To solve this optimization problem, we need to formulate it as a linear programming (LP) problem. We have three decision variables: hours worked by Mary (M), hours worked by Dale (D), and hours worked by Laura (L). 

The objective function is to minimize the total cost, which is given by 8.72M + 5.55D + 1.19L.

We have several constraints:

- The likelihood to quit index for each employee is given, but these are not directly used as constraints. Instead, we have constraints on the combined likelihood to quit indices.
- The total combined likelihood to quit index from hours worked by Dale and Laura should be at least 34.
- The total combined likelihood to quit index from hours worked by Mary and Laura must be at least 29.
- There are two constraints that seem to repeat the requirement that the total combined likelihood to quit index from hours worked by Mary, Dale, and Laura has to be as much or more than 29. We'll consider this as one constraint since they appear to be identical.
- The expression -4D + 8L must be at least zero.
- The total combined likelihood to quit index from hours worked by Mary and Dale has to be as much or less than 43.

Here's how we can translate these into Gurobi code:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define the decision variables
M = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Mary")
D = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Dale")
L = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="hours_worked_by_Laura")

# Define the objective function
m.setObjective(8.72*M + 5.55*D + 1.19*L, GRB.MINIMIZE)

# Add constraints
m.addConstr(4.77*D + 11.23*L >= 34, "D_and_L_constraint")
m.addConstr(9.26*M + 11.23*L >= 29, "M_and_L_constraint")
m.addConstr(9.26*M + 4.77*D + 11.23*L >= 29, "M_D_and_L_constraint")
m.addConstr(-4*D + 8*L >= 0, "D_and_L_relationship")
m.addConstr(9.26*M + 4.77*D <= 43, "M_and_D_constraint")

# Optimize the model
m.optimize()

# Print out the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print("Objective:", m.objVal)
```