\$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;