MRP theory is a theoretical body treating production-inventory systems, in which produced items are made up of sets of produced or purchased sub-items, required to be available a lead-time before each product is completed. The hierarchical dependence between items is captured using input matrices from input-output-analysis, the necessary advanced timing by employing Laplace transform methodology, and the economic consequences by applying the net present value. Little attention has hitherto been given to aspects of a non-empty initial state, e.g. an initial inventory position. Since such states are common in industry, there is a strong need for this theory to include such aspects, in order to gain further practical acceptance. In this paper, theoretical consequences from having a non-empty initial state are investigated. A method for finding the Lot-for-Lot solution is developed using the concept of a truncated monotonically non-decreasing time function, generalising the approach of the generalised Leontief inverse and instrumental for designing plans meeting the necessary inner-corner requirement for optimality. Also the definition of inventory-related costs needs a modification for this concept to be consistent with NPV. These findings are applied in an extensive numerical example. Immediate future research concerns investigating principles for the optimal removal of initial backlogs.