Aggregation Operators Research Articles

Interval-valued (p,q,r)-spherical fuzzy sets and their applications in MCGDM and MCDM based on TOPSIS method and aggregation operators

The (p,q,r)-Spherical fuzzy set is a useful generalization of other fuzzy structures such as fuzzy, intuitionistic fuzzy, Pythagorean fuzzy, picture fuzzy, q-rung orthopair fuzzy, spherical fuzzy, and T-spherical fuzzy sets. In this paper, we define concept of interval-valued (p,q,r)-spherical fuzzy set (IVpqr-SFS) and Set-theoretical and arithmetic operations of them. Two aggregation operators named interval-valued (p,q,r)-spherical fuzzy weighted arithmetic (IVpqr-SFWAA) and interval-valued (p,q,r)-spherical fuzzy weighted geometric (IVpqr-SFWGA) aggregation operators are introduced for IVpqr-SF numbers (IVpqr-SFN) and their properties were examined. The proposed operators are explained with examples. Furthermore, the score and accuracy functions of an IVpqr-SFN are introduced. We also define distance measures between two same types IVpqr-SFSs based on Hamming, Euclidean, and Hausdorff distance measures. Moreover, a multi-criteria group decision-making (MCGDM) method is developed based on the TOPSIS method by using the proposed distance measures and score function, and a numerical example is given to show the process of the proposed MCGDM method. In addition, a multi-criteria decision-making (MCDM) method is developed using the proposed IVpqr-SFWAA, IVpqr-SFWGA operators, and an illustrative example is given to show the process of the proposed MCDM method. Finally, a comparative analysis is presented between the proposed methods and the existing methods.

Intuitionistic Fuzzy Gram-Schmidt Orthogonalized Artificial Neural Network for Solving MAGDM Problems

Objectives: To propose a suitable decision-making model based on Intuitionistic Fuzzy sets (IFSs) and Gram-Schmidt orthogonalization process for Artificial Neural Network (ANN). Methods: The IFS data sets appearing in the form of matrices are aggregated using the available aggregation operators in the literature and then the collective aggregated information is processed through Gram-Schmidt orthogonalization for the revised input vectors which is then fed into the ANN algorithm following Delta Learning Rule for the next phase. The weight updation is performed through the ANN and the output is improvised. Findings: The proposed Gram-Scmidt Orthogonalization process is utilized in Intuitionistic Fuzzy Artificial Neural Network model. The Delta learning rule is utilized in the process of the Neural Network, where the Intuitionistic Fuzzy nature of the input data is transformed into a fuzzy data and then the ranking of the alternatives is done based on the weights updation through the learning phase of the ANN. Once the vector is trained out of the learning phase, it is then processed through the activation function for the final selection of the best alternative required of the Multiple Attribute Group Decision Making (MAGDM) problem posed in this work. To demonstrate the usefulness and applicability of this new method with the Gram-Schmidt process, the numerical example also adds more insight to the proposed methodology of ANN with the application of some Linear Space techniques. Novelty: Most of the research done on Intuitionistic Fuzzy Artificial Neural Network model are based on learning rules or using some other calculations. The proposed Gram-Schmidt Orthogonalization process is used to find the orthogonal basis that are used as input training vectors in the Delta learning rule for ANN. Keywords: MAGDM, ANN, Aggregation operators, Learning Rules, Intuitionistic Fuzzy sets, Gram-Scmidt Orthogonalization

TOPSIS Method Based on Intuitionistic Fuzzy Soft Set and Its Application to Diagnosis of Ovarian Cancer

Fuzzy set theory is a mathematical method for dealing with uncertainty and imprecision in decision-making. Some of the challenges and complexities involved in medical diagnosis can be addressed with the help of fuzzy set theory. Ovarian cancer is a disease that affects the female reproductive system's ovaries, which also make the hormones progesterone and estrogen. The ovarian cancer stages demonstrate how far the disease has spread from the ovaries to other organs. The TOPSIS technique (Technique for Order Preference by Similarity to Ideal Solution) aids in selecting the best option from a selection of choices by taking into account a number of variables. It provides a ranking or preference order after weighing the benefits and drawbacks of each solution. Intuitionistic fuzzy soft set (IFSS) is the framework to deal with the uncertain information with the help of the parameters. The goal of this article is to develop some basic aggregation operators (AOs) based on the IFSS and then use them to diagnose the stages of the ovarian cancer using the TOPSIS technique. Furthermore, the variation of the parameters used in the developed model AOs is also observed and graphically represented.

Group decision making based on dice similarity measure and 2-dimension linguistic fuzzy weighted arithmetic aggregation operator for 2-dimension linguistic fuzzy variables

In the process of decision-making, it usually applies 2-dimension linguistic fuzzy variable (2DLFV) to describe the uncertainty caused by the decision objects and the decision makers. This proposal aims to design group decision making method based on dice similarity measure (DSM) and 2-dimension linguistic fuzzy weighted arithmetic aggregation (2DLFWAA) operator for 2-dimension linguistic fuzzy framework. Based on the psychological theory and prospect theory, 2-dimension linguistic scale function (2DLSF) about 2DLFV is firstly proposed, and new operational rules of 2DLFV are introduced. Two forms of DSM are proposed to measure the closeness between two 2-dimension linguistic fuzzy sets (2DLFSs). A definition is given for 2DLFWAA operator and some of its properties are studied. Based on the DSM and the 2DLFWAA operator for 2DLFSs, the technique for order of preference by similarity to ideal solution (TOPSIS) method is extended to handle the multiple attribute group decision making (MAGDM) problems under 2-dimension linguistic fuzzy environments. Finally, an illustrative example is provided to show the concrete process of the proposed method. Comparative analysis and sensitivity analysis are conducted to test the rationality and robust of the proposed method. The results suggest that the proposed method offers a flexible and reliable approach in 2-dimension linguistic fuzzy scenarios.

Hierarchical representation learning for next basket recommendation

The task of next basket recommendation is pivotal for recommender systems. It involves predicting user actions, such as the next product purchase or movie selection, by exploring sequential purchase behavior and integrating users’ general preferences. These elements may converge and influence users’ subsequent choices. The challenge intensifies with the presence of varied user purchase sequences in the training set, as indiscriminate incorporation of these sequences can introduce superfluous noise. In response to these challenges, we propose an innovative approach: the Selective Hierarchical Representation Model (SHRM). This model effectively integrates transactional data and user profiles to discern both sequential purchase transactions and general user preferences. The SHRM’s adaptability, particularly in employing nonlinear aggregation operations on user representations, enables it to model complex interactions among various influencing factors. Notably, the SHRM employs a Recurrent Neural Network (RNN) to capture extended dependencies in recent purchasing activities. Moreover, it incorporates an innovative sequence similarity task, grounded in a k-plet sampling strategy. This strategy clusters similar sequences, significantly mitigating the learning process’s noise impact. Through empirical validation on three diverse real-world datasets, we demonstrate that our model consistently surpasses leading benchmarks across various evaluation metrics, establishing a new standard in next-basket recommendation.

Aggregation operators of complex fuzzy Z-number sets and their applications in multi-criteria decision making

Fuzzy sets (FSs) are a flexible and powerful tool for reasoning about uncertain situations that cannot be adequately expressed by classical sets. However, these sets fall short in two areas. The first is the reliability of this tool. Z-numbers are an extension of fuzzy numbers that improve the representation of uncertainty by combining two important components: restriction and reliability. The second is the problems that need to be solved simultaneously. Complex fuzzy sets (CFSs) overcome this problem by adding a second dimension to fuzzy numbers and simultaneously adding connected elements to the solution. However, they are insufficient when it comes to problems involving these two areas. We cannot express real-life problems that need to be solved at the same time and require the reliability of the information given with any set approach given in the literature. Therefore, in this study, we propose the complex fuzzy Z-number set (CFZNS), a generalization of Z-numbers and CFS, which fills this gap. We provide the operational laws of CFZNS along with some properties. Additionally, we define two essential aggregation operators called complex fuzzy Z-number weighted averaging (CFZNWA) and complex fuzzy Z-number weighted geometric (CFZNWG) operators. Then, we present an illustrative example to demonstrate the proficiency and superiority of the proposed approach. Thus, we process multiple fuzzy expressions simultaneously and take into account the reliability of these fuzzy expressions in applications. Furthermore, we compare the results with the existing set operations to confirm the advantages and demonstrate the efficiency of the proposed approach. Considering the simultaneous expression of fuzzy statements, this study can serve as a foundation for new aggregation operators and decision-making problems and can be extended to many new applications such as pattern recognition and clustering.

Pricing Strategies for Distribution Network Electric Vehicle Operators Considering the Uncertainty of Renewable Energy

In the future, the active load of the distribution network side will be dominated by electric vehicles (EVs), showing that the charging power demand of electric vehicles will change with the change in charging electricity price. With the popularity of electric vehicles in the distribution network, their aggregation operators will play a more prominent role in pricing management and charging behavior, and setting an appropriate charging price can achieve a win–win situation for operators and electric vehicle users. At the same time, the proportion of scenery in the distribution network is relatively high, and the uncertainty of self-output has a certain impact on the pricing strategy of operators and the charging behavior of electric vehicle users, which has become an important research topic. Based on the above background, an EV operator pricing strategy considering the landscape uncertainty is proposed, a Stackelberg game model is established to maximize the respective benefits of operators and EV users, and the two-layer model is further transformed into a single-layer model through the Karush–Kuhn–Tucker (KKT) condition and duality theorem. Finally, the IEEE 33 system is simulated with the CPLEX solver, and the global optimal pricing strategy is obtained. Simulation results prove that electric vehicle operators experience a maximum profit increase of 2.6% due to the impact of maximum capacity of energy storage equipment and the uncertainty of renewable energy output can result in electric vehicle operators losing approximately 20% of their profits at most.

Evaluation and selection of E-learning websites using intuitionistic fuzzy confidence level based Dombi aggregation operators with unknown weight information

In recent years, the surge in online learning has eclipsed traditional classroom teaching, elevating the prominence of e-learning platforms. However, the quest for a reliable methodology to evaluate these platforms has remained pressing. While several studies have explored this domain, none have integrated decision experts’ confidence levels. This study bridges this gap by seamlessly incorporating confidence levels into Dombi aggregation operators within an intuitionistic fuzzy framework. The study introduces two innovative operators: the confidence-level intuitionistic fuzzy Dombi weighted averaging operator (CIFDWA) and the confidence-level intuitionistic fuzzy Dombi weighted geometric operator (CIFDWG). These operators are employed to form a novel multi-attribute group decision-making (MAGDM) approach, which is demonstrated by applying them to select the optimal e-learning website in a real-world scenario. The “Stepwise Weight Assessment Ratio Analysis (SWARA)” method is utilized to determine criteria weights. The findings highlight the critical importance of sub-criteria such as topic diversity, pricing, content credibility, data security, and user interface simplicity. The analysis reveals Unacademy as the leading platform, closely followed by Byju’s. Through meticulous comparative analysis and sensitivity evaluations, we underscore the stability and coherence of the proposed model. The study’s novelty lies in its innovative approach to integrating confidence levels in multi-attribute group decision-making scenarios, promising broader applications across various real-world contexts. Educational advisers and management can leverage this methodology to empower students with informed choices.

Selecting optimal celestial object for space observation in the realm of complex spherical fuzzy systems

The sensible selection of celestial objects for observation by the James Web Space Telescope (JWST) is pivotal for the precise decision-making (DM) process, aligning with scientific priorities and instrument capabilities to maximize valuable data acquisition to address key astronomical questions within the constraints of limited observation time. Aggregation operators are valuable models for condensing and summarizing a finite set of data of imprecise nature. Utilization of these operators is critical when addressing multi-attribute decision-making (MCDM) challenges. The complex spherical fuzzy (CSF) framework effectively captures and represents the uncertainty that arises in a DM problem with more precision. This paper presents two novel aggregation operators, namely the complex spherical fuzzy Yager weighted averaging (CSFYWA) operator and the complex spherical fuzzy Yager weighted geometric (CSFYWG) operator. Many fundamental structural properties of these operators are delineated, and thereby an improved score function is suggested that addresses the limitations of the existing score function within the CSF system. The newly defined operators are applied to formulate an algorithm for MADM problems to tackle the challenges of ambiguous data in the selection process. Moreover, these strategies are effectively applied to handle the MADM problem of selecting the optimal astronomical object for space observation within the CSF context. Additionally, a comparative analysis is also performed to demonstrate the validity and superiority of the proposed techniques compared to the existing strategies.

Enhancing healthcare supply chain management through artificial intelligence-driven group decision-making with Sugeno–Weber triangular norms in a dual hesitant q-rung orthopair fuzzy context

The healthcare industry faces numerous challenges in managing its supply chain efficiently, where critical decisions must be made promptly to ensure the availability of essential medical resources. This research introduces a novel artificial intelligence (AI) approach, utilizing the “Sugeno–Weber (SW) t-conorms and t-norms” (SWt-CNs&t-Ns) for decision-making in a Dual Hesitant q-Rung Orthopair Fuzzy (DHq-ROF) context. The SWt-CNs&t-Ns are chosen for their adaptability in data unification, serving as prominent operations for union and intersection processes. Developing a set of fundamental operations is imperative to effectively utilize SWt-CNs&t-Ns and hybrid aggregation operators in DHq-ROF settings. Following the introduction of these processes, several aggregating operators have been provided. These operators include DHq-ROF SW weighted averaging, ordered weighted averaging, hybrid averaging, and their geometric counterparts utilizing DHq-ROF data. The SW triangular norm-based approach aggregates group preferences, facilitating a systematic decision-making process. Triangular norms ensure a realistic representation of interrelationships among decision criteria, leading to optimal healthcare supply chain management solutions. Furthermore, the SW triangular norm-based approach aggregates group preferences, enabling a systematic and comprehensive decision-making process. Choosing the best healthcare supply chain management solutions is easier when you use triangular norms because they give a more accurate picture of how the decision criteria affect each other. The effectiveness of the proposed AI framework is demonstrated through a series of experiments and case studies, showcasing its ability to enhance decision accuracy, reduce uncertainty, and improve overall supply chain performance. The research findings underscore the potential of AI-driven solutions to revolutionize healthcare supply chain management, ultimately leading to better resource allocation, cost efficiency, and improved patient care.

Some weighted aggregation operators of quadripartitioned single-valued trapezoidal neutrosophic sets and their multi-criteria group decision-making method for developing green supplier selection criteria

Some weighted aggregation operators of quadripartitioned single-valued trapezoidal neutrosophic sets and their multi-criteria group decision-making method for developing green supplier selection criteria

T-Spherical fuzzy TOPSIS method based on distance measures and Hamacher Heronian mean averaging aggregation operators and its application to waste management

T-Spherical fuzzy sets provide a robust framework for handling uncertainty in real-world decision-making. This paper addresses the construction of T-Spherical fuzzy Heronian mean averaging aggregation operators using novel Hamacher operations. The study mathematically verifies the properties of these operators and develops new distance measures such as Hamming, Euclidean and Jaccard for quantifying dissimilarity between T-Spherical fuzzy numbers. Furthermore, the TOPSIS method is implemented using the proposed operators and distance measures to solve a practical decision-making problem in selecting the best transportation firm for medical waste disposal. A sensitivity analysis is made to assesses the efficacy of the proposed operators, while a comparison study with existing methods is exhibited to establish their efficiency.

A novel multi-criteria group decision making algorithm for enhancing supply chain efficiency under high uncertainty during crisis based on q-rung orthopair fuzzy information

In recent years, q-rung orthopair fuzzy sets (q-ROFSs) have been becoming gradually more advantageous in handling information with high uncertainty. Several multi-criteria group decision-making (MCGDM) algorithms under q-ROFS information have been proposed in literature and used to solve various real-life problems. However, several shortcomings of some existing MCGDM algorithms under certain circumstances also emerged which limit their applicability. To overcome these challenges and deal with information under high uncertainty more accurately and reasonably, we propose a novel MCGDM algorithm under q-ROF information, i.e., (q-ROF-MCGDM), that retains the advantages of currently available methods, but extend its applicability by introducing a new q-rung orthopair fuzzy weighted averaging aggregation operator (q-ROFWAAO) along with a new entropy measure. The proposed q-ROF-MCGDM algorithm involves the steps: firstly, the input requirement in terms of alternatives, criteria and expert evaluations are required. Secondly, aggregating expert opinions using weights and normalizing decision matrices, and finally, calculating entropy weights and aggregating overall evaluations for final prioritization. Further, new operational laws have been developed and several necessary properties of the proposed q-ROFWAAO are also proved. Moreover, a sensitivity and comparative analysis has been carried out for validity and effectiveness of the proposed q-ROF-MCGDM algorithm. The proposed q-ROF-MCGDM algorithm has been implemented to enhance the efficiency of the United Arab Emirates (UAE) food industry under high uncertainty during the recent crisis. Finally, an evaluation of identifying and prioritizing the most severe food SC disruptions and appropriate mitigation strategies under crisis is provided to demonstrate applicability of the proposed q-ROF-MCGDM algorithm, and the obtained real case results confirm its usefulness. Findings of the study offer valuable insights to both food industry researchers and managers in developing effective recovery strategies, mitigating risks, and improving overall efficiency to ensure the survival of food businesses during the crisis.

Multiple attribute decision making based on score function of q-connection numbers, q-CNPWG aggregation operator of q-connection numbers, and set pair analysis theory in the environments of q-rung orthopair fuzzy numbers

This paper proposes a new multiple attribute decision making (MADM) method in the environments of q-rung orthopair fuzzy numbers (q-ROFNs) based on the proposed score function of q-connection numbers (q-CNs), the proposed q-connection number power weighted geometric (q-CNPWG) aggregation operator (AO) of q-CNs, and the set pair analysis (SPA) theory. Firstly, we propose a score function of q-CNs based on the SPA theory, where we also present some properties of the proposed score function of q-CNs. Then, we propose the q-CNPWG AO for aggregating q-CNs, where we also present some properties of the proposed q-CNPWG AO of q-CNs. Finally, we propose a new MADM method in the environments of q-ROFNs based on the proposed score function of q-CNs, the proposed q-CNPWG AO of q-CNs, and the SPA theory. The proposed MADM method can overcome the drawbacks of existing MADM methods in the environments of q-ROFNs, where they cannot distinguish preference orders of alternatives in some situations. The proposed MADM method is very useful for MADM in the environments of q-ROFNs.

Interactive Optimization for Cartographic Aggregation of Building Features

AbstractAggregation, as an operation of cartographic generalization, provides an effective means of abstracting the configuration of building features by combining them according to the scale reduction of the 2D map. Automating this design process effectively helps professional cartographers design both paper and digital maps, but finding the best aggregation result from the numerous combinations of building features has been a challenge. This paper presents a novel approach to assist cartographers in interactively designing the aggregation of building features in scale‐aware map visualization. Our contribution is to provide an appropriate set of candidates for the cartographer to choose from among a limited number of possible combinations of building features. This is achieved by collecting locally optimal solutions that emerge in the course of aggregation operations, formulated as a label cost optimization problem. Users can also explore better aggregation results by interactively adjusting the design parameters to update the set of possible combinations, along with an operator to force the combination of manually selected building features. Each cluster of aggregated building features is tightly enclosed by a concave hull, which is later adaptively simplified to abstract its boundary shapes. Experimental design examples and evaluations by expert cartographers demonstrate the feasibility of the proposed approach to interactive aggregation.

Fredholm theory of the Toeplitz algebra on the space of all entire functions

AbstractWe prove that an aggregate Toeplitz operator on the Fréchet space of all entire functions is a Fredholm operator if and only if its symbol does not vanish. The result is motivated by and closely resembles the classical result of Gohberg and Douglas from the Hardy space theory of Toeplitz operators. There are however some subtle differences which we also discuss.

A type-2 neutrosophic entropy-based group decision analytics model for sustainable aquaculture engineering

Fish cages are crucial for the establishment of sustainable fish farms. The selection of fish cages is an essential subject to be examined for ensuring sustainability. This research introduces an advanced decision support model for the selection of fish cage types used in fish farming in reservoirs (R-FFs). The decision support model is based on multi-criteria decision-making (MCDM) approach and utilizes type-2 neutrosophic numbers (T2NNs). The importance levels of criteria are determined using the T2NN-Entropy method, and the ranking of fish cage types is achieved through the alternative ranking order method accounting for two-step normalization (AROMAN). The developed model is referred to as the T2NN-Entropy-AROMAN hybrid method and relies on expert opinions. Two advanced aggregation operators based on Yager t-norm and t-conorm operations, named T2NN Yager weighted arithmetic mean and T2NN Yager weighted geometric mean, are developed for aggregating T2NNs. The validity of these two aggregation operators is demonstrated. A real-life case study is developed to confirm the applicability of the T2NN-Entropy-AROMAN hybrid method. This case study focuses on the selection of fish cage types for the Artvin-Borçka R–FF in Artvin province, Turkey. The research results indicate that floating cages are the best alternative type for sustainable fish farming. Sensitivity analysis scenarios confirm the robustness of the research model and findings. The research findings, coupled with the developed model, offer valuable insights and guidance to decision-makers, researchers, and practitioners within the sector.

Assessment of environment-conscious propulsion technologies for road freight distribution based on T-spherical fuzzy Schweizer-Sklar power operators

This paper explores the viability of environmentally conscious propulsion technologies for road freight distribution, specifically using T-spherical fuzzy information. With the increasing need for sustainable transportation, the study investigates the potential of various propulsion technologies, including electric, hybrid, solar, and diesel vehicles, and evaluates their environmental impact in terms of emissions reduction and energy efficiency. The T-spherical fuzzy information is utilized to analyze and compare the performance of these technologies in different scenarios. In this paper, we proposed hybrid aggregation operators (AOs) namely T-spherical fuzzy Schweizer-Sklar power AOs for the aggregation of T-spherical fuzzy information. Moreover, the T-spherical fuzzy aggregation method provides a comprehensive and effective tool for evaluating the viability of different propulsion technologies. This article exhibits some characteristics of the proposed AOs. Using suggested AOs with numerous evaluations by decision-makers and partial weight information under T-spherical fuzzy information, a method for multi-criteria decision-making is constructed. The results show that the adoption of environmentally conscious propulsion technologies can significantly reduce carbon emissions and improve energy efficiency in road freight distribution. In addition, the research includes a sensitivity analysis and a comparison between the proposed strategy and current methods.

Enhancing breast cancer treatment selection through 2TLIVq-ROFS-based multi-attribute group decision making.

Breast cancer is an extremely common and potentially fatal illness that impacts millions of women worldwide. Multiple criteria and inclinations must be taken into account when selecting the optimal treatment option for each patient. The selection of breast cancer treatments can be modeled as a multi-attribute group decision-making (MAGDM) problem, in which a group of experts evaluate and rank alternative treatments based on multiple attributes. MAGDM methods can aid in enhancing the quality and efficacy of breast cancer treatment selection decisions. For this purpose, we introduce the concept of a 2-tuple linguistic interval-valued q-rung orthopair fuzzy set (2TLIVq-ROFS), a new development in fuzzy set theory that incorporates the characteristics of interval-valued q-rung orthopair fuzzy set (IVq-ROFS) and 2-tuple linguistic terms. It can express the quantitative and qualitative aspects of uncertain information, as well as the decision-makers' level of satisfaction and dissatisfaction. Then, the 2TLIVq-ROF weighted average (2TLIVq-ROFWA) operator and the 2TLIVq-ROF weighted geometric (2TLIVq-ROFWJ) operator are introduced as two new aggregation operators. In addition, the multi-attribute border approximation area comparison (MABAC) method is extended to solve the MAGDM problem with 2TLIVq-ROF information. To demonstrate the efficacy and applicability of the suggested model, a case study of selecting the optimal breast cancer treatment is presented. The results of the computations show that the suggested MAGDM model is able to handle imprecision and subjectivity in complicated decision-making scenarios and opens new research scenarios for scholars.

Revolutionizing diabetes care with innovative decision-making using cubic intuitionistic fuzzy Schweizer and Sklar power aggregation operators

The cubic intuitionistic fuzzy set is an expansion of the cubic fuzzy set that displays massive information to demonstrate interval-valued intuitionistic fuzzy sets and intuitionistic fuzzy sets. This increment informs limitations essential in existing frameworks, primarily focusing on the significance of embracing our access for more accurate decisions in compound and unresolved structures. The Schweizer and Sklar (SS) operations are engaged in promoting strong aggregation operators for cubic intuitionistic fuzzy sets through this research. Operators such as cubic intuitionistic fuzzy Schweizer and Sklar power weighted average (CIFSSPWA) and cubic intuitionistic fuzzy Schweizer and Sklar power weighted geometric (CIFSSPWG) are offered that enhance the workability of data aggregation within the cubic intuitionistic fuzzy (CIF) environment when compared to surviving methods. The proposed operators may assist in patient treatment and handling by upgrading decision-making in medical sectors like diabetes care. Moreover, to determine the stability and reliance of the outcomes, sensitivity and comparison studies are richly absorbed by this approach.