Slim-Down Manufacturing makes a line of nutritionally complete, weight-reduction beverages. One of their products is a strawberry shake which is designed to be a complete meal. The strawberry shake consists of several ingredients. Some information about each of these ingredients is given below. Calories Total Vitamin Calories (per tbsp) 50 100 0 120 80 Ingredient Strawberry flavoring Cream Vitamin supplement Artificial sweetener Thickening agent from fat (per tbsp) 1 75 0 0 30 Content (mg/tbsp) 20 0 50 0 2 Thickeners (mg/tbsp) 3 8 1 2 25 Cost (c/tbsp) 10 8 25 15 6 The nutritional requirements are as follows. The beverage must total between 380 and 420 calories (inclusive). No more than 20% of the total calories should come from fat. There must be at least 50 milligrams (mg) of vitamin content. For taste reasons, there must be at least two tablespoons (tbsp) of strawberry flavoring for each tbsp of artificial sweetener. Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage. Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements. Formulate a linear programming model for this problem.
Slim-Down Manufacturing makes a line of nutritionally complete, weight-reduction beverages. One of their products is a strawberry shake which is designed to be a complete meal. The strawberry shake consists of several ingredients. Some information about each of these ingredients is given below. Calories Total Vitamin Calories (per tbsp) 50 100 0 120 80 Ingredient Strawberry flavoring Cream Vitamin supplement Artificial sweetener Thickening agent from fat (per tbsp) 1 75 0 0 30 Content (mg/tbsp) 20 0 50 0 2 Thickeners (mg/tbsp) 3 8 1 2 25 Cost (c/tbsp) 10 8 25 15 6 The nutritional requirements are as follows. The beverage must total between 380 and 420 calories (inclusive). No more than 20% of the total calories should come from fat. There must be at least 50 milligrams (mg) of vitamin content. For taste reasons, there must be at least two tablespoons (tbsp) of strawberry flavoring for each tbsp of artificial sweetener. Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage. Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements. Formulate a linear programming model for this problem.
Practical Management Science
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ISBN:9781337406659
Author:WINSTON, Wayne L.
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Transcribed Image Text:**Slim-Down Manufacturing: Strawberry Shake Nutritional Overview**
Slim-Down Manufacturing produces a range of nutritionally complete, weight-reduction beverages. Among these is a strawberry shake designed as a complete meal. Below is detailed information on the ingredients used in this shake.
| Ingredient | Calories from fat (per tbsp) | Total Calories (per tbsp) | Vitamin Content (mg/tbsp) | Thickeners (mg/tbsp) | Cost (¢/tbsp) |
|------------------------|-----------------------------|--------------------------|--------------------------|---------------------|--------------|
| Strawberry flavoring | 1 | 50 | 20 | 3 | 10 |
| Cream | 75 | 100 | 0 | 8 | 8 |
| Vitamin supplement | 0 | 0 | 50 | 1 | 25 |
| Artificial sweetener | 0 | 120 | 0 | 2 | 15 |
| Thickening agent | 30 | 80 | 2 | 25 | 6 |
**Nutritional Requirements:**
1. **Caloric Needs:**
- Total calories for the beverage must be between 380 and 420 calories (inclusive).
- No more than 20% of the total calories should derive from fat.
2. **Vitamin Content:**
- The shake must contain at least 50 milligrams of vitamins.
3. **Flavor and Texture:**
- For taste, there must be a minimum of two tablespoons of strawberry flavoring for each tablespoon of artificial sweetener.
- To maintain proper thickness, the shake needs exactly 15 milligrams of thickeners.
**Objective:**
Management aims to determine the optimal quantity of each ingredient to meet the above requirements while minimizing costs.
**Task:**
Formulate a linear programming model to address this optimization problem, ensuring all nutritional and cost conditions are satisfied.
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