$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.

Keywords: linear programming, transportation problem, scheduling, solver option feasOpt,
          computing minimal relaxation, infeasible problem
$offText

$ifI %system.lp% == cplex  $goTo cont
$ifI %system.lp% == gurobi $goTo cont
$ifI %system.lp% == copt   $goTo cont
$exit

$label cont
Set
   i 'canning plants' / seattle,  san-diego /
   j 'markets'        / new-york, chicago, topeka /;

Parameter
   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;

Variable
   x(i,j) 'shipment quantities in cases'
   z      'total transportation costs in thousands of dollars';

Positive Variable x;

Equation
   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;
$ifI %system.lp% == copt   File fslv 'solver option file' / copt.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$(sum((i,j), x.infeas(i,j)) + sum(i,supply.infeas(i)) > 1e-6) x.infeas, supply.infeas, demand.infeas;
abort$(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$(sum((i,j), x.infeas(i,j)) + sum(i,supply.infeas(i)) > 1e-6) x.infeas, supply.infeas, demand.infeas;
abort$(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$(sum((i,j), x.infeas(i,j)) > 1e-6) x.infeas, supply.infeas, demand.infeas;
abort$(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$(sum((i,j), x.infeas(i,j)) > 1e-6) x.infeas, supply.infeas, demand.infeas;
abort$(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 (copt: dual Simplex)
$ifI %system.lp% == cplex  putClose fslv / 'advind 2'   / 'lpmethod 1';
$ifI %system.lp% == gurobi putClose fslv / 'usebasis 1' / 'method 0';
$ifI %system.lp% == copt   putClose fslv / 'presolve 0' / 'lpmethod 1';

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;