## Problem Description and Symbolic Representation

The problem involves a bike factory that produces two types of bikes: road bikes and mountain bikes. The goal is to maximize profit given the constraints on machine hours available for tooling on the grinder and polisher.

### Symbolic Variables:
- $x_1$ = Number of road bikes produced
- $x_2$ = Number of mountain bikes produced

### Objective Function:
The profit per road bike is $70, and per mountain bike is $100. Therefore, the objective function to maximize profit ($P$) is:
\[ P = 70x_1 + 100x_2 \]

### Constraints:
1. Grinder constraint: $3x_1 + 5x_2 \leq 12$
2. Polisher constraint: $2x_1 + 2.5x_2 \leq 12$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'road bikes'), ('x2', 'mountain bikes')],
    'objective_function': '70*x1 + 100*x2',
    'constraints': [
        '3*x1 + 5*x2 <= 12',
        '2*x1 + 2.5*x2 <= 12',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define variables
x1 = model.addVar(name="road_bikes", lb=0, vtype=gp.GRB.CONTINUOUS)
x2 = model.addVar(name="mountain_bikes", lb=0, vtype=gp.GRB.CONTINUOUS)

# Objective function: Maximize profit
model.setObjective(70*x1 + 100*x2, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(3*x1 + 5*x2 <= 12, name="grinder_constraint")
model.addConstr(2*x1 + 2.5*x2 <= 12, name="polisher_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal Solution:")
    print(f"Road Bikes: {x1.varValue}")
    print(f"Mountain Bikes: {x2.varValue}")
    print(f"Max Profit: {model.objVal}")
else:
    print("The model is infeasible or has no solution.")
```