A highway capacity function in Korea: Measurement and calibration

Tmnspn. Res.-A. Vol. DA. Printed in Great Britain.

0191-260790 s3.oo+.w 0 1990 Rr$amon Press plc

No. 3. pp. 177-186. 1990

A HIGHWAY CAPACITY FUNCTION IN KOREA: MEASUREMENT AND CALIBRATION SUNDUCK SUH Korea Transport Institute, Seoul, Korea CHANG-HO PARK Department of Civil Engineering, Seoul National University, Seoul, Korea TSCHANGHOJOHN KIM Department of Urban and Regional Planning and Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received 14 February

1989; in revisedform

15 August 1989)

Capacity functions are important in the model that accounts for the user’s route choice behavior based on the traveller’s perception of the travel time. This is because a capacity function represents the relationship between the traffic volume and the travel time on the link. The capacity function developed by the U.S. Bureau of Public Roads (BPR) has been used in many countries, including Korea, without much effort to calibrate the parameters for its own transportation environment. Countries other than the United States, however, have distinctive demographic, economic, cultural, and behavioral characteristics: and they might need unique capacity functions for their own environments. Thus, it is important for Korea to have its own capacity function that can appropriately represent the Korean highway environment. Any attempt to model the Korean highway system without using a suitable capacity function might result in inappropriate solutions, because most modeling activities are crucially based on link travel time, and it is the capacity function that furnishes those link travel time. A link capacity function for Korea is calibrated based on a BPR type formula utilizing an alternative method. The alternative method is developed in a bilevel programming framework that uses link volume counts instead of link flow and travel time data. Detailed calibration results are reported. Abstract-

1. INTRODUCTION Travel time, or in a more general sense, travel impedance, on a given highway link usually increases and speed decreases as the traffic flow increases. As den-

sity increases beyond a certain level, both traffic flow and speed decreases. In modeling a highway network, it is necessary to have a correct capacity function to account for increases in travel time as traffic flow increases for the given highway environment. This highway environment includes driver characteristics, roadway conditions, and roadside activities. Different countries with distinctive demographic, economic, cultural, and behavioral characteristics might have unique capacity functions for their own environments. The capacity function developed by the Bureau of Public Roads (BPR 1964) of the United States has been used in Korea. To date there has not been much effort in calibrating the parameters of the function. Korea, however, has traffic characteristics distinctive from those of the United States. Thus, it is desirable for Korea to have its own capacity function that can represent appropriately the Korean highway environment. Any attempt to model the Korean highway system without using a suitable capacity function might result in inappropriate solutions, because most modeling activities are crucially based on link travel time, and it is the capacity function that furnishes those link travel time.

This capacity function is calibrated for the Korean highway environment in this paper. Toward this end, reviews on the capacity function are described in Section 2. Because, the definition of capacity plays a vital role in the development of capacity functions, a brief description of highway capacity is included in Section 3. Also included in Section 3 is a review of current highway capacity estimation practice in Korea. A bilevel programming model for estimating the parameters of the highway capacity function is introduced in Section 4. The bilevel programming model is applied to the Korean highway system to calibrate parameters of the highway capacity function that are appropriate for Korean highway system in Section 5. A summary and conclusions follow. 2. LINK CAPACITYFUNCTION

2. I Introduction For any given link, it has been generally observed that speed decreases as the flow increases up to a point of critical density (the density where the capacity of the link is obtained). Beyond this point, both flow and speed decrease (Fig. la) while travel time increases (Fig. lb). This function, which represents the relationship between flow and travel time on a link, has many different names: capacity function, capacity restraint function, link capacity function, link performance function, congestion function, or 177

178

S. SUH, C-H. PARKand T. J. KIM

capacity P

Flow

_I

Fig. 2.

A typical monotonically increasing capacity function.

ment (Beckmann et al., 1956), which follows the Wardrop’s (1952) first principle, the requirements for the desirable capacity function to meet become more conspicuous:

capacity

Flow

Lb1 Fig. 1. Speed and time vs. flow relationship.

link cost function. In this paper the function is referred to as a link capacity function or capacity function. The link capacity function becomes important in the assignment procedure (also called route choice procedure) because it is this function that determines increases in travel time as flow increases in a link. Without an accurate link capacity function, it is impossible to accurately model the user’s route choice behavior, which is based on a traveler’s perception of travel time or cost. The actual capacity function, such as that shown in Fig. lb, has limitations, because this capacity function will not give a unique flow solution. A capacity function has to be a monotonically increasing function of flow for a mathematical programming assignment model to have a unique flow solution (see, for example, Sheffi 1985). Therefore, most research typically utilizes a monotonically increasing link capacity function which theoretically extends the actual capacity function (Fig. 2) beyond capacity (see, for example, BPR (1964)). One notable exception is the work of Okutani (1984), who utilized two strictly convex functions to approximate the original shape (Fig. lb) in the assignment procedure. Multiple local solutions, however, have resulted. With the popular adoption of mathematical programming formulation of the equilibrium assign-

1. The function should be a non-negative, monotonically increasing function of flow for a unique solution. 2. The function should be continuous and derivatives should exist. 3. The function should be easily calculatable. Based on these points, a capacity function that is applicable for all types of highways is developed for the Korean environment in this paper. The development is based on the actual link volume counts. A bilevel programming model is used and the parameters are systematically searched. 2.2 Review Although many different capacity functions have been proposed and utilized, there has been no consensus about the type of capacity function necessary for any particular links. Branston (1976) classified two main approaches taken in defining the link capacity function, the mathematical function approach and the theoretical approach. In the mathematical function approach, a simple mathematical function replicating observed data is devised. Because of its simplicity, it is usually hard to incorporate network or link characteristics. In the theoretical approach, network characteristics such as signal spacing, signal settings, and/or link characteristics are well represented based on the queuing theory. Therefore, the resulting capacity function becomes more complex than that of the mathematical function approach. The earlier research development of capacity function is summarized largely based on Branston’s work (1976) and is listed next for the sake of completeness. The detailed description of most of the work mentioned here can be found in Branston (1976).

179

Highway capacity function in Korea 2.2.1 Mathematical function approach.

A hyperbolic capacity function is defined as

,+4To-8) Q-a

Irwin, Dodd, and Von Cube (1961) proposed a function of two straight line segments. T= TO+a(Q’-Cp’) T= T,+P(Q’-Cp’)

where for Q’
(1)

where

Q
developed the most widely used capacity function, usually known as the BPR formula.

To= T,,+cxC’, Q: flow on a link

T= To[1+ a(Q/C,M

T: travel time at flow Q T,,: travel time at zero flow C,: practical capacity cr,p: parameters Q’: flow on a link per lane CL: practical capacity per lane

(6)

where rY=o.15,

p=4

5. In the Pittsburgh Area Transportation

This function was applied to the Toronto network. This two-line segment function evolved to a threeline segment function later by Irwin and Von Cube (1962).

Study, Soltman (1965) developed another nonlinear capacity function. T=

T&Q/G

(7)

where T =T,+a(Q’-C’) T =T,+p(Q’-C’& T =T&+r(Q’-C’)

for Q’C;

Q/C, s 2

(2)

6. The Traffic Research Corporation

(1966) used the following function for the Winnipeg area:

where T, = T,+aC; T6 = T,+@(C’-C,) C’ : Level of Service (LOS) E capacity

T=a+/3(QP-y)+d@2(Q’-y)2+6]1

Equation (7) was generalized by Overgaard (1967) as

It has been reported that the predicted flows from the assignment procedure with these capacity functions agreed reasonably well with the observed data. However, these functions cannot be used in the mathematical programming assignment problem because of the discontinuities at C’, and C’. The development of these two-line segment and three-line segment linear capacity functions parallels to more recent interests on piece-wise linearlization (approximation) of nonlinear capacity function (eq. 6) (Ben-Ayed 1988; LeBlanc and Boyce 1986; Morlok et al., 1973). A nonlinear function used in the Detroit Area Transportation Study was proposed by Smock (1962). T= TOexp[Q/q

Mosher’s (1963) logarithmic fined as follows:

link function

T=T,+Irz(a)-In(a-Q)

where QSCY

(8)

(3)

is de-

7-z ~,~@‘cp’~

(9)

eqn (7) is the special case of eqn (9) when a=2 and p=1. Dafermos (1968) proposed the following function: To= a&W&, + Z,NuaX,2 + Bdr,

(10)

where

Z, X, B,(.)

capacity added to link a flow on link a function

In 1974, Steenbrink (1974) substituted the practical capacity of BPR formula with LOS E capacity (C) and calibrated a and /3 for Dutch environment.

(4) T = To[l +a(Q/C)q a =2.62 p =5

(11)

S. SUH,

180

C-H. PARK and T. J. KIM

2.2.2 Theoretical approach. 1. Campbell, Keefer, and Adams (1959) reported a function known as CATS (Chicago Area Transportation Study) function based on a theoretical approach. r= r,

for-$b0.6 5

T=TO+u($-0.6) 5

for>0.6

(12)

2. Another popular link capacity function belonging to this category is that of Davidson (1966). T=t(l+J--

(13)

5 1-c

C= Q/S

(14)

where S: Saturation parameter J: Delay parameter t: zero flow time

3. HIGHWAY CAPACITY

This function attracted interests from many researchers (Boyce, Janson, and Eash, 1981; Daganzo, 1977a). 3. Wardrop (1968) developed a network capacity function that shows the relationship between overall travel speed in the network and flow. The corresponding link capacity function is

To

43

T=(l-ye)+(o!-Q)D

data’ and statistical insignificance,2 these are the only efforts done for the Korean environment thus far. The work in this paper is systematic and presents a large scale endeavor to calibrate parameters of BPR type (eq. 6) link capacity function for the Korean environment. As can be identified from the review of previous works on capacity functions, there are two different capacities involved in the mathematical programming approach for capacity function development. The first capacity is the practical capacity. The 1950 Highway Capacity Manual (HCM) (BPR 1950) refers the practical capacity. The second capacity, the Level of Service (LOS) E capacity, has been introduced in the 1965 HCM and reinforced in the 1985 HCM (TRB 1985). The next section briefly summarizes the changes in the definitions of capacities from 1950 HCM to 1985 HCM. A description of the current Korean practices on highway capacity calculation follows.

(15)

where D link length lX>C

2.3 Capacity fmction: Current practice in Korea The BPR formula has been the de facto standard capacity function for Korea since its introduction in later part of 1970s. The function has been widely used even without any efforts to calibrate the parameters (a&) for the Korean environment. Recently, some efforts to calibrate the parameters for different capacity functions have been reported. Choi (1987) tried to calibrate the J factor of the Davidson formula (es. 13) for the Korean urban environment. Lee (1987) attempted to calibrate the BPR formula’s (eq. 6) parameters for the Korean environment with ad hoc procedures. Both Choi and Lee tried a couple of numbers for the parameters for the assignment procedure and chose the best among the numbers that yield the best statistics. Even though both works suffered from the lack of

3. I Introduction The concept of highway capacity changed twice since the 1950 HCM (BPR 1950). The 1950 Highway Capacity Manual (BPR 1950) defined three different levels of highway capacity concepts; basic, possible, and practical capacity, noting that the word capacity itself is just a generic expression for the ability of a roadway to accommodate traffic. Practical capacity is defined as the maximum number of vehicles that can pass a given point on a roadway or in a designated lane during one hour without the traffic density being so great as to cause unreasonable delay, hazard, or restriction to the driver’s freedom to maneuver under the prevailing roadway and traffic conditions. (BPR 1950)

The maximum practical capacity was 1,500 passenger cars per lane per hour for urban areas and 1,000 passenger cars per lane per hour for rural areas for multi-lane highways. Calculations for two-lane highways used 1,500 passenger cars per hour for urban areas and 600 passenger cars per hour for rural areas as standards. That is between 70 to 75% of the basic capacity when the operating speed is assumed to be 35 to 40 miles per hour. It was this practical capacity that was used in the link capacity function developed by the Bureau of Public Roads (BPR 1964). These notions of basic, possible, and practical capacity, were not found in the 1965 HCM. The definition of capacity in the 1965 HCM corresponds to the possible capacity in the 1950 HCM. The concept of IData for each work come from one small subarea. 2For example, Choi’s regression equation for J is significant only at a 24% significance level.

181

Highway capacity function in Korea capacity under ideal conditions replaced basic capacity. Furthermore, the notion of practical capacity was replaced by several service volumes that correspond to each level of service (LOS). Six service levels, A to F, were then defined. Under ideal roadway conditions, these capacities for each service level can be summarized by passenger car and by the operating speed (Table 1). These capacity concepts become more sophisticated in the 1985 HCM (TRB 1985). Most notably, flow rate becomes the unit of capacity, replacing the old unit of volume. By using a flow rate of a peak fifteenminute period, variations in flow during an hour can be addressed. Furthermore, maximum flow can be analyzed by utilizing the flow rate concept. The concept of six levels of service is retained even though the specific criteria to define these levels of service have been modified. Service volume was also changed to service rate following the change in the definition of capacity. Level-of-service criteria for freeways with different design speeds are described in Table 2. The corresponding criteria for multilane highways are summarized in Table 3 while the criteria for twolane highways are listed in Table 4. The LOS C capacity is the closest to the 1950 definition of the practical capacity in terms of volume/capacity (V/C) ratio (Table 2). LOS C that corresponds to a V/C ratio of 0.77 and 0.69, for design speed of 70 miles per hour and sixty miles per hour, respectively, is the most probable choice to replicate the practical capacity, for which V/C ratio was defined in 1950 as 0.7-0.75. For multilane highways, volume between LOS C and LOS D corresponds to the practical capacity based on both the V/C ratio and operating speed (Table 3). For two-lane highways (Table 4), the 1950 practical capacity corresponds to the volume between the capacities of LOS D and LOS E in terms of V/C ratio and LOS E in terms of the operating speed. It is important to note that these are only rough comparisons between practical capacity and LOS volume, because practical capacity is a subjective entity, despite the fairly objective guidelines. Based on this summary, it is assumed that the practical capacity that has been used in the original BPR capacity function most likely corresponds to the LOS C, D, and E of 1985 HCM (TRB 1985) depending on types of highways. With this conclusion in mind, the next subsection summarizes the Korean highway capacity analysis practice. 3.2 High way capacity estimation practice in Korea Before the advent of 1985 HCM (TRB 1985), the highway capacity calculation practice of Korea has been based on 1965 HCM (TRB 1965). As stated in Section 1, the Korean highway environment is different from that of the United States. For example, the size of cars, driver characteristics, high heavy-vehicle (such as bus and truck) mix, and land use characteristics are unique in Korea. Cha and Kim (1986) proposed a series of capacililtA)24:3-n

ties of selected highways for Korea.3 Accepting the headway and, subsequently, density figures of 1985 HCM (TRB 1985),4 and applying shorter car length (five meters) and different design speed, they came up with slightly higher ideal capacity5 figures than those of 1985 HCM (TRB 1985). After obtaining the ideal capacity in this way, they applied 1985 HCM procedures based on the national Korean highway environment to calculate the representative capacities for different kinds of highways. Cha and Kim (1986) tested their calculation against actual traffic flow rate observed on the various highway segments and reported successful goodness-of-fit between their estimation and actual volume counts. Their results are used as a basis for capacity estimation in this paper pending more comprehensive research on the capacity calculation for the Korean highway environment. The next section analyzes the procedures involved in developing a link capacity function. A bilevei programming model is introduced to implement the parameter calibration procedures discussed. 4. A BILEVEL

PROGRAMMING

MODEL FOR ESTIMATING

THE HIGHWAY CAPACITY FUNCTIOK

4.1 Introduction The following assumptions and observations are made toward developing a link capacity function in Korea. Three different service levels (LOS C, D, and E) of capacity are chosen for the analysis based on the result of the previous section, which suggested that the practical capacity might be closest to the LOS C, D, and E capacity depending on types of highways. In deciding the form of capacity function, the following points are considered: (i) Function should be non-negative and monotonically increasing as the flow increases; (ii) Function should be able to be decided with the minimum number of parameters; (iii) Function should replicate the original capacity function (Fig. 1b) well. Many reported satisfactory performance of the BPR formula (eq. 6) in the U.S. For example, Branston (1976) reported how the BPR formula (eq. 6), which has only two parameters, replicates the TRC formula (eq. S), which has five parameters. Based on all these points and performance, the BPR type funcJHighways included are expressway, two-lane expressway, multilane national highways, two-lane national highway, two-lane unpaved highway. 4Vehiclespacing of four car-lengths is used in the HCM, producing 67 pc/mi/ln assuming car length is twenty feet. Cha and Kim used a car length of five meters, producing 50 pc/km/ln. This corresponds to the capacity under ideal condition for 1965 HCM and basic capacity for 1950 HCM.

S. SUH, C-H. PARKand T. J. JCN

182

Table 1. Six levels of service based on the 1965 HCM Rural Highway

LOS A LOS B LOS c LOS D LOS E LOS F

Multilane

Two-lane

Freeway 700 pc/ln/hr b 60 mph 1000 pc/ln/hr L 55 mph 1500 pc/ln/hr 50 mph 1800 pc/ln/hr 40 mph 2000 pc/ln/hr 30-35 mph O-2000 pc/ln/hr < 30 mph

400 pc/hr 900 pc/hr 1400 pc/hr 1700 pc/hr 2000 pc/ln O-2000 pc/hr

600 pc/ln/hr 60 mph loo0 pc/ln/hr 55 mph 1500 pcllnihr 45 mph 1800 pc/ln/hr 35 mph 2000 pc/ln/hr 30 mph O-2000 pc/ln/hr c 30 mph

Sauce: HCM 1965 (TRB 1965). tion is chosen as the base form of capacity function for the Korean environment. The different functional forms, such as the Davidson (1966) function (eq. 13),

can be calibrated using the same basic procedure utilized in this paper. The usual procedure of the link capacity function development is summarized in four steps. First, a function that can replicate the observed link travel time and volume data is devised. Second, the parameters of the function are calibrated with observed link travel time and volume data, usually based on some statistical means or other mathematical technique (see, for example, Taylor, 1977). Third, this function is utilized in an assignment process. And fourth, the predicted link volume from the assignment process is tested against the actual link volume count. If the test results are unsatisfactory, the whole process might be repeated. The alternative procedure used in this paper is different from the previous approach in the data requirements. The observed data of link travel flows at various speeds are not readily available and are very expensive to obtain. Therefore, the alternative process utilizes only the observed link volume as input data. The annual report on link volume count by the Ministry of Construction (MOC) of Korea is utilized. In fact, the method used here actually combines the second to fourth step mentioned before and can be summarized in two steps. First, a functional form is proposed (in this case, the BPR formula (eq. 6)).

Second, a bilevel programming model is used to calibrate the parameters that can minimize the sum of the least square of the differences between observed link flows and predicted link flows. The advantage of using bilevel programming is that it can specifically restrict the flows to follow the user-optimal principle during the systematic parameter estimation process. Figure 3 shows the general steps that will be followed in this paper. The next subsection introduces the bilevel programming model utilized; but first, a brief introduction to the bilevel programming problem is in order. 4.2 Model In general, a bilevel programming problem (Boyce and Kim 1987; Kolstad 1985) is defined as follows: Pl:

Ul) s.t.

“,‘” F(x,y) G(x)50

where Y(x) is implicitly defined by Ll) s.t

“yi” f(x,Y) g(x,y)sO

LOS A B C D E F

Speed 260 257 254 ~46 230 530

V/C

6It is also called an outer, out-side, level one, leader, or policy problem.

0.35 0.54 0.77 0.93 1.00

60 MPH MSF 700 1100 1550 1850 2000

Speed 250 247 242 230 530

(18) (19)

Ul) is defined as an upper levels problem and Ll)

Table 2. Levels of service for freeway 70 MPH

(16) (17)

VIC

0.49 0.69 0.84 1.00

50 MPH iMSF 1000 1400 1700 2000

MSF: Maximum service flow rate per lane under ideal conditions. Source: HCM 1985 (TRB 1985).

Speed 243 240 228 528

V/C

MSF

0.67 0.83 1.00

_ 1300 1600 1900

la3

Highway capacity function in Korea Table 3. Levels of service for multilane highways

LOS A B C D E F

50 MPH

60 MPH

70 MPH Speed

v/c

MSF

Speed

v/c

MSF

Speed

v/c

illSF

257 253 250 240 230 130

0.36 0.54 0.71 0.87 1.00

700 1100 1400 1750 2000

250 248 244 240 230 s30

0.33 0.50 0.65 0.80 1.00

650 1000 1300 1600 2000

242 239 235 228 s2a

0.45 0.60 0.76 1.00

a50 1150 1450 1900

MSF: Maximum service flow rate per lane under ideal conditions. Source: HCM 1985 (TRB 1985). a lower level.7 The decision maker at the upper level influences the lower level decision maker by setting X, thus restricting the feasible constraints set for the lower level decision maker. The upper level decision maker also interacts with the lower level decision maker via the objective function of the lower level decision maker. Also note that the decision variable of the lower level problem is expressed as a function of the decision variable of the upper level (v(x)). For a more detailed description of bilevel programming, see Kim and Suh (1988) or Suh (1989). Specifically, the model utilized in this paper to implement the alternative approach is as follows:

W

U2

min,,;, ~(f,“-fob,P))2

(20)

0

I+320

(21)

where f solves L2

min, C ~~c&,c~,~)& ll

(22)

(23) P Xijp>O

~,cw,P) = 4J[l+4.wq to = free flow travel time; observed travel time times 0.87

i j

P

JQ: observed link volumes volume of path p connecting zone i and j Tii : traffic volume between zone i and j

XijP

:

&Up

=

1 if link CYis on the path p 0 otherwise

U2) utilizes a least square measurement (eq. 20) to minimize the sum of the squared differences between the observed link flow and predicted link flow by adjusting CYand 0 of the capacity function. Different values of cu,/3 passed to L2) from the U2) cause different link travel time and, subsequently, different user-optimal link flow patterns in the L2). L2) is, in fact, a useroptimal assignment programming model. An equilibrium link flow pattern satisfies the Wardrop’s (1952) two equilibrium conditions. These conditions are: 1. All routes which are used between each O-D pair have equal cost, and

(24)

Function Proposed

‘It is also called an inner, in-side, level two, follower, or behavioral problem.

Initial Parsmeters

where

Table 4. Levels of service criteria for two-lane highways Level Terrain LOS A B C D E

Rolling Terrain

Uee Function in Aerignment

Mountainous Terrain

Speed

v/c

Speed

v/c

Speed

v/c

258 255 ~52 250 245

0.15 0.27 0.43 0.64 1.00

257 254 251 249 240

0.15 0.26 0.42 0.62 0.97

~56 254 249 245 235

0.14 0.25 0.39 0.58 0.91

Maximum flow rate is 2800 passenger car per hour in both directions. Source: HCM 1985 (TRB 1985).

J Test Predicted VolameagainstVolumeCount I New Parameters

1 Fig. 3.

Procedures utilized in this paper.

S. SUH, C-H.

184

2.

PARK and T. J. KIM

no unused route has a lower cost than a used route.

This means that at equilibrium no one can reduce his or her travel time by unilaterally changing to an alternative route. Recent reports from many feasibility studies for rural highway improvement in Korea showed good applicability of a user-optimal assignment programming model even for rural roads. Measurements other than the least square measurement can also be utilized as objective functions in U2). One example is a x2 measurement (fo-p/f). 1 Maximum likelihood formulation is also possible (for example, see Lee, 1986). For this research, the least square measurement is utilized. Daganzo (1977b) reported that the least square measurement performed as well as x* or maximum likelihood measurement in a small stochastic network problem. The next section describes a case study of the model. The model is applied to the Korean highway system to calibrate CY,~of the BPR function. 5. Er’UMERICAL

ANALYSIS: A CASE STUDY OF KOREA

The proposed model has unique structure in the sense that there are only two upper level variables. Despite the fact that there are only two variables involved, it is beyond the scope of this paper to describe detailed algorithms for solving bilevel programming problems. Detailed description on solution procedures are omitted here, since Suh and Kim (1988a. 1988b) and Suh (1989) report such description in detail. The lower level problem L2) is solved with the Frank-Wolfe algorithm (LeBlanc ef al., 1975). The network shown in Figure 4 and Origin/Destination (O/D) tables of 1985 are used for the lower level problem. There are 25 O/D zones, 212 nodes, and 706 links in the network. Table 5 shows the computation results. The computation was done on the Cray XMP/48 at the National Center for Supercomputing Applications (NCSA) at the University of Illinois at UrbanaChampaign. The starting point (~=0.15, /3=4 (BPR formula, 1964), and 01=2.62, @=S (Steenbrink, 1974) are used with LOS E, C, and D capacities in the model. The existence of multiple local optima is identified from the solution. The parameter values of cy=2.72 and 6=6 are found to be the best fit when LOS D capacity is used with given network and O/D table. The least square measurement is reduced by about thirty percent using new parameter values over old BPR parameters. Figure 5 compares parameter values thus found with those of Steenbrink (1974) and BPR when free flow travel time (r,) is 1.

Fig. 4.

The network

Korean highway system. Toward this end, the paper first identified that LOS C, D, and E service volumes in the 1985 HCM corresponds to the practical capacity used in the BPR capacity formula, depending on the type of highway. The link capacity function is calibrated based on a BPR type formula utilizing an alternative method called a bilevel programming model. The alternative method uses link volume counts instead of link flows at various speeds, which are usually not available. Based on the several computational analyses using Table 5. Calibration Starting Pt.

a=0.15 13=4

F

Iter.

CPU (set)

LOS D

(u=o.45 p=7 cu=o.41 p=7.21 (~=I.65 p=9 a=2.72 p=6 1x=2.7 P=6 1x=2.92 p=2

8465ElO

111

42.00

7165610

151

49.73

8685ElO

122

34.00

6843ElO

108

44.00

6943ElO

108

46.41

6578E10

8

LOS c

LOS D LOS c

AND CONCLUSIONS

The purpose of this paper has been to calibrate the parameters of the BPR capacity function for the

function

Sol.

LOS E

01=2.62 p=5

results for link capacity

Cap.

LOS E 6. SUMMARY

for the case study with 25 O/D zones.

*: This calibration the search process.

16.00*

is stopped when ol,@ become negative in The value is not a local optimal solution.

Highway

capacity

function

185

in Korea

22 7 21 20

V/C ratio o

This

Research Fig. 5.

different and

starting

points

@=6 are found

and

+ BPR

Comparison

capacity

to be the

best

levels, fit

for

of capacity

functions

Steenbrink

with Tc= 1.

cu=2.72 Choi, K. J. (1987). rl Study on Comparative Analysis of Korean

highways.

Capacify Restraint Function and Method of Application. Master’s thesis, Seoul National University, Seoul, Korea. Dafermos,

Acknowledgement-Partial ence Foundation

support by the National Sci(SES-8718146) is gratefully acknowledged.

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