## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_{ij} \): Number of visits student \( i \) makes to restaurant \( j \).
  - \( x_{ij} \geq 0 \) and integer for all students \( i \) and restaurants \( j \).

#### Objective Function
Maximize the total satisfaction score across all students and restaurants:
\[ 
\text{Maximize } \sum_{i} \sum_{j} \text{Rating}_{j} \times x_{ij} 
\]
Where:
- \(\text{Rating}_{j}\) is the rating of restaurant \( j \) from the `Restaurant` table.

#### Constraints
1. Budget Constraints for each student:
   \[
   \sum_{j} \text{Cost}_{ij} \times x_{ij} \leq \text{Budget}_{i} \quad \forall i
   \]
   Where:
   - \(\text{Cost}_{ij}\) is the cost per visit for student \( i \) to restaurant \( j \) from the `Cost_Per_Visit` table.
   - \(\text{Budget}_{i}\) is the budget for student \( i \) from the `Student_Budget` table.

2. Non-negativity and integer constraints:
   \[
   x_{ij} \geq 0 \quad \text{and integer} \quad \forall i, j
   \]

Data Source Verification:
- \(\text{Rating}_{j}\) from `Restaurant.Rating`
- \(\text{Cost}_{ij}\) from `Cost_Per_Visit.Cost`
- \(\text{Budget}_{i}\) from `Student_Budget.Budget`

This formulation is a Mixed-Integer Linear Programming (MILP) model due to the integer constraints on the decision variables \( x_{ij} \). The objective function and constraints are linear, adhering to the requirements for linear optimization.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def restaurant_optimization():
    """Optimize student visits to restaurants for maximum satisfaction."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("restaurant_optimization")
    
    # Data from the problem statement
    restaurants = [1, 2, 3]
    students = [101, 102, 103]
    
    ratings = {1: 4.5, 2: 3.8, 3: 5.0}
    budgets = {101: 120, 102: 180, 103: 150}
    costs = {(101, 1): 12.0, (102, 2): 15.0, (103, 3): 20.0}
    
    # Validate data lengths
    assert len(ratings) == len(restaurants), "Ratings length mismatch"
    assert len(budgets) == len(students), "Budgets length mismatch"
    assert len(costs) == len(students) * len(restaurants), "Costs length mismatch"
    
    # 2. VARIABLES
    # Decision variables: number of visits each student makes to each restaurant
    visits = model.addVars(students, restaurants, vtype=GRB.INTEGER, name="visits", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction score
    model.setObjective(gp.quicksum(ratings[j] * visits[i, j] for i in students for j in restaurants), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Budget constraints for each student
    for i in students:
        model.addConstr(gp.quicksum(costs[i, j] * visits[i, j] for j in restaurants if (i, j) in costs) <= budgets[i], name=f"budget_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in students:
            for j in restaurants:
                if visits[i, j].x > 1e-6:
                    print(f"Student {i} visits Restaurant {j}: {visits[i, j].x:.0f} times")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
restaurant_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp_v7eeqr1.py", line 55, in <module>
    restaurant_optimization()
  File "/tmp/tmp_v7eeqr1.py", line 23, in restaurant_optimization
    assert len(costs) == len(students) * len(restaurants), "Costs length mismatch"
AssertionError: Costs length mismatch

**Analysis**: Gurobipy encountered an error due to a mismatch in the length of the costs array, indicating a potential issue with data preparation or input consistency.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def restaurant_optimization():
    """Optimize student visits to restaurants for maximum satisfaction."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="restaurant_optimization")
    
    # Data from the problem statement
    restaurants = [1, 2, 3]
    students = [101, 102, 103]
    
    ratings = {1: 4.5, 2: 3.8, 3: 5.0}
    budgets = {101: 120, 102: 180, 103: 150}
    costs = {(101, 1): 12.0, (102, 2): 15.0, (103, 3): 20.0}
    
    # Validate data consistency
    assert len(ratings) == len(restaurants), "Ratings length mismatch"
    assert len(budgets) == len(students), "Budgets length mismatch"
    assert len(costs) == len(students) * len(restaurants), "Costs length mismatch"
    
    # Safe range for indexing
    safe_students = range(len(students))
    safe_restaurants = range(len(restaurants))
    
    # 2. VARIABLES
    # Decision variables: number of visits each student makes to each restaurant
    visits = {(i, j): mdl.integer_var(name=f"x_{i}_{j}", lb=0) for i in students for j in restaurants}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total satisfaction score
    objective = mdl.sum(ratings[j] * visits[i, j] for i in students for j in restaurants)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Budget constraints for each student
    for i in students:
        mdl.add_constraint(mdl.sum(costs[i, j] * visits[i, j] for j in restaurants) <= budgets[i], ctname=f"budget_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in students:
            for j in restaurants:
                value = solution.get_value(visits[i, j])
                if value > 0:
                    print(f"Student {i} visits Restaurant {j}: {value} times")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Run the optimization
restaurant_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpo10g2c37.py", line 57, in <module>
    restaurant_optimization()
  File "/tmp/tmpo10g2c37.py", line 22, in restaurant_optimization
    assert len(costs) == len(students) * len(restaurants), "Costs length mismatch"
AssertionError: Costs length mismatch

**Analysis**: DOCplex also faced a costs length mismatch error, similar to Gurobipy, suggesting a systemic issue with the input data.

## 7. Pyomo Implementation

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

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

def restaurant_optimization():
    """Optimize student visits to restaurants for maximum satisfaction."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define the data based on the provided SQL data
    students = [101, 102, 103]
    restaurants = [1, 2, 3]
    
    ratings = {1: 4.5, 2: 3.8, 3: 5.0}
    budgets = {101: 120, 102: 180, 103: 150}
    costs = {(101, 1): 12.0, (102, 2): 15.0, (103, 3): 20.0}
    
    # CRITICAL: Validate array lengths before indexing
    assert len(costs) == len(students), "Costs length mismatch"
    
    # 3. SETS
    model.I = pyo.Set(initialize=students)
    model.J = pyo.Set(initialize=restaurants)
    
    # 4. PARAMETERS
    model.rating = pyo.Param(model.J, initialize=ratings)
    model.budget = pyo.Param(model.I, initialize=budgets)
    model.cost = pyo.Param(model.I, model.J, initialize=costs, default=0)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.NonNegativeIntegers)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.rating[j] * model.x[i, j] for i in model.I for j in model.J)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    def budget_constraint_rule(model, i):
        return sum(model.cost[i, j] * model.x[i, j] for j in model.J) <= model.budget[i]
    model.budget_constraint = pyo.Constraint(model.I, rule=budget_constraint_rule)
    
    # 8. SOLVING WITH GUROBI
    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("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            for j in model.J:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0:  # Only print non-zero values
                    print(f"x[{i},{j}] = {x_val}")
        
    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

# Execute the optimization
restaurant_optimization()
```

### Execution Results
**Status**: UNBOUNDED
**Error**: Problem is unbounded
**Analysis**: Pyomo reported the problem as unbounded, which may indicate missing constraints or incorrect formulation, especially given the errors in other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.57s | N/A | 4 |
| Docplex | ERROR | N/A | 7.53s | N/A | 4 |
| Pyomo | UNBOUNDED | N/A | 3.12s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Inconsistent Solvers**: gurobipy, docplex, pyomo
**Potential Issues**:
- Mismatch in costs data length
- Potential missing or incorrect constraints leading to unbounded solution
- Data preparation errors affecting all solvers
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: LOW
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers encountered significant issues, indicating a need to revisit the data and formulation before a reliable solver preference can be established.

### Business Interpretation
**Overall Strategy**: The current results do not provide a valid solution due to data and formulation issues.
**Objective Value Meaning**: The optimal objective value would represent the maximum total satisfaction score achievable within the given budget constraints.
**Resource Allocation Summary**: Currently unavailable due to errors; requires correction of data and formulation.
**Implementation Recommendations**: Review and correct the input data, particularly the costs array, and ensure all constraints are properly defined and implemented.