$title Transportation model as equilibrium problem (TRANSEQL,SEQ=45)

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Dantzig's original transportation model TRNSPORT (in GAMS Model Library) is
reformulated using EMP's equilibrium system with multiple agents. It reproduces
the results of TRANSMCP (in GAMS Model Library) which uses a linear
complementarity approach.

As in TRANSMCP, we solve four models:
   1. original fixed-demand LP to get calibration data
   2. reproduce fixed-demand results with flexible-demand model
   3. counter-factual with fixed-demand model
   4. counter-factual with flexible-demand model

Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.

Contributor: Michael Ferris & Steven Dirkse, January 2010

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Parameter report(*,*,*)  summary report;

$call gamslib -q trnsport

* --- 1. now we solve the original fixed-demand trnsport model
$include trnsport
report(i,j,'fixed')       = x.l(i,j);
report(i,"price",'fixed') = supply.m(i);
report("price",j,'fixed') = demand.m(j);

* now we introduce a flexible demand function
parameters esub(j) price elasticity of demand (at prices equal to unity)
                   / new-york 1.5, chicago 1.2, topeka 2.0  /
           pbar(j) reference price at demand node j;

variable  p(j)             shadow price at demand node j;
Equations flexdemand(j)  price-responsive demand at market j
          flexprofit     profit definition ;

flexdemand(j)..   sum(i, x(i,j)) =l= b(j)*(pbar(j)/p(j))**esub(j);

flexprofit.. z =e= sum((i,j), (p(j)-c(i,j))*x(i,j));

model emp trnsport model with flexible demand / flexprofit,supply,flexdemand /;

p.lo(j) = 1e-3; option limcol=0,limrow=0;

* use the EMP info file to define the price to be equal to the
* dual of the flexible demand equation
file fx / '%emp.info%' /;
put fx     'equilibrium';
put      / 'max z x flexprofit supply flexdemand';
putclose / 'dualVar p flexdemand';

*  calibrate the demand functions:
pbar(j) = demand.m(j);

* --- 2. replicate the fixed demand equilibrium
solve emp us emp;
report(i,j,"flex")        = x.l(i,j);
report(i,"price",'flex') = supply.m(i);
report("price",j,"flex")  = p.l(j);
report("profit",'',"flex") = sum((i,j), (p.l(j)-c(i,j))*x.l(i,j));

*  prepare data for counter-factual
c("seattle","chicago") = 0.5 * c("seattle","chicago");

* --- 3. counter-factual with fixed demand
Solve transport min z us lp;
report(i,j,'fixed CF')       = x.l(i,j);
report(i,"price",'fixed CF') = supply.m(i);
report("price",j,'fixed CF') = demand.m(j);

* --- 4. counter-factual with flexible demand
Solve emp us emp;
report(i,j,"flex CF")        = x.l(i,j);
report(i,"price",'flex CF') = supply.m(i);
report("price",j,"flex CF")  = p.l(j);
report("profit",'',"flex CF") = sum((i,j), (p.l(j)-c(i,j))*x.l(i,j));

Display report;
execute_unload 'eqlReport.gdx', report;