A Mathematical Model for Organizational Knowledge Flow Planning Based on Vector Space Theory

Knowledge flow represents the movement of knowledge between individuals, groups, and organizational levels, encompassing both tacit and explicit knowledge types [۳]. Unlike information flow, knowledge flow is inherently multidimensional and dynamic, requiring sophisticated modeling approaches that capture its complexity [۴]. Our previous research has established foundational frameworks for knowledge flow in various organizational contexts. Tajgardoon et al. [۵] proposed a knowledge flow framework for e-government in Iran, identifying thirty knowledge flows required to support the successful implementation of e-government roadmap. Subsequent work [۶] developed a model for estimating knowledge-based trust in organizational knowledge flows, highlighting the critical role of trust dynamics in knowledge transfer effectiveness. Furthermore, our comparative studies [۷] examined knowledge management tools for project-based organizations, identifying the lack of mathematical models for knowledge flow planning as a significant gap in the literature. Despite extensive research on knowledge flow modeling, existing approaches exhibit several limitations: (۱) most models focus predominantly on explicit knowledge, neglecting tacit knowledge components [۸]; (۲) quantitative measurement of knowledge gaps remains underdeveloped [۹]; (۳) prioritization mechanisms for knowledge flow interventions are largely absent [۱۰]; and (۴) integration with organizational tasks and roles is often superficial [۱۱]. This paper addresses these gaps by proposing a mathematical model for organizational knowledge flow planning based on vector space theory. The core innovation lies in representing knowledge as vectors within an organizational task space, enabling quantitative measurement of knowledge gaps through vector subtraction. Recent studies have demonstrated the growing application of vector space concepts in knowledge modeling. Liang et al. [۱۲] used topic-citation fusion to identify knowledge flow paths, while Zahedi et al. [۱۳] identified and prioritized factors affecting knowledge flow in high-tech industries. Our model complements these approaches by providing a task-integrated mathematical framework with explicit prioritization capabilities. The proposed model provides: A rigorous mathematical framework for representing current and desired knowledge states, Quantitative measurement of organizational knowledge gaps through vector subtraction, A systematic prioritization algorithm for identifying and ranking knowledge flow interventions, and Visualization capabilities for organizational knowledge status. The remainder of this paper is organized as follows: Section ۲ reviews related work on knowledge flow modeling and identifies research gaps. Section ۳ presents the mathematical model and prioritization algorithm. Section ۴ illustrates the model through a hypothetical example in a software development organization. Section ۵ compares the proposed model with existing approaches using eleven evaluation criteria. Section ۶ concludes the paper and suggests future research directions.