Respuesta :
Answer:
The company should produce 1980 units of product B and 840 units of product C to maximize its profit (no units should be produced of product A)
Step-by-step explanation:
Each department time cost for each product is a number divisible of 6. We can see how much profit each department obtains with each product in 6 hours making only that.
Department 1:
Product A: since it takes 2 hours to make this product, then in 6 hours it would make 3. The profit is 3*24 = $72
Product B: It takes only 1 hour to make, thus in 6 hours dep 1 can produce 6. The profit is 6*16 = $96
Product C: it takes 2 hours to make, thus dep 1 makes 3 units in 6 hours, giving a profit of 3*20 = $60.
Department 2:
Product A: It takes 3 hours to make, therefore in 6 hours it will be able to make just 2 units, giving a profit of 2*24 = $48
Product B: The profit is the same than in Dep 1, since the time cost per unit is equal. The profit is $96
Product C: The profit is the same than in Dep 1, since the time cost per unit is equal. The profit is &60
Department 3:
Product A: The profit is the same than in Dep 1, since the time cost per unit is equal. The prodit is $72
Product B: This department makes 3 units of product B in 6 hours, because it takes 2 hours to make one unit instead of 1. The profit is 3*16 = $48
Product C: Since it takes only 1 hour to make 1 unit, then in 6 hours it can make 6. The profit is 6*20 = $120
Whit this information, we conclude that department 1 and department 2 are more cost efficient at producing product B and department 3 is more cost efficient at producing product C. In all the cases, each department takes 1 hour to produce one unit of its most cost efficient product, as a result
- Department 1 should use its 900 hours to produce product B, producing 900 units
- Department 2 should use its 1080 hours to produce product B, producing 1080 units
- Department 3 should use its 840 hours to produce product C, producing 840 units.
We conclude that the company should produce 900+1080 = 1980 units of product B and 840 units of product C to maximize its profit.
The company should produce 180 units of product A, 140 units of product B and 20 units of product C to maximize profit
From the table, the constraints are:
[tex]\mathbf{2A + B + 2C \le 900}[/tex]
[tex]\mathbf{3A + B + 2C \le 1080}[/tex]
[tex]\mathbf{2A + 2B + C \le 840}[/tex]
The objective function is:
[tex]\mathbf{Max\ Z \ =24A + 16B + 20C}[/tex]
Express the inequalities as an equation
[tex]\mathbf{2A + B + 2C = 900}[/tex]
[tex]\mathbf{3A + B + 2C = 1080}[/tex]
[tex]\mathbf{2A + 2B + C = 840}[/tex]
Make C the subject in [tex]\mathbf{2A + 2B + C = 840}[/tex]
[tex]\mathbf{C = 840 - 2A - 2B}[/tex]
Substitute [tex]\mathbf{C = 840 - 2A - 2B}[/tex] in the other equations
[tex]\mathbf{2A + B + 2(840 - 2A - 2B) = 900}[/tex]
[tex]\mathbf{2A + B + 1680 - 4A -4B = 900}[/tex]
[tex]\mathbf{-2A - 3B = -780}[/tex]
[tex]\mathbf{2A + 3B = 780}[/tex]
[tex]\mathbf{3A + B + 2(840 - 2A - 2B) = 1080}[/tex]
[tex]\mathbf{3A + B + 1680 - 4A -4B = 1080}[/tex]
[tex]\mathbf{-A - 3B = -600}[/tex]
[tex]\mathbf{A + 3B = 600}[/tex]
Make A the subject
[tex]\mathbf{A = 600 - 3B}[/tex]
Substitute [tex]\mathbf{A = 600 - 3B}[/tex] in [tex]\mathbf{2A + 3B = 780}[/tex]
[tex]\mathbf{2(600-3B) + 3B = 780}[/tex]
[tex]\mathbf{1200-6B + 3B = 780}[/tex]
[tex]\mathbf{-6B + 3B = 780-1200}[/tex]
[tex]\mathbf{- 3B = -420}[/tex]
Divide through by -3
[tex]\mathbf{B = 140}[/tex]
Substitute [tex]\mathbf{B = 140}[/tex] in [tex]\mathbf{A = 600 - 3B}[/tex]
[tex]\mathbf{A = 600 - 3(140)}[/tex]
[tex]\mathbf{A = 600 - 420}[/tex]
[tex]\mathbf{A = 180}[/tex]
Substitute [tex]\mathbf{A = 180}[/tex] and [tex]\mathbf{B = 140}[/tex] in [tex]\mathbf{C = 840 - 2A - 2B}[/tex]
[tex]\mathbf{C = 840 - 2(180) - 2(140)}[/tex]
[tex]\mathbf{C = 840 - 360 - 280}[/tex]
[tex]\mathbf{C = 200}[/tex]
Hence, the company should produce 180 units of product A, 140 units of product B and 20 units of product C to maximize profit
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