# Complete Optimization Problem and Solution: station_weather

## 1. Problem Context and Goals

### Context  
The primary objective is to optimize train scheduling by minimizing the total travel time, taking into account delays caused by weather conditions at each station. The decision-making process involves adjusting the delay factors for each station, which are continuous variables representing the impact of weather on train schedules. The operational parameters are aligned with the goal of minimizing the total travel time, calculated as the sum of adjusted travel times for each train and station, multiplied by the respective delay factors.

The business configuration includes specific coefficients that quantify the impact of weather conditions on delay factors. These are:
- A coefficient for precipitation, which influences the delay factor by reflecting the increased travel time due to rain or snow.
- A coefficient for wind speed, which accounts for the additional delays caused by high winds, particularly in open areas.

These coefficients are scalar parameters, determined by domain experts, and are crucial for calculating the delay factors. The business logic ensures that these parameters are applied consistently across all stations, maintaining a linear relationship between weather conditions and delay factors. The focus is on precise operational decision-making that adheres to linear formulations, avoiding any nonlinear relationships such as variable products or divisions.

### Goals  
The optimization goal is to minimize the total travel time for all trains. This is achieved by adjusting the delay factors at each station to account for weather-induced delays. The metric for optimization is the total travel time, which is the sum of the adjusted travel times for each train and station, multiplied by the corresponding delay factors. Success is measured by the ability to reduce this total travel time, using the coefficients for precipitation and wind speed as key inputs in the calculation of delay factors. The goal is articulated in natural language, emphasizing the linear nature of the optimization process.

## 2. Constraints    

The primary constraint ensures that the delay factors are adjusted based on the impact of weather conditions, using predefined coefficients. Specifically, the delay factor for each station is determined by adding a base value to the product of the precipitation coefficient and the precipitation level at the station, plus the product of the wind speed coefficient and the wind speed at the station. This constraint is expressed in business terms, highlighting the linear relationship between weather conditions and delay factors, without involving any nonlinear operations.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include creating new tables for travel times and delay factors, modifying existing tables to include weather data, and updating configuration logic for weather impact coefficients.

CREATE TABLE train_travel_times (
  train_id INTEGER,
  station_id INTEGER,
  adjusted_travel_time FLOAT
);

CREATE TABLE station_delay_factors (
  station_id INTEGER,
  delay_factor FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive overview of the tables and columns, explaining their business purposes and roles in the optimization process:

- **Train Travel Times Table**: This table stores the adjusted travel times for trains between stations. Each entry includes:
  - **Train ID**: A unique identifier for each train, used to index travel times.
  - **Station ID**: A unique identifier for each station, used to index travel times.
  - **Adjusted Travel Time**: The travel time for a train, adjusted for weather-related delays, serving as a coefficient in the objective function.

- **Station Delay Factors Table**: This table records the delay factors due to weather conditions at each station. Each entry includes:
  - **Station ID**: A unique identifier for each station, used to index delay factors.
  - **Delay Factor**: A continuous variable representing the delay due to weather, used as a decision variable in the optimization model.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical weather conditions and their impact on train schedules, ensuring a realistic and solvable optimization problem.

-- Realistic data for train_travel_times
INSERT INTO train_travel_times (train_id, station_id, adjusted_travel_time) VALUES (1, 101, 125.0);
INSERT INTO train_travel_times (train_id, station_id, adjusted_travel_time) VALUES (2, 102, 135.0);
INSERT INTO train_travel_times (train_id, station_id, adjusted_travel_time) VALUES (3, 103, 115.0);

-- Realistic data for station_delay_factors
INSERT INTO station_delay_factors (station_id, delay_factor) VALUES (101, 1.15);
INSERT INTO station_delay_factors (station_id, delay_factor) VALUES (102, 1.25);
INSERT INTO station_delay_factors (station_id, delay_factor) VALUES (103, 1.1);
```