# Complete Optimization Problem and Solution: epinions_1

## 1. Problem Context and Goals

### Context  
The platform operates in a business environment where users provide reviews, and other users rely on these reviews to make informed decisions. To enhance the reliability of these reviews, the platform aims to assign trust weights to reviews based on the trust levels between users. These trust weights reflect the confidence a user has in the reviewer's opinion. The platform's goal is to maximize the overall trust-weighted ratings of reviews, ensuring that the weights are non-negative and do not exceed the trust levels between users. This approach ensures that reviews from highly trusted users have a more significant impact on the overall rating.

The business configuration includes a maximum allowable trust weight for a review, which is set to 1.0. This parameter ensures that the trust weights remain realistic and feasible, aligning with the highest possible trust level between users. The decision variables in this optimization problem are the continuous trust weights assigned to each review, and the objective is to maximize the sum of the trust-weighted ratings of all reviews.

### Goals  
The primary goal of this optimization problem is to maximize the overall trust-weighted ratings of reviews on the platform. This is achieved by assigning optimal trust weights to each review based on the trust levels between users. The success of this optimization is measured by the total sum of the trust-weighted ratings, where each review's rating is multiplied by its corresponding trust weight. The platform aims to ensure that the trust weights are non-negative and do not exceed the trust levels between users, thereby maintaining the integrity and reliability of the reviews.

## 2. Constraints    

The optimization problem is subject to two main constraints:

1. **Non-Negative Trust Weights**: The trust weight assigned to each review must be non-negative. This ensures that the impact of a review on the overall rating is always positive or neutral, reflecting the platform's commitment to maintaining the integrity of the review system.

2. **Trust Weight Upper Bound**: The trust weight assigned to each review must not exceed the trust level between the reviewer and the user. This constraint ensures that the trust weights are realistic and feasible, aligning with the actual trust levels between users. The maximum allowable trust weight for a review is set to 1.0, as defined in the business configuration.

These constraints ensure that the trust weights are within acceptable limits and that the optimization problem remains linear, avoiding any nonlinear relationships such as variable products or divisions.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 1 Database Schema
-- Objective: Schema changes include adding a trust_weight table for decision variables, updating the review table to include optimization-relevant data, and moving scalar parameters to business_configuration_logic.json.

CREATE TABLE trust_weight (
  weight FLOAT
);

CREATE TABLE review (
  rating FLOAT
);

CREATE TABLE trust (
  trust FLOAT
);
```

### Data Dictionary  
- **trust_weight Table**:  
  - **Business Purpose**: Stores the trust weights assigned to reviews.  
  - **Optimization Role**: Decision variables in the optimization model.  
  - **Columns**:  
    - **weight**: The weight assigned to a review based on trust. This is a continuous variable that directly influences the optimization objective.

- **review Table**:  
  - **Business Purpose**: Stores review data, including ratings provided by users.  
  - **Optimization Role**: Provides the coefficients for the objective function.  
  - **Columns**:  
    - **rating**: The rating of the review by the author. This value is used as a coefficient in the objective function to calculate the trust-weighted ratings.

- **trust Table**:  
  - **Business Purpose**: Stores the trust levels between users.  
  - **Optimization Role**: Provides the bounds for the constraints.  
  - **Columns**:  
    - **trust**: The trust level between the source and target user. This value is used to set the upper bound for the trust weights in the constraints.

### Current Stored Values  
```sql
-- Iteration 1 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on realistic user interactions, trust levels, and review ratings, ensuring they align with the business context of maximizing trust-weighted ratings.

-- Realistic data for trust_weight
INSERT INTO trust_weight (weight) VALUES (0.5);
INSERT INTO trust_weight (weight) VALUES (0.7);
INSERT INTO trust_weight (weight) VALUES (0.9);

-- Realistic data for review
INSERT INTO review (rating) VALUES (4.5);
INSERT INTO review (rating) VALUES (3.8);
INSERT INTO review (rating) VALUES (5.0);
INSERT INTO review (rating) VALUES (3.0);
INSERT INTO review (rating) VALUES (4.0);

-- Realistic data for trust
INSERT INTO trust (trust) VALUES (0.6);
INSERT INTO trust (trust) VALUES (0.8);
INSERT INTO trust (trust) VALUES (1.0);
INSERT INTO trust (trust) VALUES (0.7);
```

## 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.