Safari Bookstores want sell an exclusive line of backpacks. The line has four mo
ID: 3146132 • Letter: S
Question
Safari Bookstores want sell an exclusive line of backpacks. The line has four models of backpacks: canvas, plastic, nylon, and leather. The bookstores want no fewer than 15 and no more than 40 of any one model. The bookstores have agreed to pay $35.50 for each canvas backpack, $39.50 for each plastic backpack, $42.50 for each nylon backpack, and $69.50 for each leather backpack.
The manufacturer, a local company with limited manufacturing capabilities, has to deliver the backpacks by the end of the week. The company has three part-time workers available to do the sewing. One employee can work on either canvas or plastic backpacks, can complete a backpack in 1.5 hours, and charges $7/hour. Because it’s fall break, she can commit to 90 hours of work in the next week. A second employee can work on nylon backpacks, completes one unit every 1.7 hours, charges $8/hour and can commit to 42.5 hours. The third worker can sew leather, can complete a backpack in 1.9 hours, charges $9/hour, and can work 80 hours.
The following table provides additional information about each backpack:
Canvas
Plastic
Nylon
Leather
Square yards per pack
2.25
2.40
2.10
2.60
Square yards available
200
350
700
550
Cost per square yard
$4.50
$4.25
$7.65
$9.45
Formulate as a linear program to determine the combination of backpacks that maximizes the total profit.
Canvas
Plastic
Nylon
Leather
Square yards per pack
2.25
2.40
2.10
2.60
Square yards available
200
350
700
550
Cost per square yard
$4.50
$4.25
$7.65
$9.45
Explanation / Answer
let w be number of canvas backpacks
x be number of plastic backpacks
y be number of Nylon back packs
z be number of leather back packs
unit profit on canvas backpacks= selling price -material cost-labour cost=35.5-(2.25*4.5)-7*1.5=$14.875
unit profit on plastic backpacks=39.5-(2.4*4.25)-7*1.5=$18.8
unit profit on nylon backpacks =42.5-(2.10*7.65)-8*1.7=12.835
unit profit on leather backpacks=69.5-(2.6*9.45)-(9*1.9)=27.83
objective function is to maximize profit
max P:14.875w+18.8x+12.835y+27.83z
subjected to
1.5w+1.5x<=90
1.7y<=42.5
w,x,y,z>=15
w,x,y,z<=40
2.25w<=200
2.4x<=350
2.1y<=700
2.6z<=550
w,x,y,z>=0
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