Question

Bob makes three different silverware - spoons, forks and knives. Each unit (dozen) of spoon gives him a profit of $9, forks of $20 and knives of $17.50. He has 183 hours of machine time, 250 hours of labor and 185 pounds of steel. It takes him 4 hours of machine time, 4 hours of labor time and 3 pounds of steels to make one unit of spoons. Each unit of forks take 4 hours of machine time, 9 hours of labor time and 2 pounds of steel, while knives take 5 hours of machine time, 5 hours of labor and 4 pounds of steel. It is also found that he should make at least 16 units of forks in the process. The solution is

x1 = ________
x2 = ________
x3 = ________

with an objective value of ________
The resources that are used up are________

If it was decided that as many as 20 units of forks could be made instead of 16, how much would profit increase?

Answer 1

Solution is:

z (max) = 687.5

x₁ = 0

x₂ = 16

x₃ = 21

Explanation:

From the problem statement we have:

Resources: machine time ( 183 h) labor (250 h) steel 185 (pounds)

Unit

Spoons 4 4 3

forks 4 9 2

knives5 5 5 4

Profit $ 9 20 17.5

Objective function z = 9*x₁ + 20*x₂ + 17.5 *x₃ to maximize

Subject to:

Availability of machine time : 183 h

4*x₁ + 4*x₂ + 5*x₃ ≤ 183

Availability of labor : 250 h

4*x₁ + 9*x₂ + 5*x₃ ≤ 250

Availability of steel : 185 pounds

3*x₁ + 2*x₂ + 4*x₃ ≤ 185

Requirement:

x₂ ≥ 16

General constraints:

x₁ ≥ 0 x₃ ≥ 0 all integers

After 6 iteration the solution using AtomZmath on-line solver

z (max) = 687.5

x₁ = 0

x₂ = 16

x₃ = 21

Resources used:

Machine time: 16* 4 + 21*5 = 64 + 105 = 169

remains 183 - 169 = 14 h

Labor: 16*9 + 21* 5 = 144 + 105 = 249

remains 250 - 249 = 1 h

Steel : 16*2 + 21*4 = 32 + 84 = 116

remains 185 - 116 = 69 pounds.

If it is decided that 20 units of forks are to be made then

we will need 4*4 = 16 h of machine time

9*4 = 36 h of labor

2*4 = 8 pounds of steel

We can get that from abandom to make one unit of x₃ ??

No because as we said we need 36 hours of labor ( we still have 1 we need 35 more hours ) if we make 20 x₃ insted of 21 we get only 5 hours.

z we got is maximum