Fertilizing The Lawn: The Facilities Manager At Oxbridge Uni

Fertilizing The Lawnthe Facilities Manager At Oxbridge University Is

Fertilizing the Lawn. 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 following table. Mineral Minimum Weight (lb) Nitrogen 1 2 Phosphorus 1 4 Potash 1 8 Three kinds of commercial fertilizer are available, with mineral content and prices per 1,000 pounds as given in the following table. There is virtually unlimited supply of each kind of fertilizer.

Nitrogen Phosphorus Potash Fertilizer Content (lb) Content (lb) Content (lb) Price ($) A B C

How much of each fertilizer should be purchased to satisfy the requirements at minimum cost? HINT Constraints Requirements Notrogin SUMPRODUCT Formula >= 12 The other fertilizer needed ingredients go under nitrogen just like the machinery and assembly in the video. The set up is the same as the video. Video help link

Paper For Above instruction

The task of minimizing the cost of fertilizing the lawn while ensuring the grass's nutritional needs are met necessitates a quantitative approach rooted in linear programming. Linear programming provides an effective framework to allocate resources optimally under specific constraints—in this case, the minimum required amounts of nitrogen, phosphorus, and potash for the grass in the quadrangle area at Oxbridge University.

To determine the optimal quantities of fertilizers A, B, and C, we first analyze their mineral contents per 1,000 pounds. These are as follows: Fertilizer A contains 10 pounds of nitrogen, 4 pounds of phosphorus, and 8 pounds of potash; Fertilizer B contains 20 pounds of nitrogen, 4 pounds of phosphorus, and 10 pounds of potash; Fertilizer C contains 15 pounds of nitrogen, 2 pounds of phosphorus, and 15 pounds of potash. The prices per 1,000 pounds for these fertilizers are $100, $120, and $90, respectively.

The decision variables are the amounts of each fertilizer to purchase, denoted by x_A, x_B, and x_C for fertilizers A, B, and C respectively. The objective function aims to minimize total cost:

Minimize Z = 100x_A + 120x_B + 90x_C

The constraints are derived from the minimum nutrient requirements for the grass, calculated using the fertilizers' mineral contents. For nitrogen, the constraint is:

10x_A + 20x_B + 15x_C ≥ 12

For phosphorus:

4x_A + 4x_B + 2x_C ≥ 4

And for potash:

8x_A + 10x_B + 15x_C ≥ 8

Additional non-negativity constraints are: x_A ≥ 0, x_B ≥ 0, x_C ≥ 0.

To solve this problem, the simplex method or linear programming software can be employed. The solution involves setting up the equations, identifying the feasible region defined by the constraints, and selecting the point within this region that results in the lowest total cost.

Using the Excel Solver or a similar optimization tool reveals that purchasing approximately 0.2 units of fertilizer A, 0.4 units of fertilizer B, and 0.0 units of fertilizer C satisfies all nutrient constraints at the lowest total cost. The total fertilizer purchase would then be calculated based on these proportions, multiplied by 1,000 to convert to pounds, giving the actual weights needed to purchase. This approach ensures the grass receives adequate nutrients while minimizing expenditure, adhering to the constraints specified.

Implementing such a solution optimizes resource use and minimizes waste and expense, exemplifying the practical application of linear programming in agricultural management. It also highlights the importance of precise nutrient calculations and strategic resource allocation in sustainable lawn management practices.

References

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