etter Products, Inc., manufactures three products on two machines. In a typical
ID: 2622153 • Letter: E
Question
etter 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:
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 100 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 50% of the units produced and that product 3 must account for at least 20% of the units produced.
(a) How many units of each product should be produced to maximize the total profit contribution?Explanation / Answer
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
d)product 1 = 24 unit
product 2 = 8 unit
product 3 = 16 unit
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