Since $b+d>a+d$, then $b>a$. This means that $a$ does not have the greatest value. Since $c+e>b+e$, then $c>b$. This means that $b$ does not have the greatest value. Since $b+d=c$ and each of $b, c, d$ is positive, then $d<c$, which means that $d$ does not have the greatest value. Consider the last equation $a+c=b+e$ along with the fact that $a<b<c$. From this, we see that $e=c+(a-b)$. Since $a<b$, then $a-b$ is negative and so $e<c$. This means that $c$ has the greatest value.