Below is our staffing model in its entirety:

MODEL:

SETS:

  DAYS: REQUIRED, START;

ENDSETS

 

DATA:

  DAYS =     MON TUE WED THU FRI SAT SUN;

  REQUIRED =  20  16  13  16  19  14  12;

ENDDATA

 

MIN = @SUM( DAYS( I): START( I));

 

@FOR( DAYS( J):

  @SUM( DAYS( I) | I #LE# 5:

     START( @WRAP( J - I + 1, 7)))

        >= REQUIRED( J)

);

END

Model: STAFFDEM

Solving the model, we get the solution report:

Global optimal solution found.

Objective value:                              22.00000

Infeasibilities:                              0.000000

Total solver iterations:                             5

 

   Variable           Value       Reduced Cost

START( MON)        8.000000           0.000000

START( TUE)        2.000000           0.000000

START( WED)        0.000000           0.000000

START( THU)        6.000000           0.000000

START( FRI)        3.000000           0.000000

START( SAT)        3.000000           0.000000

START( SUN)        0.000000          0.3333333

 

       Row     Slack or Surplus     Dual Price

         1        22.00000           -1.000000

         2        0.000000          -0.3333333

         3        0.000000            0.000000

         4        0.000000          -0.3333333

         5        0.000000            0.000000

         6        0.000000          -0.3333333

         7        0.000000          -0.3333333

         8        0.000000            0.000000

Solution to STAFFDEM

The objective value of 22 means we need to hire 22 workers.

We start our workers according to the schedule:

If we look at the surpluses on our staffing requirement rows (rows 2 - 7), we see that the slack values are 0 on all of the days. This means there are no more workers than required and we just meet staffing requirements on every day. Even though this is a small model, trying to come up with a solution this efficient "by hand" would be a difficult task.