1. **Problem Statement:** A farmer wants to mix two types of food, A and B, to meet minimum daily nutritional requirements at minimum cost. 2. **Define variables:** Let $x$ = number of bags of Food A Let $y$ = number of bags of Food B 3. **Given data:** Cost per bag: Food A = 20000, Food B = 22000 Nutrient contents per bag: - Food A: 40 protein, 20 minerals, 10 vitamins - Food B: 30 protein, 20 minerals, 30 vitamins 4. **Constraints (minimum daily requirements):** $$40x + 30y \geq 150 \quad \text{(protein)}$$ $$20x + 20y \geq 90 \quad \text{(minerals)}$$ $$10x + 30y \geq 60 \quad \text{(vitamins)}$$ 5. **Objective function (minimize cost):** $$C = 20000x + 22000y$$ 6. **Solve constraints for equality to find feasible points:** From minerals: $$20x + 20y = 90 \Rightarrow x + y = 4.5 \Rightarrow y = 4.5 - x$$ Substitute into protein: $$40x + 30(4.5 - x) = 150$$ $$40x + 135 - 30x = 150$$ $$10x = 15 \Rightarrow x = 1.5$$ Then, $$y = 4.5 - 1.5 = 3$$ Check vitamins: $$10(1.5) + 30(3) = 15 + 90 = 105 \geq 60$$ (satisfied) 7. **Check cost at $(x,y) = (1.5,3)$:** $$C = 20000(1.5) + 22000(3) = 30000 + 66000 = 96000$$ 8. **Check other corner points:** At $x=0$, minerals require $20y \geq 90 \Rightarrow y \geq 4.5$ Check protein: $$40(0) + 30(4.5) = 135 < 150$$ (not satisfied) At $y=0$, minerals require $20x \geq 90 \Rightarrow x \geq 4.5$ Check protein: $$40(4.5) + 30(0) = 180 \geq 150$$ (satisfied) Check vitamins: $$10(4.5) + 30(0) = 45 < 60$$ (not satisfied) Try $x=3$, $y=1.5$: Protein: $$40(3) + 30(1.5) = 120 + 45 = 165 \geq 150$$ Minerals: $$20(3) + 20(1.5) = 60 + 30 = 90 \geq 90$$ Vitamins: $$10(3) + 30(1.5) = 30 + 45 = 75 \geq 60$$ Cost: $$20000(3) + 22000(1.5) = 60000 + 33000 = 93000$$ 9. **Compare costs:** At (1.5,3): 96000 At (3,1.5): 93000 (lower cost) 10. **Optimal solution:** The farmer should use 3 bags of Food A and 1.5 bags of Food B daily. --- **Part b) Savings calculation:** Current purchase: 5 bags A and 4 bags B Current cost: $$20000(5) + 22000(4) = 100000 + 88000 = 188000$$ Recommended cost: $$93000$$ Savings: $$188000 - 93000 = 95000$$ --- **Part c) Limitations and solutions:** 1. Assumes linearity and constant nutrient content, which may vary. 2. Ignores other nutritional factors and animal preferences. 3. Assumes fractional bags can be purchased, which may not be practical. 4. Real-world constraints like storage, availability, and cost fluctuations are not considered. **Overcoming limitations:** - Use integer programming to handle whole bags. - Incorporate more nutrients and constraints. - Use stochastic models to handle variability. - Regularly update data and constraints based on real conditions.