Question

A fabric firm has received an order for cloth specified to contain at least 45 pounds of cotton and 25 pounds of silk. The cloth can be woven out on any suitable mix of two yarns, A and B. Material A costs $3 per pound, and B costs $2 per pound. They contain the following proportions of cotton and silk (by weight):

Yarn	Cotton (%)	Silk (%)
A	30	50
B	60	10

Let X: A yarn pounds

   Y: B yarn pounds

Note: Report two digits after the decimal point. Do NOT use thousands-separators (,)

What is an appropriate objective function?

Maximize 3X + 2Y

Minimize 3X + 2Y

Maximize 2X + 3Y

Minimize 2X+ 3Y

a- What are the appropriate constraints?

0.3 X + 0.6 Y ≥ 45	AnswerYesNo
0.3 X + 0.6 Y ≤ 45	AnswerYesNo
0.5 X + 0.1 Y ≤ 25	AnswerYesNo
0.5 X + 0.1 Y ≥ 25	AnswerYesNo
0.6 X + 0.1 Y  ≥ 45	AnswerYesNo
0.3 X + 0.5 Y  ≥ 25	AnswerYesNo
X, Y ≥ 0	AnswerYesNo
X, Y ≤ 0	AnswerYesNo

b- By using the Graphical Approach or Excel Solver,

X =	Answer
Y =	Answer
Z =	Answer

c- If the generated costs of each model are changed to $2 for each pound of yarn and $2.5 for each pound of yarn B, is the optimal solution different or still the same?

AnswerYESNO the optimal solution would AnswerStill SameChange because that the new ratio is Answer which is Answerwithinnot within the range of optimality.

d- If the generated profits of each model are changed to $1,350 for each unit of Alpha 4s and $2,050 for each unit of Betas, the total new profit Z would = Answer $

Answer 1

Q1.Objective function

Min ( 3X+2Y)

Ans. Option 2 is right.

Q2 Constraints

0.3X+0.6Y<=45

0.5X+0.1Y<=25

Ans. 2,3 and 7 are right.

Points of optimality are (0,250) ( 150.0) ( 38.88,55.55)

Value of OF at these points

At (0.250) = 500
At (150,0)=450

at (38,55) = 3x38+2x55 =224

The optimal solution is X= 38 Y=55. The value of Z is 224.

If the objective function changes to 2X+2.5Y, he values

at (0.250) = 625

(150,0) = 300

(38,55) = 76+137.5 = 213.5

The optimal solution still remains the same because the new value of Z is 213.5 which is within range of optimality.

Note: As per policies, I can answer first 4 parts of a problem. Inconvenience is regretted.