$380$ CHAPTER $10$ InTEgER PRogRAMMIng, goAL PRogRAMMIng, And nonLInEAR PRogRAMMIngthe planes purchased should be the longer$-$range $787$;$(4)$ the annual maintenance budget is to be no more than$$8$ million; $(5)$ the annual maintenance cost per $787$ isestimated to be $$800,000,$ and it is $$500,000$ for each$767$; and $(6)$ each $787$ can carry $125,000$ passengersper year, whereas each $767$ can fly $81,000$ passengersannually. Formulate this as an integer programmingproblem to maximize the annual passenger$-$carryingcapability. What category of integer programmingproblem is this? Solve this problem$\mathrm{.}10-14$ Trapeze Investments is a venture capital firm that iscurrently evaluating six different investment oppor$-$tunities. There is not sufficient capital to invest inall of these, but more than one will be selected. A$01$ integer programming model is planned to helpdetermine which of the six opportunities to choose.Variables X$1,$ X$2,$ X$3,$ X$4,$ X$5,$ and X$6$ represent the sixchoices. For each of the following situations, write aconstraint $($or several constraints$)$ that would be used.$($a$)$ At least three of these choices are to be selected.$($b$)$ Either investment $1$ or investment $4$ must beundertaken but not both.$($c$)$ If investment $4$ is selected, then investment$6$ must also be selected. However, if investment$4$ is not selected, it is still possible to selectinvestment $6.($d$)$ Investment $5$ cannot be selected unless invest$-$ments $2$ and $3$ are both also selected.$($e$)$ Investment $5$ must be selected if investments$2$ and $3$ are both also selected$\mathrm{.}10-15$ Horizon Wireless, a cellular telephone company, isexpanding into a new era. Relay towers are neces$-$sary to provide wireless telephone coverage to thedifferent areas of the city. A grid is superimposed ona map of the city to help determine where the tow$-$ers should be located. The grid consists of $8$ areaslabeled A through H$.$ Six possible tower locations$($numbered $16)$ have been identified, and each loca$-$tion could serve several areas. The following tableindicates the areas served by each of the towers.TOWERLOCATION$123456$AREASSERVEDA, B$,$ D B$,$ C$,$ G C$,$ D$,$ E$,$ F E$,$ F$,$ H E$,$ G$,$ H A$,$ D$,$ FFormulate this as a $01$ programming model to mini$-$mize the total number of towers required to cover allthe areas. Solve this using a computer.