Discover the dynamic world of power distribution optimization in this comprehensive sample solution. This analysis delves into two distinct scenarios – one with no power plant constraints and another with a cap of 4,000 MW per plant. In Scenario A, we find an optimal solution that minimizes costs while supplying power to various cities. Transitioning to Scenario B, we investigate the impact of constraints on the distribution cost, revealing an annual increase of approximately $100,566.41. This case study showcases the critical balance between power supply, demand, and cost-effectiveness, highlighting the importance of optimizing energy allocation.

## Problem Description:

The problem description for this statistical analysis assignmentrevolves around the optimization of power distribution from multiple power plants to a network of cities. The primary objective is to minimize the total annual power distribution cost while addressing two key scenarios:

**Scenario A:**No Power Plant Constraints - In this scenario, the assignment aims to find the optimal solution where there are no restrictions on the amount of power supplied. The task is to determine the cities supplied by each power plant while minimizing the total distribution cost.**Scenario B:**Power Plant Constraints - In this scenario, the problem is expanded by introducing constraints that limit each power plant to supply at most 4,000 MW of power. The assignment seeks to identify the new optimal solution, calculate the total annual power distribution cost under these constraints, and quantify the annual increase in distribution cost compared to Scenario A.

### Scenario A: No Power Plant Constraints

This involves optimizing the allocation of power from multiple plants to various cities, without any restrictions on power supply. The goal is to find the most cost-effective solution, minimizing the total annual power distribution cost, while ensuring that each city's demand is met efficiently.

**Optimal Solution:**

For this scenario, there are no restrictions on the amount of power that can be supplied by any of the power plants. The optimal solution, cities supplied by each power plant, and the total annual power distribution cost are as follows:

**Optimal Solution:**

- Los Angeles (LA) supplies 6,412.50 MW.
- Tulsa supplies 1,543.75 MW.
- Seattle supplies 2,968.75 MW.

**Cities Supplied by Each Power Plant:**

**Denver:**Supplied by Seattle (950 MW).**Portland:**Supplied by Seattle (831.25 MW).**San Francisco:**Supplied by San Francisco (2,375 MW).**Boise:**Supplied by Seattle (593.75 MW).**Reno:**Supplied by Denver (950 MW).**Bozeman:**Supplied by Seattle (593.75 MW).**Laramie:**Supplied by Denver (1,187.50 MW).**Park City:**Supplied by Denver (712.50 MW).**Flagstaff:**Supplied by San Francisco (1,187.50 MW).**Durango:**Supplied by Tulsa (1,543.75 MW).

**Total Annual Power Distribution Cost:** The total annual power distribution cost for this solution is $2,552,382.81.

This solution optimally meets the demand in each city while minimizing the total distribution cost.

**Scenario B:** Power Plant Constraints

This scenario introduces a constraint where each power plant is limited to supplying a maximum of 4,000 MW. The objective is to find a new optimal solution under these constraints and calculate the total annual power distribution cost. Additionally, it quantifies the annual increase in distribution cost compared to Scenario 1, showcasing the impact of these limitations on cost-effectiveness.

**Optimal Solution:**

In this scenario, the constraint is introduced that at most 4,000 MW of power can be supplied by any one of the power plants. The optimal solution and the annual increase in power distribution cost resulting from these constraints are as follows:

- Los Angeles (LA) supplies 4,000 MW.
- Tulsa supplies 2,925 MW.
- Seattle supplies 4,000 MW.

Total Annual Power Distribution Cost with Constraints: The total annual power distribution cost with the new constraints is $2,652,949.22.

Annual Increase in Power Distribution Cost: The annual increase in power distribution cost is calculated as the difference between the total cost with constraints and the total cost without constraints.

**Total cost with constraints:**$2,652,949.22**Total cost without constraints:**$2,552,382.81

Annual Increase = Total cost with constraints - Total cost without constraints

Annual Increase ≈ $100,566.41

Hence, the annual increase in power distribution cost resulting from adding the constraints (at most 4,000 MW per plant) to the original formulation is approximately $100,566.41.

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