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He was born as the son of an egg trader from The Hague. In 1915 he joined the mobilized Dutch airforce as a professional officer, at the time still called the "militaire luchtvaartafdeling" (military aviation department), in Soesterberg. After World War I in which the Netherlands remained neutral, he began with the organization of the ELTA, the "Eerste Luchtverkeer Tentoonstelling Amsterdam" (First Aviation Exhibition Amsterdam), held from 1 August till 14 September 1919. For that occasion enormous halls (hangars) were built. Right after the exhibition these were used by Anthony Fokker, for his new company Nederlandse Vliegtuigenfabriek (Dutch Aircraft Factory), subsequently the Fokker airplane factory.
All these activities led to the establishment of the Royal Dutch Airlines (KLM), of which Plesman became director and made it a flourishing company. After the Second World War Plesman was appointed president-director of KLM. After the recovery from the war the company became a renowned airline company under his leadership. Plesman pleaded in vain for a displacement of Schiphol Airport to a location near Burgerveen.
For his achievements for the international civil aviation he received in 1959 posthumously the first Edward Warner Award.