Immediate Parcel to Vehicle Assignment for Cross Docking in City

Logistics: A Dynamic Assignment Vehicle Routing Problem

F. Phillipson

and S. E. de Koff

Netherlands Organisation for Applied Scientiﬁc Research (TNO), Den Haag, The Netherlands

Keywords:

Parcel Distribution, Cross Docking, Assignment, Dynamic Assignment Vehicle Routing Problem.

Abstract:

In this paper we present the Dynamic Assignment Vehicle Routing Problem. This problem arises in parcel

to vehicle assignment where the destination of the parcels is not known up to the assignment of the parcel

to a delivering vehicle. The assignment has to be done immediately without the possibility of re-assignment

afterwards. The problem is deﬁned and various methods are proposed to come to an efﬁcient solution method.

Three cases are presented to test this efﬁciency. An approach using minimum cost insertion with penalty

performs best.

1 INTRODUCTION

City logistics focuses on the efﬁcient and effec-

tive transportation of goods in urban areas while

taking into account the negative effects on con-

gestion, safety, and environment (Savelsbergh and

Van Woensel, 2016). In city logistics two main

transport strategies are used: full truckload or less-

than-truckload (Cattaruzza et al., 2017). Less-than-

truckload examples are found typically in parcel de-

livery services, express services and supermarkets

distribution. Here, consolidation and transshipment

ask for satellite locations with cross docking, redis-

tributing the incoming freight into other, possibly

smaller vehicles to serve customers. This results in

2 echelon distribution and vehicle routing problems

(2E-VRP), which are described in the survey of (Cuda

et al., 2015). The authors of this survey consider

strategic planning decisions, including decisions con-

cerning the infrastructure of the network, and tactical

planning decisions, including the routing of freight

through the network and the allocation of customers

to the intermediate facilities. At the tactical level, the

customer locations are considered known.

In the case the customer locations are not known

before operations, we come in the range of dynamic

vehicle routing problems. The review of (Pillac et al.,

2013) gives a separation of those problems between

static and dynamic on one axis and deterministic and

stochastic on the other axis. In ‘static and determinis-

https://orcid.org/0000-0003-4580-7521

tic’ problems, all input is known beforehand and ve-

hicle routes do not change once they are in execution,

see for an overview of these classic vehicle routing

problems (VRP) (Baldacci et al., 2007). ‘Static and

stochastic’ problems are characterised by input par-

tially known as random variables, which realisations

are only revealed during the execution of the routes,

see for example (Bertsimas and Simchi-Levi, 1996).

Here also clustering techniques for stochastic data can

be used (Ngai et al., 2006).

In ‘dynamic and deterministic’ problems, part or

all of the input is unknown and revealed dynamically

during the design or execution of the routes. These are

also called online VRP problems (Bjelde et al., 2017;

Jaillet and Wagner, 2008). Similarly, ‘dynamic and

stochastic’ problems have part or all of their input un-

known. This input is revealed dynamically during the

execution of the routes, but in contrast with the latter

category, exploitable stochastic knowledge is avail-

able on the dynamically revealed information. See for

a survey (Ritzinger et al., 2016). In addition, methods

based on anticipation can be used (Ulmer et al., 2015).

In this paper, we look at a satellite location where

the incoming parcels have to be distributed over a

number of vehicles to deliver them to the customers.

The satellite location has no, or a rather small, space

for storage. This means that the parcels have to be

assigned directly, after unloading and scanning, to an

outgoing vehicle. There is no possibility to reassign

on a later moment in time. The parcels are delivered

to the assigned vehicle instantaneously. The destina-

tion of the parcels is not known beforehand and is re-

138

Phillipson, F. and E. de Koff, S.

Immediate Parcel to Vehicle Assignment for Cross Docking in City Logistics: A Dynamic Assignment Vehicle Routing Problem.

DOI: 10.5220/0008871801380142

In Proceedings of the 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020), pages 138-142

ISBN: 978-989-758-396-4; ISSN: 2184-4372

vealed only at arrival at the satellite location. This

gives a problem that is, to the author’s knowledge, not

studied earlier. We call it the Dynamic Assignment

Vehicle Routing Problem (DA-VRP). This looks like

a ‘dynamic deterministic’ VRP, however, the assign-

ment of the parcels is done at the same time the des-

tination is revealed. This means that the planning is

done dynamically, but in contrast to the common dy-

namic case, it is done in upfront, where the route is not

being executed yet, giving the possibility to change

the order per vehicle, but not to interchange between

the vehicles. We will present and compare approaches

for this problem.

Furthermore, we will extend this problem with the

possibility to reveal the destination of a certain per-

centage (x%) of the parcels beforehand. This can be

revealed by using an information system or by the

possibility to store a certain percentage after reveal-

ing the destination, before the assignment has to be

ﬁxed. Also for this extension we will compare vari-

ous approaches.

The remaining of this paper is organised as fol-

lows. First, in Section 2 the problem will be de-

ﬁned. Next, in Section 3, we will present approaches

to come to a dynamic assignment of the parcels to the

vehicle and we will discuss the assumptions under-

lying the model. In Section 4, we will elaborate on

the cases we use to show the performance of the var-

ious approaches, followed by the results in Section

5. Conclusions and recommendations can be found in

Section 6.

2 DYNAMIC ASSIGNMENT

VEHICLE ROUTING PROBLEM

We assume a situation where parcels have to be

delivered, called ‘the demand’, to customers in a

certain region. In the methodology and the example

cases, we assume a homogeneous distribution of the

demand over all potential customers, characterised

by a group of houses, for example a postal code

area. Parcels are delivered at a satellite location from

various directions, where the destination of each

parcel is revealed at arrival by scanning the parcel.

In the base case, the parcel is, simultaneously with

revealing its destination, assigned to an outgoing

vehicle.

Deﬁnition: DA-VRP - In a Dynamic Assign-

ment Vehicle Routing Problem (DA-VRP) k parcels

arrive at a location in a speciﬁc order. In that speciﬁc

order, each of the parcels reveal their destination

and have to be assigned immediately to one of the m

vehicles, that will deliver the parcel to its destination.

In the basic version of this problem, each parcel

requires one capacity unit of the vehicles and all

vehicles have capacity C. When assigning the jth

parcel, vehicle i can only be regarded iff n

< C,

where n

equals the load of vehicle i at that current

moment.

In the extended case, a certain percentage, ran-

domly drawn from all parcels, reveals its destination

earlier, for example coming from an other distribu-

tion point, or can be stored some time, after which

the stored parcels can be assigned in one step to the

empty vehicles.

3 METHOD

In this section we indicate some approaches that can

be used to assign the parcels to the vehicles. We dis-

tinguish two steps in our approach:

1. Initial assignment of direction to vehicles;

2. Dynamic assignment of arriving parcels;

The ﬁrst step gives a potential direction to each of the

vehicles, or none if empty vehicles are used, by as-

signing a base load, a certain region or some initial

direction. The second step assigns directly the incom-

ing parcels to the vehicles. Both the initial assignment

and the dynamic assignment will be discussed in the

next section in more detail. In Section 3.2 the math-

ematical solution techniques that are used in the as-

signment steps are presented. After these two steps

the load of all vehicles is known and for each vehicle

a regular TSP can be solved.

3.1 Detailed Steps

It might be helpful to give an initial load or assign-

ment of a certain area to each of the available vehicles.

This is done in step 1 of the approach, the initial as-

signment of direction to vehicles. Here we distinguish

four basic methods:

1. No load – the vehicles stay empty until the ﬁrst

dynamic assignment.

2. Basic load – In the case that a certain percent-

age (x%) of the parcels reveal their destination

before assignment, we will assign these parcels to

the available vehicles evenly using a VRP solution

method. We assume that the revealed parcels are

randomly distributed over all potential customers.

3. Separation by dummy location - We perform a

k-Means clustering over all potential customers

Immediate Parcel to Vehicle Assignment for Cross Docking in City Logistics: A Dynamic Assignment Vehicle Routing Problem

139

and assign a dummy parcel with one of the clus-

ter means to each vehicle. K-means clustering

(James et al., 2013) aims to partition observations

into k clusters, in which each observation belongs

to the cluster with the nearest mean.

4. Total geographical separation - Again we per-

form a (k-Means) clustering over all potential cus-

tomers (postal codes) and assign each of those

(postal codes) clusters to a vehicle.

Next, three methods for the second step, the dynamic

assignment of arriving parcels, are proposed:

1. Based on smallest distance to cluster mean – for

all vehicles we can calculate the geographical

mean of all assigned parcel destinations. We as-

sign the arriving parcel to that vehicle for which

the distance from the parcel destination to the ge-

ographical mean is minimal.

2. Based on minimal insertion costs – for all vehi-

cles we calculate the minimal cost of inserting the

arriving parcel destination to the route. As we as-

sume that the assigned parcels are ordered in the

routing of the vehicle, we can calculate the cost

by trying to insert the parcel between each pair of

consecutive parcels in the vehicle. The insertion

that is cheapest will be selected.

3. Based on ﬁxed clusters – here the parcel is simply

assigned to the cluster it belongs to, using the total

geographical separation of the initial stage, based

on customer or postal code of the customer.

When, by one of these methods, the assignment is de-

termined, the parcel is inserted on the right place in

the route of the selected vehicle, meeting the assump-

tion of ordering.

If the vehicles have a ﬁxed capacity, there are two

aspects we have to consider. First is what to do when

a vehicle is loaded to its capacity. In assignment strat-

egy one and two, this means that this vehicle cannot

be selected anymore for assignment. The cost of as-

signment to a fully loaded vehicle will be set to inﬁn-

ity. For the third assignment strategy, we propose to

switch to the second strategy if the vehicle that was

selected is full.

The second aspect is asymmetry in loading. If the

vehicles start empty, it is likely that the ﬁrst two ve-

hicles will be loaded until one reaches its maximum

capacity. Then a new vehicle is started. This strat-

egy does not account for a more even distribution of

the parcels over the vehicles, giving the possibility to

each get a (relatively) restricted area and it does not

account for the possibility that parcels will show up

that clearly should have been assigned to a vehicle,

but cannot, due to capacity restrictions. For these rea-

sons we add a fourth dynamic assignment strategy, as

alternative to the second, where the price of insertion

increases when the vehicle has more load. Precise de-

tails follow in the next section.

4. Based on minimal insertion costs – for all vehi-

cles we calculate the minimal cost of inserting the

arriving parcel destination to the route. As we as-

sume that the assigned parcels are ordered in the

routing of the vehicle, we can calculate the cost

by trying to insert the parcel between each pair

of consecutive parcels in the vehicle. The cost is

multiplied by a penalty factor, depending on the

load of the vehicle. The insertion that is cheapest

will be selected.

If we assume no initial load, this leads to 12 combi-

nation of methods, of which 7 are sound and will be

discussed:

1. Empty trucks (1) – ﬁlled by cluster approach (1)

2. Separation (3) – ﬁlled by cluster approach (1)

3. Empty trucks (1) – ﬁlled by insertion approach (2)

4. Separation (3) – ﬁlled by insertion approach (2)

5. Empty trucks (1) – ﬁlled by insertion approach

with penalty (4)

6. Separation (3) – ﬁlled by insertion approach with

penalty (4)

7. Total Separation (4) – ﬁlled based on ﬁxed clus-

ters (3)

If there is initial load, in the extended case, we use

the second method for the initial assignment (2. Ba-

sic load) and we could use methods 2-4 at the dynamic

assignment phase. Next to these scenarios we use for

comparison a ’full information solution’. Here all in-

formation is known beforehand and a VRP method

can be used for solving this.

3.2 Solution Techniques

We use four mathematical solution techniques within

our approaches: VRP solution method, k-Means clus-

tering method, Insertion method and a Dynamic clus-

tering method. The two former are used from other

sources, the two latter are described in more detail.

For the VRP method, which will be used for

’Basic load’ and for the ’Full information’ solution

we used a VRP solver, based on a Simulated An-

nealing method, implemented in Matlab by Yarpiz

(www.yarpiz.com). We use this implementation with

the following parameters: number of iterations 2500;

number of inner iterations: 250. For the k-Means

clustering method in the Separation methods we use

the basic Matlab implementation, ’kmeans(X,k)’

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

140

The Insertion method works as follows. We as-

sume that already each vehicle i has a tour indicated

by the (x,y) coordinates of the destination of the

parcels, starting and ending at the satellite location

with coordinates (x

i,0

) = (x

i,0

),(x

i,1

),. ..,(x

i,n

),(x

i,0

)

Now, a parcel with coordinates (x,y) has to be as-

signed to a cluster. For each vehicle i, determine the

pair of consecutive points k, l such that

= min

k,l

d((x

i,k

),(x, y)) + d((x,y), (x

i,l

))

−d((x

i,k

),(x

i,l

))

is minimal for all pairs k,l, where d((x

),(x

))

denotes the distance between two destinations, noted

by their (x, y) coordinates. The parcel will now be

inserted, on the spot between k

∗

and l

∗

, in the tour

i that minimises d

for all i. In case of the Insertion

method with penalty, this distance is multiplied by a

penalty factor (1 + p

) where

= P · n

/C,

where P is the chosen penalty value, n

the current

load of vehicle i and C the capacity of the vehicle. The

value of P is case dependent. In all cases used later,

we use the ﬁrst 10 problem instances for learning the

optimal value of P for that case.

The Dynamic Clustering methods works as fol-

lows. Again a parcel with coordinates (x,y) has to

be assigned to a cluster. Cluster i consists of parcels

with coordinates (x

i,1

),. .. ,(x

i,n

) and a clus-

ter point (

), where x

∑

k=1

i,k

and y

∑

k=1

i,k

. Now assign the parcel to cluster i such

that d((x, y),(x

)) is minimised.

4 CASES

The cases are all situated in the city of The Hague,

The Netherlands. The city has circa 530,000 inhabi-

tants and circa 255,000 houses. The total area of the

city consists of 98 square kilometres. The houses are

divided into 13,297 postal code areas. For each case

we draw a number of destination, uniformly over all

postal codes and assume that a parcel has to be deliv-

ered to that location. The number of parcels and the

available number of trucks vary in the three cases:

• Case 1: 350 parcels, 10 vehicles;

• Case 2: 350 parcels, 5 vehicles;

• Case 3: 200 parcels 5 vehicles.

For each case we sampled 100 days or instances, in-

dependently. We assume that the only costs are the

variable costs based on the total distance driven by the

vehicles. A parcel is assumed to have capacity 1 and

the capacity of the vehicles is in number of parcels.

5 RESULTS

As deﬁned earlier we have seven scenarios if there is

no initial load assigned to the vehicles. These scenar-

ios are used on the three cases. A ’full information

solution’ is used as reference point, where all parcels

are known and the tours are constructed using the

VRP solver. This is not a guaranteed optimal solution,

where the used method is a meta-heuristic. We see in

Table 1 that methods 1, 2 and 3 give an overall bad

solution. The average scores over all 100 instances

are all more than double the VRP solution. These are

both the clustering approaches and the insertion ap-

proach without penalty using empty vehicles. Meth-

ods 4 and 7 work reasonably on the two cases with

5 vehicles. These are the insertion approaches with-

out penalty and an initial separation and the method

with strict separation, ﬁxed areas of delivery. Best

performing are the methods 5 and 6 based on the in-

sertion approach with penalty. Resulting in a 14-17%

higher cost that the VRP solution.

Table 1: Performance of the 7 combinations of methods

compared to the VRP solution.

Case 1 Case 2 Case 3

1 282% 229% 230%

2 286% 231% 233%

3 283% 229% 230%

4 237% 146% 146%

5 115% 117% 115%

6 117% 115% 114%

7 224% 130% 127%

VRP 100% 100% 100%

If we assume the possibility to reveal the destina-

tion of a certain percentage, e.g., 50%, of the parcels

beforehand, we can compare this with the situation

where there is no information about the destination of

the parcels, and with the situation with full informa-

tion. We use case 2 in this example. For the situation

with no information we use methods 6 and 7 as ref-

erence, for the full information situation we use the

VRP solution. The information can be revealed by the

use of an information system or by the possibility to

store a certain percentage after revealing the destina-

tion and before the assignment. We assume here that

for this part the VRP approach can be used, using the

number of vehicles (here 5) and 50% of the capacity.

Immediate Parcel to Vehicle Assignment for Cross Docking in City Logistics: A Dynamic Assignment Vehicle Routing Problem

141

Now an evenly distributed base load for all vehicles

is known. Next the insertion method with penalty is

used for the dynamic assignment of the next 50% of

the parcels. Starting with the average solution of the

VRP approach over all 100 instances and calling this

100%, we see that the ﬁxed areas of delivery gives a

30% increase in costs and the insertion with penalty

15%. Revealing 50% of the destinations leads to de-

crease of costs of 9%-point, compared to the 0% so-

lution, and is 6% higher that the 100% information

VRP solution. The results are summarised in Table 2.

Table 2: Results.

Method Score

Fixed clusters 0% 130%

Insertion 0% 115%

Insertion 50% 106%

VRP 100% 100%

6 CONCLUSIONS AND

RECOMMENDATIONS

In this paper we discussed a problem in parcel dis-

tribution where the destination of the parcels is re-

vealed only after arrival at the satellite location: the

Dynamic Assignment Vehicle Routing Problem. In

the case that there is no, or limited, space for storage,

the parcels have to be assigned directly and moved to

one of the available distribution vehicles. We showed

that use of an insertion method, with an increasing

penalty function with the occupancy rate, gives the

best results. The initial assignment to trucks has no or

low effect on this result.

If an initial load is assumed, which is distributed

equally over the vehicles using the VRP solver, fur-

ther assigning using the insertion method leads to a

decrease in cost. From this we can conclude that as-

signing incoming parcels dynamically, using the in-

sertion method is preferable to ﬁxed clusters. Using

information for even a part of the parcels improves the

solution even more in the direction of a full informa-

tion based solution.

For further research we propose to look at the ef-

fect of higher capacities, more dense demand distri-

butions and variable demand, leading to an unknown

number of required vehicles per day. Also a more de-

tailed deﬁnition of capacity, in volume, and time re-

strictions at customers side can be added.

ACKNOWLEDGEMENTS

This work has been carried out within the project

‘Self-Organising Logistics in Distribution (SOLiD)’,

supported by NWO (the Netherlands Organisation for

Scientiﬁc Research).

REFERENCES

Baldacci, R., Toth, P., and Vigo, D. (2007). Recent advances

in vehicle routing exact algorithms. 4OR, 5(4):269–

298.

Bertsimas, D. J. and Simchi-Levi, D. (1996). A new

generation of vehicle routing research: robust algo-

rithms, addressing uncertainty. Operations Research,

44(2):286–304.

Bjelde, A., Disser, Y., Hackfeld, J., Hansknecht, C., Lip-

mann, M., Meißner, J., Schewior, K., Schl

oter, M.,

and Stougie, L. (2017). Tight bounds for online tsp

on the line. In Proceedings of the Twenty-Eighth An-

nual ACM-SIAM Symposium on Discrete Algorithms,

pages 994–1005. Society for Industrial and Applied

Mathematics.

Cattaruzza, D., Absi, N., Feillet, D., and Gonz

alez-Feliu,

J. (2017). Vehicle routing problems for city logis-

tics. EURO Journal on Transportation and Logistics,

6(1):51–79.

Cuda, R., Guastaroba, G., and Speranza, M. G. (2015). A

survey on two-echelon routing problems. Computers

& Operations Research, 55:185–199.

Jaillet, P. and Wagner, M. R. (2008). Generalized online

routing: New competitive ratios, resource augmenta-

tion, and asymptotic analyses. Operations research,

56(3):745–757.

James, G., Witten, D., Hastie, T., and Tibshirani, R. (2013).

An introduction to statistical learning, volume 112.

Springer.

Ngai, W. K., Kao, B., Chui, C. K., Cheng, R., Chau, M.,

and Yip, K. Y. (2006). Efﬁcient clustering of uncertain

data. In Data Mining, 2006. ICDM’06. Sixth Interna-

tional Conference on, pages 436–445. IEEE.

Pillac, V., Gendreau, M., Gu

eret, C., and Medaglia, A. L.

(2013). A review of dynamic vehicle routing prob-

lems. European Journal of Operational Research,

225(1):1–11.

Ritzinger, U., Puchinger, J., and Hartl, R. F. (2016). A sur-

vey on dynamic and stochastic vehicle routing prob-

lems. International Journal of Production Research,

54(1):215–231.

Savelsbergh, M. and Van Woensel, T. (2016). 50th anniver-

sary invited article—city logistics: Challenges and op-

portunities. Transportation Science, 50(2):579–590.

Ulmer, M. W., Brinkmann, J., and Mattfeld, D. C. (2015).

Anticipatory planning for courier, express and parcel

services. In Logistics Management, pages 313–324.

Springer.

ICORES 2020 - 9th International Conference on Operations Research and Enterprise Systems

142