To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables and formulating the objective function and constraints using these variables.

Let's denote:
- $x_1$ as the number of days the eastern location operates per week.
- $x_2$ as the number of days the western location operates per week.

The objective is to minimize the total cost. The eastern location costs $1000 per day, and the western location costs $2000 per day. Therefore, the objective function can be represented as:
\[ \text{Minimize:} \quad 1000x_1 + 2000x_2 \]

The constraints are based on the production requirements:
- Cheap caviar: $5x_1 + 2x_2 \geq 25$
- Regular caviar: $7x_1 + 6x_2 \geq 35$
- Expensive caviar: $x_1 + 4x_2 \geq 15$

Additionally, since the locations cannot operate for a negative number of days and the problem implies they can only operate for whole days, we have:
- $x_1 \geq 0$
- $x_2 \geq 0$
- $x_1$ and $x_2$ must be integers.

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of days the eastern location operates per week'), 
                      ('x2', 'number of days the western location operates per week')],
    'objective_function': '1000*x1 + 2000*x2',
    'constraints': ['5*x1 + 2*x2 >= 25', '7*x1 + 6*x2 >= 35', 'x1 + 4*x2 >= 15', 
                    'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a model
m = Model("caviar_production")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="eastern_location_days")
x2 = m.addVar(vtype=GRB.INTEGER, name="western_location_days")

# Set the objective function
m.setObjective(1000*x1 + 2000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x1 + 2*x2 >= 25, "cheap_caviar_constraint")
m.addConstr(7*x1 + 6*x2 >= 35, "regular_caviar_constraint")
m.addConstr(x1 + 4*x2 >= 15, "expensive_caviar_constraint")
m.addConstr(x1 >= 0, "non_negative_eastern")
m.addConstr(x2 >= 0, "non_negative_western")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Eastern location operates for {x1.x} days")
    print(f"Western location operates for {x2.x} days")
else:
    print("No optimal solution found")
```