As commercial airliner makes stops j $=1,...,$ n of its daily routine and returns to where it started it takes on fuel for the next leg. Fuel is added at stop j to assure that the plane will arrive at stop j $+1$ with at least the required safety reserve rj$+1.$ Fuel unit costs cj $($Rupees per Kg$)$ vary considerably from stop to stop, so it is sometimes economical to carry more fuel than the minimum required in order to buy less at high$-$cost stops. However, the take$-$off fuel load at any j must not exceed safety limit tj$.$ The amount of fuel at takeoff also affects the weight of the aircraft and thus its fuel consumption during flight. For each leg from stop j to j $+1,$ the fuel required can be estimated as constant $\backslash$alpha j plus a slope $\backslash$beta j times the onboard fuel at takeoff from j$.$

Formulate this fuel management problem as a linear program. Assume that stop $1$ is the successor of stop n$,$ and use non$-$zero lower bounds if needed