Andrews took over as second permanent secretary at the MoD, the second highest-ranking civil servant in the department, at the start of the month.
Defence Estates said it had yet to find a replacement for Andrews, who became chief executive in 1998, the year the prime contracting initiative was launched.
Andrews takes over from Sir Roger Jacking as second permanent secretary. Jacking was made head of the new Defence Academy last month.
A spokesperson emphasised that Andrews would still be involved in Defence Estates' procurement strategy, despite his move.
The spokesperson said: "He will still have a connection and interest in terms of how the agency brings its initiatives forward. He has not severed his links with Defence Estates."
The industry was divided over the effect of Andrews' departure on the prime contracting initiative, which moved Defence Estates away from traditional procurement to partnering style arrangements with contractors.
One contractor said: "Ian has had a crucial role in the roll-out of prime contracting. It's a blow to see him leave."
However, Major Contractors Group director Bill Tallis played down the move. He said: "He has been there for some time. He was probably ready for a move – the civil service tends to move people a lot."
Defence Estates' prime contracting initiative has attracted industry criticism since its launch for creating too much risk for contractors and being too lengthy.
Tallis said the MCG was due to meet Defence Estates next month over complaints about a recent MoD prime contract.
Contractors claimed that five bidders vying for the £70m Catterick Garrison project in North Yorkshire would have to spend up to £1m each on bidding costs for the project.
Tallis said: "We are meeting with them regularly. The tendering process takes too long." One contractor added: "Something has to change. The attitude of the MoD is not in line with the culture and ethos of prime contracting."