## 4. Mathematical Optimization Formulation

#### Decision Variables
Let \( x_{i,j} \) be the number of phones produced with chip model \( i \) and screen mode \( j \), where:
- \( i \in \{1, 2, 3\} \) (chip models)
- \( j \in \{1, 2, 3\} \) (screen modes)

#### Objective Function
Minimize the total cost of production:
\[
\text{Minimize } \sum_{i=1}^{3} \sum_{j=1}^{3} (c_i + s_j) \cdot x_{i,j}
\]
where:
- \( c_i \) is the cost of chip model \( i \) (from `chip_models.cost`)
- \( s_j \) is the cost of screen mode \( j \) (from `screen_modes.cost`)

#### Constraints
1. **Total RAM Constraint**:
\[
\sum_{i=1}^{3} \sum_{j=1}^{3} r_i \cdot x_{i,j} \geq 8000
\]
where \( r_i \) is the RAM capacity of chip model \( i \) (from `chip_models.RAM_MiB`).

2. **Total ROM Constraint**:
\[
\sum_{i=1}^{3} \sum_{j=1}^{3} o_i \cdot x_{i,j} \geq 128000
\]
where \( o_i \) is the ROM capacity of chip model \( i \) (from `chip_models.ROM_MiB`).

3. **Total Accreditation Level Constraint**:
\[
\sum_{i=1}^{3} \sum_{j=1}^{3} a_j \cdot x_{i,j} \geq 5
\]
where \( a_j \) is the accreditation level of screen mode \( j \) (from `screen_modes.Accreditation_level`).

4. **Non-Negativity Constraint**:
\[
x_{i,j} \geq 0 \quad \forall i, j
\]

#### Data Source Verification
- \( c_i \): `chip_models.cost`
- \( s_j \): `screen_modes.cost`
- \( r_i \): `chip_models.RAM_MiB`
- \( o_i \): `chip_models.ROM_MiB`
- \( a_j \): `screen_modes.Accreditation_level`

#### Numerical Coefficients
Based on the provided data:
- \( c_1 = 50.0 \), \( c_2 = 75.0 \), \( c_3 = 100.0 \)
- \( s_1 = 30.0 \), \( s_2 = 45.0 \), \( s_3 = 60.0 \)
- \( r_1 = 2048 \), \( r_2 = 4096 \), \( r_3 = 8192 \)
- \( o_1 = 32768 \), \( o_2 = 65536 \), \( o_3 = 131072 \)
- \( a_1 = 3 \), \( a_2 = 4 \), \( a_3 = 5 \)

This formulation provides a complete, immediately solvable LINEAR mathematical model for the optimization problem.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy 12.0.2 Implementation for Phone Production Optimization
"""

import gurobipy as gp
from gurobipy import GRB

def phone_production_optimization():
    """Optimize phone production to minimize total cost while meeting hardware and accreditation requirements."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("phone_production")
    
    # Chip models data
    chip_models = {
        1: {"cost": 50.0, "RAM_MiB": 2048, "ROM_MiB": 32768},
        2: {"cost": 75.0, "RAM_MiB": 4096, "ROM_MiB": 65536},
        3: {"cost": 100.0, "RAM_MiB": 8192, "ROM_MiB": 131072}
    }
    
    # Screen modes data
    screen_modes = {
        1: {"cost": 30.0, "Accreditation_level": 3},
        2: {"cost": 45.0, "Accreditation_level": 4},
        3: {"cost": 60.0, "Accreditation_level": 5}
    }
    
    # Minimum requirements
    min_RAM = 8000
    min_ROM = 128000
    min_Accreditation = 5
    
    # Validate data lengths
    assert len(chip_models) == 3, "Chip models data length mismatch"
    assert len(screen_modes) == 3, "Screen modes data length mismatch"
    
    # 2. VARIABLES
    x = model.addVars(chip_models.keys(), screen_modes.keys(), vtype=GRB.CONTINUOUS, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(
        gp.quicksum((chip_models[i]["cost"] + screen_modes[j]["cost"]) * x[i, j] 
                    for i in chip_models.keys() for j in screen_modes.keys()), 
        GRB.MINIMIZE
    )
    
    # 4. CONSTRAINTS
    
    # Total RAM Constraint
    model.addConstr(
        gp.quicksum(chip_models[i]["RAM_MiB"] * x[i, j] 
                    for i in chip_models.keys() for j in screen_modes.keys()) >= min_RAM, 
        name="total_RAM"
    )
    
    # Total ROM Constraint
    model.addConstr(
        gp.quicksum(chip_models[i]["ROM_MiB"] * x[i, j] 
                    for i in chip_models.keys() for j in screen_modes.keys()) >= min_ROM, 
        name="total_ROM"
    )
    
    # Total Accreditation Level Constraint
    model.addConstr(
        gp.quicksum(screen_modes[j]["Accreditation_level"] * x[i, j] 
                    for i in chip_models.keys() for j in screen_modes.keys()) >= min_Accreditation, 
        name="total_Accreditation"
    )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in chip_models.keys():
            for j in screen_modes.keys():
                if x[i, j].x > 1e-6:
                    print(f"x[{i},{j}] = {x[i, j].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 158.4375
**Execution Time**: 0.18 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution with the lowest execution time, indicating high efficiency and reliability.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex implementation for phone production optimization problem
"""

from docplex.mp.model import Model

def phone_production_optimization():
    """Optimize phone production to minimize total cost while meeting hardware and accreditation requirements."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="phone_production_optimization")
    
    # Data from the problem
    chip_models = [
        {"id": 1, "cost": 50.0, "RAM_MiB": 2048, "ROM_MiB": 32768},
        {"id": 2, "cost": 75.0, "RAM_MiB": 4096, "ROM_MiB": 65536},
        {"id": 3, "cost": 100.0, "RAM_MiB": 8192, "ROM_MiB": 131072}
    ]
    
    screen_modes = [
        {"id": 1, "cost": 30.0, "Accreditation_level": 3},
        {"id": 2, "cost": 45.0, "Accreditation_level": 4},
        {"id": 3, "cost": 60.0, "Accreditation_level": 5}
    ]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(chip_models) == 3 and len(screen_modes) == 3, "Array length mismatch"
    safe_range_chip = range(len(chip_models))
    safe_range_screen = range(len(screen_modes))
    
    # 2. VARIABLES
    # Decision variables: x_{i,j} = number of phones produced with chip model i and screen mode j
    x = {(i, j): mdl.continuous_var(name=f"x_{i}_{j}", lb=0) 
         for i in safe_range_chip for j in safe_range_screen}
    
    # 3. OBJECTIVE FUNCTION
    # Minimize total cost: sum_{i,j} (c_i + s_j) * x_{i,j}
    total_cost = mdl.sum((chip_models[i]["cost"] + screen_modes[j]["cost"]) * x[(i, j)] 
                         for i in safe_range_chip for j in safe_range_screen)
    mdl.minimize(total_cost)
    
    # 4. CONSTRAINTS
    
    # Total RAM Constraint: sum_{i,j} r_i * x_{i,j} >= 8000
    total_ram = mdl.sum(chip_models[i]["RAM_MiB"] * x[(i, j)] 
                        for i in safe_range_chip for j in safe_range_screen)
    mdl.add_constraint(total_ram >= 8000, ctname="total_ram")
    
    # Total ROM Constraint: sum_{i,j} o_i * x_{i,j} >= 128000
    total_rom = mdl.sum(chip_models[i]["ROM_MiB"] * x[(i, j)] 
                        for i in safe_range_chip for j in safe_range_screen)
    mdl.add_constraint(total_rom >= 128000, ctname="total_rom")
    
    # Total Accreditation Level Constraint: sum_{i,j} a_j * x_{i,j} >= 5
    total_accreditation = mdl.sum(screen_modes[j]["Accreditation_level"] * x[(i, j)] 
                                  for i in safe_range_chip for j in safe_range_screen)
    mdl.add_constraint(total_accreditation >= 5, ctname="total_accreditation")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range_chip:
            for j in safe_range_screen:
                value = solution.get_value(x[(i, j)])
                if value > 1e-6:
                    print(f"x_{i}_{j} = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

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

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 158.4375
**Execution Time**: 1.00 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, matching Gurobipy's result, but with a longer execution time.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def phone_production_optimization():
    """Optimize phone production to minimize total cost while meeting hardware and accreditation requirements."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Chip models data
    chip_models_data = {
        1: {'cost': 50.0, 'RAM_MiB': 2048, 'ROM_MiB': 32768},
        2: {'cost': 75.0, 'RAM_MiB': 4096, 'ROM_MiB': 65536},
        3: {'cost': 100.0, 'RAM_MiB': 8192, 'ROM_MiB': 131072}
    }
    
    # Screen modes data
    screen_modes_data = {
        1: {'cost': 30.0, 'Accreditation_level': 3},
        2: {'cost': 45.0, 'Accreditation_level': 4},
        3: {'cost': 60.0, 'Accreditation_level': 5}
    }
    
    # Validate array lengths
    assert len(chip_models_data) == 3, "Chip models data length mismatch"
    assert len(screen_modes_data) == 3, "Screen modes data length mismatch"
    
    # 3. SETS
    model.ChipModels = pyo.Set(initialize=chip_models_data.keys())
    model.ScreenModes = pyo.Set(initialize=screen_modes_data.keys())
    
    # 4. PARAMETERS
    model.chip_cost = pyo.Param(model.ChipModels, initialize={i: chip_models_data[i]['cost'] for i in chip_models_data})
    model.chip_RAM = pyo.Param(model.ChipModels, initialize={i: chip_models_data[i]['RAM_MiB'] for i in chip_models_data})
    model.chip_ROM = pyo.Param(model.ChipModels, initialize={i: chip_models_data[i]['ROM_MiB'] for i in chip_models_data})
    
    model.screen_cost = pyo.Param(model.ScreenModes, initialize={j: screen_modes_data[j]['cost'] for j in screen_modes_data})
    model.screen_accred = pyo.Param(model.ScreenModes, initialize={j: screen_modes_data[j]['Accreditation_level'] for j in screen_modes_data})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.ChipModels, model.ScreenModes, within=pyo.NonNegativeIntegers)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum((model.chip_cost[i] + model.screen_cost[j]) * model.x[i, j] for i in model.ChipModels for j in model.ScreenModes)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.minimize)
    
    # 7. CONSTRAINTS
    def ram_constraint_rule(model):
        return sum(model.chip_RAM[i] * model.x[i, j] for i in model.ChipModels for j in model.ScreenModes) >= 8000
    model.ram_constraint = pyo.Constraint(rule=ram_constraint_rule)
    
    def rom_constraint_rule(model):
        return sum(model.chip_ROM[i] * model.x[i, j] for i in model.ChipModels for j in model.ScreenModes) >= 128000
    model.rom_constraint = pyo.Constraint(rule=rom_constraint_rule)
    
    def accred_constraint_rule(model):
        return sum(model.screen_accred[j] * model.x[i, j] for i in model.ChipModels for j in model.ScreenModes) >= 5
    model.accred_constraint = pyo.Constraint(rule=accred_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
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nProduction quantities:")
        for i in model.ChipModels:
            for j in model.ScreenModes:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0:
                    print(f"Chip Model {i} with Screen Mode {j}: {int(x_val)} units")
        
    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__":
    phone_production_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 160.0
**Execution Time**: 0.83 seconds
**Reliability**: medium
**Analysis**: Pyomo found an optimal solution, but the objective value is slightly higher than Gurobipy and DOCplex, suggesting potential numerical precision issues.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 158.44 | 0.18s | N/A | N/A |
| Docplex | OPTIMAL | 158.44 | 1.00s | N/A | N/A |
| Pyomo | OPTIMAL | 160.00 | 0.83s | N/A | N/A |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Consistent Solvers**: gurobipy, docplex
**Inconsistent Solvers**: pyomo
**Potential Issues**:
- Numerical precision differences in Pyomo
- Solver-specific optimization algorithms
**Majority Vote Optimal Value**: 158.4375

### Final Recommendation
**Recommended Optimal Value**: 158.4375
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time and consistent optimal value with DOCplex. Pyomo's slightly higher objective value suggests it may not be as precise.

### Business Interpretation
**Overall Strategy**: The optimal production cost is $158.44, achieved by Gurobipy and DOCplex. This represents the most cost-efficient allocation of resources.
**Objective Value Meaning**: The optimal objective value of $158.44 represents the minimum total production cost while meeting all constraints.
**Resource Allocation Summary**: Resources should be allocated to produce phones with the most cost-effective combination of chip models and screen modes, ensuring RAM, ROM, and accreditation level requirements are met.
**Implementation Recommendations**: Use Gurobipy for solving the optimization problem. Allocate production based on the optimal decision variables to minimize costs while meeting all technical and accreditation requirements.