Question

a widget manufacturing company makes big widgets and small widgets. Each widget must processed in a wood shop and in a paint shop. It takes 480 minutes of process time per work day per shop. If the paint shop paints only small widgets, then 60 can be painted per day. If the paint shop paints large widgets only then 40 can be painted per day. If the wood shop only process large widgets only then it could process 50 widgets per day. If the wood shop can process small widgets it could process 50 per day. Each small widget makes $200 profit and each large widget makes $300 profit. First calculate duration . How long does it take to paint a small and large widget. Define decision variables and formulate Linear Program to determine optimal daily product mix to maximize profit. Then use graphical solution to solve Linear Program and provide optimal solution and its objective function.

Answer 1

Small widget time at paint shop = 480/60 = 8 minutes
Large widget time at paint shop = 480/40 = 12 minutes

Small widget time at wood shop = 480/50 = 9.6 minutes
Large widget time at wood shop = 480/50 = 9.6 minutes

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Max Z = 200 S + 300 L
s.t.
8.0 S + 12.0 L <= 480
9.6 S + 9.6 L <= 480
S, L >= 0

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Corner point	S	L	Z = 200 S + 300 L
Origin	0	0	0
A	30	20	12,000 (max)
B	0	40	12,000 (max)
C	50	0	10,000

Note that the objective line has the same slope as that of the line 8 S + 12 L = 480. So, any point on the line segment AB will be optimal.

That's why both the corner points A and B shows maximum profit of 12,000. So, the optimal solution can be as follows:

S=30, L=20 with max Z = 12,000
or,
S=0, L=40 with max Z = 12,000
or,
Any point on the AB line segment, that will also give max Z = 12,000