5 Beerco manufactures ale and beer from corn, hops, and malt. Currently, 40 lb of corn, 30 lb of hops, and 40 lb of malt are available. A barrel of ale sells for $40 and requires 1 lb of corn, 1 lb of hops, and 2 lb of malt. A barrel of beer sells for $50 and requires 2 lb of corn, 1 lb of hops, and 1 lb of malt. Beerco can sell all ale and beer that is produced.

Assume that Beerco’s goal is to maximize total sales revenue and solve the following LP:

max z = 40ALE + 50BEER s.t. 2ALE + 2BEER 40 (Corn constraint)

s.t. 2ALE + 2BEER 30 (Hops constraint)

s.t. 2ALE + 2BEER 40 (Malt constraint)

s.t. 2 ALE, BEER >= 0 ALE = barrels of ale produced, and BEER barrels of beer produced

a Graphically find the range of values for the price of ale for which the current basis remains optimal.
b Graphically find the range of values for the price of beer for which the current basis remains optimal.
c Graphically find the range of values for the amount of available corn for which the current basis remains optimal.
What is the shadow price of the corn constraint?
d Graphically find the range of values for the amount of available hops for which the current basis remains optimal.
What is the shadow price of the hops constraint?
e Graphically find the range of values for the amount of available malt for which the current basis remains optimal.
What is the shadow price of the malt constraint?
f Find the shadow price of each constraint if the constraints were expressed in ounces instead of pounds.
g Draw a graph of the optimal z-value as a function of the price of ale.

h Draw a graph of the optimal z-value as a function of the amount of available corn.
i Draw a graph of the optimal z-value as a function of the amount of available hops.
j Draw a graph of the optimal z-value as a function of the amount of available malt.

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MAX 400 X1 + 500 X2 - 100 R SUBJECT TO 2) 2 X1 + 3 X2 - R = 25 END LP OPTIMUM FOUND AT STEP OBJECTIVE FUNCTION VALUE 1) 19000.0000 VARIABLE VALUE X1 20.000000 X2 25.000000 R 15.000000 2 REDUCED COST 0.000000 0.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 3) 10.000000 4) 0.000000 5) 0.000000 6) 0.000000 100.000000 0.000000 200.000000 0.000000 -200.000000 2 NO. ITERATIONS= RANGES IN WHICH THE BASIS IS UNCHANGED: OBJ COEFFICIENT RANGES VARIABLE CURRENT COEF X1 400.000000 X2 500.000000 R -100.000000 ALLOWABLE INCREASE INFINITY 200.000000 ALLOWABLE DECREASE 100.000000 INFINITY 100.000000 100.000000 RIGHTHAND SIDE RANGES ROW CURRENT RHS ALLOWABLE INCREASE ALLOWABLE DECREASE 2 100.000000 15.000000 INFINITY 3 120.000000 INFINITY 10.000000 456 4 70.000000 3.333333 0.000000 20.000000 0.000000 INFINITY 6 25.000000 0.000000 2.500000