-cuts. We added one artificial constructed example to show that our proposal is different than to similarity measure defined by Hwang et al. Moreover, we added some more benchmark examples to show the efficacy of the proposed similarity measure.

A Fuzzy Data Envelopment Analysis (FDEA) is a popular technique to measure the relative efficiency of decisionmaking units (DMUs) with imprecise and vague data for multiple inputs and outputs. In real-life applications, there are two types of outputs: desirable outputs and undesirable outputs. In this paper, we have proposed a new version of FDEA model, named as Parabolic based Fuzzy Data Envelopment Analysis (PFDEA) model that computes parametric efficiency of a DMU in the presence of undesirable outputs. The inputs and outputs are represented in the form of asymmetric parabolic fuzzy numbers in the proposed model. A new technique is introduced to convert PFDEA model into a linear programming problem using α-cut approach with a novel section formula based method, named as Ratio Division Method. This method is used to perform the complete ranking of the DMUs in a numerical example using Cross-Efficiency Method to provide a complete ranking of the DMUs.

In this paper, Interval-valued intuitionistic fuzzy set (IVIFS) is exploited to propose a generalization of fuzzy TOPSIS method. We have termed the proposed TOPSIS method as Interval-valued intuitionistic fuzzy TOPSIS method (IVIFSTOPSIS). Here, the IVIFS-TOPSIS handles the Supplier Selection problem in which linguistic variables based criteria description is given. The best supplier obtained by IVIFS-TOPSIS is in concurrence with the other well-known fuzzy TOPSIS methods. It is possible to use IVIFS-TOPSIS over other types of linguistic variable multi-criteria decision making (MCDM) problems.

Recently proposed Probabilistic Intuitionistic Fuzzy c-Means Algorithm (PIFCM) is a Probabilistic Euclidian distance measure (PEDM) based clustering technique, which incorporate computation of probabilistic intervals (Pij, Qij) for each of the data point. PIFCM algorithm employs a random membership function $\frac{1}{{\left| x \right|}}$ and discards a data point if its membership value is uniformly distributed in the clusters. Fuzzy clustering always gets affected by the choice of the membership function. Accordingly, in PIFCM algorithm, membership function changes the properties of the data limiting its capabilities in giving consistent clustering results. Moreover, PIFCM algorithm incorporates computation of redundant matrices while finding Pijand Qij. In this paper, we propose some novel changes in the existing PIFCM algorithm, and hence introduce our Improved PIFCM algorithm. The improved PIFCM algorithm considers the min-max normalization as membership function, and also removes the redundant matrix computation that was used to find the Pijand Qijin the original PIFCM. Results over various UCI datasets validates the superiority of our improved PIFCM algorithm over FCM algorithm, IFCM algorithm and PIFCM algorithm.

Most of the time membership value in the fuzzy set cannot be exactly defined. Interval-valued fuzzy set (IVFS) is a special type of type-2 fuzzy sets which represents the membership value of the fuzzy set as an interval. IVFS assumes that membership interval can better represent the uncertainty in the data. Accordingly, IVFS can be used to obtain good clustering results since it can represent the uncertainty more appropriately. Thus, this paper proposes the interval-valued fuzzy c-means algorithm (IVFCM) which uses IVFSs to represent the data. The concept of the proposed IVFCM is then extended to introduce the interval-valued density based fuzzy c-means (IVDFCM) algorithm based on the distance measure of IVFSs. Both IVFCM and IVDFCM are simulated over various UCI benchmark datasets to show their suitability and supremacy over their existing counterparts.

Atanassov's intuitionistic fuzzy set (A-IFS) is a more general form of Zadeh's fuzzy set, and it is defined to deal with the uncertainty more accurately. The closeness of A-IFS to the reality attracted researcher from multidisciplinary areas. Regarding A-IFS, a lot of work has been done in both: theoretical and practical aspects. Nowadays, intuitionistic fuzzy integrals is a very hot topic of Atanassov intuitionistic fuzzy (A-IF). Moreover, the existence of the integral is essential to define aggregation operator and process the information, that is, necessary to check whether for a given data, the desired integral exists or not. Therefore, in this paper, our main focus will be to ensure the existence of the additive definite integrals (ADI). To do so, we will use bounded variation based functions (BV function), that are the functions whose approximate length are finite. Finally, we have constructed an example for application purpose and have shown that aggregation operator induced from ADI with BV function is a excellent choice to aggregate a huge amount of data.

This paper proposes the novel way to estimate the gross domestic product (GDP) of a country from its carbon emission (CO2) data. This alternative method to predict GDP is required for the war affected and non-accessible nations as the macroeconomic data available for those nations is highly unpredictable or insufficient. However, first we need to train and develop a reliable model for which we have used the Transfer learning (TL) approach. TL is applied in a way that a neural network (NN) or machine learning (ML) model is trained on the developed nation's data and used for testing with developing nation data. The NN models used in this paper are Extreme Learning Machine (ELM) and Generalized Regression Neural Network (GRNN), while ML model is Support Vector Regression (SVR). Since the data is measurement data collected from several devices, it contains noise and hence, uncertain. Thus, first we have modelled the dataset with type-1 fuzzy sets and then with the interval type-2 fuzzy sets. The results are then compared with the crisp input data values.

Image steganography is the art of hiding secret data into an image in such a way that it cannot be detected by any intruder. For steganography, image is a good carrier because it contains a higher redundancy of pixel values, making it less sensitive to the human visual system. Therefore, embedding in those pixels of the image leads to high visual quality, payload capacity and security. This paper proposes an interval type-2 fuzzy logic-based system to detect those pixels of the image which are less sensitive to human eyes. The embedding is performed on the selected pixels using the least significant bit (LSB) method. We have termed this method the interval type-2 fuzzy logic system based LSB (IT2FLS-LSB) steganographic method. The experimental simulations were performed on a collection of image datasets to show the efficacy of IT2FLS-LSB embedding. Quality index metrics such as PSNR (peak signal-to-noise ratio), UQI (universal quality index), and SSIM (structural similarity measure) are used to assure the visual quality of the stego images. We have also evaluated the high payload capacity of our proposed method.

We often have many datasets where hard clustering algorithms do not deliver satisfactory clustering results. It is found that many times fuzzy clustering technique improves the clustering results obtained by hard clustering algorithms. Fuzzy c-means (FCM) is the most prominent fuzzy clustering techniques whose improvement was proposed through the introduction of intuitionistic fuzzy set (IFS) based c-means algorithm. In order to implement IFS based c-means algorithm over a real valued dataset, data points were first converted into IFSs by employing a highly popular technique known as Yager's generating function. The Yager's generating function tunes only the non-membership and hesitancy component of an IFS. Therefore, IFS based c-means algorithm produces compromised clustering results. In this paper, we have generalized the Yager's generating function in such a manner that our IFS generation function tunes all the three components of the IFSs. We have utilized the proposed IFS generation function in two highly used IFS based c-means algorithms of clustering known as intuitionistic fuzzy c-means (IFCM) and Novel intuitionistic fuzzy c-means (Novel-IFCM) algorithms on the UCI datasets. Our results obtained using the proposed function are better than the results obtained using YGF.

The CO2emission due to industrialization is a crucial parameter which is directly proportional to the economic growth of any country/nation. However, for the non-accessible nations and war-torn nations with highly unreliable or insufficient macroeconomic data, the prediction of gross domestic product (GDP) is a challenging task. Thus, in this paper, we have proposed a novel approach for the reliable GDP estimation utilizing only the CO2emission data. For this purpose, transfer learning (TL) is applied which learns on the previously acquired information and solve the new task. The training is performed on the GDP data of the developed nations and then prediction is estimated for the developing nations. This is implemented using kernel extreme learning machine (KELM) in which the output weights are modelled using interval type-2 fuzzy sets (IT2 Fss) for the effective transferal of knowledge from developed to developing nation. Experimental results have shown that the proposed IT2F-KELM provide much-improved RMSE as compared with the traditional KELM.

This paper presents a novel probabilistic similarity measure (PSM) for Atanassov intuitionistic fuzzy sets. It then exploits PSM to propose an adaptive probabilistic similarity degree and develops the novel probabilistic λ-cutting algorithm for clustering. Further, the probabilistic distance measure (obtained from the PSM) is used to develop a new clustering technique, which we have named “probabilistic intuitionistic fuzzy c-mean (PIFCM) algorithm”. Simulation experiments have been conducted over a variety of datasets including UCI machine learning datasets and realworld car dataset. The results obtained have been thoroughly compared with other well-known clustering techniques such as fuzzy c-mean (FCM), intuitionistic fuzzy c-mean, association coefficient method, and λ-cutting method. Based upon the experimental results, it can be concluded that our probabilistic λ-cutting algorithm and PIFCM algorithm outperform their existing counterparts.

The multiobjective reliability redundancy allocation problem (MORRAP) aims to ensure high system reliability in the presence of optimally redundant components. This is one of the most important design considerations for system designers. Due to the associated uncertainty in component parameters, precise computations of overall system reliability, cost, and weight, etc., are difficult during design time. Hence, these parameters are befitting to be modeled as fuzzy quantities. As type-1 fuzzy numbers have limitations in representing higher order uncertainties, so this paper models the component parameters viz., reliability, cost, and weight with interval type-2 fuzzy numbers. Thus, we propose a novel formulation of MORRAP, termed as interval type-2 fuzzy multiobjective optimization problem (IT2FMORRAP). A popular multiobjective evolutionary algorithm, viz., nondominated sorting genetic algorithm II, is used to solve the proposed IT2FMORRAP, for which we have developed two novel algorithms in this paper. Numerical examples are included to demonstrate the solution approach. On comparing the outcomes with earlier results, we have found that the proposed IT2FMORRAP outperforms classical as well as other type-1 fuzzy-number-based approaches.

In the modern era of industrialization and globalization, distribution and control of goods are essential aspects for multinational corporations and strategic partners. Vendor managed inventory (VMI) is one of the well-known strategies of merchandizing between supplier and retailer. In this paper, we consider different number of suppliers and retailers to perform business under VMI system and formulate three: single-supplier and single-retailer, single-supplier and multi-retailer, and multi-supplier and multi-retailer VMI systems. The objective is to minimize the total cost of VMI system. Since it is a non-linear integer programming problem, this paper proposes a novel hybrid biogeography-based optimization algorithm to solve it. We enhance the proposed algorithm by embedding stochastic fractal search (SFS) in biogeography-based optimization (BBO). SFS algorithm is a newly developed powerful evolutionary algorithm to find global optimum much faster and efficiently. The diffusion process of SFS improved the exploitation ability of search in BBO. Our proposed algorithm is applied on all three versions of VMI systems under different constraints. We have considered suitable input data for all the different problems and obtained the results. By comparison, we show that the results outperformed for all VMI systems.

Reliability redundancy allocation problem (RRAP) is an optimization problem with objective to maximize the system reliability considering component reliability and redundancies as decision variables. RRAP was mostly solved as a single level optimization problem. However, the nature of the problem fits quite well in the framework of bilevel optimization. In this paper, we have proposed two novel bilevel formulations for the RRAP and solve them using a latest bilevel optimization algorithm called BLEAQ (bilevel evolutionary algorithm based on quadratic approximations). So far we knew no other research has been reported till date, where RRAP was addressed with bilevel optimization algorithm. Here, optimization is needed at two separate levels, where one problem is encircled within another problem. The inner problem is known as lower-level problem and the external problem is called upper-level problem. Here, we have presented two mixed-integer non-linear bilevel formulations for the RRAP of series-parallel system in a competitive environment. The purpose of the upper-level problem is to determine the component reliability that maximizes the total system reliability; whereas, lower-level problem minimizes the total cost (or weight) needed. We demonstrate the applicability of our approach with a suitable numerical example and show that our proposed approach works quite well than existing single level optimization tools.

Vendor managed inventory (VMI) is one of the well-known strategies of merchandizing between suppliers and retailers. The essential aspect in VMI model is to fulfill the demand of the retailers by the suppliers. Since, the costs of items are highly volatile, it is not possible to design a VMI model with deterministic demand. In this paper, we have considered one supplier and retailer to perform business of multi-product under VMI model with uncertain demand. Thus, it proposes a novel VMI model, termed as `interval type-2 fuzzy vendor managed inventory (IT2 FVMI) model' that considers interval type-2 fuzzy number to represent the uncertain demand. The objective is to minimize the total cost of VMI by finding the optimum ordered quantity and backordered level. We develop a particle swarm optimization algorithm based solution approach to solve it. A suitable real-world application is considered to perform the simulation. We have considered different number of instances of the products to obtain the results. By comparing the results with the deterministic demand and type-1 fuzzy number, we show the efficacy of our proposed IT2 FVMI model.

Transfer learning framework is designed to use previously acquired knowledge to solve a new but somewhat related task (like humans do). Non-availability of sufficient and relevant information in building a learning model is a major bottleneck in this research area. However, such models are highly susceptible to negative transfer learning (NTL) during transferral of knowledge due to the hesitancy in the decision making. Negative transfer learning may cause chaotic learning and have a profound effect on their predictive precision. In this paper, we have proposed a novel Intuitionistic Fuzzy Domain Adaptation (IFDA) algorithm, which uses Yager-generating function over Atanassov's Intuitionistic fuzzy set theory in conjunction with modified Hausdorff Intuitionistic similarity metric to build a fuzzy domain adaptation algorithm which is independent of supervised machine learning technique. It exploits the hesitancy margin in intuitionistically fuzzified features by eradicating similar looking but useless instances. Therefore, it selects optimal source instances from a previous problem in bridging the knowledge gap, in order to solve a new target problem, by containing negative transfer learning.

Supplier selection is a process in which one supplier is selected out of given suppliers on the basis of certain features such as reliability, maintenance, delivery performance, quality etc. Nowadays, supplier selection problem is a big issue because selection of the best supplier is a multi-criteria decision making (MCDM) problem under many combating criteria. The knowledge of the decision makers (DMs) is incomplete as well as imprecise. To deal this complex situation (uncertainty and hesitation), Intuitionistic fuzzy sets (IFSs) are used to select better DMs preferences. IFS is a powerful tool as it deals with membership function, non-membership function together with hesitancy. In this paper, we propose an intuitionistic fuzzy TOPSIS decision making method using correlation coefficient to deal with MCDM problems using IFS. Intuitionistic fuzzy weighted averaging (IFWA) operator is used to aggregate each DMs opinions for valuating the importance of alternatives and criteria. The proposed method is implemented on the numerical example given in F. E. Boran, S. Genc, M. Kurt and D. Akay, [Expert Systems with Applications, 8(2009), 11 363-11 368] to demonstrate about our procedure. The results obtained by our method matches with the results of Boran. Hence, our logic is proper for handling supplier selection problem.

In this paper, we have addressed the reliability-redundancy allocation problem with a particle swam optimization based technique. The parameters of the system components are actually imprecise or uncertain quantity since those are generally guessed by the designers during the design-time. Thus, important features of the designed system, viz. reliability, costs, weight etc very suitably qualifies to be considered as fuzzy quantity. Our problem formulation considers these parameters as type-2 fuzzy quantity. There are few reports where the problem has been studied under type-1 fuzzy uncertainty. As far as we know, no research has been reported where the problem has been addressed with a particle swam optimization based approach in a type-2 fuzzy environment. Suitable examples are included to demonstrate our approach. Results are compared showing that the type-2 fuzzy uncertainty based approach outperforms other recently reported results.

In Intuitionistic fuzzy sets(IFSs), experts assign both membership value and non-membership value to each fuzzy element x with a certain degree of hesitation. The hesitancy in the opinion of the experts appear due to incomplete information available regarding x. Therefore, precise estimation of its both membership value and non-membership value becomes highly difficult. Hence, there is a high chance that both membership value and the non-membership value assigned to x by the expert may not be absolutely correct. So, whenever we try to measure similarity between the IFSs using the various distance measures involving all the components of IFSs like membership value, non-membership value together with hesitation, then we often notice that all of them fails to describe the underlying situation completely. Therefore, the similarity measures derived from these distance measures also fails to produce good results. So, we introduce a new similarity measure by properly defining a similarity degree through the result established in this paper. The similarity measure has a central role in developing a modified λ-cutting algorithm for clustering. Here we also establish the efficacy of our modified λ-cutting algorithm while implementing it on a real world data set.

Reliability is the measure of the result of the quality of the system over a long run. The reliability-redundancy allocation problem (RRAP) aims to ensure high systems reliability in the presence of optimally redundant systems components. This is one of the most important design considerations for the systems designers. Several researchers have addressed this important issue during last few decades. However, due to the embedded uncertainty in the parameters of the system components, reliability as well as the costs of the whole system fits very well to be modeled as fuzzy quantity. We therefore modeled this problem as a fuzzy multi-objective optimization problem (MORRAP) that is addressed using the popular multi-objective evolutionary algorithm, NSGA-II (non-dominated sorting genetic algorithm-II). We have considered the based MORRAP with fuzzy type-2 uncertainty. As far as we know, no research has been reported where MORRAP was considered under type-2 fuzzy uncertainty. A typical numerical example is included and results are compared showing that our approach outperforms other recently reported results.

In real-time systems, energy efficiency is a vital issue since usually such systems run on battery and are remotely placed. Another important aspect of these systems is their capabilities to produce timely results. In this paper, we have reported how these two conflicting issues of embedded real-time systems can be addressed with the help of an efficient evolutionary algorithm viz. NSGA-II (Non-Dominated Sorting Algorithm-II). Moreover, during the system design time, the timing parameters in real-time systems are all designers' approximation since those can hardly be predicted before runtime. This means that there exists some uncertainty and hence it is appropriate to consider fuzzy numbers to model these timing parameters. Although type-I fuzzy numbers were used by a number of researchers to model the timing parameters of real-time embedded systems, they suffer from the interpretability issues. To address this, we thus propose here to consider type-2 fuzzy numbers to model real-time tasks timing parameters. Few numerical examples are included to demonstrate our proposed technique.