104

historical traffic data with a limited increase in computational timeand memory, and (b) utilize Google Maps open-source applicationprogramming interface (API) and network data to produce distanceand travel time matrices. To the authors’ knowledge, no researcheffort has integrated time-dependent routing algorithms, historicaltraffic data, real-world road network data, and public open-sourceAPIs to incorporate the impact of congestion on delivery routes.

LITERATURE REVIEW

This section covers two main areas of research: (a) the effects ofcongestion and travel time variability on vehicle routes and logisticsoperations and (b) TDVRP solution algorithms and their applicationto urban areas.

Direct and indirect costs of congestion on passenger travel time,shipper travel time and market access, production, and labor produc-tivity have been widely studied and reported in the literature. Thework of Weisbrod et al. provides a comprehensive review of this lit-erature (1). Substantial progress has been made in the developmentof econometric techniques to study the joint behavior of carriers andshippers relative to congestion (2, 3).

Survey results suggest that the type of freight operation has a sig-nificant influence on how congestion affects carriers’ operations andcosts. Survey data from California indicate that congestion is per-ceived as a serious problem for companies specializing in LTL, refrig-erated, and intermodal cargo (4). Similar conclusions are reached byreports analyzing the effects of highway limitations and traffic con-gestion in the Portland, Oregon, region (5, 6). Golob and Regan iden-tified a positive relationship between the level of local congestion andthe purchase of routing software (7). Carriers that do not follow reg-ular routes, such as for-hire carriers, tend to place a higher value onthe use of real-time information to mitigate the effects of congestionand logistical services to plan fleet deployments (8). Other researchersattribute the scant usage of TDVRP algorithms to the lack of reliabletime-dependent travel time data, which can be particularly expensiveor difficult for small carriers to obtain (9). These authors recommendthe implementation of open-access online TDVRP and data servicesto increase the efficiency of routes in congested urban areas.

Another line of research investigated carrier reactions to toll mea-sures intended to shift freight traffic to off-peak hours. Holguin-Veraset al. investigated the effects of congestion charges in New York Cityand found that delivery times are heavily dictated by customer timewindows (10). Congestion charges increase carriers’ operating costswhile inducing little shifting of deliveries from peak to off-peak hours.This suggests an inelastic relationship between freight congestioncharges and routes with time-definite delivery times. Quak and Koster

Algorithms to Quantify Impact ofCongestion on Time-Dependent Real-WorldUrban Freight Distribution Networks

Ryan G. Conrad and Miguel Andres Figliozzi

Urban congestion presents considerable challenges to time-definite trans-portation service providers. Package, courier, and less than truckloadoperations and costs are severely affected by growing congestion levels.With congestion increasing at peak morning and afternoon periods, pub-lic policies and logistics strategies that avoid or minimize deliveries dur-ing congested periods have become crucial for many operators and publicagencies. However, in many cases these strategies or policies can intro-duce unintended side effects, such as higher labor costs, shorter workinghours, and tighter customer time windows. Research efforts to analyzeand quantify the impact of congestion are hindered by the complexities ofvehicle routing problems with time-dependent travel times and the lackof networkwide congestion data. Research used real-world road networkdata to estimate travel distance and time matrices, land use and customerdata to localize and characterize demand patterns, congestion data froman extensive archive of freeway and arterial street traffic sensor data toestimate time-dependent travel times, and an efficient time-dependentvehicle routing (TDVRP) solution method to design routes. Novel algo-rithms were developed to integrate real-world road network and traveldata to TDVRP solution methods. Results show the impact of congestionon depot location, fleet size, and distance traveled.

Congested urban areas present considerable challenges for less-than-truckload (LTL) carriers, courier services, and industries that requirefrequent and time-sensitive deliveries. With congestion increasing atpeak morning and afternoon periods, public policies and logisticsstrategies that avoid or minimize deliveries during congested periodshave become crucial for many operators and public agencies. How-ever, in many cases, these strategies and policies can introduceunintended side effects, such as higher labor costs, shorter workinghours, and tighter customer time windows.

Although current research on vehicle routing algorithms is exten-sive, much less attention has been given to investigation of the impactof congestion on carrier operations. Furthermore, most algorithmsfor solving the time-dependent vehicle routing problem (TDVRP)found in the existing literature do not deal with the estimation ofdistance and time-dependent travel time matrices. Thus, this researchfocuses on two primary objectives: (a) develop efficient algorithmsto apply TDVRP solution methods to actual road networks by using

Department of Civil and Environmental Engineering, Maseeh College of Engineeringand Computer Science, Portland State University, P.O. Box 751, Room 301-D,1930 SW Fourth Avenue, Portland, OR 97207. Corresponding author: M. A. Figliozzi,figliozzi@pdx.edu.

Transportation Research Record: Journal of the Transportation Research Board,No. 2168, Transportation Research Board of the National Academies, Washington,D.C., 2010, pp. 104–113.DOI: 10.3141/2168-13

presented a methodology to quantify the impact of delivery con-straints and urban policies by using a fractional factorial regression(11). They found that vehicle restrictions and delivery curfews havea compounding effect on customer costs, whereas vehicle restrictionsalone are costlier only when vehicle capacity is limited. There is littleresearch into the effects of congestion on vehicle route characteristics.Figliozzi analytically modeled routes (12), extending Daganzo’s con-tinuous approximations (13), and analyzed how routing constraintsand customer service durations affect route characteristics by usinga classification based on supply-chain characteristics. This analysisshowed that a decrease in travel speed severely affects total distancetraveled for routes with time window constraints, whereas capacityconstrained routes are less affected. The impact of travel time reliabil-ity on LTL delivery was also analyzed by using continuous approxi-mations and real-world data (14). This research concludes that traveltime variability has a significant impact on carrier costs when averagedistance to delivery areas increases and average travel speed decreases.

Classic versions of the vehicle routing problem (VRP) such asthe capacitated VRP or VRP with time windows (VRPTW) have beenwidely studied. However, time-dependent problems have receivedconsiderably less attention. Figliozzi presented a comprehensivereview of TDVRP approaches and an efficient TDVRP algorithm(15); that work also creates benchmark problems for the TDVRPaltering the classical VRPTW Solomon instances. Fleischmann et al.reviewed the adaptation of the VRP algorithms to time-dependentdata from traffic information systems in the city of Berlin (16). Egleseet al. analyzed the construction of a time-dependent travel time data-base (17 ) by using Dijkstra’s algorithm for time-dependent links.Eglese et al. applied their methodology to a real-world network inEngland. However, these research efforts do not incorporate into theiranalyses the influence of time windows and recurring bottlenecks orthe impact of congestion on fleet size and total distance traveled.

PORTLAND, OREGON, CASE STUDY

Considered a gateway to international sea and air freight transport, thecity of Portland, Oregon, has established itself as an important hub forinternational and domestic freight movements. Its favorable geogra-phy for both international ocean and domestic river freight via theColumbia River is complemented by its highway connections. I-5 isthe most important freeway connecting the West Coast from Mexicoto Canada as well as southern California ports and main West Coastpopulation centers (5). The I-5 freeway is also used by many carriersdelivering in Portland and the city’s surrounding suburbs becauseit provides the main north–south freight corridor through the cityof Portland.

Recent increases in regional traffic congestion have negativelyaffected freight operations. A recent report investigated the effects ofcongestion on Portland-area businesses and LTL deliveries (5). Thatreport provides insightful yet qualitative information on various strate-gies used by businesses to cope with congestion, additional deliverycosts, and uncertainty. The report indicates that congestion has madesome afternoon deliveries infeasible, which requires that deliveries bemade during nonbusiness hours early in the morning. However, avoid-ing congesting by shifting deliveries to early morning generates addi-tional costs by reducing route durations. In some cases, early deliveriesare not feasible near residential areas where parking problems andnoise can lead to sound or traffic ordinance violations and conflictswith residents (5).

The recurrent effects of traffic congestion at peak periods presentdaily challenges to LTL carriers in the Portland metropolitan area,

Conrad and Figliozzi 105

represented by a numerical analysis presented in a later section.Customer data and depot locations are generated by using a land usezoning map of the Portland metropolitan area. Network and conges-tion data sources, including recurrent bottlenecks, are described, as isa methodology for applying TDVRP algorithms to real-world net-works. The methodologies and algorithms developed in this researchassume that customers’ demands and time windows are known a pri-ori, for example, the night before delivery. Congestion data relatedto nonrecurrent conditions, such as due to accidents, is not analyzedand is left as a future research topic.

DATA SOURCES

Two main data sources were used in this research: Google MapsAPI for implementation of the TDVRP algorithm and the PortlandTransportation Archived Listing (PORTAL) for obtaining historicaltravel time data.

Overview of Google Maps API

The Google Maps API allows access to the up-to-date street networkin the studied region with a high level of geographical detail. Theopen-source application allows for considerable freedom in modify-ing the program and user interface (18). Figure 1 shows the processof creating customer distributions and obtaining optimized routesfrom the TDVRP algorithm as implemented with the API. The APIconsists of several interfaces:

• A customer selection screen where a set of customers and asingle depot can be created by clicking on locations on the map. Acoordinate output is provided that is then copied into a text (.txt) file.

• An interface that calculates the shortest paths between pairsof customers and constructs the distance and travel time origin–destination matrices. Distance and travel time matrices are estimatedand stored as text files.

• Travel speed, occupancy and vehicle flow data from trafficsensors used to incorporate the impact of congestion on traveltimes.

• A solution interface where solution sets output from the TDVRPalgorithm can be loaded and plotted to provide a visual verification ofresults.

Perhaps the greatest advantage of the API is that the open-sourcesoftware and high-quality network data can be accessed free ofcharge from http://code.google.com/apis/maps. This together withthe TDVRP solution algorithm developed to interface with the APIoffers very-low-cost solutions for route planning and optimizationwhile allowing access to detailed and accurate network data such asroad hierarchy and restrictions (e.g., one-way streets or no-left-turnmovements at intersections). The effects of congestion are includedby modifying the travel times initially calculated by Google Maps.After the TDVRP algorithm design the routes, the API interface canbe used to obtain detailed driving directions.

Simulating Congestion Effects

Google Maps provides reasonable travel time estimations duringuncongested periods. However, to increase the accuracy of travel timeestimations, highway sensor data are used. For example, segments

106 Transportation Research Record 2168

PORTAL Traffic Data Tables

• Travel Speed

• Occupancy

• Vehicle Flow

OutputCustomer Coordinates

SelectCustomers

Customers can be selected by clicking once on a location on the map. The first selection is the depot. Alternatively, customer locations in decimal latitude-longitude format can be uploaded.

OutputDistance O-D Matrix

OutputTravel Time O-D Matrix

VRP Algorithm

Kk ji KkA Cj

kj

kknt

kij

kijd

Kk Cj

kj

xyycxd

x

,001

0

Minimize c

Minimize

Speed Function

Free-flow Speeds(O-D Matrices)

ij

ijij

t

du

s �uij , vkm �

dij

tij

Optimized Routesand Performance

Measures

Adjustable Parameters

•Speed Parameter

•Occupancy Threshold

•Avg. Vehicle Spacing

•Queuing Parameter

O-DMatrices

The customer coordinate text file is uploaded to the Driving Distance screen where Google Maps calculates the free-flow travel time and distance between each pair of customers.

CalculateResults

Travel time data from PORTAL for the bottleneck locations and the free-flow speed O-D matrix are inserted into the speed function. Bottleneck coordinates are entered separately into the VRP algorithm along with the speed function. Optimized routes can be displayed in a route plotting interface in Google Maps. Results are also provided for several performance criteria including number of required vehicles, total driving distance and travel

Display Routes in Google Maps Interface

Map data © Tele Atlas

Map data © Tele Atlas

pnV

pnO

p1nU

nv

nO

nR

nL

FIGURE 1 Overview of TDVRP solution methodology and integration of Google Maps API.

Conrad and Figliozzi 107

Data obtained from PORTAL are also used to model the impactof traffic queuing on the surrounding network. The areas of reducedtravel speed for each bottleneck location are assumed as a functionof the measured occupancy and vehicle inflow and outflow ratesat each bottleneck location. Research has shown that traffic queuesoften begin to form at occupancies approximately equal to or greaterthan 20% (20), but according to speed flow data, queues may format occupancies as low as 13%. From these queuing concepts andassumptions, the radius of the area of travel speed reduction aroundeach bottleneck where vehicle travel speed reduced is varied in pro-portion to the difference in the inflow and outflow rates multiplied byaverage vehicle spacing when the occupancy is above a certain thresh-old value. Strictly, this assumes that there is conservation of vehicles(i.e., no vehicles enter or exit the road segment in question) andignores the presence of moving traffic queues.

The travel speeds used in this research are calculated from 15-minarchived travel time data averaged for the year 2007 along the I-5 free-way corridor from the Portland suburb of Wilsonville to Vancouver,Washington. These data are sufficient to demonstrate the proposedmethodology, but consideration of seasonal or monthly variability intravel time is important for many LTL carriers and is feasible viaPORTAL. In this research, it is assumed that carriers account for onlyrecurrent congestion and plan their routes the night before making thedeliveries.

METHODOLOGY

TDVRP Algorithm Overview

Figliozzi presented a description of the TDVRP algorithm used in thisexperiment, along with a full TDVRP formulation (15). Because ofhard time window constraints, the primary objective is the minimiza-tion of the number of vehicles or routes; the secondary objective isminimization of the travel time or distance. The TDVRP solution algo-rithm consists of a route construction phase and a route improvementphase, each using two separate algorithms (Figure 3). During routeconstruction, the auxiliary routing algorithm Hr determines feasibleroutes with the construction algorithm Hc assigning customers andsequencing the routes. Route improvement is done first with theroute improvement algorithm Hi, which compares similar routes and

FIGURE 2 Example with bottleneck locations and areas of effective travel speed reduction.

AuxiliaryRouting

Algorithm

rH

RouteConstruction

Algorithm

cH

RouteImprovement

Algorithm

iH

Service TimeImprovement

Algorithm

yH

Arrival TimeAlgorithm

Departure TimeAlgorithm

PORTAL Data

Google Maps APITravel Speed Data

Route Construction Route Improvement

yfH

ybH

ycH

FIGURE 3 TDVRP solution method.

along I-5 near traffic bottlenecks are selected to represent areas ofdecreased travel speed. The selected segments are between free-way interchanges or on and off ramps where vehicle detector loopsare located.

Detailed traffic data are obtained from PORTAL, Portland’s imple-mentation of an archived data user service, which coordinates andobtains data from approximately 436 inductive loop detectors alongInterstate freeways in the Portland metropolitan area. Bertini et al.give a description of this transportation data archive (19). Bottlenecksare modeled as point locations surrounded by areas of reduced travelspeed. Travel in proximity to a bottleneck is expressed as a percentreduction in travel speed proportional to the speed reduction at thebottleneck location. Figure 2 shows the bottleneck locations andareas of effective travel speed reduction.

consolidates customers into a set of improved routes. The servicetime improvement algorithm Hy eliminates early time windowviolations and then reduces the route duration without introduc-ing additional early or late time window violations; these tasks areaccomplished by using the arrival time and departure time algo-rithms Hyf and Hyb, respectively, and customers are subsequentlyresequenced as necessary. It is with these algorithms that the POR-TAL data and shortest-path travel speeds generated by the GoogleMaps API are inserted into the solution algorithm.

Notation

For the following travel time algorithms, the total depot working time[e#, l#] is partitioned into a set of p time periods Tp = {T1, T2, . . . , Tp}.Each traffic bottleneck location βm ∈ βn = {β1, β2, . . . , βn} is assignedthe following data at each time partition Tk ∈ Tp:

Opn = [Okm]p×n, table of occupancy values for each time period

Tk ∈ Tp and bottleneck βm ∈ βn;Up

n�1 = [Ukm]p×n+1, table of vehicle flow inflow and outflow ratesfor each time period and bottleneck location; the inflowand outflow rates at time period Tk for bottleneck βm areUkm and Uk,m+1, respectively; and

vpn = [vkm]p×n, table of congested travel speeds obtained from

PORTAL.

All data are collected from PORTAL, and the point source locationof each traffic bottleneck is assumed to be midway between detectorloops. The algorithms include the following adjustable parameters foreach bottleneck location:

108 Transportation Research Record 2168

R_

m ∈ R_

n = {R_

1, R_

2, . . . , R_

m, . . . , R_

n}; set of initial radius valuesat time t = 0;

L_

m ∈ L_

n = {L_

1, L_

2, . . . , L_

m, . . . , L_

n}; set of average vehiclespacing values;

O_

m ∈ O_

n = {O_

1, O_

2, . . . , O_

m, . . . , O_

n}; set of threshold occupancypercentages that determine the expected onset oftraffic queuing; and

v_

m ∈ v_

n = {v_

1, v_

2, . . . , v_

m, . . . , v_

n}; set of free-flow speeds.

Table 1 contains a complete listing of variable and functiondefinitions.

Traffic Queuing Algorithm

The following is a summary of the Hyc algorithm that assembles a tableof bottleneck radii Rkm for each bottleneck βm and time period Tk.The algorithm requires the input data arrays Op

n and Upn�1 as well as the

adjustable parameters R_

n, L_

n, and O_

n. The output table Rpn contains

the radius value for each time period Tk at each bottleneck βm in ap × n array. Beginning with the conditional statement within thenested for loop for a particular βm and starting at t = 0, the algorithmcan be described as follows:

1. First assign the variable R the base parameter value R_

m at t = 0.2. Begin the k iteration; if the occupancy Okm at a given k iter-

ation is greater than the threshold value O_

m, add the differences inthe outflow and inflow traffic volumes multiplied by the durationof the time partition tk

_ − tk by the average vehicle spacing L_

m tothe variable R.

TABLE 1 Notation

Definition

Variables

i, j, m

βi = (xi, yi); βj = (xj, yj); βm = (xm, ym)

aj

bi

ei

gi

d

dij

tij

uij =

Array and Vector Quantities

Tk ≡ [tk, tk] ∈ Tp = {T1, T2, . . . , Tp},–Rm ∈–

Rn = {–R1,

–R2, . . . ,

–Rn}

–Lm ∈––

Ln = {–L1,

–L2, . . . ,

–Ln}

–Om ∈ –

On = {–O1,

–O2, . . . ,

–On}

–vm ∈ –

vn = {–v1,

–v2, . . . ,

–vn]

Ukm, Uk,m+1 ∈Upn+1 = [Ukm]p×n+1

Opn = [Okm]p×n

vpn = [vkm]p×n

Function

f(xµ, xv, yµ, yv)

d

tij

ij

Indices for set of consecutive customers (i, j) and bottlenecks (m)

Geographic coordinates of customer i, customer j and bottleneck m, respectively

Arrival time at customer j

Departure time from customer i

Lower time window for customer i

Service time at customer i

Iterated driving distance variable

Driving distance between customers i and j calculated by the Google Maps API

Free-flow travel time between customers i and j calculated by the Google Maps API

Free-flow speed used in TDVRP algorithm

Set of time periods as fraction of depot working time

Set of initial radius values at each bottleneck location at time t = 0

Set of average vehicle spacing values for each bottleneck location

Set of threshold occupancy percentages that determine the expected onset of traffic queuing

Bottleneck speed parameters

Table of vehicle flow inflow and outflow rates for each time period and bottleneck

Table of occupancy values for each time period and bottleneck

Speed at bottleneck Bm for the kth time period entered as a p × n array

Euclidean distance between two sets of x and y coordinates

3. If the occupancy Okm is less than O_

m and the radius variable Ris greater than the base parameter R

_m, then subtract the quantity from

Step 2 from R.4. Take the maximum of the set [R, R

_m]; this and the second con-

dition of Step 3 prevent R from being assigned a negative value andensure that R

_m is a lower bound for the variable R when the predicted

traffic queue is dispersing.5. Otherwise, retain R = R

_m.

6. Construct a column vector Rp of R values obtained from eachk iteration.

7. Repeat Steps 1 through 6 n times and construct the outputmatrix Rp

n from the column vectors Rp obtained from each iteration.

The Hyr algorithm adds or subtracts expected lengths of trafficqueues to the radius of the effective area of each bottleneck that isdependent on whether the measured occupancy is above or beloweach threshold value contained in O

_n. The table of values in Rp

n isreferenced by the Hyf and Hyb algorithms described in detail in thefollowing section. The objective is to extrapolate travel time trendsfrom the data that are available and apply them to the surroundingroad network.

Arrival and Departure Time Algorithms

The following is a summary of the arrival time and departure timealgorithms Hyf and Hyb adapted from Figliozzi (15) that estimatetravel times between pairs of customers βi and βj by using the traveltime data. The Hyf algorithm calculates the expected arrival time ata customer βj when departing from a previous customer βi by usinga forward-iterative process. Similarly, the Hyb algorithm uses a back-ward iterative process and simultaneously calculates the requireddeparture time from customer βi to reach customer βj.

The impact of bottlenecks as vehicles are moving through differentperiods is a function of the estimated distance between the vehicle andthe bottleneck at the beginning of each time period. A linear approx-imation of the vehicle location is used to reduce computationalcomplexity because shortest-path and Euclidean distances are highlycorrelated. High levels of correlation between Euclidian and shortest-path distances are usually found in urban areas (21). The distance trav-

Conrad and Figliozzi 109

eled along the Euclidean connecting line is calculated as a percentageof the actual route traversed such that

where

Dij = Euclidean distance between customers i and j, dij = shortest-path driving distance from customer i to customer j

calculated by the API, andd = iterated distance from i to j along the actual driving route (a

derivation of this function is given in the appendix).

With the law of cosines (see Figure 4), the distance from a pointon the Euclidian connecting line to each bottleneck at a given timeiteration in the forward iterative calculation can be shown to be

where Dim and Djm are Euclidean distances between customer i andbottleneck βm and customer j and bottleneck βm, respectively.

Similarly, for the backward iterative process of the departure timealgorithm, the distance from the nearest bottleneck is

The travel speed function sm is applied at each time iteration Tk andcalculates a speed value for each bottleneck. This function calculatescongested travel speeds sm as reductions in the API-derived speed uij

proportional to the speed reduction measured at the traffic bottleneckssuch that sm/uij = vkm/v

_m if the virtual location on the Euclidean connect-

ing line is within the radius Rkm. Here vkm is the time-varying speedobtained from PORTAL, and v

_m is an adjustable parameter that may

represent the freeway free-flow speed. In other words, the reduction intravel speed due to congestion in the surrounding network is assumedto be proportional to the reduction observed from the PORTAL

rd

dD D

d D D D

dmij

ij im

ij im jm

ij

=⎛⎝⎜

⎞⎠⎟

+ −+ −( )2

22 2

(33)

rd

dD D

d D D D

dmij

ij jm

ij jm im

ij

=⎛⎝⎜

⎞⎠⎟

+ −+ −( )2

22 2

(22)

′ =dd

dD

ijij ( )1

dij : Actual route Dij : Euclidean connecting line

dij

dij

d

dij

d dij

d

d Dij

Dij

Dij

Dij 1−dij

d Dij 1

Dim

Dim

rm

rm

rm

Djm

Djm

D2jm

2

Dim2 Dij

2 Djm2 (xi,yi)

(xj,yj)

(xm,ym)

FIGURE 4 Illustration of the method to approximate bottleneck influence.

freeway data at the bottleneck (detector station) with the slowest travelspeed. This function can be expressed as

where rm is the distance from a point along the Euclidean connectingline to a bottleneck βm.

The following is a summary of the Hyf algorithm pseudocode:

1. First determine if the arrival time ai is less than the lower timewindow ei at customer i

– If so, then the vehicle waits and the expected departure timeis ei plus the service time gi.

– If not, then the departure time is simply the arrival time plusthe service time2. Determine k for the discrete time period Tk with bounds [tk–

, tk_]

that the expected departure time bi lies in. This is the initial value forthe iterator in the while loop.

3. Determine the Euclidean distance of each traffic bottleneck tothe location βi = (xi, yi) of customer i; the speed function is calculatedfor each value m and a row vector Sn of speeds is assembled. The ini-tial travel speed of the vehicle in the subsequent forward-iterativeprocess is calculated as the minimum value of Sn, that is, the travelspeed is only as fast as that imposed by the bottleneck with the worsttravel speed (only among the subset of bottlenecks whose area ofinfluence affects the path between customers at a given time).

4. Terminate the while loop when the vehicle has reached its des-tination. In each period, speeds are recalculated and distances accu-mulated until the vehicle has reached its destination. Output: theexpected arrival time aj at customer j when departing from customeri at time bi.

The Hyb algorithm works in a similar fashion: given a customer j atlocation vj with an expected arrival time aj obtained from the Hyf algo-rithm, determine the required departure time bi from customer i atlocation βi to make the trip between βi and βj without allowing for latetime window violations.

Calibration

Travel times can be calibrated by adjusting R_

n, L_

n, O_

n, v_

n parametersas well as the time-dependent travel speeds provided by PORTAL

s

u v

vr R

u r R

m

ij km

mm km

ij m km

=≤

⎧

⎨⎪

⎩⎪ >

( )4

110 Transportation Research Record 2168

(vpn). Directional and time-of-day effects can be incorporated. Mem-

ory requirements are reduced because the algorithms work with onetravel time and distance matrix. Simple linear functions and intuitiveparameters are used to adapt free-flow travel times to congestedconditions.

EXPERIMENTAL SETTING

To test the model with real-world constraints, two delivery periodsare modeled and analyzed: (a) an early morning delivery period thatavoids most of the morning peak hour traffic congestion but withtighter time windows, and (b) an extended morning delivery time thatincreases the feasible working time but with increased travel duringthe morning peak. Figure 5 provides a qualitative comparison of thesimulated delivery times.

A total of 50 customer locations are used (Figure 6) with constraintsassigned according to the zoning criteria. All customers normallyserved after 9:00 a.m. are assumed to be able to shift delivery timesbefore this hour. Time windows of 15 min are randomly assigned toall customer types. Additionally, deliveries to all customers in mixed-use and residential areas are prohibited before 7:00 a.m. to modelrequired compliance with local noise ordinances. In the early-morningdelivery option, this reduces the effective depot working time tojust 2 h for these customers. The extended morning delivery optionprovides a 4-h working time for these customers but includes theeffects of the morning peak-hour congestion to a greater degree. Thecalibration of the model was tested by varying the travel speed param-eters v

_n to alter the simulated travel speed derived from the PORTAL

travel time data and contained in the travel speed table vpn.

EXPERIMENTAL RESULTS

Results comparing the number of vehicles and total distance trav-eled during the morning and extended morning delivery periods arepresented in this section. In addition, to incorporate the impact oftravel time reliability, time-varying travel speeds from PORTAL aredecreased by a coefficient δ. This adjustment maintains the overalltrend in travel speed variation throughout the delivery period butallows for adjustments to the travel time to more accurately reflectreal-world differences between average travel speeds and the actualdistribution of travel speeds. A value δ = 1 utilizes average time-varying travel speed PORTAL data and assumes that no hard time-window violations take place if realized travel times are at leastthe average travel speed. However, if the carriers would like to

Constrained Customers in (Residential areas)

Early Morning Delivery Extended Early Morning Delivery

03:00 07:00 09:0005:00

Congestion intensifies

03:00 07:00 09:00 11:00

15 min. timewindows

15 min. timewindows

FIGURE 5 Modeled delivery periods, constrained customers, and time window constraints.

account for travel time unreliability, a value of δ < 1 can be used inthe calculations as follows:

A value of δ < 1 guarantees a higher value of customer service (14).The sensitivity to travel time unreliability and buffer times was testedby setting the parameter δ = {0.4, 0.6, 0.8, 1}.

Impact of Congestion on Number of Vehicles

For the number of required vehicles (Figure 7), the central depotshowed less sensitivity to changes in travel time reliability than didthe suburban depot. As expected (14), reduced travel speed appearsto have a greater impact on fleet size when the depot has a suburbanlocation. The number of vehicles required is consistently less for theextended early-morning delivery period, and a larger fleet is stillrequired when the depot has a suburban location.

Impact of Congestion on Total Distance Traveled

Figure 8 compares total vehicle miles traveled (VMT). Similar to therequired number of vehicles, total VMT is significantly higher for

s

u v

vr R

u r R

m

ij km

mm km

ij m km

=≤

⎧

⎨⎪

⎩⎪

δ

>

( )5

Conrad and Figliozzi 111

tours originating at the suburban depot location. Constrained ser-vice times for customers in the early-morning delivery period alsoappear to affect total VMT to a slightly greater extent than doestravel speed.

CONCLUSIONS

This research proposed a new methodology for integrating real-worldroad networks and travel data to time-dependent vehicle routingmethods. The use of traffic sensor data and Google Maps API pro-vides a unique approach to interface routing algorithms, travel time,and congestion data. Intuitive algorithms and parameters are usedto incorporate the effects of congestion on time-dependent traveltime matrices. The proposed methodology is a significant improve-ment for representing the impact of congestion in congested urbanareas leveraging on existing open-source data and applications.

The results show the dramatic effects of congestion on carrier fleetsizes and distance traveled. The results also suggest that congestionhas a significant impact on fleet size, particularly for depots locatedin suburban areas outside the customer service area.

APPENDIX A

The following is the derivation of the bottleneck distance function rfor the forward-iterative calculation in the arrival time algorithm.An identical argument with the distance d iterated in the backward

CentralDepot

Customers with servicetime constraints

CentralDepot

SuburbanDepot

FIGURE 6 Customer service area and depot locations.

(a)

(b)

Travel Time Reliability (δδ)

40%0

2

4

6

8

10

12

14

16

60% 80% 100%

Nu

mb

er o

f V

ehic

les

Morning

Ext. Morning

Travel Time Reliability (δ)

40%0

5

10

15

20

25

60% 80% 100%

Nu

mb

er o

f V

ehic

les Morning

Ext. Morning

FIGURE 7 Effects of congestion on fleet size: (a) central depot and (b) suburban depot.

(a)

(b)

Travel Time Reliability (δδ)

40% 60% 80% 100%0

50100150200250300350400450500

To

tal D

ista

nce

Tra

vele

d (

mile

s)

CentralDepot

SuburbanDepot

0

100

200

300

400

500

600

700

800

900

To

tal D

ista

nce

Tra

vele

d (

mile

s)

CentralDepot

SuburbanDepot

Travel Time Reliability (δ)

40% 60% 80% 100%

FIGURE 8 Effects of congestion on total VMT: (a) extended morning deliveryperiod and (b) early morning delivery period.

direction from a customer j to i obtains the bottleneck distancefunction for the departure time algorithm in a trivial manner.

Let θim be the angle opposite Dim, the Euclidean distance fromcustomer i to bottleneck Bm. With the law of cosines, D2

im = D2ij +

D2jm − 2DijDjmcos(θim),

θim is also the angle opposite to r. Equating r and Equation A-1, andwith the law of cosines again,

ACKNOWLEDGMENTS

The authors thank the Oregon Transportation, Research and Educa-tion Consortium, the Port of Portland, and Portland State UniversityResearch Administration for sponsoring this research. The authorsalso thank Myeonwoo Lim of the Computer Science Department atPortland State University for assisting with the computer program-ming aspects of this project and Nikki Wheeler of the Departmentof Civil and Environmental Engineering at Portland State Univer-sity for compiling and aggregating the PORTAL travel time dataused in the final computer model.

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The Urban Freight Transportation Committee peer-reviewed this paper.