## Step 1: Define the symbolic representation of the problem
The problem involves four variables: black pens, headsets, mechanical pencils, and lanyards, which we can denote as $x_0, x_1, x_2, x_3$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $4x_0 + 4x_1 + 8x_2 + 9x_3$.

## 3: List the constraints in symbolic notation
The constraints given are:
- $20x_1 + 2x_3 \geq 51$
- $9x_0 + 20x_1 + 10x_2 \geq 90$
- $9x_0 + 20x_1 + 10x_2 + 2x_3 \geq 90$
- $16x_1 + 7x_3 \geq 28$
- $16x_1 + 17x_2 \geq 45$
- $17x_2 + 7x_3 \geq 66$
- $10x_0 + 16x_1 + 17x_2 + 7x_3 \geq 66$
- $16x_0 + 9x_3 \geq 73$
- $16x_0 + x_2 \geq 72$
- $16x_0 + 12x_1 \geq 56$
- $12x_1 + x_2 \geq 62$
- $16x_0 + 12x_1 + x_2 + 9x_3 \geq 62$
- $-7x_0 + 3x_1 \geq 0$
- $9x_0 + 2x_3 \leq 261$
- $10x_2 + 2x_3 \leq 341$
- $20x_1 + 2x_3 \leq 153$
- $20x_1 + 10x_2 \leq 136$
- $9x_0 + 10x_2 \leq 100$
- $9x_0 + 20x_1 + 2x_3 \leq 182$
- $9x_0 + 20x_1 + 10x_2 \leq 208$
- $10x_0 + 17x_2 \leq 180$
- $10x_0 + 16x_1 \leq 281$
- $10x_0 + 7x_3 \leq 197$
- $10x_0 + 16x_1 + 17x_2 \leq 107$
- $10x_0 + 16x_1 + 7x_3 \leq 216$
- $16x_0 + 9x_3 \leq 332$
- $12x_1 + x_2 \leq 309$
- $16x_0 + 12x_1 \leq 235$
- $x_0, x_1, x_2, x_3$ are integers.

## 4: Create the symbolic representation dictionary
```json
{
    'sym_variables': [
        ('x0', 'black pens'), 
        ('x1', 'headsets'), 
        ('x2', 'mechanical pencils'), 
        ('x3', 'lanyards')
    ], 
    'objective_function': '4*x0 + 4*x1 + 8*x2 + 9*x3', 
    'constraints': [
        '20*x1 + 2*x3 >= 51',
        '9*x0 + 20*x1 + 10*x2 >= 90',
        '9*x0 + 20*x1 + 10*x2 + 2*x3 >= 90',
        '16*x1 + 7*x3 >= 28',
        '16*x1 + 17*x2 >= 45',
        '17*x2 + 7*x3 >= 66',
        '10*x0 + 16*x1 + 17*x2 + 7*x3 >= 66',
        '16*x0 + 9*x3 >= 73',
        '16*x0 + x2 >= 72',
        '16*x0 + 12*x1 >= 56',
        '12*x1 + x2 >= 62',
        '16*x0 + 12*x1 + x2 + 9*x3 >= 62',
        '-7*x0 + 3*x1 >= 0',
        '9*x0 + 2*x3 <= 261',
        '10*x2 + 2*x3 <= 341',
        '20*x1 + 2*x3 <= 153',
        '20*x1 + 10*x2 <= 136',
        '9*x0 + 10*x2 <= 100',
        '9*x0 + 20*x1 + 2*x3 <= 182',
        '9*x0 + 20*x1 + 10*x2 <= 208',
        '10*x0 + 17*x2 <= 180',
        '10*x0 + 16*x1 <= 281',
        '10*x0 + 7*x3 <= 197',
        '10*x0 + 16*x1 + 17*x2 <= 107',
        '10*x0 + 16*x1 + 7*x3 <= 216',
        '16*x0 + 9*x3 <= 332',
        '12*x1 + x2 <= 309',
        '16*x0 + 12*x1 <= 235'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x0 = model.addVar(name='black_pens', vtype=gurobi.GRB.INTEGER)
    x1 = model.addVar(name='headsets', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='mechanical_pencils', vtype=gurobi.GRB.INTEGER)
    x3 = model.addVar(name='lanyards', vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(4*x0 + 4*x1 + 8*x2 + 9*x3, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(20*x1 + 2*x3 >= 51)
    model.addConstr(9*x0 + 20*x1 + 10*x2 >= 90)
    model.addConstr(9*x0 + 20*x1 + 10*x2 + 2*x3 >= 90)
    model.addConstr(16*x1 + 7*x3 >= 28)
    model.addConstr(16*x1 + 17*x2 >= 45)
    model.addConstr(17*x2 + 7*x3 >= 66)
    model.addConstr(10*x0 + 16*x1 + 17*x2 + 7*x3 >= 66)
    model.addConstr(16*x0 + 9*x3 >= 73)
    model.addConstr(16*x0 + x2 >= 72)
    model.addConstr(16*x0 + 12*x1 >= 56)
    model.addConstr(12*x1 + x2 >= 62)
    model.addConstr(16*x0 + 12*x1 + x2 + 9*x3 >= 62)
    model.addConstr(-7*x0 + 3*x1 >= 0)
    model.addConstr(9*x0 + 2*x3 <= 261)
    model.addConstr(10*x2 + 2*x3 <= 341)
    model.addConstr(20*x1 + 2*x3 <= 153)
    model.addConstr(20*x1 + 10*x2 <= 136)
    model.addConstr(9*x0 + 10*x2 <= 100)
    model.addConstr(9*x0 + 20*x1 + 2*x3 <= 182)
    model.addConstr(9*x0 + 20*x1 + 10*x2 <= 208)
    model.addConstr(10*x0 + 17*x2 <= 180)
    model.addConstr(10*x0 + 16*x1 <= 281)
    model.addConstr(10*x0 + 7*x3 <= 197)
    model.addConstr(10*x0 + 16*x1 + 17*x2 <= 107)
    model.addConstr(10*x0 + 16*x1 + 7*x3 <= 216)
    model.addConstr(16*x0 + 9*x3 <= 332)
    model.addConstr(12*x1 + x2 <= 309)
    model.addConstr(16*x0 + 12*x1 <= 235)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objval)
        print('Black pens: ', x0.varValue)
        print('Headsets: ', x1.varValue)
        print('Mechanical pencils: ', x2.varValue)
        print('Lanyards: ', x3.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```