Application of Fuzzy Set Theory to the Analysis of Capacity and

Level of Service of Highways

Partha Chakroborty Shinya Kikuchi
Civil Engineering Department
University of Delaware
Newark, Delaware 19716, U.S.A.

Abstract by the 1985 HCM. The procedure of determining the

Applicability of fuzzy set theory to the analysis of ca- capacity of an existing facility involves two steps:
pacity and level of service of highway facilities is dis- 1. Identifying the capacity under ideal conditions
cussed. Many rigid assumptions used i n the existing (ideal capacity);
methods suggested b y the Highway Capacity
2. Adjusting the ideal capacity for the
can be relaxed and a subjective case b y case judgement
conditions.
of the engineer can be incorporated i f fuzzy set theory
is used. This paper presents how fuzzy set theory can The ideal capacity is the maximum expected hourly
be incorporated i n the present analysis procedure. rate of flow under some pre-defined ideal condition
with respect to the state of roadway, traffic, and con-
Introduction trol conditions. The value of the ideal capacity is as-
The concept and procedure to determine capacity sumed fixed and it is based on empirical analysis of
and level of service of highway facilities have been the observations, but it is rarely achieved since the pre-
subject of traffic engineering for many years. Capacity sumed conditions are too idealistic.
is considered as a measure of supply and level of service The capacity under the prevailing conditions is de-
as a measure of user satisfaction of a transportation fa- termined by modifying the ideal capacity using ad-
cility. Unlike other engineering topics, measurements justment factors for roadway, traffic, and control con-
of these parameters depend to a large degree on sub- ditions; each accounts for the deviation from the ideal
jective judgement of the engineers and drivers. condition. Their values vary between zero and one.
The Highway Capacity Manual (HCM, 1985), pub- They are used in the product form to give the actual
lication of Transportation Research Board, National capacity when multiplied by the ideal capacity.
Research Council, has established the standards for In reality, the capacity of any highway facility can
determining these parameters. In the manual, con- hardly be a single number, as suggested by the HCM,
cept of capacity and level of service is stated as be- but should be a range. Roess and McShane [3] stress
ing subjective and flexible, yet in its quantitative pro- the fact that the capacity predicted by the HCM is
cedure many rigid assumptions and relationships are not a definitive measure, and that higher flows can be
used. The method often lacks flexibility in adapting observed on any facility. The 1985 HCM [5] also lists
the judgements and feelings of engineers and highway a number of observations where volumes on freeways
users. Fuzzy set theory can by its very nature, over- are higher than the predicted capacity. Wortman [7],
come many of the present shortcomings which arise also notes that several instances when capacity was
due to the use of traditional mathematics to deal with greater than the predicted value. Baass [l]points out
uncertain and vague phenomena. some serious drawbacks in the calculation of potential
capacity of unsignalized intersections, in particular, he
Concept of Capacity and Level of Service
questions the use of a single value for the critical gap
Capacity and suggests that the present concept of critical gap
The capacity of a highway facility is defined as the should be modified.
“maximum hourly flow rate at which vehicles can rea-
sonably be ezpected to traverse a point or uniform sec- Level of Service
tion of lane or roadway during a given time period un- The level of service (LOS) of a highway facility is a
der prevailing roadway, traffic and control conditions,” subjective measure of the quality of service provided.

146
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 It is defined as qualitative measure describing op-
erational conditions within a traffic stream, and their Capacity
perception by motorists and / or passengers.” The The problems of the current procedure are generally
level of service measure attempts to take the following related to the use of crisp and simplified input variables
perceived factors into account: and relations between influencing factors.
First, the ideal capacity of a given facility is fixed
Speed and travel time, at a single constant value. For example, it is 2000
passenger cars per hour per lane for a basic freeway
Freedom to maneuver,
segment.
Traffic interruptions, Second, crisp values of the adjustment factors, which
account for the difference between the actual and ideal
Comfort and convenience, conditions, are assumed. The value of the factors are
determined based on crisp relations explaining the ef-
Safety. fect of departure from the ideal conditions.
Third, the HCM method assumes that accurate data
The HCM defines six of levels of service in terms of
are known before the analysis. In reality this is not al-
“A” through “F.” LOS(A) is assumed to be the best
ways possible and therefore allowance for uncertainty
and LOS(F) to be the worst. Definition of each level
in the input data should be provided.
of service is rather general and the boundary between
two levels of service is vague. For example, LOS(C) is
Level of Service
defined as the condition when “ .... operation of each
In determining the level of service we face similar
individual users become significantly affected by inter-
problems caused by establishing a rigid criteria for a
action with others in the traffic stream. The selection
qualitative aspect of traffic flow.
of speed is affected by the presence of others, and ma-
First, the current HCM method suggests the use
neuvering within the traffic stream requires substantial
of one criterion for determining the level of service.
vzgilance .... . The level of comfort and convinience de-
This is based on the assumption that one quantitaive
cline noticeably at this level. ”
criterion can represent the combined effect of all the
The LOS is a subjective and perceived measure and
influencing factors of the quality of traffic flow. Deter-
thus establishing a specific LOS for a given condition
mining the level of service based on one criterion could
with absolute certainty is a difficult task. To point out
exclude many qualitative aspects of traffic flow which
the difference in perception among drivers Webster [6]
influence the level of service.
cites the following example; only 8% of the drivers sur-
Second, specific levels of service (A through F) are
veyed could predict their speed correctly. Incidentally,
defined based on rigid ranges of values for the criterion.
speed is the only measurable quantity among the user
It would be natural to assume that the transition be-
perceived factors on which the level of service is based
tween two levels of service is likely to be gradual rather
in the HCM [5].
than abrupt.
Roess and McShane [4]mentions that for any facil-
In summary, most of the problems of the current
ity the boundary values of the criterion for each LOS
procedure originate from the attempt to quantify com-
category were determined based on the judgement and
plex and qualitative relations and parameters by a lim-
experience of traffic engineers. This process has ren-
ited number of rigid measurements.
dered the rigidity of the boundaries debatable. In a
specific case cited in this paper it is mentioned that Applications of Fuzzy Set Theory to Capacity
the boundary between LOS(A) and LOS(B) for signal- and Level of Service Determination
ized intersections was very much debated before fixing
it at its current value of 5 second delay per vehicle. In In this section, application of fuzzy set theory is
some drafts of the current manual it even had values proposed in order to overcome some of the draw-
of 10 seconds per vehicle. backs which were pointed out in the previous sections
. Fuzzy numbers can be used to represent the values
Limitations of the Current Procedure to of many input and output variables which are involved
Determine Capacity and Level of Service in calculating capacity and level of service. Input vari-
ables, such as ideal capacity, sight distance, volume
The limitations of the current procedure are dis- of traffic, headway between cars, can be represented
cussed as the background for proposing fuzzy approach by fuzzy numbers. Likewise, output variables such
for the analysis of capacity and level of service of high- as adjustment factors, actual capacity, level of service
way facilities. criteria like the volume to capacity ratio or reserved

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 capacity, can also be represented by fuzzy numbers. the LOS of a basic freeway segment. In the figure the
This section is divided into three subsections each of vertical axis Z represents , U L O S ( ~ ) ( : )and ,US(:). The
which highlights the application of fuzzy set theory to dotted line, in Figure 1, represents the fuzzy set S for
selected topics in the capacity and level of sewice anal- the f value. The level of service of the freeway segment
ysis. can then be determined in terms of certainty factors,
< ~ o s ( ~ ) (ItS )is. defined as the certainty with which
Fuzzy Sets for Level of Service Categories a particular condition S belongs to a level of service
The definition of fuzzy set and membership func- category z, where z is A, B, C, D, E or F:
tion makes it amenable to application in determining
the level of service of a highway facility. Level of ser-
vice is a qualitative measure of traffic flow and, thus
boundary between two levels of service cannot be de-
termined clearly. In the HCM volume to capacity ra-
tio, f , is used as the criterion for determining the LOS
of a basic freeway segment. A f of 0.54, for example,
represents a LOS(B). It is therefore logical to assume
that a f of 0.55 would not depict a very different situa-
tion. Yet, according to the HCM procedure this value
indicates a LOS(C), which, by definition, has quite dif-
ferent service chracteristics from that of LOS(B). This
suggests that the current method may not be suited
as an indicator of LOS when the value of the criterion
lies at the boundary of two LOS categories.
Let each LOS category be defined as a set; for exam-
2
1.0 ’
I A B C

\
D
- ”
E F

ple, in set notation the current definitions of LOS(B)

1.
0.4
and LOS(C) of a basic freeway segment can be pre-
sented as, 0. 1

LOS(B) = 10.35 5 E 5 0.54},

LOS(C) = {E 10.55 5 5 0.77).
Figure 1: Level of service categories defined as fuzzy
Since the two sets are mutually exclusive, a condi- sets
tion with a given value cannot belong to both sets.
The problem of abrupt change in LOS arises when the In the case of the example shown in Figure 1, the con-
value of f is near the boundary of the sets. If the dition is a LOS(B) with a certainty factor of 1, and
two sets are made intersecting, then it is possible for LOS(C) with a certainty factor of 0.4.
a condition to belong to both LOS(B) and LOS(C).
This is paradoxical, since now the same facility under Fuzzy Composition to Represent Parametric
the same condition has two different service character- Relationships
istics. The present procedure cannot handle approximate
If the boundary of two adjacent sets is not rigid and relations. For example, it cannot describe situations in
crisp, but somewhat flexible and “hazy,” then a given which one foot reduction in average lane width causes
condition can belong to more than one LOS category a reduction in capacity by “approximately” 5%. The
with varying degree. This can be accommodated by highway and traffic characteristics are so complex that
defining each LOS category as a fuzzy set. Further- relationships between parameters can normally be de-
more, the value of the LOS criterion for a given con- scribed only in approximate terms. Fuzzy relations
dition is normally a fuzzy number. For instance, the can portray exactly such a situation. In many cases
value of f , in this case, is a fuzzy number, since the the input variables to a fuzzy relation are also fuzzy.
capacity (c) is a fuzzy concept as described before. In such situations, the output can be derived using
When the LOS categories are fuzzy sets and the fuzzy composition.
value of the LOS criterion is also given as fuzzy number For example, in determining the adjustment factor
(fuzzy set), the LOS can be determined by examining to the ideal capacity when the average lane width is
the intersection of the two types of sets. Figure 1 illus- around 10 feet instead of 12 feet (which is the ideal
trates the procedure using the example of evaluating condition), fuzzy composition technique can be used.

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 Let A be the vector representing the fuzzy lane width, wishing to cross it will accept, is considered to be con-
and R be the matrix representing the relation between stant for a given speed. However, the driver's accep-
lane width and the values of the adjustment factors. tance of a gap depends on his perception of the gap
Then the adjustment factor is a fuzzy number B , ob- and the speed of the oncoming traffic. The decision
tained as: of a driver is also rarely consistent. The same driver
may reject a gap, and, at other times, accept a smaller
A o R = B. gap. The decision of a driver to accept a gap is in fact
based on an approximate reasoning process.
In this example, A represents the fuzzy number around
The approximate reasoning method can model such
10 feet, for example,
situations. A set of fuzzy inference rules, whose input
and output variables are fuzzy sets, can be formulated
Lane width (f.) to simulate more closely the human perception and
decision making process. For example, a set of n rules
8.5 9 9.5 10 10.5 11 11.5 of the following type can be formulated to estimate the
approximate critical gap which a driver would choose
A = ( 0 .01 .49 1 .49 .01 0 ) for a given condition:
R represents the relation between the different lane
widths and the values of the adjustment factors, for I f Speed is Si + Critical G a p is Ti
example,
where Si and Ti may be triangular fuzzy numbers and
are given as,
Adj. factors
0.6 0.7 0.8 0.9 1.0
and
8.5 ' 211 212 213 214 215 Ti = [ T i T2i, 'Gi].
9.0 221 . When the speed S is given, the rules from the above
9.5 231 . set are activated simultaneously and a fuzzy critical
R = 10 241 . gap, T, can be obtained by using the following stan-
10.5 251 . dard procedure:
11 261 .
11.5 \ 271 . . 275

Fuzzy adjustment factor B (a fuzzy number) for

around 10 feet lane width is obtained by the max-min Tvi is called the truth value of rule i or the certainty
composition method, with which the implication i is valid for an input value
Adj. factors
S . It can be obtained by using the following relation,

0.6 0.7 0.8 0.9 1.0 for

B = ( Yl1 Y12 Y13 Y14 Y15 ) for

The fuzzy relations and composition techniques can
otherwise
be used for many similar situations, including relations
between the type of drivers in the traffic stream and Discussion
reduction in capacity, between grade and passenger car
equivalencies of heavy vehicles. This paper has discussed the possibility of applying
fuzzy set theory to the analysis of highway capacity
Approximate Reasoning to Model driver De- and the level of service using examples of its applica-
cision Patterns tion to selected topics. Traffic engineering and control
There are cases in which the adjustment factors are is one of the areas where fuzzy set theory is suited for
dependent directly on human perception and decision analysis, since design and control of highway traffic fa-
making process. The current procedure, however, as- cilities highly depends on the drivers perception and
sumes that the values of such factors are constant judgement which are often subjective and qualitative.
among drivers. For example, the critical gap, which Among the topics in traffic engineering, the analysis
is the minimum gap in a traffic stream which a driver of capacity and the level of service has traditionally

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 been the central issue, since these are important indi- References
cators on which decision to build and improve high-
way facilities depend upon. Capacity is the maximum 1. Baass, Karsten G. (1987), “The Potential Capac-
number of vehicles the facility can accommodate, while ity of Unsignalized Intersections,” ITE Journal,
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the facility under the prevailing condition based on NO. 10, pp 43-46.
the perception of users. Methods to determine these
2. Highway Research Board (1965), Highway
have been compiled in the Highway Capacity Man-
Capacity Manual, Highway Research Board Spe-
ual. Although the manual specifically warns that the
cial Report 87.
method and the procedure must be used with discre-
tion, the current procedure relies on the determinstic 3. Roess, Roger P., and McShane, William R.
binary logic and does not provide how to incorporate (1987), “Capacity and Level-of-Service Concepts
the discretion and judgement of the engineers in the in the Highway Capacity Manual,” ITE Journal,
analysis procedure. Institute of Transportation Engineers, Vol. 57,
Determination of capacity and level of service is a NO. 4, pp. 27-30.
complex process and their values are influenced by
many factors and site-specific conditions. Obviously, 4. Roess, Roger P., and McShane, William R.
it is reasonable to determine the capacity as a range (1987), “Changing Concepts of Level-of-Service in
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vice should be treated as a fuzzy measure of the pre- ples,” ITE Journal, Institute of Transportation
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relations, fuzzy reasoning, and fuzzy measures would 6. Webster, Lee A. (1966), “Driver Opinions
greatly facilitate the explanation of the capacity and and Characteristics Related Rural Speed,”
the level of sevice to the public. Since the percep- ITE Journal, Institute of Transportation Engi-
tion of such quantities are generally fuzzy, the public neers, Vol. 58, No. 7, pp. 23-25.
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7. Wortman, Robert H. (1988), “The Highway Ca-
ables in fuzzy quantities and the decisions made on
pacity Manual Open Meetings: A Summary Re-
its basis, thus resulting in less friction between the
port,” ITE Journal, Institute of Transportation
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