ASSIGNED PROBLEM 2: Mrs. Bear has a home business in which she makes and sells stuffed animals from her home. She primarily makes two products – dolphins and manatees. Each dolphin requires 2 square feet of material, 4 ounces of stuffing and 3 hours of labor, while each manatee requires 4 feet of material, 8 ounces of stuffing and 1 hour of labor. Mrs. Bear purchases 80 feet of material and 120 ounces of stuffing each week and allots 60 hours for her business. If she earns a profit of $5 for each dolphin and $7 for each manatee, how many of each should she make in order to maximize her profit and what is that profit? Set up the linear programming problem and answer the following:
2a. How much would the profit increase with an additional ounce of stuffing?
2b. How much of the 80 feet of material are actually used in the optimal
Welcome to the heartwarming tale of Mrs. Bear, a dedicated home business owner with a passion for crafting adorable stuffed animals – dolphins and manatees. Despite facing resource limitations, Mrs. Bear’s unwavering determination to succeed drove her to optimize her profits using the powerful tool of linear programming. In this story, we explore how she made informed decisions to achieve the perfect mix of products, maximizing her profits while effectively utilizing available resources.
As Mrs. Bear embarked on her entrepreneurial journey, she quickly realized the importance of striking the right balance between costs and profits. To achieve this, she set her primary goal as maximizing profit, a critical metric that would determine the success of her home business.
Mrs. Bear, ever resourceful, discovered the magic of linear programming, a mathematical technique designed to optimize decisions in the face of constraints. Linear programming allowed her to represent her production process as a set of linear equations and inequalities, with each stuffed animal’s material, stuffing, and labor requirements carefully factored in.
The key to Mrs. Bear’s success lay in formulating an objective function that captured her profit aspirations. She defined P = 5D + 7M, where D represented the number of dolphins and M the number of manatees sold. The objective: to maximize P within the bounds of resource constraints.
With the objective function in place, Mrs. Bear confronted her resource limitations head-on. She had 80 feet of material, 120 ounces of stuffing, and 60 hours of labor per week at her disposal. To ensure efficient resource allocation, she expressed the following constraints:
Material Constraint: 2D + 4M ≤ 80 feet
Stuffing Constraint: 4D + 8M ≤ 120 ounces
Labor Constraint: 3D + M ≤ 60 hours
As Mrs. Bear let the linear programming algorithm work its magic, she marveled at the insights it provided. The results were astonishing! The algorithm revealed that the optimal solution was to produce approximately 10 dolphins (D ≈ 10) and 15 manatees (M ≈ 15) each week. This combination of products promised the highest profit imaginable.
With the optimal solution in hand, Mrs. Bear’s excitement soared. The calculations demonstrated that by producing 10 dolphins and 15 manatees weekly, she could achieve a remarkable profit of $155. This new knowledge empowered her to make data-driven decisions that would positively impact her business’s bottom line.
The linear programming adventure did not end there. Mrs. Bear realized that she could further enhance her profits by delving into marginal analysis. By calculating the marginal profit for stuffing, she discovered that each additional ounce of stuffing used in the production process would boost her profit by $7. Armed with this insight, she could fine-tune her production choices to maximize returns.
Mrs. Bear’s eyes sparkled with joy when she realized the true efficiency of her optimized production plan. The linear programming analysis revealed that her optimal solution utilized the entire 80 feet of available material. This meant that she made the most of her weekly material supply, crafting the ideal mix of dolphins and manatees while minimizing waste and maximizing resource utilization.
Mrs. Bear’s journey with linear programming transformed her home business into a flourishing success story. By leveraging this powerful technique, she identified the optimal mix of products that maximized her profits while effectively utilizing available resources. Armed with data-driven insights and a strategic approach, Mrs. Bear could confidently expand her business, delighting customers with her charming stuffed animals and securing long-term success in her entrepreneurial venture.
Through the magic of linear programming, Mrs. Bear proved that even in the world of crafting stuffed animals, data and optimization could lead to a heartwarming tale of triumph and profitability. Her story serves as an inspiring reminder that embracing technology and data-driven decision-making can unlock untapped potential, paving the way for success in any business endeavor.
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