# Single Period Inventory Models: The Ultimate Guide in 2023

4.5 Minutes

## What are Single Period Inventory Models?

Single-period inventory models are often referred to as newsboy models. They are inventory strategies used for items that have a one-time selling opportunity such as perishable goods or seasonal items. They aim to determine the optimal order quantity that balances the costs of underordering and overordering for a single sales period.

## Importance of Single Period Inventory Models

The single period inventory model is crucial for a number of reasons. Some of the most common include:

### 1. Minimization of Costs:

The primary objective of the single period inventory model is to determine the optimal order quantity that minimizes the total costs. These costs can include overordering costs and underordering costs which ensure businesses don't overspend on excess inventory or miss potential sales.

### 2. Enhanced Customer Satisfaction:

By optimizing inventory levels, businesses can better meet customer demand. This reduces instances of stockouts or excessive waiting times for products which leads to improved customer satisfaction and loyalty.

### 3. Ideal for Perishable Items:

For products with a limited shelf life, like fresh produce or fashion items, it's crucial to get the order quantity right. The single period inventory model helps companies optimize their orders for such products. It reduces waste and ensures product freshness for the end customer.

### 4. Facilitates Decision Making Under Uncertainty:

Businesses often face uncertain demand due to various unpredictable factors. The model provides a structured way to make inventory decisions under such uncertainties as it relies on historical data, costs, and other relevant factors.

## Types of Single Period Inventory Models

While there are a number of inventory models used across industries, here are some of the most common single period inventory models:

### 1. Newsvendor Model:

Optimizes order quantity for perishable goods or items with a short selling season to minimize costs from overordering and underordering.

Calculation:

Where:

• P(D ≤ Q*): Probability that demand (D) is less than or equal to the optimal order quantity (Q*).
• Cu: Cost of underordering a unit.
• Co: Cost of overordering a unit.

Example:

Given:

Cu= \$10 and Co = \$5.

Calculation: Optimal probability P(D ≤ Q*) =[(10)/(10 + 5)] = 0.67  or 67%.

### 2. Partial Backordering:

Adjusts for situations where part of unmet demand can be back ordered. This is useful when temporary stockouts are acceptable.

Calculation: Specifics vary, but it's about balancing costs of underordering, overordering, and back ordering.

Example: An item has different associated costs. Based on historical data, an order quantity that allows 10% back ordering minimizes overall costs.

### 3. Service Level Approach:

Determines order quantity based on a set service level or fill rate. This is ideal for businesses prioritizing customer satisfaction through product availability.

Calculation:

Where:

• Forecasted Demand: The predicted demand for the product over a period.
• Safety Stock: Additional inventory kept as a buffer against uncertainties.

Example:

Given: Forecasted Demand = 400 units and Safety Stock = 50 units

Calculation: Order Quantity = 400 + 50 = 450 units.

### 4. Marginal Analysis:

Finds order quantity where the benefit of one more unit matches the cost of that unit. This is used in situations with noticeable cost/revenue shifts from small order quantity changes.

Calculation: Principle is Marginal Profit = Marginal Cost.

Where:

• Marginal Profit: Additional profit from selling one more unit.
• Marginal Cost: Additional cost of ordering and potentially not selling one more unit.

Example: By analyzing costs and revenues, it's found that ordering 510 units maximizes the profit since the profit from the 511th unit matches its cost.

### 5. Profit Maximization Approach:

Seeks the order quantity to maximize expected profit over a selling season. This accounts factors like salvage value and costs of shortages.

Calculation:

Where:

• Selling Price: The price at which the product is sold to customers.
• Expected Sales: Predicted number of units to be sold.
• Purchase Cost: Cost of purchasing one unit of product.
• Order Quantity: Number of units ordered.
• Salvage Value: Value recovered from unsold units.
• Expected Unsold Units: Predicted number of units that will not be sold.
• Shortage Cost: Cost incurred for not having enough stock to meet demand.
• Expected Shortage: Predicted shortfall in inventory.

Example:

Given:

• Selling Price: \$50
• Expected Sales: 600 units
• Purchase Cost: \$30 per toy
• Order Quantity: The retailer decides to order 700 units initially.
• Salvage Value: \$20 per toy (maybe they can be sold at a discount or returned to the supplier)
• Expected Unsold Units: Given past trends, the retailer expects that 100 toys might remain unsold.
• Shortage Cost: \$10 per toy (This accounts for lost sales, customer dissatisfaction, etc.)
• Expected Shortage: Given the demand, the retailer expects to be short of 50 toys if only 700 units are ordered.

Plugging these values into the formula:

Expected Profit = (\$50 x 600) - (\$30 x 700) + (\$20 x 100) - (\$10 x 50)

Expected Profit = \$10,500

### 6. Simulation:

Uses computational simulations for demand scenarios and evaluates the performance of varying order quantities. This is ideal for complex situations with multiple uncertainties.

Calculation: Computational algorithms and iterative scenarios (no direct formula). The approach uses simulations to account for various potential scenarios and their outcomes.

Example: A simulation run for a product with varying demand scenarios indicates that an order quantity of 520 units minimizes costs and risks of stockouts over a given period.

## 6 Step Process for Choosing the Right Models

Choosing the right single period inventory model depends on the specific characteristics of the business. Here’s our simple 6 step process:

### Step 1. Assess the Product's Perishability

Products with short shelf life or limited selling season should lean towards the Newsvendor model due to the significant costs associated with unsold inventory.

Example: A store expects to sell 300 copies of the newly released book "Mystical Mornings" in the first month, with demand dropping to 50 in the next. Using the Newsvendor model, and given costs of overordering (\$10 per unsold copy) and underordering (\$20 for each missed sale) they can compute the optimal order quantity.

### Step 2. Evaluate Customer Tolerance to Waiting

When selling products for which customers show some willingness to wait if it's temporarily out of stock the Partial Backordering model becomes relevant.

Example: "Elegant Timepieces" sells custom-designed watches at \$500 each. If they order 100 pieces but demand surges to 120, they could backorder the 20 extra. They could save on potential storage costs while ensuring customer needs are eventually met.

### Step 3. Prioritize Service Level Commitments

For businesses emphasizing a strict service level or when customer loyalty depends heavily on product availability, adopt the Service Level Approach.

Example: "MediCare Supplies" assures a 99% service level for a specific heart monitor priced at \$2,000. If they forecast sales of 150 units, they might order an extra 10 units as safety stock to maintain this level. This ensures hospitals receive timely deliveries.

### Step 4. Analyze Marginal Benefits

For products where minor adjustments in order quantity have significant implications on costs or revenues, the Marginal Analysis model is appropriate.

Example: "TechVerse" sells a new software CD at \$100 each. Producing one more CD costs \$30, but not having an extra CD when demanded can mean a lost sale. Through Marginal Analysis, they can determine the point where the benefit of producing an extra CD equals the potential lost sale.

### Step 5. Focus on Profit Potential

If a business deals with items having significant post-seasonal value or alternative uses, the Profit Maximization Approach should be employed.

Example: "Fashion Elites" stocks winter jackets priced at \$150 each. Unsold items at the season's end can be sold at a clearance price of \$60. Considering storage, missed sales, and salvage values, this model will help decide the optimal initial order quantity to maximize profits.

### Step 6. Navigate Complex Demand Patterns with Simulation

For intricate scenarios with multiple uncertainties or non-standard demand patterns, consider the Simulation method.

Example: "Global Harvest" imports exotic fruits like "Zanzibar Oranges" at \$5 each, but demand fluctuates based on factors like weather and cultural events. They can run 10,000 simulated scenarios with different order quantities and choose the one that averages best across these simulations.

## Example

Bella's Bakery sells various baked goods. Let’s explore below which single period inventory model is best for two of their products: "Golden Sun Cupcakes" and "Rustic Rye Bread.

### 1. Assess the Product's Perishability:

Bella's "Golden Sun Cupcakes" have a shelf life of just 2 days. Bella estimated a demand of 150 cupcakes per day with overordering costs at \$1 per cupcake and underordering costs at \$3 per missed sale. With this, she utilized the Newsvendor formula to pinpoint the optimal daily order quantity which is 164 cupcakes.

### 2. Evaluate Customer Tolerance to Waiting:

For Bella's "Rustic Rye Bread," some customers are content with waiting a day for their preferred loaf. This indicates the Partial Backordering model's relevance. With usual sales at 50 loaves daily and occasional surges to 70, Bella can backorder the extra 20 loaves. This ensures a blend of demand fulfillment and inventory management.

### 3. Prioritize Service Level Commitments:

Although neither the cupcakes nor the bread has strict service level agreements tied to them, Bella ensures that she has a consistent stock to uphold her bakery's reputation and customer trust.

### 4. Analyze Marginal Benefits:

While Bella's baked goods have variable demand, the cost or revenue implications of slight adjustments in order quantities aren't substantial enough to warrant the Marginal Analysis model as a primary strategy.

### 5. Focus on Profit Potential:

Unsold "Golden Sun Cupcakes" are bundled for \$2 instead of \$4. Leftover "Rustic Rye Bread" is repurposed into croutons, recovering \$3 of the \$5 price. The Profit Maximization Approach helps Bella determine the best initial order, considering salvage opportunities.

### 6. Navigate Complex Demand Patterns with Simulation:

Demand for Bella's "Golden Sun Cupcakes" can fluctuate due to local events or weather. Running 1,000 simulated scenarios, Bella finds that an order quantity of 120 consistently yields optimal results. Using the Simulation approach, she identifies this as her ideal order quantity amidst variable demands.

We hope that you now have a better understanding of what single period inventory models are and how to choose suitable models for your business.