To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote:

- $x$ as the number of boat trips.
- $y$ as the number of cargo plane trips.

The objective is to maximize the total number of boxes of litchis delivered, which can be represented as $500x + 200y$, since each boat trip can take 500 boxes and each cargo plane trip can take 200 boxes.

Given constraints are:

1. The cost constraint: $5000x + 3000y \leq 200000$, since each boat trip costs $5000 and each cargo plane trip costs $3000, and the total budget is $200000.
2. The constraint on the number of boat trips relative to cargo plane trips: $x \leq y$, because the number of boat trips cannot exceed the number of cargo plane trips.

Both $x$ and $y$ must be non-negative integers since they represent the number of trips, which cannot be negative or fractional.

Now, let's express this problem in Gurobi code using Python:

```python
from gurobipy import *

# Create a model
m = Model("Litchi_Shipment")

# Define decision variables
x = m.addVar(vtype=GRB.INTEGER, name="boat_trips")
y = m.addVar(vtype=GRB.INTEGER, name="cargo_plane_trips")

# Set the objective function: Maximize the total boxes of litchis delivered
m.setObjective(500*x + 200*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5000*x + 3000*y <= 200000, "budget_constraint")
m.addConstr(x <= y, "trip_constraint")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of boat trips: {x.x}")
    print(f"Number of cargo plane trips: {y.x}")
    print(f"Total boxes delivered: {500*x.x + 200*y.x}")
else:
    print("No optimal solution found")
```