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Logic and Logistics : Crossing a Desert and Supply Chain

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This is a workbook example called crossing a desert.

Suppose you and your truck are all set to leave the town of Ucnestahn on the edge of Ucnahara desert. you want to travel across the desert as quickly as possible. You have calculated that all the provisions you need for one day's travel (food, fuel, lager, chewing gum, 'war and peace DVD etc) can be contained in 1 Bigbox and the truck can carry exactly 10 Bigboxes. there are no provisions available in the desert, but there is a limitless supply at Ucnestahn.

How do you plan your journey, and how many days will it take you, in each of the following situations?
[you may assume that Bigboxes are perfectly secure and so they may be dumped at any point and recovered later with no loss of usefulness. Time taken to load and unload is negligible.]

Situation 1: you know the distance across the desert is exactly 10 days travelling.
Situation 1: you know that the distance across the desert is exactly 12 days travelling
Situation 3: you know that the distance across the desert is exactly 20 days travelling
Situation 4: you know the distance across the desert is exactly 100 days travelling

What difference would it make to your answers if you can leave part Bigboxes to pick up later? What should your strategy be if you did not know the size of the desert?

NB there are several ways of approaching this problem, formulating terms to limit, etc. answer can vary accordingly, it is the approach, correct application of mathematical logic etc which is important.

I need the help with the approach to this answer thank you!

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This is a logic problem. There are no simple rules or exact formulas; you have to play it out in your mind or on paper and see what comes out.
First ignore all the details like the name of the town and the idea of Bigboxes; these are distractors.
The essential fact of this situation is that you can take ten days of supplies with you. On Day 11 you would be out of gas and stuck with no water, so best to plan ahead.
If the trip is ten days or less (your Situation 1), no problem; you just pack ten days of supplies and go.
If the trip is more than ten days, you have to go part way, leave a stack of supplies, come back, reload, and go a bit farther next time, picking up your stack on the way. The question is what is the most efficient way to do this.
For the twelve-day trip (Situation 2), you need to add on two extra days. The quickest way I see to do this is to pack the truck with six days of supplies, drive two days out, leave two in the desert, and drive two days back. Then you reload with ten days, drive two days out, refresh with the stack you left, and go on to the end.
The twenty-day trip (Situation 3) is going to be more complicated, because you can't leave a ten-day-stack (you need some supplies to get back with); you're going to have to do it in multiple stages. The first thought that comes to me is that you could drive five days out and five days back, but on reflection I realize that is not too good because there would be nothing to leave for the next stage. So I start to think out various patterns: one day out, leave eight, one day back; two days out, leave six, two days back; three days out, leave four, three days back; four days out, leave two, four days back. It's always 2a + b = 10, where a is the distance from your supply depot and b is the amount you leave for the next time. This means you could go out the second time and reload up to ten days. Still not quite enogh to make twenty; you've now got up to thirteen ...

Solution Summary

A logic and logistics problem is solved. The solution is detailed and well presented.

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