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The pymcdm-reidentify tool addresses the challenge of reconstructing multi-criteria decision analysis (MCDA) and decision-making (MCDM) models when original parameters are unavailable, but rankings are known. This Python package integrates with existing MCDA libraries and uses stochastic optimization to determine model parameters such as criterion weights and reference objects. Built on the pymcdm and Mealpy libraries, pymcdm-reidentify offers advanced methods for model re-identification, including visualization and fuzzy normalization. Its capabilities facilitate the update and adaptation of decision models, enhancing accuracy and efficiency in both academic and practical applications.

This paper presents the pyrepo-mcda Python package upgrade with the implementation of the Preference Vector Method (PVM) multi-criteria method. This upgrade extends the scope of multi-criteria decision analysis offered by this package. Several advantages of the PVM method, such as the reduction of the participation of decision-makers, the possibility of giving individual preference vectors, and the possibility of modification and further development, are in the interest of decision-makers in various multi-criteria decision analysis problems, particularly in the sustainability assessment.

The presented programming package introduces an innovative tool for sensitivity analysis within Multi-Criteria Decision Analysis (MCDA), offering a modular and flexible structure for versatile applications. The PySensMCDA package is developed in Python and ensures adaptability by adjusting the behavior of sensitivity analysis methods to the specific needs of users. Robust input data validation enhances reliability by minimizing errors. The versatility of the proposed package allows for both personal and research applications within the MCDA domain, making it an important contribution to enhancing the comprehensiveness of the assessment. The eight modules encompass a variety of sensitivity analysis approaches, including alternative modification, criteria weights adjustment, probabilistic approaches, ranking alterations, compromise solutions, preference calculations, graph visualizations, and data validation. The main aim is to equip decision-makers with predefined sensitivity analysis methods, thereby facilitating more comprehensive and detailed analyses in multi-criteria decision scenarios.

The Evaluation based on Distance from Average Solution (EDAS) is a multi-criteria decision analysis (MCDA) technique that uses various distances from average values to make decisions. It bears resemblance to other distance-based approaches like SPOTIS, VIKOR or TOPSIS, except that instead of positive and negative ideal solutions, it uses an average solution. For hesitant intuitionistic fuzzy linguistic term sets (HIFLTSs), we first define several operational laws and aggregation operators. As a follow-up, the hesitating intuitionistic fuzzy linguistic Einstein weighted averaging (HIFLEWA) operator is introduced. In this study, the EDAS technique is used to address a multi-criteria group decision making (MCGDM) issue by utilizing the suggested operational laws and aggregation operators for HIFLTSs, aiming to mitigate the uncertainties of decision makers (DMs). A representative numerical example is employed to illustrate the strength and logical soundness of our proposed method in identifying the optimal choice under the given limitations. A comparison examination with the existing TOPSIS approach is also performed to ensure that the results generated with EDAS are accurate.

Using decision-support systems for experts in multi-criteria problems is a common practice to ensure that the choices are rational and based on precise analysis. To facilitate the design process of these systems, it is necessary to provide access to tools designed to simplify the design process of decision-making models. To this end, we propose a MakeDecision tool designed for the graphical design of structured decision models based on crisp and fuzzy data, where the fuzzy data is represented as Triangular Fuzzy Numbers. The application allows the rapid prototyping and development of decision-making models, using predefined components that do not require own implementations of techniques from the decision-making field. Open access and customizable configuration contribute to a flexible design process for decision model structures.

This paper analyzes the behavior of the well-known Spearman's footrule distance (F-distance) to measure the distance between two rankings over the same set of objects. We show that F-distance is not invariant to labeling, and therefore, it suffers from a serious drawback for its use in applications. To circumvent this problem, we propose a new distance between rankings which is invariant under indexing (i.e., labeling) and appears as a good alternative to the direct use of F-distance between rankings, and also the invariant-under-indexing Kemeny's distance as well. We also show how our new distance can work with importance weights. Some simple examples are given to show the interest of our method with respect to the classical one based on F-distance and Kemeny's distance.

The paper introduces the update of the Python Intuitionistic Fuzzy Decision Making (PyIFDM) package, emphasizing its focus on Multi-Criteria Decision Analysis (MCDA) with uncertain data modeled as Intuitionistic Fuzzy Sets (IFS). The update enhances PyIFDM’s capabilities, introducing five new IF-MCDA methods and basic mathematical operations on IFS. The update includes the implementation of Max normalization, a Hausdorff measure-based Euclidean distance, and a module for IFS-similarity measures. The release contributes to a significant library upgrade, providing advanced tools for modeling fuzzy data in and outside the MCDA space. PyIFDM now supports a greater variety of popular IF-MCDA methods, offering flexibility and applicability in decision-making scenarios. The added visualization module facilitates the graphical representation of IFS, aiding the interpretation of complex decision models involving intuitionistic fuzzy data. The enhanced version of the PyIFDM library expands its applicability in research methodologies, making it a more efficient tool for conducting multi-criteria decision analysis in an intuitionistic fuzzy environment.

This paper describes an extension of the pyFDM library by extending the functionalities of the programming package with new Fuzzy Multi-Criteria Decision Analysis methods such as Combined Compromise Solution (CoCoSo), Stable Preference Ordering Towards Ideal Solution (SPOTIS), Weighted Aggregated Sum Product ASSessment (WASPAS), Weighted Product Model (WPM), and Weighted Sum Model (WSM). Furthermore, the proposed update includes new normalization techniques, new distance metrics, and defuzzification methods to allow users to examine the greater variety of research approaches in a Triangular Fuzzy environment. Furthermore, the update also extends with the implementations allowing for fuzzy calculations outside the multi-criteria field, which is allowed by the implementation of selected mathematical operations for Triangular Fuzzy Numbers. The opportunity to visualize the data within the fuzzy extension considered is also included. The extension of the pyFDM package broadens the scope of its usage in research approaches, making it a more suitable tool for multi-criteria decision analysis within the fuzzy environment.

This paper looks into the extension of the pymcdm library, focusing on reference point-based techniques. It introduces the implementations of methods such as the Reference Ideal Method (RIM), Preference Ranking On the Basis of Ideal-average Distance (PROBID), and Election based on Relative Value Distances (ERVD). This update is intended to meet the increasing demand for solutions tailored to decision makers’ expertise and experience. Furthermore, it introduces techniques related to sensitivity analysis and weight comparison factors of criteria. By expanding this library, the range of MCDA/MCDM tools is broadened and further progress is encouraged in the use of expert knowledge and the implementation of compromise methods.

This paper presents an update of the pyrepo-mcda Python package, providing a broad set of multi-criteria methods and methods supporting decision analysis, such as determining compromise rankings, criteria weights, distance metrics, and performing sensitive analysis. The pyrepo-mcda package was extended with the Vector Measure Construction Method (VMCM). This update expands the scope of multi-criteria analysis approaches offered by the proposed package. Several features of the VMCM method, such as considering non-typical objects, the possibility of individual defining patterns, and classifying evaluated objects, are sought by decision-makers in many decision-making problems, especially sustainability assessment.

This paper presents an extension of the developed crispyn Python package providing weighting methods for multi-criteria decision analysis. The crispyn package, including until now eleven objective methods for determining criteria weights, was supplemented with four subjective weighting methods. The extension of the crispyn package significantly expands its scope of applications for decision problems requiring considering subjective criteria preferences of decision-makers’ for example, in reliable sustainability assessment.

The extension of intuitionistic fuzzy sets (IFS) to Pythagorean fuzzy sets (PFS) is a significant advancement, addressing the inherent limitations of IFS. This study introduces a novel entropy measure specifically designed for Pythagorean fuzzy sets, establishing its axiomatic definition and presenting key properties. Decision making guided by entropy is advantageous, as it effectively mitigates ambiguity with increasing entropy values. Furthermore, a numerical example is provided to facilitate a comparative assessment of our newly introduced entropy measure in contrast to existing PFS entropy measures. The validation of our findings is achieved through the application of the COPRAS method, which determines decision outcomes based on a multitude of influencing factors. Notably, the determination of weights in this method is underpinned by the utilization of our innovative entropy measure.

The ever-increasing demand for high-quality solutions drives research toward more sophisticated decision-making solutions. In the field of decision making, the ability to solve complex real-world problems is of paramount importance. To this end, fuzzy sets are used, which offer the possibility of incorporating uncertainty into the values describing decision options. This study focuses on Pythagorean fuzzy sets, an extension of classical fuzzy sets, providing even more tools for modeling real-world problems by presenting a distance measure for these specific sets. A verification of the characteristics of the proposed distance measure has been carried out, proving its validity. The proposed measure is characterized by a more straightforward formula and thus simplifies the calculations. Furthermore, to confirm its usability, a multi-criteria decision-making methodology is presented, the results of which are compared with two multi-criteria decision-making methods, namely, PF-TOPSIS and PF-VIKOR, and another distance measure previously presented in the literature. The comparative analysis highlights lower variability in terms of preference values calculated using the proposed distance measure, which confirms the stability and reliability of the newly proposed distance measure while maintaining low computational complexity. Moreover, a high correlation with rankings calculated using PF-TOPSIS ensures its utility in terms of decision making.

The introduction of fuzzy logic assumptions into Multi-Criteria Decision Analysis (MCDA) methods has broadened the application area of these techniques, increasing their practical potential. Due to their popularity, decision-makers need to have access to open and comprehensive tools to assist them in their decision-making process. In this paper, we propose a software library written in Python for multi-criteria computing in an uncertain environment. PyIFDM (Python Intuitionistic Fuzzy Decision Making) offers users access to selected Intuitionistic Fuzzy MCDA methods, normalization techniques, distance measures, score functions, or objective weighting methods. It enables users to build complex decision-making models regarding the uncertainties occurring in the problem.

Multi-criteria decision-making/analysis is a vast field designed for solving decision-making problems. Due to the widespread use of many relevant MCDA/MCDM approaches and their frequent appearance in scientific papers, there is a particular gap regarding comprehensive software that should have relevant components. Therefore, this paper proposes a flexible library written in Python 3 for multi-criteria analysis/decision-making. It includes tools related to evaluating alternative options, determination of Pareto optimal solutions, determination of criteria relevance, comparative analysis, and visualization. The novelty that distinguishes the library from other software is the provision of a wide range of approaches, high performance, consistency of the software, and its small size. The research presented in this paper presents its high potential as a tool that decision-makers and domain experts can use in the decision-making process.

Decision-making is an integral part of life. Many situations require us to make rational choices among numerous decision options. In addition, real-world problems are characterized by uncertainties that hinder the entire process. Multi-Criteria Decision Analysis (MCDA) comes with help. Many evaluation techniques have been developed along with fuzzy logic assumptions, classified as fuzzy MCDA tools. These methods allow efficient and effective evaluation of decision variants in an uncertain environment. Existing tools facilitate the work of these methods, but they are limited or not updated with newly established technologies. Therefore, this article proposes the pyFDM (Python Fuzzy Decision Making): a new Python 3 software library to facilitate calculations using methods that operate on uncertain data. In addition to evaluation methods, various techniques for normalization, defuzzification, distance measure, or objective criteria weighting are also included. It provides a comprehensive set of tools for using available techniques to perform calculations in this field. The operation of the proposed library is demonstrated through two practical examples. The first is directed to selecting the most appropriate stock, while the second is devoted to the ERP system selection firms. The presented examples show that the implemented library can be effectively used for fuzzy decision-making problems based on Triangular Fuzzy Numbers. The modular structure allows the interchangeable use of available methods, making the tool a comprehensive environment for calculations in this area.

The purpose of this paper is to present the new hybrid multi-criteria decision analysis (MCDA) method CODAS-COMET for evaluating sustainable electricity generation considering renewable energy sources (RES). The introduced approach combines the advantages of two known MCDA methods: Characteristic Object METhod (COMET) and COmbinative Distance-based ASsessment (CODAS). Due to the advantages of the COMET method, the proposed approach enables the identification of the whole domain model, is resistant to the rank reversal phenomenon, is easy to apply, and allows to obtain an objective and reliable recommendation based on the gathered data. Meanwhile, combining the COMET method with CODAS eliminates the limitations of COMET by automatizing and accelerating the characteristic object comparison procedure. The presented method’s practical application includes evaluating Poland’s sixteen voivodeships, considering the sustainable use of RES in the electricity generation domain. An additional contribution of this research is to investigate the impact on results of distance metrics other than the default distance metric used by the method that is the focus of this paper and demonstrate the design and practical application of a research framework for selecting the most appropriate metrics. A comparative analysis of the CODAS-COMET method with the TOPSIS-COMET reference approach was performed to confirm the results’ reliability. The results showed a high correlation among all final rankings and confirmed the robustness of the CODAS-COMET approach to changing metrics in the algorithm. Furthermore, the results proved that our approach is a suitable and valuable tool for sustainability assessment in renewable energy domains and other complex issues requiring consideration of multiple criteria.

The subject of electric vehicles (EVs) is constantly relevant from the perspective of climate change and sustainability. Multi-Criteria Decision Analysis (MCDA) methods can be successfully used to evaluate models of such vehicles. In many cases, the MCDA methods are modified to account for uncertainty in the data. There are many ways to express uncertainty, including more advanced ones, such as fuzzy sets, for example, but expressing attributes in terms of interval numbers remains a popular method because it is an easy-to-implement and easy-to-understand technique. This study focuses on interval extensions of the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method. It aims to compare the most popular extension proposed by Jahanshahloo and the proposed new modification, which returns the result in an interval form. Certain inconsistencies of the Jahanshahloo extension are discussed, and it is explained how the new extension avoids them. Both extensions are applied to an EV evaluation problem taken from the literature as an example for sustainable assessment. The results are then analyzed, and the question of whether the input data of the interval should receive an evaluation in the form of interval results is addressed.

The aim of this paper is to present the use of an innovative approach based on MCDM methods as the main component of a consumer Decision Support System (DSS) by recommending the most suitable products among a given set of alternatives. This system provides a reliable recommendation to the consumer in the form of a compromise ranking constructed from the five MCDM methods: the hybrid approach TOPSIS-COMET, COCOSO, EDAS, MAIRCA, and MABAC. Each of the methods used contributes significantly to the final compromise ranking built with the Copeland strategy. Chosen MCDM methods were combined with the objective CRITIC weighting method, and their performance was presented on the illustrative example of choosing the most suitable mobile phone. A sensitivity analysis involving the rw and WS correlation coefficients was performed to determine the match between the compromise ranking of the candidates and the rankings provided by each MCDM method. Sensitivity analysis demonstrated that all investigated compromise candidate rankings show high convergence with the rankings provided by the particular MCDM methods. Thus, the performed study proved that the proposed approach shows high potential to be successfully used as a central component of DSS for recommending the most suitable product. Such DSS could be a universal and future-proof solution for e-commerce sites and websites, providing advanced product comparison capabilities in delivering a recommendation to the user as a final ranking of alternatives.

Multi-Criteria Decision Analysis (MCDA) methods have gained popularity among practitioners and researchers in recent years. MCDA methods based on measuring the distance to reference objects are particularly noteworthy since their suitability for most decision problems, comparable and straightforward use, interpretation, and wide post-analytical possibilities. However, software implementations devoted to the MCDA domain show the lack of solutions dedicated to this family containing a sufficient number of methods, providing additional distance metrics and post-analytical tools such as sensitivity analysis. Therefore, this article presents a Python 3 based library that addresses this gap. The research demonstrating the functionalities of the proposed library includes a comparative analysis of the rankings provided by the different methods implemented in the library, a sensitivity analysis of the alternatives to criteria weights modifications, and a robustness analysis of the alternatives to changes in the performance values. The applicability of the proposed library is demonstrated in two real-life numerical examples. The first illustrative example involves the recommendation of renewable energy resources (RES) for development focusing on increasing the significance of RES. The other example involves the evaluation of material suppliers for a steel manufacturing company. Data for both examples were acquired from research papers. Based on the research results, it can be concluded that the proposed library is helpful in the process of supporting the solution of multi-criteria decision problems, and the implementation of a set of methods provides opportunities to search for the most reliable alternative.

This paper presents the consolidation and extension of two developed Python packages for multi-criteria decision analysis (MCDA). The pyrepo-mcda package providing a set of MCDA methods and supporting techniques such as compromise rankings and sensitivity analysis has been consolidated with the Crispyn package providing a broad set of objective criteria weighting methods. Then, the package was supplemented with seven additional MCDA methods. The extension of the pyrepo-mcda package significantly expands its analytical capabilities. Additional MCDA methods, especially those supporting the strong sustainability paradigm, increase the scope of applications of the package for reliable sustainability assessment.

In this study we propose a measure of the entropy of the norm (R, S) for q-row orthopair fuzzy sets (qROFS). The proposed entropy measure is validated both theoretically and practically to ensure validity. We also propose a simple methodology for the purpose of solving a multi-criteria decision-analysis problems using the introduced entropy measure. This method takes into account different circumstances of criteria weights, such as unknown weights, as well as other cases when the weights are not fully known. Finally, a demonstration with numerical examples for the proposed entropy has been provided to show how to apply the novel methodologies.

Correlation is considered the most important factor in analyzing the data in statistics. It is used to measure the movement of two different variables linearly. The concept of correlation is well-known and used in different fields to measure the association between two variables. The hesitant 2-tuple fuzzy linguistic set (H2FLS) comes out to be valuable in addressing people's reluctant subjective data. The purpose of this paper is to analyze new correlation measures between H2FLSs and apply them in the decision-making process. First and foremost, the ideas of mean and variance of hesitant 2-tuple fuzzy linguistic elements (H2FLEs) are introduced. Then, a new correlation coefficient between H2FLSs is established. In addition, considering that different H2FLEs may have different criteria weights, the weighted correlation coefficient and ordered weighted correlation coefficient are further investigated. A practical example concerning the detailed procedure of solving problems is exemplified to feature the reasonableness and attainability of the proposed technique in situations where the criteria weights are either known or unknown. When the weight vector is unknown, the best-worst method (BWM) is used to acquire the criteria weights in the context of a hesitant 2-tuple fuzzy linguistic environment. Furthermore, a comparative study is undertaken with current techniques to provide a vision into the design decision-making process. Finally, it is verified that the proposed correlation coefficient between H2FLSs is more satisfactory than the extant ones, and the correlation coefficient with the weights of criteria being either known or unknown is applicable.

This paper proposes some operations on the cubic intuitionistic set along with useful properties. We propose the internal cubic intuitionistic set (ICIS), the external cubic intuitionistic set (ECIS), P-order, R-order order (P-(R-) order), P-union, R-union (P-(R-) union), P-intersection, and R-intersection (P-(R-) intersection). We further investigate several properties of the P-(R-) union and P-(R-) intersection of ICISs and ECISs, and present some examples in this context. Some important theorems related to ICISs and ECISs are also presented with proof. Finally, an application example is given to measure the effectiveness and significance of the proposed operations by solving a multi-criteria decision-making (MCDM) problem.

Sustainable transportation has a significant impact on factors related to urban development and economic development. Therefore, much research is being undertaken to select the best strategies to manage sustainable transportation. Transportation requires a carefully designed method to manage the development of mobility modes in terms of the pollution they produce or the use of renewable energy sources. However, due to numerous preferences of decision-makers and data uncertainty problems, it is challenging to select the optimal strategy. In this paper, we focus on creating a framework for determining the best strategy for sustainable transportation management. For this purpose, T-spherical fuzzy graphs will be used, which, together with the combination of Laplacian Energy, can accurately represent decision-makers’ preferences in an uncertain environment. Due to the lack of limitations of T-spherical fuzzy graphs and its numerous membership functions, decision-makers can decide which factor seems most important for selecting the optimal sustainable transportation strategy. Additionally, due to the applicability, the SFS TOPSIS approach has been used in this approach. The obtained results demonstrate the high performance of the proposed approach and the applicability of the approach in management and sustainable transport problems.

A fuzzy set extension known as the hesitant fuzzy set (HFS) has increased in popularity for decision making in recent years, especially when experts have had trouble evaluating several alternatives by employing a single value for assessment when working in a fuzzy environment. However, it has a significant problem in its uses, i.e., considerable data loss. The probabilistic hesitant fuzzy set (PHFS) has been proposed to improve the HFS. It provides probability values to the HFS and has the ability to retain more information than the HFS. Previously, fuzzy regression models such as the fuzzy linear regression model (FLRM) and hesitant fuzzy linear regression model were used for decision making; however, these models do not provide information about the distribution. To address this issue, we proposed a probabilistic hesitant fuzzy linear regression model (PHFLRM) that incorporates distribution information to account for multi-criteria decision-making (MCDM) problems. The PHFLRM observes the input–output (IPOP) variables as probabilistic hesitant fuzzy elements (PHFEs) and uses a linear programming model (LPM) to estimate the parameters. A case study is used to illustrate the proposed methodology. Additionally, an MCDM technique called the technique for order preference by similarity to ideal solution (TOPSIS) is employed to compare the PHFLRM findings with those obtained using TOPSIS. Lastly, Spearman’s rank correlation test assesses the statistical significance of two rankings sets.

The Pythagorean fuzzy sets conveniently capture unreliable, ambiguous, and uncertain information, especially in problems involving multiple and opposing criteria. Pythagorean fuzzy sets are one of the popular generalizations of the intuitionistic fuzzy sets. They are instrumental in expressing and managing hesitant under uncertain environments, so they have been involved extensively in a diversity of scientific fields. This paper proposes a new Pythagorean entropy for Multi-Criteria Decision-Analysis (MCDA) problems. The entropy measures the fuzziness of two fuzzy sets and has an influential position in fuzzy functions. The more comprehensive the entropy, the more inadequate the ambiguity, so the decision-making established on entropy is beneficial. The COmplex PRoportional ASsessment (COPRAS) method is used to tackle uncertainty issues in MCDA and considers the singularity of one alternative over the rest of them. This can be enforced to maximize and minimize relevant criteria in an assessment where multiple opposing criteria are considered. Using the Pythagorean sets, we represent a decisional problem solution by using the COPRAS approach and the new Entropy measure.

A graph is a tool for designing a system’s required interconnection network. The topology of such networks determines their compatibility. For the first time, in this work we construct subdivided ηζ network S(ηζΓ) and discussed their topology. In graph theory, there are a variety of invariants to study the topology of a network, but topological indices are designed in such a way that these may transform the graph into a numeric value. In this work, we study S(ηζΓ) via Zagreb connection indices. Due to their predictive potential for enthalpy, entropy, and acentric factor, these indices gain value in the field of chemical graph theory in a very short time. ηζΓ formed by ζ time repeated process which consists ηζ copies of graph Γ along with η2|V(Γ)|ζηζ−1 edges which used to join these ηζ copies of Γ. The free hand to choose the initial graph Γ for desired network S(ηζΓ) and its relation with chemical networks along with the repute of Zagreb connection indices enhance the worth of this study. These computations are theoretically innovative and aid topological characterization of S(ηζΓ).

In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: Gϵ∗=sup∫ΩG(u)ϵq(x)dx:∥∇u∥Lp(.)≤ϵ,u=0 on ∂Ω, with the help of Γ-convergence, where G:R→R is upper semicontinuous, non zero in the L1 sense, 0≤G(u)≤c|u|q(.), Ω is a bounded open subset of Rn, n≥3, 1

In this paper, we propose a new intuitionistic entropy measurement for multi-criteria decision-making (MCDM) problems. The entropy of an intuitionistic fuzzy set (IFS) measures uncertainty related to the data modelling as IFS. The entropy of fuzzy sets is widely used in decision support methods, where dealing with uncertain data grows in importance. The Complex Proportional Assessment (COPRAS) method identifies the preferences and ranking of decisional variants. It also allows for a more comprehensive analysis of complex decision-making problems, where many opposite criteria are observed. This approach allows us to minimize cost and maximize profit in the finally chosen decision (alternative). This paper presents a new entropy measurement for fuzzy intuitionistic sets and an application example using the IFS COPRAS method. The new entropy method was used in the decision-making process to calculate the objective weights. In addition, other entropy methods determining objective weights were also compared with the proposed approach. The presented results allow us to conclude that the new entropy measure can be applied to decision problems in uncertain data environments since the proposed entropy measure is stable and unambiguous.

The purpose of this paper is to propose a new Pythagorean fuzzy entropy for Pythagorean fuzzy sets, which is a continuation of the Pythagorean fuzzy entropy of intuitionistic sets. The Pythagorean fuzzy set continues the intuitionistic fuzzy set with the additional advantage that it is well equipped to overcome its imperfections. Its entropy determines the quantity of information in the Pythagorean fuzzy set. Thus, the proposed entropy provides a new flexible tool that is particularly useful in complex multi-criteria problems where uncertain data and inaccurate information are considered. The performance of the introduced method is illustrated in a real-life case study, including a multi-criteria company selection problem. In this example, we provide a numerical illustration to distinguish the entropy measure proposed from some existing entropies used for Pythagorean fuzzy sets and intuitionistic fuzzy sets. Statistical illustrations show that the proposed entropy measures are reliable for demonstrating the degree of fuzziness of both Pythagorean fuzzy set (PFS) and intuitionistic fuzzy sets (IFS). In addition, a multi-criteria decision-making method complex proportional assessment (COPRAS) was also proposed with weights calculated based on the proposed new entropy measure. Finally, to validate the reliability of the results obtained using the proposed entropy, a comparative analysis was performed with a set of carefully selected reference methods containing other generally used entropy measurement methods. The illustrated numerical example proves that the calculation results of the proposed new method are similar to those of several other up-to-date methods.

An expert may experience difficulties in decision making when evaluating alternatives through a single assessment value in a hesitant environment. A fuzzy linear regression model (FLRM) is used for decision-making purposes, but this model is entirely unreasonable in the presence of hesitant fuzzy information. In order to overcome this issue, in this paper, we define a hesitant fuzzy linear regression model (HFLRM) to account for multicriteria decision-making (MCDM) problems in a hesitant environment. The HFLRM provides an alternative approach to statistical regression for modelling situations where input–output variables are observed as hesitant fuzzy elements (HFEs). The parameters of HFLRM are symmetric triangular fuzzy numbers (STFNs) estimated through solving the linear programming (LP) model. An application example is presented to measure the effectiveness and significance of our proposed methodology by solving a MCDM problem. Moreover, the results obtained employing HFLRM are compared with the MCDM tool called technique for order preference by similarity to ideal solution (TOPSIS). Finally, Spearman’s rank correlation test is used to measure the significance for two sets of ranking.

Graph theory can be used to optimize interconnection network systems. The compatibility of such networks mainly depends on their topology. Topological indices may characterize the topology of such networks. In this work, we studied a symmetric network θϕ formed by ϕ time repetition of the process of joining θ copies of a selected graph Ω in such a way that corresponding vertices of Ω in all the copies are joined with each other by a new edge. The symmetry of θϕ is ensured by the involvement of complete graph Kθ in the construction process. The free hand to choose an initial graph Ω and formation of chemical graphs using θϕΩ enhance its importance as a family of graphs which covers all the pre-defined graphs, along with space for new graphs, possibly formed in this way. We used Zagreb connection indices for the characterization of θϕΩ. These indices have gained worth in the field of chemical graph theory in very small duration due to their predictive power for enthalpy, entropy, and acentric factor. These computations are mathematically novel and assist in topological characterization of θϕΩ to enable its emerging use.

Decision support systems (DSS) are currently developing rapidly and are increasingly used in various fields. More often, those systems are inseparable from information-based systems and computer systems. Therefore, from a methodical point of view, the algorithms implemented in the DSS play a critical role. In this aspect, multi-criteria decision support (MCDA) methods are widely used. As research progresses, many MCDA methods and algorithms for the objective identification of the significance of individual criteria of the MCDA models were developed. In this paper, an analysis of available objective methods for criteria weighting is presented. Additionally, the authors presented the implementation of the system that provides easy and accessible weight calculations for any decision matrix with the possibility of comparing results of different weighting methods. The results of weighting methods were compared using carefully selected similarity coefficients to emphasise the correlation of the resulting weights. The performed research shows that every method should provide distinctive weights considering input data, emphasising the importance of choosing the correct method for a given multi-criteria decision support model and DSS.

This paper aims to present an innovative approach based on two newly developed Multi-Criteria Decision-Making (MCDM) methods: COMET combined with TOPSIS and SPOTIS, which could be the basis for a decision support system (DSS) in the problem of selecting solar panels. Solar energy is one of the most promising and environmentally friendly energy sources because of the enormous potential of directly converting available solar radiation everywhere into electricity. Furthermore, ever-lower prices for photovoltaic systems make solar electricity more competitive with power from conventional energy sources, increasing interest in solar panels among companies and households. This fact generates the need for a user-friendly, objective, fully automated DSS to support the multi-criteria selection of solar panels. Both MCDM methods chosen for this purpose are rank-reversal-free and precise. First, the objective entropy weighting method was applied for determining criteria weights. Final rankings were compared by two ranking correlation coefficients: symmetrical rw and asymmetrical WS. Then the sensitivity analysis providing local weights of alternatives for decision criteria was performed. The obtained results prove the adequacy and practical usefulness of the presented approach in solving the problem of solar panels selection.

Ambiguous and uncertain facts can be handled using a hesitant 2-tuple linguistic set (H2TLS), an important expansion of the 2-tuple linguistic set. The vagueness and uncertainty of data can be grabbed by using aggregation operators. Therefore, aggregation operators play an important role in computational processes to merge the information provided by decision makers (DMs). Furthermore, the aggregation operator is a potential mechanism for merging multisource data which is synonymous with cooperative preference. The aggregation operators need to be studied and analyzed from various perspectives to represent complex choice situations more readily and capture the diverse experiences of DMs. In this manuscript, we propose some valuable operational laws for H2TLS. These new operational laws work through the individual aggregation of linguistic words and the collection of translation parameters. We introduced a hesitant 2-tuple linguistic weighted average (H2TLWA) operator to solve multi-criteria group decision-making (MCGDM) problems. We also define hesitant 2-tuple linguistic Bonferroni mean (H2TLBM) operator, hesitant 2-tuple linguistic geometric Bonferroni mean (H2TLGBM) operator, hesitant 2-tuple linguistic Heronian mean (H2TLHM) operator, and a hesitant 2-tuple linguistic geometric Heronian mean (H2TLGHM) operator based on the novel operational laws proposed in this paper. We define the aggregation operators for addition, subtraction, multiplication, division, scalar multiplication, power and complement with their respective properties. An application example and comparison analysis were examined to show the usefulness and practicality of the work.

Artificial Neural Networks (ANNs) have proven to be a powerful tool for solving a wide variety of real-life problems. The possibility of using them for forecasting phenomena occurring in nature, especially weather indicators, has been widely discussed. However, the various areas of the world differ in terms of their difficulty and ability in preparing accurate weather forecasts. Poland lies in a zone with a moderate transition climate, which is characterized by seasonality and the inflow of many types of air masses from different directions, which, combined with the compound terrain, causes climate variability and makes it difficult to accurately predict the weather. For this reason, it is necessary to adapt the model to the prediction of weather conditions and verify its effectiveness on real data. The principal aim of this study is to present the use of a regressive model based on a unidirectional multilayer neural network, also called a Multilayer Perceptron (MLP), to predict selected weather indicators for the city of Szczecin in Poland. The forecast of the model we implemented was effective in determining the daily parameters at 96% compliance with the actual measurements for the prediction of the minimum and maximum temperature for the next day and 83.27% for the prediction of atmospheric pressure.

The multi-criteria decision-making (MCDM) problem has a solution whose quality can be affected by the experts’ inclinations. Under essential conditions, the fuzzy MCDM method can provide more acceptable and efficient outcomes to select the best alternatives. This work consists of a consensus-based technique for selecting and evaluating suppliers in an incomplete fuzzy preference relations (IFPRs) environment utilizing TL-transitivity (Lukasiewicz transitivity). The suggested method is developed based on the criteria of the Analytical Hierarchy Process (AHP) Fframework, and the decision matrix is construtced using consistent fuzzy preference relations (FPRs). We use the symmetrical decisional matrix approach. A variety of numerical explanations and an analysis of quantitative results illustrate the suggested methodology’s logic and effectiveness.

Decision support systems often involve taking into account many factors that influence the choice of existing options. Besides, given the expert’s uncertainty on how to express the relationships between the collected data, it is not easy to define how to choose optimal solutions. Such problems also arise in sport, where coaches or players have many variants to choose from when conducting training or selecting the composition of players for competitions. In this paper, an objective fuzzy inference system based on fuzzy logic to evaluate players in team sports is proposed on the example of football. Based on the Characteristic Objects Method (COMET), a multi-criteria model has been developed to evaluate players on the positions of forwards based on their match statistics. The study has shown that this method can be used effectively in assessing players based on their performance. The COMET method was chosen because of its unique properties. It is one of the few methods that allow identifying the model without giving weightings of decision criteria. Symmetrical and asymmetrical fuzzy triangular numbers were used in model identification. Using the calculated derivatives in the point, it turned out that the criteria weights change in the problem state space. This prevents the use of other multi-criteria decision analysis (MCDA) methods. However, we compare the obtained model with the Technique of Order Preference Similarity (TOPSIS) method in order to better show the advantage of the proposed approach. The results from the objectified COMET model were compared with subjective rankings such as Golden Ball and player value.

The paper undertakes the problem of proper structuring of multi-criteria decision support models. To achieve that, a methodological framework is proposed. The authors’ framework is the basis for the relevance analysis of individual criteria in any considered decision model. The formal foundations of the authors’ approach provide a reference set of Multi-Criteria Decision Analysis (MCDA) methods (TOPSIS, VIKOR, COMET) along with their similarity coefficients (Spearman correlation coefficients and WS coefficient). In the empirical research, a practical MCDA-based wind farm location problem was studied. Reference rankings of the decision variants were obtained, followed by a set of rankings in which particular criteria were excluded. This was the basis for testing the similarity of the obtained solutions sets, as well as for recommendations in terms of both indicating the high significance and the possible elimination of individual criteria in the original model. When carrying out the analyzes, both the positions in the final rankings, as well as the corresponding values of utility functions of the decision variants were studied. As a result of the detailed analysis of the obtained results, recommendations were presented in the field of reference criteria set for the considered decision problem, thus demonstrating the practical usefulness of the authors’ proposed approach. It should be pointed out that the presented study of criteria relevance is an important factor for objectification of the multi-criteria decision support processes.

Human activity is moving steadily to virtual reality. More and more, people from all over the world are keen on growing fascination with e-sport. In practice, e-sport is a type of sport in which players compete using computer games. The competitions in games, like FIFA, Dota2, the League of Legends, and Counter-Strike, are prestigious tournaments with a global reach and a budget of millions of dollars. On the other hand, reliable player ranking is a critical issue in both classic and e-sport. For example, the “Golden Ball” is the most valuable prize for an individual football player in the whole football history. Moreover, the entire players’ world wants to know who the best player is. The position of each player in the ranking depends on the assessment of his skills and predispositions. In this paper, we studied identification of players evaluation and ranking obtained using the multiple-criteria decision-making based method called Characteristic Objects METhod (COMET) on the example of the popular game Counter-Strike: Global Offensive (CS: GO). We present a range of advantages of the player evaluation model created using the COMET method and, therefore, prove the practicality of using multi-criteria decision analysis (MCDA) methods to build multi-criteria assessment models in emerging areas of eSports. Thus, we provide a methodical and practical background for building a decision support system engine for the evaluation of players in several eSports.

To answer the question stated in the title, we present and compare two approaches: first, a standard approach for solving dual fuzzy nonlinear systems (DFN-systems) based on Newton’s method, which uses 2D FN representation and second, the new approach, based on multidimensional fuzzy arithmetic (MF-arithmetic). We use a numerical example to explain how the proposed MF-arithmetic solves the DFN-system. To analyze results from the standard and the new approaches, we introduce an imprecision measure. We discuss the reasons why imprecision varies between both methods. The imprecision of the standard approach results (roots) is significant, which means that many possible values are excluded.

Multi-Criteria Decision-Analysis (MCDA) methods are successfully applied in different fields and disciplines. However, in many studies, the problem of selecting the proper methods and parameters for the decision problems is raised. The paper undertakes an attempt to benchmark selected Multi-Criteria Decision Analysis (MCDA) methods. To achieve that, a set of feasible MCDA methods was identified. Based on reference literature guidelines, a simulation experiment was planned. The formal foundations of the authors’ approach provide a reference set of MCDA methods ( Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), Complex Proportional Assessment (COPRAS), and PROMETHEE II: Preference Ranking Organization Method for Enrichment of Evaluations) along with their similarity coefficients (Spearman correlation coefficients and WS coefficient). This allowed the generation of a set of models differentiated by the number of attributes and decision variants, as well as similarity research for the obtained rankings sets. As the authors aim to build a complex benchmarking model, additional dimensions were taken into account during the simulation experiments. The aspects of the performed analysis and benchmarking methods include various weighing methods (results obtained using entropy and standard deviation methods) and varied techniques of normalization of MCDA model input data. Comparative analyses showed the detailed influence of values of particular parameters on the final form and a similarity of the final rankings obtained by different MCDA methods.

Many scientific papers are devoted to solving multi-criteria problems. Researchers solve these problems, usually using methods that find discrete solutions and with the collaboration of domain experts. In both symmetrical and asymmetrical problems, the challenge is when new decision-making variants emerge. Unfortunately, discreet identification of preferences makes it impossible to determine the preferences for new alternatives. In this work, we propose a new approach to identifying a multi-criteria decision model to address this challenge. Our proposal is based on stochastic optimization techniques and the characteristic objects method (COMET). An extensive work comparing the use of hill-climbing, simulated annealing, and particle swarm optimization algorithms are presented in this paper. The paper also contains preliminary studies on initial conditions. Finally, our approach has been demonstrated using a simple numerical example.

A significant challenge in the current trend in decision-making methods is the problem’s class in which the decision-maker makes decisions based on partially incomplete data. Classic methods of multicriteria decision analysis are used to analyze alternatives described by using numerical values. At the same time, fuzzy set modifications are usually used to include uncertain data in the decision-making process. However, data incompleteness is something else. In this paper, we show two approaches to identify fuzzy models with partially incomplete data. The monolithic approach assumes creating one model that requires many queries to the expert. In the structured approach, the problem is decomposed into several interrelated models. The main aim of the work is to compare their accuracy empirically and to determine the sensitivity of the obtained model to the used criteria. For this purpose, a study case will be presented. In order to compare the proposed approaches and analyze the significance of the decision criteria, we use two ranking similarity coefficients, i.e., symmetric rw and asymmetric WS. In this work, the limitations of each approach are presented, and the results show great similarity despite the use of two structurally different approaches. Finally, we show an example of calculations performed for alternatives with partially incomplete data.

Over the past few decades, several researchers and professionals have focused on the development and application of multi-criteria group decision making (MCGDM) methods under a fuzzy environment in different areas and disciplines. This complex research area has become one of the more popular topics, and it seems that this trend will be increasing. In this paper, we propose a new MCGDM approach combining intuitionistic fuzzy sets (IFSs) and the Characteristic Object Method (COMET) for solving the group decision making (GDM) problems. The COMET method is resistant to the rank reversal phenomenon, and at the same time it remains relatively simple and intuitive in practical problems. This method can be used for both symmetric and asymmetric information. The Triangular Intuitionistic Fuzzy Numbers (TIFNs) have been used to handle uncertain data. This concept can ensure the preference information about an alternative under specific criteria more comprehensively and allows for easy modelling of symmetrical or asymmetrical linguistic values. Each expert provides the membership and non-membership degree values of intuitionistic fuzzy numbers (IFNs). So this approach deals with a different kind of uncertainty than with hesitant fuzzy sets (HFSs). The proposed combination of COMET and IFSs required an adaptation of the matrix of expert judgment (MEJ) and allowed to capture the behaviour aspects of the decision makers (DMs). Therefore, we get more reliable solutions while solving MCGDM problems. Finally, the proposed method is presented in a simple academic example.

A q-rung orthopair fuzzy set (q-ROFS), an extension of the Pythagorean fuzzy set (PFS) and intuitionistic fuzzy set (IFS), is very helpful in representing vague information that occurs in real-world circumstances. The intention of this article is to introduce several aggregation operators in the framework of q-rung orthopair fuzzy numbers (q-ROFNs). The key feature of q-ROFNs is to deal with the situation when the sum of the qth powers of membership and non-membership grades of each alternative in the universe is less than one. The Einstein operators with their operational laws have excellent flexibility. Due to the flexible nature of these Einstein operational laws, we introduce the q-rung orthopair fuzzy Einstein weighted averaging (q-ROFEWA) operator, q-rung orthopair fuzzy Einstein ordered weighted averaging (q-ROFEOWA) operator, q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG) operator, and q-rung orthopair fuzzy Einstein ordered weighted geometric (q-ROFEOWG) operator. We discuss certain properties of these operators, inclusive of their ability that the aggregated value of a set of q-ROFNs is a unique q-ROFN. By utilizing the proposed Einstein operators, this article describes a robust multi-criteria decision making (MCDM) technique for solving real-world problems. Finally, a numerical example related to integrated energy modeling and sustainable energy planning is presented to justify the validity and feasibility of the proposed technique.

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

Niniejsza praca omawia istotny problem związany z klasyfikacją i właściwym zdefiniowaniem wielkości pompowni melioracyjnych. Podstawowa definicja wielkości jest bardzo szeroka, zatem ze względu na liczbę objaśniających ją parametrów wymaga precyzyjnego uściślenia. W dotychczasowych ocenach rozmiarów pompowni melioracyjnych posługiwano się na ogół wrażeniami osobistymi opartymi na subiektywnym odczuciu, które z kolei jest oparte na nieznanej liczbie parametrów. Pojawiające się przy tym trudności z określeniem przynależności obiektów do danej grupy wielkości wynikały głównie z luk informacyjnych dotyczących właściwej granicy podziału pomiędzy poszczególnymi grupami. W niniejszej pracy, do klasyfikacji i podziału pompowni według ich wielkości zastosowano takie ich parametry jak: moc nominalna pompowni P w kW, wydajność całkowita Q w m3 · s oraz zasięg oddziaływania pompowni Fp w ha, dla 270 losowo wybranych obiektów melioracyjnych w Polsce. Do rozwiązania problemu wykorzystano jedną z wielokryterialnych metod podejmowania decyzji (MCDM), opartą na mechanizmach logiki rozmytej, noszącą nazwę metody obiektów charakterystycznych (COMET).

The preference of one alternative over another is a useful way to express the opinion of the decision-maker. In the process of group decision-making, preference relations are used in preference modeling of the alternatives under given criteria. The probability is an important tool to deal with uncertainty and, in many scenarios of decision-making problems, the probabilities of different events affect the decision-making process directly. In order to deal with this issue, the hesitant probabilistic multiplicative preference relation (HPMPR) is defined in this paper. Furthermore, consistency of the HPMPR and consensus among decision makers are studied here. In this respect, many algorithms are developed to achieve consistency of HPMPRs, reasonable consensus between decision-makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision-making. Lastly, we present a case study with complete illustration of the proposed model and discuss the effects of probabilities on decision-making validating the importance of the introduction of probability in hesitant multiplicative preference relations.