Editing Route assignment (section)

===Equilibrium assignment===
To assign traffic to paths and links we have to have rules, and there are the well-known [[John Glen Wardrop|Wardrop equilibrium]] conditions.<ref>{{Cite conference|last=Wardrop|first=J. G.|year=1952|title=Some Theoretical Aspects of Road Traffic Research|url=https://trid.trb.org/view/120659|conference=Institution of Civil Engineers|pages=325–378|volume=1}}</ref>  The essence of these is that travelers will strive to find the shortest (least resistance) path from origin to destination, and network equilibrium occurs when no traveler can decrease travel effort by shifting to a new path. These are termed user optimal conditions, for no user will gain from changing travel paths once the system is in equilibrium.

The user optimum equilibrium can be found by solving the following [[nonlinear programming]] problem

<math>
\min \sum_a {\int_0^{v_a} {S_a \left( x \right)} } dx
</math>
						

subject to:

<math> v_a  = \sum_i {\sum_j {\sum_r {\alpha _{ij}^{ar} x_{ij}^r } } }  </math>

<math>
\sum_r {x_{ij}^r  = T_{ij} } 
</math>

<math>
v_a \geq 0,\;x_{ij}^r  \geq 0
</math>

where
<math>
x_{ij}^r 
</math>
is the number of vehicles on path ''r'' from origin ''i'' to destination ''j''.  So constraint (2) says that all travel must take place –''i = 1 ... n; j = 1 ...  n''

<math>\alpha _{ij}^{ar} </math> = 1  if link  a is on path  r  from  i  to  j ; zero otherwise.  So constraint (1) sums traffic on each link.  There is a constraint for each link on the network.  Constraint (3) assures no negative traffic.