Inventory Management - Reorder Point (ROP) Model

Inventory Management

Inventory management includes stock, warehouse management. There are two main decisions: how much to order and when to order.

Decisions

1. How much to order: Techniques include EOQ (Economic Order Quantity) and quantity discount models.

2. When to order: This involves the ROP (Reorder Point) technique.

Reorder Point (ROP) Model

The ROP model addresses the decision of when to order. It's the inventory level at which a new order should be placed. In an ideal scenario (EOQ), the reorder point would be zero, meaning an order is placed when the inventory level reaches zero, and the new order arrives exactly when the inventory is depleted.

Lead Time

Lead time is the time it takes for a new order to arrive after it is placed. The reorder point must account for the demand during this lead time. For example, if the lead time is two days, you must place a new order when you have enough product to cover the demand for those two days.

Calculation

The basic rule for calculating the reorder point is:

$\text{Reorder Point} = \text{Lead Time} \times \text{Daily Demand}$

Cases: Certainty vs. Uncertainty

The ROP calculation differs based on whether the demand and lead time are certain or uncertain.

• Under Certainty: If the demand is fixed and known with 100% certainty, the ROP is simply the lead time multiplied by the daily demand.

  $\text{ROP} = \text{Lead Time} \times \text{Daily Demand}$

• Under Uncertainty: When there is variability in demand or lead time, a safety stock is needed to account for the uncertainty.

  • Variability in demand or lead time introduces the possibility of a shortage (stock out), where demand exceeds available supply.

  • Safety stock is extra inventory held to mitigate the risk of stock outs due to variable demand or lead time.

ROP Under Uncertainty

The general formula for ROP under uncertainty is:

$\text{ROP} = (\text{Lead Time} \times \text{Daily Demand}) + \text{Safety Stock}$

Models for Uncertainty

There are three models to handle uncertainty:

1. Model 1: Demand is variable, but lead time is fixed.

2. Model 2: Demand is fixed, but lead time is variable.

3. Model 3: Both demand and lead time are variable (uncertain).

Model 1: Variable Demand, Fixed Lead Time

In this model, the ROP is calculated as:

$\text{ROP} = (\text{Daily Demand} \times \text{Lead Time}) + \text{Safety Stock}$

The safety stock is determined by:

$\text{Safety Stock} = z \times \text{Standard Deviation of Demand} \times \sqrt{\text{Lead Time}}$

Where:

• $z$ is the Z-score corresponding to the desired service level.

• $\text{Standard Deviation of Demand}$ represents the variability in demand.

Model 2: Fixed Demand, Variable Lead Time

In this model, the ROP is calculated considering the variability in lead time:

$\text{Safety Stock} = z \times \text{Daily Demand} \times \text{Standard Deviation of Lead Time}$

The ROP is determined by:

$\text{ROP} = (\text{Daily Demand} \times \text{Lead Time}) + \text{Safety Stock}$

Model 3: Variable Demand and Variable Lead Time

This is the most complex scenario, where both demand and lead time are uncertain. The safety stock calculation involves both the standard deviation of demand and the standard deviation of lead time.

$\text{Safety Stock} = z \times \sqrt{(\text{Lead Time} \times \text{Standard Deviation of Demand}^2) + (\text{Daily Demand}^2 \times \text{Standard Deviation of Lead Time}^2)}$

And:

$\text{ROP} = (\text{Daily Demand} \times \text{Lead Time}) + \text{Safety Stock}$

Service Level and Stock Out Risk

The service level represents the probability of not stocking out during the lead time. It is directly related to the stock out risk:

$\text{Service Level} = 100\% - \text{Stock Out Risk}$

For example, a 90% service level means there is a 10% risk of stocking out. The Z-score (z) is obtained from the normal distribution table based on the desired service level.

Example

A restaurant uses an average of 50 jars of special sauce each week. The weekly usage has a standard deviation of three jars. The lead time is two weeks, and the manager is willing to accept no more than a 10% risk of stock out. Determine the ROP and safety stock.

• Demand (average weekly usage): 50 jars

• Standard deviation of demand: 3 jars

• Lead time: 2 weeks

• Stock out risk: 10%

• Service level: 90%

• Z-score for 90% service level: 1.28

Since the demand is variable and lead time is constant, we use Model 1.

$\text{ROP} = (50 \times 2) + (1.28 \times 3 \times \sqrt{2}) = 100 + 5.4 = 106 \text{ units}$

$\text{Safety Stock} = 1.28 \times 3 \times \sqrt{2} = 5.4 \text{ units}$

Example

A model replaces broken glasses at a rate of 25 per day. This quantity has tended to vary normally with a standard deviation of three glasses per day. Lead time is normally distributed with a mean of ten days and a standard deviation of two days. The service level is 95%. Determine the ROP and safety stock.

• Demand (daily): 25 glasses

• Standard deviation of demand: 3 glasses

• Lead time (average): 10 days

• Standard deviation of lead time: 2 days

• Service level: 95%

• Z-score for 95% service level: 1.65

Since both demand and lead time are variable, we use Model 3.

$\text{ROP} = (25 \times 10) + 1.65 \times \sqrt{(10 \times 3^2) + (25^2 \times 2^2)}$

$\textbf{Safety Stock} = 1.65 \times \sqrt{(10 \times 9) + (625 \times 4)} = 1.65 \times \sqrt{90 + 2500} \approx 1.65 \times \sqrt{2590} \approx 1.65 \times 50.89 \approx 83.97$

Therefore,

$\text{ROP} = 250 + 83.97 \approx 334$

$\text{Safety Stock} \approx 84$