4. Consider a glass factory that can produce four different kinds of glasses: beer, wine, drinking, and schnapps. Glass manufacturing uses machine hours, labour, and sand as its main inputs.
The demand for each type of glass depends on the price set by the factory. To make matters simple, assume that demand is as follows:
Beer up to 700 units at price ≤ 500, and nothing for price > 500
Wine up to 1100 units at price ≤ 500, and nothing for price > 500
Drinking up to 900 units at price ≤ 350, and nothing for price > 350
Schnapps up to 1400 units at price ≤ 320, and nothing for price > 320
Hence, the factory would simply charge 500 for a beer glass, 500 for wine, 350 for drinking and 320 for Schnapps. Production technology is described by the following table:
Machine Hours
(hrs/unit)
Labour Hours
(hrs/unit)
Sand
(ton/unit)
Beer
1.0
10
0.5
Wine
0.7
12
0.4
Drinking
0.8
6
0.5
Schnapps
0.5
7
0.3
We consider two scenarios. In the first scenario (A), labour and sand are available in unlimited supply at the prices of $20/hr and $100/ton, while machine hours are in fixed supply: a maximum of 2,000 hours. In the second scenario (B), also labour hours are in fixed supply: a maximum of 25,000 hours.
a) Set up a linear program for each of the two scenarios (A and B) to determine the optimal product mix for the glass factory.
b) Explain what the shadow prices mean in each case. In scenario A, explain what the reduced cost for drinking glasses mean.
c) In scenario B, find out how high (at least) the factory should set the price of a champagne glass that uses inputs as follows: 1.2 hours of machine time, 11 hours of labour time and 0.4 tons of sand per unit of production. Hint: Add champagne glass as an additional product that the factory can produce and initially charge a price of zero for it. What does the reduced cost for champagne glass mean?