## Step 1: Define the optimization problem
The goal is to minimize the objective function: $6 \times \text{green beans} + 6 \times \text{cherry pies} + 4 \times \text{fruit salads} + 1 \times \text{kiwis}$, subject to various constraints on umami index and cost.

## 2: List all variables and their attributes
- Variables: green beans, cherry pies, fruit salads, kiwis
- Resources/Attributes:
  - $r0$: umami index
    - green beans: 1
    - cherry pies: 14
    - fruit salads: 5
    - kiwis: 12
  - $r1$: dollar cost
    - green beans: 5
    - cherry pies: 12
    - fruit salads: 14
    - kiwis: 7

## 3: Formulate constraints
### Umami Index Constraints
1. $14 \times \text{cherry pies} + 12 \times \text{kiwis} \geq 34$
2. $1 \times \text{green beans} + 14 \times \text{cherry pies} \geq 40$
3. $1 \times \text{green beans} + 5 \times \text{fruit salads} \geq 78$
4. $1 \times \text{green beans} + 12 \times \text{kiwis} \geq 62$
5. $1 \times \text{green beans} + 14 \times \text{cherry pies} + 5 \times \text{fruit salads} + 12 \times \text{kiwis} \geq 62$
6. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 34$
7. $12 \times \text{cherry pies} + 7 \times \text{kiwis} \geq 35$
8. $12 \times \text{cherry pies} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 34$
9. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} \geq 34$
10. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 7 \times \text{kiwis} \geq 34$
11. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 34$
12. $12 \times \text{cherry pies} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 39$
13. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} \geq 39$
14. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 7 \times \text{kiwis} \geq 39$
15. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 39$
16. $12 \times \text{cherry pies} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 23$
17. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} \geq 23$
18. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 7 \times \text{kiwis} \geq 23$
19. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 23$
20. $12 \times \text{cherry pies} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 36$
21. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} \geq 36$
22. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 7 \times \text{kiwis} \geq 36$
23. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 36$
24. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \geq 36$

### Other Constraints
25. $-10 \times \text{green beans} + 6 \times \text{fruit salads} \geq 0$
26. $-4 \times \text{green beans} + 8 \times \text{cherry pies} \geq 0$
27. $5 \times \text{fruit salads} + 12 \times \text{kiwis} \leq 142$
28. $1 \times \text{green beans} + 5 \times \text{fruit salads} \leq 294$
29. $14 \times \text{cherry pies} + 12 \times \text{kiwis} \leq 277$
30. $1 \times \text{green beans} + 14 \times \text{cherry pies} \leq 230$
31. $1 \times \text{green beans} + 14 \times \text{cherry pies} + 12 \times \text{kiwis} \leq 184$
32. $1 \times \text{green beans} + 5 \times \text{fruit salads} + 12 \times \text{kiwis} \leq 270$
33. $12 \times \text{cherry pies} + 14 \times \text{fruit salads} \leq 58$
34. $5 \times \text{green beans} + 14 \times \text{fruit salads} + 7 \times \text{kiwis} \leq 45$
35. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 14 \times \text{fruit salads} \leq 79$
36. $5 \times \text{green beans} + 12 \times \text{cherry pies} + 7 \times \text{kiwis} \leq 63$

## 4: Implement in Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()
    
    # Define variables
    green_beans = model.addVar(name="green_beans", lb=0)
    cherry_pies = model.addVar(name="cherry_pies", lb=0)
    fruit_salads = model.addVar(name="fruit_salads", lb=0)
    kiwis = model.addVar(name="kiwis", lb=0)
    
    # Objective function
    model.setObjective(6 * green_beans + 6 * cherry_pies + 4 * fruit_salads + kiwis, gurobi.GRB.MINIMIZE)
    
    # Constraints
    model.addConstr(14 * cherry_pies + 12 * kiwis >= 34)
    model.addConstr(green_beans + 14 * cherry_pies >= 40)
    model.addConstr(green_beans + 5 * fruit_salads >= 78)
    model.addConstr(green_beans + 12 * kiwis >= 62)
    model.addConstr(green_beans + 14 * cherry_pies + 5 * fruit_salads + 12 * kiwis >= 62)
    model.addConstr(5 * fruit_salads + 7 * kiwis >= 34)
    model.addConstr(12 * cherry_pies + 7 * kiwis >= 35)
    model.addConstr(12 * cherry_pies + 14 * fruit_salads + 7 * kiwis >= 34)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads >= 34)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 7 * kiwis >= 34)
    model.addConstr(5 * green_beans + 14 * fruit_salads + 7 * kiwis >= 34)
    model.addConstr(12 * cherry_pies + 14 * fruit_salads + 7 * kiwis >= 39)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads >= 39)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 7 * kiwis >= 39)
    model.addConstr(5 * green_beans + 14 * fruit_salads + 7 * kiwis >= 39)
    model.addConstr(12 * cherry_pies + 14 * fruit_salads + 7 * kiwis >= 23)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads >= 23)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 7 * kiwis >= 23)
    model.addConstr(5 * green_beans + 14 * fruit_salads + 7 * kiwis >= 23)
    model.addConstr(12 * cherry_pies + 14 * fruit_salads + 7 * kiwis >= 36)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads >= 36)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 7 * kiwis >= 36)
    model.addConstr(5 * green_beans + 14 * fruit_salads + 7 * kiwis >= 36)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads + 7 * kiwis >= 36)
    
    model.addConstr(-10 * green_beans + 6 * fruit_salads >= 0)
    model.addConstr(-4 * green_beans + 8 * cherry_pies >= 0)
    model.addConstr(5 * fruit_salads + 12 * kiwis <= 142)
    model.addConstr(green_beans + 5 * fruit_salads <= 294)
    model.addConstr(14 * cherry_pies + 12 * kiwis <= 277)
    model.addConstr(green_beans + 14 * cherry_pies <= 230)
    model.addConstr(green_beans + 14 * cherry_pies + 12 * kiwis <= 184)
    model.addConstr(green_beans + 5 * fruit_salads + 12 * kiwis <= 270)
    model.addConstr(12 * cherry_pies + 14 * fruit_salads <= 58)
    model.addConstr(5 * green_beans + 14 * fruit_salads + 7 * kiwis <= 45)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 14 * fruit_salads <= 79)
    model.addConstr(5 * green_beans + 12 * cherry_pies + 7 * kiwis <= 63)
    
    model.optimize()
    
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Green Beans: ", green_beans.varValue)
        print("Cherry Pies: ", cherry_pies.varValue)
        print("Fruit Salads: ", fruit_salads.varValue)
        print("Kiwis: ", kiwis.varValue)
    else:
        print("No solution found")

optimize_problem()
```