行ってらっしゃい
I know that 行ってらっしゃい is a formal response to 行って来ます and English translations could be "Have a good day".
But what does 行ってらっしゃい actually mean? It has 行 in it, and maybe 行って is the て form of 行. But where does らしゃい come from?
I'm sorry if this is a stupid question.

Question

行ってらっしゃい
I know that 行ってらっしゃい is a formal response to 行って来ます and English translations could be "Have a good day". 
But what does  行ってらっしゃい actually mean? It has 行 in it, and maybe  行って is the て form of 行. But where does  らしゃい  come from? 
I'm sorry if this is a stupid question.

Answer

いってらっしゃい consists of 行って＋（い）らっしゃい

いらっしゃい　is probably the 連用形 form of いらっしゃる 
which comes fromいらせらる 
which consists of いら, せ, and らる

いら　comes from いる（入る: enter)
せ and られる both show respect.

いらっしゃる is a respect form of 行く(go), 来る（come), or 居る(be)

どこへいらっしゃるのですか？(where are you going?)
先生がいらっしゃった（The teacher has arrived)
明日は家にいらっしゃいますか？（Will you be home tomorrow?)

FYI: おっしゃる(say) comes from おおせある consisting of おおせ and ある

Answer

according to Google  , 行って来ます= I'll go and come 　　
行ってらっしゃい（ 行っていらっしゃいませ）=you can go and come        
いらっしゃい= welcome

when I use them ,Ifeel like I'm leaving for now（行って来ます）and the person is leaving for now（行ってらっしゃい）In both case, come back finaly