Problem 2:•BIG produces four types of yogurts A, B, C, and D. Each box of yogurt carries protein, sugar, and Fat. The following table provides information about the content of each yogurt box, together with each box's production cost.BIG wants to produce yogurts in such a way that will provide at least 300 grams of protein with minimum production cost. However, BIG also must ensure they provide at most 200 gm of sugar and 200 gm of Fat while providing the required protein. Moreover, BIG has to produce at least one unit of each type of yogurt, and the number of yogurt A produced should not exceed the number of yogurt C produced. ### Yogurt Nutritional and Cost AnalysisThis table provides a detailed comparison of different yogurt products based on their nutritional content and production cost. The yogurts are labeled A, B, C, and D, with varying amounts of protein, sugar, fat, and associated production costs per box.#### Nutritional Content and Cost Breakdown:| Yogurt Type | Protein/Box (g) | Sugar/Box (g) | Fat/Box (g) | Production Cost/Box ($) ||-------------|-----------------|---------------|-------------|-------------------------|| Yogurt A | 15 | 3 | 1.5 | 2 || Yogurt B | 20 | 2 | 1 | 3 || Yogurt C | 30 | 2.5 | 1.5 | 5 || Yogurt D | 10 | 2 | 1.5 | 1.5 |#### Analysis:- **Yogurt A** offers a moderate protein content and sugar level with a mid-range cost.- **Yogurt B** provides a higher protein content with less sugar, costing slightly more.- **Yogurt C** maximizes protein per box with moderate sugar and one of the highest production costs.- **Yogurt D** offers the lowest protein and cost, with the same fat content as others except Yogurt B.This information aids consumers in selecting yogurt based on nutritional needs and budget considerations.

Let the amounnt of Yogurt A, B, C, D produced by BIG is x1, x2 , x3, x4 to minimize the production…

Answered: Problem 2: •BIG produces four types…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 2:

•BIG produces four types of yogurts A, B, C, and D. Each box of yogurt carries protein, sugar, and Fat. The following table provides information about the content of each yogurt box, together with each box's production cost.

BIG wants to produce yogurts in such a way that will provide at least 300 grams of protein with minimum production cost. However, BIG also must ensure they provide at most 200 gm of sugar and 200 gm of Fat while providing the required protein. Moreover, BIG has to produce at least one unit of each type of yogurt, and the number of yogurt A produced should not exceed the number of yogurt C produced.

### Yogurt Nutritional and Cost Analysis

This table provides a detailed comparison of different yogurt products based on their nutritional content and production cost. The yogurts are labeled A, B, C, and D, with varying amounts of protein, sugar, fat, and associated production costs per box.

#### Nutritional Content and Cost Breakdown:

| Yogurt Type | Protein/Box (g) | Sugar/Box (g) | Fat/Box (g) | Production Cost/Box ($) |
|-------------|-----------------|---------------|-------------|-------------------------|
| Yogurt A | 15 | 3 | 1.5 | 2 |
| Yogurt B | 20 | 2 | 1 | 3 |
| Yogurt C | 30 | 2.5 | 1.5 | 5 |
| Yogurt D | 10 | 2 | 1.5 | 1.5 |

#### Analysis:
- **Yogurt A** offers a moderate protein content and sugar level with a mid-range cost.
- **Yogurt B** provides a higher protein content with less sugar, costing slightly more.
- **Yogurt C** maximizes protein per box with moderate sugar and one of the highest production costs.
- **Yogurt D** offers the lowest protein and cost, with the same fat content as others except Yogurt B.

This information aids consumers in selecting yogurt based on nutritional needs and budget considerations.

Expert Solution

Step 1: Formulation of the problem:

Let the amounnt of Yogurt A, B, C, D produced by BIG is x_1, x₂, x₃, x₄ to minimize the production cost. Then total amount of protien required to produce 15x₁+20x₂+30x₃+10x₄. As they have produce yogurts such that they produce minimum of 300 grams of protien we have, $15 x subscript 1 plus 20 x subscript 2 plus 30 x subscript 3 plus 10 x subscript 4 greater or equal than 300.$ Now the total amount of sugar required will be $3 x subscript 1 space plus 2 x subscript 2 space plus 2.5 x subscript 3 space plus 2 x subscript 4.$ Since they can at most use 200 gms of sugar we have $3 x subscript 1 plus 2 x subscript 2 plus 2.5 x subscript 3 plus 2 x subscript 4 less or equal than 200.$

Now the amount of fats to produce these yogurts is $1.5 x subscript 1 plus 1 x subscript 2 plus 1.5 x subscript 3 plus 1.5 x subscript 4.$ Since the total amount of fats used can be at most 200 grams, we have $1.5 x subscript 1 plus 1 x subscript 2 plus 1.5 x subscript 3 plus 1.5 x subscript 4 less or equal than 200.$ As BIG has to produce at least one unit of each yogurts we have $x subscript 1 comma end subscript space x subscript 2 comma space x subscript 3 space comma x subscript 4 greater or equal than 1.$ Also the number of Yogurt A produced should not exceed the number of Yogurt C produced we have , $x subscript 1 less or equal than x subscript 3. end subscript$ Now the total produuction cost is $2 x subscript 1 plus 3 x subscript 2 plus 5 x subscript 3 plus 1.5 x subscript 4.$ So the LPP is

MIN Z = $2 x subscript 1 plus 3 x subscript 2 plus 5 x subscript 3 plus 1.5 x subscript 4.$
subject to
$15 x subscript 1 plus 20 x subscript 2 plus 30 x subscript 3 plus 10 x subscript 4 greater or equal than 300.$
$3 x subscript 1 plus 2 x subscript 2 plus 2.5 x subscript 3 plus 2 x subscript 4 less or equal than 200.$
$1.5 x subscript 1 plus 1 x subscript 2 plus 1.5 x subscript 3 plus 1.5 x subscript 4 less or equal than 200.$
$x subscript 1 minus x subscript 3 less or equal than 0$
$x subscript 1 comma end subscript space x subscript 2 comma space x subscript 3 space comma x subscript 4 greater or equal than 1.$
and $x subscript 1 comma end subscript space x subscript 2 comma space x subscript 3 space comma x subscript 4 greater or equal than 0.$

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