Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm

Multi-attribute group decision making method for sponge iron factory location selection problem using multi-polar fuzzy EDAS approach

This paper presents new averaging operators, such as mpF Einstein weighted averaging (mpFEWA), mpF Einstein ordered weighted averaging (mpFEOWA), mpF Einstein hybrid weighted averaging (mpFEHWA), mpF Einstein weighted geometric (mpFEWG), and mpF Einstein hybrid weighted geometric (mpFEHWG), as well as new Einstein operations (mpFNs) for handling multi-polar fuzzy numbers. We evaluate these operators for idempotency, boundedness, monotonicity, and commutativity, and we design them to deal with multi-polar fuzzy numbers (mpFNs). Furthermore, the study investigates the use of these operators in MAGDM settings, namely mpFEWA and mpFEWG operators, to expand on this. Additionally, it proposes a procedure for determining the best site for a sponge iron production plant by use of the created MAGDM method. The EDAS method, which stands for "Evaluation based on Distance from Average Solution," verifies that the solutions are effective. Finally, the suggested model highlights the benefits and possible improvements provided by these creative strategies by comparing the new approach to conventional methods and evaluating its efficiency and practicality.

Archimedean Heronian mean operators based on complex intuitionistic fuzzy sets and their applications in decision-making problems

In this article, we derive the Archimedean aggregation operators for complex intuitionistic fuzzy sets, for this, first, we evaluate some Archimedean operational laws based on complex intuitionistic fuzzy values and then we discuss their special cases because the Archimedean norms are the general form of all existing norms, for instance, algebraic, Einstein, Hamacher, and Frank operational laws. Furthermore, we present the complex intuitionistic fuzzy Archimedean Heronian aggregation operator and complex intuitionistic fuzzy weighted Archimedean Heronian aggregation operator. Several special cases and the basic properties of the above-proposed operators are also diagnosed, because proposing the Heronian mean operators based on Archimedean norms are very challenging and complicated tasks, because of their features and structure. Additionally, a decision-making process is developed under the identified operators by using complex intuitionistic fuzzy information. Finally, we illustrate several examples to show the multi-attribute decision-making technique is more flexible than the prevailing works with the help of sensitive analysis between explored and certain prevailing works.

Partitioned dual Maclaurin symmetric mean operators based on picture fuzzy sets and their applications in multi-attribute decision-making problems

The partitioned Dual Maclaurin symmetric mean (PDMSM) operator has the supremacy that can justify the interrelationship of distinct characteristics and there are a lot of exploration consequences for it. However, it has not been employed to manage "multi-attribute decision-making" (MADM) problems represented by picture fuzzy numbers. The basic inspiration of this identification is to develop the novel theory of picture fuzzy PDMSM operator, and weighted picture fuzzy PDMSM operator and to identify their important results (Idempotency, Monotonicity, and Boundedness). Further, to identify the best decision, every expert realized that they needed the best way to find the beneficial optimal using the proper decision-making procedure, for this, we diagnosed the MADM tool in the consideration of deliberated approaches based on PF information. Finally, to drive the characteristics of the invented work, several examples are utilized to test the manifest of the comparative analysis with various more existing theories, which is a fascinating and meaningful technique to deeply explain the features and exhibited of the proposed approaches.

An efficient spherical fuzzy MEREC-CoCoSo approach based on novel score function and aggregation operators for group decision making

The major objective of the current investigation is to build an integrated multiple criteria group decision-making (MCGDM) methodology based on combined compromise solution (CoCoSo) and spherical fuzzy set for determining the optimal solar power station. To begin with, an innovative spherical fuzzy score function is brought forward to strengthen the efficiency of the comparison for spherical fuzzy number (SFN). Secondly, several newly operational laws for SFN are defined and some novel aggregation operation based on them are propounded. The corresponding excellent properties of the novel operators are also explored at length. Further, the spherical fuzzy method on the removal effects of criteria (MEREC) technique is presented by the proposed score function to work out the importance of the criteria. Lastly, an MCGDM approach is propounded based on improved spherical fuzzy CoCoSo to obtain the ranking of the solar power station locations. The feasibility and practicability of the proposed SF-MEREC-CoCoSo method are investigated through the comparison study with the extant methods. The sensibility analysis is also executed to discuss the robustness and stability of the propounded methodology.

WASPAS method and Aczel-Alsina aggregation operators for managing complex interval-valued intuitionistic fuzzy information and their applications in decision-making

Aczel-Alsina t-norm and t-conorm are a valuable and feasible technique to manage ambiguous and inconsistent information because of their dominant characteristics of broad parameter values. The main theme of this analysis is to explore Aczel-Alsina operational laws in the presence of the complex interval-valued intuitionistic fuzzy (CIVIF) set theory. Furthermore, we derive the theory of aggregation frameworks based on Aczel-Alsina operational laws for managing the theory of CIVIF information. The CIVIF Aczel-Alsina weighted averaging (CIVIFAAWA), CIVIF Aczel-Alsina ordered weighted averaging (CIVIFAAOWA), CIVIF Aczel-Alsina hybrid averaging (CIVIFAAHA), CIVIF Aczel-Alsina weighted geometric (CIVIFAAWG), CIVIF Aczel-Alsina ordered weighted geometric (CIVIFAAOWG) and CIVIF Aczel-Alsina hybrid geometric (CIVIFAAHG) operators are proposed, and their well-known properties and particular cases are also detailly derived. Further, we derive the theory of the WASPAS method for CIVIF information and evaluate their positive and negative aspects. Additionally, we demonstrate the multi-attribute decision-making (MADM) strategy under the invented works. Finally, we express the supremacy and dominancy of the invented methods with the help of sensitive analysis and geometrical shown of the explored works.

A study on the Heronian mean operators for managing complex picture fuzzy uncertain linguistic settings and their application in decision making

The major influence of this manuscript is to diagnose a valuable and considerable technique of Complex Picture Fuzzy Uncertain Linguistic setting and described its useful and valuable operational laws. The theory of Complex Picture Fuzzy Uncertain Linguistic information is massive modified and generalized than the Complex intuitionistic Fuzzy Uncertain Linguistic, Complex Fuzzy Uncertain Linguistic, Fuzzy Uncertain Linguistic and Uncertain Linguistic information. Keeping the supremacy and dominancy of the Complex Picture Fuzzy Uncertain Linguistic information, we investigated the Complex Picture Fuzzy Uncertain Linguistic Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Weighted Arithmetic Heronian Mean, Complex Picture Fuzzy Uncertain Linguistic Geometric Heronian Mean and Complex Picture Fuzzy Uncertain Linguistic Weighted Geometric Heronian Mean operators. The property of idempotency, boundedness, monotonicity, and various well-known results with certain specific cases of the invented work are also deliberated. Furthermore, in the availability of the above-proposed analysis, we constructed a multi-attribute decision-making technique by considering the diagnosed operators for complex picture fuzzy uncertain linguistic information to enhance the worth and rationality of the invented theory. Finally, we illustrated the merits and restrictions of the novel operators by comparing them with certain prevailing operators based on fuzzy generalization. Finally, in the presence of evaluated examples, we compared the pioneered operators with various existing operators to enhance the feasibility and worth of the invented operators.

Picture Fuzzy Einstein Hybrid-Weighted Aggregation Operator and Its Application to Multicriteria Group Decision Making

As an extension of intuitionistic fuzzy sets (IFSs), picture fuzzy sets (PFSs) can better model and represent the hesitancy and uncertainty of decision makers’ preference information. In this study, we propose a multicriteria group decision making (MCGMD) method based on picture fuzzy sets. We first define some basic picture Einstein operations with closed properties among PFSs on the basis of the Einstein t-norms and t-conorms. Then, utilizing the hybrid-weighted operator and the developed picture Einstein operations laws, we put forward a picture fuzzy Einstein hybrid-weighted aggregation operator for aggregating PFSs and discuss its several important properties. Furthermore, we present a new MCGMD method based on the proposed picture fuzzy Einstein hybrid-weighted aggregation operator. Finally, an example is conducted to validate the effectiveness of the proposed MCGMD method.

A study of an EOQ model where the demand depends on time and varying number of tourists using fuzzy triangular norms

The fundamental prerequisite of decision making is how to aggregate individual expert's preference information. For constructing various aggregation operators on intuitionistic fuzzy set, various kinds of t-norms and co-norms are the most significant tools. This work takes a closer look on various fuzzy triangular norms and uses them for the first time in an economic order quantity (EOQ) model. Traditional economic theory has a paradigm shift on objective factors affecting demand such as price and income. Recently, behavioural economics gives more weightage to psychological and social factors that affect our preferences and choices in markets differently. As per literature survey concern, this paper may indeed be the first to adopt the concept of the number of variations of a tourist's dependent demand in inventory management problems. Utilizing several triangular norms, we have characterized Attanassov (standard) and non-Attanassov's (non-standard) feasible region. A case study has been performed for numerical illustration and model validation. A solution algorithm, has also been developed which proves Attanassov's solution space is quite inferior to non-Attanassov's domain. Finally, sensitivity analysis and graphical illustrations are also provided to justify the model.

Graphical abstract

Covering-based compound mean operators arising from Heronian and Bonferroni mean operators in fuzzy and intuitionistic fuzzy environments

Both Heronian mean (HM) operators and Bonferroni mean (BM) operators can capture the interrelationship between input arguments and have been a hot research topic as a useful aggregation technique in fuzzy and intuitionistic fuzzy environments. In this paper, associated with the common characters of these operators we propose the covering-based compound mean operators in fuzzy environments to capture various interrelationships between input arguments, some desirable properties and special cases of the proposed mean operators are provided. Then, conditions under which these covering-based compound mean operators can be directly used to aggregate the membership degrees and nonmembership degrees of intuitionistic fuzzy information, are provided. In particular, novel intuitionistic fuzzy HM operators and intuitionistic fuzzy BM operators are directly derived from the classical ones. We list the detailed steps of multiple attribute decision making with the developed aggregation operators, and give a comparison of the new extensions of BM operators by this paper with the corresponding existing ones to prove the rationality and effectiveness of the proposed method.

Intuitionistic fuzzy-valued neutrosophic multi-sets and numerical applications to classification

A single-valued neutrosophic multi-set is characterized by a sequence of truth membership degrees, a sequence of indeterminacy membership degrees and a sequence of falsity membership degrees. Nature of a single-valued neutrosophic multi-set allows us to consider multiple information in the truth, indeterminacy and falsity memberships which is pretty useful in multi-criteria group decision making. In this paper, we consider sequences of intuitionistic fuzzy values instead of numbers to define the concept of intuitionistic fuzzy-valued neutrosophic multi-set. In this manner, such a set gives more powerful information. We also present some set theoretic operations and a partial order for intuitionistic fuzzy-valued neutrosophic sets and provide some algebraic operations between intuitionistic fuzzy-valued neutrosophic values. Then, we develop two types of weighted aggregation operators with the help of intuitionistic fuzzy t-norms and t-conorms. By considering some well-known additive generators of ordinary t-norms, we give the Algebraic weighted arithmetic and geometric aggregation operators and the Einstein weighted arithmetic and geometric aggregation operators that are the particular cases of the weighted aggregation operators defined via general t-norms and t-conorms. We also define a simplified neutrosophic valued similarity measure and we use a score function for simplified neutrosophic values to rank similarities of intuitionistic fuzzy-valued neutrosophic multi-values. Finally, we give an algorithm to solve classification problems using intuitionistic fuzzy-valued neutrosophic multi-values and proposed aggregation operators and we apply the theoretical part of the paper to a real classification problem.

A Multi-MOORA decision making method based on Muirhead mean operators and complex spherical fuzzy uncertain linguistic setting

The Muirhead mean (MM) operators offer a flexible arrangement with its modifiable factors because of Muirhead’s general structure. On the other hand, MM aggregation operators perform a significant role in conveying the magnitude level of options and characteristics. In this manuscript, the complex spherical fuzzy uncertain linguistic set (CSFULS), covering the grade of truth, abstinence, falsity, and their uncertain linguistic terms is proposed to accomplish with awkward and intricate data in actual life dilemmas. Furthermore, by using the MM aggregation operators with the CSFULS, the complex spherical fuzzy uncertain linguistic MM (CSFULMM), complex spherical fuzzy uncertain linguistic weighted MM (CSFULWMM), complex spherical fuzzy uncertain linguistic dual MM (CSFULDMM), complex spherical fuzzy uncertain linguistic dual weighted MM (CSFULDWMM) operators, and their important results are also elaborated with the help of some remarkable cases. Additionally, multi-attribute decision-making (MADM) based on the Multi-MOORA (Multi-Objective Optimization Based on a Ratio Analysis plus full multiplicative form), and proposed operators are developed. To determine the rationality and reliability of the elaborated approach, some numerical examples are illustrated. Finally, the supremacy and comparative analysis of the elaborated approaches with the help of graphical expressions are also developed.

Heronian Mean Operators Based on Novel Complex Linear Diophantine Uncertain Linguistic Variables and Their Applications in Multi-Attribute Decision Making

In this manuscript, we combine the notion of linear Diophantine fuzzy set (LDFS), uncertain linguistic set (ULS), and complex fuzzy set (CFS) to elaborate the notion of complex linear Diophantine uncertain linguistic set (CLDULS). CLDULS refers to truth, falsity, reference parameters, and their uncertain linguistic terms to handle problematic and challenging data in factual life impasses. By using the elaborated CLDULSs, some operational laws are also settled. Furthermore, by using the power Einstein (PE) aggregation operators based on CLDULS: the complex linear Diophantine uncertain linguistic PE averaging (CLDULPEA), complex linear Diophantine uncertain linguistic PE weighted averaging (CLDULPEWA), complex linear Diophantine uncertain linguistic PE Geometric (CLDULPEG), and complex linear Diophantine uncertain linguistic PE weighted geometric (CLDULPEWG) operators, and their useful results are elaborated with the help of some remarkable cases. Additionally, by utilizing the expounded works dependent on CLDULS, I propose a multi-attribute decision-making (MADM) issue. To decide the quality of the expounded works, some mathematical models are outlined. Finally, the incomparability and relative examination of the expounded approaches with the assistance of graphical articulations are evolved.

Mixed aggregation functions for outliers detection

A scenario that often encounters in the event of aggregating options of different experts for the acquisition of a robust overall consensus is the possible existence of extremely large or small values termed as outliers in this paper, which easily lead to counter-intuitive results in decision aggregation. This paper attempts to devise a novel approach to tackle the consensus outliers especially for non-uniform data, filling the gap in the existing literature. In particular, the concentrate region for a set of non-uniform data is first computed with the proposed searching algorithm such that the domain of aggregation function is partitioned into sub-regions. The aggregation will then operate adaptively with respect to the corresponding sub-regions previously partitioned. Finally, the overall aggregation is operated with a proposed novel consensus measure. To demonstrate the working and efficacy of the proposed approach, several illustrative examples are given in comparison to a number of alternative aggregation functions, with the results achieved being more intuitive and of higher consensus.

An Algorithm for Fuzzy Negations Based-Intuitionistic Fuzzy Copula Aggregation Operators in Multiple Attribute Decision Making

In this paper, we develop a novel computation model of Intuitionistic Fuzzy Values with the usage of fuzzy negations and Archimedean copulas. This novel computation model’s structure is based on the extension of the existing operations of intuitionistic fuzzy values with some classes of fuzzy negations. Many properties of the proposed operations are investigated and proved. Additionally, in this paper we introduce the concepts of intuitionistic fuzzy Archimedean copula weighted arithmetic and geometric aggregation operators based on fuzzy negations, including a further analysis of their properties. Finally, using a case study from an already published paper we found that our method has many advantages.

The Linguistic Interval-Valued Intuitionistic Fuzzy Aggregation Operators Based on Extended Hamacher T-Norm and S-Norm and Their Application

Linguistic interval-valued intuitionistic fuzzy sets, as an extension of interval-valued intuitionistic fuzzy sets, have strong practical value in the management of complex uncertainty system with qualitative evaluation information. This study focuses on the development of several linguistic interval-valued intuitionistic fuzzy Hamacher (LIVIFH) aggregation operators based on the extended Hamacher t-norm and s-norm. First, the extended Hamacher t-norm and s-norm, which are applicable to linguistic information environment, are applied to define the linguistic interval-valued intuitionistic fuzzy Hamacher operational laws. Second, based on the proposed operational laws, this study defines the linguistic interval-valued intuitionistic fuzzy Hamacher weighted average (LIVIFHWA) operator and the linguistic interval-valued intuitionistic fuzzy Hamacher weighted geometric (LIVIFHWG) operator, and then investigates their properties. Furthermore, the degeneracy and monotonicity of the proposed operators with respect to the adjustable parameter are explored. Finally, a multiple attribute group decision-making (MAGDM) approach is developed based on the proposed LIVIFH aggregation operators, and then this approach is applied to a supplier selection problem. Parameter analysis indicates that the adjustable parameter in the proposed LIVIFH aggregation operators could reflect the attitudes of decision makers. The LIVIFHWA operator would be more appropriate to optimistic decision makers, and the LIVIFHWG operator to pessimistic decision makers. In addition, as the adjustable parameter increasing, both attitudes tend to be neutral. The proposed method is also compared with two other approaches to show its feasibility and efficiency.

Novel Multiple Attribute Group Decision-Making Methods Based on Linguistic Intuitionistic Fuzzy Information

As an effective technique to qualitatively depict assessment information, a linguistic intuitionistic fuzzy number (LIFN) is more appropriate to portray vagueness and indeterminacy in actual situations than intuitionistic fuzzy number (IFN). The prominent feature of a Muirhead mean (MM) operator is that it has the powerful ability to capture the correlations between any input-data and MM operator covers other common operators by assigning the different parameter vectors. In the article, we first analyze the limitations of the existing ranking approaches of LIFN and propose a novel ranking approach to surmount these limitations. Secondly, we propound several novel MM operators to fuse linguistic intuitionistic fuzzy (LIF) information, such as the LIF Muirhead mean (LIFMM) operator, the weighted LIF Muirhead mean (WLIFMM) operator and their dual operators, the LIFDMM operator and the WLIFDMM operator. Subsequently, we discuss several desirable properties along with exceptional cases of them. Moreover, two novel multiple attribute group decision-making approaches are developed based upon these operators. Ultimately, the effectuality and practicability of the propounded methods are validated through dealing with a global supplier selection issue, and the comparative analysis and the merits of the presented approaches are demonstrated by comparing them with existing approaches.

Multiple criteria decision making based on weighted Archimedean power partitioned Bonferroni aggregation operators of generalised orthopair membership grades

In this paper, a multiple criteria decision making (MCDM) method based on weighted Archimedean power partitioned Bonferroni aggregation operators of generalised orthopair membership grades (GOMGs) is proposed. Bonferroni mean operator, geometric Bonferroni mean operator, power average operator, partitioned average operator, and Archimedean T-norm and T-conorm operations are introduced into generalised orthopair fuzzy sets to develop the Bonferroni aggregation operators. Their formal definitions are provided, and generalised and specific expressions are constructed. On the basis of the specific operators, a method for solving the MCDM problems based on GOMGs is designed. The working process, characteristics, and feasibility of the method are, respectively, demonstrated via a numerical example, a qualitative comparison at the aspect of characteristics, and a quantitative comparison using the example as benchmark. The demonstration results show that the proposed method is feasible that has desirable generality and flexibility in the aggregation of criterion values and concurrently has the capabilities to deal with the heterogeneous interrelationships of criteria, reduce the negative influence of biased criterion values, and capture the risk attitudes of decision makers.

Linguistic Pythagorean Einstein Operators and Their Application to Decision Making

Linguistic Pythagorean fuzzy (LPF) set is an efficacious technique to comprehensively represent uncertain assessment information by combining the Pythagorean fuzzy numbers and linguistic variables. In this paper, we define several novel essential operations of LPF numbers based upon Einstein operations and discuss several relations between these operations. For solving the LPF numbers fusion problem, several LPF aggregation operators, including LPF Einstein weighted averaging (LPFEWA) operator, LPF Einstein weighted geometric (LPFEWG) operator and LPF Einstein hybrid operator, are propounded; the prominent characteristics of these operators are investigated as well. Furthermore, a multi-attribute group decision making (MAGDM) approach is presented on the basis of the developed operators under an LPF environment. Ultimately, two application cases are utilized to demonstrate the practicality and feasibility of the developed decision approach and the comparison analysis is provided to manifest the merits of it.

Power Average Operators of Trapezoidal Cubic Fuzzy Numbers and Application to Multi-attribute Group Decision Making

Trapezoidal cubic fuzzy numbers (TzCFNs) are an extraordinary cubic fuzzy set on a real number set. TzCFNs are useful for dealing with well-known quantities in decision data and decision making problems themselves. This paper is about multi-attribute group decision making problems in which the attribute values are stated with TzCFNs, which are solved by developing a new decision method based on power average operators of TzCFNs. The new operation laws for TzCFNs are given. Hereby, the power average operator of real numbers is extended to four kinds of power average operators of TzCFNs, involving the power average operator of TzCFNs, the weighted power average operator of TzCFNs, the power ordered weighted average operator of TzCFNs, and the power hybrid average operator of TzCFNs. In the proposed group decision method, the individual overall evaluation values of alternatives are generated by using the power average operator of TzCFNs. Applying the hybrid average operator of TzCFNs, the specific general evaluation standards of alternatives are then combined into the collective ones, which are used to rank the alternatives. The example analysis shows the practicality and effectiveness of the proposed method.

Pythagorean Hesitant Fuzzy Hamacher Aggregation Operators in Multiple-Attribute Decision Making

The Hamacher product is a t-norm and the Hamacher sum is a t-conorm. They are good alternatives to the algebraic product and the algebraic sum, respectively. Nevertheless, it seems that most of the existing hesitant fuzzy aggregation operators are based on algebraic operations. In this paper, we utilize Hamacher operations to develop some Pythagorean hesitant fuzzy aggregation operators: Pythagorean hesitant fuzzy Hamacher weighted average operator, Pythagorean hesitant fuzzy Hamacher weighted geometric operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric operator, Pythagorean hesitant fuzzy Hamacher hybrid average operator, and Pythagorean hesitant fuzzy Hamacher hybrid geometric operator. The prominent characteristics of these proposed operators are studied. Then, we utilize these operators to develop some approaches for solving the Pythagorean hesitant fuzzy multiple-attribute decision-making problems. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Generalized Hamacher Aggregation Operators for Intuitionistic Uncertain Linguistic Sets: Multiple Attribute Group Decision Making Methods

In this paper, we consider multiple attribute group decision making (MAGDM) problems in which the attribute values take the form of intuitionistic uncertain linguistic variables. Based on Hamacher operations, we developed several Hamacher aggregation operators, which generalize the arithmetic aggregation operators and geometric aggregation operators, and extend the algebraic aggregation operators and Einstein aggregation operators. A number of special cases for the two operators with respect to the parameters are discussed in detail. Also, we developed an intuitionistic uncertain linguistic generalized Hamacher hybrid weighted average operator to reflect the importance degrees of both the given intuitionistic uncertain linguistic variables and their ordered positions. Based on the generalized Hamacher aggregation operator, we propose a method for MAGDM for intuitionistic uncertain linguistic sets. Finally, a numerical example and comparative analysis with related decision making methods are provided to illustrate the practicality and feasibility of the proposed method.

Linguistic Neutrosophic Numbers Einstein Operator and Its Application in Decision Making

Linguistic neutrosophic numbers (LNNs) include single-value neutrosophic numbers and linguistic variable numbers, which have been proposed by Fang and Ye. In this paper, we define the linguistic neutrosophic number Einstein sum, linguistic neutrosophic number Einstein product, and linguistic neutrosophic number Einstein exponentiation operations based on the Einstein operation. Then, we analyze some of the relationships between these operations. For LNN aggregation problems, we put forward two kinds of LNN aggregation operators, one is the LNN Einstein weighted average operator and the other is the LNN Einstein geometry (LNNEWG) operator. Then we present a method for solving decision-making problems based on LNNEWA and LNNEWG operators in the linguistic neutrosophic environment. Finally, we apply an example to verify the feasibility of these two methods.

Granular Data Aggregation: An Adaptive Principle of the Justifiable Granularity Approach

The design of information granules assumes a central position in the discipline of Granular Computing and its applications. The principle of justifiable granularity offers a conceptually and algorithmically attractive way of designing information granule completed on a basis of some experimental evidence (especially present in the form of numeric data). This paper builds upon the existing principle and presents its significant generalization, referred here as an adaptive principle of justifiable information granularity. The method supports a granular data aggregation producing an optimal information granule (with the optimality expressed in terms of the criteria of coverage and specificity commonly used when characterizing quality of information granules). The flexibility of the method stems from an introduction of the adaptive weighting scheme of the data leading to a vector of weights used in the construction of the optimal information granule. A detailed design procedure is provided along with the required optimization vehicle (realized with the aid of the population-based optimization techniques, such as particle swarm optimization and differential evolution). Two direct application areas in which the principle becomes of direct usage include prediction of time series and prediction of spatial data. In both cases, it is advocated that the results formed by the principle are reflective of the precision (quality) of the prediction process.