He was first elected to the Armenian National Assembly in 2012 as a member of the Prosperous Armenia Party.
He was re-elected in the 2017 Armenian parliamentary election and subsequently chosen as one of three Deputy Speakers of the National Assembly.
In 1983, Melkumyan graduated from Armenian State University of Economics, receiving the qualification of an economist.
Also in 2000, Melkumyan was appointed a deputy minister of state property management, a post which he held until 2003.
Melkumyan was elected to the Armenian National Assembly three times: on May 6, 2012, as a proportional representation candidate from the Prosperous Armenia party, on April 2, 2017, as a member of Tsarukyan Alliance, a political bloc led by Gagik Tsarukyan, the leader of the Prosperous Armenia party, and on December 9, 2018, again as a representative of Prosperous Armenia.