# 11.4.4 Extensions of the EOQ Formula: Lead-Time Orientation and Discount Levels

### Intended learning outcomes: Present lead-time-oriented batch sizing. Describe batch sizing considering discount levels.

1. Lead-time-oriented batch sizing is a generalization of the simplified approach using the EOQ formula for production, taking the cost of work in process into consideration.

As a complement to the variables in Figure 11.4.2.1, we add the variables shown in Figure 11.4.4.1. Most of these data come from the route sheet.

The EOQ has results according to the formula given in Figure 11.4.4.2. For details of the derivation, see [Nyhu91], p. 103. The denominator under the radical is significantly larger than the one in classic batch sizing only for a long manufacturing lead time.

Fig. 11.4.4.2       Lead-time-oriented batch sizing: determination of the minimum.

2. Batch sizing considering discount levels is a generalization of the simplified approach using the EOQ formula.

Figure 11.4.4.3 illustrates the decreasing batch-size-dependent unit costs as a function of the lot size, as well as the resulting total costs curves.

Fig. 11.4.4.3       Total costs curves, taking discount levels into consideration.

Batch-size-dependent unit costs CU are dependent on the purchased quantity. This is particularly valid for procured goods.

A quantity discount is a price reduction allowance on orders over a certain minimal order quantity or value.

For example, a supplier may offer a quantity discount for the whole order quantity with three discount levels; that is, reduced unit costs CU2 as soon as the quantity exceeds Xm2, CU3 as soon as the quantity exceeds Xm3, CU4, as soon as the quantity exceeds Xm4.

Every total costs curve for the various values of cost per piece demon­strates a minimum within the range of its validity. This is either the minimum of the corresponding total costs curve (X02 in Figure 11.4.4.3), or it lies on the border of a discount level curve (Xm3 in Figure 11.4.4.3). If dis­counts are not large, we may also argue that the batch sizes for the diffe­rent discount levels according to the EOQ formula will lie very close to each other. We may thus calculate the optimum batch size by selecting a partic­ular mean cost per piece, and then rounding it up to the next discount level.

A similar line of thinking is followed when evaluating economic efficiency and batch siz­ing in the case of alternative (less expensive) production processes using larger batch sizes.

Continuation in next subsection (11.4.4b).

## 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, and Setup and Ordering Costs

Intended learning outcomes: Differentiate between batch-size-dependent production or procurement costs and batch-size-independent production or procurement costs.

• ##### 11.4.1b Carrying Cost

Intended learning outcomes: 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: The Classic Economic Order Quantity (EOQ)

Intended learning outcomes: Explain the concept of the economic order quantity (EOQ). Explain variables for the EOQ formula.

• ##### 11.4.2b The Economic Order Quantity (EOQ) Formula

Intended learning outcomes: Explain the economic order quantity (EOQ) formula. Describe the cost curves as a function of batch size.

• ##### 11.4.2c Optimum Length of Order Cycle

Intended learning outcomes: Present the optimum length of order cycle.

• ##### 11.4.3 Economic Order Quantity (EOQ) Formula: Sensitivity Analysis

Intended learning outcomes: Present in detail the sensitivity analysis of the EOQ calculation.

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

Intended learning outcomes: 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 EOQ Formula: Lead-Time Orientation and Discount Levels

Intended learning outcomes: Present lead-time-oriented batch sizing. Describe batch sizing considering discount levels.

• ##### 11.4.4b Extensions of the EOQ Formula: Joint Replenishment

Intended learning outcomes: Produce an overview on joint replenishment: kit materials management and collective materials management.