Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Refer to Example 6 in this section. Suppose Ace Novelty received an order from another amusement park for 1200 Pink Panthers, 1800 Giant Pandas, and 1400 Big Birds. The quantity of each type of stuffed animal to be produced at each plant is shown in the following production matrix: Each Panther requires \(1.3 \mathrm{yd}^{2}\) of plush, \(20 \mathrm{ft}^{3}\) of stuffing, and 12 pieces of trim. Assume the materials required to produce the other two stuffed animals and the unit cost for each type of material are as given in Example 6 . a. How much of each type of material must be purchased for each plant? b. What is the total cost of materials that will be incurred at each plant? c. What is the total cost of materials incurred by Ace Noyelty in filling the order?

Step-by-Step Solution

Answer

In order to fill the order, Ace Novelty must purchase the following amounts of materials for each plant: 4060 yd² of plush, 63500 ft³ of stuffing, and 33300 pieces of trim for Plant 1; 1125 yd² of plush, 20250 ft³ of stuffing, and 9750 pieces of trim for Plant 2; and 555 yd² of plush, 9800 ft³ of stuffing, and 5400 pieces of trim for Plant 3. The total cost of materials incurred at each plant is $12,281 for Plant 1, $3,391.25 for Plant 2, and $2,446.75 for Plant 3, resulting in a total cost of $17,119 for Ace Novelty to fill the order.

Step 1: Set up the matrices

Set up the order matrix (quantities of each product) and the material requirements matrix (amounts of each material per product per plant).

Step 2: Multiply matrices

Multiply the order vector by the material requirements matrix to get the total material needed for each plant.

Step 3: State the result

The required materials are: 4060 yd\(^2\) of plush, 63500 ft\(^3\) of stuffing, and the appropriate amounts of other materials as determined by the matrix product.

Key Concepts

Production Matrix

Imagine a factory where different products are being assembled, and there are multiple plants within this factory that specialize in various stages of each product's production. A production matrix effectively organizes and displays the quantity of each product manufactured at every plant.

Let's look at Ace Novelty's example, where the company has received an order for various stuffed animals. The order consists of 1200 Pink Panthers, 1800 Giant Pandas, and 1400 Big Birds. The production matrix is a three-by-three grid, representing each animal type down the columns and each plant across the rows. The numbers in the matrix determine how many of each stuffed animal each plant needs to produce to meet the order.

Understanding this matrix is crucial because it sets the groundwork for determining the materials necessary and, ultimately, the cost of production.

Material Requirements Matrix

If the production matrix tells us 'how much' we need to produce, the material requirements matrix tells us 'what' and 'how much of it' we need to make those products. This matrix contains rows for each product and columns for the different materials required, such as plush, stuffing, and trim for stuffed animals.

In the case of Ace Novelty, by multiplying the production matrix by the material requirements matrix, we can determine the total amount of each material needed for each plant to fulfill the order. It is an essential tool for managing inventory and identifying the material needs across all plants. Understanding this concept helps in planning and purchasing decisions, ensuring that each plant has exactly what it needs without excess, which can save the company money and avoid waste.

Cost Calculation

Once we know the quantity of materials needed for production, the next step is to calculate the cost. This calculation is essential for budgeting, pricing, and financial planning. We can determine the cost by multiplying the amount of each material needed by its unit cost and then summing up these products.

For each plant within Ace Novelty, we perform this calculation separately to understand the specific costs incurred at each location. We must add up the costs across all plants to understand the total expense the company will bear in fulfilling the order. Appreciating this process allows students and businesses alike to grasp the financial implications of production and encourages efficient use of resources to control costs.