❖ Linear Programming Word Problem: Minimize the Objective Function ❖

Summary

In this linear programming example, a rancher seeks to minimize costs while meeting specific nutritional requirements for his cattle. Using two types of cattle feed, Brand X and Brand Y, he must ensure that the total intake consists of at least 60 grams of protein and 30 grams of fat. By defining the variables, setting up a cost function, and graphing the constraints, the rancher finds that using 2.4 units of Brand X and 1.2 units of Brand Y results in the minimum cost of $2.52, satisfying all nutritional needs.

Takeaways

• 🐄 A rancher needs to determine the optimal amounts of two cattle feed types (Brand X and Brand Y) to minimize costs while meeting nutritional requirements.
• 📊 The nutritional requirements include at least 60 grams of protein and 30 grams of fat for the cattle.
• 🍽️ Brand X contains 15 grams of protein and 10 grams of fat per unit, costing $0.80 per unit.
• 🍽️ Brand Y contains 20 grams of protein and 5 grams of fat per unit, costing $0.50 per unit.
• 💡 The objective is to minimize the cost function: C = 0.8X + 0.5Y.
• 📈 The protein constraint is represented by the inequality 15X + 20Y ≥ 60.
• 📉 The fat constraint is represented by the inequality 10X + 5Y ≥ 30.
• 🛑 Non-negativity constraints ensure that both X and Y must be greater than or equal to zero.
• 📐 Graphing the constraints helps identify feasible regions and corner points for optimization.
• 💰 The minimum cost is achieved at the point (2.4, 1.2), where the rancher should use 2.4 units of Brand X and 1.2 units of Brand Y, resulting in a total cost of $2.52.

Q & A

What is the main objective of the rancher in this linear programming problem?

The rancher's main objective is to minimize costs while providing adequate nutrition for his cattle.

What are the nutritional requirements for the cattle's diet?

The cattle's diet requires at least 60 grams of protein and 30 grams of fat.

What are the nutritional values for Brand X?

Brand X contains 15 grams of protein and 10 grams of fat per unit and costs $0.80 per unit.

What are the nutritional values for Brand Y?

Brand Y contains 20 grams of protein and 5 grams of fat per unit and costs $0.50 per unit.

How are the variables defined in the problem?

The variables are defined as X for the number of units of Brand X and Y for the number of units of Brand Y.

What is the cost function that needs to be minimized?

The cost function is C = 0.8X + 0.5Y, which represents the total cost of purchasing the food.

What are the inequalities that represent the nutritional constraints?

The inequalities are 15X + 20Y ≥ 60 for protein and 10X + 5Y ≥ 30 for fat.

How do we determine the feasible region for this problem?

The feasible region is determined by graphing the inequalities and identifying the area that satisfies all constraints.

What are the corner points identified in the feasible region?

The corner points identified are (0, 6), (4, 0), and (2.4, 1.2).

What is the minimum cost solution found for the rancher's food requirements?

The minimum cost solution is to purchase 2.4 units of Brand X and 1.2 units of Brand Y, resulting in a cost of $2.52.