## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be a binary decision variable indicating whether medicine \( i \) is selected for the clinical trial.  
  \( x_i = 1 \) if medicine \( i \) is selected, and \( x_i = 0 \) otherwise.  
  \( i \in \{1, 2, 3\} \) (based on the provided data).

#### Objective Function
Maximize the total effectiveness score of the selected medicines:  
\[
\text{Maximize } Z = 0.85x_1 + 0.75x_2 + 0.9x_3
\]  
**Data Source Verification**:  
- Coefficients \( 0.85, 0.75, 0.9 \) are from `medicine_effectiveness.effectiveness_score`.

#### Constraints
1. **Adverse Interaction Limit**: The total adverse interaction score of the selected medicines must not exceed 0.5:  
\[
0.15x_1 + 0.25x_2 + 0.1x_3 \leq 0.5
\]  
**Data Source Verification**:  
- Coefficients \( 0.15, 0.25, 0.1 \) are from `medicine_adverse_interaction.adverse_interaction_score`.

2. **Medicine Count Limit**: The total number of selected medicines must not exceed 3:  
\[
x_1 + x_2 + x_3 \leq 3
\]  
**Data Source Verification**:  
- This constraint is based on the business requirement.

3. **FDA Approval Requirement**: Only FDA-approved medicines can be selected:  
\[
x_2 = 0
\]  
**Data Source Verification**:  
- Medicine 2 is not FDA-approved, as per `medicine.FDA_approved`.

4. **Binary Decision Variables**:  
\[
x_i \in \{0, 1\} \quad \forall i \in \{1, 2, 3\}
\]  
**Data Source Verification**:  
- This is a standard requirement for binary decision variables.

This is a complete, immediately solvable Mixed-Integer Linear Programming (MILP) model with all numerical coefficients derived from the provided data.

## 5. Gurobipy Implementation

```python
#!/usr/bin/env python3
"""
Gurobipy implementation for medicine_enzyme_interaction optimization problem
"""

import gurobipy as gp
from gurobipy import GRB

def optimize_medicine_selection():
    """Optimize medicine selection for clinical trial"""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("medicine_selection")
    
    # Data from the problem
    medicine_ids = [1, 2, 3]
    effectiveness_scores = {1: 0.85, 2: 0.75, 3: 0.9}
    adverse_interaction_scores = {1: 0.15, 2: 0.25, 3: 0.1}
    fda_approved = {1: True, 2: False, 3: True}
    
    # Validate array lengths
    assert len(medicine_ids) == len(effectiveness_scores) == len(adverse_interaction_scores) == len(fda_approved), "Array length mismatch"
    
    # 2. VARIABLES
    x = {i: model.addVar(vtype=GRB.BINARY, name=f"x_{i}") for i in medicine_ids}
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(effectiveness_scores[i] * x[i] for i in medicine_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Adverse Interaction Limit
    model.addConstr(gp.quicksum(adverse_interaction_scores[i] * x[i] for i in medicine_ids) <= 0.5, name="adverse_interaction_limit")
    
    # Medicine Count Limit
    model.addConstr(gp.quicksum(x[i] for i in medicine_ids) <= 3, name="medicine_count_limit")
    
    # FDA Approval Requirement
    for i in medicine_ids:
        if not fda_approved[i]:
            model.addConstr(x[i] == 0, name=f"fda_approval_{i}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in medicine_ids:
            if x[i].x > 1e-6:
                print(f"Medicine {i} is selected: {x[i].x:.0f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Execute the optimization
optimize_medicine_selection()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.75
**Execution Time**: 0.17 seconds
**Reliability**: high
**Analysis**: Gurobipy found the optimal solution quickly with a low execution time.

## 6. DOCplex Implementation

```python
#!/usr/bin/env python3
"""
DOCplex implementation for medicine_enzyme_interaction optimization problem
"""

from docplex.mp.model import Model

def optimize_medicine_selection():
    """Optimize medicine selection for clinical trial"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="medicine_enzyme_interaction")
    
    # Data from the problem statement
    medicine_ids = [1, 2, 3]
    effectiveness_scores = [0.85, 0.75, 0.9]
    adverse_interaction_scores = [0.15, 0.25, 0.1]
    fda_approved = [True, False, True]
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(medicine_ids) == len(effectiveness_scores) == len(adverse_interaction_scores) == len(fda_approved), "Array length mismatch"
    safe_range = range(min(len(medicine_ids), len(effectiveness_scores), len(adverse_interaction_scores), len(fda_approved)))  # Safe indexing
    
    # 2. VARIABLES
    # Binary decision variables for medicine selection
    x = {i: mdl.binary_var(name=f"x_{medicine_ids[i]}") for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total effectiveness score
    objective = mdl.sum(effectiveness_scores[i] * x[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Constraint 1: Adverse Interaction Limit
    adverse_interaction_limit = mdl.sum(adverse_interaction_scores[i] * x[i] for i in safe_range)
    mdl.add_constraint(adverse_interaction_limit <= 0.5, ctname="adverse_interaction_limit")
    
    # Constraint 2: Medicine Count Limit
    medicine_count_limit = mdl.sum(x[i] for i in safe_range)
    mdl.add_constraint(medicine_count_limit <= 3, ctname="medicine_count_limit")
    
    # Constraint 3: FDA Approval Requirement
    for i in safe_range:
        if not fda_approved[i]:
            mdl.add_constraint(x[i] == 0, ctname=f"fda_approval_{medicine_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"Medicine {medicine_ids[i]} selected: {value:.0f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Execute the optimization
if __name__ == "__main__":
    optimize_medicine_selection()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.75
**Execution Time**: 1.10 seconds
**Reliability**: high
**Analysis**: DOCplex also found the optimal solution but took longer compared to Gurobipy.

## 7. Pyomo Implementation

```python
#!/usr/bin/env python3
"""
Pyomo 6.9.2 Implementation for Medicine Selection Optimization Problem
"""

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

def medicine_selection_optimization():
    """Optimize medicine selection for clinical trial"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    medicine_ids = [1, 2, 3]
    effectiveness_scores = [0.85, 0.75, 0.9]
    adverse_interaction_scores = [0.15, 0.25, 0.1]
    FDA_approved = [True, False, True]
    
    # CRITICAL: Validate array lengths before indexing
    assert len(medicine_ids) == len(effectiveness_scores) == len(adverse_interaction_scores) == len(FDA_approved), "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.Set(initialize=medicine_ids)  # Medicine IDs
    
    # 4. PARAMETERS
    model.effectiveness = pyo.Param(model.I, initialize={i: s for i, s in zip(medicine_ids, effectiveness_scores)})
    model.adverse_interaction = pyo.Param(model.I, initialize={i: s for i, s in zip(medicine_ids, adverse_interaction_scores)})
    model.FDA_approved = pyo.Param(model.I, initialize={i: a for i, a in zip(medicine_ids, FDA_approved)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.Binary)  # Selection decision variables
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.effectiveness[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Adverse Interaction Limit
    def adverse_interaction_rule(model):
        return sum(model.adverse_interaction[i] * model.x[i] for i in model.I) <= 0.5
    model.adverse_interaction_constraint = pyo.Constraint(rule=adverse_interaction_rule)
    
    # Medicine Count Limit
    def medicine_count_rule(model):
        return sum(model.x[i] for i in model.I) <= 3
    model.medicine_count_constraint = pyo.Constraint(rule=medicine_count_rule)
    
    # FDA Approval Requirement
    def FDA_approval_rule(model, i):
        if not model.FDA_approved[i]:
            return model.x[i] == 0
        return pyo.Constraint.Skip
    model.FDA_approval_constraint = pyo.Constraint(model.I, rule=FDA_approval_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)  # tee=True shows solver output
    
    # 9. RESULT PROCESSING
    # Check solver status
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nSelected Medicines:")
        for i in model.I:
            if pyo.value(model.x[i]) > 0.5:  # Only print selected medicines
                print(f"Medicine {i} is selected")
        
    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

if __name__ == "__main__":
    medicine_selection_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 1.75
**Execution Time**: 0.90 seconds
**Reliability**: high
**Analysis**: Pyomo found the optimal solution but had the longest execution time among the three solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 1.75 | 0.17s | N/A | N/A |
| Docplex | OPTIMAL | 1.75 | 1.10s | N/A | N/A |
| Pyomo | OPTIMAL | 1.75 | 0.90s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 1.75

### Final Recommendation
**Recommended Optimal Value**: 1.75
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is recommended due to its faster execution time while still providing the same optimal solution as the other solvers.

### Optimal Decision Variables
- **x_1** = 1.000
  - *Business Meaning*: Medicine 1 is selected for the clinical trial, contributing 0.85 to the effectiveness score.
- **x_2** = 0.000
  - *Business Meaning*: Medicine 2 is not selected due to FDA approval constraints.
- **x_3** = 1.000
  - *Business Meaning*: Medicine 3 is selected for the clinical trial, contributing 0.9 to the effectiveness score.

### Business Interpretation
**Overall Strategy**: The optimal selection of medicines for the clinical trial includes Medicine 1 and Medicine 3, maximizing the total effectiveness score while adhering to all constraints.
**Objective Value Meaning**: The total effectiveness score of 1.75 indicates the combined effectiveness of the selected medicines in the clinical trial.
**Resource Allocation Summary**: Resources should be allocated to include Medicine 1 and Medicine 3 in the clinical trial, ensuring compliance with adverse interaction limits and FDA approval requirements.
**Implementation Recommendations**: Proceed with the clinical trial using Medicine 1 and Medicine 3. Ensure all regulatory and safety constraints are met during the trial.