Mahabharata text states that Vyasa composed The Mahabharata text, 18 years after the Mahabharata War.
Assuming this to be the time of completion of the composition of Mahabharata text, we can estimate,
Year of the composition of the Mahabharata = 5561 BCE + 18 = 5543 BCE
Mahabharata text also states that it took Vyasa three years to compose the Mahabharata.
If two statements (with their corresponding assumptions are combined)
Vyasa began composing the Mahabharata text, around 5546 BCE.