To solve this problem, we first need to define the decision variables and the objective function. Let's denote the number of servings of orange flavored drink as $x_1$ and the number of servings of apple flavored drink as $x_2$. The objective is to minimize the total cost, which can be represented as $8x_1 + 5x_2$.

The constraints are based on the minimum requirements for fiber and iron:
- For fiber: $4x_1 + 6x_2 \geq 13$
- For iron: $5x_1 + 3x_2 \geq 13$

Since we cannot have negative servings of drinks, we also have non-negativity constraints:
- $x_1 \geq 0$
- $x_2 \geq 0$

Now, let's formulate the linear programming (LP) problem:

Minimize: $8x_1 + 5x_2$
Subject to:
- $4x_1 + 6x_2 \geq 13$ (Fiber constraint)
- $5x_1 + 3x_2 \geq 13$ (Iron constraint)
- $x_1, x_2 \geq 0$ (Non-negativity constraints)

This problem can be solved using Gurobi in Python. Below is the code to solve this LP:

```python
from gurobipy import *

# Create a new model
m = Model("Fiber_and_Iron")

# Define variables
x1 = m.addVar(name="Orange_Flavored", vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name="Apple_Flavored", vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(8*x1 + 5*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(4*x1 + 6*x2 >= 13, name="Fiber_Constraint")
m.addConstr(5*x1 + 3*x2 >= 13, name="Iron_Constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Orange Flavored: {x1.x} servings")
    print(f"Apple Flavored: {x2.x} servings")
    print(f"Total Cost: ${8*x1.x + 5*x2.x}")
else:
    print("No optimal solution found.")
```