(a) Explanation Given Information: There are three flavours that an ice cream factory offers, Creamy Vanilla, Continental mocha and Succulent Strawberry. There are 200 eggs in stock. Each batch of Creamy Vanilla takes 2 eggs, of Continental Mocha takes 1 egg and of Succulent Strawberry takes 1 egg. There are 120 cups of milk available in stock. Each batch of Creamy Vanilla takes 1 cup of milk, of Continental Mocha takes 1 cup of milk and of Succulent Strawberry takes w cups of milk. There are 200 cups of cream available. Each batch of Creamy Vanilla takes 2 cups of cream, of Continental Mocha takes 2 cups of cream and of Succulent Strawberry takes 2 cups of cream. Formula used: According to the Simplex method, a linear programming problem can be solved using the following steps, Step 1: Use slack variables and write the inequalities in equation form. Step 2: Write the equations in an augmented matrix. Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Step 4: Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column. Step 6: Repeat the above steps till all the entries in the bottom row are non negative. Use row operations to clear the column of the augmented matrix. Calculation: Consider x represent number of batches of Creamy Vanilla, y represents the number of sections of Continental Mocha and z represents the number of sections of Succulent Strawberry. Thus, the maximum profit is given by p = 3 x + 2 y + 4 z . And, the constraints are 2 x + y + z ≤ 200 x + y + 2 z ≤ 120 2 x + 2 y + 2 z ≤ 200 x ≥ 0 , y ≥ 0 , z ≥ 0 Consider the maximisation problem. Maximize p = 3 x + 2 y + 4 z Subject to 2 x + y + z ≤ 200 x + y + 2 z ≤ 120 2 x + 2 y + 2 z ≤ 200 x ≥ 0 , y ≥ 0 , z ≥ 0 Step 1: Use slack variables and write the inequalities in equation form. Add the slack variables s , t and u . 2 x + y + z + s = 200 x + y + 2 z + t = 120 2 x + 2 y + 2 z + u = 200 Step 2: Write the equations in an augmented matrix. The equations are 2 x + y + z + s = 200 x + y + 2 z + t = 120 2 x + 2 y + 2 z + u = 200 − 3 x − 2 y − 4 z + p = 0 Write the equations in augmented matrix. x y z s t u p s 2 1 1 1 0 0 0 200 t 1 1 2 0 1 0 0 120 u 2 2 2 0 0 1 0 200 p − 3 − 2 − 4 0 0 0 1 0 Step 3: Choose the smallest negative number on the left side of the bottom row and pivot the column. Since, the smallest negative number on the left side of the bottom row is − 4 . Thus, choose the left most entry and pivot the z column. Step 4: Select the pivot which is the smallest of the test ratios a b , where a is the entry in the right most column and b is the corresponding entry in the pivot column. Since, the test ratios are neither 0 nor negative. Thus, the test ratios are 200 1 = 200 , 120 2 = 60 and 200 2 = 100 . Since, 60 is smallest. Thus, the pivot is 2 in column z . x y z s t u p s 2 1 1 1 0 0 0 200 t 1 1 2 0 1 0 0 120 u 2 2 2 0 0 1 0 200 p − 3 − 2 − 4 0 0 0 1 0 Step 5: Clear the pivot column using the pivot and label the pivot row with the label of the pivot column (b) To determine To calculate: The amount of ingredients left if all the ingredients have not been used up. (c) To determine To calculate: The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit, if due to poor strawberry harvest, the factory cannot process more than 10 batches of succulent Strawberry.

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 5.3, Problem 26E

(a)

To determine

To calculate: The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit.

(b)

To determine

To calculate: The amount of ingredients left if all the ingredients have not been used up.

(c)

To determine

To calculate: The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit, if due to poor strawberry harvest, the factory cannot process more than 10 batches of succulent Strawberry.

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Chapter 5.3, Problem 25E

Chapter 5.3, Problem 27E

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Finite Mathematics

Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 6E Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - In Exercises 1-26, sketch the region that...Ch. 5.1 - Prob. 9E Ch. 5.1 - In Exercises 1-26, sketch the region that...

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The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit.

Solution Summary: The author explains how a linear programming problem can be solved using the Simplex method.

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To calculate: The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit.

To calculate: The amount of ingredients left if all the ingredients have not been used up.

To calculate: The number of batches of the flavour Creamy Vanilla, Continental Mocah and Succulent Strawberry to maximize the profit, if due to poor strawberry harvest, the factory cannot process more than 10 batches of succulent Strawberry.

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Chapter 5 Solutions