## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit of a video game store by determining the optimal number of regular games and collector's edition games to stock, given certain constraints.

### Variables

- \(R\): The number of regular games to stock.
- \(C\): The number of collector's edition games to stock.

### Objective Function

The profit per regular game sold is $20, and the profit per collector's edition game sold is $30. The objective is to maximize the total profit \(P\), which can be represented as:

\[P = 20R + 30C\]

### Constraints

1. **Sales Limit**: The store can sell at most 100 video games of either type per month.
\[R + C \leq 100\]

2. **Budget Constraint**: Each regular game costs $30, and each collector's edition game costs $50. The store wants to spend at most $4000.
\[30R + 50C \leq 4000\]

3. **Non-Negativity Constraint**: The number of games cannot be negative.
\[R \geq 0, C \geq 0\]
\[R, C \in \mathbb{Z}\] (Since we cannot stock a fraction of a game, both \(R\) and \(C\) should be integers.)

## Gurobi Code

To solve this problem, we will use the Gurobi Optimizer in Python. We will first define the problem, add the variables, the objective function, and the constraints, and then solve it.

```python
import gurobi as gp

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

# Define variables
R = model.addVar(vtype=gp.GRB.INTEGER, name="Regular_Games")
C = model.addVar(vtype=gp.GRB.INTEGER, name="Collectors_Edition_Games")

# Objective function: Maximize profit
model.setObjective(20*R + 30*C, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(R + C <= 100, name="Sales_Limit")
model.addConstr(30*R + 50*C <= 4000, name="Budget_Constraint")
model.addConstr(R >= 0, name="Non_Negativity_R")
model.addConstr(C >= 0, name="Non_Negativity_C")

# Optimize
model.optimize()

# Print solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal stock of regular games: {R.varValue}")
    print(f"Optimal stock of collector's edition games: {C.varValue}")
    print(f"Maximal profit: {model.objVal}")
else:
    print("No optimal solution found.")
```