An optimisation problem for a transport engineer

During 2019, Tina and I had two holidays cycling in France.  (One was a circular ride around the Loire valley, the second was from Paris to Le Mont St Michel.)   It was "lazy" cycling, around 60-70 kilometres per day and our luggage was carried between hotels.  There was time for sightseeing and leisurely days - though we encountered gales and heavy rain on a few days, so on those days, we weren't tempted to linger over our open-air picnics. 

Some of our rides were along former railway tracks, which are pleasant because they are smooth and don't have steep gradients.  One day I noticed that our gentle ascent through a cutting would have posed an interesting problem for the engineer - and I am sure that it was one met numerous times in the heyday of railway construction in the 19th century. 

Suppose that the railway has to ascend or descend a ridge.  The line could be engineered to run over the summit of the ridge with cuttings and embankments to create a steady gradient, or it could be engineered to cross the summit of the ridge through a cutting.  The depth of the cutting (D) can be a variable, from 0 metres (summit - no cutting) to some maximum.  For any value of D, the engineer can make an estimate of the cost of construction - the cuttings and embankments that need to be included.  There will be constraints - especially the maximum gradient of the trackbed.  So, for this simple model, the cost will be a function of D, which can be optimised.

This is simplistic, of course.  Not every ridge is going to be as simple as this.  A railway crosses other obstacles, but the engineer's problem is still one of optimisation - how to create a route which has (perhaps almost) minimal cost - with cuttings, bridges, embankments and tunnels, and acceptable gradients for the trains to negotiate.  Famously, Isambard Brunel designed the line from London to Bristol with no gradient steeper than 1 in 50.  The problems for those crossing the Alps, Rockies and Andes were more serious. 



The same design problems are encountered for road-building, and canal-building.  The Tiverton Canal (Grand Western Canal) follows a contour around a valley rather than cross on an embankment.

The Grand Western Canal  
The canal stretch between Dudley Weatherley Jubilee Bridge and Greenway Bridge is known as the "Swan's Neck" and is about twice as long as the straight line distance.  But much cheaper to build!

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