$Title Transportation model with variable demand function using bilevel programming (TRANSBP,SEQ=26)
$Ontext
Dantzig's original transportation model TRNSPORT (in GAMS Model Library) is
reformulated using EMP's bilevel programming.
Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions.
Princeton University Press, Princeton, New Jersey, 1963.
Additional features:
The fixed demand b(j) is replaced by a function:
g(j) = max(b(j),y(j))
with an outer objective function that tries to force y to be close to a
target value (400). The max function is modeled using Variational Inequalties.
The original trnsport model is the lower level problem.
Contributor: Michael Ferris, December 2009
$Offtext
Parameter report(*,*,*) summary report;
Sets
i canning plants / seattle, san-diego /
j markets / new-york, chicago, topeka / ;
Parameters
a(i) capacity of plant i in cases
/ seattle 350
san-diego 600 /
b(j) demand at market j in cases
/ new-york 325
chicago 300
topeka 275 / ;
Table d(i,j) distance in thousands of miles
new-york chicago topeka
seattle 2.5 1.7 1.8
san-diego 2.5 1.8 1.4 ;
Scalar f freight in dollars per case per thousand miles /90/ ;
Parameter c(i,j) transport cost in thousands of dollars per case ;
c(i,j) = f * d(i,j) / 1000 ;
Variables
x(i,j) shipment quantities in cases
z total transportation costs in thousands of dollars
g(j) generated demand;
Positive Variable x ;
Equations
cost define objective function
supply(i) observe supply limit at plant i
demand(j) satisfy demand at market j ;
cost .. z =e= sum((i,j), c(i,j)*x(i,j)) ;
supply(i) .. sum(j, x(i,j)) =l= a(i) ;
* --- Note: the fixed the demand b(j) was replaced by a variable demand g(j)
demand(j) .. sum(i, x(i,j)) =g= g(j) ;
Model transport /all/ ;
* --- Now we solve the original fixed price trnsport model
g.fx(j) = b(j);
Solve transport using lp minimizing z ;
report(i,j,'fixed') = x.l(i,j);
Variables
obj,
y(j) 'external demand function';
Equation
outerobj,
extdem(j);
outerobj.. obj =e= sum(j, sqr((y(j) - 400)/b(j)));
extdem(j).. g(j) =g= y(j);
model emp bilevel trnsport model / all /;
g.up(j) = inf; g.lo(j) = b(j);
* use the EMP info file to define the model
file fx / '%emp.info%' /;
put fx 'bilevel' /;
put ' min z x cost supply demand' /;
putclose ' vi extdem g' /;
* --- Now we solve the bilevel trnsport model using EMP
solve emp us emp min obj;
report(i,j,"bilevel") = x.l(i,j);
Display report;