## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{ij} \) be the number of devices allocated to shop \( i \) for device \( j \).
  - \( x_{ij} \) is a non-negative integer.

#### Objective Function
Maximize the total potential sales across all shops:
\[ \text{Maximize } Z = \sum_{i,j} \text{potential\_sales}_{ij} \times x_{ij} \]

Where:
- \(\text{potential\_sales}_{ij}\) is the estimated sales potential for device \( j \) at shop \( i \).

#### Constraints
1. Shop Capacity Constraints:
   - For each shop \( i \), the total number of devices allocated must not exceed the shop's capacity:
   \[ \sum_{j} x_{ij} \leq \text{capacity}_i \]
   - Where \(\text{capacity}_i\) is the maximum capacity of shop \( i \).

2. Non-negativity and Integer Constraints:
   - \( x_{ij} \geq 0 \) and \( x_{ij} \) is an integer for all \( i, j \).

Data Source Verification:
- \(\text{potential\_sales}_{ij}\) comes from the `PotentialSales` table, specifically `PotentialSales.potential_sales`.
- \(\text{capacity}_i\) comes from the `ShopCapacity` table, specifically `ShopCapacity.capacity`.

### Numerical Coefficients from Provided Data
Using the provided data, the model can be instantiated as follows:

Objective Function:
\[ \text{Maximize } Z = 600x_{1,101} + 850x_{2,102} + 950x_{3,103} \]

Constraints:
1. For shop 1:
   \[ x_{1,101} \leq 120 \]

2. For shop 2:
   \[ x_{2,102} \leq 180 \]

3. For shop 3:
   \[ x_{3,103} \leq 220 \]

4. Non-negativity and Integer Constraints:
   - \( x_{1,101}, x_{2,102}, x_{3,103} \geq 0 \) and are integers.

This formulation is ready to be solved using a linear or mixed-integer programming solver, ensuring that the allocation of devices maximizes the total potential sales while respecting shop capacities.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation - Retry Attempt 3

import gurobipy as gp
from gurobipy import GRB

def optimize_device_allocation():
    # 1. MODEL & DATA SETUP
    model = gp.Model("device_allocation")
    
    # Data from the problem statement
    shop_ids = [1, 2, 3]
    device_ids = [101, 102, 103]
    
    # Potential sales data
    potential_sales = {
        (1, 101): 600.0,
        (2, 102): 850.0,
        (3, 103): 950.0
    }
    
    # Shop capacity data
    shop_capacity = {
        1: 120,
        2: 180,
        3: 220
    }
    
    # Validate data lengths
    assert len(potential_sales) == len(shop_ids) == len(device_ids), "Data length mismatch"
    
    # 2. VARIABLES
    # Decision variables: number of devices allocated to each shop
    x = model.addVars(shop_ids, device_ids, vtype=GRB.INTEGER, name="x", lb=0)
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total potential sales
    model.setObjective(
        gp.quicksum(potential_sales[i, j] * x[i, j] for i in shop_ids for j in device_ids if (i, j) in potential_sales),
        GRB.MAXIMIZE
    )
    
    # 4. CONSTRAINTS
    # Shop capacity constraints
    for i in shop_ids:
        model.addConstr(
            gp.quicksum(x[i, j] for j in device_ids if (i, j) in potential_sales) <= shop_capacity[i],
            name=f"capacity_{i}"
        )
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in shop_ids:
            for j in device_ids:
                if (i, j) in potential_sales and 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")

# Run the optimization
optimize_device_allocation()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 434000.0
**Execution Time**: 0.18 seconds
**Reliability**: high
**Retry Attempt**: 3
**Analysis**: Gurobipy successfully found an optimal solution with the expected objective value. The execution time was efficient, and no errors were encountered.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation - Retry Attempt 3

from docplex.mp.model import Model

def device_allocation_optimization():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="device_allocation")
    
    # Data from the problem statement
    shop_ids = [1, 2, 3]
    device_ids = [101, 102, 103]
    potential_sales = {
        (1, 101): 600.0,
        (2, 102): 850.0,
        (3, 103): 950.0
    }
    shop_capacity = {
        1: 120,
        2: 180,
        3: 220
    }
    
    # Validate data lengths
    assert len(potential_sales) == len(shop_ids) == len(device_ids), "Data length mismatch"
    
    # 2. VARIABLES
    x = {(i, j): mdl.integer_var(name=f"x_{i}_{j}", lb=0) for i in shop_ids for j in device_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(potential_sales[i, j] * x[i, j] for i, j in potential_sales)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    # Shop capacity constraints
    for i in shop_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for j in device_ids if (i, j) in potential_sales) <= shop_capacity[i], ctname=f"capacity_{i}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i, j in potential_sales:
            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}")

# Execute the optimization
device_allocation_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 434000.0
**Execution Time**: 2.33 seconds
**Reliability**: high
**Retry Attempt**: 3
**Analysis**: DOCplex also found an optimal solution matching Gurobipy's results. The execution time was longer, but the solution is consistent and reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation - Retry Attempt 3

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

def device_allocation_optimization():
    """Optimize device allocation to maximize sales potential across shops."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Define the data based on the provided database schema and current stored values
    shops = [1, 2, 3]
    devices = [101, 102, 103]
    
    potential_sales_data = {
        (1, 101): 600.0,
        (2, 102): 850.0,
        (3, 103): 950.0
    }
    
    shop_capacity_data = {
        1: 120,
        2: 180,
        3: 220
    }
    
    # 3. SETS
    model.Shops = pyo.Set(initialize=shops)
    model.Devices = pyo.Set(initialize=devices)
    
    # 4. PARAMETERS
    model.potential_sales = pyo.Param(model.Shops * model.Devices, initialize=potential_sales_data, within=pyo.NonNegativeReals)
    model.capacity = pyo.Param(model.Shops, initialize=shop_capacity_data, within=pyo.NonNegativeIntegers)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.Shops, model.Devices, within=pyo.NonNegativeIntegers)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.potential_sales[i, j] * model.x[i, j] for i in model.Shops for j in model.Devices)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    def capacity_constraint_rule(model, i):
        return sum(model.x[i, j] for j in model.Devices) <= model.capacity[i]
    model.capacity_constraint = pyo.Constraint(model.Shops, rule=capacity_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)
    
    # 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("\nDevice allocation:")
        for i in model.Shops:
            for j in model.Devices:
                x_val = pyo.value(model.x[i, j])
                if x_val > 0:
                    print(f"Shop {i}, Device {j}: {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

# Execute the optimization
device_allocation_optimization()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpd7pw90ic.py", line 84, in <module>
    device_allocation_optimization()
  File "/tmp/tmpd7pw90ic.py", line 43, in device_allocation_optimization
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
  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/objective.py", line 336, in construct
    ans = self._setitem_when_not_present(index, rule(block, index))
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/initializer.py", line 485, in __call__
    return self._fcn(parent)
  File "/tmp/tmpd7pw90ic.py", line 42, in obj_rule
    return sum(model.potential_sales[i, j] * model.x[i, j] for i in model.Shops for j in model.Devices)
  File "/tmp/tmpd7pw90ic.py", line 42, in <genexpr>
    return sum(model.potential_sales[i, j] * model.x[i, j] for i in model.Shops for j in model.Devices)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 1012, in __getitem__
    return super().__getitem__(args)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/indexed_component.py", line 662, in __getitem__
    return self._getitem_when_not_present(index)
  File "/dccstor/nl2opt/miniforge3/envs/nl2opt_optim/lib/python3.10/site-packages/pyomo/core/base/param.py", line 627, in _getitem_when_not_present
    raise ValueError(
ValueError: Error retrieving immutable Param value (potential_sales[(1, 102)]):
	The Param value is undefined and no default value is specified.

**Analysis**: Pyomo encountered an error due to an undefined parameter value, which prevented it from finding a solution. This suggests a potential issue with data input or model setup.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 434000.00 | 0.18s | N/A | 3 |
| Docplex | OPTIMAL | 434000.00 | 2.33s | N/A | 3 |
| Pyomo | ERROR | N/A | 0.83s | N/A | 3 |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex
**Majority Vote Optimal Value**: 434000.0
**Solver Retry Summary**: gurobipy: 3 attempts, docplex: 3 attempts, pyomo: 3 attempts

### Final Recommendation
**Recommended Optimal Value**: 434000.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy provided a reliable and efficient solution with the shortest execution time and no errors.

### Optimal Decision Variables
- **x_1,101** = 120.000
  - *Business Meaning*: Resource allocation for x_1,101
- **x_2,102** = 180.000
  - *Business Meaning*: Resource allocation for x_2,102
- **x_3,103** = 220.000
  - *Business Meaning*: Resource allocation for x_3,103

### Business Interpretation
**Overall Strategy**: The optimal allocation of devices maximizes potential sales while respecting shop capacities.
**Objective Value Meaning**: The optimal objective value of 434000.0 represents the maximum potential sales achievable with the given device allocations.
**Resource Allocation Summary**: All available shop capacities are fully utilized to achieve maximum sales potential.
**Implementation Recommendations**: Ensure accurate data input for all parameters, particularly when using Pyomo. Implement the recommended device allocations to maximize sales.