3.4: Linear Programming

1. 3.4: Linear Programming Basic procedure Unbounded vs. bounded solution regions word problems

2. Linear Programming: The Basic • Our goal is to find the point(s) in the solution region of a system of inequalities that minimizes or maximizes a quantity whose value depends on x and y • Ingredients: - 2 variables - a system of inequalities (constraints) - Objective function: ex. F(x,y) = 2x + 3y

3. Basic Steps (abstract problems) • Graph your system of inequalities • Locate the vertex points • Substitute coordinates of each vertex into objective function • List the vertices which minimize or maximize the objective function

4. Why substitute coords from vertices? After all, solution region contains an infinite # of points! • Theorem: IF there IS a point which minimizes or maximizes the objective function, it will be a vertex of the solution region • Note the qualifier IF in the theorem… we will see that for UNBOUNDED solution regions, there may not be a point which minimizes/maximizes the objective function

5. Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Example 4-1a Step 1 Find the vertices of the region. Graph the inequalities. The polygon formed is a triangle with vertices at (–2, 4), (5, –3), and (5,4).

6. Example 4-1a Step 2 Use a table to find the maximum and minimum values of f(x, y). Substitute the coordinates of the vertices into the function. Answer:The vertices of the feasible region are (–2, 4), (5, –3), and (5, 4). The maximum value is 21 at (5, –3). The minimum value is –14 at (–2, 4).

7. Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Example 4-1b Answer:vertices: (1, 5),(4, 5)(4, 2); maximum: f(4, 2)= 10, minimum: f(1, 5) = –11

8. Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Example 4-2a Graph the system of inequalities. There are only two points of intersection, (–2, 0)and (0, –2).

9. Example 4-2a The minimum value is –6 at (0, –2). Although f(–2, 0) is–4, it is not the maximum value since there are other points that produce greater values. For example, f(2,1) is 7 and f(3, 1) is 10. It appears that because the region is unbounded, f(x, y) has no maximum value. Answer:The vertices are at (–2, 0) and (0, –2). There is no maximum value. The minimum value is –6 at (0, –2).

10. Graph the following system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the function for this region. Example 4-2b Answer:vertices: (0, –3), (6, 0); maximum: f(6, 0)= 6; no minimum

11. Word problem notes • With word problems, often you have to do the following steps yourself: - define variables - write the system of inequalities (constraints) - write the objective function • The objective function is what you are trying to minimize or maximize (ex. Maximize profit, or minimize cost) • Usually you need a constraint (inequality) for each limitation the problem imposes (ex. Only a certain number of hours available, or a certain number of employees available) • Often one constraint is that the variables must be greater than or equal to 0 (non-negative) • in such cases, the x and y axes form boundaries of solution region

12. Example 4-3a Landscaping A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of its crews. Step 1 Define the variables. m= the number of mowing jobs p = the number of pruning jobs

13. Example 4-3a Step 2 Write a system of inequalities. Since the number of jobs cannot be negative, m and p must be nonnegative numbers. m 0, p 0 Mowing jobs take 1 hour. Pruning jobs take 3 hours. There are 9 hours to do the jobs. There are no more than 2 pruning jobs a day. p 2

14. Example 4-3a Step 3 Graph the system of inequalities.

15. The function that describes the income is We want to find the maximum value for this function. Example 4-3a Step 4 Find the coordinates of the vertices of the feasible region. From the graph, the vertices are at (0, 2),(3, 2), (9, 0), and (0, 0). Step 5 Write the function to be maximized.

16. Example 4-3a Step 6 Substitute the coordinates of the vertices into the function. Step 7 Select the greatest amount.

17. Example 4-3a Answer:The maximum values are 360 at (3, 2) and 360 at (9, 0). This means that the company receives the most money with 3 mows and 2 prunings or 9 mows and 0 prunings.

18. Example 4-3b Landscaping A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs and 4 hours for raking jobs. Each crew is scheduled for no more than2 raking jobs per day. Each crew’sschedule is set up for a maximum of 8 hours per day. On the average, the charge forraking a lawn is $50 and the charge for mulching is$30. Find a combination of raking leaves andmulching that will maximize the income the company receives per day from one of its crews.

19. Example 4-3b Answer: 0 raking jobs and 4 mulching jobs