7 The Complete Mesh Topology, Routing, and Spare Capacity Problem
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11.5 Optimal Design of Ring-Mesh Hybrids
This section is devoted to establishing an optimal model (in the sense of cost) for a completely generalized ring-mesh hybrid. It is a
general model in that no a priori principle (such as an access/core partitioning) is assumed. The practical difficulty with the model is its
computational complexity for network sizes of practical interest, hence the need for an effective heuristic, which follows. From a research
standpoint, however, the optimal model serves to define the problem and, where solutions are feasible, it allows us to check our later
hypothesis about what constitutes a synergistic set of rings and co-designed mesh. The model that follows envisages a span-restorable
mesh in conjunction with BLSR or UPSR type rings. Demands are routed before the optimization of ring placements and mesh capacity.
All wi capacity crossing a span is either included within a ring or is protected by restoration flows in the mesh component.
11.5.1 Concept of a Single-Layer Ring-Mesh Hybrid Transport Network
An important initial clarification is to stress that the generalized hybrid of rings and mesh that we now consider does not involve any
escalation or a multi-layer successive response to failures. This is a common misunderstanding. While it will be natural to speak of the
rings overlying the residual mesh in this type of hybrid, they are really embedded in the mesh and do not constitute a multi-layer
restoration scheme in the usual sense of works such as [Dem99] for example. In this hybrid architecture each length-wise segment of an
end-to-end path is protected by either the mesh or ring components of the hybrid. The action of both hybrid components is logically and
temporally concurrent, not successive, and each acts independently and autonomously for protection of the working capacity in their
domain. Restoration in this type of hybrid is not first attempted in one component and then handed "up to" another layer. These hybrids
still constitute a single layer restoration scheme, but have a mixture of ring and mesh logical components. Figure 11-12 illustrates. Faults
occurring in the physical layer are restored in the transport layer before reaching the service layer. The realization of the transport layer
considers a generalized mixture of rings amidst a corresponding mesh. There is, however, no new interlayer communication, coordination,
or escalation of responses required.
Figure 11-12. A generalized single-layer ring-mesh hybrid transport network.
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11.5.2 Cost Modeling for Ring-Mesh Hybrids
A challenge in developing an optimization model for assemblies of ring and mesh network elements is that the cost structure for the two
types of equipment are not the same. In comparing one mesh approach against another mesh alternative, details of the cost structure may
not be so important. As long as nodal equipment types are the same then total capacity indicates which alternative is economically
superior, regardless of absolute costs. But to consider hybrid architectures, we need a more detailed consideration of the cost structure for
ADMs, cross-connects, and transmission capacity. The modeling of cost for ring systems particularly requires attention. The most
important way that the cost structure of ADM rings differs from mesh network models is that ring systems are fundamentally modular. In
contrast to a cross-connect that can manipulate single channels as the basic unit of spare capacity, an ADM is quite modular in its
protection capacity. In comparing mesh designs we often adopt an integer channel capacity model but the fundamentally modular nature
of rings requires that in approaching ring-mesh hybrids we must adopt a fully modular capacity view for the rings, and preferably for the
mesh domain as well.
Let us define the following parameters for cost modeling in a ring-mesh hybrid environment. First are the attributes of pure optical line
transmission, where there are no differences between ring or mesh. These are characterized as follows and as indicated in Figure 11-13:
=This is the cost of a transmission module of the m
a ring.
th
capacity size, regardless of its use as part of a mesh span or to help form
M is a set of modular transmission capacity sizes, for example OC-48,OC-192, OC-768 for SONET ADMs or 40, 80, 160 l for
OADMs, index m.
Z
m
= The number of capacity units for the m
th
module size.
Figure 11-13. Generic Equipment Cost Model for a Add-Drop Multiplexer (ADM).
The following attributes reflect the type of nodal equipment that terminates the optical line:
mesh,m
th
T
= cost of terminating a transmission module of them capacity on a cross-connect, including a fully-allocated cost of
the OXC core. For instance, if a fiber pair bearing 80 (bidirectional) ls terminates on an OXC of ultimate size of 4096 fibers then
mm
T
reflects the direct cost to interface the fiber pair plus an allocated cost of the overall OXC core capacity that it consumes.
ring,m
th
T
= cost of terminating a transmission module of the m capacity on a ring ADM including half the cost of the ADM node
core of the corresponding total capacity.
In practice, the allocation of core cost to each line system terminating on the OXC may involve separate consideration of both the number
of fiber ports consumed and the number of wavelengths the fiber brings to the core for switching. This recognizes that the OXC may have
separate ultimate capacity limits in terms of the number of fiber terminations and the ultimate number of ls that can be switched. The cost
accounting for consumption of the ADM common core infrastructure is much simpler. It is a discrete investment that is completely
consumed by its two optical line interfaces. Building up from these primary parameters we also define:
.
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th
if the k member of a set of candidate ring systems includes an ADM at noden, 0 otherwise. This recognizes that
there may not necessarily be an ADM at every node of a ring. When no ADM is present a glass through simply continues the
fiber continuity through that location.
r(k) = the cycle of the graph on which ring candidatek is formed.
th
= Complete cost of constructing the r candidate ring at module size m, including the one-time cost of the ring
ring,m
ADM cores and line-rate terminations (in T
) plus line transmission costs, but excluding any per-channel add/drop
interface costs (which are dependent on actual capacity assigned to the ring):
Equation 11.1
= cost of add/drop for each demand that enters or leaves a ring at an ADM of modular capacitym. This cost may be
essentially zero under the integrated OADM/OXC equipment model of Section 11.1.
As written, Equation 11.1 is for a 4-fiber ring where the ring needs two of the basic line capacity modules for each of its spans; one for
working, one for protection. The leading factor of two can be dropped from the cost coefficient for a 2-fiber BLSR but then only half the
corresponding line-rate capacity is obtained as working capacity. It should be noted at this point that the strict representation of ring
candidates for the hybrid design is complicated by the fact that associated with each cycle on the network graph there are
Equation 11.2
distinct ring systems that could be built, where n is the number of physical node locations on the ring andh is the number of these that have
ADM terminals (as opposed to glass-throughs) on the ring. M is the set of different line rates for ring systems.
At least in the case where demand routing follows the shortest path over the graph prior to any ring or mesh placement decisions, the
latter consideration is not so explosive in the combinatorial sense as it may first seem. In this case we know ahead of time which demands
will be loaded into any prospective ring if the ring is placed and we can base its cost as a candidate ring on this information. More
specifically, we can assume that if a ring is placed on cycle r of the graph, it will be used to displace directly underlying working capacity
from the graph to its fullest extent. We therefore know which underlying path segments would be displaced into the ring and are able to
deduce the total for add/drop interface costs (if any) for each ring by noting where the route of a demand enters and leaves the ring of
interest. Thus, a preprocessing produces the cost coefficients to set up the MIP tableau. In particular,
represents the total
th
cost for the proposition of building a specific ring type, of the m modularity, over cycle k on the graph, loaded with maximal amount of
working capacity flows from the underlying spans of the original shortest-path routing plan, and accounting for add/drop costs at each
ADM, if relevant. Any site where an ADM core is not implied by the routing of demands over the graph, is considered to be a glass-through
[2]
site. It follows therefore, that although it is a detailed process, the
coefficients are precomputable for each candidate ring.
Later, we use a heuristic to predetermine a relatively small number of high-promise candidate rings for the hybrid, thereby limiting the total
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workload required for
evaluation.
[2]
Even the view of having either a glass-through or an ADM terminal is over simplified here. Locations on an optical
ring may actually have a splice (a glass-through), an EDFA, an OADM configured in pass-through mode serving as
a regenerator, or a full OADM node with one or more local add/drops.
Two other cost components can be acknowledged for completeness, but dismissed because they provide constant terms in the total cost
function for any hybrid that we consider. These are the first cost of establishing a cross-connect infrastructure in every node, plus the cost
of one pair of tributary rate add/drop interfaces on a DCS for every demand at its origin and destination. These contribute a constant total
cost that does not change across the hybrid designs here and are therefore omitted from the optimization model.
11.5.3 Ring-Mesh Hybrid Design Model
Having above acknowledged some of the more difficult aspects of a precisely modeling cost for accurate optimization of ring-mesh
hybrids, we can nonetheless realize a fairly useful model for generalized hybrids. By way of summary, the following assumptions apply:
1.
Demands are shortest-path routed over the graph and will stay on these routes whether associated with either ring or mesh
components in the final design.
2.
The mesh component operates on the basis of span restoration.
3.
There is no cost to make a transition from ring to mesh en route of a demand, by virtue of the integrated ADM functionality
hosted on the cross-connects (as in Section 11.1).
4.
Under the assumption that all inter-ring transitions are DCS-managed, and the assumption of integrated ADM functionality,
there is also no cost for a demand to make a transition from one ring to another.
In addition to the cost modeling parameters defined above, the following parameters are common to both ring and mesh components of a
hybrid:
S is the set of spans in the network, individual spans are indexed byi and, where a second span must be simultaneously
specified, by j.
is the logical working capacity on spani if the demands are shortest path routed (or routed by any other method) over the
facilities graph prior to the hybrid design.
Z
m
is the number of usable units of working capacity provided by them
th
size of transmission module. In the mesh this capacity
m
is fully usable to support accumulations of both working and spare channels. In a ring Z actually represents half of the line
capacity, i.e., the working capacity of the ring. A matching amount of protection capacity is reflected in the corresponding ring
cost coefficients, but only the working capacity of the ring appears explicitly in the model. Furthermore, the ring working
capacity is only the same as the corresponding module size in the mesh if the ring is a 4-fiber BLSR. For 2-fiber BLSRs, the
m
ring use of the corresponding transmission capacity is reduced to Z /2.
Parameters specific to the mesh component design are:
Pi is the set of all eligible routes for mesh restoration of span failurei, indexed by p.
th
if the p restoration route for span i uses span j, 0 otherwise.
Solution variables that describe the mesh network component of the hybrid are:
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=Number of modules of the m
th
size added on span j (integer variable).
wj =Mesh logical working capacity needed on spanj to serve demands.
th
=Restoration flow through thep route for failure of span i.
sj =Mesh logical spare capacity needed on spanj to support restoration flows in the mesh.
The ring-related design parameters are:
R= Set of all distinct simple cycles of the graph (i.e., templates for rings), indexr.
if the r
th
cycle includes span j.
And the ring-related solution variables are:
=Number of instances of an m-capacity ring placed on cycler in the design.
The variables specify the ring-mesh hybrid structure in terms of the number and size of self-contained ring systems to place, the number
and size of modular transmission systems between OXC of the mesh component, the logical partitioning of this capacity into working and
spare, and the routes to be used for restoration of the working capacity in the mesh component. The ring restoration routes and protection
capacity are implicit in specification of the ring cycle, r and modularity m. The model for a minimum cost ring-mesh hybrid in terms of the
above is:
Hyb-1:
Minimize:
Equation 11.3
Subject to:
1.
Mesh restoration flows protect all working capacities in the mesh component:
Equation 11.4
2.
Spare capacity on spans of the mesh suffice to support the restoration flows:
Equation 11.5
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3.
Any working capacity on a span not covered by the mesh must be in an overlying ring:
Equation 11.6
4.
Modular capacity in the mesh network:
Equation 11.7
All variables are nominally integer although the relaxation of mesh-restoration flow variables
is reasonable. Additionally, depending
on demand volumes and ring capacities, it might be reasonable to assume at most one instance of any ring candidate in the solution,
thereby replacing
by a binary decision variable, i.e., instead of deciding how many copies of anm-modular ring to commission on
cycle r, we decide simply whether such a ring is—or is not—in the solution.
The objective function is the total cost of ring systems placed plus the total cost for modular transmission systems and line rate termination
mesh,m
on OXC. Ring OADM node costs are included in
and allocated costs of OCX cores in the mesh are reflected byT
. As
mentioned, routing-dependent add/drop costs are ignored as either a constant or zero cost item under the assumptions above. The rings
are inherently modular and their cost includes the built-in protection capacity for ring-type survivability. For this reason the model needs no
explicit expression of survivability considerations for the ring component of the hybrid. Any working capacity referred into a ring is
automatically protected.
Set R contains all the simple cycles on the network graph, up to some limit on hop length. The significance is thatR contains all the possible
topological layouts for rings that could be considered in the design. Each cycle is notionally a template for the layout of a possible ring (or
stacked ring(s)) of any size or type considered. The preprocessing for both the MIP formulation and the later heuristics require finding the
set R, or a working set of candidate cycles that is limited by some criteria such as hop limit. The algorithm by JohnsonJohn75]
[
or the related
algorithm in Section 4.10.2 are suitable for this.
The first set of constraints ensures that the working channels on each span in the mesh are restorable by asserting that there are enough
individual paths assigned to eligible routes for restoration of the span to cover its working capacity. This assertion of required restoration
paths is countered by the second constraint system, which ensures that the assignment of paths to routes is feasible under the spare
capacity assigned to each span. The spare capacity variables influence the cost of mesh capacity in the objective through the modularity
constraint, Equation 11.7.
In Equation 11.6 we assert that any working capacity traversing spanj that is not handled in the mesh must be covered by the capacity of
overlying rings on that span. This involves the ring modular capacity sizes, the ring placement decision variables and the interception of
ring capacity that overlies span j. This constraint system interacts with both parts of the objective function: through
it has a drive to
effect the fewest and smallest rings. But any ring capacity placed can displace working capacity from the mesh and so also has the
prospect of reducing
through increasing
.
What makes this trade-off quite intricate, however, is that an increment to an
variable implies an entire ring addition to the network,
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whereas
speaks only of the capacity on one specific span of the mesh. Intuitively, it may seem that there would be only relatively rare
opportunities, with a high degree of coordination between ring and mesh components, to justify triggering a ring addition. Any such
synergy would have to arise at the network level as a whole because placing a whole ring to displace capacity from one mesh span would
rarely, if ever, make sense in its own right. In fact a particular insight about just when this formulation would find it advantageous to
displace capacity from the mesh into overlying rings follows, and is the basis for the heuristic in the next section. Results with the optimal
design model follow in Section 11.6.4 where it is used as a reference model to assess performance of this heuristic.
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11.6 Forcer Clipping Ring-Mesh Hybrids
In the Hyb-1 model above it is interesting to think further about the interactions between constraint systems. In particular, we need to ask
when would the solver trade an increase in
for a decrease in
on some span(s) of the mesh? After all an increase in
a decision to completely place one additional ring system, whereas decrementing
means
saves mesh capacity on only one span. A
reasonable hypothesis would be that trading
for
only happens when some new ring placement would have a more than
proportionate impact on the design of the mesh counterpart. The concept of forcers, introduced in Section 5.5, and the idea of forcer
clipping gives us just such a view of how certain rings can be especially synergistic with the surrounding mesh in which they are
embedded as a hybrid. The key idea is that by clipping off strong forcers of the mesh, certain rings can contribute to enhancing the
efficiency of the mesh and, if they themselves are well loaded by the capacity they have "clipped off" of the mesh, then the placement of
such a ring may be of overall economic benefit. In the access / core partitioned hybrids the use of ring and mesh principles is more or less
separated spatially. In contrast, the heuristics that we now develop based on the forcer clipping idea are much more general and have no a
priori bias as to how and where to use rings. Here, rings and mesh will not be topologically separated, but selected rings will be
commissioned within the body of a mesh-restorable network that have a forcer clipping effect on the efficiency mesh network design.
11.6.1 The Forcer Clipping Hypothesis
Chapter 5 introduced the forcer concept and inChapter 8 it was applied to issues involved with nodal bypass and survivable design with
express routes. The idea now is to use forcer analysis to tell us which spans are most advantageous for ring placement by virtue of having
the greatest network-wide effect of reducing spare capacity in the remaining mesh component. The direct reduction of working capacity in
the mesh by ring placement is another beneficial effect to the mesh component but, by itself is generally detrimental to total cost because
the ring use of protection capacity is less efficient. Good hybrid ring placements must therefore be at least cost neutral in terms of the
direct shift of working capacity from one domain to the other, but have some more than proportional cost saving effect on the spare
capacity of the mesh from which the working capacity was displaced. This is where the forcer concept comes into play.
Recall that a basic implication of the forcer concept is that reduction of the working capacity on certain spans of a mesh-restorable network
would yield no corresponding reduction of mesh spare capacity. Even if these non-forcer spans are reduced, working flows across other
spans, i.e., the forcers, require retention of the spare capacity to ensure restoration. Measures such asFx* let us pinpoint which spans most
drive the spare capacity requirements of the surrounding restorable mesh. In effect Fx* reflects the "height" by which spanx's working
capacity is above the point at which it would no longer be a forcer (i.e., other spans would become forcers, halting the relief of spare
capacity).
What can we do with knowledge of how the spare capacity of the mesh would be relieved if some specific working span quantities were
reduced? One use could be to reroute working paths, detouring them from their shortest paths so as to reduce wi quantities on the
strongest forcers. This is conceptually valid and is essentially the explanation of why joint optimization of working path routes and spare
capacity placement in a mesh network is beneficial. But another way that strong forcers of a mesh network might be reduced to improve
the overall efficiency of the mesh would be to try clipping these forcers down with ring systems. Especially if one ring can group together
several strong forcers and provide for their survivability within one ring, there might be a more than proportionate cost reduction in the
remaining mesh network. Hence the term "forcer clipping" for the hypothesis that a ring might be placed on the mesh network to clip the
tops off of one or more of the forcer spans, thereby more than proportionally reducing its total working and spare capacity cost. Figure
11-14 is a conceptual illustration. Net cost reductions would arise if the cost of the forcer-clipping ring is less than the net savings in the
underlying mesh layer after its working capacity is reduced and its spare capacity is re-optimized.
Figure 11-14. The concept behind "forcer clipping": rings that make a mesh more efficient.
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This is the central idea from which we proceed toward construction of heuristic for generalized ring-mesh hybrid design. A priori, there are
several reasons to expect that certain rings could pay in on the basis of forcer-clipping effects. One is a similarity to the concept of express
rings already known in ring network planning. An express ring is one that is sized and situated, typically with relatively few add/drop
terminals, to handle just a few of the largest demand flows in the network. In fact a 1+1 DP can be viewed at a two-terminal express ring
and is sometimes used as such to handle single particularly large demands in a ring network design. By getting these dominant flows out
of the way in their own semi- or fully-dedicated express rings, the rest of the multi-ring design problem can be more effectively solved for
the remaining more similarly sized demand bundles. In that sense the forcer clipping ring concept is like proposing a special kind of
express ring to relieve the effects of certain demands on the mesh network design.
Another line of reasoning that supports the forcer clipping hypothesis in principle is the nonlinear dependence of the redundancy of a mesh
network on the diversity of its wi values. Consider one node in isolation that hasd spans. A necessary (but not by itself sufficient) set of
conditions for full restoration of every span cut is that at any node adjacent to a span failure i:
Equation 11.8
which further implies that
Equation 11.9
is a lower bound on the total sparing required at the node. With a few more steps this leads again to the lower bound on the redundancy of
a node being 1/(d - 1) (Section 5.4.2).
Given Equation 11.8 and Equation 11.9 it is possible to show how important it is to "level" thewi quantities. Consider for example, a node of
degree d=4, let wi= (3,15,3,15). The minimal sparing vector will besi = (5,5,5,5) for a node redundancy of 55.5%. Now consider the node
being overlaid with a 12-channel ring to "clip off" most of the high flow of 15 units going through this node. In the residual mesh this node
has wi= (3,3,3,3) and requires si = (1,1,1,1) for a redundancy of 33%, which is actually at the lower bound ford=4. This illustrates the notion
of "forcer leveling": in the latter case, every span is an equal co-forcer of all other spans at the node. The potential economic benefit is in
the trade off of the ring cost against the direct savings in displaced working mesh capacity plus the benefits of redundancy reduction in the
residual restorable mesh. In this isolated node example, the ring provides 12 working capacity units on two spans but the total (wi+si) that
it displaced from the mesh is 24 + 16 = 40 units. Even if the exchange of the forcer clipping working capacity from the mesh to the ring by
itself was cost neutral, net savings still arise whenever the ring placement has the added leverage of reducing spares in the mesh due to
forcer-leveling effects which inherently increase the mesh efficiency.
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Figure 11-15 moves up from the isolated node considerations to show a small quantitatively exact example of effects from rings embedded
in a mesh. Figure 11-15(a) shows a 7-node, 10-span, pure mesh span-restorable network of unit capacity modularity. The w
( i, si) note by
each span gives the working capacity on the span (generated by shortest path mapping of the demand matrix onto the topology), and the
spare capacity assigned (by SCA) for 100% span restorability with a hop limit of five. The pure mesh reference design of Figure 11-15(a)
has a total of 283 (working plus spare) channels.
Figure 11-15. Illustrating the effects of various forcer-clipping ring candidates.
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Figure 11-15(b) shows the effect of overlaying one ring that turns out to be fairly effective in forcer clipping. On a simple bulk capacity
basis this 4-span 12-channel ring represents an investment of 4(2*12) = 96 (working plus spare) capacity units. Its placement alters four of
the wi values in the residual mesh, and the corresponding changes in spare capacity are shown (underlined values). The residual mesh
has a revised total of 190 (wi+si) channels. Thus, the placement of this particular ring reduces the mesh by 93 channels in return for a bulk
equivalent capacity of 96 channels invested in the ring. The ratio, called the capacity return ratio (CRR) is = 93/96 = 0.969 in this case. The
high CRR in this case is an indication of a ring placement that is most likely economic because, on a bulk bandwidth basis including nodal
costs, most ring systems cost considerably less that 97% of the cost for the equivalent point-to-point capacity and termination in a mesh
technology environment. Thus, the CRR numerically represents the discount factor for ring capacity (relative to mesh) below which a given
ring placement would yield a net savings. More detailed inspection of the example in Figure 11-15(b) shows that it is a combination of
good working fill on the ring spans and good forcer clipping effects that make this ring effective. Figure 11-15(c) is a counterexample to
show that by no means will all possible ring placements necessarily be economic. Through the same process it can be seen that the CRR
of the ring in Figure 11-15(c) is 0.60. Thus, if the cost of the ring was more than 60% of the cost of the corresponding capacity in the
mesh, the placement of this ring would be a losing proposition. The subsequent heuristics mechanize the process of finding near optimal
sets of several such rings to embed in a mesh residual.
11.6.2 Forcer Clipping Heuristics
[3]
Two closely related heuristics are now discussed to place rings within a mesh network based on the forcer clipping hypothesis. Both
use information from forcer analysis of the initial pure mesh design to identify a set of potentially good forcer clipping ring placements. After
having identified this set of potentially good candidates the algorithms make trial ring placements, redesign the residual mesh network in
detail, and apply a cost model to assess the net benefit of the particular ring trial placement. The ring with the best economic return is
placed in the design and iteration repeats until there are no remaining ring candidates that produce an economic return. The basic
algorithm is outlined in Figure 11-16. First, the following preliminary processing is performed:
[3]
Work in Section 11.6.2 to 11.6.4 was in collaboration with R. Martens and presented in preliminary form at
[GrMa00].
1.
Initial mesh design: From the given graph topology and demand matrix a pure mesh restorable design is created as a starting
point. In this case it is an SCA design using the methods of Chapter 5. Demands are mapped onto the network spans by
shortest path routing, generating the wi quantities and SCA is then solved giving the initial meshsi values.
2.
Forcer analysis of mesh: Forcer analysis of the network is done as described inChapter 8. We obtain the forcer skeleton and
Fx* magnitudes of the forcers in it.
3.
Cycle finding: All elemental cycles of the network graph, up to a maximum hop size, are generated. This step identifies all the
topological possibilities for ring placement on the network graph, populating the set R. The procedure and data is identical to
that used in the optimal MIP model discussed in Section 11.5.3.
4.
Build working set: Since the set R is typically large, it is at this stage that we use our ideas about forcer clipping to select a
smaller set of "high promise" cycles that will act as ring system templates. This is also where the two heuristics differ. In both
methods every cycle in R is inspected at this stage with respect to the forcer information from step 2 to obtain a forcer clipping
figure of merit.
a.
In the first heuristic the metric used is the sum of the Fx* magnitudes on spans that the cycle overlies. Promising
rings are those that group together many forcers (or at least a few strong forcers), producing large positive Fx*
totals.
b. The second heuristic looks only to see what proportion of all spans in a cycle have logical forcer status.
Figure 11-16. Basic heuristic for ring-mesh hybrid design based on forcer-clipping