## Symbolic Representation

Let's denote:
- \(x_1\) as the number of large packages
- \(x_2\) as the number of small packages

The objective is to maximize profit. Given that the profit per large package is $3 and per small package is $0.50, the objective function can be represented as:
\[ \text{Maximize:} \quad 3x_1 + 0.5x_2 \]

The constraints are:
1. Budget constraint: \(3x_1 + x_2 \leq 2000\)
2. Shelf space constraint: \(3x_1 + x_2 \leq 400\)
3. Percentage of small packages: \(x_2 \geq 0.7(x_1 + x_2)\)
4. Non-negativity constraints: \(x_1 \geq 0, x_2 \geq 0\)

## Algebraic Formulation

- **Variables:** \((x_1, x_2)\) corresponding to (large packages, small packages)
- **Objective Function:** \(3x_1 + 0.5x_2\)
- **Constraints:**
  1. \(3x_1 + x_2 \leq 2000\)
  2. \(3x_1 + x_2 \leq 400\)
  3. \(0.3x_1 - 0.7x_2 \leq 0\)
  4. \(x_1 \geq 0, x_2 \geq 0\)

Correcting the constraints for accuracy:
- The budget and shelf space constraints are correctly represented.
- The percentage constraint is algebraically manipulated for clarity.

## Gurobi Code

```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(lb=0, name="large_packages")
    x2 = model.addVar(lb=0, name="small_packages")

    # Objective function
    model.setObjective(3*x1 + 0.5*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(3*x1 + x2 <= 2000, name="budget_constraint")
    model.addConstr(3*x1 + x2 <= 400, name="shelf_space_constraint")
    model.addConstr(0.3*x1 - 0.7*x2 <= 0, name="small_packages_percentage")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Large packages: {x1.varValue}")
        print(f"Small packages: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```

However, upon closer inspection, there seems to be an inconsistency in the problem constraints as presented: the budget constraint ($2000) seems to be much larger than what would be necessary given the shelf space constraint (400 units). The shelf space constraint $3x_1 + x_2 \leq 400$ will be much more limiting than the budget constraint $3x_1 + x_2 \leq 2000$. The code reflects the problem as stated but solving it may yield infeasibility or results that reflect the dominance of one constraint over the others.

## Symbolic Representation in JSON

```json
{
  "sym_variables": [
    ["x1", "large packages"],
    ["x2", "small packages"]
  ],
  "objective_function": "3x1 + 0.5x2",
  "constraints": [
    "3x1 + x2 <= 2000",
    "3x1 + x2 <= 400",
    "0.3x1 - 0.7x2 <= 0",
    "x1 >= 0",
    "x2 >= 0"
  ]
}
```