will explore that "branch" first. With d(6,7) assigned a discrete diameter one size
lower than our original design, we can use the constraint h1 for consumers 2 and 3
to find the lower bounding continuous values for d(7,2) and d(7,3). We obtain
d(7,2) = 0.0778
d(7,3) = 0.0892.
Thus, any combinations with discrete pipe diameters less than these need not be
19, 20, 21, 25, 26 and 27 because these would violate the h1 constraint for both
consumers. Also, we see that combinations 22, 23 and 24 would all violate the h1
constraint for consumer 3, so they are infeasible as well. Thus, we have eliminated
all the combinations in this branch as originally proposed. As noted earlier, there are
combinations that deviate by more than one pipe size from our original design that
were not considered. Before exploring any of these, we compute the cost of the
design above with continuous diameters to see if it is an improvement on our
original design. When doing so we find that the variable cost portion of the heat loss
and pipe capital costs is slightly less than our original design: a 0.77% reduction. At
this point we could decide not to further explore this branch, since it offers such a
small potential for improvement; however, we will continue since it illustrates the
method to be used. From combinations 22, 23 and 24, we know that if we increase
the pipe size of d(7,3) to the next discrete pipe size greater than 0.0892, the h1
constraint for consumer 3 will be satisfied as well. Thus, we propose the discrete
design
d(6,7) = 0.0825
d(7,2) = 0.0825
d(7,3) = 0.1071.
We know that this design is feasible, so now we need to compute its cost to see if
it's an improvement over our original design. When the variable portion of the heat
loss and pipe capital cost is computed, we see that it's 7.77% greater than the original
design. Thus, we dismiss this design as well as any other feasible designs in this
branch, since all other feasible designs would need to have larger discrete diameters
and would thus be more costly yet.
We have two other major branches yet to explore: one where d(6,7) remains the
same as in the original design and one where it is increased one discrete pipe size.
The latter branch has four combinations remaining, one more than the other branch,
so we will explore it first. We proceed as before by 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) = 0.0544
d(7,3) = 0.0624.
As before, we also compute the total variable cost portion of the heat losses and pipe
capital costs for this design. We find that this cost is 3.88% greater than our original
design. Thus, we need not look at any discrete designs in this branch, since all will
require larger discrete diameters than those continuous diameters found above and
thus they will be more costly. Note that the two feasible combinations 16 and 17 in
this branch identified in Table 13 do in fact have costs in excess of 3.88% above the
original design.
Now we explore the remaining branch, where d(6,7) is the same discrete pipe size
as found in our original design. As before, we compute the minimum continuous
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