Case Study: Arc Routing
The Bridges of Konigsberg
1) Was there ever a date at which an Eulerian walk over the steel bridges of Konigsberg were possible, without using any wooden bridges? If so, when? If not, show that it was never possible. For a Eulerian walk to be possible two conditions must hold
a) The undirected graph must be connected
b) Each node must be of even degree.
Name of the bridge| Date built|
Honey Bridge (7)| 1879-1882|
Slaughter Bridge (3)| 1886|
Blacksmith’s Bridge (4)| 1896|
Salesman Bridge (1)| 1900|
Timber Bridge (5)| 1904|
Emperor’s bridge (8)| 1905|
Green Bridge (2)| 1907|
High Bridge (6)| 1937|
At no point in time do the above two conditions hold, therefore it is never possible to do an Eulerian walk in the steel bridges of Konigsberg.
2) Consider the current situation in Kaliningrad (with the Estacada connecting Altstadt and Vorstadt, but not Kant Island, and with bridge 10 not connected yet). What bridge(s) would you construct to make an Eulerian walk possible? All the nodes are connected as of now. We have to add one more bridge between nodes A and L to make all the nodes have even degree. So, add a bridge between Altstadt and Kneiphof.
Roadway Snow and Ice Control
1) What factors contribute to the complexity of roadway snow and ice control problems? There were tremendous variations in the RSIC operations themselves due to differences in climate, level of service, network complexity and size. a) Weather conditions (temperature, wind precipitation type and amount , accumulation rate, total accumulation etc.). b) Traffic conditions were also dynamic and had a strong effect on the snow removal operations. c) Types of roadways encountered – different mixes of roads – one lane roads, two lane roads, multi lane roads, alleys, gravel roads etc. d) Policy decisions and operational decisions...