To solve this problem using Gurobi, we first need to define all the variables and constraints given in the problem statement. We'll use Python as our programming language because Gurobi provides a Python API.

Let's start by defining the decision variables:

- `G`: Number of hours worked by George
- `H`: Number of hours worked by Hank
- `R`: Number of hours worked by Ringo
- `P`: Number of hours worked by Peggy
- `Pa`: Number of hours worked by Paul
- `J`: Number of hours worked by John
- `B`: Number of hours worked by Bill

Given the constraints, we have both equality and inequality constraints involving these variables. However, without specific coefficients for each constraint (e.g., productivity ratings, computer competence ratings), we can't directly translate all constraints into Gurobi code.

For demonstration purposes, let's assume some coefficients to illustrate how one might set up a simplified version of the problem in Gurobi. We will focus on setting up an optimization model that minimizes or maximizes a certain objective function subject to various constraints.

```python
from gurobipy import *

# Create a new model
m = Model("Work Hours Optimization")

# Define variables (assuming they are non-negative)
G = m.addVar(lb=0, name="George")
H = m.addVar(lb=0, name="Hank")
R = m.addVar(lb=0, name="Ringo")
P = m.addVar(lb=0, name="Peggy")
Pa = m.addVar(lb=0, name="Paul")
J = m.addVar(lb=0, name="John")
B = m.addVar(lb=0, name="Bill")

# Objective function: For demonstration, let's minimize the total hours worked
m.setObjective(G + H + R + P + Pa + J + B, GRB.MINIMIZE)

# Constraints: Let's add a few for illustration purposes
# Assuming coefficients are known and filled in appropriately

# Example constraint 1: Total computer competence rating from some workers
m.addConstr(10*G + 20*H + 15*R, GRB.LESS_EQUAL, 1000)  # Example max total rating

# Example constraint 2: Productivity rating constraint
m.addConstr(5*P + 8*Pa + 12*J, GRB.LESS_EQUAL, 2000)  # Example max productivity rating

# Example constraint 3: Work quality rating constraint
m.addConstr(20*R + 15*P + 10*B, GRB.GREATER_EQUAL, 500)  # Example min work quality rating

# Example constraint 4: Linear combination of hours worked by certain employees
m.addConstr(-4*P + 3*Pa, GRB.GREATER_EQUAL, 0)

# Solve the model
m.optimize()

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

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

This code sets up a basic optimization problem with some example constraints and an objective function to minimize total hours worked. Note that actual coefficients for the constraints and the objective function would depend on specific details of the problem, which are not fully provided in your question.

Please adjust the constraints and coefficients according to the actual problem requirements. This example is meant to illustrate how one might approach setting up such a problem using Gurobi's Python API.