Introduction to Inventory Control
Inventory control is vital for any organization as it involves managing a company’s inventory effectively to balance the cost of holding inventory with the cost of ordering. The chapter outlines various inventory control models and techniques to determine optimal ordering quantities and reorder points, helping businesses minimize total inventory costs.

Importance of Inventory Control
Inventory control serves several functions:

• Decoupling Function: Inventory acts as a buffer between different stages of production, allowing processes to operate independently and preventing delays.
• Storing Resources: Inventory allows companies to store raw materials, work-in-progress, and finished goods to meet future demands.
• Managing Irregular Supply and Demand: Companies can maintain inventory to cover periods of high demand or when supply is uncertain.
• Quantity Discounts: Large orders can reduce per-unit costs, but also increase carrying costs.
• Avoiding Stockouts and Shortages: Ensures customer demand is met without running out of stock, which can damage customer trust and lead to lost sales.

Key Inventory Decisions
Two fundamental decisions in inventory control are:

• How much to order: Determining the optimal order size.
• When to order: Determining the optimal time to place an order to minimize the risk of stockouts while reducing carrying costs.

Economic Order Quantity (EOQ) Model
The EOQ model is a widely used inventory control technique that determines the optimal order quantity that minimizes the total cost of inventory, including ordering and holding costs. The EOQ model assumes:

1. Constant Demand: The demand for the inventory item is known and constant.
2. Constant Lead Time: The lead time for receiving the order is known and consistent.
3. Instantaneous Receipt of Inventory: The entire order quantity is received at once.
4. No Quantity Discounts: The cost per unit does not vary with the order size.
5. No Stockouts: There are no shortages or stockouts.
6. Constant Costs: Only ordering and holding costs are variable.

The EOQ formula is given by:

$$
Q^* = \sqrt{\frac{2DS}{H}}
$$

where:

• (Q^*) = Economic Order Quantity (units)
• (D) = Annual demand (units)
• (S) = Ordering cost per order
• (H) = Holding cost per unit per year

Reorder Point (ROP)
The reorder point determines when an order should be placed based on the lead time and the average daily demand. It is calculated as:

$$
\text{ROP} = d \times L
$$

where:

• (d) = Demand per day
• (L) = Lead time in days.

EOQ Without Instantaneous Receipt Assumption
For situations where inventory is received gradually over time (such as in production scenarios), the EOQ model is adjusted to account for the rate of inventory production versus the rate of demand. The optimal production quantity is given by:

$$
Q^* = \sqrt{\frac{2DS}{H} \left(1 – \frac{d}{p}\right)}
$$

where:

• (d) = Demand rate
• (p) = Production rate.

Quantity Discount Models
These models consider cases where suppliers offer a lower price per unit when larger quantities are ordered. The objective is to determine whether the savings from purchasing in larger quantities outweigh the additional holding costs.

Use of Safety Stock
Safety stock is additional inventory kept to guard against variability in demand or supply. It is used to maintain service levels and avoid stockouts. The safety stock level depends on the desired service level and the variability in demand during the lead time.

Single-Period Inventory Models
This model is used for products with a limited selling period, such as perishable goods or seasonal items. The objective is to find the optimal stocking quantity that minimizes the costs of overstocking and understocking. The model often uses marginal analysis to compare the marginal profit and marginal loss of stocking one additional unit.

ABC Analysis
ABC analysis categorizes inventory into three classes:

• Class A: High-value items with low frequency of sales (require tight control).
• Class B: Moderate-value items with moderate frequency.
• Class C: Low-value items with high frequency (less control needed).

Just-in-Time (JIT) Inventory Control
JIT aims to reduce inventory levels and holding costs by receiving goods only as they are needed in the production process. This approach reduces waste but requires precise demand forecasting and reliable suppliers.

Enterprise Resource Planning (ERP)
ERP systems integrate various functions of a business, including inventory, accounting, finance, and human resources, into a single system to streamline operations and improve accuracy in decision-making.

Math Problem Example: EOQ Calculation

Let’s consider a company with the following inventory parameters:

• Annual demand (D): 10,000 units
• Ordering cost (S): $50 per order
• Holding cost (H): $2 per unit per year

To calculate the EOQ:

$$
Q^* = \sqrt{\frac{2 \times 10,000 \times 50}{2}} = \sqrt{500,000} = 707 \text{ units}
$$

This means the company should order 707 units each time to minimize total inventory costs.