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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 no 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.

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 no 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.

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter4: Linear Programming Models
Section: Chapter Questions
Problem 107P
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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 = $ Зх1 + 2.40х2 + 2.80хз Subject to: X1 + x2 + X3 3.5x1 + 3.7x2 + 4.2xa < 27,000 (1) vat capacity < 45,000 (2) Grain 2 5,000 (3) premium < 12,000 (5) combined premium/light X1 X1 +X3 X1, X2, Хз
Transcribed Image Text: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 = $ Зх1 + 2.40х2 + 2.80хз Subject to: X1 + x2 + X3 3.5x1 + 3.7x2 + 4.2xa < 27,000 (1) vat capacity < 45,000 (2) Grain 2 5,000 (3) premium < 12,000 (5) combined premium/light X1 X1 +X3 X1, X2, Хз
Chart I. Variable Value Reduced Cost Original Val Lower Bound Upper Bound Premium X1 12000 3 2.35 Infinity Regular X2 Light X3 810.81 2.4 3.17 0.65 2.8 -Infinity| 3.45 Chart II. Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound Vat Capacity 14189.19 27000 12810.81 Inifinity Grain 0.6486487 45000 42000 97500 Premium 7000 5000 -Infinity 12000 Premium/Light 0.7297297 12000 5000 12857.14 -How many gallons of each type of beer should be produced to maximize profit? -Has the vat capacity been entirely used? If not, how much is left over? -What would happen to the optimal solution and the optimal profit (be specific) if the profit per gallon of regular increased by $ .50 per gallon? 2) What would happen to the optimal solution and optimal profit (be specific) if the profit per gallon of premium decreased by $1 per gallon?
Transcribed Image Text:Chart I. Variable Value Reduced Cost Original Val Lower Bound Upper Bound Premium X1 12000 3 2.35 Infinity Regular X2 Light X3 810.81 2.4 3.17 0.65 2.8 -Infinity| 3.45 Chart II. Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound Vat Capacity 14189.19 27000 12810.81 Inifinity Grain 0.6486487 45000 42000 97500 Premium 7000 5000 -Infinity 12000 Premium/Light 0.7297297 12000 5000 12857.14 -How many gallons of each type of beer should be produced to maximize profit? -Has the vat capacity been entirely used? If not, how much is left over? -What would happen to the optimal solution and the optimal profit (be specific) if the profit per gallon of regular increased by $ .50 per gallon? 2) What would happen to the optimal solution and optimal profit (be specific) if the profit per gallon of premium decreased by $1 per gallon?
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