## Step 1: Define the optimization problem and identify the variables, objective function, and constraints.
The problem involves minimizing an objective function subject to various constraints. The variables are 'lemons', 'ham sandwiches', 'cherry pies', and 'fruit salads'. The objective function to minimize is $7.26 \times \text{lemons} + 6.62 \times \text{ham sandwiches} + 1.42 \times \text{cherry pies} + 2.56 \times \text{fruit salads}$.

## 2: List all the constraints based on the problem description.
The constraints are as follows:
1. $5.8 \times \text{lemons} + 8.4 \times \text{fruit salads} \geq 27$
2. $2.9 \times \text{cherry pies} + 8.4 \times \text{fruit salads} \geq 37$
3. $3.85 \times \text{ham sandwiches} + 8.4 \times \text{fruit salads} \geq 30$
4. $5.8 \times \text{lemons} + 3.85 \times \text{ham sandwiches} + 2.9 \times \text{cherry pies} \geq 31$
5. $5.8 \times \text{lemons} + 2.9 \times \text{cherry pies} + 8.4 \times \text{fruit salads} \geq 31$
6. $5.8 \times \text{lemons} + 3.85 \times \text{ham sandwiches} + 2.9 \times \text{cherry pies} \geq 41$
7. $5.8 \times \text{lemons} + 2.9 \times \text{cherry pies} + 8.4 \times \text{fruit salads} \geq 41$
8. $5.8 \times \text{lemons} + 3.85 \times \text{ham sandwiches} + 2.9 \times \text{cherry pies} + 8.4 \times \text{fruit salads} \geq 41$
9. $5.15 \times \text{lemons} + 7.62 \times \text{fruit salads} \geq 27$
10. $5.15 \times \text{lemons} + 3.63 \times \text{ham sandwiches} \geq 24$
11. $3.3 \times \text{cherry pies} + 7.62 \times \text{fruit salads} \geq 47$
12. $3.63 \times \text{ham sandwiches} + 3.3 \times \text{cherry pies} \geq 38$
13. $5.15 \times \text{lemons} + 3.3 \times \text{cherry pies} + 7.62 \times \text{fruit salads} \geq 32$
14. $5.15 \times \text{lemons} + 3.63 \times \text{ham sandwiches} + 3.3 \times \text{cherry pies} + 7.62 \times \text{fruit salads} \geq 32$
15. $-6 \times \text{cherry pies} + 7 \times \text{fruit salads} \geq 0$
16. $2 \times \text{ham sandwiches} - \text{cherry pies} \geq 0$
17. $5.8 \times \text{lemons} + 2.9 \times \text{cherry pies} \leq 130$
18. $2.9 \times \text{cherry pies} + 8.4 \times \text{fruit salads} \leq 169$
19. $3.85 \times \text{ham sandwiches} + 2.9 \times \text{cherry pies} \leq 181$
20. $3.85 \times \text{ham sandwiches} + 8.4 \times \text{fruit salads} \leq 73$
21. $5.15 \times \text{lemons} + 3.3 \times \text{cherry pies} + 7.62 \times \text{fruit salads} \leq 68$

## 3: Implement the optimization problem using Gurobi.
We will use the Gurobi library in Python to model and solve this optimization problem.

```python
import gurobi as gp

# Create a new model
m = gp.Model("optimization_problem")

# Define the variables
lemons = m.addVar(name="lemons", lb=0)  # No lower bound specified, assuming 0
ham_sandwiches = m.addVar(name="ham_sandwiches", lb=0)  # No lower bound specified, assuming 0
cherry_pies = m.addVar(name="cherry_pies", lb=0)  # No lower bound specified, assuming 0
fruit_salads = m.addVar(name="fruit_salads", lb=0)  # No lower bound specified, assuming 0

# Define the objective function
m.setObjective(7.26 * lemons + 6.62 * ham_sandwiches + 1.42 * cherry_pies + 2.56 * fruit_salads, gp.GRB.MINIMIZE)

# Add constraints
m.addConstr(5.8 * lemons + 8.4 * fruit_salads >= 27)
m.addConstr(2.9 * cherry_pies + 8.4 * fruit_salads >= 37)
m.addConstr(3.85 * ham_sandwiches + 8.4 * fruit_salads >= 30)
m.addConstr(5.8 * lemons + 3.85 * ham_sandwiches + 2.9 * cherry_pies >= 31)
m.addConstr(5.8 * lemons + 2.9 * cherry_pies + 8.4 * fruit_salads >= 31)
m.addConstr(5.8 * lemons + 3.85 * ham_sandwiches + 2.9 * cherry_pies >= 41)
m.addConstr(5.8 * lemons + 2.9 * cherry_pies + 8.4 * fruit_salads >= 41)
m.addConstr(5.8 * lemons + 3.85 * ham_sandwiches + 2.9 * cherry_pies + 8.4 * fruit_salads >= 41)
m.addConstr(5.15 * lemons + 7.62 * fruit_salads >= 27)
m.addConstr(5.15 * lemons + 3.63 * ham_sandwiches >= 24)
m.addConstr(3.3 * cherry_pies + 7.62 * fruit_salads >= 47)
m.addConstr(3.63 * ham_sandwiches + 3.3 * cherry_pies >= 38)
m.addConstr(5.15 * lemons + 3.3 * cherry_pies + 7.62 * fruit_salads >= 32)
m.addConstr(5.15 * lemons + 3.63 * ham_sandwiches + 3.3 * cherry_pies + 7.62 * fruit_salads >= 32)
m.addConstr(-6 * cherry_pies + 7 * fruit_salads >= 0)
m.addConstr(2 * ham_sandwiches - cherry_pies >= 0)
m.addConstr(5.8 * lemons + 2.9 * cherry_pies <= 130)
m.addConstr(2.9 * cherry_pies + 8.4 * fruit_salads <= 169)
m.addConstr(3.85 * ham_sandwiches + 2.9 * cherry_pies <= 181)
m.addConstr(3.85 * ham_sandwiches + 8.4 * fruit_salads <= 73)
m.addConstr(5.15 * lemons + 3.3 * cherry_pies + 7.62 * fruit_salads <= 68)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Lemons: ", lemons.varValue)
    print("Ham sandwiches: ", ham_sandwiches.varValue)
    print("Cherry pies: ", cherry_pies.varValue)
    print("Fruit salads: ", fruit_salads.varValue)
else:
    print("The model is infeasible")
```