## 4. Mathematical Optimization Formulation

#### Decision Variables
- Let \( x_{i,j} \) be a binary decision variable where \( x_{i,j} = 1 \) if faculty member \( i \) is assigned to activity \( j \), and \( x_{i,j} = 0 \) otherwise.

#### Objective Function
Maximize the total participation score:
\[ \text{Maximize } \sum_{i,j} \text{Participation\_Score}(i,j) \times x_{i,j} \]

Where:
- \(\text{Participation\_Score}(i,j)\) is the participation score for faculty member \( i \) and activity \( j \) from the `Participation_Score` table.

#### Constraints
1. **Faculty Availability Constraint**: Each faculty member can participate in a limited number of activities.
   \[
   \sum_{j} x_{i,j} \leq \text{Max\_Activities}(i) \quad \forall i
   \]
   - \(\text{Max\_Activities}(i)\) is the maximum number of activities faculty member \( i \) can participate in, derived from business configuration.

2. **Activity Staffing Constraint**: Each activity must have a minimum number of faculty members assigned.
   \[
   \sum_{i} x_{i,j} \geq \text{Min\_Faculty}(j) \quad \forall j
   \]
   - \(\text{Min\_Faculty}(j)\) is the minimum number of faculty members required for activity \( j \), derived from business configuration.

3. **Binary Constraint**: Each decision variable is binary.
   \[
   x_{i,j} \in \{0, 1\} \quad \forall i, j
   \]

Data Source Verification:
- Participation scores \(\text{Participation\_Score}(i,j)\) are sourced from the `Participation_Score` table.
- Maximum activities per faculty \(\text{Max\_Activities}(i)\) and minimum faculty per activity \(\text{Min\_Faculty}(j)\) are derived from business configuration parameters, which are not explicitly detailed in the provided data but are assumed to be part of the business rules.

This formulation provides a complete, immediately solvable linear mathematical model using the given data and constraints.

## 5. Gurobipy Implementation

```python
# Complete GUROBIPY implementation

import gurobipy as gp
from gurobipy import GRB

def optimize_faculty_allocation():
    """Optimize faculty allocation to activities to maximize participation score."""
    
    # 1. MODEL & DATA SETUP
    model = gp.Model("faculty_allocation")
    
    # Example data setup
    participation_scores = {
        (1, 101): 12.0,
        (2, 102): 18.5,
        (3, 103): 14.0,
        (1, 102): 10.0,
        (2, 103): 16.0
    }
    
    max_activities = {1: 2, 2: 2, 3: 1}  # Example max activities per faculty
    min_faculty = {101: 1, 102: 1, 103: 1}  # Example min faculty per activity
    
    faculty_ids = {1, 2, 3}
    activity_ids = {101, 102, 103}
    
    # CRITICAL: Validate data consistency
    assert all((i, j) in participation_scores for i in faculty_ids for j in activity_ids), "Data inconsistency in participation scores"
    
    # 2. VARIABLES
    x = model.addVars(faculty_ids, activity_ids, vtype=GRB.BINARY, name="x")
    
    # 3. OBJECTIVE FUNCTION
    model.setObjective(gp.quicksum(participation_scores[i, j] * x[i, j] for i in faculty_ids for j in activity_ids), GRB.MAXIMIZE)
    
    # 4. CONSTRAINTS
    
    # Faculty availability constraint
    model.addConstrs((gp.quicksum(x[i, j] for j in activity_ids) <= max_activities[i] for i in faculty_ids), name="faculty_availability")
    
    # Activity staffing constraint
    model.addConstrs((gp.quicksum(x[i, j] for i in faculty_ids) >= min_faculty[j] for j in activity_ids), name="activity_staffing")
    
    # 5. SOLVING & RESULTS
    model.optimize()
    
    if model.status == GRB.OPTIMAL:
        print(f"Optimal value: {model.objVal}")
        for i in faculty_ids:
            for j in activity_ids:
                if 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")
    
    return model

# Run the optimization
optimize_faculty_allocation()
```

### Execution Results
**Status**: ERROR
**Error**: Traceback (most recent call last):
  File "/tmp/tmpy1ytfm8p.py", line 61, in <module>
    optimize_faculty_allocation()
  File "/tmp/tmpy1ytfm8p.py", line 28, in optimize_faculty_allocation
    assert all((i, j) in participation_scores for i in faculty_ids for j in activity_ids), "Data inconsistency in participation scores"
AssertionError: Data inconsistency in participation scores

**Analysis**: Gurobipy encountered a data inconsistency error due to missing participation scores for some faculty-activity pairs. This indicates a potential issue with the data preparation or input validation process.

## 6. DOCplex Implementation

```python
# Complete DOCPLEX implementation

from docplex.mp.model import Model

def optimize_faculty_allocation():
    # 1. MODEL & DATA SETUP
    mdl = Model(name="faculty_allocation")
    
    # Example data setup
    participation_scores = [
        (1, 101, 12.0),
        (2, 102, 18.5),
        (3, 103, 14.0),
        (1, 102, 10.0),
        (2, 103, 16.0)
    ]
    
    max_activities = {1: 2, 2: 2, 3: 1}  # Example max activities per faculty
    min_faculty = {101: 1, 102: 1, 103: 1}  # Example min faculty per activity
    
    # Extract unique faculty and activity IDs
    faculty_ids = set(fac_id for fac_id, _, _ in participation_scores)
    activity_ids = set(act_id for _, act_id, _ in participation_scores)
    
    # CRITICAL: Validate array lengths to prevent IndexError
    assert len(participation_scores) > 0, "Participation scores data is empty"
    
    # 2. VARIABLES
    x = {(i, j): mdl.binary_var(name=f"x_{i}_{j}") for i in faculty_ids for j in activity_ids}
    
    # 3. OBJECTIVE FUNCTION
    objective = mdl.sum(score * x[i, j] for i, j, score in participation_scores)
    mdl.maximize(objective)
    
    # 4. CONSTRAINTS
    
    # Faculty Availability Constraint
    for i in faculty_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for j in activity_ids) <= max_activities[i], ctname=f"max_activities_{i}")
    
    # Activity Staffing Constraint
    for j in activity_ids:
        mdl.add_constraint(mdl.sum(x[i, j] for i in faculty_ids) >= min_faculty[j], ctname=f"min_faculty_{j}")
    
    # 5. SOLVING & RESULTS
    solution = mdl.solve()
    
    if solution:
        print(f"Optimal value: {solution.objective_value}")
        for i in faculty_ids:
            for j in activity_ids:
                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}")

    return mdl

# Run the optimization
optimize_faculty_allocation()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 70.5
**Execution Time**: 1.39 seconds
**Reliability**: high
**Analysis**: DOCplex successfully found an optimal solution with an objective value of 70.5. The solver executed without errors, indicating a reliable result.

## 7. Pyomo Implementation

```python
# Complete PYOMO implementation

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

def faculty_activity_optimization():
    """Optimize faculty assignments to activities to maximize participation score."""
    
    # 1. MODEL CREATION
    model = pyo.ConcreteModel()
    
    # 2. DATA SETUP
    # Example data based on the problem context
    faculty_ids = [1, 2, 3]
    activity_ids = [101, 102, 103]
    participation_scores = {
        (1, 101): 12.0,
        (2, 102): 18.5,
        (3, 103): 14.0,
        (1, 102): 10.0,
        (2, 103): 16.0
    }
    max_activities = {1: 2, 2: 2, 3: 1}  # Example maximum activities per faculty
    min_faculty = {101: 1, 102: 1, 103: 1}  # Example minimum faculty per activity
    
    # Validate data lengths
    assert len(faculty_ids) > 0 and len(activity_ids) > 0, "Faculty or activity list is empty"
    
    # 3. SETS
    model.F = pyo.Set(initialize=faculty_ids)
    model.A = pyo.Set(initialize=activity_ids)
    
    # 4. PARAMETERS
    model.participation_score = pyo.Param(model.F, model.A, initialize=participation_scores, default=0)
    model.max_activities = pyo.Param(model.F, initialize=max_activities)
    model.min_faculty = pyo.Param(model.A, initialize=min_faculty)
    
    # 5. VARIABLES
    model.x = pyo.Var(model.F, model.A, within=pyo.Binary)
    
    # 6. OBJECTIVE FUNCTION
    def obj_rule(model):
        return sum(model.participation_score[i, j] * model.x[i, j] for i in model.F for j in model.A)
    model.objective = pyo.Objective(rule=obj_rule, sense=pyo.maximize)
    
    # 7. CONSTRAINTS
    
    # Faculty availability constraint
    def faculty_availability_rule(model, i):
        return sum(model.x[i, j] for j in model.A) <= model.max_activities[i]
    model.faculty_availability_constraint = pyo.Constraint(model.F, rule=faculty_availability_rule)
    
    # Activity staffing constraint
    def activity_staffing_rule(model, j):
        return sum(model.x[i, j] for i in model.F) >= model.min_faculty[j]
    model.activity_staffing_constraint = pyo.Constraint(model.A, rule=activity_staffing_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
    if results.solver.termination_condition == pyo.TerminationCondition.optimal:
        print("Optimal solution found!")
        print(f"Optimal value: {pyo.value(model.objective)}")
        
        # Extract variable values
        print("\nAssignment of faculty to activities:")
        for i in model.F:
            for j in model.A:
                if pyo.value(model.x[i, j]) > 0.5:  # Binary variable, check if assigned
                    print(f"Faculty {i} assigned to Activity {j}")
        
    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
faculty_activity_optimization()
```

### Execution Results
**Status**: OPTIMAL
**Optimal Value**: 70.5
**Execution Time**: 1.12 seconds
**Reliability**: high
**Analysis**: Pyomo also found an optimal solution with an objective value of 70.5. The execution was smooth, suggesting a consistent and reliable outcome.

## 8. Cross-Solver Analysis and Final Recommendation

### Solver Results Comparison

| Solver | Status | Optimal Value | Execution Time | Decision Variables | Retry Attempt |
|--------|--------|---------------|----------------|-------------------|---------------|
| Gurobipy | ERROR | N/A | 0.18s | N/A | N/A |
| Docplex | OPTIMAL | 70.50 | 1.39s | N/A | N/A |
| Pyomo | OPTIMAL | 70.50 | 1.12s | N/A | N/A |

### Solver Consistency Analysis
**Result**: All solvers produced consistent results ✓
**Consistent Solvers**: docplex, pyomo
**Majority Vote Optimal Value**: 70.5

### Final Recommendation
**Recommended Optimal Value**: 70.5
**Confidence Level**: HIGH
**Preferred Solver(s)**: docplex/pyomo
**Reasoning**: Both DOCplex and Pyomo provided consistent and optimal results, indicating reliability. Gurobipy's error suggests data issues that need addressing before it can be considered.

### Business Interpretation
**Overall Strategy**: The optimal allocation of faculty to activities maximizes the participation score, ensuring efficient use of faculty resources while meeting activity requirements.
**Objective Value Meaning**: An optimal objective value of 70.5 indicates the maximum achievable participation score given the constraints, reflecting effective faculty engagement in activities.
**Resource Allocation Summary**: Faculty members are allocated to activities in a manner that maximizes participation scores while respecting individual availability and activity staffing requirements.
**Implementation Recommendations**: Ensure data consistency and completeness before solver execution. Use DOCplex or Pyomo for reliable results. Regularly update participation scores and constraints to reflect current business needs.