transvi.gms : VI version of the transport model

**Description**

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

**Small Model of Type : ** VI

**Category :** GAMS EMP library

**Main file :** transvi.gms

```
$title VI version of the transport model (TRANSVI,SEQ=2)
$ontext
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
$offtext
Sets
i canning plants / seattle, san-diego /
j markets / new-york, chicago, topeka / ;
Parameters
a(i) capacity of plant i in cases
/ seattle 350
san-diego 600 /
b(j) demand at market j in cases
/ new-york 325
chicago 300
topeka 275 / ;
Table d(i,j) distance in thousands of miles
new-york chicago topeka
seattle 2.2 1.7 1.8
san-diego 2.5 1.8 1.4 ;
Scalar f freight in dollars per case per thousand miles /90/ ;
Parameter c(i,j) transport cost in thousands of dollars per case ;
c(i,j) = f * d(i,j) / 1000 ;
Variables
x(i,j) shipment quantities in cases
z total transportation costs in thousands of dollars ;
Positive Variable x ;
Equations
cost define objective function
supply(i) observe supply limit at plant i
demand(j) satisfy demand at market j ;
cost .. z =e= sum((i,j), c(i,j)*x(i,j)) ;
supply(i) .. sum(j, x(i,j)) =l= a(i) ;
demand(j) .. sum(i, x(i,j)) =g= b(j) ;
Model transport /all/ ;
* Solve the LP
option lp = pathnlp;
Solve transport using lp minimizing z ;
abort$[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$[mcpTransport.objVal > 1e-6] 'Input for model mcpTransport should be optimal, was not';
*-------------------------------------------------------------------------------
* Now use EMP to this reformulation
* F(x) = c for our VI: LP models yield a linear VI
equations
grad(i,j) 'dcost/dx(i,j)';
grad(i,j).. c(i,j) =N= 0;
model viTransport / grad, supply, demand /;
file myinfo / '%emp.info%' /;
put myinfo '* complementarity pairs for grad.x' /;
putclose 'vi grad x';
*Verify the solution we already have.
viTransport.iterlim = 0;
solve viTransport using emp;
abort$[viTransport.objVal > 1e-6] 'Input for model viTransport should be optimal, was not';
```