$title VI version of the transport model (TRANSVI,SEQ=2)

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Example showing how to write a VI as an MCP

We want to write the VI using EMP to avoid manual translation to MCP. We use
the definitions of VI and MCP from

Steven P. Dirkse,  Ph.D. Dissertation
    Robust Solution of Mixed Complementarity Problems.
    Mathematical Programming Technical Report 94-12, August 1994.
    ftp://ftp.cs.wisc.edu/math-prog/tech-reports/94-12.ps
    Pages 4-6

In this case, the VI to start with is what we get by letting F(x) = df/dx,
where f is the objective in the transport model.

We adjusted the data to get nonzero supply marginals.

Contributor: Steven Dirkse and Jan-H. Jagla, January 2009

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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          2.2           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/ ;

* Solve the LP
option lp = pathnlp;
Solve transport using lp minimizing z ;
abort$[transport.modelstat <> %modelStat.optimal%] 'LP not solved';

*-------------------------------------------------------------------------------
* That's how the VI looks like

positive variable
     dPrice(j)   'demand price';
negative variable
     sPrice(i)   'supply price*(-1)';
equations
     ggrad(i,j)  'MCP version of grad from VI';

ggrad(i,j)..  c(i,j) - sPrice(i) - dPrice(j) =N= 0;

model mcpTransport / ggrad.x, supply.sPrice, demand.dPrice /;

*Adopt solution from LP solve and verify it is a solution of the MCP
sPrice.l(i) = supply.m(i);
dPrice.l(j) = demand.m(j);
mcpTransport.iterlim = 0;
solve mcpTransport using mcp;
abort$[mcpTransport.objVal > 1e-6] 'Input for model mcpTransport should be optimal, was not';

*-------------------------------------------------------------------------------
* Now use EMP to this reformulation

* F(x) = c for our VI: LP models yield a linear VI
equations
  grad(i,j) 'dcost/dx(i,j)';

grad(i,j)..  c(i,j) =N= 0;

model viTransport / grad, supply, demand /;

file myinfo / '%emp.info%' /;
put myinfo '* complementarity pairs for grad.x' /;
putclose   'vi grad x demand supply';

*Verify the solution we already have.
viTransport.iterlim = 0;
solve viTransport using emp;
abort$[viTransport.objVal > 1e-6] 'Input for model viTransport should be optimal, was not';