## Step 1: Define the optimization problem
The problem is to minimize the objective function: $8.66x_0 + 6.28x_1 + 1.06x_2 + 4.68x_3$, where $x_0, x_1, x_2, x_3$ represent the hours worked by Bobby, Hank, Jean, and Laura, respectively.

## Step 2: List all the constraints
The constraints are:
- $x_0 \geq 0, x_1 \geq 0, x_2 \geq 0, x_3 \geq 0$ (non-negativity constraints, but not explicitly stated, we assume they are non-negative)
- $2x_0 = 2$ (Bobby's likelihood to quit index)
- $13x_0 = 13$ (Bobby's productivity rating)
- $5x_0 = 5$ (Bobby's paperwork competence rating)
- $13x_1 = 13$ (Hank's likelihood to quit index)
- $4x_1 = 4$ (Hank's productivity rating)
- $10x_1 = 10$ (Hank's paperwork competence rating)
- $9x_2 = 9$ (Jean's likelihood to quit index)
- $5x_2 = 5$ (Jean's productivity rating)
- $10x_2 = 10$ (Jean's paperwork competence rating)
- $1x_3 = 1$ (Laura's likelihood to quit index)
- $1x_3 = 1$ (Laura's productivity rating)
- $12x_3 = 12$ (Laura's paperwork competence rating)
- $13x_1 + 9x_2 + 1x_3 \geq 13$ (total combined likelihood to quit index from Hank, Jean, and Laura)
- $2x_0 + 13x_1 + 9x_2 + 1x_3 \geq 13$ (total combined likelihood to quit index from all)
- $13x_0 + 5x_2 \geq 28$ (total combined productivity rating from Bobby and Jean)
- $4x_1 + 1x_3 \geq 21$ (total combined productivity rating from Hank and Laura)
- $5x_2 + 1x_3 \geq 20$ (total combined productivity rating from Jean and Laura)
- $13x_0 + 4x_1 \geq 28$ (total combined productivity rating from Bobby and Hank)
- $4x_1 + 5x_2 + 1x_3 \geq 23$ (total combined productivity rating from Hank, Jean, and Laura)
- $13x_0 + 4x_1 + 5x_2 + 1x_3 \geq 23$ (total combined productivity rating from all)
- $10x_2 + 12x_3 \geq 17$ (total combined paperwork competence rating from Jean and Laura)
- $5x_0 + 10x_2 \geq 15$ (total combined paperwork competence rating from Bobby and Jean)
- $5x_0 + 10x_1 + 10x_2 + 12x_3 \geq 15$ (total combined paperwork competence rating from all)
- $4x_1 - 2x_2 \geq 0$ (linear constraint)
- $2x_0 + 9x_2 \leq 51$ (total combined likelihood to quit index from Bobby and Jean)
- $4x_1 + 5x_2 \leq 61$ (total combined productivity rating from Hank and Jean)
- $5x_2 + 1x_3 \leq 98$ (total combined productivity rating from Jean and Laura)
- $13x_0 + 5x_2 \leq 64$ (total combined productivity rating from Bobby and Jean)
- $13x_0 + 4x_1 + 1x_3 \leq 77$ (total combined productivity rating from Bobby, Hank, and Laura)
- $4x_1 + 5x_2 + 1x_3 \leq 91$ (total combined productivity rating from Hank, Jean, and Laura)
- $5x_0 + 12x_3 \leq 81$ (total combined paperwork competence rating from Bobby and Laura)
- $10x_1 + 10x_2 \leq 74$ (total combined paperwork competence rating from Hank and Jean)
- $10x_1 + 12x_3 \leq 60$ (total combined paperwork competence rating from Hank and Laura)
- $10x_2 + 12x_3 \leq 77$ (total combined paperwork competence rating from Jean and Laura)
- $5x_0 + 10x_1 + 12x_3 \leq 70$ (total combined paperwork competence rating from Bobby, Hank, and Laura)
- $5x_0 + 10x_1 + 10x_2 \leq 43$ (total combined paperwork competence rating from Bobby, Hank, and Jean)

## 3: Simplify constraints
Upon inspection, we see that many constraints are equality constraints for individual variables:
- $2x_0 = 2 \Rightarrow x_0 = 1$
- $13x_0 = 13 \Rightarrow x_0 = 1$
- $5x_0 = 5 \Rightarrow x_0 = 1$
- $13x_1 = 13 \Rightarrow x_1 = 1$
- $4x_1 = 4 \Rightarrow x_1 = 1$
- $10x_1 = 10 \Rightarrow x_1 = 1$
- $9x_2 = 9 \Rightarrow x_2 = 1$
- $5x_2 = 5 \Rightarrow x_2 = 1$
- $10x_2 = 10 \Rightarrow x_2 = 1$
- $1x_3 = 1 \Rightarrow x_3 = 1$
- $1x_3 = 1 \Rightarrow x_3 = 1$
- $12x_3 = 12 \Rightarrow x_3 = 1$

## 4: Apply simplifications to constraints and objective
Given $x_0 = 1, x_1 = 1, x_2 = 1, x_3 = 1$, let's apply these to the constraints and objective function:
- Objective function: $8.66*1 + 6.28*1 + 1.06*1 + 4.68*1 = 8.66 + 6.28 + 1.06 + 4.68 = 20.68$

## 5: Check feasibility
Given $x_0 = 1, x_1 = 1, x_2 = 1, x_3 = 1$, let's check if these values satisfy all constraints:
- $13*1 + 9*1 + 1*1 = 23 \geq 13$ (satisfied)
- $2*1 + 13*1 + 9*1 + 1*1 = 25 \geq 13$ (satisfied)
- $13*1 + 5*1 = 18 \geq 28$ (not satisfied)
Since one constraint is not satisfied, the problem as defined with these equality constraints for individual variables does not have a feasible solution that meets all requirements.

## 6: Formulate the problem in Gurobi
However, to follow instructions and provide Gurobi code, we recognize that the previous steps indicate a specific solution but not a general optimization problem solution due to the equality constraints. Let's formulate the optimization problem without assuming $x_0, x_1, x_2, x_3 = 1$:
```python
import gurobi as gp

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

# Define variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Bobby
x1 = m.addVar(name="x1", lb=0)  # hours worked by Hank
x2 = m.addVar(name="x2", lb=0)  # hours worked by Jean
x3 = m.addVar(name="x3", lb=0)  # hours worked by Laura

# Objective function
m.setObjective(8.66*x0 + 6.28*x1 + 1.06*x2 + 4.68*x3, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(2*x0 == 2)
m.addConstr(13*x0 == 13)
m.addConstr(5*x0 == 5)
m.addConstr(13*x1 == 13)
m.addConstr(4*x1 == 4)
m.addConstr(10*x1 == 10)
m.addConstr(9*x2 == 9)
m.addConstr(5*x2 == 5)
m.addConstr(10*x2 == 10)
m.addConstr(x3 == 1)
m.addConstr(x3 == 1)
m.addConstr(12*x3 == 12)
m.addConstr(13*x1 + 9*x2 + x3 >= 13)
m.addConstr(2*x0 + 13*x1 + 9*x2 + x3 >= 13)
m.addConstr(13*x0 + 5*x2 >= 28)
m.addConstr(4*x1 + x3 >= 21)
m.addConstr(5*x2 + x3 >= 20)
m.addConstr(13*x0 + 4*x1 >= 28)
m.addConstr(4*x1 + 5*x2 + x3 >= 23)
m.addConstr(13*x0 + 4*x1 + 5*x2 + x3 >= 23)
m.addConstr(10*x2 + 12*x3 >= 17)
m.addConstr(5*x0 + 10*x2 >= 15)
m.addConstr(5*x0 + 10*x1 + 10*x2 + 12*x3 >= 15)
m.addConstr(4*x1 - 2*x2 >= 0)
m.addConstr(2*x0 + 9*x2 <= 51)
m.addConstr(4*x1 + 5*x2 <= 61)
m.addConstr(5*x2 + x3 <= 98)
m.addConstr(13*x0 + 5*x2 <= 64)
m.addConstr(13*x0 + 4*x1 + x3 <= 77)
m.addConstr(4*x1 + 5*x2 + x3 <= 91)
m.addConstr(5*x0 + 12*x3 <= 81)
m.addConstr(10*x1 + 10*x2 <= 74)
m.addConstr(10*x1 + 12*x3 <= 60)
m.addConstr(10*x2 + 12*x3 <= 77)
m.addConstr(5*x0 + 10*x1 + 12*x3 <= 70)
m.addConstr(5*x0 + 10*x1 + 10*x2 <= 43)

# Solve the model
m.optimize()

# Print the objective value
print(m.objVal)

# Print the variable values
print(x0.varName, x0.x)
print(x1.varName, x1.x)
print(x2.varName, x2.x)
print(x3.varName, x3.x)
```

The final answer is: 
```python
import gurobi as gp

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

# Define variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Bobby
x1 = m.addVar(name="x1", lb=0)  # hours worked by Hank
x2 = m.addVar(name="x2", lb=0)  # hours worked by Jean
x3 = m.addVar(name="x3", lb=0)  # hours worked by Laura

# Objective function
m.setObjective(8.66*x0 + 6.28*x1 + 1.06*x2 + 4.68*x3, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(2*x0 == 2)
m.addConstr(13*x0 == 13)
m.addConstr(5*x0 == 5)
m.addConstr(13*x1 == 13)
m.addConstr(4*x1 == 4)
m.addConstr(10*x1 == 10)
m.addConstr(9*x2 == 9)
m.addConstr(5*x2 == 5)
m.addConstr(10*x2 == 10)
m.addConstr(x3 == 1)
m.addConstr(x3 == 1)
m.addConstr(12*x3 == 12)
m.addConstr(13*x1 + 9*x2 + x3 >= 13)
m.addConstr(2*x0 + 13*x1 + 9*x2 + x3 >= 13)
m.addConstr(13*x0 + 5*x2 >= 28)
m.addConstr(4*x1 + x3 >= 21)
m.addConstr(5*x2 + x3 >= 20)
m.addConstr(13*x0 + 4*x1 >= 28)
m.addConstr(4*x1 + 5*x2 + x3 >= 23)
m.addConstr(13*x0 + 4*x1 + 5*x2 + x3 >= 23)
m.addConstr(10*x2 + 12*x3 >= 17)
m.addConstr(5*x0 + 10*x2 >= 15)
m.addConstr(5*x0 + 10*x1 + 10*x2 + 12*x3 >= 15)
m.addConstr(4*x1 - 2*x2 >= 0)
m.addConstr(2*x0 + 9*x2 <= 51)
m.addConstr(4*x1 + 5*x2 <= 61)
m.addConstr(5*x2 + x3 <= 98)
m.addConstr(13*x0 + 5*x2 <= 64)
m.addConstr(13*x0 + 4*x1 + x3 <= 77)
m.addConstr(4*x1 + 5*x2 + x3 <= 91)
m.addConstr(5*x0 + 12*x3 <= 81)
m.addConstr(10*x1 + 10*x2 <= 74)
m.addConstr(10*x1 + 12*x3 <= 60)
m.addConstr(10*x2 + 12*x3 <= 77)
m.addConstr(5*x0 + 10*x1 + 12*x3 <= 70)
m.addConstr(5*x0 + 10*x1 + 10*x2 <= 43)

# Solve the model
try:
    m.optimize()
    if m.status == gp.GRB.Status.OPTIMAL:
        print("Optimal solution found.")
        print("Objective value:", m.objVal)
        print("Hours worked by Bobby:", x0.x)
        print("Hours worked by Hank:", x1.x)
        print("Hours worked by Jean:", x2.x)
        print("Hours worked by Laura:", x3.x)
    elif m.status == gp.GRB.Status.INFEASIBLE:
        print("The model is infeasible.")
    else:
        print("The model has a non-optimal status:", m.status)
except gp.GRB.Error as e:
    print("Gurobi error:", e)
```