ages, it may still be more economical to replace the tractor after $1$ or $2$ years. The following table gives the total net discounted cost associated with purchasing a tractor $($purchase price minus trade$-$in allowance, plus running and maintenance costs$)$ at the end of year i and trading it in at the end of year $j($where year $0$ is now$).$

The problem is to determine at what times $($if any$)$ the tractor should be replaced to minimize the total cost for the tractors over $3$ years.

$($a$)$ Formulate this problem as a shortest$-$path problem.

$($b$)$ Use the algorithm described in Sec. $10\mathrm{.}3$ to solve this shortestpath problem.

C $($c$)$ Formulate and solve a spreadsheet model for this problem.