A Matrix- and Tensor-Based Reformulation of Time-Phased Material Requirements Planning

Material Requirements Planning (MRP) is traditionally implemented through recursive, level-oriented, and time-sequential procedures. The practical logic of classical MRP is well established, yet its computational representation is usually procedural rather than structural. This paper develops a rigorous reformulation of MRP in which product-structure explosion, time-phased propagation, lead-time offsetting, inventory netting, and planned-order generation are expressed through matrix operators, and then generalized through a tensor representation in which structural and temporal dependencies are embedded within a single multilinear object. The matrix formulation makes explicit the algebraic nature of multilevel explosion and the connection between bill-of-material (BOM) structures, graph reachability, and Leontief-style input-output systems. The tensor formulation extends this perspective by incorporating delay directly into the production operator, thereby eliminating the conceptual separation between structural propagation and temporal shifting. A fully reproducible proof-of-concept implementation is provided, including all source files and data needed to recreate the prototype. The resulting framework is theoretically coherent, computationally feasible, and suitable for further development toward scalable, analytically transparent, and optimization-ready planning systems.