## Problem Description and Formulation

The problem is a classic example of a linear programming problem. Banana sells two types of software packages: a student version and a professional version. The goal is to maximize profits given certain constraints.

### Decision Variables

- Let \(S\) be the number of student version software packages produced.
- Let \(P\) be the number of professional version software packages produced.

### Objective Function

The profit per student version is $400, and the profit per professional version is $1500. The objective is to maximize the total profit:

\[ \text{Maximize:} \quad 400S + 1500P \]

### Constraints

1. **Production Cost Constraint**: The cost to produce a student version is $750, and to produce a professional version is $3000. The total production cost should not exceed $500,000.

\[ 750S + 3000P \leq 500,000 \]

2. **Sales Constraint**: The marketing department estimates that they can sell at most 350 licenses for both versions combined a month.

\[ S + P \leq 350 \]

3. **Non-Negativity Constraint**: The number of software packages produced cannot be negative.

\[ S \geq 0, P \geq 0 \]

## Gurobi Code

Here is the Gurobi code in Python that captures the problem description and provides a solution:

```python
import gurobi

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

    # Define the decision variables
    S = model.addVar(lb=0, name="Student_Version")
    P = model.addVar(lb=0, name="Professional_Version")

    # Objective function: Maximize profit
    model.setObjective(400 * S + 1500 * P, gurobi.GRB.MAXIMIZE)

    # Production cost constraint
    model.addConstr(750 * S + 3000 * P <= 500000, name="Production_Cost_Constraint")

    # Sales constraint
    model.addConstr(S + P <= 350, name="Sales_Constraint")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: Student Version = {S.varValue}, Professional Version = {P.varValue}")
        print(f"Max Profit: ${model.objVal}")
    else:
        print("The model is infeasible.")

solve_production_problem()
```

This code defines the optimization problem using Gurobi's Python interface, solves it, and prints out the optimal production quantities and the maximum profit achievable. If the problem is infeasible, it indicates that there is no solution that satisfies all the constraints.