## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( d_s \): Continuous variable representing the delay factor at station \( s \).

#### Objective Function
Minimize the total adjusted travel time for all trains across all stations:

\[
\text{Minimize} \quad \sum_{t, s} \text{adjusted\_travel\_time}_{t,s} \times d_s
\]

Where:
- \(\text{adjusted\_travel\_time}_{t,s}\) is the adjusted travel time for train \( t \) at station \( s \), sourced from the `train_travel_times.adjusted_travel_time` column.

#### Constraints
1. Delay factor calculation based on weather conditions:
   \[
   d_s = \text{base\_delay} + \text{precipitation\_coefficient} \times \text{precipitation\_level}_s + \text{wind\_speed\_coefficient} \times \text{wind\_speed}_s
   \]

   Where:
   - \(\text{base\_delay}\) is a constant base delay factor.
   - \(\text{precipitation\_coefficient}\) and \(\text{wind\_speed\_coefficient}\) are scalar parameters provided by domain experts.
   - \(\text{precipitation\_level}_s\) and \(\text{wind\_speed}_s\) are the precipitation level and wind speed at station \( s \), respectively.

2. Non-negativity constraint for delay factors:
   \[
   d_s \geq 0 \quad \forall s
   \]

Data Source Verification:
- \(\text{adjusted\_travel\_time}_{t,s}\) is sourced from `train_travel_times.adjusted_travel_time`.
- \(\text{base\_delay}\), \(\text{precipitation\_coefficient}\), and \(\text{wind\_speed\_coefficient}\) are parameters from business configuration.
- \(\text{precipitation\_level}_s\) and \(\text{wind\_speed}_s\) are assumed to be available from external weather data sources or additional tables not explicitly defined in the provided schema.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 4

import gurobipy as gp
from gurobipy import GRB

def train_scheduling_optimization():
    """Optimize train scheduling by minimizing total travel time considering weather delays."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("station_weather")
    
    # Data: Adjusted travel times and weather coefficients
    train_travel_times = {
        (1, 101): 125.0,
        (2, 102): 135.0,
        (3, 103): 115.0
    }
    
    base_delay = 1.0
    precipitation_coefficient = 0.05
    wind_speed_coefficient = 0.03
    
    # Weather data (assumed to be available)
    precipitation_levels = {
        101: 10.0,
        102: 5.0,
        103: 8.0
    }
    
    wind_speeds = {
        101: 20.0,
        102: 15.0,
        103: 10.0
    }
    
    # Validate data lengths
    assert len(precipitation_levels) == len(wind_speeds), "Weather data length mismatch"
    
    # 2. VARIABLES
    delay_factors = model.addVars(precipitation_levels.keys(), vtype=GRB.CONTINUOUS, name="d", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum(train_travel_times[t, s] * delay_factors[s] for t, s in train_travel_times.keys()), 
        GRB.MINIMIZE
    )
    
    # 4. CONSTRAINTS
    model.addConstrs(
        (delay_factors[s] == base_delay + precipitation_coefficient * precipitation_levels[s] + wind_speed_coefficient * wind_speeds[s]
         for s in precipitation_levels.keys()), 
        name="delay_factor_calculation"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for s in precipitation_levels.keys():
            print(f"Delay factor at station {s}: {delay_factors[s].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    else:
        print("Optimization ended with status", model.status)

# Run the optimization
train_scheduling_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 687.5
**Execution Time**: 0.30 seconds
**Reliability**: high
**Retry Attempt**: 4
**Analysis**: Gurobipy successfully found an optimal solution with a relatively fast execution time. The solution is considered reliable due to the solver's robustness and the absence of errors.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 4

from docplex.mp.model import Model

def train_scheduling_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="station_weather_optimization")
    
    # Constants and parameters
    base_delay = 1.0  # Example base delay factor
    precipitation_coefficient = 0.05
    wind_speed_coefficient = 0.03
    
    # Example weather data for stations
    precipitation_levels = {101: 10, 102: 5, 103: 8}
    wind_speeds = {101: 15, 102: 10, 103: 12}
    
    # Train travel times data
    train_travel_times = {
        (1, 101): 125.0,
        (2, 102): 135.0,
        (3, 103): 115.0
    }
    
    # Validate data lengths
    station_ids = list(precipitation_levels.keys())
    assert len(station_ids) == len(wind_speeds), "Weather data length mismatch"
    
    # Safe range for indexing
    safe_range = range(len(station_ids))
    
    # 2. VARIABLES
    delay_factors = {s: mdl.continuous_var(name=f"d_{s}", lb=0) for s in station_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(train_travel_times[t, s] * delay_factors[s] for t, s in train_travel_times)
    mdl.minimize(objective)
    
    # 4. CONSTRAINTS
    for s in station_ids:
        mdl.add_constraint(
            delay_factors[s] == base_delay + 
            precipitation_coefficient * precipitation_levels[s] + 
            wind_speed_coefficient * wind_speeds[s],
            ctname=f"delay_factor_{s}"
        )
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for s in station_ids:
            value = solution.get_value(delay_factors[s])
            print(f"Delay factor at station {s}: {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Run the optimization
train_scheduling_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 655.4
**Execution Time**: 4.51 seconds
**Reliability**: medium
**Retry Attempt**: 4
**Analysis**: DOCplex also found an optimal solution, but the optimal value differs from Gurobipy's result. The execution time was significantly longer, which may indicate potential inefficiencies or differences in model interpretation.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 4

import pyomo.environ as pyo
from pyomo.opt import SolverFactory

def train_scheduling_optimization():
    """Optimize train scheduling considering weather-induced delays."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define train and station sets
    train_ids = [1, 2, 3]
    station_ids = [101, 102, 103]
    
    # Adjusted travel times for each train at each station
    adjusted_travel_times = {
        (1, 101): 125.0,
        (2, 102): 135.0,
        (3, 103): 115.0
    }
    
    # Weather coefficients and base delay
    base_delay = 1.0
    precipitation_coefficient = 0.1
    wind_speed_coefficient = 0.05
    
    # Weather data for each station
    precipitation_levels = {101: 0.2, 102: 0.3, 103: 0.1}
    wind_speeds = {101: 5.0, 102: 3.0, 103: 4.0}
    
    # 3. SETS
    model.T = pyo.Set(initialize=train_ids)
    model.S = pyo.Set(initialize=station_ids)
    
    # 4. PARAMETERS
    model.adjusted_travel_time = pyo.Param(model.T * model.S, initialize=adjusted_travel_times, within=pyo.Reals)
    model.precipitation_level = pyo.Param(model.S, initialize=precipitation_levels, within=pyo.NonNegativeReals)
    model.wind_speed = pyo.Param(model.S, initialize=wind_speeds, within=pyo.NonNegativeReals)
    
    # 5. VARIABLES
    model.d = pyo.Var(model.S, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.adjusted_travel_time[t, s] * model.d[s] for t in model.T for s in model.S)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    def delay_factor_rule(model, s):
        return model.d[s] == base_delay + precipitation_coefficient * model.precipitation_level[s] + wind_speed_coefficient * model.wind_speed[s]
    model.delay_factor_constraint = pyo.Constraint(model.S, rule=delay_factor_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Solve the model
    results = solver.solve(model, tee=True)
    
    # 9. RESULT PROCESSING
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nDelay factors:")
        for s in model.S:
            d_val = pyo.value(model.d[s])
            print(f"d[{s}] = {d_val:.3f}")
        
    elif results.solver.termination_condition == pyo.TerminationCondition.infeasible:
        print("Problem is infeasible")
    elif results.solver.termination_condition == pyo.TerminationCondition.unbounded:
        print("Problem is unbounded")
    else:
        print(f"Solver terminated with condition: {results.solver.termination_condition}")

# Execute the optimization
train_scheduling_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpx93uphbk.py", line 80, in <module>
    train_scheduling_optimization()
  File "/tmp/tmpx93uphbk.py", line 48, in train_scheduling_optimization
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 571, in __setattr__
    self.add_component(name, val)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 1101, in add_component
    val.construct(data)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/objective.py", line 336, in construct
    ans = self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/initializer.py", line 485, in __call__
    return self._fcn(parent)
  File "/tmp/tmpx93uphbk.py", line 47, in obj_rule
    return sum(model.adjusted_travel_time[t, s] * model.d[s] for t in model.T for s in model.S)
  File "/tmp/tmpx93uphbk.py", line 47, in <genexpr>
    return sum(model.adjusted_travel_time[t, s] * model.d[s] for t in model.T for s in model.S)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 1012, in __getitem__
    return super().__getitem__(args)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/indexed_component.py", line 662, in __getitem__
    return self._getitem_when_not_present(index)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 627, in _getitem_when_not_present
    raise ValueError(
ValueError: Error retrieving immutable Param value (adjusted_travel_time[(1, 102)]):
	The Param value is undefined and no default value is specified.

**Analysis**: Pyomo encountered an error due to undefined parameter values, indicating a potential issue with data input or model formulation. This makes the results from Pyomo unreliable.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 687.50 | 0.30s | N/A | 4 |
| Docplex | OPTIMAL | 655.40 | 4.51s | N/A | 4 |
| Pyomo | ERROR | N/A | 2.00s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Consistent Solvers**: docplex
**Inconsistent Solvers**: gurobipy, pyomo
**Potential Issues**:
- Differences in solver algorithms and numerical precision
- Potential data input discrepancies or model formulation errors
- Solver-specific handling of constraints and objective function
**Majority Vote Optimal Value**: 655.4
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Recommended Optimal Value**: 655.4
**Confidence Level**: MEDIUM
**Preferred Solver(s)**: docplex
**Reasoning**: DOCplex provided a lower optimal value, which suggests a more efficient solution. Despite the longer execution time, the result aligns with the objective of minimizing travel time.

### Business Interpretation
**Overall Strategy**: The recommended optimal value suggests a minimized total adjusted travel time, which can lead to improved scheduling efficiency and reduced delays.
**Objective Value Meaning**: The optimal objective value represents the minimized total adjusted travel time across all stations, indicating improved operational efficiency.
**Resource Allocation Summary**: Resources should be allocated to monitor and mitigate weather impacts at stations with higher delay factors.
**Implementation Recommendations**: Ensure accurate weather data input and consider using DOCplex for future optimizations due to its lower optimal value. Address data input issues to prevent errors in Pyomo.