$TITLE  BOTTLENECK PROBLEM
$OFFUPPER
* This problem finds a least cost shippingmum weighted sum of
* productive capacities at the end of a 5 year plan.
*
*  References: 
*        The problem is taken from the book on Linear Programming
*        in Infinite Dimensional Spaces. The problem is originally
*        from a paper of Bellman and is the first instance of
*        an infinite dimensional LP.

  SETS
       I   different good produced   / G1*G5 /
       J   different plants   / P1*P3 /
       T   time periods  / 1989*1994 / ;

  TABLE B(I,J)  fraction Gi invested for additional capacity Pj
         P1      P2     P3
G1      .15     .13     .23
G2      .09     .23     .13
G3      .10     .11     .05
G4      .08     .08     .09
G5      .10     .04     .09 ;

  TABLE D(I,J)  rate of production Gi per unit of Pj
        P1      P2      P3
G1      1.2     1.3     1.4
G2      .9      1.1     .8
G3      1.1     1.5     1.9
G4      1.9     .2      .7
G5      .6      .4      .9 ;

VARIABLES
  X(J,T)   rate of production of new capacity of type j at time t
  Z(J,T)   total productive capacity of type j at time t
  W        total productive capacity for 5 year plan ;

POSITIVE VARIABLE X ;

*PARAMETER BX(I,T)  total amount of goods I consumed for new capacity;
*    BX(I,T)= SUM(J,B(I,J)*X(J,T)) ;
*PARAMETER DZ(I,T)  total rate of production of goods available at t;
*    DZ(I,T)= SUM(J,D(I,J)*Z(J,T)) ;
*PARAMETER DX(I,T)  rate of production of goods at time t ;
*    DX(I,T)= SUM(J,D(I,J)*X(J,T)) ;
PARAMETER
  CO(J)  initial production capacities
   /P1  350
    P2  700
    P3  250 /
  A(J)   weights for objective function
   /P1  2
    P2  5
    P3  3 / ;
PARAMETER C(I,T)   current production capacities;
  C(I,T) = SUM(J,D(I,J)*CO(J)) ;

EQUATIONS
  COST    weighted sum of production capacities at end period
  CAPACITY(I,T)  restriction on capacities ;

COST ..   W =E= SUM(T,SUM(J,A(J)*X(J,T)));

CAPACITY(I,T) ..  SUM(J,B(I,J)*X(J,T))-SUM(J,D(I,J)*X(J,T)) =L= C(I,T) ;


  MODEL BOTTLENECK /ALL/ ;

  SOLVE BOTTLENECK USING LP MAXIMIZING W ;

  DISPLAY X.L, X.M ;