# Complete Optimization Problem and Solution: browser_web

## 1. Problem Context and Goals

### Context  
The company is focused on optimizing the deployment of web client accelerators to various browsers to enhance market share coverage. Each accelerator has the potential to be compatible with multiple browsers, and the objective is to select a combination of accelerators that maximizes the total market share of the browsers they support. The decision-making process involves determining whether a browser is covered by an accelerator, represented by the binary decision variable 'x_b', and whether an accelerator is selected, represented by the binary decision variable 'y_a'. The operational goal is to maximize the total market share by considering the market share of each browser and ensuring that the number of selected accelerators does not exceed the predefined limit. The business configuration specifies the total number of accelerators that can be selected, which serves as a constraint in the optimization model. This approach ensures that the decision-making process is linear, focusing on maximizing market share without involving complex nonlinear relationships.

### Goals  
The primary goal of the optimization is to maximize the total market share covered by the selected accelerators. This involves maximizing the sum of the market shares of the browsers that are covered by the accelerators. The success of this optimization is measured by the total market share achieved, which is directly linked to the market share data of each browser. The objective is clearly defined in linear terms, focusing on maximizing the total market share without involving any complex mathematical expressions.

## 2. Constraints    

The optimization problem is subject to specific constraints that ensure the solution is feasible and aligns with business requirements. The first constraint is the compatibility constraint, which limits the total number of accelerators that can be selected to the predefined maximum. This ensures that the selection of accelerators is within the operational limits set by the business. The second constraint is the coverage constraint, which ensures that a browser can only be considered covered if there is at least one selected accelerator that is compatible with it. These constraints are expressed in business terms that naturally lead to linear mathematical forms, avoiding any nonlinear relationships.

## 3. Available Data  

### Database Schema  
```sql
-- Iteration 2 Database Schema
-- Objective: Incorporated market share data into the schema, updated configuration logic for scalar parameters, and ensured all tables meet the 3-row minimum rule.

CREATE TABLE accelerator_compatible_browser (
  accelerator_id INTEGER,
  browser_id INTEGER
);

CREATE TABLE decision_variables (
  x_b BOOLEAN,
  y_a BOOLEAN
);

CREATE TABLE browser_market_share (
  browser_id INTEGER,
  market_share FLOAT
);
```

### Data Dictionary  
The data dictionary provides a comprehensive mapping of tables and columns to their business purposes and roles in the optimization process:

- **Accelerator Compatible Browser Table**: This table defines the compatibility between accelerators and browsers. Each entry indicates which accelerators can work with which browsers, serving as critical business data for determining compatibility in the optimization model.

  - **Accelerator ID**: A unique identifier for each accelerator, used to determine compatibility with browsers.
  - **Browser ID**: A unique identifier for each browser, used to link market share data and compatibility information.

- **Decision Variables Table**: This table stores the decision variables used in the optimization process.

  - **x_b**: A binary variable indicating whether a browser is covered by a selected accelerator.
  - **y_a**: A binary variable indicating whether an accelerator is selected for deployment.

- **Browser Market Share Table**: This table contains market share data for each browser, which serves as the coefficient in the objective function.

  - **Browser ID**: A unique identifier for each browser, linking market share data to the optimization process.
  - **Market Share**: The market share percentage of each browser, used as a coefficient in the objective function to maximize total market share.

### Current Stored Values  
```sql
-- Iteration 2 Realistic Data
-- Generated by triple expert (business + data + optimization)
-- Values were determined based on typical market share distributions and compatibility scenarios in the web browser and accelerator industry. The goal was to ensure a diverse set of browsers with varying market shares and a realistic number of accelerators that can cover these browsers.

-- Realistic data for accelerator_compatible_browser
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id) VALUES (1, 101);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id) VALUES (1, 102);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id) VALUES (2, 103);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id) VALUES (3, 101);
INSERT INTO accelerator_compatible_browser (accelerator_id, browser_id) VALUES (3, 104);

-- Realistic data for decision_variables
INSERT INTO decision_variables (x_b, y_a) VALUES (True, True);
INSERT INTO decision_variables (x_b, y_a) VALUES (False, False);
INSERT INTO decision_variables (x_b, y_a) VALUES (True, False);

-- Realistic data for browser_market_share
INSERT INTO browser_market_share (browser_id, market_share) VALUES (101, 0.4);
INSERT INTO browser_market_share (browser_id, market_share) VALUES (102, 0.3);
INSERT INTO browser_market_share (browser_id, market_share) VALUES (103, 0.15);
INSERT INTO browser_market_share (browser_id, market_share) VALUES (104, 0.1);
```

## 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.