A Data-thrifty Approach to Routing Optimization

Thomas Fayolle, Renaud de Landtsheer, Gustavo Ospina and Fabian Germeau

CETIC Research Centre, Charleroi, Belgium

Keywords:

Local Search, PDPTW, Data, Routing Optimization, OscaR, Cbls.

Abstract:

Solving a routing optimisation problem often requires to know the travel times between each pair of points of

the problem. Usually, when solving a routing optimisation problem, the travel time is assumed to be constant.

However, in real life problems, it can vary a lot due to trafﬁc jams, especially near big cities. Some map data

providers can provide accurate travel time estimations, but the data are expensive and querying such a data

provider is time consuming. In this paper, we present a method to solve the Pickup and Delivery Problem with

Time Windows (PDPTW) that reduces the number of calls to paid data providers while preserving the quality

of the solution.

1 INTRODUCTION

Given a set of pickup points and, for each pickup

point, an associated delivery point, solving the Pickup

and Delivery Problem (PDP) (Toth and Vigo, 2014)

consists in ﬁnding the shortest path that serves all the

pickup and the delivery points. The pickup point and

its associated delivery point must be served in that or-

der by the same vehicle. In the Pickup and Delivery

Problem with Time Windows (PDPTW), a time win-

dow can be associated to each pickup point and each

delivery point: the point must be served before the

latest time of the time window and it can be reached

before the earliest time of the time window, but in that

case the vehicle must wait until the earliest time be-

fore leaving.

Solving the PDPTW requires to know the travel

times between each pair of points. In the classical ver-

sion of the problem, the travel times are often given

as a matrix of constant values representing the travel

times between each pair of points. However, in real

life problems, travel times may signiﬁcantly change

depending on the time of the day, especially near big

cities. This problem is not new and has been deﬁned

in the most general form of Vehicle Routing Problem

by (Malandraki and Daskin, 1992). This paper and

many others that propose solutions for Time Depen-

dent Vehicle Routing Problem variants assume that

the time-dependent travel time is known when the op-

timization starts.

Unfortunately, the data representing the travel

throughout the day is a very large histogram. The eas-

iest solution is to buy those data from a map provider,

as main map data providers include accurate travel

times that take trafﬁc into account. The problem with

map providers is that they are ﬁnancially expensive

and computationally slow. For instance, getting a

100x100 data matrix for every 30 minutes of the day

costs hundreds of euros and takes more than an hour.

On the other side, free data providers (e.g. Open-

StreetMap (OpenStreetMap, 2004)) combined with

source free path ﬁnding algorithms (e.g. (Luxen and

Vetter, 2011) or (GraphHopper, 2018)) can provide

travel times without taking trafﬁc into account.

In this paper, we introduce a method that uses ap-

proximated data and a local search engine in order to

solve a PDPTW while reducing the amount of data

that is needed to perform the optimization. The ap-

proach is to repeatedly perform optimization on ap-

proximate data that are both cheap and fast to obtain.

This optimization leads to a route and high-quality

data is queried only for the travels belonging to this

route. The approximate data is then corrected with a

high-quality data that takes trafﬁc into account. The

process is started again until the generated route only

contains high quality data. The method has been ap-

plied to a real life PDPTW problem with real life data.

In this example, it was able to signiﬁcantly reduce the

amount of data used while preserving the quality of

the solution, the downside being that the optimization

–so far– seems much slower with our approach.

Fayolle, T., De Landtsheer, R., Ospina, G. and Germeau, F.

A Data-thrifty Approach to Routing Optimization.

DOI: 10.5220/0010308504330437

In Proceedings of the 10th International Conference on Operations Research and Enter prise Systems (ICORES 2021), pages 433-437

ISBN: 978-989-758-485-5

433

2 RELATED WORKS

In a survey on vehicle routing problems for city lo-

gistics, (Cattaruzza et al., 2017) identify four main

challenges in the optimization of vehicle routes in ur-

ban area. One of the challenges is the optimization

with time-dependent travel times. They conclude that

“considering time-dependent travel times is essential

when solving VRP in cities” and that it should “be-

come a must-have attribute in the future”.

Effectively, (Donati et al., 2008) compute solu-

tions to a PDP using constant travel time on one hand

and time-dependent travel time on the other hand.

Then they evaluated the solution computed with con-

stant travel time in the context of time dependent

travel time. They show that approximating trafﬁc-

dependent travel time through constant travel time de-

livers a solution roughly 8% worse (up to 12%) than

if the problem was solved with accurate data.

In their paper, (Sun et al., 2018) solve PDPTW

with time-dependent travel times using a branch and

price algorithm that give an exact solution. Like many

papers that address a VRP with time-dependent travel

times, they assume that the travel time function is

known at the beginning of the route optimization.

To the best of our knowledge, we do not know any

paper that solves a Vehicle Routing Problem while

facing the lack of accurate data.

3 OPTIMIZATION WITH

(UNDER-)APPROXIMATED

DATA

This section presents our approach. Section 3.1

presents the requirements that shall be fulﬁlled by

the method, section 3.2 presents an overview of the

method and sections 3.3 and 3.4 discuss the effect of

data enrichment methods and stop conditions on the

approach.

3.1 Requirements

The method aims at solving a routing optimization

problem while respecting two main requirements: (1)

the method shall reduce the amount of trafﬁc-aware

data compared to an approach requiring the com-

plete information prior to the optimization and (2) the

method shall ﬁnd a solution that is as good as the so-

lution found when knowing all the data.

The method presented in section 3.2 repeatedly

uses a routing optimization engine. In the exam-

ple detailed in section 4, the local search engine Os-

caR.cbls (OscaR Team, 2012) has been used. Thus,

getting the optimal solution is not guaranteed, but re-

quirement (2) means that the search method presented

in this paper should not give a worse solution than a

solution obtained with the same routing optimization

engine and all the needed data.

3.2 Overview of the Method

The whole process of the search method is depicted

in Figure 1.

The search starts with an under-approximated

time matrix. Such a matrix is obtained using a free

map provider and open source path ﬁnding algorithm.

The travel times obtained using this method do not

take trafﬁc data into account. We assume that the real

travel time is greater or equal to the travel time com-

puted this way. The method is based on the assump-

tion that the use of an under-approximated time ma-

trix guarantees the requirement (2). The idea behind

this assumption is that an under-approximated travel

time makes time windows constraints more permis-

sive; any solution of the problem that is acceptable

wrt. the strong constraints with a real travel time data

shall be accepted with the under-approximated time.

One of the purpose of the experimentation presented

in section 4 is to test if this claim is reasonable.

The under-approximated time matrix is used in

a local search engine to obtain a solution, which is

computed using approximated travel time. In order to

have a more accurate solution, the travel times of this

solution are updated using a high-quality data. Af-

ter the update, the solution has signiﬁcantly changed,

and may even not be feasible (for example because

some time window constraint is not respected any-

more). However, updating the solution gives us more

information about the travel times. The new infor-

mation can be used to enrich the knowledge about the

travel times. Several solutions for the enrichment pro-

cess are presented in section 3.3.

After enriching the knowledge, a new search can

be launched. Since now we have more data, this

search will ﬁnd either a different solution or a solu-

tion with accurate travel times (or more likely an hy-

brid solution with both different and similar parts with

accurate travel times). The new solution is updated

using a paid data provider, the knowledge is enriched,

and a new search begin.

The process is repeated until the solution obtained

from the local search engine is satisfying. The stop-

ping criterion of the search is discussed in section 3.4.

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434

Get Approximate

Data

Free Provider

Paying ProviderPaying Provider

(Accurate Data)

Data Providers

Routing

Optimization

Solution

Stop?

Enrich

knowledge

Yes

Figure 1: Solving the PDPTW with approximated travel times.

3.3 Enriching the Knowledge

When a solution is obtained after a local search, the

data are enriched with a paying provider, in order to

have the accurate travel times for that solution. The

new data obtained from the paying provider can be

used to enrich the knowledge. The most naive idea

to enrich the knowledge is to store the data received

from the paying provider in order to use the real data

the next time the travel time is needed in the same

hour. In less naive versions, the data can be used to

get a better approximation of the travel time on other

points of the problem: we can propose landmark-

based and learning-based approaches.

Algorithms like landmarks (Goldberg and Harrel-

son, 2003) can further improve the quality of the ap-

proximate data by performing data extrapolations. It

uses the triangle inequality to deduce sound under-

approximate travel time out of known high-quality

travel times. This idea is that given the distance from

a landmark L to point A and a point B, we know that

(A → B) ≥ (L → A) − (L → B) so (L → A) − (L →

B) is an under approximation of A → B. Since the

landmark generates under-approximated values, they

should preserve the quality of the solution. On the

other side, the under-approximation may be really low

(0 if L is equidistant from A and B)

Data learning methods could be used too to get

a better approximation than the travel time without

trafﬁc data. Since the search is started with a bet-

ter approximation, it should take a smaller number

of iterations and thus less time to arrive to a solu-

tion that is computed with real travel time, but learn-

ing methods do not guarantee to provide sound under-

approximated values and the impact on the quality of

the solution should be leveraged.

3.4 Stop Condition

The stop condition can affect the computation time

and the quality of the solution. The strongest stop

condition will be to verify that the solution only relies

on exact travel time data. Weaker conditions could be

used, like checking that the solution is still feasible

after updating the travel times. As usual when using

meta-heuristics, there is a trade-off between computa-

tion time and solution quality. This trade-off usually

depends on the problem speciﬁc situation where the

tool has to be implemented.

4 PRELIMINARY RESULTS

In order to verify the claim that the method presented

in section 3 fulﬁlls the requirements introduced in the

same section, it has been applied on a PDPTW prob-

lem. In this section, we present the problem on which

it has been applied and the preliminary results ob-

tained.

4.1 Benchmark Settings

The PDPTW on which the method has been applied

contains real GPS positions around Brussels, Bel-

gium. It is a shared cab problem, which has about

300 nodes within 98 different geographic locations.

Among the 300 nodes, 14 are vehicle depots; each

vehicle has a capacity that should not be exceeded.

Since it is a pickup and delivery problem, the pickup

node and the delivery shall be served in that order (the

pickup node before the delivery node) and shall be

served by the same vehicle.

A Data-thrifty Approach to Routing Optimization

435

Statistics (mean on 10 runs)

Objective Time

Iteration Free Provider calls Paying Provider calls

Entire Matrix 104 051 534 5’02” - 9 604 451 388

Request when needed 104 429 079 3’28” - 9 604 18 483

Search with approximation 101 848 245 49’12” 10,9 9 604 662

Since the API is simulated, this time does not take into account the time required to get the data from the map provider

Figure 2: Benchmark results.

A time window is associated to each node. The

latest time of the time window is a strong constraint:

the solution cannot be accepted if the node is served

after the latest time of the time window. The earliest

time of the time window is not a constraint that can be

used to reject a solution. However, if a vehicle reaches

a node before the earliest time of the time window, it

shall not leave the node before the earliest time of the

time window. The objective function is the length of

the route (in meters) plus a penalty for the unrouted

nodes.

4.2 The Results

The main purpose of the example described here is

to verify if the method presented in section 3 reduces

the number of calls to a paid data provider and does

not deteriorate the value of the objective function. A

real travel time function matrix has been acquired to

a paid map provider. For each pair of points in the

98 points of the problem, the travel time is asked for

each 30 minutes in the clock of the day (this means we

have 98*98*48 = 460 992 travel times). As a cost and

computation time indication, getting the travel times

costed about 200C and took about 20 hours. The use

of asynchronous calls to the API could reduce this

time to a time between 1 or 2 hours.

With this data, we simulated the map provider

API. The number of calls to the simulated provider

is recorded during the search. The search with ap-

proximation is compared with two different methods.

In the ﬁrst method, the entire data matrix is asked to

the provider. The problem is solved using the entire

matrix. In the second method, the map provider is

called each time a travel time is needed. The search

with approximation is used without a smart enrich-

ment method: when the travel time is requested to the

map provider, it is stored; in further local searches,

the real travel time for a given departure time is used

if it is already known and the travel time at midnight

is used otherwise. The stop condition is strong: the

solution is considered satisfying if the arrival times at

each point of the solution are exactly the same before

getting the real times and after getting the real times.

In other words, the solution is considered as satisfy-

ing if it has been calculated only with real travel times.

For each method, the solution is found using the Very

Large-Scale Neighborhood (VLSN) (Mouthuy et al.,

2011) implemented in the local search framework Os-

caR.cbls.

The results are shown in the table in Figure 2. The

ﬁgures in this table are the average ﬁgures of 10 runs.

The ﬁrst column contains the value of the objective

function for each method. The value of the objective

function is almost identical for the ﬁrst two methods,

and is a bit better with the search with approxima-

tion. As local search engines may fall into a local

minimum, one of the easiest meta-heuristics used to

escape local minima is to restart the search after per-

turbing the solution. This is what is (involuntarily)

done in the search with approximation, and it could

explain why the objective is better.

The second column contains the run time for ev-

ery method. The third column contains the number

of time the loop of Figure 1 is run. The search with

approximation takes signiﬁcantly more time than the

two other methods. It is explained by the fact that it

requires more than 10 iterations of local search on av-

erage where the two other methods only require one

local search. However, the API of the map provider is

simulated, and it does not consider the time required

to request and receive the data from the map provider.

The last column contains the number of API calls.

The calls are separated between the calls for the travel

time at midnight and the calls for the travel time at

other time of the day. The calls at midnight are con-

sidered as free: they can be acquired from a free

data provider. This number is identical, no matter the

method. But the number of calls to a paid provider

is signiﬁcantly reduced when using the search with

approximation: only 3.58% of the calls needed for

the second method are needed for the search with ap-

proximation. On the provider from which we got the

data for this benchmark, it represents a cost of about

0.25C, instead of about 6.7C for the second method

and about 200C for the ﬁrst method.

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436

5 CONCLUSION AND FUTURE

WORK

In this paper, we presented a method to solve a

PDPTW with approximated data that reduces the need

for accurate data from a paid provider. The appli-

cation of the method on an example shows that it

allows to signiﬁcantly reduce the need for data and

that the solution obtained is not worse than a solution

obtained with a full access to the data, even though

this result should be consolidated by giving the same

search time to both method.

The development of the method is an ongoing

work and a there is more to be done. Firstly, the run

time is quite high. This is due to the number of iter-

ations that are needed to obtain a solution computed

with only exact travel times. In the example presented

in this paper, when the travel time is not known, it is

approximated by the time without trafﬁc information,

which can be a bad approximation. Using a better ap-

proximation should reduce the number of iterations.

Approximation using landmarks and learning meth-

ods are mentioned in the paper and should be experi-

mented.

Secondly, the method has been tested with a sim-

ulated API which makes the presented example rather

theoretical. The access to the data is immediate and

there is no network access problem. The method

should be tested with a real API in order to leverage

the impact of getting data from a real web service.

Finally, the results presented here have been ac-

quired with one example and one data set. This data

set is a particular set of points and does not allow to

compare the results with state-of-the-art results. More

examples should be tested in order to validate the con-

clusions of the paper.

ACKNOWLEDGEMENTS

This research was supported by the SAMOBIGrow

CWALITY research project from the Walloon Region

of Belgium (nr. 1910032).

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