Multi-start Approach for Solving an Asymmetric Heterogeneous

Vehicle Routing Problem in a Real Urban Context

José Cáceres-Cruz

, Daniel Riera

, Roman Buil

, Angel A. Juan

and Rosa Herrero

IN3-Computer Science Department, Open University of Catalonia, Barcelona, Spain

Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, Bellaterra, Spain

Keywords: Heterogeneous Vehicle Routing Problem, Asymmetric Cost Matrix, Clarke and Wright, Randomized

Algorithms, Heuristics.

Abstract: Urban transportation is a strategic domain that has become an important issue for client satisfaction in

distribution companies. In academic literature, this problem is categorized as a Vehicle Routing Problem, a

popular research stream that has undergone significant theoretical advances but has remained far from

practice implementations. Most Vehicle Routing Problems usually assume homogenous fleets, that is, all

vehicles are considered of the same type and size. In reality, this is usually not the case as most companies

use different types of trucks to distribute their products. Also, researchers consider symmetric distances

between customers. However, in intra-urban distribution it is more appropriate to consider asymmetric

costs. In this study, we address the Heterogeneous Fixed Fleet Vehicle Routing Problem with some

additional constraints: (a) Asymmetric Cost matrix, (b) Service Times and (c) Routes Length restrictions.

Our objective function is to reduce the total routing costs. We present an approach using a multi-start

algorithm that combines a randomized Clarke & Wright’s Savings heuristic and a local search procedure.

We execute our algorithm with data from a company that distributes food to more than 50 customers in

Barcelona. The results reveal promising improvements when compared to an approximation of the

company’s route planning.

1 INTRODUCTION

In the last years, logistics and transportation

companies are facing growingly demanding

situations with fewer available resources. Market

instability and the competitive business environment

have caused an increasing optimization of logistic

processes. Several fields of research have directed

their efforts to conceive techniques to fulfil this

purpose, like applied mathematics, operations

management and computer sciences. The main

challenge for these theoretical domains is the

consideration of real contexts including real

constraints into their approaches.

Vehicle routing is a complex logistics

management problem and represents a key phase for

the logistic optimization. There are many variations

for the routing problem. Particularly, we have

considered a special variant where several

restrictions are considered at the same time. The set

of defined constraints are taken from a real case

provided by a food distribution company located in

Barcelona, Spain. The distribution inside cities has

special conditions like little time for delivery,

congestion, traffic lights, and different types of

vehicles related to the size and velocity issues. Also,

there are many possible configurations (routes) to

visit a customer because the street direction creates a

special network of available arcs. The purpose of

this study is to develop and apply a randomized

multi-start algorithm based on a Clarke & Wright

savings heuristic for the Asymmetric Heterogeneous

Fleet Vehicle Routing Problem (AHVRP) with

service times and routes length restrictions. The

main advantage of the proposed approach is to

design a simple algorithm that does not need any

special fine-tuning.

The paper is organized as follows: Section 2

describes the theoretical background and previous

works. In Section 3 we develop the details of the

proposed algorithm. Section 4 presents the data

instances from the distribution company. Section 5

shows the results of applying the proposed

methodology to a real context case. To conclude,

Section 6 summarizes with some final remarks and

Cáceres-Cruz J., Riera D., Buil R., Juan A. and Herrero R..

Multi-start Approach for Solving an Asymmetric Heterogeneous Vehicle Routing Problem in a Real Urban Context.

DOI: 10.5220/0004209101680174

In Proceedings of the 2nd International Conference on Operations Research and Enterprise Systems (ICORES-2013), pages 168-174

ISBN: 978-989-8565-40-2

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

future research lines.

2 THE VRP

The Vehicle Routing Problem (VRP) has been

studied for over 50 years (Laporte, 2009). The

simplest version is known as the Capacitated

Vehicle Routing Problem (CVRP), defined by

(Dantzig and Ramser, 1959). In CVRP, a directed

graph G = (V, A) is given, where V = {0, 1, …, n) is

the set of n + 1 nodes and A is the set of arcs. Node 0

represents the depot, while the remaining nodes V’ =

V \ {0} correspond to the n customers. Each

customer i ϵ V’ requires a known supply of q

units,

i.e., its demand, from a single depot (assume q

= 0).

This demand is going to be served by exactly one

visit of a single vehicle. In this basic form, there is a

homogeneous fleet of m identical vehicles with

capacity Q to serve these n customers. Each vehicle

has also a time limit L for their single trip. A

vehicle’s trip is a sequence of customers, whose total

demand cannot exceed Q that starts from and

finishes at the depot with duration no greater than L

(used to be a really big value in order to ignore its

effect). CVRP aims at finding m trips (vehicles) so

that all customers are serviced and the total distance

travelled by the fleet is minimized.

2.1 Heterogeneous VRP

VRP’s basic version is very theoretical and

restrictive. In practice, there exist some other

constraints on customers, depot(s), vehicles, etc. that

may have a significant impact on solutions. In

particular, since companies try to use their resources

efficiently and as needed, constraints regarding the

type and size of vehicles as well as the number of

trips they can make limit the application of basic

VRP models considerably. The Heterogeneous Fleet

Vehicle Routing Problem (HVRP) overcomes some

of these issues.

Different variants of HVRP have been proposed

in literature. (Baldacci et al., 2008), for instance,

present a comprehensive description of some of

them. One of the most relevant works on this area is

(Li et al., 2007). The realist aspect of this research

line has produced several recent studies, like that in

(Subramanian et al., 2012). In general HVRP

context, there is a heterogeneous vehicle fleet M

composed by m different vehicle types, i.e., M =

{1,…, m}. For each vehicle type, there are m

vehicles, a number that might be very large or,

essentially, unlimited. The m

vehicles of type k ϵ M

have capacity Q

, fixed cost F

, and variable cost per

arc (i, j) travelled c

(i ≠ j). The number of trips

performed by type k vehicles must not be greater

than m

. The cost of a route results from adding the

costs of arcs included in the route and the vehicle’s

fixed cost F

In this paper, we consider the HVRP with the

following additional considerations regarding the

available fleet and its costs:

 The number of vehicles of each type, m

, is

limited (fixed fleet) and their use must be

determined. This is known in literature as

Fixed Fleet HVRP, and;

 For each vehicle type: (1) its fixed costs are

ignored (i.e.





); (2) its

routing costs are vehicle-independent

(

12 1 2

,, ,

ij ij ij

ccckkMkk



  ).

2.2 Asymmetric VRP

Also notice that the cost, c

, of each travelled arc

(i, j) could not be the same for inverse direction

(j, i), i.e.

,ij



; i ≠ j; c

≠ c

. This is the basic

definition of Asymmetric CVRP (ACVRP), where a

directed-graph is created and the cost of each arc is

independent. (Laporte et al., 1986) develop an exact

algorithm for the asymmetrical CVRP. The authors

use a Branch-and-Bound tree in which sub-problems

are modified assignment problems subjected to some

restrictions. Computational results for problems

involving up to 260 cities are reported. (Vigo, 1996)

proposed a heuristic algorithm using additive

bounding procedures for the ACVRP. Randomly test

problems involving 300 customers are used to show

the promising performance of his approach. (Toth

and Vigo, 1999) addressed a different problem, the

symmetric and asymmetric VRP with Backhauls.

The authors proposed a Cluster-first-Route-second

heuristic. Randomly generated instances are used to

produce computational results. (Rodríguez and Ruiz,

2012) have made experiments to study the effect of

asymmetric matrix on CVRP instances. On this, the

authors have considered classical heuristics and

current state-of-the-art metaheuristics. They

highlighted that “a higher asymmetry degree in the

instances affects in a statistically significant way the

CPU time needed by the algorithms and deteriorates

the quality of the solutions obtained”.

However, the combination of these two

restrictions, Heterogeneous Fleet and Asymmetric

Cost matrix, is not frequent in the literature. In

summary, the original problem we consider in this

paper is the Asymmetric Heterogeneous Fixed Fleet

Multi-startApproachforSolvinganAsymmetricHeterogeneousVehicleRoutingProbleminaRealUrbanContext

VRP (AHVRP). We also assume that (a) any vehicle

type can visit any individual customer (the smallest

vehicle capacity is bigger than the biggest demand);

(b) there are independent service times for each node

(the delivery time spent in each client for unloading

of merchandise) that follows a specific statistical

distribution; and (c) the length of routes is controlled

by a maximum value. The objective function is

focused on minimizing the total routing costs,

considering travelling plus service times and a

duration restriction of routes.

3 OUR APPROACH

Our approach is based on the algorithm called

Simulation in Routing via the Generalized Clarke

and Wright Savings heuristic (SR-GCWS) proposed

by (Juan et al., 2010). This randomized procedure

was originally made for solving the CVRP. Figure 1

presents an overview of our approach, where a

multi-start process is started during a specific period

of time, and, at each iteration, a solution is

constructed using a randomization version of the

classical parallelized Clarke and Wright Savings

(CWS) heuristic (Clarke and Wright, 1964). CWS is

probably one of the most cited heuristic to solve the

CVRP. This procedure uses the concept of savings.

On general, at each step of the solution construction

process, the edge with the most savings is selected if

and only if the two corresponding routes can

feasibly be merged using the selected edge. The

CWS algorithm usually provides relatively good

solutions in less than a second, especially for small

and medium-size problems. In the literature, there

are several variants and improvements of the CWS.

The original version of CWS is based on the

estimation of possible savings originated from

merging routes, i.e., for unidirectional or symmetric

edges sav(i, j) = c(0, i) + c(0, j) – c(i, j). These

savings are estimated between all nodes, and then

decreasingly sorted. Then the bigger saving is

always taken, and used to merge the two associated

routes. On the randomized version of this algorithm,

we use a pseudo-geometric distribution to induce a

biased randomization selection of savings.

Moreover, this selection probability is coherent with

the savings value associated with each edge, i.e.,

edges with higher savings will be more likely to be

selected from the list than those with lower savings.

Therefore, each combination of edges has a chance

of being selected and merged with previously built

routes. This allows obtaining different outputs at

each iteration of the multi-start procedure.

However, the savings construction is modified for

being applied to the AHVRP, because the inversed

edges are also considered in the set of options

(multiplying the original quantity on the symmetric

version by two), i.e., for two different nodes i and j:

sav(i, j) = c(i, 0) + c(0, j) – c(i, j) and also sav(j, i)

= c(0, i) + c(j, 0) – c(j, i). Therefore, all savings will

be competing to be taken in the biased randomized

process, and those with higher savings will define

the orientation of routes.

Likely the routes construction process will

consider the direction of savings edges. Once a route

takes a direction then all considered candidate routes

to be merged with the first one must follow the same

direction.

Just before the construction process, the total

route duration (travelling plus service times) and the

candidate vehicle taking care of the new route are

validated. The bigger vehicle between the two

processing routes will be responsible of the new

route. This vehicle assignment promotes the merging

of routes as possible (Cáceres-Cruz et al., 2012). If a

route does not have an assigned vehicle, then the

first vehicle on the available vehicle list

(decreasingly sorted by capacity) is selected. For

this, several fictitious vehicles will be required

mainly at the beginning of the CWS process. The

fictitious vehicle should be defined using the

minimum possible capacity on the instance. At the

end, the fictitious vehicles must be discarded, if not

the solution is unfeasible. This vehicle assignment

rule does not add any computational time on to the

algorithm execution keeping the overall complexity

of the algorithm controlled. However there is a

remark: any individual demand can be carried out by

any truck (even the smallest and fictitious).

After construction, the solution is improved with

a local search method based on a memory cache

(Juan et al., 2011). This technique keeps in memory

the best known routes so far with the different

combination of customers. This procedure compares

and saves the best order for visiting the nodes on all

solutions generated so far. The previously assigned

vehicle to each route remains unchanged during this

process. At the end, the best solution is recorded.

4 COMPANY INSTANCES

With the analysis based on (Pessoa et al., 2008);

(Baldacci et al., 2008), we have identified standard

benchmarks such as the ACVRP and HVRP. We

could not find a general accepted dataset for the

combination of these two problems. The most

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

Compute initial dummy solution

and list of savings’ edges for

both edge directions

Sort the list of savings’ edges

with a biased random criterion

Assign bigger available vehicles

to each route

Unify routes

Is mergedCost = maxRouteLength?

Each route load can be assigned to a

candidate vehicle? (load = vCap)

Is empty the savings’ edge list?

Apply cache-based Local Search

Update best found solution

Is time < maxTime?

Each route has a candidate vehicle?

Extract a saving edge from the

list

End

Figure 1: Overview of our approach.

appropriate dataset presented in (Marmion et al.,

2010) is related to our specific problem. The

experiments are based on a set of real instances

related to ACVRP (Fischetti et al., 1994). The

authors simulate the heterogeneous fleet over a

range of values for testing some operators on four

different algorithms. However, the proposed

benchmarks have only considered the effect of

variable cost on vehicles selection by ignoring the

different capacities. As a result, there is no specific

dataset for the above studied problem.

As the case of study, we used the information of

a food distribution company located in Barcelona,

Spain. The company has provided us with the

delivery address of their customers in six

independent days along with their demands for those

days. The transportation limits are defined inside of

the city borders (urban distribution).

The main interest of the company is to apply the

proposed approach to bigger datasets using a web

information tool. For this reason, the company just

compile the information during a short period (as a

sample) in order to produce a preliminary result. In

addition, the compiling process represented an

important investment of resources considering the

size of the company. Therefore on a daily basis, this

company receives requests from around 50

customers. This information serves as input to

manually design the company’s routing planning.

According to the size of the company it is not

possible to employ a person specialized in

mathematical software in order to apply exact

methods. Therefore they prefer to have an

approximated solution algorithm embed in a web

tool which could be used to give automatic solution

in little time.

There is a specific constraint: each vehicle must

visit all customers of a route in a maximum period

of 180 minutes. This route length restriction must to

include the travelling time and the service time. So

far, the company uses two types of vehicles, which

are described in the Table 1. The columns of this

table show the capacity (Q

) and quantity (m

) of

available vehicles for each type (k). Actually the

company used four vehicles, but they needed to

determine if it is possible to reduce the total routing

costs and also execute the same deliveries with

fewer routes.

Table 1: Composition of the current company fleet.

Vehicle Type

20 2

30 2

We have used a map-location service, like

Google Maps to generate the asymmetric cost matrix

Multi-startApproachforSolvinganAsymmetricHeterogeneousVehicleRoutingProbleminaRealUrbanContext

between every pair of nodes (50 x 50 maximum

cells). Even when this kind of routing considers all

possible streets of the city, the cost matrix will only

represent the best travelling time between each two

nodes.

The main features of given six data instances are

summarized in the Table 2. On the first column, we

present the identification of each instance that

represents a day. The second column shows the

number of customers with demands. Third column is

the total demand. And the last column represents the

total service time of all the nodes on the instance.

As commented before, the company provides us

with the historic data of some of their service times

and routes. We have randomly generated the

respective values for the instances, using simulation

theory (Monte Carlo Simulation) and the provided

data. Then, we have defined that the service time for

each client follows a triangular distribution with min

= 1, max = 12 and mode = 3 minutes. This

distribution is often used to represent time in general

simulation models. However, the routes used differ

among all days. Notice that the company did not

save exact information of all their routes, even

within a whole day. Likely they do not apply any

specific routing method. A person in charge, who

tries to assign routes to all drivers, designs the

routing planning.

Table 2: General features of real instances.

Instance

(day)

Number of

Customers

Total

Requested

Demand

Total

Service

Time

(min)

40 53

163

50 75

213

40 60

163

39 54

159

40 57

162

18 28

5 NUMERICAL RESULTS

Our algorithm was implemented as a Java

application and used to run the six instances

described above on an Intel Xeon E5603 at 1.60 Ghz

and 8 GB of RAM. For each instance, a single run

with a total maximum time of 500 seconds was

employed. The limitation in computing time is due

to the fact that we wanted to obtain results in a

‘reasonable’ amount of time. We employ the

Random Number Generator (RNG) library for

Stochastic Simulation developed by researchers of

the Montreal University

(http://www.iro.umontreal.ca/~simardr/ssj/).

Table 3 shows the results obtained in

experiments. The first column shows the instance id;

the second, the number of routes defined in the

solution; the third column, the total travelling times

of routes; the fourth column, the total routing costs

considering the travelling times plus the service

times of the instance; and the last column, the

computational time needed to find the best solution.

The travelling costs on instances B and E

represent the higher values obtained. Both of them

travelling costs are bigger than the previously

commented restriction of 180 minutes. However,

this restriction is applied to the route duration and

also it considers the service time on each node. On

these two instances, the average total routing cost of

routes has to be considered. For this, the total

routing cost is divided by the number of routes on

the solution producing 134 and 174 minutes

respectively.

Notice that even when the running time is set to a

maximum limit of 500 seconds, the average time for

finding the best solutions is less than 131 seconds.

Table 3: Results of Best Solutions after 500 seconds

running.

Instance

(day)

Routes

Total

Travelling

Cost

(min)

Total

Routing

Cost

(min)

Time

(sec)

173 336 1.14

189 402 114.76

170 333 137.52

172 331 275.90

186 348 253.42

116 191 0.25

Average 2.17

167.67 323.50 130.50

In order to validate the solution quality of our

approach, we have compared our results against an

approximated value of the current total routing costs.

As we said before, the company does not have the

exact values of routing costs. However, they tend to

use all four vehicles as an attempt to reduce delivery

times, in an intuitive way. Therefore we have forced

our algorithm to use four vehicles in order to

produce a near value of current company solutions.

The output represents the best solution found in 500

seconds. We delivered the forced four-route solution

to the company in order to validate it with the real

planning, and we obtained a positive confirmation.

Table 4 presents the travelling times for each

ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems

scenario and the gap between these two solutions.

The difference between the approximated

company solutions and our approach results is

around 13%. In the next two images, we have

illustrated both routing solutions of the

approximated planning (Figure 2), and the new

proposed solution (Figure 3) for the instance B,

where the number of routes was reduced to 3. Notice

that the average number of routes of our approach is

around 2 which represents a considerable reduction

of the amount of routes.

Table 4: Comparison with extreme case using whole fleet

(four vehicles).

Instance

(day)

Best Costs

using 4 routes

(min)

(2)

Best Costs

(min)

(1)

GAP

(2-1)

192 173 -9.90%

205 189 -7.80%

206 170 -17.48%

190 172 -9.47%

211 186 -11.85%

153 116 -24.18%

Average

192.83 167.67 -13.45%

Figure 2: Approximated routing planning of the company

for instance B, using Google Maps.

6 CONCLUSIONS

In this paper, we have presented a multi-start

approach for solving the Asymmetric Heterogeneous

Vehicle Routing Problem (AHVRP) with service

time consideration and routes length restrictions.

The proposed approach integrates a randomized

heuristic approach with a local search. Our results

Figure 3: Designed routes in the proposed solution for

instance B, using Google Maps.

are based on data obtained from a distribution

company and we compare our solutions with an

approximation value of the actual ones implemented

by the company. These results revealed promising

improvements.

Through this experience it was possible to

support a food distribution company to: (a) realize

the current situation with quantitative methods; and

(b) improve their routing planning with a simple

approach. We used Monte Carlo Simulation to

complete the missing data from the company, and

obtain the information required for testing.

A popular way to evolve a study related to

savings algorithms is to propose new savings

definitions. The proposed definition of savings for

asymmetric VRPs could change in order to promote

other types of route constructions. Likely the

inclusion of other real constraints for urban

distribution is also being considered in the next steps

of our research, such as manage open routes and

balanced loads on routes. In fact this last restriction

is important because there are some routes with

fewer planned visits whereas others with more.

ACKNOWLEDGEMENTS

This work has been partially supported by the

Spanish Ministry of Science and Innovation

(TRA2010-21644-C03). It has been developed in the

context of the IN3-ICSO program and the CYTED-

HAROSA network (http://dpcs.uoc.edu).

Multi-startApproachforSolvinganAsymmetricHeterogeneousVehicleRoutingProbleminaRealUrbanContext

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ICORES2013-InternationalConferenceonOperationsResearchandEnterpriseSystems