## Problem Description and Formulation

The problem is an optimization problem where we need to maximize the objective function:

\[ 8.96 \times \text{hours worked by Peggy} + 1.78 \times \text{hours worked by Paul} + 9.49 \times \text{hours worked by Bill} \]

subject to several constraints related to the organization scores from hours worked by Peggy, Paul, and Bill.

## Constraints

1. Individual organization scores: 
   - Peggy's organization score is 20.
   - Paul's organization score is 17.
   - Bill's organization score is 4.

2. Combined organization scores:
   - Peggy + Paul ≥ 38
   - Peggy + Bill ≥ 35
   - Paul + Bill ≥ 50
   - Peggy + Paul + Bill ≥ 83
   - Paul + Bill ≤ 198
   - Peggy + Paul ≤ 166
   - Peggy + Paul + Bill ≤ 166

## Gurobi Code Formulation

We will use Gurobi's Python API to formulate and solve this problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("organization_score_optimization")

# Define variables
peggy_hours = m.addVar(name="peggy_hours", lb=0)  # hours worked by Peggy
paul_hours = m.addVar(name="paul_hours", lb=0)   # hours worked by Paul
bill_hours = m.addVar(name="bill_hours", lb=0)   # hours worked by Bill

# Objective function coefficients
obj_coef_peggy = 8.96
obj_coef_paul = 1.78
obj_coef_bill = 9.49

# Set objective function
m.setObjective(obj_coef_peggy * peggy_hours + obj_coef_paul * paul_hours + obj_coef_bill * bill_hours, gp.GRB.MAXIMIZE)

# Constraints
peggy_score = 20
paul_score = 17
bill_score = 4

# Peggy + Paul >= 38
m.addConstraint(peggy_score * peggy_hours + paul_score * paul_hours >= 38, name="peggy_paul_score")

# Peggy + Bill >= 35
m.addConstraint(peggy_score * peggy_hours + bill_score * bill_hours >= 35, name="peggy_bill_score")

# Paul + Bill >= 50
m.addConstraint(paul_score * paul_hours + bill_score * bill_hours >= 50, name="paul_bill_score")

# Peggy + Paul + Bill >= 83
m.addConstraint(peggy_score * peggy_hours + paul_score * paul_hours + bill_score * bill_hours >= 83, name="total_score")

# Paul + Bill <= 198
m.addConstraint(paul_score * paul_hours + bill_score * bill_hours <= 198, name="paul_bill_score_upper")

# Peggy + Paul <= 166
m.addConstraint(peggy_score * peggy_hours + paul_score * paul_hours <= 166, name="peggy_paul_score_upper")

# Peggy + Paul + Bill <= 166
m.addConstraint(peggy_score * peggy_hours + paul_score * paul_hours + bill_score * bill_hours <= 166, name="total_score_upper")

# Solve the model
m.optimize()

# Print solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Peggy hours: {peggy_hours.varValue}")
    print(f"Paul hours: {paul_hours.varValue}")
    print(f"Bill hours: {bill_hours.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```