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Formulating Linear Programming Model to Maximize Farms’ Profits Using STATA

In this homework solution, we tackle the complex decision-making process of crop planting for four distinct farms using STATA. Through linear programming, we develop a model to help these farms make strategic choices to maximize their expected profits. This model accounts for factors like usable acreage, labor resources, crop-specific constraints, and even a uniform workload distribution to ensure that each farm operates optimally. By determining the ideal allocation of crops, the solution offers a practical and efficient approach to crop management, benefiting both the farms and the agricultural industry as a whole.

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Problem Statement:

The objective of this STATA homework is to develop a linear programming model to optimize the allocation of crops across four different farms (Farm 1, Farm 2, Farm 3, and Farm 4) while maximizing the expected profit. The decision variables, $x_{ij}$, represent the number of acres of each of the three crops (A, B, and C) planted on each farm. The goal is to determine the allocation that maximizes the expected profit while adhering to various constraints.

Solution

Decision Variables:

To formulate this as a linear programming model, we introduce decision variables:

  • $x_{ij}$ represents the number of acres of crop $j$ planted on Farm $i$, where:
$i$ ranges from 1 to 4 (for the four farms).

$j$ ranges from A, B, to C (for the three crops).

Objective Function: Our primary goal is to maximize the expected profit. The profit for each crop is calculated by multiplying the number of acres of that crop with its expected profit per acre. Therefore, our objective function is to maximize $Z$: Z=2x1A+4x1B+3x1C+2x2A+4x2B+3x2C+2x3A+4x3B+3x3C+2x4A+4x4B+3x4C

Constraints: To ensure the optimization reflects the real-world constraints of each farm, we have the following limitations:

Usable Acreage Constraints:

  • $x_{1A} + x_{1B} + x_{1C} \leq 500$
  • $x_{2A} + x_{2B} + x_{2C} \leq 900$
  • $x_{3A} + x_{3B} + x_{3C} \leq 300$
  • $x_{4A} + x_{4B} + x_{4C} \leq 700$

Labor Constraints:

  • $700x_{1A} + 500x_{1B} + 300x_{1C} \leq 1700$
  • $700x_{2A} + 500x_{2B} + 300x_{2C} \leq 3000$
  • $700x_{3A} + 500x_{3B} + 300x_{3C} \leq 900$
  • $700x_{4A} + 500x_{4B} + 300x_{4C} \leq 2200$

Maximum Acreage Constraints for Each Crop:

  • $x_{1A} + x_{2A} + x_{3A} + x_{4A} \leq 800$
  • $x_{1B} + x_{2B} + x_{3B} + x_{4B} \leq 500$
  • $x_{1C} + x_{2C} + x_{3C} + x_{4C} \leq 200$

Uniform Workload Constraint: Let $p$ represent the percentage of usable acreage planted on each farm. This constraint ensures that the percentage of usable acreage planted is consistent across all farms.

  • $\frac{x_{1A} + x_{1B} + x_{1C}}{500} = p$
  • $\frac{x_{2A} + x_{2B} + x_{2C}}{900} = p$
  • $\frac{x_{3A} + x_{3B} + x_{3C}}{300} = p$
  • $\frac{x_{4A} + x_{4B} + x_{4C}}{700} = p$ The value of $p$ is a derived variable determined by the linear programming model.

Non-negativity Constraints: All $x_{ij}$ values must be greater than or equal to zero.

This linear programming model can be solved using appropriate methods or software to determine the optimal number of acres for each crop to be planted on each farm, ensuring the maximum expected profit is achieved. The value of $p$ will be a key outcome of this optimization, ensuring that the workload is evenly distributed among the farms.