LESSON 6–5 Linear Optimization

Five-Minute Check (over Lesson 6-4) Then/Now New Vocabulary Key Concept: Vertex Theorem for Optimization Key Concept: Linear Programming Example 1: Maximize and Minimize an Objective Function Example 2: Real-World Example: Maximize Profit Example 3: Optimization at Multiple Points Example 4: Real-World Example: Unbounded Feasible Region Lesson Menu

Find the partial fraction decomposition of . B. C. D. 5–Minute Check 1

Find the partial fraction decomposition of . B. C. D. 5–Minute Check 2

Find the partial fraction decomposition of . B. C. D. 5–Minute Check 3

Find the partial fraction decomposition of . B. C. D. 5–Minute Check 4

Which of the following represents the form of the partial fraction decomposition for B. C. D. 5–Minute Check 5

You solved systems of linear inequalities. (Lesson 0-4) Use linear programming to solve applications. Recognize situations in which there are no solutions or more than one solution of a linear programming application. Then/Now

multiple optimal solutions unbounded optimization linear programming objective function constraints feasible solutions multiple optimal solutions unbounded Vocabulary

Key Concept 1

Key Concept 1

Maximize and Minimize an Objective Function Find the maximum and minimum values of the objective function f(x, y) = 2x + y and for what values of x and y they occur, subject to the following constraints. x + y ≤ 5 2x + 3y ≤ 12 x ≥ 0 y ≥ 0 Begin by graphing the given system of four inequalities. The solution of the system, which makes up the set of feasible solutions for the objective function, is the shaded region, including its boundary segments. The polygonal region of feasible solutions has four vertices. One vertex is located at (0, 0). Example 1

Maximize and Minimize an Objective Function Solve each of the three systems below to find the coordinates of the remaining vertices. Example 1

f(0, 0) = 2(0) + 0 or 0 Minimum value of f(x, y) Maximize and Minimize an Objective Function Find the value of the objective function f(x, y) = 2x + y at each of the four vertices. f(0, 0) = 2(0) + 0 or 0 Minimum value of f(x, y) f(3, 2) = 2(3) + 2 or 8 f(5, 0) = 2(5) + 0 or 10 Maximum value of f(x, y) f(0, 4) = 2(0) + 4 or 4 So, the maximum value of f is 10 when x = 5 and y = 0. The minimum value of f is 0 when x = 0 and y = 0. Answer: max of 10 at f(5, 0), min of 0 at f(0, 0) Example 1

A. max of 12 at f(0, 4), min of 0 at f(0, 0) Find the maximum and minimum values of the objective function f(x, y) = x + 3y and for what values of x and y they occur, subject to the following constraints. 2x + 3y ≤ 12 2x + y ≤ 6 x ≥ 0 y ≥ 0 A. max of 12 at f(0, 4), min of 0 at f(0, 0) B. max of 3 at f(3, 0), min of 0 at f(0, 0) C. max of 10.5 at f(1.5, 3), min 3 of at f(3, 0) D. max of 12 at f(0, 4), min of 3 at f(3, 0) Example 1

Maximize Profit A. AUTOMOTIVE Mechanics at a repair garage carry two name brands of tires United, x, and Royale, y. The number of Royale tires sold is typically less than or equal to twice the number of United tires sold. The shop can store at most 500 tires at one time. Due to factory capacity, the number of Royale tires produced is greater than or equal to 50 more than 0.25 times the number of United tires. The garage earns $25 profit for each United tire and $20 profit for each Royale tire. Write an objective function and a list of constraints that model the given situation. Example 2

Maximize Profit Let x represent the number of United tires the garage sells and y represent the number of Royale tires the garage sells. The objective function is given by f(x, y) = 25x + 20y. The constraints are given by the following. x + y ≤ 500 Garage storage capacity constrain y ≤ 2x Demand constraint y ≥ 0.25x + 50 Production constraint Example 2

Maximize Profit Because x and y cannot be negative, additional constraints are x  0 and y  0. Answer: y ≥ 0.25x + 50 Example 2

Maximize Profit B. AUTOMOTIVE Mechanics at a repair garage carry two name brands of tires United, x, and Royale, y. They typically sell at least twice the number of Royale tires as United. The shop can store at most 500 tires at one time. Due to factory capacity, the number of Royale tires produced is greater than or equal to 50 more than 0.25 times the number of United tires. The garage earns $25 profit for each United tire and $20 profit for each Royale tire. Sketch a graph of the region determined by the constraints to find how many of each tire the garage should sell to maintain optimal profit. Example 2

Maximize Profit The shaded trianglular region has there vertex points at about (167, 333), (360, 140), and about (29, 57). Find the value of f(x, y) = 25x + 20y at each of the three vertices. Example 2

f(360, 140) = 25(360) + 20(140) or 11,800 ←Maximum value of f(x, y) Maximize Profit f(167, 133) = 25(167) + 20(333) or 10,835 f(360, 140) = 25(360) + 20(140) or 11,800 ←Maximum value of f(x, y) f(29, 57) = 25(29) + 20(57) or 1865 Because f is greatest at (360, 140), the garage should sell 360 United tires and 140 Royale tires to earn a maximum profit of $11,800. Answer: 360 United tires and 140 Royale tires Example 2

THEATER A local high school drama club is selling tickets to their spring play. A student ticket, x, costs $5 and a nonstudent ticket, y, costs $7. The auditorium has 522 seats. Based on current ticket sales the number of nonstudent tickets sold is less than or equal to half the number of student tickets sold. Write an objective function and a list of constraints that model the given situation. Determine how many of each kind of ticket the drama club needs to sell to maximize it’s profit. What is the maximum profit? Example 2

A. B. C. D. Example 2

Optimization at Multiple Points Find the maximum value of the objective function f(x, y) = 2x + 2y and for what values of x and y it occurs, subject to the following constraints. y + x ≤ 7 x ≤ 5 y ≤ 4 x ≥ 0 y ≥ 0 Graph the region bounded by the given constraints. The polygonal region of feasible solutions has five vertices at (0, 0), (5, 0), (5, 2), (3, 4), and (0, 4). Find the value of the objective function f(x, y) = 2x + 2y at each vertex. Example 3

Optimization at Multiple Points xxx-new art Example 3

f(0, 0) = 2(0) + 2(0) or 0 f(5, 0) = 2(5) + 2(0) or 10 Optimization at Multiple Points f(0, 0) = 2(0) + 2(0) or 0 f(5, 0) = 2(5) + 2(0) or 10 f(5, 2) =2(5) + 2(2) or 14 f(3, 4) = 2(3) + 2(4) or 14 f(0, 4) = 2(0) + 2(4) or 8 Because f(x, y) = 14 at (5,2) and (3, 4), the problem has multiple optimal solutions. An equation of the line through these two vertices is y = –x + 7. Therefore, f has a maximum value of 14 at every point on y = –x + 7 for 3  x  5. In other words, every point on the segment from (3, 4) to (5, 2) is a maximum. Example 3

Optimization at Multiple Points Answer: f(x, y) = 14 at (3, 4), (5, 2), and every point on the line y = –x + 7 for 3  x  5. Example 3

Find the maximum value of the objective function f(x, y) = 3x + 3y and for what values of x and y it occurs, subject to the following constraints. y + x ≤ 9 x ≤ 6 y ≤ 4 x ≥ 0 y ≥ 0 A. f(x, y) = 18 at (6, 0) B. f(x, y) = 27 at (5, 4) and (6, 3) C. f(x, y) = 27 at (5, 4), (6, 3), and every point on the line y = –x + 9 for 3 ≤ x ≤ 4 D. f(x, y) = 27 at (5, 4), (6, 3), and every point on the line y = –x + 9 for 5 ≤ x ≤ 6. Example 3

Unbounded Feasible Region A. WAREHOUSE The employees of a warehouse work 8-hour shifts. There are two different shifts the employees can work, the day shift from 8 A.M. to 4 P.M. or the second shift from 2 P.M. to 10 P.M. Employees earn $11.50 per hour for the day shift and $13 for second shift. The day shift must have at least 35 employees. The second shift must have at least 25 employees. For the overlapping time, from 2 P.M. to 4 P.M., there must be at least 65 employees working. Write an objective function and list the constraints that model the given situation. Example 4

Answer: f(x, y) = 92x + 104y; x ≥ 35, y ≥ 25, x + y ≥ 65 Unbounded Feasible Region Let x represent the number of employees on the day shift and y represent the number of employees on the second shift. The objective function is given by f(x, y) = 92x + 104y. The constraints on the number of workers are given by x  35 Number of workers on the day shift y  25 Number of workers on the second shift x + y  65 Number of workers from 2 P.M. to 4 P.M. Because x and y cannot be negative, there are also constraints of x  0 and y  0. Answer: f(x, y) = 92x + 104y; x ≥ 35, y ≥ 25, x + y ≥ 65 Example 4

Unbounded Feasible Region B. WAREHOUSE The employees of a warehouse work 8-hour shifts. There are two different shifts the employees can work, the day shift from 8 A.M. to 4 P.M. or the second shift from 2 P.M. to 10 P.M. Employees earn $11.50 per hour for the day shift and $13 for second shift. The day shift must have at least 35 employees. The second shift must have at least 25 employees. For the overlapping time, from 2 P.M. to 4 P.M., there must be at least 65 employees working. Sketch a graph of the region determined by the constraints to find how many day-shift and second-shift employees should be scheduled to optimize labor costs. Example 4

The shaded polygonal region has two vertices at (35, 30) and (40, 25). Unbounded Feasible Region The shaded polygonal region has two vertices at (35, 30) and (40, 25). The optimal cost would be the minimum value of f(x, y) = 92x + 104y. Find the value of the objective function at each vertex. Example 4

Answer: 40 day shift and 25 second shift employees; $6280 Unbounded Feasible Region f(35, 30) = 92(35) + 104(30) or 6340 f(40, 25) =92(40) + 104(25) or 6280 Therefore, to minimize the warehouse costs, the warehouse should have 40 day shift employees and 25 second shift employees to have a minimal cost of $6280. Answer: 40 day shift and 25 second shift employees; $6280 Example 4

YEARBOOK A high school’s yearbook must contain at least 100 pages YEARBOOK A high school’s yearbook must contain at least 100 pages. At least 15 pages must contain color, x, and at least 30 pages must be black and white, y. Pages with color cost $9 each to format and pages that are black and white cost $8 dollars each to format. Write an objective function and a list of constraints that model the given situation. Determine how many of each kind of page is needed to minimize the cost of formatting the yearbook. What is the minimum cost? Example 4

A. f(x, y) = 8x + 9y; x + y ≥ 100, x ≥ 15, y ≥ 30; 15 pages with color, 85 black and white pages, $885 B. f(x, y) = 9x + 8y; x + y ≥ 100, x ≥ 15, y ≥ 30; 15 pages with color, 85 black and white pages; $815 C. f(x, y) = 9x + 8y; x + y ≥ 100, x ≥ 15, y ≥ 30; 15 pages with color, 85 black and white pages, $870 D. f(x, y) = 9x + 8y; x + y ≥ 100, x ≥ 15, y ≥ 30; 85 pages with color, 15 black and white pages, $885 Example 4

LESSON 6–5 Linear Optimization