5.3 Linear Programming in Two Dimensions: A Geometric Approach

Question 1

What is Linear programming?

Answer 1

Linear programming is a
mathematical process that has been developed to help management in decision making.

Question 2

Getting an objective function from given information

Answer 2

Question 3

Using the tent manufactuing example, what are decision variables?

Answer 3

Question 4

What is the objective?

Answer 4

The objective is to find values of the decision variables that produce the optimal value (in this case, maximum value) of the objective function.

Question 5

What are problem constraints?

Answer 5

The form of the objective function indicates that the profit can be made as large as we like, simply by producing enough tents. But any manufacturing company has limits imposed by available resources, plant capacity, demand, and so on. These limits are referred to as problem constraints.

Question 6

What are non-negative constraints?

Answer 6

Question 7

What can a mathamatical model for these problems look like?

Answer 7

Question 8

What can a feasible region for these kinds of problems look like?

Answer 8

Question 9

Example of a maximization problem?

Answer 9

Out of all possible production schedules 1x, y2 from the feasible region, which
schedule(s) produces the maximum profit? This is a maximization problem.

Question 10

What is a constant-profit line?

Answer 10

By assigning P in P=50x +80y a particular value and plotting the resulting
equation in the coordinate system shown in Figure 1, we obtain a constant-profit line.

Every point in the feasible region on this line represents a production schedule that will produce the same profit. By doing this for a number of values for P, we obtain a family of constant-profit lines (Fig. 2) that are parallel to each other, since they all have the same slope.

To see this, we write P=50x +80y in the slope-intercept form

Question 11

Graphically, where does maximum profit occur?
What is this point called?

Answer 11

Therefore, the maximum profit occurs at a point where a constant-profit line
is the farthest from the origin but still in contact with the feasible region. This is the optimal solution

Question 12

PROCEDURE Constructing a Model for an Applied Linear Programming Problem

Answer 12

Question 13

General Description of Linear Programming

Answer 13

Question 14

Fundamental Theorem of Linear Programming

Answer 14

Question 15

THEOREM 2 Existence of Optimal Solutions

Answer 15

Question 16

PROCEDURE Geometric Method for Solving a Linear Programming
Problem with Two Decision Variables

Answer 16

Question 17

What is a multiple optimal solution

Answer 17

In general, if two corner points are both optimal solutions to a linear programming problem, then any point on the line segment joining them is also an optimal solution. This is the only way that optimal solutions can occur at noncorner points

Question 18

What is a critical step in the solution of a linear programming problem?

Answer 18

Determining that an optimal solution exists is a critical step in the solution of a
linear programming problem.

If you skip this step, you may examine a corner point table like the one in the solution of Example 2B and erroneously conclude that the maximum value of the objective function is 400.