## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_i \) be the amount of resources allocated to institution \( i \).
  - \( x_1 \) for institution 1
  - \( x_2 \) for institution 2
  - \( x_3 \) for institution 3

#### Objective Function
Maximize the total sequence identity:
\[ \text{Maximize } 0.85x_1 + 0.9x_2 + 0.8x_3 \]

#### Constraints
1. Total Resource Constraint:
   \[ x_1 + x_2 + x_3 \leq 600 \]
   - This constraint ensures that the total resources allocated do not exceed the total available resources.

2. Building Capacity Constraints:
   \[ x_1 \leq 600 \]
   \[ x_2 \leq 700 \]
   \[ x_3 \leq 500 \]
   - These constraints ensure that the resources allocated to each institution do not exceed the capacity of their respective buildings.

Data Source Verification:
- Objective coefficients:
  - \( 0.85 \) from `protein.sequence_identity_coefficient` for institution 1
  - \( 0.9 \) from `protein.sequence_identity_coefficient` for institution 2
  - \( 0.8 \) from `protein.sequence_identity_coefficient` for institution 3
- Total resource constraint from business configuration (assumed total available resources = 600).
- Building capacity constraints:
  - \( 600 \) from `building.building_capacity` for institution 1
  - \( 700 \) from `building.building_capacity` for institution 2
  - \( 500 \) from `building.building_capacity` for institution 3

This linear model is immediately solvable using standard linear programming techniques, ensuring that the allocation of resources maximizes the total sequence identity while respecting the constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_protein_institute():
    """Optimize resource allocation to maximize sequence identity"""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("protein_institute")
    
    # Data: Sequence identity coefficients and building capacities
    sequence_identity_coefficients = [0.85, 0.9, 0.8]
    building_capacities = [600, 700, 500]
    total_resources_available = 600
    
    # Validate array lengths
    n_institutions = len(sequence_identity_coefficients)
    assert len(building_capacities) == n_institutions, "Array length mismatch"
    
    # 2. VARIABLES
    # Decision variables for resource allocation
    x = {i: model.addVar(vtype=GRB.CONTINUOUS, name=f"x_{i+1}", lb=0) 
         for i in range(n_institutions)}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total sequence identity
    model.setObjective(gp.quicksum(sequence_identity_coefficients[i] * x[i] 
                                   for i in range(n_institutions)), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    # Total resource constraint
    model.addConstr(gp.quicksum(x[i] for i in range(n_institutions)) <= total_resources_available, 
                    name="total_resource_constraint")
    
    # Building capacity constraints
    for i in range(n_institutions):
        model.addConstr(x[i] <= building_capacities[i], name=f"building_capacity_{i+1}")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in range(n_institutions):
            if x[i].x > 1e-6:
                print(f"x[{i+1}] = {x[i].x:.3f}")
    elif model.status == GRB.INFEASIBLE:
        print("Problem is infeasible")
    elif model.status == GRB.UNBOUNDED:
        print("Problem is unbounded")
    
    return model

# Run the optimization
optimize_protein_institute()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 540.0
**Execution Time**: 0.67 seconds
**Reliability**: high
**Analysis**: Gurobipy found an optimal solution quickly with the highest reliability due to its robust performance and industry reputation.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def protein_institute_optimization():
    """Optimize resource allocation to maximize sequence identity"""
    
    # 1. MODEL & DATA SETUP
    mdl = Model(name="protein_institute")
    
    # Data from the problem statement
    sequence_identity_coefficients = [0.85, 0.9, 0.8]
    building_capacities = [600, 700, 500]
    total_resources_available = 600
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(sequence_identity_coefficients) == len(building_capacities), "Array length mismatch"
    safe_range = range(min(len(sequence_identity_coefficients), len(building_capacities)))  # Safe indexing
    
    # 2. VARIABLES
    # Variable dictionaries for resource allocation
    allocation = {i: mdl.continuous_var(name=f"x_{i}", lb=0) for i in safe_range}
    
    # 3. OBJECTIVE FUNCTION
    # Maximize total sequence identity
    objective = mdl.sum(sequence_identity_coefficients[i] * allocation[i] for i in safe_range)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS - CORRECT SYNTAX PATTERNS
    
    # Total Resource Constraint
    total_allocation = mdl.sum(allocation[i] for i in safe_range)
    mdl.add_constraint(total_allocation <= total_resources_available, ctname="total_resource_limit")
    
    # Building Capacity Constraints
    for i in safe_range:
        mdl.add_constraint(allocation[i] <= building_capacities[i], ctname=f"capacity_limit_{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(allocation[i])
            if value > 1e-6:
                print(f"Allocation for institution {i+1}: {value:.3f}")
    else:
        print("No solution found")
        print(f"Status: {mdl.solve_details.status}")
    
    return mdl

# Run the optimization
protein_institute_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 540.0
**Execution Time**: 2.61 seconds
**Reliability**: high
**Analysis**: DOCplex also found the 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 protein_institute_optimization():
    """Optimize resource allocation to maximize sequence identity"""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    n_institutions = 3
    sequence_identity_coefficients = [0.85, 0.9, 0.8]
    building_capacities = [600, 700, 500]
    total_resources_available = 600
    
    # CRITICAL: Validate array lengths before indexing
    assert len(sequence_identity_coefficients) == len(building_capacities) == n_institutions, "Array length mismatch"
    
    # 3. SETS
    model.I = pyo.RangeSet(1, n_institutions)  # 1-based indexing
    
    # 4. PARAMETERS
    model.sequence_identity_coefficient = pyo.Param(model.I, initialize={i+1: sequence_identity_coefficients[i] for i in range(n_institutions)})
    model.building_capacity = pyo.Param(model.I, initialize={i+1: building_capacities[i] for i in range(n_institutions)})
    
    # 5. VARIABLES
    model.x = pyo.Var(model.I, within=pyo.NonNegativeReals)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.sequence_identity_coefficient[i] * model.x[i] for i in model.I)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Total Resource Constraint
    def total_resource_constraint_rule(model):
        return sum(model.x[i] for i in model.I) <= total_resources_available
    model.total_resource_constraint = pyo.Constraint(rule=total_resource_constraint_rule)
    
    # Building Capacity Constraints
    def building_capacity_rule(model, i):
        return model.x[i] <= model.building_capacity[i]
    model.building_capacity_constraint = pyo.Constraint(model.I, rule=building_capacity_rule)
    
    # 8. SOLVING WITH GUROBI
    solver = SolverFactory('gurobi')
    
    # 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):.3f}")
        
        # Extract variable values
        print("\nVariable values:")
        for i in model.I:
            x_val = pyo.value(model.x[i])
            if x_val > 1e-6:  # Only print non-zero values
                print(f"x[{i}] = {x_val:.3f}")
        
    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
protein_institute_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 540.0
**Execution Time**: 3.08 seconds
**Reliability**: high
**Analysis**: Pyomo achieved the optimal solution with the longest execution time, but the results are consistent with other solvers.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | OPTIMAL | 540.00 | 0.67s | N/A | N/A |
| Docplex | OPTIMAL | 540.00 | 2.61s | N/A | N/A |
| Pyomo | OPTIMAL | 540.00 | 3.08s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: gurobipy, docplex, pyomo
**Majority Vote Optimal Value**: 540.0

### Final Recommendation
**Recommended Optimal Value**: 540.0
**Confidence Level**: HIGH
**Preferred Solver(s)**: gurobipy
**Reasoning**: Gurobipy is preferred due to its faster execution time and high reliability, making it suitable for time-sensitive applications.

### Optimal Decision Variables
- **x_1** = 0.000
  - *Business Meaning*: Resources allocated to institution 2, which is optimal at 600.
- **x_2** = 600.000
  - *Business Meaning*: Resources allocated to institution 3, which is optimal at 0.
- **x_3** = 0.000
  - *Business Meaning*: Resource allocation for x_3

### Business Interpretation
**Overall Strategy**: Allocate all available resources to institution 2 to maximize sequence identity.
**Objective Value Meaning**: The optimal objective value of 540.0 represents the maximum achievable sequence identity given the constraints.
**Resource Allocation Summary**: All resources should be allocated to institution 2 to achieve the highest sequence identity.
**Implementation Recommendations**: Implement the solution by reallocating resources to institution 2, ensuring compliance with the building capacity and total resource constraints.