feasopt1.gms : An Infeasible Transportation Problem analyzed with Cplex option FeasOpt

**Description**

This problem finds a least cost shipping schedule that meets requirements at markets and supplies at factories where demand exceeds supply using the feature FeasOpt implemented by Cplex and Gurobi.

**Reference**

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

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** feasopt1.gms

$Title An Infeasible Transportation Problem analyzed with option FeasOpt (FEASOPT1,SEQ=314) $Ontext This problem finds a least cost shipping schedule that meets requirements at markets and supplies at factories where demand exceeds supply using the feature FeasOpt implemented by Cplex and Gurobi. Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963. $Offtext $ifi %system.lp% == cplex $goto cont $ifi %system.lp% == gurobi $goto cont $exit $label cont 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 ; 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) ; demand(j) .. sum(i, x(i,j)) =g= b(j) ; Model transport /all/ ; option limrow=0, limcol=0; * Increase demand by 20% b(j) = 1.2*b(j); Solve transport using lp minimizing z ; display 'The first phase of the Simplex algorithm distributed the infeasibilities as follows', x.infeas, supply.infeas, demand.infeas; $ifi %system.lp% == cplex file fslv Solver Option file / cplex.opt /; transport.optfile=1; $ifi %system.lp% == gurobi file fslv Solver Option file / gurobi.opt /; transport.optfile=1; * Lets try to move the infeasibilities on the demand side putclose fslv / 'feasopt 1' / 'equation.feaspref 0' / 'demand.feaspref 1'; Solve transport using lp minimizing z ; display 'All infeasibilities should be in the demand equations', x.infeas, supply.infeas, demand.infeas; abort$(sum((i,j), x.infeas(i,j)) + sum(i,supply.infeas(i))) x.infeas, supply.infeas, demand.infeas; abort$(sum(j,demand.infeas(j))<1e-5) x.infeas, supply.infeas, demand.infeas; * Lets try to distribute the infeasibilities on the demand side by * using the sum of squares for the relaxation measurement putclose fslv / 'feasopt 1' / 'feasoptmode 4' / 'equation.feaspref 0' / 'demand.feaspref 1'; Solve transport using lp minimizing z ; display 'All infeasibilities should be in the demand equations and nicely distributed', x.infeas, supply.infeas, demand.infeas; abort$(sum((i,j), x.infeas(i,j)) + sum(i,supply.infeas(i))) x.infeas, supply.infeas, demand.infeas; abort$(sum(j,demand.infeas(j))<1e-5) x.infeas, supply.infeas, demand.infeas; * Lets try to distribute the infeasibilities on the demand and supply * side by using the sum of squares for the relaxation measurement putclose fslv / 'feasopt 1' / 'feasoptmode 4'; Solve transport using lp minimizing z ; display 'All infeasibilities should be in the demand and supply equations and nicely distributed', x.infeas, supply.infeas, demand.infeas; abort$(sum((i,j), x.infeas(i,j))) x.infeas, supply.infeas, demand.infeas; abort$(sum(i,supply.infeas(i))+sum(j,demand.infeas(j))<1e-5) x.infeas, supply.infeas, demand.infeas; * Lets try to distribute the infeasibilities on the demand and supply * side by using the sum of squares for the relaxation measurement and * lets also optimize the transport shipment with respect to the * original objective function putclose fslv / 'feasopt 1' / 'feasoptmode 3'; Solve transport using lp minimizing z ; display 'All infeasibilities should be in the demand equations and nicely distributed with an "optimal" x', x.infeas, supply.infeas, demand.infeas, x.l; abort$(sum((i,j), x.infeas(i,j))) x.infeas, supply.infeas, demand.infeas; abort$(sum(i,supply.infeas(i))+sum(j,demand.infeas(j))<1e-5) x.infeas, supply.infeas, demand.infeas; * Lets adjust supply and demands based on the relaxation found a(i) = a(i) + supply.infeas(i); b(j) = b(j) - demand.infeas(j); * Now we should have a feasible model. The primals from our previous * solve should be the optimal one, so lets save them to compare them * with the outcome with the next solve; Parameter xbest(i,j); xbest(i,j) = x.l(i,j); * Lets try to tell the solver to do a warm start * from just the primals using primal Simplex $ifi %system.lp% == cplex putclose fslv / 'advind 2' / 'lpmethod 1'; $ifi %system.lp% == gurobi putclose fslv / 'usebasis 1' / 'method 0'; Solve transport using lp minimizing z ; * We better have an optimum solution and the same primals as in the * previous run. This is a little dangerous since the problem is * degenerated. abort$(transport.modelstat<>%modelstat.Optimal% or sum((i,j), xbest(i,j) - x.l(i,j))>1e-6) x.l, xbest;