## Problem Description and Formulation

The problem is an optimization problem with the goal to minimize a linear objective function subject to several linear constraints. The variables are the hours worked by Bill, John, Laura, and Jean. The objective function to minimize is $7.73 \times \text{hours worked by Bill} + 6.89 \times \text{hours worked by John} + 8.6 \times \text{hours worked by Laura} + 6.81 \times \text{hours worked by Jean}$.

The constraints include:

1. Individual work quality ratings for Bill, John, Laura, and Jean.
2. Combined work quality rating constraints for various pairs and combinations of individuals.
3. Upper bounds on combined work quality ratings for various pairs and combinations of individuals.
4. Non-negativity and fractional allowance for hours worked by all individuals.

## Gurobi Code Formulation

```python
import gurobi as gp

# Define the model
m = gp.Model("Work_Optimization")

# Define the variables
hours_worked_by_Bill = m.addVar(name="hours_worked_by_Bill", lb=0)  # No upper bound specified, so lb=0
hours_worked_by_John = m.addVar(name="hours_worked_by_John", lb=0)
hours_worked_by_Laura = m.addVar(name="hours_worked_by_Laura", lb=0)
hours_worked_by_Jean = m.addVar(name="hours_worked_by_Jean", lb=0)

# Objective function
m.setObjective(7.73 * hours_worked_by_Bill + 6.89 * hours_worked_by_John + 8.6 * hours_worked_by_Laura + 6.81 * hours_worked_by_Jean, gp.GRB.MINIMIZE)

# Constraints
# Individual work quality ratings
m.addConstr(9 * hours_worked_by_Bill <= 9 * hours_worked_by_Bill)  # This constraint is always true
m.addConstr(8 * hours_worked_by_John <= 8 * hours_worked_by_John)  # This constraint is always true
m.addConstr(14 * hours_worked_by_Laura <= 14 * hours_worked_by_Laura)  # This constraint is always true
m.addConstr(13 * hours_worked_by_Jean <= 13 * hours_worked_by_Jean)  # This constraint is always true

# Combined work quality rating constraints
m.addConstr(8 * hours_worked_by_John + 14 * hours_worked_by_Laura >= 38)
m.addConstr(9 * hours_worked_by_Bill + 13 * hours_worked_by_Jean >= 24)
m.addConstr(9 * hours_worked_by_Bill + 8 * hours_worked_by_John >= 43)
m.addConstr(9 * hours_worked_by_Bill + 8 * hours_worked_by_John + 14 * hours_worked_by_Laura + 13 * hours_worked_by_Jean >= 43)

# Other constraints
m.addConstr(-1 * hours_worked_by_Bill + 2 * hours_worked_by_John >= 0)
m.addConstr(-6 * hours_worked_by_Bill + 8 * hours_worked_by_Laura >= 0)

# Upper bounds on combined work quality ratings
m.addConstr(8 * hours_worked_by_John + 14 * hours_worked_by_Laura <= 174)
m.addConstr(9 * hours_worked_by_Bill + 13 * hours_worked_by_Jean <= 121)
m.addConstr(9 * hours_worked_by_Bill + 8 * hours_worked_by_John <= 119)
m.addConstr(8 * hours_worked_by_John + 13 * hours_worked_by_Jean <= 118)
m.addConstr(9 * hours_worked_by_Bill + 8 * hours_worked_by_John + 14 * hours_worked_by_Laura <= 72)
m.addConstr(9 * hours_worked_by_Bill + 8 * hours_worked_by_John + 13 * hours_worked_by_Jean <= 100)
m.addConstr(8 * hours_worked_by_John + 14 * hours_worked_by_Laura + 13 * hours_worked_by_Jean <= 173)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Hours worked by Bill: {hours_worked_by_Bill.varValue}")
    print(f"Hours worked by John: {hours_worked_by_John.varValue}")
    print(f"Hours worked by Laura: {hours_worked_by_Laura.varValue}")
    print(f"Hours worked by Jean: {hours_worked_by_Jean.varValue}")
    print(f"Objective: {m.objVal}")
else:
    print("No optimal solution found.")
```