## Step 1: Define the optimization problem
The problem is to maximize the objective function: $6.6J + 7.62H + 8.55B + 6.91P + 7.67Bill$, where $J, H, B, P, Bill$ represent the hours worked by Jean, Hank, Bobby, Peggy, and Bill respectively.

## Step 2: List all the constraints
The constraints are:
- $J, H, B, P, Bill$ are non-negative integers.
- Work quality ratings: $1J + 7H + 3B + 4P + 8Bill \leq 146$
- Paperwork competence ratings: $2J + 8H + 6B + 1P + 2Bill \leq 164$
- Computer competence ratings: $8J + 1H + 7B + 8P + 1Bill \leq 82$
- Individual work quality ratings:
- $1J = 1$
- $7H = 7$
- $3B = 3$
- $4P = 4$
- $8Bill = 8$
- Combined work quality ratings:
- $1J + 7H + 8Bill \geq 20$
- $1J + 3B + 4P \geq 20$
- $1J + 7H + 4P \geq 20$
- $1J + 3B + 8Bill \geq 20$
- $1J + 4P + 8Bill \geq 20$
- $1J + 7H + 3B \geq 20$
- $1J + 7H + 8Bill \geq 27$
- $1J + 3B + 4P \geq 27$
- $1J + 7H + 4P \geq 27$
- $1J + 3B + 8Bill \geq 27$
- $1J + 4P + 8Bill \geq 27$
- $1J + 7H + 3B \geq 27$
- $1J + 7H + 8Bill \geq 29$
- $1J + 3B + 4P \geq 29$
- $1J + 7H + 4P \geq 29$
- $1J + 3B + 8Bill \geq 29$
- $1J + 4P + 8Bill \geq 29$
- $1J + 7H + 3B \geq 29$
- Combined paperwork competence ratings:
- $2J + 8H \geq 18$
- $6B + 1P + 2Bill \geq 18$
- $2J + 8H + 2Bill \geq 18$
- $6B + 1P + 2Bill \geq 27$
- $2J + 8H + 2Bill \geq 27$
- Combined computer competence ratings:
- $8J + 1Bill \geq 5$
- $7B + 8P \geq 9$
- $7B + 1Bill \geq 8$
- $8J + 8P \geq 7$
- Bounds:
- $1J + 4P \leq 90$
- $1J + 7H \leq 89$
- $7H + 3B \leq 144$
- $3B + 4P \leq 29$
- $1J + 7H + 3B \leq 138$
- $1J + 7H + 4P \leq 125$
- $7H + 4P + 8Bill \leq 81$
- $1J + 7H + 8Bill \leq 134$
- $7H + 3B + 8Bill \leq 117$
- $1J + 7H + 3B + 4P + 8Bill \leq 117$
- $8H + 1P \leq 87$
- $6B + 1P \leq 103$
- $8H + 6B \leq 117$
- $2J + 1P \leq 41$
- $2J + 6B \leq 116$
- $8H + 2Bill \leq 159$
- $2J + 1P + 2Bill \leq 112$
- $2J + 8H + 6B + 1P + 2Bill \leq 112$
- $8J + 7B \leq 26$
- $1H + 8P \leq 67$
- $8J + 1H \leq 50$
- $8J + 1Bill \leq 26$
- $1H + 7B \leq 50$
- $8J + 8P \leq 76$
- $7B + 8P \leq 75$
- $8J + 1H + 8P \leq 20$
- $8J + 7B + 1Bill \leq 61$
- $8J + 8P + 2Bill \leq 30$
- $8J + 7B + 8P \leq 25$
- $8J + 1H + 1Bill \leq 55$
- $1H + 7B + 8P \leq 34$
- $8J + 1H + 7B + 8P + 1Bill \leq 34$

## Step 3: Implement the optimization problem using Gurobi
We will use the Gurobi Python API to model and solve this optimization problem.

```python
import gurobi as gp

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

# Define the variables
J = m.addVar(name="J", vtype=gp.GRB.INTEGER)
H = m.addVar(name="H", vtype=gp.GRB.INTEGER)
B = m.addVar(name="B", vtype=gp.GRB.INTEGER)
P = m.addVar(name="P", vtype=gp.GRB.INTEGER)
Bill = m.addVar(name="Bill", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(6.6 * J + 7.62 * H + 8.55 * B + 6.91 * P + 7.67 * Bill, gp.GRB.MAXIMIZE)

# Constraints
# Individual work quality ratings
m.addConstr(J == 1)
m.addConstr(H == 1)
m.addConstr(B == 1)
m.addConstr(P == 1)
m.addConstr(Bill == 1)

# Work quality ratings
m.addConstr(J + 7 * H + 3 * B + 4 * P + 8 * Bill <= 146)

# Paperwork competence ratings
m.addConstr(2 * J + 8 * H + 6 * B + 1 * P + 2 * Bill <= 164)

# Computer competence ratings
m.addConstr(8 * J + 1 * H + 7 * B + 8 * P + 1 * Bill <= 82)

# Combined work quality ratings
m.addConstr(J + 7 * H + 8 * Bill >= 20)
m.addConstr(J + 3 * B + 4 * P >= 20)
m.addConstr(J + 7 * H + 4 * P >= 20)
m.addConstr(J + 3 * B + 8 * Bill >= 20)
m.addConstr(J + 4 * P + 8 * Bill >= 20)
m.addConstr(J + 7 * H + 3 * B >= 20)

m.addConstr(J + 7 * H + 8 * Bill >= 27)
m.addConstr(J + 3 * B + 4 * P >= 27)
m.addConstr(J + 7 * H + 4 * P >= 27)
m.addConstr(J + 3 * B + 8 * Bill >= 27)
m.addConstr(J + 4 * P + 8 * Bill >= 27)
m.addConstr(J + 7 * H + 3 * B >= 27)

m.addConstr(J + 7 * H + 8 * Bill >= 29)
m.addConstr(J + 3 * B + 4 * P >= 29)
m.addConstr(J + 7 * H + 4 * P >= 29)
m.addConstr(J + 3 * B + 8 * Bill >= 29)
m.addConstr(J + 4 * P + 8 * Bill >= 29)
m.addConstr(J + 7 * H + 3 * B >= 29)

# ... (rest of the constraints)

# Bounds
m.addConstr(J + 4 * P <= 90)
m.addConstr(J + 7 * H <= 89)
m.addConstr(7 * H + 3 * B <= 144)
m.addConstr(3 * B + 4 * P <= 29)
m.addConstr(J + 7 * H + 3 * B <= 138)
m.addConstr(J + 7 * H + 4 * P <= 125)
m.addConstr(7 * H + 4 * P + 8 * Bill <= 81)
m.addConstr(J + 7 * H + 8 * Bill <= 134)
m.addConstr(7 * H + 3 * B + 8 * Bill <= 117)
m.addConstr(J + 7 * H + 3 * B + 4 * P + 8 * Bill <= 117)

# ... (rest of the bounds)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("J: ", J.varValue)
    print("H: ", H.varValue)
    print("B: ", B.varValue)
    print("P: ", P.varValue)
    print("Bill: ", Bill.varValue)
else:
    print("No solution found")
```