Table 13. Constraint satisfaction and costs for the remaining combinations. Our original design
(combination no. 1) is shown in bold, and the other feasible designs are shown in italic.
Cost
Comb. no.
d(6,7)
d(7,2)
d(7,3)
(h1)2
(h1)3
Variable
premium
(N/m2)
(N/m2)
costs ($)
(%)
and type
(m)
(m)
(m)
1 (0,0,0)
0.1071
0.0703
0.0703
96,183
50,374
38,002
0
3 (0,0,)
0.1071
0.0703
0.0545
96,183
193,207
35,435
6.76
7(0,,0)
0.1071
0.0545
0.0703
25,607
50,374
36,718
3.38
9(0,,)
0.1071
0.0545
0.0545
25,607
193,207
34,151
10.13
12(+,0,)
0.1325
0.0703
0.0545
123,482
165,908
39,480
3.89
15(+,+,)
0.1325
0.0825
0.0545
149,022
165,908
40,464
6.48
16(+,,0)
0.1325
0.0545
0.0703
1,692
77,673
40,764
7.27
17(+,,+)
0.1325
0.0545
0.0825
1,692
128,753
42,732
12.45
18(+,,)
0.1325
0.0545
0.0545
1,692
165,908
38,196
0.51
19(,0,0)
0.0825
0.0703
0.0703
18,468
64,278
34,058
10.38
20(,0,+)
0.0825
0.0703
0.0825
18,468
13,198
36,027
5.20
21(,0,)
0.0825
0.0703
0.0545
18,468
307,859
31,491
17.13
22(,+,0)
0.0825
0.0825
0.0703
7,071
64,278
35,042
7.79
23(,+,+)
0.0825
0.0825
0.0825
7,071
13,198
37,011
2.61
24(,+,)
0.0825
0.0825
0.0545
7,071
307,859
32,475
14.54
25(,,0)
0.0825
0.0545
0.0703
140,259
64,278
32,774
13.76
26(,,+)
0.0825
0.0545
0.0825
140,259
13,198
34,743
8.58
27(,,)
0.0825
0.0545
0.0545
140,259
307,859
30,207
20.51
status of the two consumer constraints that must be satisfied and the total of the
variable portions of the capital costs and heat loss costs for each combination. We see
by examining the constraint satisfaction that only two combinations are feasible, i.e.,
they satisfy the h1 constraint for both consumers 2 and 3. However, when we
calculate the cost of these feasible combinations, we find that both cost more than our
original design. Thus, we are left with the result that none of the alternatives
investigated so far are better than our original design. There are some additional
designs that we have not investigated, however. Recall that earlier we dismissed the
possible designs that would increase or decrease pipe sizes by more than one
discrete size from the original design. Depending on how many pipe sizes we are
willing to deviate from our original design, there are many alternate designs. Of
course, there is no guarantee that these designs will be feasible, let alone lower in cost
than the original design. To explore these designs without resorting to "exhaustive
enumeration," i.e., calculating the constraint satisfaction and cost of each, we can use
the branch-and-bound technique described in detail in Chapter 5. Below we apply
this technique to our example problem. In the process of doing so, we will not only
explore additional designs not considered yet, but we will show how the technique
would have allowed us to dismiss some of the alternatives in Table 13 without
computing the constraint satisfaction or total variable cost.
As noted in the previous chapter, the objective of the branch-and-bound tech-
nique is to use what is known about designs already explored to reduce the number
of remaining ones that must be examined in detail. In addition, we would like to do
so without dismissing any designs superior to the best feasible ones identified. We
have effectively already used the technique above to dismiss nine of the possible
combinations of Table 12. In that case, we used the fact that the variable portions of
the heat losses and capital pipe costs were monotonically increasing in pipe
diameter. This allowed us to dismiss cases that only increased pipe size.
After our initial elimination of nine combinations, as discussed above, we see that
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