MODEL:
! JobShop/PERT/CPM project scheduling with resource constraints;
! A jobset or project consists of a collection of activities or operations. 
   Each activity is described by:
!   a) duration, 
    b) set of predecessor activities, 
    c) set of resources or machines required.
! There is a limited number of each resource/machine.
! An activity or operation cannot be started until: 
   1) all its predecessors have completed, and 
!  2) resources/machines required are available.;
! A jobshop problem is a special case in which the precedence
   relations are only between operations within each job;
! Reference:
     Pritzker, A., L. Watters, and P. Wolfe(1969),"Multiproject Scheduling with 
  Limited Resources: a Zero-One Programming Approach", Management Science,
  vol. 16, no. 1, Sept., pp. 93-108.;
! Keywords: Job shop scheduling, Scheduling, Resource constraints, Fisher-Thompson;
SETS:
! There is a set of tasks with a given duration, and
    a start time to be determined;
   TASK: TIME, ES;
! The precedence relations, the first task in the
    precedence relationship needs to be completed before the
    second task can be started;
   PRED( TASK, TASK);
! There are a set of periods;
   PERIOD;
   RESOURCE: CAP;
! Some operations need capacity in some department;
   TXR( TASK, RESOURCE): NEED;
! SX( I, T) = 1 if task I starts in period T;
   TXP( TASK, PERIOD): SX, B;       
  ENDSETS
 
 DATA: 
! This is the six job, six machine, six operations per job scheduling problem
    of Fisher and Thompson. Ref.
    Muth, J. and G. Thompson(1963) Industrial Scheduling, Prentice-Hall, Englewood Cliffs, NJ.;

 ! Upper limit on number of periods required to complete the project;
   PERIOD = 1..60;
 ! Task names and duration, the Fisher-Thompson problem;
   TASK  TIME = 
    j11   1    j12   3    j13   6    j14   7    j15   3    j16   6
    j21   8    j22   5    j23  10    j24  10    j25  10    j26   4
    j31   5    j32   4    j33   8    j34   9    j35   1    j36   7
    j41   5    j42   5    j43   5    j44   3    j45   8    j46   9
    j51   9    j52   3    j53   5    j54   4    j55   3    j56   1
    j61   3    j62   3    j63   9    j64  10    j65   4    j66   1
    j00   0; 

! The machines,
   with capacities in each unit of time...;
   RESOURCE = M1 M2 M3 M4 M5 M6; 
        CAP =  1, 1, 1, 1, 1, 1;
! How much each task needs of each resource;
          TXR,       NEED = 
    j11 M3  1    j12 M1  1    j13 M2  1    j14 M4  1    j15 M6  1    j16 M5  1
    j21 M2  1    j22 M3  1    j23 M5  1    j24 M6  1    j25 M1  1    j26 M4  1
    j31 M3  1    j32 M4  1    j33 M6  1    j34 M1  1    j35 M2  1    j36 M5  1
    j41 M2  1    j42 M1  1    j43 M3  1    j44 M4  1    j45 M5  1    j46 M6  1
    j51 M3  1    j52 M2  1    j53 M5  1    j54 M6  1    j55 M1  1    j56 M4  1
    j61 M2  1    j62 M4  1    j63 M6  1    j64 M1  1    j65 M5  1    j66 M3  1; 
! The predecessor/successor pairs;
  PRED=  j11 j12 j12 j13 j13 j14 j14 j15 j15 j16 j16 j00  ! Job 1;
         j21 j22 j22 j23 j23 j24 j24 j25 j25 j26 j26 j00  ! Job 2;
         j31 j32 j32 j33 j33 j34 j34 j35 j35 j36 j36 j00
         j41 j42 j42 j43 j43 j44 j44 j45 j45 j46 j46 j00
         j51 j52 j52 j53 j53 j54 j54 j55 j55 j56 j56 j00
         j61 j62 j62 j63 j63 j64 j64 j65 j65 j66 j66 j00; ! Job 6;
 ENDDATA
!----------------------------------------------------------;
! Variables:
    SX(i,t) = 1 if task i starts in period t,
    B(i,t) = 1 if task i starts in period t or earlier,
  Computed parameters:
    ES(i) = earliest start of task i;

NL = @SIZE( TASK);
! Minimize start time of last (typically dummy) task;
 MIN = @SUM( PERIOD( T): T * SX( NL, T));

 @FOR( TASK( I):
!  Each task I must be started in some period T;
   [DO]  @SUM( PERIOD( T): SX( I, T)) = 1;
! The B vars are binary, i.e., 0 or 1;
   @FOR( PERIOD( T): @BIN( B( I, T));
        );
! B(I,T) = 1 if task I starts in T or earlier;
    B(I,1) = SX(I,1);
    @FOR( PERIOD(T)|T#GT#1:
       B(I,T) = B(I,T-1) + SX(I,T); 
        )
      );

! Precedence constraints;
  @FOR( PRED( I, J):
    @FOR( PERIOD(T)| T #LE# TIME(I): B(J,T)= 0;);
    @FOR( PERIOD(T)| T #GT# TIME(I):
    [PCD] B(J,T) <= B(I,T-TIME(I));
        )
    !  START( J) >= START( I) + TIME( I);
      );

! Resource usage, For each resource R and period T;
  @FOR( RESOURCE( R):
    @FOR( PERIOD( T):
! Sum over all tasks I that use resource R in period T;
    [RS] @SUM( TXR( I, R):
       @SUM( PERIOD( S)| S #GE# ( T - ( TIME( I) - 1)) #AND# S #LE# T:
               NEED( I, R) * SX( I, S))) <= CAP( R);
        )
      );

! The following makes the formulation tighter;
! Compute earliest start disregarding resource constraints;
   @FOR( TASK( J):
     ES( J) = @SMAX( 0, @MAX( PRED( I, J): ES( I) + TIME(I)));
   ! Task cannot start earlier than unconstrained early start;
     @SUM( PERIOD(T) | T #LE# ES( J): SX( J, T)) = 0;
       );
 END