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 *h*1 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

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 *h*1 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 *h*1 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

Now, with these discrete diameters, we calculate the minimum continuous value

of *d*(6,7) that would satisfy the constraint *h*1 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.

71