$title Traffic Assignment Model (TRAFFIC,SEQ=3)

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A simple traffic assignment problem:
 - Move 5 units of stuff from A to B as quickly as possible, using one or more
   of the N paths p1..pN between A and B.
 - The delay d on each path is a function of the volume of traffic on
   the path:
      d(p) = a(p) + x(p) + x(p)**1.5

This is a simple example showing how VI can be used to compute
*effective delay*, i.e. the maximum delay over all paths with nonzero
flow, or the time it takes for all the stuff to reach the
destination.  For an example involving a nontrivial network see the
traffic2 model in this library.

Contributor: Steven Dirkse, January 2009

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set
   p 'paths from A to B' / p1 * p4 /;
scalar
   b 'volume of stuff to move' / 5 /;
parameter
   a(p);

execseed = 1;
a(p)     = uniform(0,100);

positive variable
  x(p) 'path flows';
equation
  d(p) 'delay', demand;

d(p)..  a(p) + x(p) + x(p)**1.5 =N= 0;
demand..   sum{p, x(p)} =G= b;

model vi / d, demand /;

* define the complementary pairs for vi
file myinfo / '%emp.info%' /;
putclose myinfo 'vi d x demand';

solve vi using emp;
abort$[vi.modelstat <> %modelStat.locallyOptimal%] 'bad modelstat for vi';
abort$[vi.solvestat <> %solveStat.normalCompletion%] 'bad solvestat for vi';

* for the report labels, ed = effective delay
parameters
  report(*)
  report2(p,*);

report2(p,'flow') = x.l(p);
report2(p,'delay') = d.l(p);

report('ed_marg') = demand.m;
report('ed_calc') = smax(p$[x.l(p) > 1e-8], report2(p,'delay'));
report('ed_diff') = abs(report('ed_calc')-report('ed_marg'));

display report, report2;

abort$[report('ed_diff') > 1e-6] 'bad ed';