A furniture manufacturer builds tables. The cost for...

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is $\$ 240$ and the profit is $\$ 160 .$ The cost to build a dining room table is $\$ 320$ and the profit is $\$ 240$. (See Examples $2-3)$ Let $x$ represent the number of kitchen tables produced per month. Let $y$ represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling $x$ kitchen tables and $y$ dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. - The number of each type of table cannot be negative. - Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. - The company can build at most 90 dining room tables. - The company does not want to exceed a monthly cost of $\$ 48,000$. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?

A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is $\$ 240$ and the profit is $\$ 160 .$ The cost to build a dining room table is $\$ 320$ and the profit is $\$ 240$. (See Examples $2-3)$ Let $x$ represent the number of kitchen tables produced per month. Let $y$ represent the number of dining room tables produced per month.
a. Write an objective function representing the monthly profit for producing and selling $x$ kitchen tables and $y$ dining room tables.
b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints.
- The number of each type of table cannot be negative.
- Due to labor and equipment restrictions, the company can build at most 120 kitchen tables.
- The company can build at most 90 dining room tables.
- The company does not want to exceed a monthly cost of $\$ 48,000$.
c. Graph the system of inequalities represented by the constraints.
d. Find the vertices of the feasible region.
e. Test the objective function at each vertex.
f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.)
g. What is the maximum profit?

Key Concepts

Decision Variables

These are the variables that represent the quantities to be determined in the problem. They are typically defined to represent the number of units produced or sold and serve as the inputs for both the objective function and the constraints.

Objective Function

This is a linear function that represents the goal of the problem, such as maximizing profit or minimizing cost. It is formulated in terms of the decision variables and serves as the target to be optimized.

Constraints

Constraints are the restrictions or limitations that the decision variables must satisfy. They are expressed as linear equations or inequalities representing resource limitations, production capacities, budget restrictions, or other operational limits.

Feasible Region

The feasible region is the set of all possible solutions that satisfy all the constraints simultaneously. In linear programming, this region is typically defined in a graphical form as the intersection of the half-planes represented by the constraints.

Vertices (Corner Points)

Vertices are the intersection points of the constraint boundaries within the feasible region. According to the corner point theorem, the optimal solution to a linear programming problem will occur at one of these vertices.

Graphical Method

This is a technique used to solve linear programming problems by graphing the constraints to visually identify the feasible region and its vertices. It is especially useful when there are only two decision variables.

Corner Point Evaluation

This involves testing the value of the objective function at each vertex of the feasible region. By comparing these values, one can determine which vertex gives the optimal solution, either maximum or minimum, depending on the problem.

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