Aggie Apparel is selling hats, shirts, and towels at home games. Each hat takes 1 minute to cut and 2 min to sew, while each shirt takes 2 minutes to cut and 1 minute to sew. Each... Aggie Apparel is selling hats, shirts, and towels at home games. Each hat takes 1 minute to cut and 2 min to sew, while each shirt takes 2 minutes to cut and 1 minute 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 presents a linear programming problem involving maximizing profits for a company selling hats, shirts, and towels, subject to constraints on sewing time, cutting time, and budget. Using the provided initial and final tableaus from the Simplex method, we need to determine the optimal quantities of each item to produce, the maximum profit, and the unused resources. The final tableau provides the solution to linear programming problems.
Answer
Maximum profit: $480 Hats: 60 Shirts: 0 Towels: 0 Unused budget: $15 Unused cutting time: 240 minutes Unused sewing time: 80 minutes
Answer for screen readers
Aggie Apparel will have a maximum profit of $480 when they produce and sell 60 hats, 0 shirts, and 0 towels.
The company will have $15 left in the budget with 240 minutes of unused cutting time and 80 minutes of unused sewing time after producing these items.
Steps to Solve
- Identify the maximum profit from the final tableau
The maximum profit is found in the bottom right corner of the final tableau, in the 'P' column.
Therefore, the maximum profit is $480.
- Identify the optimal quantities of hats, shirts, and towels to produce from the final tableau
The optimal quantities are found under the 'h', 's', and 't' columns, respectively. Note where the columns have a 1 and the rest zeros. From the final tableau, we see that:
Hats (h): 60 Shirts (s): 0 Towels (t): 0
- Identify the unused budget from the final tableau
The unused budget is found under the 's3' column because 's3' corresponds to the slack variable for the budget constraint in the initial tableau. Since we are looking for the unused budget, look at the 's3' column in the final tableau and match the '1' in the initial 's3' column to the final 's3' column. Here the 1 is in row 3 of the initial tableau. In the final tableau, s3 column, row 3, is 0.25. This value corresponds to the $h$ row. So multiple this value by the RHS amount $60$ to get the unused budget. Note that this would be more complex if the final tableau had nonzero values in multiple rows of the 's3' column. The unused budget is $0.25 \times 60 = 15$
- Identify the unused cutting and sewing time.
The unused cutting time is found under the 's2' column because 's2' corresponds to the slack variable for the cutting time constraint in the initial tableau.
The unused sewing time is found under the 's1' column because 's1' corresponds to the slack variable for the sewing time constraint in the initial tableau.
We find the answer in the RHS column of the final tableau. From the final tableau, we see that:
Unused sewing time (s1): 80 minutes Unused cutting time(s2): 240 minutes
Aggie Apparel will have a maximum profit of $480 when they produce and sell 60 hats, 0 shirts, and 0 towels.
The company will have $15 left in the budget with 240 minutes of unused cutting time and 80 minutes of unused sewing time after producing these items.
More Information
The Simplex method is an algorithm for solving linear programming problems. The final tableau provides the optimal solution, indicating the quantities of each variable that maximize the objective function (profit) while satisfying all constraints. Slack variables represent unused resources.
Tips
- Failing to correctly identify the columns representing each variable (hats, shirts, towels, slack variables).
- Misinterpreting the values in the final tableau, especially confusing the slack variables with the primary variables.
- Incorrectly calculating the unused resources from the slack variables.
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