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## Recent Developments in MRP Theory - and Future Opportunities

Wednesday 5 May 2010, 15:00

LT11, Management School

### Robert W. Grubbström FVR RI

Linköping Institute of Technology, SE-581 83 Linköping, Sweden

Mediterranean Institute for Advanced Studies, SI- 5290, Šempeter pri Gorici, Slovenia

robert@grubbstrom.com

Robert Grubbström is also editor of the International Journal of Production Economics and a visiting professor in the Management Science Department

**Abstract:** Material Requirements Theory (MRP Theory) combines the use of Input-Output Analysis and Laplace transforms, enabling the development of a theoretical background for multi-level, multi-stage production-inventory systems together with their economic evaluation, in particular applying the Net Present Value (NPV) principle. The time scale may be continuous (“bucketless” in MRP terminology) or discrete. Since the average cost measure may be viewed as an approximation of NPV, the theory may be applied to cases, when average costs are preferred as the objective function.

Central in this theory are the fundamental equations explaining the time development of available inventory, allocated component stock (allocations), and backlogs. These are balance equations, in which the generalised input matrix plays a predominant rôle. The input matrix (next assembly quantity matrix) from Input-Output Analysis describes the Bill-Of-Materials in terms of amounts of materials and subcomponents needed on each level in the product structures. In MRP lead times are taken into account, describing the advanced timing these amounts are needed in comparison to the time of completion. Assuming the lead times to be constants, as is the case in classical MRP, using Laplace transforms, they may be modelled as operators in a diagonal lead time matrix. Multiplying the lead time and input matrices, results in the generalised input matrix, which captures all requirements as well as the needs of their advanced timing. The vector-valued balance equation for available inventory (total inventory less earmarked materials reserved for use according to the Master Production Schedule, MPS) then become very compact and captures any MRP case with arbitrary product structures and arbitrary MPS.

Also services in the sense of capacity requirements may be included as amounts in the input matrix by introducing a further row for each type of capacity. The generalised input is then extended accordingly, taking into consideration when capacities are needed compared to the completion dates.

A basic constraint, when formulating the problem of optimising the MPS, is the available inventory constraint requiring that available inventory never may be negative if the MPS is to be feasible. A corresponding constraint concerns capacities, whenever capacities are assumed limited. If backlogs are allowed in the problem, these only concern items which have an external demand from outside customers. Internally demanded items may never be backlogged.

Recently, MRP Theory has been extended in new directions. Hitherto, it has mainly dealt with assembly structures by which items produced downstream contain one or more subitems on lower levels, but at each stage, the assembly activity produces only one type of output. The material flow is thus convergent. Attention has now turned to arborescent systems, in which one input may create more than one type of output. Here the material flow becomes divergent. Typical examples of arborescent systems are transportation (distribution) and recycling.

For arborescent systems the input time, rather than the completion time, will be the natural reference of each process. Then outputs will be available according to a planned delay after input time. Introducing a new diagonal matrix, the output delay matrix, with operators as elements in its diagonal corresponding to these delays, a generalised output matrix is formed capturing output amounts as well as their delayed timing. Further generalisations of the input- and output matrices make it possible to cover cases when different inputs into one process are needed at different lead times ahead of completion, or when the same type of output from different processes are available after different delays for each process, respectively.

As a second example, we choose the classical problem of optimal dynamic lotsizing in MRP. The optimal batch sizes of various produced amounts and their timing are to be determined. New formulations within MRP Theory, using a binary representation of the decision variables (emanating from the Triple Algorithm), has enabled the development of optimal lotsize solution procedures for any deterministic assembly system. Since lead times are present in this class of problems, they also cover the so called Multi-Level Lotsizing problem (MLLS), in which all lead times are assumed zero.

This presentation intends to provide an historical overview over the progress of MRP

Theory, highlighting certain recent results, and pointing at future opportunities for analysing more general sets of problems.

**Selected references**

[1] Grubbström, R.W., Transform Methodology Applied to Some Inventory Problems,

Zeitschrift für Betriebswirtschaft, 77(3), 2007, 297-324.

[2] Grubbström, R. W., Bogataj, L., (Eds), Input-Output Analysis and Laplace Transforms in Material Requirements Planning. Storlien 1997. FPP, Portorož, 1998.

[3] Grubbström, R. W., Bogataj, L., Bogataj, M., A Compact Representation and

Optimisation of Distribution and Reverse Logistics in the Value Chain, WP,

(Mathematical economics, operational research and logistics, 5), Ljubljana: Faculty of Economics, KMOR, 2007.

[4] Grubbström, R. W., Bogataj, M., Bogataj, L., Optimal Lotsizing within MRP Theory, Preprints of the 13th IFAC Symposium on Information Control Problems in

Manufacturing (INCOM'09), Moscow, June 3-5, 2009, 15-30. Also in Annual Reviews

in Control, 34, 2010, Article in Press.

[5] Grubbström, R. W., Molinder, A., Further Theoretical Considerations on the

Relationship between MRP, Input-Output Analysis and Multi- Echelon Inventory

Systems, International Journal of Production Economics, 35, 1994, 299-311.

[6] Grubbström, R. W., Ovrin, P. Intertemporal Generalization of the Relationship between Material Requirements Planning and Input- Output Analysis, International Journal of Production Economics, 26, 1992, 311-318.

[7] Grubbström, R.W., Tang, O., An Overview of Input-Output Analysis Applied to

Production-Inventory Systems, Economic Systems Review, 12, 2000, 3-25.

[8] Grubbström, R.W. Wang, Z., A Stochastic Model of Multi-Level/Multi-Stage

Capacity-Constrained Production-Inventory Systems, International Journal of

Production Economics, 81-82, 2003, 483-494.

Key words: Material requirements planning, MRP, Laplace transform, input–output analysis, net present value.