The correct chronological order of the Achaiyas is given as option 1: Shankara-Ramanuja-Madhav-Chaitanya. Let`s break down the options to understand why this is the correct order.

Option 1 starts with Shankara, who was a philosopher and theologian from the 8th century. He is known for his Advaita Vedanta philosophy.

Next is Ramanuja, who lived in the 11th and 12th centuries. He was a philosopher and theologian as well and is associated with the Vishishtadvaita Vedanta philosophy.

Madhav is the third in the order. Unfortunately, no additional information is provided about this name, so we can`t provide further details.

The final person mentioned is Chaitanya, who was a saint and religious leader from the 16th century. He is known for his role in the Bhakti movement in India.

Therefore, the correct order is Shankara-Ramanuja-Madhav-Chaitanya (option 1).