Explanation Given: Carline’s Quality Candy Confectionery is famous for fudge, chocolate and pralines. Currently candy have shortage of chocolate so they shipped 120 pounds of chocolate in the next shipment in making the 100-pound batch. In a week of 56 work hours available for packaging and equipment available 42 machine hours. A batch of fudge requires 20 pounds of chocolate while a batch of crèmes uses 25 pounds of chocolate. The time require for cooking and processing is, 120 minutes for fudge, 150 minutes for chocolate crèmes and 200 minutes for pralines. The packaging times measured in minute per 1-pound box are 1, 2 and 3 respectively for fudge, crèmes and pralines. The profit per 1-pound box is 50 ¢ on fudge, 40 ¢ on chocolate crèmes and 45 ¢ on pralines. Calculation: Suppose x 1 is the number of 100 − lb batches of fudge, x 2 is the number of 100-lb batches of chocolate crèmes and x 3 is the number of 100 − lb batches of pralines. z Is the maximum profit. Calculation of packing time, measured in minute per 1-pound box are 1, 2 and 3 respectively. Therefore packing time in minute per 100-pound are 100, 200 and 300 respectively. Total working hours in week is 56 . Thus total working hours in minute is 56 ⋅ 60 = 3360 The profit for fudge is 50 ¢ / l b , so the profit for a 100 − lb batch is $ 50 . The profit for chocolate cremes is 40 ¢ / l b , so the profit for a 100 − lb batch is $ 40 . The profit for pralines is 4 5 ¢ / l b , so the profit for a 100 − lb batch is $ 45 . The summaries information are as follows, Chocolates Time require for cooking/processing Packing time ( in min ) Maximum profit per 100 − lb Fudge ( x 1 ) 20 120 100 $ 50 Chocolate Cremes ( x 2 ) 25 150 200 $ 40 Pralines ( x 3 ) 0 200 300 $ 45 Total availability 120 2520 3360 From the above information the constrains are as, 20 x 1 + 25 x 2 ≤ 120 120 x 1 + 150 x 2 + 200 x 3 ≤ 2520 100 x 1 + 200 x 2 + 300 x 3 ≤ 3360 Where x 1 , x 2 and x 3 are all nonnegative. The first of these constraints says that 20 x 1 + 25 x 2 ≤ 120 is less than or perhaps equal to 120 . Suppose s 1 is the slack variable, Therefore, the first equation is 20 x 1 + 25 x 2 + s 1 = 120 The second constraints 120 x 1 + 150 x 2 + 200 x 3 ≤ 2520 can be expressed by adding slack variable s 2 . Therefore, the second equation is, 120 x 1 + 150 x 2 + 200 x 3 + s 2 = 2520 The third constraints 100 x 1 + 200 x 2 + 300 x 3 ≤ 3360 can be expressed by adding slack variable s 3

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Author: Lial

Publisher: PEARSON

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Chapter 7.5, Problem 12E

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To calculation: The number of batches of each types of candy for maximum profit and also determine its profit for a week.

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Chapter 7.5, Problem 11E

Chapter 7.5, Problem 13E

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