A manufacturer produces two models of patio furniture. Model...

A manufacturer produces two models of patio furniture. Model A requires 2 hr for assembly and 1.2 hr for painting. Model $\mathrm{B}$ requires $3 \mathrm{hr}$ for assembly and 1.5 hr for painting. The production information and profit for selling each model are given in the table. $$ \begin{array}{|c|c|c|c|} \hline & \text { Assembly } & \text { Painting } & \text { Profit } \\ \hline \text { Model A } & 2 \mathrm{hr} & 1.2 \mathrm{hr} & \$ 150 \\ \hline \text { Model B } & 3 \mathrm{hr} & 1.5 \mathrm{hr} & \$ 200 \\ \hline \end{array} $$ The manufacturer has 1200 hr of labor available for assembly and 660 hr of labor available for painting. a. Determine the number of model A units and the number of model $\mathrm{B}$ units that should be produced to maximize profit assuming that all furniture will be sold. b. What is the maximum profit under these constraints? c. If the profit on model A units is $$\$ 180$$ and the profit on model $\mathrm{B}$ units remains the same, how many of each type should the manufacturer produce to maximize profit?

A manufacturer produces two models of patio furniture. Model A requires 2 hr for assembly and 1.2 hr for painting. Model $\mathrm{B}$ requires $3 \mathrm{hr}$ for assembly and 1.5 hr for painting. The production information and profit for selling each model are given in the table.
$$
\begin{array}{|c|c|c|c|}
\hline & \text { Assembly } & \text { Painting } & \text { Profit } \\
\hline \text { Model A } & 2 \mathrm{hr} & 1.2 \mathrm{hr} & \$ 150 \\
\hline \text { Model B } & 3 \mathrm{hr} & 1.5 \mathrm{hr} & \$ 200 \\
\hline
\end{array}
$$
The manufacturer has 1200 hr of labor available for assembly and 660 hr of labor available for painting.
a. Determine the number of model A units and the number of model $\mathrm{B}$ units that should be produced to maximize profit assuming that all furniture will be sold.
b. What is the maximum profit under these constraints?
c. If the profit on model A units is $$\$ 180$$ and the profit on model $\mathrm{B}$ units remains the same, how many of each type should the manufacturer produce to maximize profit?

Key Concepts

Linear Programming

Linear programming is a method used to achieve the best outcome—such as maximum profit or lowest cost—in a mathematical model whose requirements are represented by linear relationships. It involves defining decision variables, constructing a linear objective function, and setting up a series of linear constraints that represent the limitations or requirements of the real-world scenario.

Objective Function

The objective function in a linear programming problem is the function that needs to be optimized (maximized or minimized). It quantitatively represents the goal of the problem, such as maximizing profit or minimizing cost, and is defined in terms of the decision variables.

Constraints

Constraints in linear programming are the linear inequalities or equations that define the limitations or bounds on the decision variables. They represent real-world restrictions, such as resource limitations or time constraints, and are critical in determining the feasible solutions for the problem.

Feasible Region

The feasible region is the set of all possible solutions that satisfy all the constraints in a linear programming problem. It is typically represented graphically as the intersection of half-planes defined by the constraints, and the optimal solution must lie within this region.

Corner Point Theorem

The corner point theorem states that if an optimal solution exists for a linear programming problem, it will be found at one of the vertices, or 'corner points,' of the feasible region. This theorem is the basis for methods such as the graphical method and the simplex method in solving linear programming problems.

Sensitivity Analysis

Sensitivity analysis involves studying how the optimal solution to a linear programming problem is affected by changes in the parameters of the objective function or the constraints. It helps in understanding the robustness of the solution and in making informed decisions under varying conditions.

Transcript

Please give Ace some feedback