General Instructions For The Ferdinand Feed LP Problem Prese

General Instructionsfor The Ferdinand Feed LP Problem Presented In The

Set up and solve the LP for Ferdinand Feed using Solver in the specified worksheet to find the optimal feed mix. Generate the Answer and Sensitivity Reports. Use these reports to answer questions regarding minimum costs and the impact of changes in raw material costs and requirements.

Identify the minimum cost for Grain 4 before including it in the final mix, determine the final raw material cost of the 1-lb mix, evaluate the effect of reducing iron requirement by 2 units on the mix cost, and analyze how high the cost of Grain 1 can become while maintaining the current optimal solution.

Paper For Above instruction

Ferdinand Feed Company aims to produce a 1-pound pet food packet that meets specified daily nutritional requirements at the minimum possible cost. The company sources four different grains, each with known costs and nutritional content, which are blended to meet or exceed the daily goals for vitamin C, protein, and iron. This optimization problem is formalized as a linear programming (LP) model that minimizes the total cost of the mixture while satisfying nutritional constraints.

The LP formulation involves decision variables representing the amount of each grain used in the blend. The objective function seeks to minimize total cost, calculated as a linear combination of the decision variables multiplied by their respective per-pound costs. Constraints ensure the mixture sums to 1 pound, and that the nutritional contents meet or exceed the specified minimums for vitamin C, protein, and iron.

Detailed LP Model

Let x1, x2, x3, x4 denote the pounds of grains 1, 2, 3, and 4 used in the mix. The LP is formulated as follows:

  • Objective: Minimize the total cost:

    0.72x₁ + 1.00x₂ + 0.80x₃ + 0.75x₄

  • Constraints:
    • Weight constraint:

      x₁ + x₂ + x₃ + x₄ = 1

    • Vitamin C requirement:

      9x₁ + 12x₂ + 10x₃ + 10x₄ ≥ 6

    • Protein requirement:

      12x₁ + 12x₂ + 8x₃ + 8x₄ ≥ 10

    • Iron requirement:

      4x₁ + 14x₂ + 15x₃ + 10x₄ ≥ 14

    • Non-negativity:

      x₁, x₂, x₃, x₄ ≥ 0

The solution process involves inputting this LP model into Excel's Solver, setting the objective cell to minimize total cost, and defining the constraints as above. Once the optimal solution is obtained, Solver's Answer and Sensitivity Reports provide essential data for analyzing cost sensitivities and the impact of parameter changes.

Analyzing the Cost and Sensitivity Data

Using the reports, specific questions are addressed:

  1. Minimum cost for Grain 4 before including it in the mix: By examining the shadow price and current optimal levels, determine the price point at which Grain 4 would be incorporated into the mixture.
  2. Final raw material cost of the 1-lb mix: Calculate based on the optimized decision variables and respective costs.
  3. Impact of reducing iron requirement by 2 units: Adjust the constraint and observe the change in the optimal cost or mix composition.
  4. Maximum cost of Grain 1 maintaining current optimality: Use the sensitivity report’s allowable increase/decrease data to discern the upper bound for Grain 1’s unit cost.

Conclusion

This linear programming approach ensures Ferdinand Feed optimally blends grains to meet nutritional requirements at minimal cost, while sensitivity analysis highlights the robustness of the solution to parameter changes. Such modeling not only aids in cost containment but also provides strategic insights on raw material adjustments, supporting decision-making in procurement and formulation.

References

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