Question

Question 4 (3 points) Set up and solve the following simple linear optimization model: MAX: 14.9 x + 23.8 y subject to: 2x + 2y = 20 3x 2 17 5x + 1y s 78 x.y 20 What is the value of the objective function at the optimal solution? Round your answer to one decimal place. Your Answer: Answer Question 5 (8 points) -

Answer 1

Answer:

Data provided in the question:

(Maximize)

Objective function, M = 14.9.2 + 23.8y

subjected to the constraints

  $2x+2y \geq 20$

  

3.6 > 17

  $5x+y \leq 78$

  $x\geq 0$

  $y\geq 0$  

Now, in order to maximize the objective function, we need to solve the above inequalities graphically and find the feasible region and the corner points.

We can graph the above inequalities separately and then plot them on the same graph as -

1. $2x+2y \geq 20$

-60- -55 -50 45 40 -35 30- 25 20- 15 加 5- -4 -2 0 2 g 10 12 14 16 18 -10-

2.   

3.6 > 17

-60 55 50 45 10- 35 30- 25 20- 15- 10 5 =4 -2 ០ 2 6 8 10 12 14 16 18 --- -10

3.   $5x+y \leq 78$

-60 55 -50 -45 -40- -35 30 25 20- 15 10 -5 -4 -2 0 2 6 自 10 12 14 16 18 -51 -104

4.   $x\geq0$

60 55, -50 -45- -40- 35 -30 25. 200 15. 100 5- - 2 0 6 8 10 12. 14 16 18 -5- -100

5.   $y\geq0$

60 55, -50 45 40 35 30 25. 20 15 10 5 4 -2. 0 2 4 6 8 10 12 14 16 18. -5- -10

Now, we can plot all the inequalities on the same graph as -

-60 55 (5.667, 49.667) -50 45 -40 35 30 25 20 -15 10 -5 (5.667, 4.333) =4 -2 0 2 18 (10, 0) (15.6, 0) -5- -10-

As we can see from the above graph that the corner points of the feasible region are -

.

(5.667, 4.333), (10,0), (5.667, 49.667) and (15.6, 0)

Therefore, we should evaluate the objective function at all these points as -

a.

  

  .

b.

  

  .

c.

  

  $=1266.5129$.

d.

  $=232.44+0$

  .

As we can see that the objective function M is maximum when x = 5.667 and y = 49.667.

Thus, the value of the objective function at the optimal solution (x = 5.667 and y = 49.667) is 1266.5 (rounded to 1 decimal place) .