Problem $1(20$ Points$)$: An airline owns an aging fleet of Boeing $737$ jet airplanes. It is considering a major purchase of up to $17$ new Boeing $757$ and $767$ jets. The decision must take into account of numerous cost and capabilities factors, including the following: the airline can finance up to $$16$ billion in purchase; each Boeing $757$ will cost $$80$ million and $767$ will cost $$110$ million; at least one$-$third of planes purchased should be the longer$-$range $757$; the annual maintenance cost per $757$ is estimated to be $$800,000$ and it is $$500,000$ for each $767$ ;

$757$ can carry $125000$ passengers per year, where as each $767$ can fly $81000$ passengers annually.

Formulate this as an integer programming problem to maximize the annual passengerscarrying capability.