91Ó°ÊÓ

Game Sales The owner of a toy store expects to sell 500 popular handheld game units in the year following its release. It costs 6 to store one handheld game unit for one year. The cost of reordering is a fixed 20 plus 4 for each game unit ordered. To minimize costs, the shop manager must balance the ordering costs incurred when many small orders are placed with the storage costs incurred when many units are ordered at once. Assume that all orders placed will contain the same number of handheld game units. Use \(x\) to represent the number of handheld game units in each order. a. Write an expression for i. The cost for one order of \(x\) units. ii. The number of times the manager will have to order \(x\) units during the year. iii. The total ordering costs for one year. b. Assume that the average number of handheld game units stored throughout the year is half the number of handheld game units in each order. What is the total storage cost for one year? c. Write a model for the combined ordering and storage costs for one year, using \(x\) as the number of handheld game units in each order. d. What order size minimizes the total yearly cost? What will the minimum total ordering and storage costs be for the year?

Step by step solution

01

Expression for the Cost of One Order

To calculate the total cost of one order of \( x \) units, we combine the fixed reordering cost and the variable cost per unit: \( \text{Total Cost} = 20 + 4x \). This consists of a \(20 fixed cost and a \)4 cost per each handheld game unit in the order.

02

Number of Orders Per Year

To find out how many times the manager will need to order during the year, divide the total expected sales by the order size \( x \): \( \text{Number of Orders} = \frac{500}{x} \).

03

Expression for Total Ordering Costs in a Year

Multiply the cost of one order by the number of orders per year: \( \text{Total Ordering Costs} = \left(20 + 4x\right) \times \frac{500}{x} = \frac{10000}{x} + 2000 \).

04

Calculate Average Storage Cost Per Year

Given that the average number of units in storage is half of \( x \), the average storage cost per unit is \( 6 \) per year, the total storage cost would be \( 6 \times \frac{x}{2} \times \frac{500}{x} = 3x \cdot \frac{500}{x} = 1500 \).

05

Model for Total Yearly Costs

The total yearly cost is the sum of ordering and storage costs: \( C(x) = \frac{10000}{x} + 2000 + 1500 \).

06

Optimize Order Size to Minimize Costs

To find the minimum cost, take the derivative of \( C(x) \), set it to zero and solve for \( x \):\( C'(x) = -\frac{10000}{x^2} + 3 = 0 \).Solving for \( x \) gives \( x = \sqrt{\frac{10000}{3}} \approx 57.735 \). Since \( x \) must be whole, check costs at \( x = 57 \) and \( x = 58 \) to find the minimum cost.

07

Calculate Costs for Optimal Order Sizes

For \( x = 58 \), \( C(58) = \frac{10000}{58} + 2000 + 1500 \approx 3719 \).For \( x = 57 \), \( C(57) = \frac{10000}{57} + 2000 + 1500 \approx 3730 \).Thus, the minimum cost occurs at \( x = 58 \) with a total cost of approximately $3719.

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Key Concepts

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

Cost minimization

In business, cost minimization is key for achieving optimal financial outcomes. Reducing costs helps increase profitability without raising product prices. In the context of inventory management, it involves balancing between ordering costs and storage costs. Ordering costs are incurred when replenishing stock, while storage costs are the expenses linked to keeping the stock over time. The challenge is to find the order quantity that minimizes the total cost involved in managing inventory. When applied to calculus, this involves evaluating cost functions and finding where the costs are at their lowest using derivative techniques.

Order quantity

Order quantity refers to the number of units ordered at a given time. It's an important parameter in inventory management. The order quantity directly impacts both the ordering frequency and storage requirements. For instance, ordering in large quantities reduces the frequency of orders, which can lower total ordering costs. However, it increases storage costs since more items are stored. Alternatively, ordering in smaller quantities lowers storage costs but increases the frequency of ordering. Businesses strive to find an optimal order quantity where the total cost is minimized. This is often achieved through mathematical models that take both ordering and storage costs into consideration.

Economic order quantity (EOQ)

The Economic Order Quantity (EOQ) is a fundamental concept in inventory management. It's the ideal order quantity that minimizes the total cost of inventory, including both ordering and holding costs. The EOQ model makes several assumptions: demand is constant, lead time is fixed, and each order is delivered completely and immediately. Calculating EOQ involves using calculus to derive a specific formula:

• \( EOQ = \sqrt{\frac{2DS}{H}} \)

where \( D \) is the demand rate, \( S \) is the cost per order, and \( H \) is the holding cost per unit per year. This formula provides a balance between ordering too frequently and ordering too much, helping businesses minimize costs effectively.

Derivative applications

In calculus, derivatives are used to determine the rate of change of a quantity or to find the slope of a tangent line to a curve at any point. In optimization problems, derivatives help find the maximum or minimum value of a function, which is essential in optimizing costs. By taking the derivative of a cost function, we identify critical points where the total cost is minimized. This involves finding where the first derivative is zero, indicating a potential minimum or maximum. When applied to inventory problems, taking the derivative of the cost function allows us to find the optimal order quantity that minimizes total costs. Furthermore, by examining the second derivative, we can confirm whether it results in a minimum or maximum point. These mathematical techniques ensure more strategic decisions in inventory planning.