## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit from selling regular and premium pens under certain constraints.

### Decision Variables:
- \(x_1\): The number of regular pens to sell.
- \(x_2\): The number of premium pens to sell.

### Objective Function:
The profit from selling \(x_1\) regular pens at $3 each and \(x_2\) premium pens at $8 each needs to be maximized. The objective function can be represented as:
\[ \text{Maximize:} \quad 3x_1 + 8x_2 \]

### Constraints:
1. **Non-negativity constraints:** \(x_1 \geq 0\) and \(x_2 \geq 0\), as the number of pens cannot be negative.
2. **Demand constraints:**
   - Daily demand for regular pens is at most 100: \(x_1 \leq 100\).
   - Daily demand for premium pens is at most 50: \(x_2 \leq 50\).
3. **Total pens constraint:** The store can sell at most 75 pens of either type per day: \(x_1 + x_2 \leq 75\).

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    x1 = model.addVar(lb=0, ub=100, name="regular_pens")
    x2 = model.addVar(lb=0, ub=50, name="premium_pens")

    # Objective function: Maximize 3x1 + 8x2
    model.setObjective(3 * x1 + 8 * x2, gurobi.GRB.MAXIMIZE)

    # Constraint: x1 + x2 <= 75
    model.addConstr(x1 + x2 <= 75, name="total_pens")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
        print(f"Maximum profit: ${3 * x1.varValue + 8 * x2.varValue}")
    else:
        print("The model is infeasible.")

# Run the function
solve_pen_store_problem()
```

This code defines the optimization problem using Gurobi's Python interface, solves it, and prints out the optimal solution and the maximum profit achievable under the given constraints.