Problem 48 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

An electronics firm uses 600 cases of components each year. Each case costs $$\$ 1,000$$. The cost of storing one case for a year is 90 cents, and the ordering fee is $$\$ 30$$ per shipment. How many cases should the firm order each time to keep total cost at a minimum? (Assume that the components are used at a constant rate throughout the year and that each shipment arrives just as the preceding shipment is being used up.)

Step-by-Step Solution

Answer

Order 200 cases each time.

Step 1: Define the variables

Let Q be the number of cases to order each time. Given data: Annual demand (D)=600 cases, Cost per case (C)=\(\text{\$1,000}\), Annual holding cost per case (H)=\(\text{\$0.90}\), Ordering cost per order (S)=\(\text{\$30}\).

Step 2: Formulate the Total Cost (TC) function

The total cost function includes ordering cost, holding cost, and the cost of the components: \[ TC = \frac{D}{Q} \times S + \frac{Q}{2} \times H + D \times C \]Substitute the given values into the total cost function: \[ TC = \frac{600}{Q} \times 30 + \frac{Q}{2} \times 0.90 + 600 \times 1,000 \]

Step 3: Find the derivative of the Total Cost function

To minimize the total cost, take the derivative of the total cost function with respect to Q and set it to zero: \[ \frac{d(TC)}{dQ} = -\frac{600 \times 30}{Q^2} + \frac{0.90}{2} \]Set this equal to zero and solve for Q: \[ 0 = -\frac{18,000}{Q^2} + 0.45 \]

Step 4: Solve for Q

Rearrange the equation to solve for Q: \[ 0.45Q^2 = 18,000 \]Then: \[ Q^2 = \frac{18,000}{0.45} \]\[ Q^2 = 40,000 \]\[ Q = \frac{40,000}{Q} \ Q = \text{200 cases} \]

Step 5: Short Explanation

The number of cases to order each time that minimizes the total cost is determined to be 200 cases.

Key Concepts

Inventory Management

Inventory management is crucial for any company that holds stock. It involves the supervision of non-capitalized assets—inventory—and stock items. The main goals are to have the right products in the right quantity for sale at the right time. Effective inventory management can help reduce costs and ensure that companies meet consumer demand without overstocking or suffering from stockouts. Ul>

Balance stock levels to avoid overstock or shortage

Minimize holding costs while providing enough stock

Enhance cash flow by reducing excessive stock expenditures

An essential part of managing inventory is understanding demand and how it fluctuates. This understanding lets firms order the right quantities, control costs, and keep operations running smoothly.

Cost Minimization

One of the primary objectives of inventory management is cost minimization. This involves finding the delicate balance between ordering costs and holding costs. Ordered too frequently? Costs for ordering go up. Ordered too infrequently? Holding costs increase because of more inventory in storage. By minimizing total costs, a business can run more efficiently and be more profitable. Core aspects of cost minimization include:

• Reducing the frequency and cost of orders
• Reducing the amount of inventory held, thus lowering storage costs
• Ensuring there are enough items to meet demand without over-ordering

In the example, the firm aims to find an order quantity that results in the lowest total inventory cost, considering both the costs of placing orders and the costs of holding inventory.

Order Quantity Calculation

Order quantity calculation is a key component in inventory management. The Economic Order Quantity (EOQ) model is used to determine the optimal order quantity that minimizes the total cost of inventory. In the given problem, the firm needs to calculate how many cases of components to order to minimize costs. The formula for EOQ is \[ Q = \sqrt{\frac{2DS}{H}} \] where:
\[ D \ = \text{Annual demand} \ S \ = \text{Ordering cost per order} \ H \ = \text{Holding cost per unit per year} \] By substituting the given values into this formula:
\[ Q = \sqrt{\frac{2 \times 600 \times 30}{0.90}} \ Q = \sqrt{40,000} \ Q = 200 \text{ cases} \] The firm should thus order 200 cases each time.

Total Cost Function

The total cost function in inventory management includes the costs of ordering and holding inventory, as well as the cost of the goods. For the provided exercise, the total cost (TC) is formulated as:
\[ TC = \frac{D}{Q} \times S + \frac{Q}{2} \times H + D \times C \]
Each term represents a different component of the total cost:

• \(\frac{D}{Q} \times S \) is the ordering cost
• \(\frac{Q}{2} \times H\) is the holding cost
• \(D \times C\) is the cost of purchasing the inventory

By finding the derivative of the total cost function and setting it to zero, we solve for the order quantity (\(Q\)) that minimizes the total cost.
The steps from the solution are:

1. Finding the derivative
2. Setting the derivative to zero
3. Solving for \(Q\)

Through this process, the optimal order quantity is found to be 200 cases. This order quantity keeps the combined cost of holding and ordering inventory at its lowest possible level.