Towards multi-class, multi-objective trac management based onvaluesa,∗ , L.A. Tavasszya

X. Lina Department of Transport and Planning, Delft University of Technology, Delft, TheNetherlands
X.Lin@tudelft.nl, L.A.Tavasszy@tudelft.nl∗ Corresponding author

Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island
March 31, 2022

Keywords: User equilibrium; multi-class trac; multi-objective optimization; value-based tracmanagement

1 INTRODUCTION
Freight trac takes up an important part in the trac composition.

According to Statistics

Netherlands (CBS) (2020), road transport within the Netherlands carried 766 million ton of
goods in the year of 2019, which was 41% of the total freight being transported. Freight trac hasvery dierent characteristics from passenger trac. This includes dierent origin / destinations,vehicle characteristics (length, speed, etc.). In particular, the value of time (VOT) is dierent
for dierent vehicle (user) classes.
Kouwenhoven et al.

According to SP surveys done in De Jong et al.

(2014),(2014), the VOT for freight is 38 euros/hour in the Netherlands, and theVOT for passenger car is 9 euros/hour. Despite the dierences, freight and passenger are bothactive participants taking a share in the public trac infrastructures.
Trac management can steer the trac in a way that the road infrastructures are used
more eciently (Wismans (2012), Schreiter (2013). In the meantime, due to the aforementioneddierences, freight trac and passenger trac can response to the same trac management
approach in dierent ways.

However, freight trac and passenger trac are generally not

distinguished while implementing trac management, as trac managers do not actively consider
the dierences in vehicle classes. Few research consider freight trac performance as a separate
objective apart from the passenger trac performance when evaluating trac management
measures.
Literature has addressed multi-objective trac management Ahn et al.

(2020), Wismans(2012). Wismans (2012) considers environmental impacts and accessibility. Ahn et al.

(2020)

consider energy consumption as well as travel time with 2 vehicle classes: electric vehicles andconventional vehicles while optimizing trac management proles.

Both research use 2-level

optimization, with the user equilibrium (UE) trac assignment as the lower level, and theobjectives they optimize at the upper level. Neither of the papers consider the VOT of the userclasses while designing trac management measures.
Lin et al.

(2021) develop an MILP formulation that represents multi-class UE assignment.

Based on this new modeling approach, this research provides a theoretical framework of multiple
vehicle class, multi-objective optimization for trac management, which can be solved as a
one-level problem.

The framework considers the performances of freight trac and passenger

TRISTAN XI Symposium

Original abstract submittal

X. Lin and L.A. Tavasszy

2trac, evaluating the eectiveness of trac management on these two dimensions by means ofmulti-objective optimization. The remainder of the abstract presents the core formulation for
multi-class trac management. We also use a simple test case to illustrate a small fraction ofwhat can be achieved with this method.

2 MATHEMATICAL PROGRAMMING FORMULATION FOR
TRAFFIC MANAGEMENT
Based on Lin et al.

(2021), we can now eciently incorporate trac management measuresconsidering the behavioral response of dierent user classes.

Assume a network G= {N , E},with nodes n ∈ N , links i ∈ E , OD pairs w ∈ W , path p ∈ Pw between OD pair w , considering
vehicle class m∈ M.

We formulate the multi-objective trac management problem as thefollowing:

min J =x,v,cXXXmw

ξm xmwp cmwp =XXwpcar
ξ car xcar
wp cwp(1)∀m ∈ M, w ∈ W, p ∈ Pw ;(2)∀m ∈ M, w ∈ W, p ∈ Pw ;(3)∀m ∈ M, w ∈ W;

(4)∀m ∈ M, w ∈ W, p ∈ Pw ;(5)∀m ∈ M, i ∈ I;

(6)∀m ∈ M, w ∈ W, p ∈ Pw ;(7)∀m ∈ M;(8)∀m ∈ M, w ∈ W, p ∈ Pw .

(9)

truck
ξ truck xtruck
wp cwp +pXXw

psubject to:

amwp (cmwp − c∗mw ) = 0,(0, if xmwp = 0
amwp =
1, if xmwp > 0,

cmwp ≥ c∗mw ,
XXXXBPR δip fmi
cmwp =
δip′ xm′ wp′ πm′  ,m′iBPR (x) = Lifmiv1+αmiXxkiwp′

β !
,xmwp = dmw ,pξm ≥ 0,xmwp ≥ 0,

The objective function sums up the total travel cost (cmwp ) over all vehicle ows (xmwp ).

c∗mw

stands for the lowest travel cost of class m, between OD pair w . πm is the passenger car equivalent(PCE) value for class m. δip = 1 stands for that link i is part of path p, otherwise δip = 0. dmw
is the xed demand of class m between OD pair w .

The BPR function fmi is used for the is the free ow speed of class m on
Li is the length of link i. vmi
link i. ki is the capacity of link i. α = 0.15 and β = 4 are constant parameters. The above
volume-delay relationship.

formulation can be eciently solved as a mixed-integer quadratic programming (MIQP) problem
using packaged/commercial solvers. In the objective function J , vehicle classes are given dierent
weights ξm . By adjusting ξm , the UE solution gives the optimal value of decision made by speed

limits (vmi ) and lane opening/closure (ki ) that prioritize the dierent user classes accordingly.

3 A SMALL TEST CASE AND PRELIMINARY RESULTS
Next we use a small test case to illustrate the eect of having trac management as decisionvariables in the formulation. A two-path network (Fig.1) is used. Trucks and passenger cars startfrom the origin on the left to the destination on the right. It is assumed that trucks have strong

TRISTAN XI Symposium

Original abstract submittal

X. Lin and L.A. Tavasszy

3

Alternative route for passengers
Main routeOrigin

Destination
Figure 1  A network with 2 routes.

Table 1  Input for the test case
Input

Description(Lmain , Lalt )(kmain , kmain ) , v )(vmainaltdvvPm dm
ξm(200, 200)
(20, 20)
Car: (vdv , 120); Truck: (80, 0)[80 130]100

ξcar + ξtruck = 1

preference for the main route; while passenger cars can travel via the main or the alternativeroute. This also guarantees the uniqueness of the UE status in every solution. (We exclude thediscussion on the uniqueness issue of multi-class UE in this document.) The rest of the input issummarized in Table 1.
We use the SCIP solver in Optitoolbox for Matlab to solve the formulation multiple times,
with dierent ξm values in each group, with a specic truck/car ratio in demand.

We test 6

groups with the truck occupancy of total demand in number of vehicles (7.5%, 10%, 12.5%,15%, 17.5%, 20%).

The results are visualized in Fig.2, in which car / truck total travel time

(xmwp × cmwp ) are x-axis and y-axis, respectively.
For each group a Pareto line is formed. The 3 stars on each Pareto line marks the speed limitfor cars on the main route (from left to right) 130 km/h, 100 km/h, and 80 km/h respectively.
Within one group, limiting the free ow speed on the main route causes more cars to switch tothe alternative route, leaving more space for trucks on the main route. This results in higher car
total travel time and lower truck total travel time.
A comparison between the groups suggests that if the truck occupancy is lower (e.g., 7.5%), aspeed limit for cars increases cars' total travel time but does not signicantly reduce truck total
travel time. The benet given to trucks of the speed limit becomes more obvious when trucks
take a higher occupancy in the travel demand.
We also use two reference lines:ξcar : ξtruck = 1 : 1 and ξcar : ξtruck = 9 : 38 De Jonget al. (2014), Kouwenhoven et al. (2014). The reference given by the SP values can be seen
as a reference for the societal fairness between passengers and freight, in terms of the usage ofinfrastructures. Take for an example with a xed speed limit for cars (100 km/h), when truck
occupancy is at around 17.5%, the management is at its fairness. The same trac managementwith a lower truck occupancy favors trucks more than cars and vice versa. It also indicates themanagement measures (in this case, speed limit on the main route) that can be used to steer themixed trac closer to its fairness point under dierent circumstances (e.g., truck occupancy).
We point out that the analysis is preliminary based on simplied models such as an underlying
static assignment as well as the BPR function. Nevertheless, we provide a framework for moredetailed and realistic analysis on the eectiveness of trac management with the considerationof values.

TRISTAN XI Symposium

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X. Lin and L.A. Tavasszy41400
Pareto Total Time
Weight = 1:1
Weight = Ratio SP value

Truck Total Travel Time

1200

1000

800

C_spd = 130
C_spd = 80600

T_occu = 20%

400

T_occu = 15%
200

T_occu = 10%

00500

1000

1500

2000

2500

3000

3500

Car Total Travel Time
Figure 2  Results

4 SUMMARY
Trac management rarely connects values to dierent user classes. With the recent development
in modeling UE with mathematical programming, this research builds up an ecient theoreticalframework to analyze user class-specic preferences for trac management measures. A small
test case is used for illustration with preliminary analysis based on freight and passenger valueof time. The test case indicates that one set of trac management prole may favor one userclass more than another, while this may also turn around as trac conditions changes. Future
research can focus on more detailed and realistic modeling and analysis based on this framework.

References
Ahn, Kyoungho, Bichiou, Youssef, Farag, Mohamed, & Rakha, Hesham A. 2020. Multi-objective EcoRouting Model Development and Evaluation for Battery Electric Vehicles.
De Jong, Gerard, Kouwenhoven, Marco, Bates, John, Koster, Paul, Verhoef, Erik, Tavasszy, Lori, &Waremius, Pim. 2014. New SP-values of time and reliability for freight transport in the Netherlands.
Transportation Research Part E: Logistics and Transportation Review, 64, 7187.
Kouwenhoven, Marco, de Jong, Gerard C, Koster, Paul, van den Berg, Vincent AC, Verhoef, Erik T,Bates, John, & Waremius, Pim MJ. 2014. New values of time and reliability in passenger transportin The Netherlands. Research in Transportation Economics, 47, 3749.
Lin, Xiao, Leclercq, Ludovic, Tavasszy, Lorant, & van Lint, Hans. 2021. Formulating multi-class user
equilibrium using mixed-integer linear programming. Submitted to a journal.
Schreiter, Thomas. 2013. Vehicle-class specic control of freeway trac. Ph.D. thesis, Delft Universityof Technology.
Statistics Netherlands (CBS). 2020. Goods transport; modes and ows of transport to and from theNetherlands. Tech. rept. url: https://www.cbs.nl/en-gb/gures/detail/83101ENG, Accessed on: 2411-2021.
Wismans, Luc Johannes Josephus. 2012. Towards sustainable dynamic trac management. Ph.D. thesis,
University of Twente, Enschede, Netherlands.
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