Need help with the following exercise. I am unable to work out the objective function.
After months of closure due to Covid-19, a food court is preparing to re-open for business. However, safe management measures need to be implemented and one of these requires that there must be a distance of at least Dmin = $4$ between any two tables occupied by customers. The food court has a total of $12$ tables and they may not meet this requirement if all tables are occupied simultaneously. Furthermore, the tables are fixed and cannot be moved, and so there is a need to determine which of the $12$ tables can be selected for use so that the maximum number of tables are made available to customers at any time without violating the requirement, while the rest of the tables not selected will be marked with $X$s to indicate that they cannot be used. The location of each of the $12$ tables is known and specified by its respective $(X_i, Y_i)$ positional coordinates measured with respect to some reference point, for $i = 1, 2,\ldots,12$, as tabulated below. Which are the tables that should be selected for use? (It can be assumed that all distances and coordinates are expressed in consistent units.)
Table 1: $(X=3, Y=17)$
Table 2: $(X=4, Y=8)$
Table 3: $(X=5, Y=4)$
Table 4: $(X=5, Y=14)$
Table 5: $(X=7, Y=6)$
Table 6: $(X=8, Y=12)$
Table 7: $(X=13, Y=7)$
Table 8: $(X=14, Y=5)$
Table 9: $(X=16, Y=13)$
Table 10: $(X=17, Y=4)$
Table 11: $(X=17, Y=10)$
Table 12: $(X=18, Y=15)$