A cosmetics company makes a profit of 15 cents on a tube of...

A cosmetics company makes a profit of 15 cents on a tube of lipstick and a profit of 8 cents on a tube of lip gloss. To meet dealer demand, the company needs to produce between 300 and 800 tubes of lipstick and between 100 and 300 tubes of lip gloss. The maximum number of tubes of lipstick and lip gloss the company can produce per day is $800 .$ How many of each type of beauty product should be produced to maximize profit?

Key Concepts

Linear Programming

Linear programming is a mathematical method used for finding the best outcome, such as maximum profit or lowest cost, in a model whose requirements are represented by linear relationships. It involves the formulation of a problem by defining a linear objective function and a set of linear constraints that represent limitations or requirements.

Objective Function

The objective function in a linear programming problem is the function that needs to be optimized, either maximized or minimized. In the context of production and profit maximization, this function represents the total profit calculated by combining the profit contributions from each product, weighted by the number of units produced.

Constraints

Constraints are the restrictions or limitations imposed on the decision variables in a linear programming problem. These can include resource limits, production capacity, market demand requirements, or other operational boundaries that the solution must satisfy.

Feasible Region

The feasible region is the set of all possible solutions that satisfy all constraints in a linear programming problem. It is typically represented as a polygon or polyhedron in the decision variable space, and any solution outside this region is not viable.

Corner Point Principle

The corner point principle states that in a linear programming problem, if an optimal solution exists, it will be found at one of the vertices or extreme points of the feasible region. Evaluating the objective function at these points is a common method to identify the best solution.

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