The entire PERT formulation and portions of its solution appear below.

MODEL:

SETS:

  TASKS: TIME, ES, LS, SLACK;

  PRED( TASKS, TASKS);

ENDSETS

 

DATA:

TASKS,  TIME =

  DESIGN    10

  FORECAST  14

  SURVEY     3

  DUMMY      0

  PRICE      3

  SCHEDULE   7

  COSTOUT    4

  TRAIN     10

;

PRED =

   DESIGN, FORECAST,

   DESIGN, SURVEY,

   FORECAST, DUMMY

   FORECAST, SCHEDULE,

   SURVEY, PRICE,

   SCHEDULE, COSTOUT,

   PRICE, TRAIN,

   COSTOUT, TRAIN,

   DUMMY, PRICE

;

ENDDATA

 

@FOR( TASKS( J)| J #GT# 1:

ES( J) = @MAX( PRED( I, J): ES( I) + TIME( I))

);

 

@FOR( TASKS( I)| I #LT# LTASK:

LS( I) = @MIN( PRED( I, J): ES( J) - TIME( I));

);

 

@FOR( TASKS( I): SLACK( I) = LS( I) - ES( I));

 

ES( 1) = 0;

LTASK = @SIZE( TASKS);

LS( LTASK) = ES( LTASK);

 

END

Model: PERT

Feasible solution found.

Total solver iterations:                  0

 

          Variable           Value

             LTASK        7.000000

       ES( DESIGN)       0.0000000

     ES( FORECAST)        10.00000

       ES( SURVEY)        10.00000

        ES( PRICE)        24.00000

     ES( SCHEDULE)        24.00000

      ES( COSTOUT)        31.00000

        ES( TRAIN)        35.00000

       LS( DESIGN)       0.0000000

     LS( FORECAST)        10.00000

       LS( SURVEY)        21.00000

        LS( PRICE)        32.00000

     LS( SCHEDULE)        24.00000

      LS( COSTOUT)        31.00000

        LS( TRAIN)        35.00000

    SLACK( DESIGN)       0.0000000

  SLACK( FORECAST)       0.0000000

    SLACK( SURVEY)        11.00000

     SLACK( PRICE)        8.000000

  SLACK( SCHEDULE)       0.0000000

   SLACK( COSTOUT)       0.0000000

     SLACK( TRAIN)       0.0000000

Solution to PERT

The interesting values are the slacks for the tasks. Both SURVEY and PRICE have slack in their start times of 11 weeks and 8 weeks, respectively. Their start times may be delayed by as much as these slack values without compromising the completion time of the entire project. The tasks DESIGN, FORECAST, SCHEDULE, COSTOUT, and TRAIN, on the other hand, have 0 slack times. These tasks constitute the critical path for the project and, if any of their start times are delayed, the entire project will be delayed. Management will want to pay close attention to these critical path projects to be sure they start on time and are completed within the allotted amount of time. Finally, the ES( TRAIN) value of 35 tells us the estimated time to the start of the roll out of the new Solar Widget will be 45 weeks-35 weeks to get to the start of training, plus 10 weeks to complete training.