## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be a binary decision variable where \( x_{ij} = 1 \) if employee \( i \) is assigned to shop \( j \), and \( x_{ij} = 0 \) otherwise.

#### Objective Function
Maximize the total bonuses awarded to employees:
\[
\text{Maximize } \sum_{i} \sum_{j} \text{Bonus}_i \times x_{ij}
\]
where \(\text{Bonus}_i\) is the bonus for employee \( i \) from the `evaluation` table.

#### Constraints
1. **Each employee is assigned to exactly one shop:**
   \[
   \sum_{j} x_{ij} = 1 \quad \forall i
   \]
   This ensures that each employee is assigned to exactly one shop.

2. **Each shop meets its minimum staffing requirement:**
   \[
   \sum_{i} x_{ij} \geq \text{Min\_Employees}_j \quad \forall j
   \]
   where \(\text{Min\_Employees}_j\) is the minimum number of employees required for shop \( j \) from the `shop_requirements` table.

3. **Binary assignment constraint:**
   \[
   x_{ij} \in \{0, 1\} \quad \forall i, j
   \]
   This ensures that the decision variable \( x_{ij} \) is binary, indicating whether an employee is assigned to a shop or not.

Data Source Verification:
- **Bonuses (\(\text{Bonus}_i\))**: Retrieved from `evaluation.Bonus`.
- **Minimum Employees (\(\text{Min\_Employees}_j\))**: Retrieved from `shop_requirements.Min_Employees`.

This formulation provides a complete linear model that can be solved using linear programming techniques to determine the optimal assignment of employees to shops, maximizing the total bonuses while satisfying all constraints.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def employee_hire_evaluation():
    # 1. MODEL & DATA SETUP
    model = gp.Model("employee_hire_evaluation")
    
    # Data from the problem context
    employee_ids = [1, 2, 3]
    shop_ids = [101, 102, 103]
    bonuses = {1: 500.0, 2: 750.0, 3: 1000.0}
    min_employees = {101: 3, 102: 4, 103: 5}
    
    # Validate data lengths
    assert len(employee_ids) == len(bonuses), "Mismatch in employee data length"
    assert len(shop_ids) == len(min_employees), "Mismatch in shop data length"
    
    # 2. VARIABLES
    # Decision variables: x[i, j] = 1 if employee i is assigned to shop j
    x = model.addVars(employee_ids, shop_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total bonuses
    model.setObjective(gp.quicksum(bonuses[i] * x[i, j] for i in employee_ids for j in shop_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Each employee is assigned to exactly one shop
    model.addConstrs((gp.quicksum(x[i, j] for j in shop_ids) == 1 for i in employee_ids), name="employee_assignment")
    
    # Each shop meets its minimum staffing requirement
    model.addConstrs((gp.quicksum(x[i, j] for i in employee_ids) >= min_employees[j] for j in shop_ids), name="shop_staffing")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in employee_ids:
            for j in shop_ids:
                if x[i, j].x > 0.5:  # Binary variable, so check if it's 1
                    print(f"Employee {i} assigned to Shop {j}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    else:
        print("Optimization ended with status", model.status)

# Run the optimization
employee_hire_evaluation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy consistently reports the problem as infeasible, indicating a potential issue with the constraints or data.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def employee_hire_evaluation():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="employee_hire_evaluation")
    
    # Data from the database
    employee_ids = [1, 2, 3]
    bonuses = [500.0, 750.0, 1000.0]
    shop_ids = [101, 102, 103]
    min_employees = [3, 4, 5]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(employee_ids) == len(bonuses), "Employee and bonus array length mismatch"
    assert len(shop_ids) == len(min_employees), "Shop and min_employees array length mismatch"
    
    # Safe indexing
    safe_employee_range = range(len(employee_ids))
    safe_shop_range = range(len(shop_ids))
    
    # 2. VARIABLES
    # Binary decision variables for employee-shop assignments
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in safe_employee_range for j in safe_shop_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total bonuses
    objective = mdl.sum(bonuses[i] * x[i, j] for i in safe_employee_range for j in safe_shop_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Each employee is assigned to exactly one shop
    for i in safe_employee_range:
        mdl.add_constraint(mdl.sum(x[i, j] for j in safe_shop_range) == 1, ctname=f"assign_employee_{i}")
    
    # Each shop meets its minimum staffing requirement
    for j in safe_shop_range:
        mdl.add_constraint(mdl.sum(x[i, j] for i in safe_employee_range) >= min_employees[j], ctname=f"min_staff_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_employee_range:
            for j in safe_shop_range:
                if solution.get_value(x[i, j]) > 0.5:  # Check if the employee is assigned to the shop
                    print(f"Employee {employee_ids[i]} assigned to Shop {shop_ids[j]}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Execute the function
employee_hire_evaluation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also reports infeasibility, suggesting that the problem formulation or data constraints are not satisfiable.

## 7. Pyomo Implementation

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

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

def employee_hire_evaluation():
    """Optimize employee assignments to maximize bonuses."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Employee and shop data
    employees = [1, 2, 3]
    shops = [101, 102, 103]
    bonuses = {1: 500.0, 2: 750.0, 3: 1000.0}
    min_employees = {101: 3, 102: 4, 103: 5}
    
    # CRITICAL: Validate data lengths
    assert len(bonuses) == len(employees), "Mismatch in bonuses and employees"
    assert len(min_employees) == len(shops), "Mismatch in min_employees and shops"
    
    # 3. SETS
    model.I = pyo.Set(initialize=employees)
    model.J = pyo.Set(initialize=shops)
    
    # 4. PARAMETERS
    model.bonus = pyo.Param(model.I, initialize=bonuses)
    model.min_employees = pyo.Param(model.J, initialize=min_employees)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, model.J, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.bonus[i] * 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
    
    # Each employee is assigned to exactly one shop
    def employee_assignment_rule(model, i):
        return sum(model.x[i, j] for j in model.J) == 1
    model.employee_assignment = pyo.Constraint(model.I, rule=employee_assignment_rule)
    
    # Each shop meets its minimum staffing requirement
    def shop_staffing_rule(model, j):
        return sum(model.x[i, j] for i in model.I) >= model.min_employees[j]
    model.shop_staffing = pyo.Constraint(model.J, rule=shop_staffing_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)
    
    # 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("\nEmployee assignments:")
        for i in model.I:
            for j in model.J:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if assigned
                    print(f"Employee {i} assigned to Shop {j}")
        
    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
employee_hire_evaluation()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Pyomo's infeasibility result aligns with the other solvers, reinforcing the likelihood of an issue with the problem setup.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.25s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 2.15s | N/A | 4 |
| Pyomo | INFEASIBLE | N/A | 1.18s | N/A | 4 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: All solvers consistently report infeasibility, suggesting a fundamental issue with the problem constraints or data.

### Business Interpretation
**Overall Strategy**: The current problem setup cannot be solved as it stands, indicating that the staffing requirements or employee bonuses need to be re-evaluated.
**Objective Value Meaning**: The objective value cannot be interpreted as the problem is infeasible.
**Resource Allocation Summary**: Current resource allocation requirements cannot be met; reassessment of staffing needs and employee bonuses is necessary.
**Implementation Recommendations**: Review and potentially relax the minimum staffing requirements for shops. Verify the accuracy of the bonus data and ensure all constraints are logically sound.