Mathematical Model
Assumptions
Assume the distances between the schools are determined and the teachers are required to teach at different schools for the morning and afternoon sessions.
Index Sets and Variables
An alias set L is defined to be the set of schools which is identical to the set J as defined earlier.
Also, as mentioned above, an indicator parameter Y(l, j) is introduced to indicate whether teacher i is assigned to school l in the morning and school j in the afternoon. That is, Y(l, j) = 1 if teacher i is required to travel from school l to school j between the two sessions, and Y(l, j) = 0 if otherwise. Note that the values for Y’s are not predetermined and their values are dependent on the preliminary assignment of the teachers for the two sessions.
Other index sets and variables are same as the core problem.
Data
A table is added to indicate the distances between the individual school. For
example, the entry D (l, j) would give the distance between the school l and school j.
Scalar "Maxdist" is introduced to specify the maximum distance between two sessions that the teachers are willing to travel.
Other data is the same as in the core problem.
Program
The objective function is the same as that of the core problem.
The additional constraint regarding the distance between the two sessions is implemented as follows:
Y(l,j)*D(l,j) = MaxDist for all j, l
Where MaxDist denotes the maximum distance that the teachers are willing to travel in between two sessions and D(l,j) represents the distances between school l and school j. The indicator variable Y(l,j) is determined as follows:
If X(i, am, l) = 1 and X(i, pm, j) = 1,
then Y(l,j) =1,
Otherwise
Y(l,j) = 0
Note that the above constraint can be interpreted as follows:
If teacher i is required to travel from school l in the morning to school j in the afternoon, then Y(l,j) = 1, and the constraint will simplify to D(l,j) <= MaxDist. If, however, teacher i is not required to make such a trip, then Y(l,j) = 0, and the constraint will not be bounding.