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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?

Respuesta :

Answer:

Solution is:

z  (max) = 687.5

x₁   =  0

x₂   = 16

x₃   = 21

Step-by-step 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

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