PROBLEM #1 A small candy shop is preparing for the holiday season. The owner mus
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Question
PROBLEM #1
A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pound peanuts, and the standard mix has 1/2 pound raisins and 1/2 pound peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with.
Peanuts cost $.60 per pound and raisins cost $1.50 per pound. The deluxe mix will sell for $2.90 per pound, and the standard mix will sell for $2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold.
If the goal is to maximize profits, how many bags of each type should be prepared? What is the expected profit?
FORMULATE THIS AS A LINEAR PROGRAM. DO NOT SOLVE.
Decision Variables:
Objective Function:
Constraints:
PROBLEM #2
A retired couple supplement their income by making fruit pies, which they sell to a local grocery store. During the month of September, they produce apple and grape pies. The apple pies are sold for $1.50 to the grocer, and the grape pies are sold for $1.20. The couple is able to sell all of the pies they produce owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month. For the month of September, they have 1,200 cups of sugar and 2,100 cups of flour. Each apple pie requires 1½ cups of sugar and 3 cups of flour, and each grape pie requires 2 cups of sugar and 3 cups of flour.
Determine the number of grape and the number of apple pies that will maximize revenues if the couple working together can make an apple pie in six minutes and a grape pie in three minutes. They plan to work no more than 60 hours.
FORMULATE THIS AS A LINEAR PROGRAM. DO NOT SOLVE.
PROBLEM #3
A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pound peanuts, and the standard mix has 1/2 pound raisins and 1/2 pound peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with.
Peanuts cost $.60 per pound and raisins cost $1.50 per pound. The deluxe mix will sell for $2.90 per pound, and the standard mix will sell for $2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold.
If the goal is to maximize profits, how many bags of each type should be prepared? What is the expected profit?
SOLVE USING SOLVER: Save the Answer and Sensitivity Reports to Answer the following questions:
If the selling price of the deluxe bags increased by $0.30 (to $3.20), would the optimal mix change? What would the total profit?
If the selling price of the standard bags decreased by $0.30 (to $2.25, would the optimal mix change? What would the total profit?
Would you be willing to purchase an additional 20 pounds of raisins for one dollar per pound (total cost of $20)? Why or why not? What would the new optimal solution be?
PROBLEM #4
A retired couple supplement their income by making fruit pies, which they sell to a local grocery store. During the month of September, they produce apple and grape pies. The apple pies are sold for $1.50 to the grocer, and the grape pies are sold for $1.20. The couple is able to sell all of the pies they produce owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month. For the month of September, they have 1,200 cups of sugar and 2,100 cups of flour. Each apple pie requires 1½ cups of sugar and 3 cups of flour, and each grape pie requires 2 cups of sugar and 3 cups of flour.
Determine the number of grape and the number of apple pies that will maximize revenues if the couple working together can make an apple pie in six minutes and a grape pie in three minutes. They plan to work no more than 60 hours.
SOLVE USING SOLVER: Save the Answer and Sensitivity Reports to Answer the following questions:
Currently you recommend that they produce both Apple and Grape pies. If the profit for each grape pie was $1.40 (instead of $1.20), would you continue to make both pies? What would the production mix be? What would the total profit be?
Would you be willing to purchase an additional 25 cups of sugar for one dollar per pound? Why or why not? What would the new optimal solution be?
If 100 cups of flour were contaminated, would this affect your optimal production mix? (Yes or No) How would this effect the total profit?
Would you be willing to hire a temp employee for $10 per hour? Why or why not?
A small candy shop is preparing for the holiday season. The owner must decide how many bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delite to put up. The deluxe mix has 2/3 pound raisins and 1/3 pound peanuts, and the standard mix has 1/2 pound raisins and 1/2 pound peanuts per bag. The shop has 90 pounds of raisins and 60 pounds of peanuts to work with.
Peanuts cost $.60 per pound and raisins cost $1.50 per pound. The deluxe mix will sell for $2.90 per pound, and the standard mix will sell for $2.55 per pound. The owner estimates that no more than 110 bags of one type can be sold.
If the goal is to maximize profits, how many bags of each type should be prepared? What is the expected profit?
Explanation / Answer
Problem 1:
Let d be delux mix s be standard mix
Decision Variables: d & s
Objective Function: Maximum profit=1.7d+ 1.2s
Constraints: d<=100, s<=100,
(2/3)d +(1/3)<=90
(1/3)d+(1/2)s<=60
Problem 2:
Let A be the number of apple pie and G be the number of grape pie.
Decision variables : A & G
Constraint: 6*A+3*G<=60hours or 3600 minutes
1.5A+2G=1200, 3A+3G=2100
Objective Function: Maximum revenue= 1.5A+1.2G
Problem 3:
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