To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables for the quantities of desks and chairs made by each manufacturer and formulating the objective function and constraints based on these variables.

Let's define:
- $x_1$ as the number of desks made by Manufacturer X,
- $x_2$ as the number of chairs made by Manufacturer Y.

The profit per desk is $140, and the profit per chair is $120. The objective is to maximize the total profit, which can be represented as:
\[ \text{Maximize: } 140x_1 + 120x_2 \]

Given constraints are:
1. Manufacturer X can make at most 22 desks a day: $x_1 \leq 22$
2. Manufacturer Y can make at most 28 chairs a day: $x_2 \leq 28$
3. The quality check company can check at most 40 items of either type per day: $x_1 + x_2 \leq 40$

Symbolic representation:
```json
{
  'sym_variables': [('x1', 'number of desks made by Manufacturer X'), ('x2', 'number of chairs made by Manufacturer Y')],
  'objective_function': '140*x1 + 120*x2',
  'constraints': ['x1 <= 22', 'x2 <= 28', 'x1 + x2 <= 40']
}
```

To solve this problem using Gurobi in Python, we'll use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Furniture_Optimization")

# Define variables
x1 = m.addVar(name="desks_by_X", lb=0)  # Number of desks made by Manufacturer X
x2 = m.addVar(name="chairs_by_Y", lb=0)  # Number of chairs made by Manufacturer Y

# Set the objective function
m.setObjective(140*x1 + 120*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 <= 22, name="max_desks_X")
m.addConstr(x2 <= 28, name="max_chairs_Y")
m.addConstr(x1 + x2 <= 40, name="quality_check_limit")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Desks made by Manufacturer X: {x1.x}")
    print(f"Chairs made by Manufacturer Y: {x2.x}")
    print(f"Total Profit: {m.objVal}")
else:
    print("No optimal solution found")
```