Bluebird Farms is a firm in the food industry. The various divisions have submitted a total of eight projects for final approval for the year $2024.$ The total allowed capital budget for $2024$ is $$300$M$.$ The eight projects are listed below, with the project t$=0$ initial cost and project NPV listed. Projects $4$ and $5$ below are mutually exclusive. What is the optimum bundle of projects that Bluebird Farms should approve that maximizes the NPV given the $$300$M budget constraint? The profitability index must be used in the analysis.

$(1)$ Nonfat Greek Yogurt: Cost $=$ $$50$M; NPV $=9$M

$(2)$ Sugar$-$free chocolate: Cost $=$ $$50$M; NPV $=8\mathrm{.}5$M

$(3)$ Lowfat Gelato: Cost $=$ $$75$M; NPV $=12$M

$(4)$ Oat milk: Cost $=$ $$75$M; NPV $=15$M

$(5)$ Soy milk: Cost $=$ $$50$M; NPV $=9\mathrm{.}5$M

$(6)$ Granola & M$$sli: Cost $=$ $$75$M$,$ NPV $=15\mathrm{.}75$M

$(7)$ Salad kits: Cost $=$ $$50$M$,$ NPV $=11$M

$(8)$ Fruit and nut snacks : Cost $=$ $$50$M$,$ NPV $=7\mathrm{.}5$M