11.4.2 Optimum Batch Size and Optimum Length of Order Cycle: The Classic Economic Order Quantity (EOQ)

Intended learning outcomes: Explain economic order quantity (EOQ), variables for the EOQ formula and the EOQ formula. Describe the cost curves as a function of batch size. Present the optimum length of order cycle.

Most methods for determining batch sizes minimize the expected total costs. In dependency upon batch size, these are essentially composed of the costs mentioned in Section 11.4.1:

1. Batch-size-dependent unit costs. Mostly the price per produced or procured unit quanti­ty does not change with increasing batch size. However, this is not true in case of allo­wance for discounts or changes in the production process from a certain batch size upward.
2. Inventory costs. These are all the costs incurred in connection with ordering and holding inventory. Thus, inventory costs are the following costs:
a.    Setup and ordering costs: These are incurred only once per production or procurement event. In the simplest and most common case, they are independent of the batch size. Thus, the larger the batch size, the smaller is the share in such costs that accrues to each unit. However, there may be an upward jump in costs if a certain batch size requires the choice of another production procurement structure (such as a different machine or means of transport).
b.    Carrying cost: With increasing batch size, the average physical inventory in­creases, together with carrying cost. For the sake of simplicity, these costs are often set as proportional to batch size, that is, proportional to the value of goods in storage. As was shown in Section 11.4.1, this is only valid provided that the following restrictions hold: Firstly, the carrying cost must be independent of the storage duration. Secondly, an entry in stock only occurs following the issue of the last piece. Issues occur regularly along the time axis. Thus, if X is the batch size, on average, X/2 pieces are in stock. Thirdly, there must be sufficient warehouse space. This means that the size of the batch does not necessitate new installations.

In the simplest case, application of these principles leads to the so-called economic order quantity.

The economic order quantity (EOQ), or the optimum batch size, or the economic lot size, is the optimal amount of an item to be purchased or manufactured at one time.

The economic order quantity is calculated with respect to a particular planning period, such as one year. The variables for its calculation are listed in Figure 11.4.2.1.

Fig. 11.4.2.1       Variables for the EOQ formula.

The equation for calculating total costs is shown in Figure 11.4.2.2.

Fig. 11.4.2.2       EOQ formula: total costs equation.

Since the objective is to minimize the total costs, the target function is as shown in Figure 11.4.2.3.

Fig. 11.4.2.3       EOQ formula: target function.

The economic order quantity X0 is the lot size with the minimum of total costs, and it results from deriving the target function and setting it to zero, as shown in Figure 11.4.2.4.

`EOQ (economic order quantity) formula is another name for the X0 formula.`

Fig. 11.4.2.4       EOQ formula: determining the optimum batch size.

Figure 11.4.2.5 shows the cost curves that correspond to the values for C1, C2, C3, and CT as a function of batch sizes.

Fig. 11.4.2.5       Cost curves as a function of batch size.

These cost curves are typical of the EOQ formula. The minimum point for total costs lies exactly at the intersection of the curves for setup and ordering costs and carrying cost.

Exercise: Get used to the EOQ calculation by chosing different values for the parameters.

Instead of an optimum batch size, we can also calculate an optimal time period for which an order or a batch covers demand.

The optimum order interval or optimum length of order cycleis an optimum period of time for which future demand should be covered.

This length is defined according to the formula in Figure 11.4.2.6. From this formula, it is immediately apparent that the optimum length of the order cycle — and the optimum batch size in Figure 11.4.2.4 — rises less than proportionally with increasing setup costs, and declines less than propor­tionally with increasing turnover. Thus, for example, if we set the value for the root of (2 × CS/p) at 40, the characteristic figures for optimum length of order cycle as a function of the value of turnover are those in Figure 11.4.2.7.

Fig. 11.4.2.6       Optimum length of order cycle.

Fig. 11.4.2.7       Sample characteristic figures for length of order cycle as a function of the value of turnover.

Unless we can reduce setup costs decisively, a very large length of order cycle will result in low turnover. In practice, however, when the range of demand coverage is very long, the depreciation risk increases disproportion­ally. For this reason, upward limits are set for the length of the order cycle, and thus as well for the batch sizes, for items with a small turnover. This is, incidentally, the simplest and most common method in practice to control nonlinear patterns of carrying cost: for example, carrying cost that jumps steeply when inventory exceeds a particular volume. The conside­ration of the length of order cycle is also an important batch-sizing policy in deterministic materials management (see Section 12.4).

Course section 11.4: Subsections and their intended learning outcomes

• 11.4 Batch Sizing, or Lot Sizing

Intended learning outcomes: Produce an overview on production or procurement costs, batch-size-dependent unit costs, setup and ordering costs, and carrying cost. Explain optimum batch size, optimum length of order cycle, the classic economic order quantity formally and in practical application. Disclose extensions of the batch size formula.

• 11.4.1 Production or Procurement Costs: Batch-Size-Dependent Unit Costs, Setup and Ordering Costs, and Carrying Cost

Intended learning outcomes: Differentiate between batch-size-dependent production or procurement costs and batch-size-independent production or procurement costs. Explain carrying cost and carrying cost rate. Produce an overview on costs of financing or capital costs, storage infrastructure costs and the risk of depreciation.

• 11.4.2 Optimum Batch Size and Optimum Length of Order Cycle: The Classic Economic Order Quantity (EOQ)

Intended learning outcomes: Explain economic order quantity (EOQ), variables for the EOQ formula and the EOQ formula. Describe the cost curves as a function of batch size. Present the optimum length of order cycle.

• 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.

• 11.4.4 Extensions of the Economic Order Quantity (EOQ) Formula

Intended learning outcomes: Present lead-time-oriented batch sizing. Describe batch sizing considering discount levels. Produce an overview on joint replenishment: kit materials management and collective materials management.