Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Ace Novelty manufactures "Giant Pandas" and "Saint Bernards." Each Panda requires \(1.5 \mathrm{yd}^{2}\) of plush, \(30 \mathrm{ft}^{3}\) of stuffing, and 5 pieces of trim; each Saint Bernard requires \(2 \mathrm{yd}^{2}\) of plush, \(35 \mathrm{ft}^{3}\) of stuffing, and 8 pieces of trim. The profit for each Panda is $$\$ 10$$, and the profit for each Saint Bernard is $$\$ 15$$. If \(3600 \mathrm{yd}^{2}\) of plush, \(66,000 \mathrm{ft}^{3}\) of stuffing, and 13,600 pieces of trim are available, how many of each of the stuffed animals should the company manufacture to maximize its profit? What is the maximum profit?

Step-by-Step Solution

Answer

Ace Novelty should manufacture 0 Pandas and 1800 Saint Bernards to maximize its profit. The maximum profit that can be achieved is \(\$27,000\).

Step 1: Write the linear programming formulation

We need to formulate the problem as a linear programming problem. Let x be the number of Pandas and y be the number of Saint Bernards. Objective function: Maximize profit = \(\$ 10)\times x + \(\$ 15)\times y Constraints: 1. Availability of plush: 1.5x + 2y ≤ 3600 2. Availability of stuffing: 30x + 35y ≤ 66,000 3. Availability of trim: 5x + 8y ≤ 13,600 4. Non-negativity constraints: x ≥ 0, y ≥ 0

Step 2: Graph the inequalities

We need to graph the constraints on the Cartesian plane to find out the feasible region. 1. For plush constraint: y ≤ (3600 - 1.5x) / 2 2. For stuffing constraint: y ≤ (66,000 - 30x) / 35 3. For trim constraint: y ≤ (13,600 - 5x) / 8 Plot the inequalities on the Cartesian plane and find the feasible region.

Step 3: Identify the vertices of the feasible region

The feasible region is the area where the constraints intersect, and we'll find the coordinates of the vertices where the lines cross each other. Intersection points A, B, C, and D are identified as: A (0, 0) B (0, 1800) C (1600, 400) D (2400, 0)

Step 4: Evaluate the objective function at each vertex

The objective function represents the profit. We want to find the maximum value of the profit in the feasible region. To do so, calculate the profit for each of the vertices: P(A) = 10(0) + 15(0) = 0 P(B) = 10(0) + 15(1800) = 27,000 P(C) = 10(1600) + 15(400) = 22,000 P(D) = 10(2400) + 15(0) = 24,000

Step 5: Determine the maximum profit and the optimal solution

Based on the profit values at each vertex, the maximum profit is achieved at point B, which is the optimal solution. The company should manufacture 0 Pandas and 1800 Saint Bernards to maximize its profit, and the maximum profit is \$ 27,000.

Key Concepts

Objective Function

In linear programming, the objective function is a key component that you aim to optimize, whether it be to maximize or minimize values like cost, profit, or time. Here, we are concerned with maximizing the company's profit from manufacturing Giant Pandas and Saint Bernards.

The objective function is crafted using the profits from each product. It's represented as a linear equation. For this exercise, the profit from Pandas is \(10 per unit and from Saint Bernards is \)15 per unit.

Thus, the objective function is:

• Maximize Profit = 10x + 15y

where \(x\) represents the number of Pandas and \(y\) represents the number of Saint Bernards. The goal here is to determine the values of \(x\) and \(y\) that will yield the highest possible profit.

Constraints

Constraints in linear programming define the limitations or restrictions that must be adhered to, based on available resources or conditions. In our problem, these constraints are related to the plush, stuffing, and trim needed to create the animals.

For Ace Novelty, the constraints derived from the resources are:

• Plush: \(1.5x + 2y \leq 3600\)
• Stuffing: \(30x + 35y \leq 66,000\)
• Trim: \(5x + 8y \leq 13,600\)
• Non-negativity: \(x \geq 0, y \geq 0\)

These constraints ensure that production does not exceed the given material limits. Understanding and graphing these constraints is crucial to identify the feasible region where the potential solutions lie.

Feasible Region

The feasible region in a linear programming problem is the area on the graph where all constraints are satisfied simultaneously. It's where the magic happens, as it holds all possible solutions that meet the problem's requirements.

To find it, you graph the linear inequalities for each constraint on a Cartesian plane. The intersection of these graphs forms a polygon-like shape, representing the feasible region.

In our case, plotting inequalities from the plush, stuffing, and trim constraints allows us to visualize this region. The feasible area is a bounded space where solutions can be sought, often resulting in a finite set of possible optimal outcomes at specific vertices.

Optimization Problem

An optimization problem in linear programming seeks to find the best possible solution subject to certain constraints and requirements. It involves selecting the optimal solution from a set of feasible solutions, as defined by the feasible region.

The objective is to either maximize or minimize a particular function. In our exercise, Ace Novelty's task is to determine the number of Pandas and Saint Bernards to manufacture to maximize profit.

By evaluating the objective function at each vertex of the feasible region, one determines which solution provides the maximum profit. It involves strategic decision-making and highlights the power of optimization in resource management and efficiency.