$title Transportation Model as Equilibrium Problem (TRANSMCP,SEQ=126)
*   Dantzig's original transportation model (TRNSPORT) is
*   reformulated as a linear complementarity problem.  We first
*   solve the model with fixed demand and supply quantities, and
*   then we incorporate price-responsiveness on both sides of the
*   market.
*
*  References: Dantzig, G B., Linear Programming and Extensions
*              Princeton University Press, Princeton, New Jersey, 1963,
*              Chapter 3-3.



  SETS
       I   canning plants   / SEATTLE, SAN-DIEGO /
       J   markets          / NEW-YORK, CHICAGO, TOPEKA / ;

  PARAMETERS

       A(I)  capacity of plant i in cases (when prices are unity)
         /    SEATTLE     325
              SAN-DIEGO   575  /,

       B(J)  demand at market j in cases (when prices equal unity)
         /    NEW-YORK    325
              CHICAGO     300
              TOPEKA      275  /,

        ESUB(J)  Price elasticity of demand (at prices equal to unity)
         /    NEW-YORK    1.5
              CHICAGO     1.2
              TOPEKA      2.0  /;

  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 ;

  PARAMETER PBAR(J) Reference price at demand node J;

  POSITIVE
  VARIABLES
       W(I)             shadow price at supply node i,
       P(J)             shadow price at demand node j,
       X(I,J)           shipment quantities in cases;

  EQUATIONS
       SUPPLY(I)        supply limit at plant i,
       FXDEMAND(J)      fixed demand at market j,
       PRDEMAND(J)      price-responsive demand at market j,
       PROFIT(I,J)      zero profit conditions;

PROFIT(I,J)..   W(I) + C(I,J)   =G= P(J);

SUPPLY(I)..     A(I) =G= SUM(J, X(I,J));

FXDEMAND(J)..   SUM(I, X(I,J))  =G=  B(J);

PRDEMAND(J)..   SUM(I, X(I,J))  =G=  B(J) * (PBAR(J)/P(J))**ESUB(J);

*       Declare models including specification of equation-variable
*       association:

  MODEL FIXEDQTY / PROFIT.X, SUPPLY.W, FXDEMAND.P/ ;
  MODEL EQUILQTY / PROFIT.X, SUPPLY.W, PRDEMAND.P/ ;

*       Initial estimate:

  P.L(J) = 1;
  W.L(I) = 1;

  PARAMETER REPORT(*,*,*)  Summary report;

  SOLVE FIXEDQTY USING MCP;
  REPORT("FIXED",I,J) = X.L(I,J);
  REPORT("FIXED","Price",J) = P.L(J);
  REPORT("FIXED",I,"Price") = W.L(I);

*       Calibrate the demand functions:

  PBAR(J) = P.L(J);

*       Replicate the fixed demand equilibrium:

  SOLVE EQUILQTY USING MCP;
  REPORT("EQUIL",I,J) = X.L(I,J);
  REPORT("EQUIL","Price",J) = P.L(J);
  REPORT("EQUIL",I,"Price") = W.L(I);

*       Compute a counter-factual equilibrium:

  C("SEATTLE","CHICAGO") = 0.5 * C("SEATTLE","CHICAGO");

  SOLVE FIXEDQTY USING MCP;
  REPORT("FIXED",I,J) = X.L(I,J);
  REPORT("FIXED","Price",J) = P.L(J);
  REPORT("FIXED",I,"Price") = W.L(I);

*       Replicate the fixed demand equilibrium:

  SOLVE EQUILQTY USING MCP;
  REPORT("EQUIL",I,J) = X.L(I,J);
  REPORT("EQUIL","Price",J) = P.L(J);
  REPORT("EQUIL",I,"Price") = W.L(I);


  DISPLAY REPORT;