Introduction/purpose: Molodtsov introduced the concept of soft sets as a new mathematical tool for dealing with problems containing uncertainties. In the literature, different kinds of operations of soft sets are defined and used in theory and applications. Methods: This study is based on the paper "A New Operation on Soft Sets: Extended Difference of Soft Sets" by Sezgin, Ahmad and Mehmood [Journal of New Theory 27 (2019) 33-42]. Results: In this paper, we define a new operation on soft sets, called extended symmetric difference and investigate its relationship between extended symmetric difference and restricted symmetric difference and some other operations of soft sets. Conclusion: The author believes that the obtained results represent a significant improvement of many known results in the existing literature.

This paper introduces a novel numerical characteristic of hesitant fuzzy soft sets called their scored-energy. We introduce this concept as a combination of two very different tools, namely, scores of hesitant fuzzy elements and singular values of a (not- necessarily square) matrix. The latter idea replicates the concept of energy or nuclear norm of a matrix, which has excellent analytical abilities in graph theory and applications in fields such as statistics or signal processing. Natural algorithms establish applications of scored-energies to both clustering and decision-making. Through examples, we demonstrate how these new techniques are efficient and implementable in practice. Comparisons are conducted that prove their validity, reliability, and credibility.

Bipolar neutrosophic soft sets are powerful tools for modeling data under conditions of uncertainty and imprecision due to their rich parametric structure and the useful mathematical properties of the operations defined on them. In this paper, motivated by the limitations of existing decision‐making algorithms, we introduce a new numerical characteristic, the energy of a bipolar neutrosophic soft set defined using singular values, analogous to the graph energy and nuclear norm. Our goal is to develop an efficient decision‐making algorithm that successfully identifies the optimal alternative even in cases where other algorithms provide inaccurate or inconsistent results. Our research is motivated by the need for more reliable decision‐making methods in complex soft environments and the potential of the energy‐based approach to overcome the weaknesses of existing methods, which we demonstrate through a comparative analysis using concrete examples.