Question

American Auto is evaluating its marketing plan for the sedans, SUVs, and trucks the company produces. A TV ad featuring this SUV has been developed. The company estimates each showing of this commercial will cost $500,000 and increase sales of SUVs by 3% but reduce sales of trucks by 1% and have no effect on the sales of sedans. The company also has a print ad campaign developed that it can run in various nationally distributed magazines at a cost of $750,000 per title. It is estimated that each magazine title the ad runs in will increase the sales of sedans, SUVs, and trucks by 2%, 1%, and 4%, respectively. The company desires to increase sales of sedans, SUVs, and trucks by at least 3%, 14%, and 4%, respectively, in the least costly manner.

a. Formulate an LP model for this problem.

b. Sketch the feasible region.

c. What is the optimal solution? PLEASE SHOW HOW TO CALCULATE CONSTRAINT POINTS THAT NEED TO BE SHOWN ON GRAPH FOR FEASIBLE REGION THANK YOU

Answer 1

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(a)

Let

x = number of TV ads to run
y = number of magazine ads to run

Min Z = 500x + 750y

ST

3x + 1y ≥ 14
-1x + 4y ≥ 4
0x + 2y ≥ 3
and x, y≥ 0

(b)

Hint to draw constraints

1. To draw constraint 3x+y≥14→(1)

Treat it as 3x+y=14

When x=0 then y=?

⇒3(0)+y=14

⇒y=14

When y=0 then x=?

⇒3x+(0)=14

⇒3x=14

⇒x=143=4.67

x	0	4.67
y	14	0

2. To draw constraint -x+4y≥4→(2)

Treat it as -x+4y=4

When x=0 then y=?

⇒-(0)+4y=4

⇒4y=4

⇒y=44=1

When y=0 then x=?

⇒-x+4(0)=4

⇒-x=4

⇒x=-4

x	0	-4
y	1	0

3. To draw constraint 2y≥3→(3)

Treat it as 2y=3

⇒y=32=1.5

Here line is parallel to X-axis

x	0	1
y	1.5	1.5

Alternaitve diagram:

(c)

The value of the objective function at each of these extreme points is as follows:

Extreme Point Coordinates (x,y)	Lines through Extreme Point	Objective function value z=500x+750y
A(0,14)	1→3x+y≥14 4→x≥0	500(0)+750(14)=10500
B(4,2)	1→3x+y≥14 2→-x+4y≥4	500(4)+750(2)=3500

Problem has an unbounded solution.

Note: In maximization problem, if shaded area is open-ended. This means that the maximization is not possible and the LPP has no finite solution. Hence the solution of the given problem is unbounded.