Wild west produces two types of cowboy hats

1. For the Reddy Mikks model, construct each of the following constraints, and express it with a linear left-hand side and a constant right-hand side:

(a) The daily demand for interior paint exceeds that of exterior paint by at least 1 ton.

X2 – x1 >= 1 OR –X1 + X2 >=1

(b) The daily usage of raw material M1 in tons is at most 8 and at least 5.

X1 + 2X2 >= 3 and X1 + 2X2<= 6

(c) The demand for exterior paint cannot be less than the demand for interior paint.

X2 >= X1 OR X1 – X2 <=0

(d) The maximum quantity that should be produced of both the interior and the exterior paint is 15 tons.

X1 + X2 >= 3

(e) The proportion of exterior paint to the total production of both interior and exterior paints must not exceed .5.

X2 / (X1+ X2) <= 0.5 OR 0.5X1 – 0.5X2 >=0

2. Determine the best feasible solution among the following (feasible and infeasible) solutions of the Reddy Mikks model:

(a) x1 = 1, x2 = 2.

(X1,X2)= (1,4)

(X1,X2)>=0

6(1) + 4(4) = 22 <24

1(1) + 2(4) = 9 ≤ 6

IT IS INFEASIBLE

(b) x1 = 3, x2 = 1.

(X1,X2)= (2,2)

(X1,X2) >= 0

6(2)+ 4(2) = 20 <=24

1(2) + 2(2) =6 <=6

-1(2) +1(2) =0 <=1

1(2) =2<=2

IT IS FEASIBLE

Z= 5( 2) + 4(2) = $18

(c) x1 = 3, x2 = 1.5.

(X1,X2) = ( 3,1.5)

(X1,X2)>=0

6(3) = 4(1.5)= 24 <=24

1(3) + 2(1.5) = 6 <=6

-1(3) + 1(1.5)= -1.5<=1

1(1.5) = 1.5<=2

IT IS FEASIBLE

Z= 5 (3) + 4 (1.5) = $21

(d) x1 = 2, x2 = 1.

(X1,X2) = (2,1)

6(2)+ 4(1) = 16 <= 24

1(2) + 2(1) =4 <=6

-1(2) + 1(1) =-1 <=1

1(1) =1 <=2

IT IS FEASIBLE

Z= 5(2) + 4(1) = $14

(e) x1 = 2, x2 = -1.

(X1,X2) = (2,-1)

X1>=0, X2<0

IT IS INFEASIBLE

5. Determine the feasible space for each of the following independent constraints, given that x1, x2≥ 0.

*(a) -3x1 + x2 ≤ 6.

(b) x1 - 2x2 ≥ 5.

(c) 2x1 - 3x2 ≤ 12.

(d) x1 - x2 ≤ 0.

*(e) -x1 + x2 ≥ 0

6. Identify the direction of increase in z in each of the following cases:

*(a) Maximize z = x1 - x2.

(b) Maximizez = -8x1 - 3x2.

(c) Maximizez = -x1 + 3x2.

*(d) Maximizez = -3x1 + x2

12. The Continuing Education Division at the Ozark Community College offers a total of 30 courses each semester. The courses offered are usually of two types: practical and humanistic. To satisfy the demands of the community, at least 10 courses of each type must be offered each semester. The division estimates that the revenues of offering practical and humanistic courses are approximately $1500 and $1000 per course, respectively.

(a) Devise an optimal course offering for the college.

no.of courses each semister=30

practical course means more revenue

the optimal course=20

practical and humanistic= 10

X1=20 , X2=10

Maximize Z= 1500X1+1000X2

20 practical= 20*1500=30,000$

10 humanistic= 10*1000=10,000$

total income is Maximize Z= 1500X1+1000X2= 30,000$+10,000$=40,000$

= X1+X2≤ 30

X1≥ 10X2≥ 10X1,X2≥ 10

(b) Show that the worth per additional course is $1500, which is the same as the revenue per practical course. What does this result mean in terms of offering additional courses?

Z= 41,500$

Z= 41,500 – 40,000

Profit = 1,500

Additional course is practical

16. Wild West produces two types of cowboy hats. A Type 1 hat requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for Type 1 and Type 2 are 150 and 200 hats per day, respectively. The profit is $8 per Type 1 hat and $5 per Type 2 hat. Determine the number of hats of each type that maximizes profit.

X1=Type1 hat

X2=Type2 hat

Maximize

Z = 8X1 + 5X2

In constraint to

2X1 + X2 <= 400

X1 <= 150

X2 <= 200

Result:

X1 = 100 hats

X2 = 200 hats

Z = $1800

To maximize profit of $1800 they need to make 100 Type 1 hats and 200 Type 2 hats

20. The Burroughs Garment Company manufactures men’s shirts and women’s blouses for Walmark Discount Stores. Walmark will accept all the production supplied by Burroughs. The production process includes cutting, sewing, and packaging. Burroughs employs 25 workers in the cutting department, 35 in the sewing department, and 5 in the 80 Chapter 2 Modeling with Linear Programming packaging department. The factory works one 8-hr shift, 5 days a week. The following table gives the time requirements and profits per unit for the two garments. Minutes per unit Garment Cutting Sewing Packaging Unit profit ($) Shirts 20 70 12 8 Blouses 60 60 4 12 Determine the optimal weekly production schedule for Burroughs

x1: shirts produce per week and

x2: blouses produce per week.

Profit=8 x1+12 x2

Time on cutting = 20x1+60x2mts

Time on packaging= 12x1+4x2mts

Time on sewing =70x1+60x2mts

z= 8 x1+ 12 x2

solution 20x1+60x2≥25*40*60

12x1+4x2≥ 5*40*60

70x1+60x2= 35*40*60

x1, x2≥0, integers