91Ó°ÊÓ

For retailers who buy from a distributor or manufacturer and sell to the public, a major concern is the cost of maintaining unsold inventory. You must have appropriate stock to do business, but if you order too much at a time, your profits may be eaten up by storage costs. One of the simplest tools for analysis of inventory costs is the basic or der quantity model. It gives the yearly inventory expense \(E=E(c, N, Q, f)\) when the following inventory and restocking cost factors are taken into account: \- The carrying cost \(c\), which is the cost in dollars per year of keeping a single unsold item in your warehouse. \- The number \(N\) of this item that you expect to sell in 1 year. \- The number \(Q\) of items you order at a time. \- The fixed costs \(f\) in dollars of processing a restocking order to the manufacturer. (Note: This is not the cost of the order; the price of an item does not play a role here. Rather, \(f\) is the cost you would incur with any order of any size. It might include the cost of processing the paperwork, fixed costs you pay the manufacturer for each order, shipping charges that do not depend on the size of the order, the cost of counting your inventory, or the cost of cleaning and rearranging your warehouse in preparation for delivery.) The relationship is given by $$ E=\left(\frac{Q}{2}\right) c+\left(\frac{N}{Q}\right) f \text { dollars per year. } $$ A new-car dealer expects to sell 36 of a particular model car in the next year. It costs \(\$ 850\) per year to keep an unsold car on the lot. Fixed costs associated with preparing, processing, and receiving a single order from Detroit total \(\$ 230\) per order. a. Using the information provided, express the yearly inventory expense \(E=E(Q)\) as a function of \(Q\), the number of automobiles included in a single order. b. What is the yearly inventory expense if 3 cars at a time are ordered? c. How many cars at a time should be ordered to make yearly inventory expenses a minimum? d. Using the value of \(Q\) you found in part c, determine how many orders to Detroit will be placed this year. e. What is the average rate of increase in yearly inventory expense from the number you found in part \(\mathrm{c}\) to an order of 2 cars more?

Step by step solution

01

Identify the Given Information

We know the following:- Yearly demand, \(N = 36\) cars- Carrying cost per car, \(c = \\(850\)- Fixed cost per order, \(f = \\)230\)Our goal is to express the yearly inventory expense \(E\) as a function of \(Q\), and solve subsequent parts.

02

Express Inventory Expense Function

Substitute the given values into the expense formula: \[E(Q) = \left(\frac{Q}{2}\right)c + \left(\frac{N}{Q}\right)f\]Replacing \(c\), \(N\), and \(f\), we get:\[E(Q) = \left(\frac{Q}{2}\right)(850) + \left(\frac{36}{Q}\right)(230)\]This simplifies to:\[E(Q) = 425Q + \frac{8280}{Q}\]

03

Calculate Yearly Expense for 3 Cars per Order

Substitute \(Q = 3\) into the function:\[E(3) = 425(3) + \frac{8280}{3}\]Calculate each term:- \(425 \times 3 = 1275\)- \(\frac{8280}{3} = 2760\)Add the values:\[E(3)= 1275 + 2760 = 4035\]

04

Find Minimum Yearly Inventory Expense

The minimum occurs when the derivative \(E'(Q)\) is zero:\[E'(Q) = 425 - \frac{8280}{Q^2} = 0\]Rearrange and solve:\[425 = \frac{8280}{Q^2} \]\[Q^2 = \frac{8280}{425}\]\[Q^2 = 19.49 \Rightarrow Q = \sqrt{19.49} \approx 4.41\]Since \(Q\) must be a whole number, consider \(Q = 4\).

05

Determine Number of Orders When Q = 4

To find the number of orders per year, divide the total sales by the order size:\[\text{Number of orders} = \frac{N}{Q} = \frac{36}{4} = 9\]

06

Calculate Increase in Expense for Q = 6

For \(Q = 6\), compute \(E(6)\):\[E(6) = 425(6) + \frac{8280}{6} = 2550 + 1380 = 3930\]The increase from \(Q = 4\) to \(Q = 6\) is:\[E(4) = 425(4) + \frac{8280}{4} = 1700 + 2070 = 3770\]Difference:\[E(6) - E(4) = 3930 - 3770 = 160\]Thus, the average rate of increase is \( \frac{160}{2} = 80\) dollars per additional car ordered.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Order Quantity Model

The order quantity model is a fundamental tool in inventory management. It's designed to help businesses decide how much to order at a time to minimize costs. Imagine you're running a store. If you order too much stock, storage costs increase; order too little, then more orders are necessary, increasing order processing costs. The order quantity model balances these to minimize total expenses. It focuses on:

• Quantity Ordered ( Q ): This is the number of items ordered each time. This variable is crucial in determining costs.
• Yearly Demand ( N ): The number of items expected to be sold in a year.
• Carrying Cost ( c ): This is the cost of holding each item in stock per year.
• Fixed Costs ( f ): These are costs related to placing an order, not dependent on the order size.

Utilizing the order quantity model helps businesses earn more by efficiently managing their inventory.

Yearly Inventory Expense

Yearly inventory expense is a key concern for businesses, particularly in retail. It represents the total cost associated with ordering and maintaining inventory throughout the year. The goal is to express this expense efficiently as a function of the order quantity Q. The formula used is:\[E(Q) = \left(\frac{Q}{2}\right)c + \left(\frac{N}{Q}\right)f\]Breaking it down:

• The first term, \(\frac{Q}{2}c\), calculates carrying costs for holding inventory.
• The second term, \(\frac{N}{Q}f\), accounts for fixed costs based on the number of orders placed annually.

The combination of these two components gives a complete view of annual expenses and aids in making cost-effective order decisions.

Carrying Cost

Carrying cost, or holding cost, is the expense incurred to store unsold inventory. It's crucial because it directly affects profits. If the carrying cost is high, businesses must be careful about overstocking, as it can significantly reduce profit margins. Factors contributing to carrying costs include:

• Storage fees, including rent or utilities.
• Insurance for goods stored.
• Opportunity costs of capital tied in inventory.
• Spoilage and obsolescence risks, especially for perishable or easily outdated products.

Knowing how to calculate and manage carrying costs allows a business to optimize its inventory levels, ensuring each item stocked contributes to profitability rather than becoming a cost burden.

Fixed Costs

Fixed costs in inventory management are expenses that remain constant regardless of the order size. They are not linked to the number of units ordered, but rather the process of placing an order itself. Some examples include:

• Administrative costs for order processing, such as employee salaries and office supplies.
• Shipping charges often stipulated by carriers as flat fees, irrespective of the order size.
• Warehouse management tasks necessary to accommodate an order, from restructuring storage layouts to taking stock counts.

Understanding fixed costs is vital because these costs accumulate with each order. Businesses aim to minimize these by finding an optimal order quantity Q , thus maximizing efficiency and minimizing unnecessary expenses.