Question

A retailer buys 600 USB Flash Drives per year from a distributor. The retailer wants to determine how many drives to order, \(x\), per shipment so that her inventory is exhausted just as the next shipment arrives. The processing fee is \(\$ 15\) per shipment, the yearly storage cost is \(\$ 1.60 x,\) and each drive costs the retailer \(\$ 4.85 .\) (a) Express the total yearly cost \(C\) as a function of the number \(x\) of drives in each shipment. (b) Use a graphing utility to determine the minimum yearly cost and the number of drives per order that yields the minimum cost.

Short answer

The function is \[ C(x) = \frac{9000}{x} + 1.6x + 2910 \]. Use a graphing utility to find the minimum cost and the optimal order size.

Step-by-step solution

- Define Variables and Costs

Identify and define the variables in the problem: let \(x\) be the number of drives per shipment.

- Identify Given Values

The retailer buys 600 USB Flash Drives per year. The processing fee per shipment is \(\$15.\) The yearly storage cost per drive is \(\$1.60.\) Each drive costs \(\$4.85.\)

- Calculate the Number of Shipments

Calculate the number of shipments per year: \[ \text{Number of shipments} = \frac{600}{x} \]

- Express the Total Yearly Cost

Write the total yearly cost \(C\) as a function of \(x\): \[ C(x) = \left(\frac{600}{x}\right)\times15 + 1.6x + 4.85\times600 \]

- Simplify the Function

Simplify the expression for \(C(x)\): \[ C(x) = \frac{9000}{x} + 1.6x + 2910 \]

- Determine Minimum Yearly Cost

Use a graphing utility to input the function \[ C(x) = \frac{9000}{x} + 1.6x + 2910 \] and find the value of \(x\) that minimizes \(C(x)\).

- Read the Graph

From the graph, determine the minimum yearly cost and the corresponding number of drives per order.

Key concepts

Optimization Problems

Optimization problems in algebra involve finding the best solution from a set of feasible options. Here, the retailer wants to minimize the total yearly cost of ordering USB drives. To solve this, we first express the cost function, then use mathematical tools to find the minimum. This kind of problem is common in inventory management. The main steps include:

• Defining the variables and costs
• Setting up the cost function
• Using algebraic techniques or graphing tools to find the minimum cost

The goal is to balance the order size and the frequency of orders to minimize costs.

Cost Function

The cost function represents the total yearly cost based on the number of USB drives ordered per shipment. In this problem, we identify three main types of costs:

• Processing fee per shipment: \( \frac{9000}{x} \)
• Yearly storage cost: \( 1.6x \)
• Cost of the drives: \( 4.85 \times 600 \)

We combine these components into a single function:
\[ C(x) = \frac{9000}{x} + 1.6x + 2910 \]
This function helps us analyze how total cost changes with different order sizes. Our goal is to find the value of \( x \) that minimizes \( C(x) \).

Supply Chain Economics

Supply chain economics deals with the flow of goods and services and how to optimize this flow for cost efficiency. In this case, the retailer aims to optimize the ordering cycle to minimize costs related to purchasing, storing, and processing USB drives. Key factors include:

• Order size \( x \): Larger orders reduce the number of shipments but increase storage costs.
• Number of shipments: Fewer shipments lower the processing fee but require more storage expense.
• Just-In-Time (JIT) concept: Striving to order in a way that aligns new shipments with the depletion of current inventory to minimize storage.

Supply chain optimization aims to find the best balance between ordering and storage costs.

Graphical Analysis

Graphical analysis involves plotting the cost function \( C(x) = \frac{9000}{x} + 1.6x + 2910 \) on a graph to determine the minimum cost. Here's how it helps:

• Plot the function: Use a graphing tool to visualize \( C(x) \).
• Analyze the curve: Identify the turning point where the cost is at its lowest.
• Determine minimum cost: The x-coordinate at this point gives the optimal order size, and the y-coordinate shows the minimum yearly cost.

Using graphical analysis simplifies the identification of the optimal solution, making it easier to see how different order sizes impact the total cost.