## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( x_i \): Credit limit allocated to account \( i \). This is a continuous variable representing the decision to be made for each account.

#### Objective Function
Maximize the total expected transaction volume across all accounts:

\[
\text{Maximize } \sum_{i} \text{expected\_volume}_i \times x_i
\]

Where:
- \(\text{expected\_volume}_i\) is the expected transaction volume for account \( i \) from the `account_expected_transaction_volume` table.

#### Constraints
1. Total credit allocated across all accounts must not exceed the total available credit:
   \[
   \sum_{i} x_i \leq \text{Total\_Available\_Credit}
   \]

2. Each account's credit limit must not exceed its maximum allowable credit limit:
   \[
   x_i \leq \text{max\_credit\_limit}_i \quad \forall i
   \]

3. For an account to be considered active, its credit limit must meet or exceed the minimum credit limit requirement:
   \[
   x_i \geq \text{Minimum\_Credit\_Limit} \quad \text{if } \text{is\_active}_i = \text{True}
   \]

Where:
- \(\text{max\_credit\_limit}_i\) is the maximum allowable credit limit for account \( i \) from the `account_credit_limits` table.
- \(\text{is\_active}_i\) is a boolean indicator from the `account_credit_limits` table.
- \(\text{Total\_Available\_Credit}\) and \(\text{Minimum\_Credit\_Limit}\) are business configuration parameters.

Data Source Verification:
- \(\text{expected\_volume}_i\) from `account_expected_transaction_volume.expected_volume`
- \(\text{max\_credit\_limit}_i\) from `account_credit_limits.max_credit_limit`
- \(\text{is\_active}_i\) from `account_credit_limits.is_active`
- \(\text{Total\_Available\_Credit}\) and \(\text{Minimum\_Credit\_Limit}\) are assumed to be provided as business configuration parameters outside the database schema.

This linear model can be solved using standard linear programming techniques to determine the optimal credit limits for maximizing expected transaction volumes while adhering to the constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_credit_limits():
    """Optimize credit limits to maximize expected transaction volume."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("customers_card_transactions")
    
    # Data from the database
    account_ids = [1, 2, 3]
    expected_volumes = [1200.0, 1800.0, 2200.0]
    max_credit_limits = [2000.0, 2500.0, 3000.0]
    is_active = [True, True, True]
    
    # Business configuration parameters
    Total_Available_Credit = 5000.0
    Minimum_Credit_Limit = 500.0
    
    # CRITICAL: Validate array lengths before loops
    assert len(account_ids) == len(expected_volumes) == len(max_credit_limits) == len(is_active), "Array length mismatch"
    
    # 2. VARIABLES
    # Variable dictionaries for credit limits
    x = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{account_ids[i]}", lb=0) 
         for i in range(len(account_ids))}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total expected transaction volume
    model.setObjective(gp.quicksum(expected_volumes[i] * x[i] for i in range(len(account_ids))), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total credit allocated must not exceed the total available credit
    model.addConstr(gp.quicksum(x[i] for i in range(len(account_ids))) <= Total_Available_Credit, name="total_credit_limit")
    
    # Each account's credit limit must not exceed its maximum allowable credit limit
    for i in range(len(account_ids)):
        model.addConstr(x[i] <= max_credit_limits[i], name=f"max_credit_limit_{account_ids[i]}")
    
    # For an account to be considered active, its credit limit must meet or exceed the minimum credit limit requirement
    for i in range(len(account_ids)):
        if is_active[i]:
            model.addConstr(x[i] >= Minimum_Credit_Limit, name=f"min_credit_limit_{account_ids[i]}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(len(account_ids)):
            if x[i].x > 1e-6:
                print(f"x[{account_ids[i]}] = {x[i].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
optimize_credit_limits()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 9900000.0
**Execution Time**: 0.64 seconds
**Reliability**: high
**Analysis**: Gurobipy achieved an optimal solution quickly, indicating efficient handling of the problem size and constraints.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_credit_limits():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="customers_card_transactions")

    # Data from the database
    account_ids = [1, 2, 3]
    expected_volumes = [1200.0, 1800.0, 2200.0]
    max_credit_limits = [2000.0, 2500.0, 3000.0]
    is_active = [True, True, True]

    # Business configuration parameters
    Total_Available_Credit = 5000.0
    Minimum_Credit_Limit = 500.0

    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(account_ids) == len(expected_volumes) == len(max_credit_limits) == len(is_active), "Array length mismatch"
    safe_range = range(len(account_ids))  # Safe indexing

    # 2. VARIABLES
    # Variable dictionaries (most common)
    x = {i: mdl.continuous_var(name=f"x_{account_ids[i]}", lb=0) for i in safe_range}

    # 3. OBJECTIVE FUNCTION
    # Use mdl.sum() for summations (DOCplex specific)
    objective = mdl.sum(expected_volumes[i] * x[i] for i in safe_range)
    mdl.maximize(objective)

    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS

    # Total credit allocated across all accounts must not exceed the total available credit
    total_credit_usage = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(total_credit_usage <= Total_Available_Credit, ctname="total_credit_limit")

    # Each account's credit limit must not exceed its maximum allowable credit limit
    for i in safe_range:
        mdl.add_constraint(x[i] <= max_credit_limits[i], ctname=f"max_credit_limit_{account_ids[i]}")

    # For an account to be considered active, its credit limit must meet or exceed the minimum credit limit requirement
    for i in safe_range:
        if is_active[i]:
            mdl.add_constraint(x[i] >= Minimum_Credit_Limit, ctname=f"min_credit_limit_{account_ids[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(x[i])
            if value > 1e-6:
                print(f"x[{account_ids[i]}] = {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

    return mdl

# Run the optimization
optimize_credit_limits()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 9900000.0
**Execution Time**: 9.32 seconds
**Reliability**: high
**Analysis**: DOCplex also reached an optimal solution, though with a longer execution time compared to Gurobipy, suggesting it is reliable but less efficient.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def optimize_credit_limits():
    """Optimize credit limits to maximize expected transaction volume."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the problem context
    account_ids = [1, 2, 3]
    expected_volumes = {1: 1200.0, 2: 1800.0, 3: 2200.0}
    max_credit_limits = {1: 2000.0, 2: 2500.0, 3: 3000.0}
    is_active = {1: True, 2: True, 3: True}
    
    # Business configuration parameters
    Total_Available_Credit = 5000.0
    Minimum_Credit_Limit = 500.0
    
    # CRITICAL: Validate array lengths before indexing
    assert len(expected_volumes) == len(max_credit_limits) == len(is_active) == len(account_ids), "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.Set(initialize=account_ids)
    
    # 4. PARAMETERS
    model.expected_volume = pyo.Param(model.I, initialize=expected_volumes)
    model.max_credit_limit = pyo.Param(model.I, initialize=max_credit_limits)
    model.is_active = pyo.Param(model.I, initialize=is_active)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.expected_volume[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total credit constraint
    def total_credit_rule(model):
        return sum(model.x[i] for i in model.I) <= Total_Available_Credit
    model.total_credit_constraint = pyo.Constraint(rule=total_credit_rule)
    
    # Maximum credit limit constraint
    def max_credit_limit_rule(model, i):
        return model.x[i] <= model.max_credit_limit[i]
    model.max_credit_limit_constraint = pyo.Constraint(model.I, rule=max_credit_limit_rule)
    
    # Minimum credit limit for active accounts
    def min_credit_limit_rule(model, i):
        if model.is_active[i]:
            return model.x[i] >= Minimum_Credit_Limit
        else:
            return pyo.Constraint.Skip
    model.min_credit_limit_constraint = pyo.Constraint(model.I, rule=min_credit_limit_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):.2f}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"x[{i}] = {x_val:.2f}")
        
    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
optimize_credit_limits()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 9900000.0
**Execution Time**: 4.19 seconds
**Reliability**: high
**Analysis**: Pyomo found the optimal solution with a moderate execution time, demonstrating good reliability and efficiency.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 9900000.00 | 0.64s | N/A | N/A |
| Docplex | OPTIMAL | 9900000.00 | 9.32s | N/A | N/A |
| Pyomo | OPTIMAL | 9900000.00 | 4.19s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 9900000.0

### Final Recommendation
**Recommended Optimal Value**: 9900000.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its fastest execution time while maintaining high reliability, making it suitable for large-scale or time-sensitive applications.

### Business Interpretation
**Overall Strategy**: The optimal allocation of credit limits maximizes the expected transaction volume, ensuring efficient use of available credit resources.
**Objective Value Meaning**: The optimal objective value of 9900000.0 represents the maximum expected transaction volume achievable under the given constraints.
**Resource Allocation Summary**: Resources should be allocated to accounts in a way that maximizes transaction volume while respecting credit limits and activation requirements.
**Implementation Recommendations**: Implement the solution by adjusting credit limits according to the optimal values determined, ensuring compliance with all constraints and business rules.