A manufacturer produces two models of mountain bicycles. The...

A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$ \begin{array}{|c|c|c|}\hline & {\text { ModelA }} & {\text { Model B }} \\ \hline \text { Assembling } & {5} & {4} \\ \hline \text { Painting } & {2} & {3} \\ \hline\end{array} $$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are $\$ 25$ for model $A$ and $\$ 15$ for model B. How many of each type should be produced to maximize profit?

A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table:
$$
\begin{array}{|c|c|c|}\hline & {\text { ModelA }} & {\text { Model B }} \\ \hline \text { Assembling } & {5} & {4} \\ \hline \text { Painting } & {2} & {3} \\ \hline\end{array}
$$
The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are $\$ 25$ for model $A$ and $\$ 15$ for model B. How many of each type should be produced to maximize profit?

Key Concepts

Linear Programming

Linear programming is a method for optimizing a linear objective function, subject to linear equality and inequality constraints. It is widely used in operations research and resource planning to determine the best possible outcome in a given mathematical model whose requirements are represented by linear relationships.

Decision Variables

Decision variables represent the key quantities that need to be determined in an optimization problem. They serve as the inputs to both the objective function and the constraints, and finding the optimal values for these variables is central to solving any linear programming problem.

Objective Function

The objective function is a linear equation that needs to be maximized or minimized, such as maximizing profit or minimizing cost. Its coefficients represent the contribution per unit change in the decision variables, effectively summarizing the goal of the optimization problem.

Constraints

Constraints are the set of linear inequalities or equalities that define the restrictions or limitations within the linear programming problem. They model real-world limitations like resource availability, production capacities, or time restrictions, and they ensure that the solution is feasible in practical terms.

Feasible Region

The feasible region is the set of all possible solutions that satisfy all the constraints of the linear programming problem. It typically takes the form of a convex polygon (or polyhedron in higher dimensions) where the optimal solution is usually found at one of the vertex points.

Graphic Method/Corner Point Principle

The graphical method is an approach used to solve two-variable linear programming problems by plotting the constraints on a graph and identifying the feasible region. The optimal solution, according to the corner point principle, is located at one of the vertices of this region.

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