الدر المختار کا سندھ میں مطالعہ: ایک تحقیقی جائزہ

A well Known book Durr al-Mukhtār has great importance in Ḥanafī School of thought. This is mainly due to its conciseness and comprehensiveness. This is why most of Ḥanafī Scholars has worked on it by editing the manuscripts and writing scholarly footnotes annotation which numbered more than sixty.   Al-Durr al-Mukhtār has remained law book in sub-continent. Sindhi scholars have also written commentaries and footnotes on this master piece of the latter Ḥanafī school of thought. This paper attempts to introduce these standard works in detail. 

Some Contributions to Cubic Sets and Their Applications

Multiple decision-making models are undertaking the complexity of the organizations with measurable mathematical models may be very hard, particularly when there is an absence of numerical data. The idea of fuzzy set was presented by Zadeh (Zadeh, 1975). Fuzzy set theory has been applied in many fields such as, homoeopathic judgment, computer science, fuzzy algebra and decision-making problems. In real life, there are a measure of decision problems in which rank relations occur between different criteria, for example, when we buy an air ticket for traveling by considering the criteria such as "safety", "price" and "quality of service". Obviously, "safety" is with the highest priority compared with the others. Therefor in Atanassov (Atanassov, 1986) introduction the notion of intuitionistic fuzzy set (IFS) and considered the degree of membership as well as the degree of non-membership function. The situation of manufacture with indecision in real world problems has been a tough research that has produced unlike administrations and theories. Fuzzy decision making by their extensions have providing an extensive variety of tools that are bright to contract with indecision in dissimilar types of problems. Fuzzy decision-making methods have become slowly popular in decision making for personnel selection. According to Liang et al. (Liang, 2017), inadequate opinions are then finished with data vaccinated from reliable experts. In adding with respect to Liang et al. (Liang, 2017), such thoughts are more adapted by simulating their evolution due to social influence. It is also demonstrated that, under certain assumptions, the development of feelings due to effect joins to a final collective opinion. Cubic sets were introduced by Jun et al. (Jun, 2012), are the generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations, one is used for the degree of membership and other is used for the non-membership. The membership function is hold in the form of interval while non-membership over the normal fuzzy set. This thesis consists of ten chapters. In chapter one, we present some basic definitions and results which are directly used in our work. Here we discussed Fuzzy set, interval-valued fuzzy set, Intuitionistic fuzzy set, triangular intuitionistic fuzzy number, trapezoidal intuitionistic fuzzy number, intuitionistic linguistic fuzzy set, hesitant fuzzy set and cubic set. In chapter 2, we present a new idea of cubic TOPSIS method with complete explanation providing in the form of different examples. We existing a new idea of cubic grey analysis set and proposed CF-MAGDM Model. In chapter 3, we introduce some basic concepts and operation laws related to triangular cubic fuzzy numbers and crisp weighted possibility means are defined. We developed weighted average operator of TCFNs and hamming distance of the TCFN are defined. We develop an MCDM method approach based on an extended VIKOR method using TCFNS; MCDM method using TCFN’s are developed. Finally, an illustrative example is given to verify the developed approach. We discuss in comparison analyses. In chapter 4, we define some new concepts comprising the definition, operations, crisp weighted possibility means and hamming distance of the trapezoidal cubic fuzzy numbers (TrCFNs). In Section 4, we develop a MAGDM approach based on an extended VIKOR method using trapezoidal cubic fuzzy numbers (TrCFNs). We discuss implementation of the solution methodology to solve the PLS problem. A discussion of the obtained results and sensitivity analysis are also included in this section. In chapter 5, we present some Einstein operations on cubic fuzzy sets (CFSs) and analysis some desirable properties of the proposed operations. We first develop some novel arithmetic averaging operators, such as the cubic fuzzy Einstein weighted averaging (CFEWA) operator, cubic fuzzy Einstein ordered weighted averaging (CFEOWA) operator and cubic fuzzy Einstein hybrid weighted averaging (CFEHWA) operator, for aggregating a collection of cubic fuzzy numbers (CFNs). We apply the CFEHWA operator to multiple attribute decision making (MADM) with cubic fuzzy material. Gives a numerical example according to our approach. In chapter 6, we discuss the trapezoidal cubic fuzzy number (TrCFN) and operational laws. We present some Einstein operations on trapezoidal cubic fuzzy sets (TrCFSs) and analysis some desirable properties of the proposed operations. We first develop some novel arithmetic averaging operators, such as the trapezoidal cubic fuzzy Einstein weighted averaging (TrCFEWA) operator, trapezoidal cubic fuzzy Einstein ordered weighted averaging (TrCFEOWA) operator and trapezoidal cubic fuzzy Einstein hybrid weighted averaging (TrCFEHWA) operator, for aggregating a collection of trapezoidal cubic fuzzy numbers (TrCFNs). We apply the TrCFEHWA operator to multiple attribute decision making (MADM) with trapezoidal cubic fuzzy material. Gives a numerical example according to our approach. We discuss comparison analysis. In chapter 7, we present the definition and operational laws of CFLSs. The score function, accuracy function, and certainty function for CFLV are also defined and thereby a lexicographer method is established to rank the CFLVs. Three kinds of cubic fuzzy linguistic arithmetic aggregation operators are defined and their required properties are deliberated in detail. Two decision methods for MAGDM problems with CFLVs are proposed. An example of investment choice and comparison analysis is given. In chapter 8, we exhibit of triangular cubic linguistic hesitant fuzzy sets and triangular cubic linguistic hesitant fuzzy elements. We exhibit a series of aggregation operators for triangular cubic linguistic hesitant fuzzy information and watch the associations among these aggregation operators. We develop an approach to group decision makings with triangular cubic linguistic hesitant fuzzy data. The application of the developed approach in group decision-making problems is shown by an illustrative example. Results and discussion are defined. In chapter 9, we exhibit a series of aggregation operators for triangular cubic linguistic hesitant fuzzy information and watch the associations among these aggregation operators. Develops an approach to group decision makings with triangular cubic linguistic hesitant fuzzy data. The application of the developed approach in group decisionmaking problems is shown by an illustrative example. We propose the comparison method. In chapter 10, we develop trapezoidal cubic hesitant fuzzy number and operational laws. We propose trapezoidal cubic hesitant fuzzy TOPSIS method. A numerical example of the proposed model is presented. We discuss comparison to different method.