Math Problem Statement

The given problem is a diet problem where the goal is to minimize the cost of two grains (G1 and G2) while satisfying nutrient requirements. Which of the given options correctly represents the problem?

Solution

The given problem is a diet problem modeled as a linear programming problem (LPP) where the goal is to determine the quantities of two grains (G1 and G2) that satisfy the nutrient requirements at minimal cost. Let’s analyze the provided options and determine the correct one.


Problem Setup:

  1. Decision Variables:

    • x1x_1: Number of units of grain G1 consumed per day.
    • x2x_2: Number of units of grain G2 consumed per day.
  2. Cost Function to minimize:

    • z=0.6x1+0.35x2z = 0.6x_1 + 0.35x_2, which represents the cost of consuming x1x_1 units of G1 and x2x_2 units of G2.
  3. Constraints:

    • Nutrient constraints (starch, proteins, vitamins):
      • 5x1+7x285x_1 + 7x_2 \geq 8 → Starch requirement
      • 4x1+2x2154x_1 + 2x_2 \geq 15 → Protein requirement
      • 2x1+x232x_1 + x_2 \geq 3 → Vitamin requirement
    • Non-negativity constraints:
      • x10,x20x_1 \geq 0, x_2 \geq 0.

The problem is to minimize the cost function zz, subject to the constraints.


Options Analysis:

  1. Option 1: Maximization problem → Incorrect.
  2. Option 2: Maximization problem → Incorrect.
  3. Option 3: Minimization with incorrect constraints (all constraints are \leq instead of \geq) → Incorrect.
  4. Option 4: Minimization with correct constraints \geq → Correct.

Correct Answer:

The correct option is Option 4.


Do you want details on how to solve this linear programming problem (e.g., graphical or simplex method)?


Related Questions:

  1. How do you solve this LPP using the simplex method?
  2. What happens if the cost coefficients change?
  3. How does the solution change if one nutrient constraint is relaxed?
  4. Can you graphically represent this LPP and identify the feasible region?
  5. What are the shadow prices for these constraints in the simplex solution?

Tip:

In linear programming, always ensure that the objective function matches the problem goal (maximize or minimize), and the inequalities reflect the correct constraints.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization Problems
Inequalities
Objective Function

Formulas

Objective Function: z = 0.6x1 + 0.35x2
Constraints: 5x1 + 7x2 ≥ 8, 4x1 + 2x2 ≥ 15, 2x1 + x2 ≥ 3
Non-negativity: x1 ≥ 0, x2 ≥ 0

Theorems

Linear Programming Theory
Feasible Region and Objective Function
Optimization under Constraints

Suitable Grade Level

Undergraduate level or advanced high school (Grades 11-12)