Question

GreenGrass is a manufacturer of lawnmowers. GreenGrass needs 40 pounds of aluminum to manufacture one lawnmower, and there is maximum 2,000 pounds available per month (you cannot store more). The manufacturing cost is $1,200 per unit during the first month, and $1,500 during the second month. Your cost of holding 1 lawnmower in inventory is estimated at $100 per month. You have to meet a demand of 30 lawnmowers in month 1, and 60 lawnmowers in month 2. In addition, GreenGrass’ plant manager wants to produce at least 20 units per month.

GreenGrass would like to minimize its total cost while meeting demand. Formulate this problem as a linear program. By following the 4 steps, write out the linear model. (25 pts.)

Answer 1

.x1 = Lawnmowers manufactured in month 1.

y1= Inventory in month 1.

x2 = Lawnmowers manufactured in month 2.

y2= Inventory in month 2.

• (x1) +(y1)≥ 20
• (x2) +(y2)≥ 20
• x1*40 + y1*40 ≤ 2000
• x2*40 + y2*40 ≤ 2000
• Demand( M1 ) = 30
• Demand( M2 ) = 60

=> x1 + y1 ≤ 50 & x2 + y2 ≤ 50

• 20 ≤ (x1) +(y1)  ≤ 50

Total Cost = 1200(x1+y1) + 1500(x2+y2) + 100y1 + 100y2 = Z => Miniminse Z

=> 1200x1 + 1500x2 + 1300y1 + 1600y2 = Z ( Minimise Z)

We can see that in order to maintain the demand and minimize the total cost: we have to manufacture

x1 = 30 and extra 20 lawnmowers can be manufactured and kept as an inventory to be sold in the next month

x1 = 30 and y1 = 20

and hence ;

x2 = 40 have to be manufactured in the month 2 and the 20 lawnmowers stored as inventory in the month 1(y1) can be sold in the month 2 => Total demand of month 2: x2+y1 = 60 is met.

Hence Solution :

x1 = 30;y1 = 20;x2=40 , y2 = 0 ( Solutions For Minimum Cost)

Z = Minimum Cost = 1200x1 + 1500x2 + 1300y1 + 1600y2 = 1200*30 + 1500*40 + 1300*20 + 0 = 36000+60000+26000 = 122000 $