## Step 1: Define the optimization problem and the goal
The goal is to minimize the objective function $7R + 4L + 1M$, where $R$, $L$, and $M$ represent the hours worked by Ringo, Laura, and Mary, respectively. The problem is subject to various constraints related to work quality, productivity, and paperwork competence ratings.

## 2: List all the constraints
1. $4R + 7L + 2M \leq 60$ (work quality rating constraint)
2. $7R + 2L + 1M \leq 63$ (productivity rating constraint)
3. $3R + 5L + 5M \leq 59$ (paperwork competence rating constraint)
4. $4R + 7L \geq 19$ (combined work quality rating constraint for Ringo and Laura)
5. $4R + 2M \geq 18$ (combined work quality rating constraint for Ringo and Mary)
6. $7L + 2M \geq 8$ (combined work quality rating constraint for Laura and Mary)
7. $4R + 7L + 2M \geq 8$ (combined work quality rating constraint for all)
8. $7R + 2L \geq 15$ (combined productivity rating constraint for Ringo and Laura)
9. $7R + 1M \geq 8$ (combined productivity rating constraint for Ringo and Mary)
10. $7R + 2L + 1M \geq 8$ (combined productivity rating constraint for all)
11. $3R + 5L \geq 17$ (combined paperwork competence rating constraint for Ringo and Laura)
12. $5L + 5M \geq 6$ (combined paperwork competence rating constraint for Laura and Mary)
13. $3R + 5L + 5M \geq 6$ (combined paperwork competence rating constraint for all)
14. $5L - 9M \geq 0$ (constraint involving Laura and Mary)
15. $-6R + 2L \geq 0$ (constraint involving Ringo and Laura)
16. $4R + 2M \leq 46$ (upper limit on combined work quality rating for Ringo and Mary)
17. $3R + 5L \leq 20$ (upper limit on combined paperwork competence rating for Ringo and Laura)

## 3: Convert the problem into Gurobi code
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
R = m.addVar(lb=0, name="Ringo_hours")  # hours worked by Ringo
L = m.addVar(lb=0, name="Laura_hours")  # hours worked by Laura
M = m.addVar(lb=0, name="Mary_hours")  # hours worked by Mary

# Define the objective function
m.setObjective(7*R + 4*L + M, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(4*R + 7*L + 2*M <= 60)  # work quality rating constraint
m.addConstr(7*R + 2*L + M <= 63)  # productivity rating constraint
m.addConstr(3*R + 5*L + 5*M <= 59)  # paperwork competence rating constraint
m.addConstr(4*R + 7*L >= 19)  # combined work quality rating constraint for Ringo and Laura
m.addConstr(4*R + 2*M >= 18)  # combined work quality rating constraint for Ringo and Mary
m.addConstr(7*L + 2*M >= 8)  # combined work quality rating constraint for Laura and Mary
m.addConstr(4*R + 7*L + 2*M >= 8)  # combined work quality rating constraint for all
m.addConstr(7*R + 2*L >= 15)  # combined productivity rating constraint for Ringo and Laura
m.addConstr(7*R + M >= 8)  # combined productivity rating constraint for Ringo and Mary
m.addConstr(7*R + 2*L + M >= 8)  # combined productivity rating constraint for all
m.addConstr(3*R + 5*L >= 17)  # combined paperwork competence rating constraint for Ringo and Laura
m.addConstr(5*L + 5*M >= 6)  # combined paperwork competence rating constraint for Laura and Mary
m.addConstr(3*R + 5*L + 5*M >= 6)  # combined paperwork competence rating constraint for all
m.addConstr(5*L - 9*M >= 0)  # constraint involving Laura and Mary
m.addConstr(-6*R + 2*L >= 0)  # constraint involving Ringo and Laura
m.addConstr(4*R + 2*M <= 46)  # upper limit on combined work quality rating for Ringo and Mary
m.addConstr(3*R + 5*L <= 20)  # upper limit on combined paperwork competence rating for Ringo and Laura

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Ringo hours: {R.varValue}")
    print(f"Laura hours: {L.varValue}")
    print(f"Mary hours: {M.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```