# 11.4.3 Economic Order Quantity (EOQ) and Optimum Length of Order Cycle in Practical Application

### Intended learning outcomes: Present in detail the sensitivity analysis of the EOQ calculation. Produce an overview on the practical implementation of the EOQ formula. Identify several factors that influence a maximum or minimum order quantity.

Unfairly, the EOQ formula has recently been held responsible for large batches. However, a closer look at practice reveals that the formula was often used with carrying cost unit rates that were much too low, or it was applied to deterministic materials management, for which other techniques are better suited (see Section 12.4).

In any case, the EOQ formula basically provides “only” an order of magnitude, not a precise number. The total costs curve shown in Figure 11.4.2.5 is very flat in the region of the minimum, so that deviations from the optimum batch size have only a very small effect on costs. The following sensitivity analysisshows this “robust” effect. Beginning with a quantity deviation as given in Figure 11.4.3.1, and the fact that the formula in Figure 11.4.3.2 holds for the optimum batch size X0, the cost deviation formula is shown in Figure 11.4.3.3.

Fig. 11.4.3.1       Sensitivity analysis: quantity deviation.

Fig. 11.4.3.2       Sensitivity analysis: carrying cost rates for optimum batch size.

Fig. 11.4.3.3       Sensitivity analysis: cost deviation.

For example, a cost deviation of b = 10% results for v = 64% as well as for v = 156%, which means that the relationship shown in Figure 11.4.3.4 is valid:

Fig. 11.4.3.4       Sensitivity analysis: quantity deviation given a cost deviation of 10%.

This sensitivity analysis reveals the surprising robustness of the calculation technique, which indeed rests on very simplified assumptions. Extending batch size formulas to include additional influencing factors produces an improvement in results that is practically relevant only in special cases. In any event, we may round off the calculated batch size, adapt it to practical considerations, and, in particular, make it smaller if a shorter lead time is desirable.

This robustness increases even further if we include not only C2 and C3, but also the actual costs of production or procurement C1 in the division for b given in Figure 11.4.3.3. If C1 is much larger than C2 + C3 — which is usually the case — even bigger changes to batch size do not have a strong effect on the total production or procurement costs.

In a similar way, we can show that errors in determining setup and ordering costs, the carrying cost rates, or the annual consumption in the cost deviations make as little difference as a quantity deviation does. Among other things, the EOQ formula is thus not very sensitive to systematic forecast errors. This means that very simple forecasting techniques, such as moving average value calculation, will generally suffice when determining batch sizes.

In the case of produced items, the reduction in costs for in-process inventory achieved through smaller batches is thus negligible in most cases. More significant is the fact that smaller batches may lead to shorter lead time. In addition to this improvement in the target area of delivery, there are also positive effects in the target areas of flexibility and costs. The positive effects discussed in Chapter 6 are lacking in the classic EOQ formula. However, as we will show in Section 13.2, smaller batches only result in shorter lead time if — on the one hand — the run time is long in relation to the lead time, particularly in line production (in classic job shop production this pro­portion is likely to be of the order of magnitude of 1:10 and less), and — on the other hand — the saturation of a work center does not have the effect of creating longer queues for the entire collection of batches.

Thus, the longer the run times — often required when much value is added — the higher the costs for goods in process are. In such cases we should choose rather lower values for batch sizes than those recommended on the basis of the EOQ formula (see also lead-time-oriented batch sizing in Section 11.4.4). For work-intensive operations especially, shorter operation times can contribute to harmonizing the content of work, which in turn leads to a further reduction in wait times, and thus lead times, as explained in Section 13.2.2. As Figure 6.2.6.2 illustrated, at lower production structure levels a reduction in lead time is likely to result in lower safety stocks, and thus cost savings. If for some reason storage is not possible at all, shorter lead times can even achieve additional sales.

A practical implementation scheme, which takes both total costs and short lead time into account, is provided in Figure 11.4.3.5.

Fig. 11.4.3.5       Practical implementation of the EOQ formula.

The minimum order quantity (or maximum order quantity) is an order quantity modifier, applied after the lot size has been calculated, that increases (or limits) the order quantity to a pre-established minimum (or maximum) ([APIC16]).

Differentiated considerations concerning the minimum and maximum order quantity can be found in Figure 11.4.3.6, as, for example, related to item groups or even individual items.

Fig. 11.4.3.6       Several factors that influence a maximum or minimum order quantity.

In the literature, there are models that take more operating conditions into consideration. We will present several of these in Section 11.4.4. Because of its simplicity, however, the EOQ formula is used frequently in current practice. Even if the simplified model assumpti­ons that underlie it are not given in the concrete case, the formula is very robust in the face of such deviations, as we have shown. Before applying a more complicated calcula­tion method, materials management should clarify whether the more costly batch size determination truly offers crucial advantages over the simple implementation considerations outlined above.