A small business produces 3 types of cables: A, B, and C. The cost of in-house production is estimated to be $6, $12, and $10 per foot of A, B, and C respectively. The manufacturing process requires machining and finishing. The machining and finishing time needed to produce a foot of each type of cable are presented in the table below.

For the coming production period the firm is contractually obligated to produce 60,000 feet of A, 80,000 feet of B, and 90,000 feet of C. As only 800 hours of machine time and 400 hours of finishing time are available, these demands cannot be met by in-house production alone. The firm has the option of procuring these cables from an external supplier in order to meet the demand. The cost prices (from the external supplier) per foot of cable are as follows: $8 for A, $15 for B, and $12 for C.

The production manager has to decide how much of each type of cable to produce in-house and how much to purchase from the external supplier in order to meet the demands exactly and minimize total cost. Use the data below to answer question A:

Formulating the problem as a Linear Program:

Decision Variables:

x₁ is the number of feet of cable type A produced

x₂ is the number of feet of cable type B produced

x₃ is the number of feet of cable type C produced

y₁ is the number of feet of cable type A procured

y₂ is the number of feet of cable type B procured

y₃ is the number of feet of cable type C procured

Objective Function:

6x₁ + 12x₂ + 10x₃ + 8y₁ + 15y₂ + 12y₃

Constraints:

demand for A: x₁ +y₁ = 60,000

demand for B: x₂ +y₂ = 80,000

demand for C: x₃ +y₃ = 90,000

Machine time- 0.50x₁+0.50x₂ + 0.60x₃=

finishing time- 0.60x₁+0.20x₂ + 0.60x₃=

all decision variables are non-negative

QUESTION A: Using the objective and constraints function above, Show step by step how to get the optimal solution EXACTLY (either setup up in a solver (which each step must be broken down step by step) or solve by hand). Do NOT skip ANY steps. If more info is needed clearly state what information exactly is needed

A B C Cable Type Demand (ft) Production Cost/ft Procurement Cost/ft Machine time needed (mins/ft) Finishing time needed (mins/ft) 60,000 $6 $8 80,000 $12 $15 0.50 90,000 $10 $12 0.60 0.50 0.60 0.20 0.40 A B C Cable Type Demand (ft) Production Cost/ft Procurement Cost/ft Machine time needed (mins/ft) Finishing time needed (mins/ft) 60,000 $6 $8 80,000 $12 $15 0.50 90,000 $10 $12 0.60 0.50 0.60 0.20 0.40