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The SSG Company makes basketballs and footballs. Each basketball requires 3 pounds of rubber and 4 square feet of leather, while each football requires 2 pounds of rubber and 5 square feet of leather. Each basketball produced results in a profit of $12, and each football earns a profit of $16.

Given that SSG has at most 800 square feet of leather and more than 500 pounds of rubber, formulate a linear programming model to identify the number of basketballs and footballs that should be produced to maximize total profit.

Show all steps involved, but you do not need to solve it. (9 marks)

Answer :

Final answer:

To solve the problem, form a Linear Programming model. Define X as the number of basketballs and Y as the number of footballs. Maximize the profit equation: Maximize Z = 12X + 16Y. Add constraints based on the availability of resources which are created by the usage of each ball: 3X + 2Y <= 500 (Rubber) and 4X + 5Y <= 800 (Leather). Further ensure non-negative productions: X>= 0, Y >= 0. Then, graphically represent and solve for the maximum profit.

Explanation:

The subject here is based on Linear Programming, a mathematical approach typically used in Business or Economics to aid decision-making and resource allocation. To create a Linear Programming model for SSG Company's situation, we need to establish the variables, the objective function, and the constraints.

Let's define:

  • X = number of basketballs
  • Y = number of footballs

Then, the profit maximization equation (objective function) would be: Maximize Z = 12X + 16Y.

With the given company resources, our constraints are the maximum available resources they can use for production:

  • 3X + 2Y <= 500 (Rubber)
  • 4X + 5Y <= 800 (Leather)

And of course, the number of basketballs and footballs manufactured cannot be negative, we stack further constraints as X >= 0, Y >= 0.

From there, you will go through the process of graphical representation, identifying feasible regions, and then using the corner point method or the iso-profit method to find the profit-maximizing output level.

Learn more about Linear Programming here:

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