Executive summary, based on a comment posted by @ffao elsewhere:
This was a job-related question, and you were asked to recognize the assignment problem, which is a well-known combinatorial optimization problem.
The difficulty of this problem comes from the fancy term:
Option expiration: your apples at some time might rot beyond of any usefulness, when it is better to throw them away, than to try eating them.
Here is a mathematical model for this puzzle.
You have apples A1, A2, ..., A10. There is a nonnegative function v(X,t), measuring the nutrition contents of the apple X at time t. Since the nutrition content of the apples decay at the same rate, v(X,t)=v(X)-D(t), where D(t) is the amount of lost nutrition during t days, and v(X):=v(X,0) is just the nutrition content of your apples at the beginning.
Let's start with two apples so that to see what is going on. The apples worth v(A1), and v(A2). On day
t this value decays by
D(t), so the next day your apples in the fridge would worth v(A1)-D(1), v(A2)-D(1). After two days, the total value you consumed is v(A1)+(v(A2)-D(1)) or v(A2)+(v(A1)-D(1)), so v(A1)+v(A2)-D(1), irrespectively of what order you have eaten the apples. The caveat is that if at some point v(Ai)-D(1)<0, your apple expired worthless (a fancy term in quantitative analysis), at which point eating it would decrease the total amount of enjoyment, so you are better off just throwing it away.
Here are two examples, to compare: v(A1)=4, v(A2)=5, v(A3)=6, D(1)=1,D(2)=2. Each day the apples rot by decrement value of 1, so in three days you eat v(A1)+v(A2)+v(A3)-D(1)-D(2)=4+5+6-1-2=12 worth of nutritions.
On the other hand, imagine that v(A1)=4, v(A2)=5, v(A3)=6, D(1)=7. This means that on day 1 if you eat the oldest apple, you gain nutrition of 4, but the next day the two other apples will have nutrition value v(A2)=-2 and v(A3)=-1, so it is better to eat the best apple on day one, gaining a total of 6 nutrition value.
Here is how you solve the general problem: If you have just a few apples, you can try all
n! permutations of the order in which you might eat them. Otherwise you look up how to solve the assignment problem which asks for a maximal weighted matching in a bipartite graph.