Question

Better products, inc., manufactures three products on two machines. in a typical week, 40 hours are available on each machine. the profit contribution and production time in hours per unit are as follows: category product 1 product 2 product 3 profit/unit $31 $60 $16 machine 1 time/unit 0.5 1.5 1.5 machine 2 time/unit 1 1.5 1 two operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. only one operator is required for machine 2. a maximum of 120 labor-hours is available for assignment to the machines during the coming week. other production requirements are that product 1 cannot account for more than 55% of the units produced and that product 3 must account for at least 25% of the units produced. how many units of each product should be produced to maximize the total profit contribution

Answer 1

The explanation of the question is given below for the company that manufactures three products.

a) max 30x? + 50x? + 20x? //objective function

st.

0.5x? + 2.0x? + 0.75x? ? 40 //machine 1 time available

1.0x? + 1.0x? + 0.5x? ? 40 //machine 2 time available

2.0x? + 5.0x? + 2.0x? ? 100 //total labor available

-0.5x? + 0.5x? + 0.5x? ? 0 //product 1 amount constraint

0.2x? + 0.2x? - 0.8x? ? 0 //product 3 amount constraint

x?, x?, x? ? 0

The maximum profit obtainable from production of Product 1 alone is $1200, limited by time on Machine 2. The maximum profit obtainable from either Product 2 or Product 3 alone is $1000, limited by available labor.

Thus, the preferred production would be of Product 1. The constraint that other products be produced in equal or greater quantity puts a premium on production rate of other products. Product 3 is produced using less resource than product 2.

The production of a mix that consists of 1 unit of Product 3 for each unit of Product 1 is constrained by available labor, which allows production of 25 units of each, for a total profit of $1250.

_____

If it weren't for the constraint on relative production rates, a mix of 3 of Product 1 and 2 of Product 3 would fully utilize both Machine 2 and Labor, resulting in a profit of $1300.

projected weekly profit = $ 1250

b) Machine Hours Schedule:

machine 1 = 31.25 hrs.

machine 2 = 37.5 hrs

c) additional labor hour = 12.50 hr

product 2 = 8 unit

product 3 = 16 unit

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