## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{i,j} \) be the number of times movie \( j \) is recommended to reviewer \( i \), where \( i \) is the reviewer ID and \( j \) is the movie ID.  
\( x_{i,j} \) is a non-negative integer variable.

#### Objective Function
Maximize the total satisfaction derived from movie recommendations:  
\[
\text{Maximize } Z = \sum_{i} \sum_{j} (\text{star\_rating}_{i,j} \times x_{i,j})
\]  
where \( \text{star\_rating}_{i,j} \) is the star rating given by reviewer \( i \) to movie \( j \).

#### Constraints
1. **Movie Recommendation Limits**: The total number of times a movie \( j \) is recommended across all reviewers must not exceed its maximum allowed recommendations:  
\[
\sum_{i} x_{i,j} \leq \text{max\_recommendations}_j \quad \forall j
\]  
2. **Reviewer Recommendation Limits**: The total number of recommendations received by a reviewer \( i \) must not exceed their maximum allowed recommendations:  
\[
\sum_{j} x_{i,j} \leq \text{max\_recommendations}_i \quad \forall i
\]  
3. **Non-Negative Recommendations**: The number of recommendations must be non-negative integers:  
\[
x_{i,j} \geq 0 \quad \forall i, j
\]  
\[
x_{i,j} \in \mathbb{Z} \quad \forall i, j
\]

#### Data Source Verification
- \( \text{star\_rating}_{i,j} \): From `MovieRatings.star_rating`.  
- \( \text{max\_recommendations}_j \): From `MovieRecommendationLimits.max_recommendations`.  
- \( \text{max\_recommendations}_i \): From `ReviewerRecommendationLimits.max_recommendations`.  

#### Numerical Example
Using the provided data:  
- \( \text{star\_rating}_{1,1} = 5 \), \( \text{star\_rating}_{2,2} = 4 \), \( \text{star\_rating}_{3,3} = 3 \).  
- \( \text{max\_recommendations}_1 = 10 \), \( \text{max\_recommendations}_2 = 7 \), \( \text{max\_recommendations}_3 = 5 \).  
- \( \text{max\_recommendations}_1 = 5 \), \( \text{max\_recommendations}_2 = 4 \), \( \text{max\_recommendations}_3 = 3 \).  

The objective function becomes:  
\[
\text{Maximize } Z = 5x_{1,1} + 4x_{2,2} + 3x_{3,3}
\]  

The constraints are:  
1. \( x_{1,1} + x_{2,1} + x_{3,1} \leq 10 \)  
   \( x_{1,2} + x_{2,2} + x_{3,2} \leq 7 \)  
   \( x_{1,3} + x_{2,3} + x_{3,3} \leq 5 \)  
2. \( x_{1,1} + x_{1,2} + x_{1,3} \leq 5 \)  
   \( x_{2,1} + x_{2,2} + x_{2,3} \leq 4 \)  
   \( x_{3,1} + x_{3,2} + x_{3,3} \leq 3 \)  
3. \( x_{i,j} \geq 0 \) and \( x_{i,j} \in \mathbb{Z} \) for all \( i, j \).  

This is a complete, immediately solvable Mixed-Integer Linear Programming (MILP) model.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def optimize_movie_recommendations():
    # 1. MODEL & DATA SETUP
    model = gp.Model("movie_recommendation_optimization")
    
    # Data from the database
    movie_ids = [1, 2, 3]
    reviewer_ids = [1, 2, 3]
    
    # Star ratings
    star_ratings = {
        (1, 1): 5,
        (2, 2): 4,
        (3, 3): 3
    }
    
    # Movie recommendation limits
    movie_limits = {
        1: 10,
        2: 7,
        3: 5
    }
    
    # Reviewer recommendation limits
    reviewer_limits = {
        1: 5,
        2: 4,
        3: 3
    }
    
    # 2. VARIABLES
    x = model.addVars(reviewer_ids, movie_ids, vtype=GRB.INTEGER, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum(star_ratings[i, j] * x[i, j] for i in reviewer_ids for j in movie_ids if (i, j) in star_ratings),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS
    
    # Movie Recommendation Limits
    for j in movie_ids:
        model.addConstr(
            gp.quicksum(x[i, j] for i in reviewer_ids) <= movie_limits[j],
            name=f"movie_limit_{j}"
        )
    
    # Reviewer Recommendation Limits
    for i in reviewer_ids:
        model.addConstr(
            gp.quicksum(x[i, j] for j in movie_ids) <= reviewer_limits[i],
            name=f"reviewer_limit_{i}"
        )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in reviewer_ids:
            for j in movie_ids:
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].x}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
optimize_movie_recommendations()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 50.0
**Execution Time**: 0.17 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: Gurobipy achieved the optimal solution efficiently with the fastest execution time.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def movie_recommendation_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="movie_recommendation_optimization")
    
    # Data from the database
    movie_ids = [1, 2, 3]
    reviewer_ids = [1, 2, 3]
    
    # Star ratings
    star_ratings = {
        (1, 1): 5,
        (2, 2): 4,
        (3, 3): 3
    }
    
    # Movie recommendation limits
    movie_limits = {
        1: 10,
        2: 7,
        3: 5
    }
    
    # Reviewer recommendation limits
    reviewer_limits = {
        1: 5,
        2: 4,
        3: 3
    }
    
    # 2. VARIABLES
    x = {(i, j): mdl.integer_var(name=f"x_{i}_{j}", lb=0) for i in reviewer_ids for j in movie_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(star_ratings.get((i, j), 0) * x[i, j] for i in reviewer_ids for j in movie_ids)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Movie Recommendation Limits
    for j in movie_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for i in reviewer_ids) <= movie_limits[j], ctname=f"movie_limit_{j}")
    
    # Reviewer Recommendation Limits
    for i in reviewer_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for j in movie_ids) <= reviewer_limits[i], ctname=f"reviewer_limit_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in reviewer_ids:
            for j in movie_ids:
                value = solution.get_value(x[i, j])
                if value > 0:
                    print(f"x[{i},{j}] = {value}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
movie_recommendation_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 50.0
**Execution Time**: 1.32 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: DOCplex also found the optimal solution but took significantly longer compared to Gurobipy.

## 7. Pyomo Implementation

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

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

def movie_recommendation_optimization():
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define sets for reviewers and movies
    reviewers = [1, 2, 3]
    movies = [1, 2, 3]
    
    # Define star ratings
    star_ratings = {
        (1, 1): 5,
        (2, 2): 4,
        (3, 3): 3
    }
    
    # Define maximum recommendations for movies
    movie_max_recommendations = {
        1: 10,
        2: 7,
        3: 5
    }
    
    # Define maximum recommendations for reviewers
    reviewer_max_recommendations = {
        1: 5,
        2: 4,
        3: 3
    }
    
    # 3. SETS
    model.reviewers = pyo.Set(initialize=reviewers)
    model.movies = pyo.Set(initialize=movies)
    
    # 4. PARAMETERS
    model.star_ratings = pyo.Param(model.reviewers, model.movies, initialize=star_ratings, default=0)
    model.movie_max_recommendations = pyo.Param(model.movies, initialize=movie_max_recommendations)
    model.reviewer_max_recommendations = pyo.Param(model.reviewers, initialize=reviewer_max_recommendations)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.reviewers, model.movies, within=pyo.NonNegativeIntegers)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.star_ratings[i, j] * model.x[i, j] for i in model.reviewers for j in model.movies)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    # Movie Recommendation Limits
    def movie_limit_rule(model, j):
        return sum(model.x[i, j] for i in model.reviewers) <= model.movie_max_recommendations[j]
    model.movie_limit_constraint = pyo.Constraint(model.movies, rule=movie_limit_rule)
    
    # Reviewer Recommendation Limits
    def reviewer_limit_rule(model, i):
        return sum(model.x[i, j] for j in model.movies) <= model.reviewer_max_recommendations[i]
    model.reviewer_limit_constraint = pyo.Constraint(model.reviewers, rule=reviewer_limit_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    results = solver.solve(model, tee=True)
    
    # 9. RESULT PROCESSING
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print(f"Optimal value: {pyo.value(model.objective)}")
    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
movie_recommendation_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 50.0
**Execution Time**: 0.87 seconds
**Reliability**: high
**Retry Attempt**: 1
**Analysis**: Pyomo successfully found 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 | 50.00 | 0.17s | N/A | 1 |
| Docplex | OPTIMAL | 50.00 | 1.32s | N/A | 1 |
| Pyomo | OPTIMAL | 50.00 | 0.87s | N/A | 1 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 50.0
**Solver Retry Summary**: gurobipy: 1 attempts, docplex: 1 attempts, pyomo: 1 attempts

### Final Recommendation
**Recommended Optimal Value**: 50.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its superior performance in terms of execution time while still achieving the optimal solution.

### Business Interpretation
**Overall Strategy**: The optimal total satisfaction score of 50 indicates the maximum possible satisfaction derived from the movie recommendations given the constraints.
**Objective Value Meaning**: The optimal objective value of 50 represents the highest possible satisfaction score achievable under the given constraints, ensuring efficient use of recommendation limits.
**Resource Allocation Summary**: Resources (recommendations) should be allocated to maximize the satisfaction score, prioritizing movies with higher star ratings and adhering to individual and movie-specific recommendation limits.
**Implementation Recommendations**: Implement the recommended movie allocations to reviewers, ensuring that the constraints on maximum recommendations per movie and per reviewer are strictly followed to maintain the optimal satisfaction score.