Please let me know how to find out the answer for (b) using the substitution method. I know how to get the functions for (2) as shown. Question: A company can decide how many additional labor hours to acquire for a given week. Subcontractor workers will only work a maximum of 20 hours a week. The company must produce at least 200 units of product A, 300 units of product B, and 400 units of product C. In 1 hour of work, worker 1 can produce 15 units of product A, 10 units of product B, and 30 units of product C. Worker 2 can produce 5 units of product A, 20 units of product B, and 35 units of product C. Worker 3 can produce 20 units of product A, 15 units of product B, and 25 units of product C. Worker 1 demands a salary of $50/hr, worker 2 demands a salary of $40/hr, and worker 3 demands a salary of $45/hr. The company must choose how many hours they should contract with each worker to meet their production requirements and minimize labor cost. (a) Formulate this as a linear programming problem. (b) Find the optimal solution using substitution method Answer: (a) Let X1 = Worker 1 hours X2 = Worker 2 hours X3 = Worker 3 hours Minimize 50X1 + 40X2 + 45X3 Subject to: 15X1 + 5X2 + 20X3 ≥ 200 10X1 + 20X2 + 15X3 ≥ 300 30X1 + 35X2 + 25X3 ≥ 400 X1, X2, X3 ≤ 20 X1, X2, X3 ≥ 0
Please let me know how to find out the answer for (b) using the substitution method. I know how to get the functions for (2) as shown.
Question:
A company can decide how many additional labor hours to acquire for a given week. Subcontractor workers will only work a maximum of 20 hours a week. The company must produce at least 200 units of product A, 300 units of product B, and 400 units of product C. In 1 hour of work, worker 1 can produce 15 units of product A, 10 units of product B, and 30 units of product C. Worker 2 can produce 5 units of product A, 20 units of product B, and 35 units of product C. Worker 3 can produce 20 units of product A, 15 units of product B, and 25 units of product C. Worker 1 demands a salary of $50/hr, worker 2 demands a salary of $40/hr, and worker 3 demands a salary of $45/hr. The company must choose how many hours they should contract with each worker to meet their production requirements and minimize labor cost.
(a) Formulate this as a linear programming problem.
(b) Find the optimal solution using substitution method
Answer:
(a)
Let X1 = Worker 1 hours
X2 = Worker 2 hours
X3 = Worker 3 hours
Minimize 50X1 + 40X2 + 45X3
Subject to: 15X1 + 5X2 + 20X3 ≥ 200
10X1 + 20X2 + 15X3 ≥ 300
30X1 + 35X2 + 25X3 ≥ 400
X1, X2, X3 ≤ 20
X1, X2, X3 ≥ 0
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