To solve this problem using Gurobi, we'll create a linear programming model that accounts for all given constraints. Since there's no objective function provided, we will assume our goal is to minimize or maximize one of the variables (for simplicity, let's choose to minimize the total combined productivity rating from all employees). 

Here's how you could set up the problem in Python using Gurobi:

```python
from gurobipy import *

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

# Define variables for hours worked by each employee
George = m.addVar(vtype=GRB.INTEGER, name="George")
Ringo = m.addVar(vtype=GRB.INTEGER, name="Ringo")
Bobby = m.addVar(vtype=GRB.INTEGER, name="Bobby")
Jean = m.addVar(vtype=GRB.INTEGER, name="Jean")
Mary = m.addVar(vtype=GRB.INTEGER, name="Mary")
Laura = m.addVar(vtype=GRB.CONTINUOUS, name="Laura")

# Define constraints
# Productivity ratings (lower bounds)
m.addConstr(George + Ringo >= 69, "Productivity_George_Ringo")
m.addConstr(Mary + Laura >= 98, "Productivity_Mary_Laura")

# Work quality ratings (lower bounds)
m.addConstr(Bobby + Jean >= 75, "WorkQuality_Bobby_Jean")
m.addConstr(George + Ringo >= 88, "WorkQuality_George_Ringo")

# Computer competence ratings (lower bounds)
m.addConstr(George + Jean >= 65, "ComputerCompetence_George_Jean")
m.addConstr(Ringo + Bobby >= 71, "ComputerCompetence_Ringo_Bobby")

# Likelihood to quit index (upper bounds)
m.addConstr(George + Bobby <= 159, "LikelihoodToQuit_George_Bobby")
m.addConstr(Jean + Laura <= 155, "LikelihoodToQuit_Jean_Laura")

# Other constraints
m.addConstr(Ringo + Jean <= 161, "Productivity_Ringo_Jean")
m.addConstr(Mary + Laura <= 314, "Productivity_Mary_Laura")

# Objective: Minimize total combined productivity rating (example)
m.setObjective(George + Ringo + Bobby + Jean + Mary + Laura, GRB.MINIMIZE)

# Optimize model
m.optimize()

# Print solution
for v in m.getVars():
    print('%s %g' % (v.varName, v.x))
```