## Symbolic Representation

To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation.

Let's define the variables:
- $x_1$ = Number of batches of bagels
- $x_2$ = Number of batches of croissants

The objective is to maximize the total profit. The profit per batch of bagels is $7.5, and the profit per batch of croissants is $5. Therefore, the objective function can be represented as:

Maximize: $7.5x_1 + 5x_2$

The constraints are:
1. Dough mixer can run for at most 2500 hours per year. Since a batch of bagels takes 2 hours and a batch of croissants takes 1.5 hours, we have:
   $2x_1 + 1.5x_2 \leq 2500$

2. Commercial bake oven can run for at most 2500 hours per year. Since a batch of bagels takes 3.5 hours and a batch of croissants takes 2 hours, we have:
   $3.5x_1 + 2x_2 \leq 2500$

3. Non-negativity constraints, as the number of batches cannot be negative:
   $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'batches of bagels'), ('x2', 'batches of croissants')],
    'objective_function': '7.5*x1 + 5*x2',
    'constraints': [
        '2*x1 + 1.5*x2 <= 2500',
        '3.5*x1 + 2*x2 <= 2500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define variables
x1 = model.addVar(name="bagels", lb=0)  # Number of batches of bagels
x2 = model.addVar(name="croissants", lb=0)  # Number of batches of croissants

# Objective function: Maximize profit
model.setObjective(7.5 * x1 + 5 * x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(2 * x1 + 1.5 * x2 <= 2500, name="dough_mixer_constraint")
model.addConstr(3.5 * x1 + 2 * x2 <= 2500, name="bake_oven_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Number of batches of bagels: {x1.varValue}")
    print(f"Number of batches of croissants: {x2.varValue}")
    print(f"Maximum profit: ${model.objVal:.2f}")
else:
    print("The model is infeasible.")
```