Since its seminal introduction by Molodtsov in 1999, soft set theory has evolved into a pivotal mathematical paradigm for the formal representation and management of uncertainty, offering a flexible and conceptually robust alternative to classical approaches. Its methodological versatility and abstract generality have fostered an expanding body of theoretical inquiry and practical deployment across diverse scientific domains. Central to the structural framework of soft set theory are its operations, which constitute the algebraic backbone of the theory and have consequently remained an enduring focus of research interest. Within this operational landscape, several soft product variants of the soft sets have been introduced; however, the extant literature provides only fragmentary examinations of their interrelations as regard soft subsets. The present study seeks to rectify this deficiency by delivering a comprehensive and rigorous analysis of this issue, with particular reference to the fundamental conditions which soft product, starting with intersection and union, is is soft subset of another soft product, and when these soft products are equal. That is, the study has been further expanded to provide a deeper analysis of which soft product is a subset of another and the circumstances under which they become equal. In pursuing this objective, the investigation elucidates a series of structural and distributive properties that not only reinforce the algebraic coherence of the soft product variants of the soft sets but also reveal their potential to generate rich algebraic configurations. Thus, the present theoretical exposition advances the foundational algebraic framework of soft set theory and contributes to a deeper understanding of its operational architecture. Beyond its intrinsic theoretical relevance, this deepened analysis is of substantial theoretical significance, as it elucidates the hierarchical and structural relationships among distinct soft products, thereby enhancing the algebraic understanding of the internal organization of soft set theory and facilitating the identification of operational dependencies and equivalences.
Control chart is not only used in manufacturing process for monitoring quality characteristics of the output it is extensively used in the field of reliability. It can be used to monitor the number of failures or breakdown of a machine, time between failure, life time data (censored or un censored). A number of researchers invested their time and thought in developing control chart techniques applicable in the field of reliability in the past few decades. In this paper we have summarized different methodologies available in the literature for controlling reliability data and discussed their merits and demerits.
Soft set theory has developed into a crucial mathematical paradigm for the formal representation and management of uncertaintysince Molodtsov's groundbreaking 1999 introduction. It provides a versatile and conceptually sound substitute for traditionalmethods. Its abstract generality and methodological adaptability have encouraged a growing corpus of theoretical research and realworld application in a variety of scientific fields. Soft set theory's operations are essential to its structural framework since they formthe algebraic foundation of the theory and have thus continued to be a subject of intense academic interest. Several soft productversions of the soft sets have been introduced within this operational landscape; nevertheless, the existing literature only offerspartial analyses of their relationships with respect to soft subsets. By providing a thorough and rigorous examination of this problem,the current study aims to address this shortcoming. It specifically addresses the essential circumstances under which one soft product,starting with symmetric difference and symmetric difference complement, is a soft subset of another soft product and when these softproducts are equal. In other words, the study has been extended to offer a more thorough examination of which soft product is asubset of another and the conditions that render them equal. In order to achieve this goal, the study clarifies a number of distributiveand structural characteristics that support the algebraic coherence of the soft product variations of the soft sets and demonstrate theircapacity to produce complex algebraic configurations. Therefore, the current theoretical exposition adds to a better understanding ofthe operational architecture of soft set theory and advances its fundamental algebraic framework. Beyond its inherent theoreticalvalue, this expanded analysis has significant theoretical importance because it clarifies the structural and hierarchical relationshipsbetween various soft products, improving algebraic comprehension of the internal structure of soft set theory and making it easier toidentify operational dependencies and equivalencies.
Construction industry is a catalyst for holistic development that drives economic growth, enhances social equity, pioneers’ technological advancements, and promotes environmental stewardship. The execution of construction practices requires resources, which include financial, technical and human. Despite recognition of environmental degradation and economic opportunities, there remains a notable gap in understanding how construction actors reconcile cost-efficiency with sustainability, particularly in applying financial planning tools to curb cost overruns while promoting environmentally responsible practices. This paper assesses construction practices in building development projects in Ruiru Municipality, Kenya. Data from 95 building development projects sampled through stratified sampling based on project value followed by administration of questionnaires to respondents comprising of contractors, site supervisors and project managers. The data was analyzed using frequency and percentages. The findings revealed that, while material planning and site preparation are generally wellimplemented, critical deficiencies in financial and human resource planning, particularly within residential and religious projects. The study recommends introducing targeted capacity building initiatives to enhance financial and human resourceplanning by training stakeholders in budget forecasting, cost control, and workforce scheduling, and by developing standardized frameworks to ensure consistency and accountability.
Since its introduction by Molodtsov, soft set theory has developed in prominence as an innovative method for dealing with uncertainty-related problems and modeling uncertainty. Soft set operations, the theory's main concept, have served as the foundation for theoretical and practical advances in the theory; thus, deriving the algebraic properties of soft set operations and studying the algebraic structure of soft sets associated with soft set operations has piqued researchers' interest continuously. Many soft union operations have been proposed in soft set theory, but there are some differences. Even though the definition of restricted union is widely acknowledged in the literature and applied in many studies, it is still lacking in its current form owing to the fact that a specific case where the soft sets' parameter sets may be disjoint is ignored in the definition. As a result, all the cases in the theorems are not taken into account, leading to errors or deficiencies in the studies that use this operation or investigate this operation’s properties. Regarding this, a thorough examination of the properly defined restricted union operation and extended union operation, together with their appropriate distributions and properties, as well as the appropriate algebraic structures connected to these soft set operations, is conspicuously lacking in the body of current literature. This study is primarily intended to fill this critical and significant gap by first fixing the presentational flaws of the restricted union definition and updating it. Furthermore, in many works on these operations, numerous theorems were offered without their proofs, or the proofs had some incorrect parts. In this paper, all the proofs based on function equality are supplied regularly, and the relations between the concept of soft subset and restricted and extended union operations are obtained for the first time with their extensive proofs. Furthermore, we explore numerous novel properties of these operations as analogies and counterparts of the union operation in classical set theory. We show that when restricted/extended union operations are combined with other types of soft set operations, several significant algebraic structures are formed, including monoid, bounded semi-lattice, semiring, hemiring, bounded distributive lattice, Bool algebra, De Morgan Algebra, and MV-algebra with detailed explanations. In this regard, this overall study represents the most comprehensive analysis of restricted union and extended union in the literature to date, as it covers all of the earlier important research on this topic with the corrected theorems and their proofs, thus advancing the theory by bridging a significant gap in the literature, serving as a guide for newcomers to this popular theory, and providing insight for further future research.
