I am going to buy a family car at the beginning of the New Year. I am going to stay in the UK for the next 4 years. I am considering the possibility of being a customer of company A which sells BMW models. Not only I can buy a car from this company but I can also trade in my car at this company and have my car serviced by the company. Company A gives a warranty that the annual maintenance costs and trade-in prices will remain the same for the next 4 years.

They are: \begin{array}{cc}\hline\text{Age of Car}&\text{Annual Maintenance}&\text{Trade-in Price at the}\\&\text{Cost}&\text{end of the period}\\\hline0&{\it\unicode{xA3}}2,000&{\it\unicode{xA3}}6,000\\1&{\it\unicode{xA3}}4,000&{\it\unicode{xA3}}5,000\\2&{\it\unicode{xA3}}6,000&{\it\unicode{xA3}}4,000\\3&{\it\unicode{xA3}}7,000&{\it\unicode{xA3}}2,000\\\hline\end{array}

To avoid maintenance costs associated with an older car, I may trade in my car and purchase a new car. My wife prefers a Rover to a BMW. Company B which sells Rover models proposes similar conditions with figures in the table below:

\begin{array}{cc}\hline\text{Age of Car}&\text{Annual Maintenance}&\text{Trade-in Price at the}\\&\text{Cost}&\text{end of the period}\\\hline0&{\it\unicode{xA3}}2,000&{\it\unicode{xA3}}8,000\\1&{\it\unicode{xA3}}3,000&{\it\unicode{xA3}}7,000\\2&{\it\unicode{xA3}}7,000&{\it\unicode{xA3}}5,000\\3&{\it\unicode{xA3}}8,000&{\it\unicode{xA3}}3,000\\\hline\end{array}

I am trying to find the minimal net cost for these 4 years which he would be using the car, here is how I went with the solution:

So this would be the cost table for the BMW and Rover respectively: \begin{align}\text{BMW}\hspace{4cm}\text{Rover}\hspace{2cm}\\\begin{array}{|c|c|}\hline c_{i,j}&1&2&3&4&5\\\hline1&&8&13&20&29\\\hline2&&&7&12&19\\\hline3&&&&6&11\\\hline4&&&&&4\\\hline5&&&&&\\\hline\end{array}\quad\begin{array}{|c|c|}\hline c_{i,j}&1&2&3&4&5\\\hline1&&10&14&23&32\\\hline2&&&8&12&30\\\hline3&&&&5&9\\\hline4&&&&&4\\\hline5&&&&&\\\hline\end{array}\end{align}

By my sketch in my notebook, I see that if we got from $1$ to $3$ with BMW and then from $3$ to $5$ with Rover, then this would minimize the cost. Is this correct?

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    $\begingroup$ Difficult decision between BMW and Rover: first world problem. $\endgroup$ – Mark L. Stone Jan 7 '20 at 13:39
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    $\begingroup$ @MarkL.Stone, well, i'm poor, so best would be for me is to optimize $\endgroup$ – The Poor Jew Jan 7 '20 at 13:49
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    $\begingroup$ Isn't the purchase price a cost? I don't see it listed. $\endgroup$ – Richard Jan 8 '20 at 6:16
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    $\begingroup$ the cost is 12 000, @Luke599999 $\endgroup$ – The Poor Jew Jan 10 '20 at 13:27
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    $\begingroup$ I understand (and agree) with the costs in the top row of the BMW table. I don't agree with the costs in the top row of the Rover table. Shouldn't they be 6, 10, 19, 29? I also don't understand the costs in the other rows. I think they should be the same as the top row, just shifted across by one cell each time... Please explain how you got the values - maybe I'm missing something. $\endgroup$ – tomi Jan 12 '20 at 14:13

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