Math Problem Statement

Solve this problem using the simplex method. (Answers should be accurate to at least 2 decimal places.)

A farmer named Jane has a 320 acre farm on which she plants two crops: corn and soybeans. For each acre of corn planted, Jane's expenses are $80 and for each acre of soybeans planted, her expenses are $100. Each acre of corn requires 100 bushels of storage and yields a profit of $60; each acre of soybeans requires 40 bushels of storage and yields a profit of $90.

The total amount of storage space available is 19200 bushels and Jane has only $20000 on hand. (a) How many acres of each crop should Jane plant in order to maximize her profit? acres of corn acres of soybeans

(b) What will Jane's profit be if she follows this strategy?

Solution

To solve this problem using the simplex method, we will first formulate it as a linear programming (LP) problem. Here's a step-by-step breakdown:

1. Decision Variables

Let:

  • x1x_1 = number of acres of corn Jane plants.
  • x2x_2 = number of acres of soybeans Jane plants.

2. Objective Function

We are asked to maximize Jane's profit. The profit per acre of corn is $60, and the profit per acre of soybeans is $90. Thus, the objective function is:

Maximize Z=60x1+90x2\text{Maximize } Z = 60x_1 + 90x_2

3. Constraints

We now need to consider the constraints:

  1. Storage Constraint: Each acre of corn requires 100 bushels of storage, and each acre of soybeans requires 40 bushels of storage. Jane has 19,200 bushels of storage available:

100x1+40x219200100x_1 + 40x_2 \leq 19200

  1. Budget Constraint: Each acre of corn costs $80, and each acre of soybeans costs $100. Jane has $20,000 to spend:

80x1+100x22000080x_1 + 100x_2 \leq 20000

  1. Land Constraint: Jane has 320 acres of land available:

x1+x2320x_1 + x_2 \leq 320

  1. Non-negativity Constraint: She cannot plant a negative number of acres, so:

x10andx20x_1 \geq 0 \quad \text{and} \quad x_2 \geq 0

4. Formulating the LP Problem

The linear programming problem is:

Maximize Z=60x1+90x2\text{Maximize } Z = 60x_1 + 90x_2 subject to: 100x1+40x219200(Storage constraint)100x_1 + 40x_2 \leq 19200 \quad \text{(Storage constraint)} 80x1+100x220000(Budget constraint)80x_1 + 100x_2 \leq 20000 \quad \text{(Budget constraint)} x1+x2320(Land constraint)x_1 + x_2 \leq 320 \quad \text{(Land constraint)} x10,x20x_1 \geq 0, \, x_2 \geq 0

5. Setting up for the Simplex Method

We convert the inequalities to equalities by introducing slack variables:

  • Let s1s_1 be the slack variable for the storage constraint.
  • Let s2s_2 be the slack variable for the budget constraint.
  • Let s3s_3 be the slack variable for the land constraint.

This gives us the following system:

100x1+40x2+s1=19200100x_1 + 40x_2 + s_1 = 19200 80x1+100x2+s2=2000080x_1 + 100x_2 + s_2 = 20000 x1+x2+s3=320x_1 + x_2 + s_3 = 320

The objective function is:

Z=60x1+90x2Z = 60x_1 + 90x_2

6. Simplex Tableaux Setup

We can now set up the initial simplex tableau, introduce the basic variables, and perform the simplex iterations to find the optimal solution.

First Tableau:

\text{BV} & x_1 & x_2 & s_1 & s_2 & s_3 & \text{Solution} \\ \hline s_1 & 100 & 40 & 1 & 0 & 0 & 19200 \\ s_2 & 80 & 100 & 0 & 1 & 0 & 20000 \\ s_3 & 1 & 1 & 0 & 0 & 1 & 320 \\ \hline -Z & -60 & -90 & 0 & 0 & 0 & 0 \\ \end{array}$$ Now, let's perform the simplex iterations to find the optimal solution. I'll calculate the result. ### Solution: (a) To maximize profit, Jane should plant: - $$\mathbf{0}$$ acres of corn. - $$\mathbf{200}$$ acres of soybeans. (b) Jane's maximum profit will be $$\mathbf{18,000}$$ dollars. Let me know if you'd like to see more details on the simplex iterations or clarifications! ### Additional Questions: 1. How does the budget constraint affect the number of acres Jane can plant? 2. What happens if the storage space is increased by 5000 bushels? 3. How would the solution change if Jane could spend an additional $5000? 4. Can Jane maximize her profit by planting only corn instead of soybeans? 5. How would a decrease in profit per acre of soybeans affect the optimal solution? ### Tip: When using the simplex method, always check if any constraints can be binding at the optimal solution to determine whether they limit the potential profit.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Optimization
Simplex Method

Formulas

Maximize Z = 60x_1 + 90x_2
100x_1 + 40x_2 ≤ 19200 (Storage constraint)
80x_1 + 100x_2 ≤ 20000 (Budget constraint)
x_1 + x_2 ≤ 320 (Land constraint)
Non-negativity constraint: x_1 ≥ 0, x_2 ≥ 0

Theorems

Simplex Method for Linear Optimization

Suitable Grade Level

Undergraduate Level (Linear Programming/Operations Research)