diameters using the h1 constraint for consumers 2 and 3 to find the lower bounding
values for the continuous diameters of d(7,2) and d(7,3), obtaining
d(7,2) = 0.0564
d(7,3) = 0.0646.
We know that this design must have a lower cost than the original since it has the
same pipe size for segment (6,7) and smaller pipe sizes for the other two pipe
segments. However, we go ahead and compute this cost saving to see if it justifies
exploring this branch further. We find that the saving is a significant 5.40%. We have
three combinations (3, 7 and 9) from Table 13 that have not been previously
eliminated from this branch. We see, however, that each of these will violate at least
one of the h1 constraints, since at least one of the pipe sizes is smaller than the
continuous minimums found above. Thus, we can dismiss all of these combinations.
In addition, we can dismiss any other designs in this branch as well, since they would
have pipe sizes greater than those of our original and would thus be more costly.
Note that our original design is actually in this branch, using the first discrete pipe
sizes greater than those found above for d(7,2) and d(7,3).
By using the branch-and-bound technique, we have eliminated all of the com-
binations of Table 13 and have only computed the cost four times. In addition, we
have computed diameters using the h1 constraint six times. These computations
compare favorably with those required for total "exhaustive enumeration" of the
possibilities (27 cost and 54 constraint calculations) and favorably to the computa-
tions of Table 13, which eliminated nine possibilities based on monotonicity consid-
erations. We have also shown that no other discrete designs in the branches
explored, i.e., even those deviating by more than one discrete pipe size, could be both
feasible and less costly that the original discrete design. What remains to be shown
is that other branches that allow d(6,7) to deviate by more than one discrete pipe size
are either infeasible or not cost effective, or both.
To explore the branches where d(6,7) is more than one discrete diameter away from
our original design, we once again look to the constraint h1 for consumers 2 and 3.
We notice that there is a limit on how much we can decrease either d(7,2) or d(7,3) and
still find values of d(6,7) that will satisfy the constraints. Physically, what has
occurred is that we have decreased the pipe sizes of d(7,2) or d(7,3) to the point where
all of the pressure loss available between the pipe junction node 6 and either
consumer 2 or 3 is being absorbed in the pipe segment (7,2) or (7,3). To utilize this
condition, we first ignore the pressure loss of pipe segment (6,7) and calculate the
minimum continuous values for d(7,2) and d(7,3) that will satisfy h1 for consumers 2
and 3 respectively. We then find the next largest discrete diameters in each case,
since any actual design would be bounded by these. The results are
d(7,2) = 0.0536 (continuous); 0.0545 (discrete)
d(7,3) = 0.0614 (continuous); 0.0703 (discrete).
Now, with these discrete diameters, we calculate the minimum continuous value
of d(6,7) that would satisfy the constraint h1 for both consumers 2 and 3. This value
is determined to be 0.1296. The minimum discrete value of d(6,7) is then 0.1325. We
see that these discrete diameters are identical to those of combination 16 in Table 13.
It was shown earlier that for this combination the cost exceeded our original design.
Thus, any larger discrete diameters would also exceed the cost of our original
design. Since this result is for the minimum possible discrete diameters for pipe
segments (7,2) and (7,3), regardless of the size of pipe segment (6,7), no lower cost
alternatives can exist since their diameters for pipe segments (7,2) and (7,3) would
be greater and thus the designs more costly.