The Facilities Manager At Oxbridge University Is Planning To

The Facilities Manager At Oxbridge University Is Planning To Applyfert

The facilities manager at Oxbridge University is planning to apply fertilizer to the grass in the quadrangle area in the spring. The grass needs nitrogen, phosphorus, and potash in at least the amounts given in the table. There are three types of commercial fertilizers available, each with specific mineral content and prices, and an unlimited supply of each. The goal is to determine how much of each fertilizer to purchase to meet the minimum nutrient requirements at the lowest total cost.

Paper For Above instruction

In agricultural management and turf maintenance, optimizing fertilizer application is crucial for cost efficiency and environmental sustainability. Specifically, determining the optimal combination of fertilizers to meet nutrient requirements at the minimum cost involves principles of linear programming. This paper formulates the problem, builds the linear programming model, and discusses solution strategies and implications.

Problem Formulation

The scenario involves three nutrients—nitrogen, phosphorus, and potash—requiring minimum quantities of 12 lb, 14 lb, and 18 lb respectively. Three fertilizer options are available, each with known nutrient contents per 1,000 pounds and respective prices:

  • Fertilizer A: 20 lb N, 10 lb P, 5 lb K; at $10 per 1,000 lb
  • Fertilizer B: 10 lb N, 5 lb P, 15 lb K; at $8 per 1,000 lb
  • Fertilizer C: 15 lb N, 10 lb P, 5 lb K; at $7 per 1,000 lb

The decision variables are the quantities (in thousands of pounds) of each fertilizer to purchase, denoted as x_A, x_B, and x_C. The objective function is to minimize total cost:

Minimize Z = 10x_A + 8x_B + 7x_C

Subject to constraints ensuring minimum nutrient requirements:

  • Nitrogen: 20x_A + 10x_B + 15x_C ≥ 12
  • Phosphorus: 10x_A + 5x_B + 10x_C ≥ 14
  • Potash: 5x_A + 15x_B + 5x_C ≥ 18

and non-negativity constraints x_A, x_B, x_C ≥ 0.

Linear Programming Model

The model can be summarized as follows:

Minimize Z = 10x_A + 8x_B + 7x_C

Subject to:


20x_A + 10x_B + 15x_C ≥ 12


10x_A + 5x_B + 10x_C ≥ 14


5x_A + 15x_B + 5x_C ≥ 18


x_A, x_B, x_C ≥ 0

Solving this model involves methods such as the simplex algorithm or graphical methods for simpler cases. Using spreadsheet tools or linear programming software can facilitate efficient solution determination.

Solution Approach

By applying the simplex method or an LP solver, optimal values for x_A, x_B, and x_C can be found. An initial feasible solution could involve evaluating corner points or basic feasible solutions. Given the coefficients and constraints, the problem is suitable for computational algorithms that quickly identify the global minimum point.

Implications and Practical Considerations

Determining the optimal fertilizer mix achieves cost savings while ensuring the lawn's nutrient needs are met. This approach minimizes waste and environmental impact by avoiding excess fertilizer application. Implementing linear programming models in turf management promotes sustainable practices and efficient resource utilization.

Conclusion

Linear programming provides a systematic and effective method for agricultural fertilization optimization. In the case of Oxbridge University's quadrangle, solving the formulated problem yields the precise amount of each fertilizer to purchase, meeting nutrient requirements at the least cost. Such models are adaptable to various agricultural scenarios, promoting economic and environmental sustainability.

References

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  5. Hillier, F. S., & Lieberman, G. J. (2021). Introduction to Operations Research (11th ed.). McGraw-Hill Education.
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