# Dynamic Lotsizing with Random Demand in an MRP System

Standard software systems following the MRP (material requirements planning) planning environment perform the calculation of derived demand (demand explosion calculus) based on deterministic demand data. Consider the following example, where the production plan for a selected product is shown. According to the MRP approach, the gross requirements, which are assumed deterministic, have been translated into net requirements which in turn have been combined to production lot sizes.

 Period 1 2 3 4 5 6 7 8 9 10 Gross requirements 77 42 38 21 26 112 45 14 76 39 Net requirements 77 42 38 21 26 112 45 14 76 39 Additional requirements - - - - - - - - - - Lot sizes 204 - - - - 171 - - 115 - Planned order receipts(*) 204 - - - - 171 - - 115 - Inventory on hand(*) 127 85 47 26 - 59 14 - 39 -

If the demand is deterministic, immediately before the production of a new lot the inventory on hand is equal to 0. However, if gross requirements are random, then the risk of backorders exists. Assume that the period demands follow a normal distribution with the above-given gross requirements as expected values and a coefficient of variation (=standard deviation/mean) of 0.3.

For the given production plan, the following development of the system will be observed:

 Demand Expected t Mean Std.-Dev. Lotsize Stock on hand Backorders 1 77 23.1 204 127 0 2 42 12.6 0 85.0044 0.0043 3 38 11.4 0 47.6076 0.6033 4 21 6.3 0 29.0276 2.42 5 26 7.8 0 12.1193 9.0917 6 112 33.6 171 61.0513 2.0513 7 45 13.5 0 26.6776 10.6263 8 14 4.2 0 18.9307 6.2531 9 76 22.8 115 46.0157 6.8955 10 39 11.7 0 21.5149 14.4992

The last column shows the backorders that newly occurred in each period. Over the ten periods the fillrate is 89.37%. In order to reduce the amount of backorders, in the literature and from SCM software vendors it is proposed to include safety stock in the MRP calculations. In this case, the available inventory is reduced by the safety stock which results in an increase of the net requirements. With these changed net demands, the lotsizes are calculated.

Interestingly, most textbooks on Operations Management include a chapter on MRP, but omit the problems associated with uncertainty of demand in this planning environment. Moreover, SCM (APS, MRP) software vendor often use false concepts to account for unceratinty.

One proposal is to set the safety stock as a multiple of the average demand. In the above example, the average demand is 49 units. Let's set the safety stock to 49 and observe the behaviour of the system. As at the beginning of the planning horizon there is no inventory available, the net demand increases by 49 to 126. Consequently, the optimum production plan has changed, as the lotsize in period 1 is increased by 49. If the demands were deterministic, then immediately before a replenishment the stock on hand is 49.

 Period 1 2 3 4 5 6 7 8 9 10 Inventory on hand - 49 49 49 49 49 49 49 49 49 Available inventory -49 - - - - - - - - - Gross requirements 77 42 38 21 26 112 45 14 76 39 Net requirements 126 42 38 21 26 112 45 14 76 39 Additional requirements - - - - - - - - - - Lot sizes 253 - - - - 171 - - 115 - Planned order receipts(*) 253 - - - - 171 - - 115 - Inventory on hand(*) 176 134 96 75 49 108 63 49 88 49

In this case the development of the system is as follows:

 Demand Expected t Mean Std.-Dev. Lotsize Stock on hand Backorders 1 77 23.1 253 176 0 2 42 12.6 0 134 0 3 38 11.4 0 96.003 0.003 4 21 6.3 0 75.0496 0.0466 5 26 7.8 0 49.6847 0.6351 6 112 33.6 171 108.1292 0.1292 7 45 13.5 0 65.0058 1.8766 8 14 4.2 0 52.714 1.7082 9 76 22.8 115 89.0315 1.0282 10 39 11.7 0 54.3316 4.3

The overall fillrate is now 98.01%. Which of course is much higher. However, using a multiple of the average demand as a safety stock norm is false, as uncertainty is not associated with the mean demands, but with its variation. If the coefficient of variation of the demand would be, say 0.1, then according to the above stock norm the safety stock would not change. However, the system behaviour would be as follows:

 Demand Expected t Mean Std.-Dev. Lotsize Stock on hand Backorders 1 77 7.7 253 176 0 2 42 4.2 0 134 0 3 38 3.8 0 96 0 4 21 2.1 0 75 0 5 26 2.6 0 49 0 6 112 11.2 171 108 0 7 45 4.5 0 63.0001 0.0001 8 14 1.4 0 49.0043 0.0041 9 76 7.6 115 88 0 10 39 3.9 0 49.0175 0.0175

Obviously, with an uncertainty this low, almost no backorders occur and the overall fillrate is 100%. If the fillrate targeted by the management is less than 100%, then inventory levels and holding costs are too high.

Other proposals include the use of methods that basically originate in the stochastic inventory theory developped for the case of stationary demand. These methods are also not appropriate, as they neglect the dynamic character of the (random) demand.

Note that whatever method for calculating an external safety stock is used, the observed fillrate is uncontrollable.

In addition, there is a second problem: it is not taken into consideration that the lot sizes have an impact on the absorption of risk. For example, with large lot sizes, the safety stock required to guarantee a target service level may be zero or even negative. As a consequence, the target service level which should be the basis for the safety stock calculation is met only by chance.

As a systematic planning procedure, a stochastic lotsizing model can be applied when the demand is dynamic and random. Thereby simultaneously randomness and dynamic conditions are considered. In the above example, for a coefficient of variation and a target fillrate of 95% the following production plan would be optimal.

 Demand Expected t Mean Std.-Dev. Lotsize Stock on hand Backorders 1 77 23.1 208 131.0586 0 2 42 12.6 0 89.0611 0.0024 3 38 11.4 0 51.4885 0.4275 4 21 6.3 0 32.396 1.9075 5 26 7.8 0 14.2566 7.8607 6 112 33.6 298 190.4114 0.0001 7 45 13.5 0 145.425 0.0136 8 14 4.2 0 131.4513 0.0263 9 76 22.8 0 59.3759 3.9246 10 39 11.7 0 30.7091 10.3332

In the case of a coefficcient of variation of 0.1 we get:

 Demand Expected t Mean Std.-Dev. Lotsize Stock on hand Backorders 1 77 7.7 195 117.8024 0 2 42 4.2 0 75.8024 0 3 38 3.8 0 37.8025 0.0001 4 21 2.1 0 16.9741 0.1717 5 26 2.6 0 1.0016 10.0275 6 112 11.2 284 162.5589 0 7 45 4.5 0 117.5589 0 8 14 1.4 0 103.5589 0 9 76 7.6 0 27.9961 0.4372 10 39 3.9 0 2.8565 13.8604

In both cases the production plan is cost-optimal with respect to a given fillrate of 95%.

The above calculations have been performed with the educational software Production Management Trainer.