A juice manufacturer produces and sells two fruit beverages, cranapple and apple- berry. One gallon of...

   

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A juice manufacturer produces and sells two fruit beverages, cranapple and apple- berry. One gallon of cran-apple beverage is made from three quarts of cranberry juice and one quart of apple juice; one gallon of appleberry beverage is made from two quarts of cranberry juice and two quarts of apple juice. The manufacturer makes a profit of 30 cents on a gallon of cranapple and 40 cents on a gallon of appleberry. Suppose that the manufacture has 2000 quarts of cranberry juice and 1000 quarts of apple juice available. Determine the optimal production policy. In other words, de- termine how many gallons of each beverage should be produced to obtain the highest possible profit without exceeding available supplies (it is ok to leave some supplies unused). (a) Begin the analysis by identifying the objective quantity, that is the quan- tity we are seeking to maximize or to minimize. What is the objective quantity in this problem? (b) Identify the quantities that the objective quantity depends on, introduce the variables that will denote these quantities and express the objective quantity as a function of these variables; (c) Identify the constraints in this problem that limit the values the variables can take. Express these constraints as inequalities; (d) Formulate the problem mathematically without the specifics and the con- text of the practical situation. A juice manufacturer produces and sells two fruit beverages, cranapple and apple- berry. One gallon of cran-apple beverage is made from three quarts of cranberry juice and one quart of apple juice; one gallon of appleberry beverage is made from two quarts of cranberry juice and two quarts of apple juice. The manufacturer makes a profit of 30 cents on a gallon of cranapple and 40 cents on a gallon of appleberry. Suppose that the manufacture has 2000 quarts of cranberry juice and 1000 quarts of apple juice available. Determine the optimal production policy. In other words, de- termine how many gallons of each beverage should be produced to obtain the highest possible profit without exceeding available supplies (it is ok to leave some supplies unused). (a) Begin the analysis by identifying the objective quantity, that is the quan- tity we are seeking to maximize or to minimize. What is the objective quantity in this problem? (b) Identify the quantities that the objective quantity depends on, introduce the variables that will denote these quantities and express the objective quantity as a function of these variables; (c) Identify the constraints in this problem that limit the values the variables can take. Express these constraints as inequalities; (d) Formulate the problem mathematically without the specifics and the con- text of the practical situation.