A company produces three different types of wrenches: W111, W222, and W333. It has a firm order for 2,000 W111 wrenches, 3,750 W222 wrenches, and 1,700 W333 wrenches. Between now and the order delivery date, the company has only 16,500 fabrication hours and 1,600 inspection hours. The time that each wrench requires in each department is shown below. Also shown are the costs to manufacture the wrenches in-house and the costs to outsource them. For labeling considerations, the company wants to manufacture in-house at least 60% of each type of wrench that will be shipped. Wrench Fabrication Hours Inspection Hours In-House Cost Outsource Cost
W111 2.5 0.25 $17.00 $20.40
W222 3.4 0.3 $19.00 $21.85
W333 3.8 0.45 $23.00 $25.76

Required:
a. How many wrenches of each type should be made in-house and how many should be outsourced?
b. What will be the total cost to satisfy the order?

z (min) =  150090.8    ( monetary units )  ( $ )

x₁  =  2000     x₄  = 0     x₂  =  3382   x₅  =  368   x₃ = 0  x₆ = 1700

Step-by-step explanation:

wrenches produced in-house   ( W111 = x₁   W222 = x₂  W333 = x₃ )

x₁      x₂    and  x₃

wrenches produced outside  (W111 = x₄  W222 = x₅  W333 = x₆ )

x₄     x₅    and  x₆

Objective function:

z  =  17*x₁  +  20.40*x₄  +  19*x₂  + 21.85*x₅ + 23*x₃ + 25.76*x₆   to minimize

First constraint: Manufacturing hours: 16500

2.5*x₁  +  3.4*x₂  +  3.8*x₃   ≤  16500

Second constraint: Inspection hours : 1600

0.25*x₁  +  0.3*x₂  +  0.45*x₃  ≤ 1600

Three demands constraint:

x₁    +   x₄   =  2000

x₂   +  x₅    =  3750

x₃   + x₆     =  1700

x₁  ≥  0     x₂  ≥   0     x₃  ≥  0   x₄  ≥  0   x₅  ≥  0   x₆  ≥  0    all integers

After 6 iterations with on-line solver the solution is:

z (min) =  150090.8    ( monetary units )  ( $ )

x₁  =  2000     x₄  = 0     x₂  =  3382   x₅  =  368   x₃ = 0  x₆ = 1700