## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( p_{d,i} \): Price of product \( i \) in dollars.
- \( p_{e,i} \): Price of product \( i \) in euros.
- \( p_{p,i} \): Price of product \( i \) in pounds.

#### Objective Function
Maximize total revenue:
\[
\text{Maximize } Z = \sum_{i} \left( p_{d,i} \times v_i + \frac{p_{e,i}}{r_{de}} \times v_i + \frac{p_{p,i}}{r_{dp}} \times v_i \right)
\]
Where:
- \( v_i \) is the sales volume for product \( i \).
- \( r_{de} \) is the exchange rate from dollars to euros.
- \( r_{dp} \) is the exchange rate from dollars to pounds.

#### Constraints
1. Minimum price constraints:
   \[
   p_{d,i} \geq \text{min\_price\_dollars}_i
   \]
   \[
   p_{e,i} \geq \text{min\_price\_euros}_i
   \]
   \[
   p_{p,i} \geq \text{min\_price\_pounds}_i
   \]

2. Stock availability constraints:
   \[
   v_i \leq \text{stock\_available}_i
   \]

3. Price consistency constraints:
   \[
   p_{e,i} = p_{d,i} \times r_{de}
   \]
   \[
   p_{p,i} = p_{d,i} \times r_{dp}
   \]

Data Source Verification:
- \( p_{d,i} \), \( p_{e,i} \), \( p_{p,i} \) are from `Catalog_Contents.price_in_dollars`, `Catalog_Contents.price_in_euros`, `Catalog_Contents.price_in_pounds`.
- \( \text{min\_price\_dollars}_i \), \( \text{min\_price\_euros}_i \), \( \text{min\_price\_pounds}_i \) are from `Catalog_Contents.minimum_price_dollars`, `Catalog_Contents.minimum_price_euros`, `Catalog_Contents.minimum_price_pounds`.
- \( v_i \) is from `Product_Sales.sales_volume`.
- \( \text{stock\_available}_i \) is from `Stock_Availability.stock_available`.
- \( r_{de} \) and \( r_{dp} \) are business configuration parameters for exchange rates from dollars to euros and dollars to pounds, respectively.

This linear model is immediately solvable using standard linear programming techniques, ensuring that all relevant factors, such as exchange rates, are correctly incorporated into the mathematical formulation.

## 5. Gurobipy Implementation

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

import gurobipy as gp
from gurobipy import GRB

def product_catalog_optimization():
    # Exchange rates
    r_de = 0.85  # Exchange rate from dollars to euros
    r_dp = 0.75  # Exchange rate from dollars to pounds

    # Data setup
    prices_dollars = [12.99, 18.49, 25.0]
    prices_euros = [11.04, 15.72, 21.25]
    prices_pounds = [9.74, 13.87, 18.75]
    min_prices_dollars = [11.99, 16.99, 22.5]
    min_prices_euros = [10.19, 14.44, 19.13]
    min_prices_pounds = [8.99, 12.74, 16.88]
    sales_volumes = [120, 180, 90]
    stock_available = [150, 200, 100]

    # Validate array lengths
    n_products = len(prices_dollars)
    assert len(prices_euros) == len(prices_pounds) == len(min_prices_dollars) == n_products
    assert len(min_prices_euros) == len(min_prices_pounds) == len(sales_volumes) == n_products
    assert len(stock_available) == n_products, "Array length mismatch"

    # Model setup
    model = gp.Model("product_catalog")

    # Variables
    p_d = model.addVars(n_products, vtype=GRB.CONTINUOUS, name="p_d", lb=0)
    p_e = model.addVars(n_products, vtype=GRB.CONTINUOUS, name="p_e", lb=0)
    p_p = model.addVars(n_products, vtype=GRB.CONTINUOUS, name="p_p", lb=0)

    # Objective function: Maximize total revenue
    model.setObjective(
        gp.quicksum(
            (p_d[i] * sales_volumes[i] + p_e[i] * sales_volumes[i] / r_de + p_p[i] * sales_volumes[i] / r_dp)
            for i in range(n_products)
        ),
        GRB.MAXIMIZE
    )

    # Constraints
    for i in range(n_products):
        # Minimum price constraints
        model.addConstr(p_d[i] >= min_prices_dollars[i], name=f"min_price_dollars_{i}")
        model.addConstr(p_e[i] >= min_prices_euros[i], name=f"min_price_euros_{i}")
        model.addConstr(p_p[i] >= min_prices_pounds[i], name=f"min_price_pounds_{i}")

        # Stock availability constraints
        model.addConstr(sales_volumes[i] <= stock_available[i], name=f"stock_availability_{i}")

        # Price consistency constraints
        model.addConstr(p_e[i] == p_d[i] * r_de, name=f"price_consistency_euros_{i}")
        model.addConstr(p_p[i] == p_d[i] * r_dp, name=f"price_consistency_pounds_{i}")

    # Solve the model
    model.optimize()

    # Output results
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_products):
            print(f"Product {i+1}: Price in dollars = {p_d[i].x:.2f}, Price in euros = {p_e[i].x:.2f}, Price in pounds = {p_p[i].x:.2f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")

# Run the optimization
product_catalog_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: Gurobipy identified the problem as infeasible, which suggests that the constraints cannot be satisfied simultaneously. This solver is known for its robustness and reliability in identifying infeasibility.

## 6. DOCplex Implementation

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

from docplex.mp.model import Model

def product_catalog_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="product_catalog")
    
    # Exchange rates
    r_de = 0.85  # Example exchange rate from dollars to euros
    r_dp = 0.75  # Example exchange rate from dollars to pounds
    
    # Data from the database (hardcoded for this example)
    prices_dollars = [12.99, 18.49, 25.0]
    prices_euros = [11.04, 15.72, 21.25]
    prices_pounds = [9.74, 13.87, 18.75]
    min_prices_dollars = [11.99, 16.99, 22.5]
    min_prices_euros = [10.19, 14.44, 19.13]
    min_prices_pounds = [8.99, 12.74, 16.88]
    sales_volumes = [120, 180, 90]
    stock_availabilities = [150, 200, 100]
    
    # Validate array lengths
    assert len(prices_dollars) == len(prices_euros) == len(prices_pounds) == len(min_prices_dollars) == len(min_prices_euros) == len(min_prices_pounds) == len(sales_volumes) == len(stock_availabilities), "Array length mismatch"
    
    safe_range = range(len(prices_dollars))  # Safe indexing
    
    # 2. VARIABLES
    p_d = {i: mdl.continuous_var(name=f"p_d_{i}", lb=min_prices_dollars[i]) for i in safe_range}
    p_e = {i: mdl.continuous_var(name=f"p_e_{i}", lb=min_prices_euros[i]) for i in safe_range}
    p_p = {i: mdl.continuous_var(name=f"p_p_{i}", lb=min_prices_pounds[i]) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum((p_d[i] * sales_volumes[i] + p_e[i] * sales_volumes[i] / r_de + p_p[i] * sales_volumes[i] / r_dp) for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    for i in safe_range:
        # Stock availability constraint
        mdl.add_constraint(sales_volumes[i] <= stock_availabilities[i], ctname=f"stock_availability_{i}")
        
        # Price consistency constraints
        mdl.add_constraint(p_e[i] == p_d[i] * r_de, ctname=f"price_consistency_euros_{i}")
        mdl.add_constraint(p_p[i] == p_d[i] * r_dp, ctname=f"price_consistency_pounds_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in safe_range:
            print(f"Product {i}: Price in dollars = {solution.get_value(p_d[i]):.2f}, Price in euros = {solution.get_value(p_e[i]):.2f}, Price in pounds = {solution.get_value(p_p[i]):.2f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")

# Run the optimization
product_catalog_optimization()
```

### Execution Results
**Status**: INFEASIBLE
**Error**: Problem is infeasible
**Analysis**: DOCplex also found the problem to be infeasible, aligning with Gurobipy's results. This consistency between two solvers increases confidence in the infeasibility diagnosis.

## 7. Pyomo Implementation

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

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

def product_catalog_optimization():
    """Optimize product pricing strategy to maximize revenue."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Exchange rates
    r_de = 1.18  # Example exchange rate from dollars to euros
    r_dp = 1.33  # Example exchange rate from dollars to pounds
    
    # Data from the database
    prices_dollars = [12.99, 18.49, 25.0]
    prices_euros = [11.04, 15.72, 21.25]
    prices_pounds = [9.74, 13.87, 18.75]
    min_prices_dollars = [11.99, 16.99, 22.5]
    min_prices_euros = [10.19, 14.44, 19.13]
    min_prices_pounds = [8.99, 12.74, 16.88]
    sales_volumes = [120, 180, 90]
    stock_availabilities = [150, 200, 100]
    
    # Validate array lengths
    n_products = len(prices_dollars)
    assert all(len(lst) == n_products for lst in [
        prices_euros, prices_pounds, min_prices_dollars, 
        min_prices_euros, min_prices_pounds, sales_volumes, stock_availabilities
    ]), "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.RangeSet(1, n_products)
    
    # 4. PARAMETERS
    model.price_dollars = pyo.Param(model.I, initialize={i+1: prices_dollars[i] for i in range(n_products)})
    model.price_euros = pyo.Param(model.I, initialize={i+1: prices_euros[i] for i in range(n_products)})
    model.price_pounds = pyo.Param(model.I, initialize={i+1: prices_pounds[i] for i in range(n_products)})
    model.min_price_dollars = pyo.Param(model.I, initialize={i+1: min_prices_dollars[i] for i in range(n_products)})
    model.min_price_euros = pyo.Param(model.I, initialize={i+1: min_prices_euros[i] for i in range(n_products)})
    model.min_price_pounds = pyo.Param(model.I, initialize={i+1: min_prices_pounds[i] for i in range(n_products)})
    model.sales_volume = pyo.Param(model.I, initialize={i+1: sales_volumes[i] for i in range(n_products)})
    model.stock_available = pyo.Param(model.I, initialize={i+1: stock_availabilities[i] for i in range(n_products)})
    
    # 5. VARIABLES
    model.p_d = pyo.Var(model.I, within=pyo.NonNegativeReals)
    model.p_e = pyo.Var(model.I, within=pyo.NonNegativeReals)
    model.p_p = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def revenue_rule(model):
        return sum(
            model.p_d[i] * model.sales_volume[i] +
            model.p_e[i] / r_de * model.sales_volume[i] +
            model.p_p[i] / r_dp * model.sales_volume[i]
            for i in model.I
        )
    model.revenue = pyo.Objective(rule=revenue_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    def min_price_dollars_rule(model, i):
        return model.p_d[i] >= model.min_price_dollars[i]
    model.min_price_dollars_constraint = pyo.Constraint(model.I, rule=min_price_dollars_rule)
    
    def min_price_euros_rule(model, i):
        return model.p_e[i] >= model.min_price_euros[i]
    model.min_price_euros_constraint = pyo.Constraint(model.I, rule=min_price_euros_rule)
    
    def min_price_pounds_rule(model, i):
        return model.p_p[i] >= model.min_price_pounds[i]
    model.min_price_pounds_constraint = pyo.Constraint(model.I, rule=min_price_pounds_rule)
    
    def stock_availability_rule(model, i):
        return model.sales_volume[i] <= model.stock_available[i]
    model.stock_availability_constraint = pyo.Constraint(model.I, rule=stock_availability_rule)
    
    def price_consistency_euros_rule(model, i):
        return model.p_e[i] == model.p_d[i] * r_de
    model.price_consistency_euros_constraint = pyo.Constraint(model.I, rule=price_consistency_euros_rule)
    
    def price_consistency_pounds_rule(model, i):
        return model.p_p[i] == model.p_d[i] * r_dp
    model.price_consistency_pounds_constraint = pyo.Constraint(model.I, rule=price_consistency_pounds_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("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.revenue)}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            p_d_val = pyo.value(model.p_d[i])
            p_e_val = pyo.value(model.p_e[i])
            p_p_val = pyo.value(model.p_p[i])
            print(f"Product {i}: p_d = {p_d_val:.2f}, p_e = {p_e_val:.2f}, p_p = {p_p_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}")

# Run the optimization
product_catalog_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmp0o7c2jdn.py", line 112, in <module>
    product_catalog_optimization()
  File "/tmp/tmp0o7c2jdn.py", line 77, in product_catalog_optimization
    model.stock_availability_constraint = pyo.Constraint(model.I, rule=stock_availability_rule)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 571, in __setattr__
    self.add_component(name, val)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/block.py", line 1101, in add_component
    val.construct(data)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/constraint.py", line 722, in construct
    self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/indexed_component.py", line 1111, in _setitem_when_not_present
    obj.set_value(value)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/constraint.py", line 469, in set_value
    raise ValueError(
ValueError: Invalid constraint expression. The constraint expression resolved to a trivial Boolean (True) instead of a Pyomo object. Please modify your rule to return Constraint.Feasible instead of True.

Error thrown for Constraint 'stock_availability_constraint[1]'

**Analysis**: Pyomo encountered an error related to constraint formulation, indicating a potential issue in the model setup rather than the problem's feasibility. This suggests a need for model debugging.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | INFEASIBLE | N/A | 0.19s | N/A | 4 |
| Docplex | INFEASIBLE | N/A | 1.44s | N/A | 4 |
| Pyomo | ERROR | N/A | 1.04s | N/A | 4 |

### Solver Consistency Analysis
**Result**: Solvers produced inconsistent results
**Consistent Solvers**: gurobipy, docplex
**Inconsistent Solvers**: pyomo
**Potential Issues**:
- The problem formulation may have conflicting constraints, particularly with price consistency and minimum price constraints.
- Pyomo's error suggests a possible issue in constraint implementation that needs to be addressed.
**Solver Retry Summary**: gurobipy: 4 attempts, docplex: 4 attempts, pyomo: 4 attempts

### Final Recommendation
**Confidence Level**: HIGH
**Preferred Solver(s)**: multiple
**Reasoning**: Both Gurobipy and DOCplex consistently identified infeasibility, suggesting a genuine issue with the problem constraints. Pyomo's error indicates a need for model correction rather than a solver preference.

### Business Interpretation
**Overall Strategy**: The current pricing strategy cannot be implemented as it stands due to conflicting constraints. This means that the business needs to reassess its pricing or stock availability assumptions.
**Objective Value Meaning**: The optimal objective value represents the maximum achievable revenue given the constraints. However, current constraints make this unachievable.
**Resource Allocation Summary**: Re-evaluate pricing strategies and stock levels to ensure feasibility. Consider adjusting minimum price constraints or exchange rates.
**Implementation Recommendations**: Review and adjust the constraints, particularly focusing on price consistency and minimum price requirements. Ensure that the model setup in Pyomo is corrected to avoid errors.