How do you determine the best solution out of a set of feasible basic solutions/extreme points? I'm working on one of my exam sets, surrounding linear optimization, and could really use some help.
The assignment is essentially $\to$ find the $10$ basic solutions $\to$ find the $5$ feasible solutions out of these $\to$ find the best feasible solution out of these.
Now I'm stuck on this last part, because as far as I know the best feasible solution is defined by all components of it being $\ge0$, aka negative or zero. Whilst as far as I can see the feasible solutions I found are all positive.
So my assumption is that I either got confused on picking the feasible solutions or I misunderstood how to find the best feasible solution.
Any help would be really appreciated, as this is the last assignment of my exam prep for tomorrow, and I'd like to understand it better than I currently do before then.

Assignment
Introduction to the assignment

The actual assignment




Our answer
Finding our basic solution set


Finding our feasible solution set/extreme points



All components being $\ge 0$ (positive or $0$) is just the condition that makes a basic solution feasible. It has nothing to do with being the worst or the best.
The best solution is determined by the objective function, which as far as I can tell you haven't considered at all yet. You should write down $\tilde c$, figure out $\tilde c \cdot \tilde x$ at each of your five basic feasible solutions, and then (since you're minimizing) pick the one that gives the lowest value.
