# In which order the woman should bring the cats back in order to minimize the time?

A woman watches her cats leave one by one with different speeds in different directions. She took a motorcycle with one extra seat and follows the cats and picks up one cat at a time and brings them back home. Each cat moves with constant individual speed $V_i$ and left home at time $T_i$. In which order the woman should bring the cats back in order to minimize the time?

I am trying to solve this problem but do not know how to begin.

I would suggest considering simple example with two cats first. Suppose the women is deciding which cat to go after first. Currently, both cats have already left and are $D_{1}$ and $D_{2}$ far away. Suppose the women decided to go after cat $1$ first and then after $2$. What time she needs to be done? (Assume the women's speed is $V_{0}$ and is larger than any of the cats.)

To catch up with the first cat she needs $t_{1}$ that solves $t_{1}V_{0}=D_{1}+t_{1}V_{1}$, or, equivalently, $t_{1}=\frac{D_{1}}{V_{0}-V_{1}}$. By $2t_{1}$ she is back home with the first cat at which point the second cat is $D_{2}+2t_{1}V_{2}$ far away. To catch up with the second cat she needs $t_{2}$ that solves $t_{2}V_{0}=D_{2}+2t_{1}V_{2}+t_{2}V_{2}$, or, equivalently, $t_{2}=\frac{D_{2}+2t_{1}V_{2}}{V_{0}-V_{2}}$. Hence the entire operation takes $2(t_{1}+t_{2})$.

Substituting in the model parameters, the length of the operation is $$2\frac{D_{2}(V_0-V_1)+D_{1}(V_0+V_2)}{(V_0-V_1)(V_0-V_2)}=2\frac{V_{0}(D_{1}+D_{2})+D_{1}V_{2}-D_{2}V_{1}}{(V_0-V_1)(V_0-V_2)}$$ Similarly, if the women chooses the other order, the operation takes $$2\frac{D_{1}(V_0-V_2)+D_{2}(V_0+V_1)}{(V_0-V_1)(V_0-V_2)}=2\frac{V_{0}(D_{1}+D_{2})+D_{2}V_{1}-D_{1}V_{2}}{(V_0-V_1)(V_0-V_2)}$$

Therefore, the $(1,2)$ order is optimal if $\frac{D_{1}}{V_{1}}>\frac{D_{2}}{V_{2}}$. My conjecture is that the general solution goes after the cats in the decreasing order of $\frac{D_{i}}{V_{i}}$, which, note, equals $\frac{tV_{i}}{V_{i}}=t$ at time $t$ for all the cats if all the cats left at the same time. In other words, what I am conjecturing is that if all the cats left at the same time, the order does not matter. In fact (further supportive evidence, not a proof), Mathematica thinks so as well for $5$ casts:

t1 = d1/(v0 - v1);
t2 = (d2 + 2*t1*v2)/(v0 - v2);
t3 = (d3 + 2*(t1 + t2)v3)/(v0 - v3);
t4 = (d4 + 2(t1 + t2 + t3)v4)/(v0 - v4);
t5 = (d5 + 2(t1 + t2 + t3 + t4)v5)/(v0 - v5);
Simplify[t1+t2+t3+t4+t5/.d1->tv1/.d2->tv2/.d3->tv3/.d4->tv4/.d5->tv5]