## 4. Mathematical Optimization Formulation

#### Decision Variables
- \( y_a \): Binary variable indicating whether accelerator \( a \) is selected. \( y_a \in \{0, 1\} \).
- \( x_b \): Binary variable indicating whether browser \( b \) is covered by at least one selected accelerator. \( x_b \in \{0, 1\} \).

#### Objective Function
Maximize the total market share covered by the selected accelerators:
\[ \text{Maximize } \sum_{b \in B} \text{market\_share}_b \times x_b \]

Where:
- \( \text{market\_share}_b \) is the market share of browser \( b \) from the `browser_market_share` table.

#### Constraints
1. **Accelerator Selection Limit**: The total number of selected accelerators cannot exceed a predefined limit \( L \).
   \[ \sum_{a \in A} y_a \leq L \]

2. **Coverage Constraint**: A browser \( b \) can only be considered covered if there is at least one selected accelerator that is compatible with it.
   \[ x_b \leq \sum_{a \in A_b} y_a \quad \forall b \in B \]

   Where:
   - \( A_b \) is the set of accelerators compatible with browser \( b \), derived from the `accelerator_compatible_browser` table.

3. **Binary Constraints**: Ensure that decision variables are binary.
   \[ y_a \in \{0, 1\} \quad \forall a \in A \]
   \[ x_b \in \{0, 1\} \quad \forall b \in B \]

Data Source Verification:
- \( \text{market\_share}_b \) is obtained from `browser_market_share.market_share`.
- Compatibility information for \( A_b \) is derived from `accelerator_compatible_browser`.
- The limit \( L \) is a business configuration parameter specifying the maximum number of accelerators that can be selected.

This formulation provides a complete and immediately solvable linear mathematical model, ensuring that the decision-making process aligns with the business goal of maximizing market share coverage while adhering to operational constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_web_client_accelerators():
    """Optimize the deployment of web client accelerators to maximize market share coverage."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("browser_web")
    
    # Data from the database
    accelerators = [1, 2, 3]
    browsers = [101, 102, 103, 104]
    market_shares = {101: 0.4, 102: 0.3, 103: 0.15, 104: 0.1}
    compatibility = {
        1: [101, 102],
        2: [103],
        3: [101, 104]
    }
    max_accelerators = 2  # Predefined limit for the number of accelerators
    
    # Validate data lengths
    assert len(market_shares) == len(browsers), "Market shares and browsers length mismatch"
    
    # 2. VARIABLES
    y = {a: model.addVar(vtype=GRB.BINARY, name=f"y_{a}") for a in accelerators}
    x = {b: model.addVar(vtype=GRB.BINARY, name=f"x_{b}") for b in browsers}
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(market_shares[b] * x[b] for b in browsers), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Accelerator Selection Limit
    model.addConstr(gp.quicksum(y[a] for a in accelerators) <= max_accelerators, name="accelerator_limit")
    
    # Coverage Constraint
    for b in browsers:
        compatible_accelerators = [a for a in accelerators if b in compatibility[a]]
        model.addConstr(x[b] <= gp.quicksum(y[a] for a in compatible_accelerators), name=f"coverage_{b}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for a in accelerators:
            if y[a].x > 1e-6:
                print(f"Accelerator {a} is selected.")
        for b in browsers:
            if x[b].x > 1e-6:
                print(f"Browser {b} is covered.")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_web_client_accelerators()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.85
**Execution Time**: 0.78 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the shortest execution time, indicating efficient performance and reliability.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_web_client_accelerators():
    """Optimize the deployment of web client accelerators to maximize market share coverage."""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="web_client_accelerators")
    
    # Data from the problem context
    accelerators = [1, 2, 3]
    browsers = [101, 102, 103, 104]
    market_shares = {101: 0.4, 102: 0.3, 103: 0.15, 104: 0.1}
    compatibility = {
        1: [101, 102],
        2: [103],
        3: [101, 104]
    }
    max_accelerators = 2  # Predefined limit on the number of accelerators
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(browsers) == len(market_shares), "Array length mismatch"
    safe_range_browsers = range(len(browsers))  # Safe indexing for browsers
    safe_range_accelerators = range(len(accelerators))  # Safe indexing for accelerators
    
    # 2. VARIABLES
    # Binary variables for accelerators and browsers
    y = {a: mdl.binary_var(name=f"y_{a}") for a in accelerators}
    x = {b: mdl.binary_var(name=f"x_{b}") for b in browsers}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize the total market share covered by the selected accelerators
    objective = mdl.sum(market_shares[b] * x[b] for b in browsers)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Constraint: Limit the number of selected accelerators
    mdl.add_constraint(mdl.sum(y[a] for a in accelerators) <= max_accelerators, ctname="accelerator_limit")
    
    # Constraint: Coverage constraint for each browser
    for b in browsers:
        compatible_accelerators = [a for a in accelerators if b in compatibility[a]]
        mdl.add_constraint(x[b] <= mdl.sum(y[a] for a in compatible_accelerators), ctname=f"coverage_{b}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for b in browsers:
            if solution.get_value(x[b]) > 0.5:  # Check if browser is covered
                print(f"Browser {b} is covered.")
        for a in accelerators:
            if solution.get_value(y[a]) > 0.5:  # Check if accelerator is selected
                print(f"Accelerator {a} is selected.")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
optimize_web_client_accelerators()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.85
**Execution Time**: 3.14 seconds
**Reliability**: high
**Analysis**: DOCplex also found an optimal solution, albeit with a longer execution time compared to Gurobipy, but still reliable.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def optimize_browser_web():
    """Optimization for maximizing browser market share coverage"""

    # 1. MODEL CREATION
    model = pyo.ConcreteModel()

    # 2. DATA SETUP
    # Define the data based on the provided database schema and current stored values
    accelerators = [1, 2, 3]
    browsers = [101, 102, 103, 104]
    market_shares = {101: 0.4, 102: 0.3, 103: 0.15, 104: 0.1}
    compatibility = {
        1: [101, 102],
        2: [103],
        3: [101, 104]
    }
    max_accelerators = 2  # Predefined limit for the number of accelerators

    # 3. SETS
    model.A = pyo.Set(initialize=accelerators)
    model.B = pyo.Set(initialize=browsers)

    # 4. PARAMETERS
    model.market_share = pyo.Param(model.B, initialize=market_shares)

    # 5. VARIABLES
    model.y = pyo.Var(model.A, within=pyo.Binary)
    model.x = pyo.Var(model.B, within=pyo.Binary)

    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.market_share[b] * model.x[b] for b in model.B)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)

    # 7. CONSTRAINTS

    # Accelerator Selection Limit
    def accelerator_limit_rule(model):
        return sum(model.y[a] for a in model.A) <= max_accelerators
    model.accelerator_limit = pyo.Constraint(rule=accelerator_limit_rule)

    # Coverage Constraint
    def coverage_rule(model, b):
        compatible_accelerators = [a for a in model.A if b in compatibility[a]]
        return model.x[b] <= sum(model.y[a] for a in compatible_accelerators)
    model.coverage_constraint = pyo.Constraint(model.B, rule=coverage_rule)

    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')

    # Solve the model
    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.objective)}")

        # Extract variable values
        print("\nVariable values:")
        for a in model.A:
            y_val = pyo.value(model.y[a])
            print(f"y[{a}] = {int(y_val)}")
        for b in model.B:
            x_val = pyo.value(model.x[b])
            print(f"x[{b}] = {int(x_val)}")

    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_browser_web()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 0.85
**Execution Time**: 3.01 seconds
**Reliability**: high
**Analysis**: Pyomo achieved an optimal solution with a similar execution time to DOCplex, demonstrating consistent reliability.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 0.85 | 0.78s | N/A | N/A |
| Docplex | OPTIMAL | 0.85 | 3.14s | N/A | N/A |
| Pyomo | OPTIMAL | 0.85 | 3.01s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 0.85

### Final Recommendation
**Recommended Optimal Value**: 0.85
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time while maintaining the same optimal value as other solvers.

### Business Interpretation
**Overall Strategy**: The optimal solution covers 85% of the market share, maximizing the reach of selected accelerators.
**Objective Value Meaning**: Achieving an optimal objective value of 0.85 means that the selected accelerators cover 85% of the total market share, which is a significant portion.
**Resource Allocation Summary**: Resources should be allocated to the accelerators that maximize browser coverage within the given limit.
**Implementation Recommendations**: Implement the solution by selecting the accelerators identified by the solver, ensuring they are compatible with the targeted browsers to achieve maximum market coverage.