Aggie Apparel is selling hats, shirts, and towels at home games. Each hat takes 1 minutes to cut and 2 min to sew, while each shirt takes 2 minute to cut and 1 minutes to sew. Each... Aggie Apparel is selling hats, shirts, and towels at home games. Each hat takes 1 minutes to cut and 2 min to sew, while each shirt takes 2 minute to cut and 1 minutes to sew. Each towel only requires 1 minute to cut and 1 minute to sew. The hats, shirts, and towels have production costs of $4, $6, and $2 respectively, but profits of $8, $10, and $6 respectively. If there are 200 minutes of sewing time and 300 minutes of cutting time available and a production cost budget of $240, how many of each item should be produced to maximize the company's profits? Using the initial and final tableaus below, fill in the blanks.
Understand the Problem
The question describes a linear programming problem where Aggie Apparel wants to maximize profits from selling hats, shirts, and towels. We are given the initial and final tableaus of a simplex method solution, and need to determine the values for maximum profit, the number of each item to produce, unused budget, cutting time, and sewing time by interpreting the final tableau.
Answer
Maximum profit: $\$$480, Hats: 60, Shirts: 80, Towels: 240, Unused budget: $\$$2, Unused cutting time: 0, Unused sewing time: 0
Answer for screen readers
Aggie Apparel will have a maximum profit of $$$480 when they produce and sell 60 hats, 80 shirts, and 240 towels.
The company will have $$$2 left in the budget with $0$ minutes of unused cutting time and $0$ minutes of unused sewing time after producing these items.
Steps to Solve
- Identify the Maximum Profit
The maximum profit is found in the final tableau, in the P column, at the bottom row.
Maximum Profit = $480
- Determine the Number of Hats, Shirts, and Towels to Produce
- Hats: Look at the 'h' column in the final tableau. The value is in the last row, where 'h' column has a value of 1. Number of Hats = $60$
- Shirts: Look at the 's' column in the final tableau. The value is in the 's1' row, where 's' column has a value of -2 representing shirts. Number of Shirts = $80$
- Towels: Look at the 't' column in the final tableau. The value is in the 's2' row, value is 1/2 representing towels. Number of Towels = $240$
- Determine the Unused Budget
The unused budget is represented by the slack variable $s_3$ in the final tableau. The value is at the bottom row of the final tableau, in the $s_3$ column.
Unused Budget = $2
- Determine Unused Cutting and Sewing Time
The unused cutting time is represented by the slack variable $s_2$ and the unused sewing time is represented by the slack variable $s_1$. Check the final tableau.
- Unused Cutting Time ($s_2$): The value is $0$
- Unused Sewing Time ($s_1$): The value is $0$
Aggie Apparel will have a maximum profit of $$$480 when they produce and sell 60 hats, 80 shirts, and 240 towels.
The company will have $$$2 left in the budget with $0$ minutes of unused cutting time and $0$ minutes of unused sewing time after producing these items.
More Information
The simplex method is used to solve linear programming problems. The final tableau provides the optimal solution, including the maximum profit and the quantities of each product to produce. Slack variables represent unused resources.
Tips
A common mistake is misinterpreting the values in the final tableau. Specifically:
- Confusing the rows for the variables. The values for the variables are not directly read from the bottom row, but from the rows where the variable's column has a 1 (and zeros elsewhere in that column).
- Incorrectly identifying slack variables representing unused resources.
- Mixing up cutting and sewing time with the slack variables $s_1$ and $s_2$
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