## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( w_i \) be the trust weight assigned to the \( i \)-th review. This is a continuous variable, \( w_i \geq 0 \), where \( i = 1, 2, \dots, 5 \).

#### Objective Function
Maximize the sum of the trust-weighted ratings of all reviews:
\[
\text{Maximize} \quad \sum_{i=1}^{5} r_i \times w_i
\]
where \( r_i \) is the rating of the \( i \)-th review.

#### Constraints
1. **Non-Negative Trust Weights**:
\[
w_i \geq 0 \quad \forall i = 1, 2, \dots, 5
\]
2. **Trust Weight Upper Bound**:
\[
w_i \leq t_i \quad \forall i = 1, 2, \dots, 5
\]
where \( t_i \) is the trust level associated with the \( i \)-th review.

#### Data Source Verification
- **Coefficients \( r_i \)**: From `review.rating` table.
  - \( r_1 = 4.5 \), \( r_2 = 3.8 \), \( r_3 = 5.0 \), \( r_4 = 3.0 \), \( r_5 = 4.0 \).
- **Upper bounds \( t_i \)**: From `trust.trust` table.
  - \( t_1 = 0.6 \), \( t_2 = 0.8 \), \( t_3 = 1.0 \), \( t_4 = 0.7 \), \( t_5 = 0.7 \).

#### Complete Linear Programming Model
\[
\text{Maximize} \quad 4.5w_1 + 3.8w_2 + 5.0w_3 + 3.0w_4 + 4.0w_5
\]
Subject to:
\[
w_1 \geq 0, \quad w_2 \geq 0, \quad w_3 \geq 0, \quad w_4 \geq 0, \quad w_5 \geq 0
\]
\[
w_1 \leq 0.6, \quad w_2 \leq 0.8, \quad w_3 \leq 1.0, \quad w_4 \leq 0.7, \quad w_5 \leq 0.7
\]

This is a complete, immediately solvable linear programming model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy Implementation for Trust-Weighted Review Optimization
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_trust_weights():
    """Optimize trust weights for reviews to maximize trust-weighted ratings"""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("trust_weight_optimization")
    
    # Data from the problem
    ratings = [4.5, 3.8, 5.0, 3.0, 4.0]  # From review.rating table
    trust_levels = [0.6, 0.8, 1.0, 0.7, 0.7]  # From trust.trust table
    
    # Number of reviews
    n_reviews = len(ratings)
    
    # CRITICAL: Validate array lengths before loops
    assert len(ratings) == len(trust_levels) == n_reviews, "Array length mismatch"
    
    # 2. VARIABLES
    # Trust weights for each review
    w = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"w_{i}", lb=0) 
         for i in range(n_reviews)}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the sum of trust-weighted ratings
    model.setObjective(gp.quicksum(ratings[i] * w[i] for i in range(n_reviews)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Pattern A: Trust weight upper bounds
    for i in range(n_reviews):
        model.addConstr(w[i] <= trust_levels[i], name=f"trust_bound_{i}")
    
    # Pattern B: Non-negative trust weights (already handled by lb=0 in variable definition)
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    # Handle solution status
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_reviews):
            print(f"w[{i}] = {w[i].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
if __name__ == "__main__":
    optimize_trust_weights()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 15.64
**Execution Time**: 0.20 seconds
**Reliability**: high
**Analysis**: Gurobipy achieved the optimal solution efficiently with the lowest execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex implementation for maximizing trust-weighted ratings of reviews
"""

from docplex.mp.model import Model

def optimize_trust_weights():
    """Optimize trust weights to maximize trust-weighted ratings"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="epinions_1")
    
    # Data from the problem statement
    ratings = [4.5, 3.8, 5.0, 3.0, 4.0]  # Coefficients for objective function
    trust_levels = [0.6, 0.8, 1.0, 0.7, 0.7]  # Upper bounds for trust weights
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(ratings) == len(trust_levels), "Array length mismatch"
    safe_range = range(min(len(ratings), len(trust_levels)))  # Safe indexing
    
    # 2. VARIABLES
    # Decision variables: trust weights for each review
    w = {i: mdl.continuous_var(name=f"w_{i}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the sum of trust-weighted ratings
    objective = mdl.sum(ratings[i] * w[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Non-negative trust weights (already handled by lb=0 in variable definition)
    
    # Trust weight upper bound
    for i in safe_range:
        mdl.add_constraint(w[i] <= trust_levels[i], ctname=f"trust_bound_{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(w[i])
            print(f"w[{i}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
if __name__ == "__main__":
    optimize_trust_weights()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 15.64
**Execution Time**: 1.31 seconds
**Reliability**: high
**Analysis**: DOCplex also achieved the optimal solution but with a longer execution time compared to Gurobipy.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def epinions_optimization():
    """Optimization model to maximize trust-weighted ratings of reviews."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Ratings from the review table
    ratings = [4.5, 3.8, 5.0, 3.0, 4.0]
    
    # Trust levels from the trust table
    trust_levels = [0.6, 0.8, 1.0, 0.7, 0.7]
    
    # Number of reviews
    n_reviews = len(ratings)
    
    # CRITICAL: Validate array lengths before indexing
    assert len(ratings) == len(trust_levels) == n_reviews, "Array length mismatch"
    
    # 3. SETS (Pyomo way to define indices)
    model.I = pyo.RangeSet(1, n_reviews)  # 1-based indexing
    
    # 4. PARAMETERS (data containers)
    model.rating = pyo.Param(model.I, initialize={i+1: ratings[i] for i in range(n_reviews)})
    model.trust = pyo.Param(model.I, initialize={i+1: trust_levels[i] for i in range(n_reviews)})
    
    # 5. VARIABLES
    # Trust weights for each review
    model.w = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    # Maximize the sum of trust-weighted ratings
    def obj_rule(model):
        return sum(model.rating[i] * model.w[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Non-Negative Trust Weights
    def non_negative_rule(model, i):
        return model.w[i] >= 0
    model.non_negative_constraint = pyo.Constraint(model.I, rule=non_negative_rule)
    
    # Trust Weight Upper Bound
    def trust_bound_rule(model, i):
        return model.w[i] <= model.trust[i]
    model.trust_bound_constraint = pyo.Constraint(model.I, rule=trust_bound_rule)
    
    # 8. SOLVING WITH GUROBI (your available solver)
    solver = SolverFactory('gurobi')
    
    # Optional: Set solver options
    solver.options['TimeLimit'] = 300  # 5 minutes
    solver.options['MIPGap'] = 0.01    # 1% gap
    
    # Solve the model
    results = solver.solve(model, tee=True)  # tee=True shows solver output
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nTrust weights:")
        for i in model.I:
            w_val = pyo.value(model.w[i])
            print(f"w[{i}] = {w_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}")
    
    return model

# Run the optimization
if __name__ == "__main__":
    epinions_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 15.64
**Execution Time**: 1.12 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with an execution time between Gurobipy and DOCplex.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 15.64 | 0.20s | N/A | N/A |
| Docplex | OPTIMAL | 15.64 | 1.31s | N/A | N/A |
| Pyomo | OPTIMAL | 15.64 | 1.12s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 15.64

### Final Recommendation
**Recommended Optimal Value**: 15.64
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its efficiency in achieving the optimal solution with the lowest execution time, while maintaining high reliability.

### Optimal Decision Variables
- **w_1** = 0.600
  - *Business Meaning*: Trust weight for the first review, optimally set to 0.6
- **w_2** = 0.800
  - *Business Meaning*: Trust weight for the second review, optimally set to 0.8
- **w_3** = 1.000
  - *Business Meaning*: Trust weight for the third review, optimally set to 1.0
- **w_4** = 0.700
  - *Business Meaning*: Trust weight for the fourth review, optimally set to 0.7
- **w_5** = 0.700
  - *Business Meaning*: Trust weight for the fifth review, optimally set to 0.7

### Business Interpretation
**Overall Strategy**: The optimal solution suggests maximizing the trust-weighted ratings by assigning the highest possible trust weights to each review, within their respective upper bounds. This ensures the most reliable reviews contribute the most to the overall rating.
**Objective Value Meaning**: The optimal objective value of 15.64 represents the maximum achievable sum of trust-weighted ratings, ensuring the most reliable reviews are prioritized.
**Resource Allocation Summary**: Allocate trust weights to each review as per the optimal solution to maximize the overall trust-weighted rating.
**Implementation Recommendations**: Implement the recommended trust weights in the review system to ensure the most reliable reviews are given the highest priority in the overall rating calculation.