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An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods.
a. Find the optimal outputs and capacity for this problem.?
a. Find the optimal outputs and capacity for this problem.?
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An electric company is setting up a power plant in foreign country and...
Lagrangian and Kuhn-Tucker Conditions
To find the optimal outputs and capacity for the power plant, we need to maximize the total profit of the electric company. The profit can be calculated by subtracting the total cost from the total revenue. Let's denote the capacity as "C", the output in peak period as "Q1", and the output in off-peak period as "Q2".
The total revenue is given by the product of the price and the quantity for each period:
Total Revenue = (P1 * Q1) + (P2 * Q2)
The total cost is the sum of the variable cost and the capacity cost:
Total Cost = (20 * (Q1 + Q2)) + (10 * C)
The profit function can be written as:
Profit = Total Revenue - Total Cost
To solve this optimization problem, we can use the Lagrangian method along with the Kuhn-Tucker conditions.
Lagrangian:
The Lagrangian function is formed by adding the constraints to the profit function with the addition of Lagrange multipliers:
L(Q1, Q2, C, λ1, λ2, λ3) = (P1 * Q1) + (P2 * Q2) - ((20 * (Q1 + Q2)) + (10 * C)) + λ1 * (400 - Q1) + λ2 * (380 - Q2) + λ3 * (C)
where λ1, λ2, and λ3 are the Lagrange multipliers associated with the three constraints.
Kuhn-Tucker Conditions:
To find the optimal solution, we need to satisfy the Kuhn-Tucker conditions:
1. Stationarity: ∂L/∂Q1 = 0, ∂L/∂Q2 = 0, ∂L/∂C = 0
2. Primal feasibility: Q1 ≥ 0, Q2 ≥ 0, C ≥ 0
3. Dual feasibility: λ1 ≥ 0, λ2 ≥ 0, λ3 ≥ 0
4. Complementary slackness: λ1 * (400 - Q1) = 0, λ2 * (380 - Q2) = 0, λ3 * C = 0
Optimal Outputs and Capacity:
To find the optimal outputs and capacity, we need to solve the Lagrangian function by taking partial derivatives and setting them to zero:
∂L/∂Q1 = P1 - 20 - λ1 = 0
∂L/∂Q2 = P2 - 20 - λ2 = 0
∂L/∂C = -10 - λ3 = 0
From the first equation, we can solve for P1:
P1 = 20 + λ1
From the second equation, we can solve for P2:
P2 = 20 + λ2
Substituting these values into the profit function, we have:
Profit = (P1 * Q1) + (P2 * Q2) - ((20 * (Q1 + Q2)) + (10 * C)) + λ1 * (400 - Q1) + λ2 * (380 - Q2) + λ3 * C
Simplifying the profit function, we get
To find the optimal outputs and capacity for the power plant, we need to maximize the total profit of the electric company. The profit can be calculated by subtracting the total cost from the total revenue. Let's denote the capacity as "C", the output in peak period as "Q1", and the output in off-peak period as "Q2".
The total revenue is given by the product of the price and the quantity for each period:
Total Revenue = (P1 * Q1) + (P2 * Q2)
The total cost is the sum of the variable cost and the capacity cost:
Total Cost = (20 * (Q1 + Q2)) + (10 * C)
The profit function can be written as:
Profit = Total Revenue - Total Cost
To solve this optimization problem, we can use the Lagrangian method along with the Kuhn-Tucker conditions.
Lagrangian:
The Lagrangian function is formed by adding the constraints to the profit function with the addition of Lagrange multipliers:
L(Q1, Q2, C, λ1, λ2, λ3) = (P1 * Q1) + (P2 * Q2) - ((20 * (Q1 + Q2)) + (10 * C)) + λ1 * (400 - Q1) + λ2 * (380 - Q2) + λ3 * (C)
where λ1, λ2, and λ3 are the Lagrange multipliers associated with the three constraints.
Kuhn-Tucker Conditions:
To find the optimal solution, we need to satisfy the Kuhn-Tucker conditions:
1. Stationarity: ∂L/∂Q1 = 0, ∂L/∂Q2 = 0, ∂L/∂C = 0
2. Primal feasibility: Q1 ≥ 0, Q2 ≥ 0, C ≥ 0
3. Dual feasibility: λ1 ≥ 0, λ2 ≥ 0, λ3 ≥ 0
4. Complementary slackness: λ1 * (400 - Q1) = 0, λ2 * (380 - Q2) = 0, λ3 * C = 0
Optimal Outputs and Capacity:
To find the optimal outputs and capacity, we need to solve the Lagrangian function by taking partial derivatives and setting them to zero:
∂L/∂Q1 = P1 - 20 - λ1 = 0
∂L/∂Q2 = P2 - 20 - λ2 = 0
∂L/∂C = -10 - λ3 = 0
From the first equation, we can solve for P1:
P1 = 20 + λ1
From the second equation, we can solve for P2:
P2 = 20 + λ2
Substituting these values into the profit function, we have:
Profit = (P1 * Q1) + (P2 * Q2) - ((20 * (Q1 + Q2)) + (10 * C)) + λ1 * (400 - Q1) + λ2 * (380 - Q2) + λ3 * C
Simplifying the profit function, we get
Community Answer
An electric company is setting up a power plant in foreign country and...
An electric company is setting up a power plant in a foreign country, and it has to plan its capacity. The peak-period demand for power is given by P1 = 400-Q1 and the off-peak demand is given by P2 = 380-Q2. The variable cost is 20 per unit (paid in both market) and capacity costs 10 per unit which is only paid once and is used in both periods.
a. Write out the lagrangian and Kuhn-Tucker condition for the problem.
b. Find the optimal output and capacity for this problem.
c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?
d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).
a. Write out the lagrangian and Kuhn-Tucker condition for the problem.
b. Find the optimal output and capacity for this problem.
c. How much of the capacity is paid for by each market (i.e., what are the values of λ1 and λ2 )?
d. How suppose capacity cost is 30 cents per unit ( paid only once). Find quantities, capacity and how much of the capacity is paid for by each market ( i.e., λ1and λ2).
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An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.?
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An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.?.
An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An electric company is setting up a power plant in foreign country and it has no plan its capacity. The peak period demand for power is given by P1=400- Q1 and off peak demand is given by P2=380-Q2. The variable cost is 20 per unit (paid in both market) and capacity cost 10 per unit which is only paid once and used in both periods. a. Find the optimal outputs and capacity for this problem.?.
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