A Time-Expanded State-Space Formulation of Material Requirements Planning

Material Requirements Planning has long been one of the core mechanisms of manufacturing planning. Its purpose is straightforward in practical terms: starting from demand for finished products, it determines what components are needed, in what quantities, and in which periods, while accounting for product structure, lead times, inventory, and scheduled receipts. Although this logic is well established and remains deeply embedded in industrial practice, it is still most often represented as a procedural sequence of steps, based on recursive explosion, time-bucket iteration, and inventory netting. In previous work, MRP was reformulated as a structured composition of operators, making explicit the distinction between structural propagation, temporal displacement, inventory evolution, and planning response. That formulation clarified an important point: the logic of MRP is not merely algorithmic, but structural. Even so, the representation still preserved a conceptual separation between product structure and time, and it retained an implicit sequential character inherited from classical implementations. The present work develops a further step in that direction by formulating MRP over a time-expanded state space. In this representation, each item-period pair is treated as an element of a single finite dependency space. Structural relations and lead-time offsets are no longer handled as separate stages, but are embedded into one global propagation framework. This changes the conceptual status of the model. MRP is no longer viewed primarily as a recursive planning routine, but as a structured dependency system defined over an expanded production horizon. This shift is significant for both theory and implementation. From a theoretical standpoint, it places MRP in direct continuity with input-output thinking, discrete-time systems, and graph-based representations of production dependencies. From a computational standpoint, it opens the possibility of replacing recursive traversal with a unified sparse formulation that is more transparent, more analyzable, and more naturally extensible. What was previously spread across multiple loops and procedural stages becomes part of one coherent state-space model. A second contribution of this work is the explicit treatment of causality. In ordinary calendar time, MRP appears anticipative, because later demand induces earlier releases. Within the time-expanded formulation, however, this logic can be interpreted rigorously as causal when planning time is read in the appropriate direction. This helps clarify the true nature of MRP as a backward scheduling system rather than a forward simulation only. The paper also extends the framework by incorporating inventory directly into an augmented state-space representation. In this setting, inventory carry-over is treated as an internal state transition, while lot-for-lot netting appears as a planning response generated by shortage conditions. The resulting model is not only a reformulation of multilevel explosion, but a broader planning system in which requirements, availability, and order generation are connected within one structured representation. The aim of this paper is therefore not to replace the practical logic of classical MRP, but to restate it at a higher level of mathematical and conceptual clarity. By doing so, it becomes possible to analyze the system more rigorously, to compare it with related production models, and to prepare a cleaner foundation for future developments in optimization, stochastic planning, and control-oriented supply chain design.