## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable for each photo \( i \), where \( x_i = 1 \) if photo \( i \) is selected for the book, and \( x_i = 0 \) otherwise.

#### Objective Function
- Maximize the total diversity score of the selected photos:
  \[
  \text{Maximize } \sum_{i} \text{diversity\_score}_i \times x_i
  \]
  where \(\text{diversity\_score}_i\) is the diversity score of photo \( i \) from the `photos` table.

#### Constraints
1. **Total Number of Photos Constraint**:
   - The total number of photos selected cannot exceed a specified limit \( P \):
     \[
     \sum_{i} x_i \leq P
     \]
   - Data Source Verification: \( P \) is a business configuration parameter.

2. **Photos Per Mountain Constraint**:
   - Let \( M_j \) be the set of photos from mountain \( j \). The number of photos selected from each mountain \( j \) cannot exceed a specified limit \( M_j^{\text{max}} \):
     \[
     \sum_{i \in M_j} x_i \leq M_j^{\text{max}} \quad \forall j
     \]
   - Data Source Verification: \( M_j^{\text{max}} \) is a business configuration parameter.

#### Data Source Verification
- The diversity scores \(\text{diversity\_score}_i\) are sourced from the `photos.diversity_score` column.
- The constraints on the total number of photos \( P \) and the maximum number of photos per mountain \( M_j^{\text{max}} \) are derived from business configuration parameters, which are not explicitly provided in the data but are assumed to be part of the business rules.

This formulation provides a complete and immediately solvable linear mathematical model for the given business problem, ensuring that the selection of photos maximizes diversity while adhering to the specified constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 1

import gurobipy as gp
from gurobipy import GRB

def optimize_mountain_photos():
    # 1. MODEL & DATA SETUP
    model = gp.Model("mountain_photos")
    
    # Data: Photo IDs and their diversity scores
    photo_ids = [1, 2, 3, 4, 5]
    diversity_scores = [0.85, 0.75, 0.9, 0.65, 0.8]
    
    # Constraints: Total number of photos and max photos per mountain
    P = 3  # Total number of photos allowed
    M_max = [2, 2, 1, 1, 1]  # Max photos per mountain (example values)
    
    # CRITICAL: Validate array lengths before loops
    assert len(photo_ids) == len(diversity_scores) == len(M_max), "Array length mismatch"
    
    # 2. VARIABLES
    # Binary decision variables for each photo
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{i}") for i in photo_ids}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total diversity score of the selected photos
    model.setObjective(gp.quicksum(diversity_scores[i-1] * x[i] for i in photo_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total number of photos constraint
    model.addConstr(gp.quicksum(x[i] for i in photo_ids) <= P, name="total_photos_limit")
    
    # Photos per mountain constraint
    # Assuming each photo belongs to a unique mountain for simplicity
    for i in photo_ids:
        model.addConstr(x[i] <= M_max[i-1], name=f"mountain_limit_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in photo_ids:
            if x[i].x > 1e-6:
                print(f"x[{i}] = {x[i].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_mountain_photos()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.55
**Execution Time**: 0.24 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: Gurobipy found an optimal solution quickly with a short execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 1

from docplex.mp.model import Model

def mountain_photos_optimization():
    """Optimize the selection of mountain photos for a photography book."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="mountain_photos")
    
    # Data: Photo IDs and their diversity scores
    photo_ids = [1, 2, 3, 4, 5]
    diversity_scores = [0.85, 0.75, 0.9, 0.65, 0.8]
    
    # Constraints: Total number of photos and max photos per mountain
    P = 3  # Total number of photos allowed
    M_max = [2, 2, 1, 1, 1]  # Max photos per mountain (example values)
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(photo_ids) == len(diversity_scores) == len(M_max), "Array length mismatch"
    safe_range = range(len(photo_ids))  # Safe indexing
    
    # 2. VARIABLES
    # Binary decision variables for each photo
    x = {i: mdl.binary_var(name=f"x_{photo_ids[i]}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total diversity score of the selected photos
    objective = mdl.sum(diversity_scores[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Total number of photos constraint
    mdl.add_constraint(mdl.sum(x[i] for i in safe_range) <= P, ctname="total_photos_limit")
    
    # Photos per mountain constraint
    for i in safe_range:
        mdl.add_constraint(x[i] <= M_max[i], ctname=f"mountain_limit_{photo_ids[i]}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            value = solution.get_value(x[i])
            if value > 1e-6:
                print(f"Photo {photo_ids[i]} selected with diversity score {diversity_scores[i]}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the optimization
mountain_photos_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.55
**Execution Time**: 4.79 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy, but still reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 1

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

def mountain_photos_optimization():
    """Optimize the selection of mountain photos for maximum diversity."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Assuming the data is provided as lists for simplicity
    photo_ids = [1, 2, 3, 4, 5]
    diversity_scores = [0.85, 0.75, 0.9, 0.65, 0.8]
    
    # Constraints parameters
    P = 3  # Total number of photos allowed
    M_max = {1: 2, 2: 1, 3: 1, 4: 1, 5: 1}  # Max photos per mountain (assuming each photo is from a different mountain)
    
    # CRITICAL: Validate array lengths before indexing
    assert len(photo_ids) == len(diversity_scores), "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.Set(initialize=photo_ids)
    
    # 4. PARAMETERS
    model.diversity_score = pyo.Param(model.I, initialize={photo_ids[i]: diversity_scores[i] for i in range(len(photo_ids))})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.diversity_score[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total number of photos constraint
    def total_photos_rule(model):
        return sum(model.x[i] for i in model.I) <= P
    model.total_photos_constraint = pyo.Constraint(rule=total_photos_rule)
    
    # Photos per mountain constraint
    def photos_per_mountain_rule(model, i):
        return model.x[i] <= M_max[i]
    model.photos_per_mountain_constraint = pyo.Constraint(model.I, rule=photos_per_mountain_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # Solve the model
    results = solver.solve(model, tee=True)
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective):.2f}")
        
        # Extract variable values
        print("\nSelected photos:")
        for i in model.I:
            if pyo.value(model.x[i]) > 0.5:  # Binary variable, check if selected
                print(f"Photo ID {i} selected with diversity score {model.diversity_score[i]:.2f}")
        
    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}")
    
    return model

# Run the optimization
mountain_photos_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 2.55
**Execution Time**: 5.08 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: Pyomo achieved the optimal solution with the longest execution time among the solvers, but the result is consistent and reliable.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 2.55 | 0.24s | N/A | 1 |
| Docplex | OPTIMAL | 2.55 | 4.79s | N/A | 1 |
| Pyomo | OPTIMAL | 2.55 | 5.08s | N/A | 1 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 2.55
**Solver Retry Summary**: gurobipy: 1 attempts, docplex: 1 attempts, pyomo: 1 attempts

### Final Recommendation
**Recommended Optimal Value**: 2.55
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time while maintaining the same optimal value as other solvers, making it efficient for large-scale problems.

### Business Interpretation
**Overall Strategy**: The optimal diversity score of 2.55 indicates the best selection of photos that maximizes diversity within the given constraints.
**Objective Value Meaning**: An optimal objective value of 2.55 reflects the maximum achievable diversity score for the selected photos, aligning with business goals of showcasing diverse mountain landscapes.
**Resource Allocation Summary**: Resources should be allocated to ensure the selected photos meet the diversity and quantity constraints, optimizing the visual appeal of the photo book.
**Implementation Recommendations**: Implement the solution using Gurobipy for efficient computation, and ensure business rules are adhered to in the selection process.