## Problem Description and Formulation

The problem involves minimizing the cost of producing standing desks and office chairs using a special machine, subject to several constraints. Let's denote the number of standing desks to be produced as \(x\) and the number of office chairs as \(y\).

### Constraints:

1. **Machine Operation Time**: The machine must be operated for at least 2000 minutes per week. Given that each standing desk takes 60 minutes and each office chair takes 35 minutes, we have:
\[ 60x + 35y \geq 2000 \]

2. **Minimum Production**: The factory must make a minimum of 100 items total:
\[ x + y \geq 100 \]

3. **Non-Negativity**: The number of standing desks and office chairs cannot be negative:
\[ x \geq 0, y \geq 0 \]

### Objective Function:

The cost for producing a standing desk is $500, and for an office chair, it is $230. The objective is to minimize the total cost:
\[ \text{Minimize:} \quad 500x + 230y \]

## Gurobi Code

To solve this linear programming problem using Gurobi, we can use the following Python code:

```python
import gurobi

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

    # Define variables
    x = model.addVar(name="standing_desks", lb=0, vtype=gurobi.GRB.INTEGER)
    y = model.addVar(name="office_chairs", lb=0, vtype=gurobi.GRB.INTEGER)

    # Objective function: Minimize cost
    model.setObjective(500*x + 230*y, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(60*x + 35*y >= 2000, name="machine_time")
    model.addConstr(x + y >= 100, name="min_production")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal Solution: standing desks = {x.varValue}, office chairs = {y.varValue}")
        print(f"Minimum Cost: ${model.objVal:.2f}")
    else:
        print("The model is infeasible.")

solve_optimization_problem()
```

This code defines the optimization problem using Gurobi's Python interface, sets up the objective function and constraints, and then solves the model. If an optimal solution exists, it prints out the number of standing desks and office chairs to produce and the minimum cost. If the model is infeasible, it indicates that there is no solution satisfying all the constraints.