A linear problem typically involves finding the best outcome (such as maximum profit or minimum cost) within a linear framework, often constrained by various conditions. One common type of linear problem is a Linear Programming problem.
Example: Diet Problem
Suppose a nutritionist wants to determine the optimal combination of two foods to meet daily nutritional requirements at the minimum cost.
Variables:
- Let \( x_1 \) be the number of servings of Food A.
- Let \( x_2 \) be the number of servings of Food B.
Objective Function: Minimize total cost: \[ \text{Cost} = 2x_1 + 3x_2 \] (where Food A costs $2 per serving and Food B costs $3 per serving)
Constraints:
- Caloric Intake: The total calories should be at least 2000 calories. \[ 400x_1 + 500x_2 \geq 2000 \]
- Protein Intake: The total protein should be at least 100 grams. \[ 30x_1 + 20x_2 \geq 100 \]
- Carbohydrate Intake: The total carbohydrates should be at least 300 grams. \[ 50x_1 + 60x_2 \geq 300 \]
- Non-negativity: Both variables must be non-negative. \[ x_1 \geq 0, \quad x_2 \geq 0 \]
In this example, the problem is to minimize the cost while satisfying the nutritional constraints. This is a linear problem because both the objective function and constraints are linear equations.
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