1.(33 points) A local brewery produces three types of beer:premium, regular and light.The brewery has enough vat capacity to produce 27,000 gallons of beer per month.A gallon of premium beer requires 3.5 pounds of grain, a gallon of regular requires 3.7 pounds of grain, and a gallon of light requires 4.2 pounds of grain.The brewery is able to acquire 45,000 pounds of grain every month.While the brewery's largest seller is regular beer, it wants to have a competitive market mix of beer.Thus, the brewery wishes to produce at least 5,000 gallons of premium beer, but not more than 12,000 gallons of light and premium beer combined.The brewery makes a profit of $3 per gallon of premium beer, $2.40 per gallon of regular beer, and $2.80 per gallon of light beer.The brewery manager wants to know how much of each type of beer should be produced in order to maximize profit.Use the following model and accompanying POM printout to answer the following questions.

Note:

x₁ = gallons of premium

x₂ = gallons of regular

x₃ = gallons of light

maximize Z = $

3x₁ + 2.40x₂ + 2.80x₃

Subject to:

x₁ + x₂ + x₃ 27,000 (1) vat capacity

3.5x₁ + 3.7x₂ +4.2x₃ 45,000 (2) Grain

x₁ 5,000 (3) premium

x₁ + x₃ 12,000 (5) combined premium/light

x₁, x₂, x₃ 0

Chart I.

Variable Value Reduced Cost Original Val Lower Bound Upper Bound Premium X1 12000 0 3 2.35 Infinity Regular X2 810.81 2.4 0 3.17 Light X3 0 0.65 2.8 -Infinity 3.45Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound Vat Capacity 0 14189.19 27000 12810.81 Inifinity Grain 0.6486487 45000 42000 97500 Premium 0 7000 5000 -Infinity 12000 Premium Light 0.7297297 0 12000 5000 12857.14