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 *h*1 for consumers 2 and 3

to find the lower bounding continuous values for *d*(7,2) and *d*(7,3). We obtain

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 *h*1 constraint for both

consumers. Also, we see that combinations 22, 23 and 24 would all violate the *h*1

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 *h*1

constraint for consumer 3 will be satisfied as well. Thus, we propose the discrete

design

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 *h*1 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

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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