A new distance between rankings.

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.

Making Group Decisions within the Framework of a Probabilistic Hesitant Fuzzy Linear Regression Model.

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.

Hesitant 2-tuple fuzzy linguistic multi-criteria decision-making method based on correlation measures.

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.

The Group Decision-Making Using Pythagorean Fuzzy Entropy and the Complex Proportional Assessment.

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.

New Pythagorean Entropy Measure with Application in Multi-Criteria Decision Analysis.

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.

Alarming decline in recognition of anatomical structures amongst medical students and physicians.

BACKGROUND: Insufficient anatomical training can put patients' safety at risk. The aim of this study was to assess the proficiency of medical students and physicians in identifying labeled anatomical structures. The second aim of the study was to evaluate factors that can affect this recognition. METHODS: An internet-based survey where participants had to correctly identify labeled anatomical structures on cadaveric specimens was designed. RESULTS: The study group included 1186 participants (58.7% females): 931 medical students and 255 medical graduates from all twelve Polish medical schools. The mean total survey score for the entire study group was 65.6%. Students gained significantly higher results than graduates (total: 67.3% vs. 59.5%, P<0.001); 331 (27.9%) participants did not pass the test (<60). There was a correlation observed between points gained in this survey and grade obtained in the gross anatomy course (P<0.001). Multivariable logistic regression found that participation in cadaver laboratory classes most strongly increases anatomical competencies (OR=5.30, 95%CI=1.20-23.40, P=0.03). Other significant factors boosting anatomical proficiency were membership in students' scientific clubs, being male, and having a high grade (≥80%) in initial gross anatomy course. The time since anatomy course completion was negatively correlated with the total survey score (OR=0.86, 95%CI=0.81-0.92, P<0.001). CONCLUSIONS: Anatomical knowledge of Polish medical students is moderate (<70%) and it significantly decreases with time. Anatomical structure recognition can be up to 25% lower in highly trained physicians when compared to pre-clinical medical students. This trend may be reversed by replacing subject-based anatomy courses with system-based (integrated) curricula at the undergraduate level or introducing short refresher anatomical courses during postgraduate training.