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The concept of complex fuzzy set (CFS) and complex intuitionistic fuzzy set (CIFS) is two recent developments in the field of fuzzy set (FS) theory. The significance of these concepts lies in the fact that these concepts assigned membership grades from unit circle in plane, i.e., in the form of a complex number instead from [0, 1] interval. CFS cannot deal with information of yes and no type, while CIFS works only for a limited range of values. To deal with these kinds of problems, in this article, the concept of complex Pythagorean fuzzy set (CPFS) is developed. The novelty of CPFS lies in its larger range comparative to CFS and CIFS which is demonstrated numerically. It is discussed how a CFS and CIFS could be CPFS but not conversely. We investigated the very basic concepts of CPFSs and studied their properties. Furthermore, some distance measures for CPFSs are developed and their characteristics are studied. The viability of the proposed new distance measures in a building material recognition problem is also discussed. Finally, a comparative study of the proposed new work is established with pre-existing study and some advantages of CPFS are discussed over CFS and CIFS.

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In this manuscript, two generalizations of fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets, known as spherical fuzzy sets and T-spherical fuzzy sets, are discussed and a numerical and geometrical comparison among them is established. A T-spherical fuzzy set can model phenomena like voting using four characteristic functions denoting the degree of vote in favor, abstinence, vote in opposition, and refusal with an infinite domain, whereas an intuitionistic fuzzy set can model only phenomena of yes or no types. First, in this manuscript, some similarity measures in the frameworks of intuitionistic fuzzy sets and picture fuzzy sets are discussed. With the help of some numerical results, it is discussed that existing similarity measures have some limitations and could not be applied to problems where information is provided in T-spherical fuzzy environment. Therefore, some new similarity measures in the framework of spherical fuzzy sets and T-spherical fuzzy sets are proposed including cosine similarity measures, grey similarity measures, and set theoretic similarity measures. With the help of some results, it was proved that the proposed similarity measures are a generalization of existing similarity measures. The newly-defined similarity measures were subjected to a well-known problem of building material recognition and the results are discussed. A comparative study of new and existing similarity measures was established and some advantages of the proposed work are discussed.

Low carbon and digitalization are the general trends of manufacturing upgrading and transformation. Digital technology enables the whole process of green manufacturing and breaks down the spatial barrier. To achieve the dual carbon goals, the pressure-state-response (PSR) model, in which digital technology enables the green innovation of the manufacturing industry, was theoretically analyzed in this study. The measurement system of the digital green innovation (DGI) in the manufacturing industry was constructed according to the PSR framework. An evaluation model based on the analytic hierarchy process and the deviation maximization technique for order preference by similarity to an ideal solution method was constructed to measure the level of DGI. The results of this study from Chinese manufacturing are as follows. (i) The measurement system of the level of DGI in manufacturing industry includes a pressure system, state system and response system. (ii) In the past five years, the comprehensive index of the DGI in manufacturing industry has generally shown a trend of fluctuating rise. There are overall low and unbalanced phenomena in all regions. The gap decreased from 0.1320 to 0.1187, showing a gradually narrowing trend. (iii) Compared with other regions, the composite index of DGI is generally higher in the regions with a better ecological environment in the east and a more developed economy in the north. State parameters are higher than pressure and response parameters in most areas. (iv) Compared with other regions, the composite index of DGI in western and southern regions is lower, and the parameters of pressure, status and response are basically coordinated. (v) The application degree of digital technology, the emission intensity of waste water/exhaust gas of output value of one hundred million yuan and the expenditure intensity of digital technology adopted by enterprises are the key influencing factors of DGI in the manufacturing industry. This study not only proposed an evaluation index system of the digital green innovation level, but also puts forward policy guidance and practical guidance of digital technology to accelerate the green and intelligent manufacturing industry.