Update schedule_one_machine.md

vxw77 · web-flow · commit 5b54bf6800da · 2021-09-26T16:40:02.000-07:00
It should be non-descending as the optimal schedule. Consider the example of two tasks, one very long and another one very short, the penalty incurred will be larger if the short one starts after the long one.
diff --git a/src/schedules/schedule_one_machine.md b/src/schedules/schedule_one_machine.md
@@ -27,7 +27,7 @@ $$\begin{align}
 \end{align}$$
 
 It is easy to see, that if the schedule $\pi$ is optimal, than any change in it leads to an increased penalty (or to the identical penalty), therefore for the optimal schedule we can write down the following condition:
-$$c \cdot t_{\pi_{i+1}} - c_{\pi_{i+1}} \cdot t_{\pi_i} \ge 0 \quad \forall i = 1 \dots n-1$$
+$$c_{\pi_{i}} \cdot t_{\pi_{i+1}} - c_{\pi_{i+1}} \cdot t_{\pi_i} \ge 0 \quad \forall i = 1 \dots n-1$$
 And after rearranging we get:
 $$\frac{c_{\pi_i}}{t_{\pi_i}} \ge \frac{c_{\pi_{i+1}}}{t_{\pi_{i+1}}} \quad \forall i = 1 \dots n-1$$
 
@@ -49,7 +49,7 @@ $$v_i = \frac{1 - e^{\alpha \cdot t_i}}{c_i}$$
 
 In this case we consider the case that all $f_i(t)$ are equal, and this function is monotone increasing.
 
-It is obvious that in this case the optimal permutation is to arrange the jobs by non-ascending processing time $t_i$.
+It is obvious that in this case the optimal permutation is to arrange the jobs by non-descending processing time $t_i$.
 
 ## The Livshits-Kladov theorem
 
