You sell small widgets for $10 each. Each small widget costs $4 each. Each small
ID: 3120812 • Letter: Y
Question
You sell small widgets for $10 each. Each small widget costs $4 each. Each small widget needs 3 ounces of plastic and 5 minutes of labor. You must make at least 100 small widgets. You sell large widgets for $20 each. Each large widget costs $ 17 each. Each large widget needs 8 ounces of plastic and 13 minutes of widgets. You must make at least 200 large widgets You can sell all widgets you make. You have 150 pounds of plastic. You have 8 workers who work 7 hours. What is the optimal product mix?
Explanation / Answer
This is an LP (Linear Programming) Problem.
Formulation of LP
Let x = number of small widget produced and y = number of large widget produced.
Then, the LPP is:
Max z = 6x + 3y [profit function]
[You sell small widgets for $10 each. Each small widget costs $4 each. => profit is $6. You sell large widgets for $20 each. Each large widget costs $ 17 each. => profit is $3]
Subject to:
3x + 8y 2400 [constraint on plastic. Note: requirements are given in ounces, but availability is given in pounds and hence convert 1 pound = 16 ounces.]
5x + 13y 3360 [constraint on time. Note: requirements are given in minutes, but availability is given in hours and hence convert 1 hour = 60 minutes.]
x 100 [because, must make at least 100 small widgets]
y 200 [because, must make at least 200 large widgets]
x, y 0 [non-negativity condition]
The above can be solved by graphical method.
The corner points on the feasible region are: A =(100, 200), B = (100, 2800/13) and C = (140, 200).
Substituting these points on the objective function, z = 6x + 3y, we get
ZA = 1200, ZB = 1246.15, ZC = 1400,
ZC being the maximum, the optimum mix is (140, 200).
Thus,
optimum product mix is: produce 140 small widgets and 200 large widgets and the resulting profit is $1400 ANSWER
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