The Effect of Cooperation in Pickup and Multiple Delivery Problems
Philip Mourdjis
1
, Fiona Polack
1
, Peter Cowling
1
, Yujie Chen
1
and Martin Robinson
2
1
YCCSA, University of York, York, U.K.
2
Transfaction Ltd., Cambridge, U.K.
Keywords:
Vehicle routing, Pickup-and-delivery, Real-world Problem.
Abstract:
Small logistics companies operate in many towns and cities across the UK, and need to be able to compete with
larger delivery companies who can leverage economies of scale to provide lower costs to customers. If small
companies were willing to work together, all could benefit from reduced operating costs, enabling them to
compete and survive against larger delivery companies. In cooperation with Transfaction Ltd., we investigate
dynamic scheduling of shared loads for real-world, long distance truck haulage in the UK. We model the
problem as a dynamic pickup and multiple delivery problem (PMDP). The PMDP is a one-many problem (one
pickup, many drop-offs), unlike the more widely researched one-one (pickup and delivery problem, PDP) and
one-many-one (vehicle routing problem, VRP) problems.
1 INTRODUCTION
With over six thousand hauliers in the UK alone
(Dff International Ltd, 2015), competition is fierce.
Hauliers face the orthogonal demands of short notice
from customers, an expectation of low-cost service,
and environmental sustainability concerns (McLeod
et al., 2012; Nahum, 2013; Demir et al., 2014). Be-
cause larger carriers can leverage economies of scale
to benefit in routing and scheduling, competition is
getting ever stronger. If smaller carriers could work
together, they could increase scheduling efficiency,
save on mileage costs, and improve flexibility. In this
paper we quantify the savings possible when carri-
ers outsource some of their customer consignments to
other carriers, working independently or as a group.
As a real-world problem, there are constraints that
must be satisfied, such as vehicle capacity, soft time
windows and driver working hour rules. The problem
is defined in terms of consignments which include a
single pickup location and one or more delivery loca-
tions. Consignments vary in size, and may be able to
share one delivery vehicle, to save cost. A key con-
straint is that each vehicle must be unloaded in the re-
verse order to the loading order: deliveries from one
vehicle are constrained to a last-in, first-out (LIFO)
order. Concretely, consignment A may be interrupted
by another if all of the second consignment’s deliver-
ies are serviced before continuing with consignment
As deliveries. We call this a pickup and multiple de-
livery problem (PMDP). This paper investigates the
cost savings which are possible if carriers distributed
across a country share consignments.
2 RELATED WORK
Research on PDPs usually concentrates on static mod-
els of small scale problems such as servicing taxi re-
quests, or ride sharing schemes (Toth and Vigo, 1997)
dial-a-ride problems (DARPs). Desaulniers et al.
(2002) present a widely accepted mathematical for-
mulation for the generic PDP, which they refer to as
the vehicle routing problem with pickup and delivery
and time windows.
Variations of the PDP handle constraints on the
number of vehicles used, time windows on requests,
capacities and number of depots. However, most of
the existing research is on static problems, in which
all requests are known in advance (Berbeglia et al.,
2007). Exact solutions to static PDPs favour branch-
and-cut-and-price algorithms using column genera-
tion techniques, for example, Dumas et al. (1991) uses
this approach to solve a multi-depot PDP for prob-
lems with up to 55 requests. No indication is given of
whether their approach scales to larger problem sizes.
Exact solutions to dynamic problems include
a variation of the column generation approach
(Gschwind et al., 2012), used to solve DARPs of up
to 96 requests, with either static or dynamic time win-
Mourdjis, P., Polack, F., Cowling, P., Chen, Y. and Robinson, M.
The Effect of Cooperation in Pickup and Multiple Delivery Problems.
DOI: 10.5220/0005748902870295
In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 287-295
ISBN: 978-989-758-171-7
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
287
dows. Xu et al. (2003) solve a PDP based on real-
world logistics with multiple carriers, vehicle types
and LIFO constraints using a set partitioning formu-
lation containing an exponential number of columns.
However, in general, exact methods do not scale
well, so heuristic and hyper-heuristic approaches that
can quickly find near-optimal solutions, have become
popular for large-scale, real-world problems. La-
porte (2009) provides a good overview of exact and
heuristic methods for vehicle routing problems. More
recently, heuristic approaches have been applied to
scheduling with LIFO loading constraints (Cherkesly
et al., 2015; Crainic et al., 2015; Benavent et al.,
2015).
Cherkesly et al. (2015) use a three phase ap-
proach. First, multiple routes are created using a
greedy randomised adaptive search procedure; next
variable neighbourhood descent (VND) applies local
search to derive new solutions using a diversification
strategy derived from Rochat and Taillard (1995). Fi-
nally, crossover is used to combine solutions to form
further candidate solutions. Benavent et al. (2015) use
a multi-start tabu search approach that uses Clarke
and Wright savings (Clarke and Wright, 1964) as
well as two random schedule heuristics to build seed
routes. The tabu search improves solutions by repeat-
edly removing and re-inserting consignments, using
traditional strategies to prevent cycling and promote
diversification.
Existing approaches to dynamic scheduling of
PDPs (summarised in Br
¨
aysy and Gendreau (2005b))
often use a two-phase hyper-heuristic (Berbeglia
et al., 2010): requests are first inserted into a schedule,
then optimisation is performed, either on a route that
has been changed or on an entire schedule. Research
has focused on different insertion, removal and local
search operators, and on the heuristics that choose be-
tween operators at any point. For example, Gendreau
et al. (2006) use neighbourhood search heuristics
and ejection chains to tackle same-day courier PDP.
Mitrovi
´
c-Mini
´
c et al. (2004) use a double horizon ap-
proach with routing and scheduling sub-problems to
schedule similar problems of a larger size. Albareda-
Sambola et al. (2014) use probabilistic information to
inform their routing of a multi-period VRP.
We are concerned with efficient solution of
scheduling under just-in-time logistics, where the
customer expectation is that hauliers respond quickly
to delivery requests, and where same-day delivery of-
ten attracts premium payment rates. In the traditional
approach used by small haulage companies, static
scheduling is re-run daily. However, static schedul-
ing cannot be used for real-time response to orders,
and does not take account of the existing schedule and
loading. We propose a dynamic scheduler that intelli-
gently adapts to incoming requests, a novel variant of
dynamic PDP (Berbeglia et al., 2010).
Our model of the PMDP is based on the generic
PDP model of Desaulniers et al. (2002). Our vari-
able neighbourhood descent with memory scheduling
(VNDM) takes inspiration from the hybrid variable
neighbourhood tabu search (VNTS, Belhaiza et al.
(2013)), which outperforms tabu and variable neigh-
bourhood approaches for static VRPs. A schedule
is built up by repeatedly inserting requests then per-
forming optimisation. The strict LIFO constraint in
PMDP, along with constraints such as the vehicle ca-
pacity, makes it difficult to find improving moves in
PMDP, so we develop a descent based algorithm and
local search operators tailored to PMDP, with roots
in classic VRP and PDP solutions. Once a solution
has been built, we perform optimisation whilst aim-
ing to minimise ordering inversions within a vehi-
cle’s schedule, as these are unlikely to improve results
in problems with tight time windows and LIFO con-
straints on deliveries. Local search techniques that
affect delivery order, such as those presented by Tail-
lard et al. (1997) and Br
¨
aysy and Gendreau (2005a),
and the GENI technique (Gendreau et al., 1992), are
unsuitable for direct use on our problem because they
cause large changes in schedule ordering.
3 MODEL
The PMDP is defined on a directed graph DG =
(N, A) where A is the arc set and N is the node
set. Each request r is identified by (n
r
, l
r
t
start
r
, t
end
r
,
tt
service
r
) where n
r
is the location, l
r
is the load (where
the summation of pickup load and delivery loads for
a consignment is equal to zero). t
start
r
, t
end
r
, represent
the start and end time of the arrival window respec-
tively where the service time tt
service
r
must begin (for
clarity we use double letters to represent quantities).
R is the set of requests where R = P D O, P being
the set of pickup-requests and D the set of delivery-
requests. O is the set of origins which are dummy re-
quests used to represent the multiple depot locations
of the problem. The arc between two requests r and
u (that is, between nodes (n
r
, n
u
)) is the arc (r, u). A
consignment c is identified by (p
c
, D
c
, t
c
) where p
c
is the pickup-request and D
c
= d
1
c
, . . ., d
nc
c
is the se-
quence of delivery-requests. Each consignment has a
received time t
c
, which is the time at which the order
is entered in the system. C is the set of consignments.
A
k
A represents the feasible arcs for vehicle k. The
binary flow variable b
ruk
is set to one if arc (r, u) A
k
is used by the vehicle k, and to zero otherwise. ll
rk
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
288
is the load of vehicle k at request r and is not fixed
but dependent on the other arcs in the vehicle’s route.
It is calculated as a running sum where each request
either adds to the load (pickup) or subtracts from the
load (delivery). A vehicle starts and ends its route at
one of the depots with load equal to zero.
The goal is to minimise the total cost of servicing
all requests r R:
min
kK
(r,u)A
k
C
ruk
b
ruk
(1)
where:
C
ruk
= nc(nn
ru
, ll
rk
) +tc(ruk) + dc(tt
delay
uk
) (2)
subject to the constraints in the appendix. C
ruk
is the
cost of vehicle k servicing (r, u), calculated using run-
ning cost estimations for a 44-tonne articulated truck
in 2014 based on data from the UK Road Haulage As-
sociation (RHA) (Dff International Ltd, 2014). The
component costs are: nc(nn
ru
, ll
rk
), the cost of travel-
ling distance nn
ru
(the length of arc (r, u)) with load
ll
rk
; tc(ruk), the cost of the time taken by vehicle k to
travel arc (r, u), and dc(tt
delay
rk
), the cost of the penalty
for arriving at request u late, we use a stepwise func-
tion (increasing every hour) after an initial grace pe-
riod, in line with industry practice. Consignments
may be either customer orders or backhauls (post-
delivery return to pickup location, for instance to dis-
pose of packaging), these differ only in that backhauls
are usually mostly empty loads.
4 SOLUTION APPROACH
Our PMDP solution approach has three components:
discrete event simulation (DES), the variable neigh-
bourhood descent with memory (VNDM) hyper-
heuristic, and the low level heuristics (LLHs).
4.1 Discrete Event Simulation (DES)
In collaboration with Transfaction Ltd., we have ac-
cess to real scheduling data and manually-scheduled
consignments for small UK hauliers (referred to as
real data). The real data are insufficient, in quantity
and quality, for our scheduling research, but provide
us with indicative distributions and other information.
By using the parameterised DES, we generate larger,
realistic, data sets on requests and consignments (re-
ferred as generated data).
We use the real data and its distributions to param-
eterise a DES of the dynamic receipt of consignment
requests. In particular, the arrival times of orders in
the system is not recorded by manual schedulers, ei-
ther for customer orders or for backhauls; the distribu-
tions for these are estimated from the typical pattern
of diurnal and weekly orders.
4.2 The VNDM Heuristic
Like other hyper-heuristic approaches, VNDM is a
two-phase approach, in which an initial set of routes
is created and then optimised. The hyper-heuristic
search framework uses a heuristic to select from a set
of LLHs (Section 4.3, below).
In PMDP, each haulage company (carrier) is as-
sumed to have an unlimited number of vehicles and
is represented by a depot, randomly located within
the area encompassing the consignments. A schedule
consists of a number of routes, each beginning and
terminating at one of the depots. To prepare an ini-
tial set of routes, consignments are sorted according
to the time that they arrive in the system, and inserted
greedily into a schedule, earliest arrival first. Con-
signments are assigned to a carrier from those carriers
with fewest consignments that is geographically clos-
est to the midpoint between a consignment’s pickup
and final delivery locations. Thus, the initial schedule
systematically distributes consignments evenly across
many carriers.
In order to simulate dynamic request arrivals, in
which new consignments are added to a schedule that
is already being serviced, we keep track of simulation
time (an internal representation of current time, stored
so that requests which in reality would have already
happened cannot be modified by our optimisation pro-
cedure). If the scheduled start time of any request
is before the current simulation time, it is marked as
“fixed”. Additional requests cannot be inserted before
these fixed requests, and the routing of a fixed request
cannot be altered in any optimising moves. The inser-
tion heuristic treats consignments atomically, finding
the lowest cost insertion location across all routes for
a pickup and all its deliveries (guaranteeing LIFO),
such that no previously inserted consignment incurs a
delay. After the insertion of each new consignment,
VNDM is used for optimisation, running for a con-
stant amount of CPU time.
VNDM is a descent-based first-improvement
heuristic. Routes are first ordered by length, then
each LLH generates a list of potential moves. Since
the majority of a schedule is unaltered after a modi-
fication, VNDM limits revisiting parts of the search
space by maintaining a record of LLHs that give no
improvement on each route (pairs of route and LLH
identifiers are stored). If a LLH fails to produce an
improving move, it is added to a tabu list. The tabu
The Effect of Cooperation in Pickup and Multiple Delivery Problems
289
list is re-initialised when a route is subsequently mod-
ified, as a LLH may now be able to find improvement
where none was previously possible.
VNDM differs from other published PDP solution
approaches in a number of ways, notably in the choice
of local moves used (specific to the PMDP), the use
of route ordering to focus the search on promising ar-
eas, and the use of a route memory to reduce repeated
searching. The search space is further reduced by im-
posing distance and time limits on nodes chosen for
potential moves, which are different for each LLH and
determined through extensive testing.
4.3 Low-level Heuristics
The nature of PMDP, with strict LIFO ordering of
consignments, guides our selection of LLHs to apply
to route optimisation. Since a pickup request must
occur before its delivery requests, reversing a section
of a schedule and repairing infeasible pickup / deliv-
ery ordering will significantly alter the distance of the
route. Because time windows are usually tight, in-
creased distance may result in significant delay in ser-
vicing requests.
Highly disruptive LLHs have been found inca-
pable of improving our schedules, ruling out LLHs
that use partial route inversions, such as GENI (Gen-
dreau et al., 1992) and iCROSS (Br
¨
aysy, 2003). How-
ever, we can use the CROSS exchange of Savelsbergh
(1992) (used by Taillard et al. (1997)) as it does not
reverse chains of requests. Additional LLHs, such as
GENI-PO (Mourdjis et al., 2014), have been chosen
or developed to preserve existing schedule ordering as
much as possible. By keeping the pickup and deliv-
eries of one consignment in the same schedule (rather
than splitting the consignment across loads and us-
ing precedence constraints), we facilitate the use of
LLHs from the widely-researched area of one-many-
one VRPs.
In selecting LLHs to modify routes, a consign-
ment may only be rescheduled if the modification
results in a valid schedule. A consignment may be
scheduled such that other pickups or deliveries oc-
cur between the consignment’s pickup and final de-
livery, providing load and LIFO constraints are not
violated. However, if the consignment is rescheduled,
the nested pickups or deliveries from other consign-
ments remain in the original schedule, thus allowing
modifications to undo nested consignments.
We provide four LLHs that can be applied to a sin-
gle route. If a single route operator can generate more
than one resulting route, that which is least disrup-
tive to existing schedule ordering is used. LLHs that
would reverse the order of a chain of requests are not
allowed, hence we do not use 2-Opt.
3-Opt moves one consignment to a different po-
sition in the route schedule, whilst 4-Opt swaps the
positions of two consignments in the route schedule.
Nest Consignment moves a whole consignment to a
position within the delivery schedule of another con-
signment, thus nesting the first consignment within
the second. Finally, Nest Two Consignments nests
two consignments inside other consignments, a use-
ful move where single-level nesting produces no im-
provement.
We provide four further LLHs that operate on
more than one route at a time. GENI-PO (Mourd-
jis et al., 2014) is a non-inverting variant of GENI
(Gendreau et al., 1992). The other three LLHs are
from Savelsbergh (1992). Relocate moves one con-
signment to a valid position in a different route sched-
ule, which may introduce nesting. Geni-PO is a vari-
ation of relocate that preserves as much previous or-
dering as possible by moving a consignment to be ge-
ographically close to other consignments: all possible
insertion position pairs are considered to find the most
improving relocation. Swap exchanges consignments
from two different routes, whilst Cross exchanges two
chains of consignments between routes, preserving
the existing ordering within each chain. Cross con-
siders chains of all lengths when used.
4.3.1 Use of Local Moves
Of the eight LLHs, three consume only small amounts
of CPU time for problems of the size we study (3-
Opt, 4-Opt and Nest consignment), whilst the others
(Nest two consignments, Relocate, Geni-PO, Swap
and Cross) are considered hard and take a signifi-
cant amount of time. However, the hard LLHs gener-
ate several orders of magnitude more potential moves
than the computationally trivial moves. There is no
intuitive reason to prefer one hard LLH to another,
and there is little advantage to running more than one
hard LLH at a time. Thus, to prevent VNDM op-
timisation simply running out of time whilst apply-
ing too many hard LLHs, each call of VNDM uses
a neighbourhood structure comprising the three low-
CPU LLHs in the order above, then one hard LLH, se-
lected at random. The random selection ensures that
all the hard LLHs are used over a series of optimisa-
tions, and thus provides ample diversification.
5 COMPUTATIONAL RESULTS
In order to investigate the effects of cooperation, we
generate 100 scenarios from data on a set of 27,153
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
290
All Contracted
Out-sourcing
Out-sourcing to coop
Cooperative
75
80
85
90
95
Cost per request (£)
Sample carrier Other carriers
Figure 1: Average savings for a single carrier (sample car-
rier) and a group of carriers (other carriers), with four dif-
ferent models of cooperation.
real-world consignments. The scenarios are built by
selecting 200 real consignments at random from this
set and building pairs of consignments representing
outbound linehaul and return backhaul legs. Each
consignment consists of at least two requests.
The scenarios examined use five carriers, and ex-
plore the effect on one carrier (the sample) under four
different configurations of cooperation with the other
four carriers. All Contracted has all consignments as-
signed to a single carrier. Optimisation is only pos-
sible between vehicles belonging to the same carrier.
Out-sourcing starts with a contracted model, but al-
lows re-assignment of consignments from the sam-
ple to any of the other carriers, if cost savings can
be made. Out-sourcing to coop(erative) adds the out-
sourcing model for the sample carrier into a model in
which the other carriers can exchange consignments
if savings can be made; the sample carrier does not
accept any additional consignments. Finally, the co-
operative model initially assigns all consignments to
individual carriers (as in Contracted) but allows unre-
stricted re-allocation during optimisation, if cost sav-
ings are possible.
We simulate one dynamic scheduling week, and
limit optimisations to 5 minutes of CPU time. Each of
the four configurations is run 30 times, using a hetero-
geneous cluster of Intel Xeon based servers, totalling
72 cores and 120GB of RAM. The results presented
here thus represent one thousand hours of CPU time.
Figure 1 shows that for the sample carrier, an av-
erage 9% saving can be made by out-sourcing to the
All Contracted
Out-sourcing
Out-sourcing to coop
Cooperative
70
75
80
85
Scheduled (%)
Across all carriers
Figure 2: Percentage of contracted consignments serviced
across the four different models of cooperation.
four other carriers, whilst the configuration that al-
lows other carriers to also cooperate results in aver-
age savings of nearly 14%, because the cooperation
allows more efficient routing across the carriers. If the
sample carrier also cooperates in efficient scheduling,
the total average saving for the sample carrier rises
to 18%. Cooperation is also beneficial for the other
carriers: accepting orders from the single carrier can
produce benefits of 3%, whilst cross-group coopera-
tion produces savings to averaging 15%.
The results shown should drive all carriers to-
wards cooperation. Competition favours carriers with
the lowest costs; the sample carrier achieves this in
configuration 2, by outsourcing to other carriers who
are not cooperating. However, rational competitors
would be expected to copy this behaviour, moving
the system towards a reallocation of consignments as
seen in configuration 3; here, the competitors are co-
operating, and the sample carrier is at a competitive
disadvantage. However, if all carriers cooperate, as
in configuration 4, the lowest costs for all carriers are
observed.
Increasing cooperation allows a greater number
of consignments to be handled. Figure 2 shows that
the schedule in which all carriers operate alone cov-
ers on average less than 70% of their contracted con-
signments. However, the fully cooperative model
can schedule over 85% of consignments. (Note that
random scenario generation means that there is no
guarantee that all consignments are feasible given the
number of carriers, their locations and that even with
an infinite number of vehicles, some consignments
The Effect of Cooperation in Pickup and Multiple Delivery Problems
291
are too far apart to be serviced whilst adhering to
driver working hour rules: since we do not consider
driver sleeping arrangements and all routes must be-
gin and end at the depot, these consignments are im-
possible in our current model.)
Table 1: Percentage of the sample carrier’s consignments
re-allocated in different configurations.
Config. Cooperation Mode Re-allocated
1 All contracted 0%
2 Out-sourcing 65.6%
3 Out-sourcing to coop 67.2%
4 Cooperative 57.2%
Table 1 shows the percentage of consignments
that are re-allocated from the sample carrier in each
configuration. Both out-sourcing and out-sourcing
to a cooperative allow almost two-thirds of the car-
rier’s consignments to be assigned to others: because
our scheduling algorithm minimises cost, these re-
allocations can be interpreted as being carried more
cheaply, due to more efficient use of resources, when
assigned to other carriers. We are most interested in
the percentage of consignments that are re-allocated
away from the sample carrier. When outsourcing and
cooperation are combined (configuration 3), the sam-
ple carrier’s re-assigned loads are most cost-effective,
as, in this configuration, the other carriers can also re-
allocate loads among themselves (but not to the sam-
ple carrier). In the fully cooperative model, the sam-
ple carrier’s consignments are less cost-effectively
reassigned than in other reallocation configurations.
However, the overall cost-effectiveness of the 5 carri-
ers is significantly better than in other configurations:
62.5% of other carriers’ consignments were reallo-
cated in this model, leading to the reduction in cost
observed for cooperation in figure 1. These results
also strongly support the contention that savings can
accrue to small hauliers who cooperate to carry each
others’ consignments efficiently.
5.1 More Group Configurations
We seek to further investigate the effects of different
sized groups of carriers on both cost and network ca-
pacity.
Using the same 100 scenarios as investigated pre-
viously, we now investigate how efficiently 10 carri-
ers can service the consignments, split into a num-
ber of different group configurations. Cooperation
is allowed within but not between these groups. In
the All Contracted configuration each of the 10 carri-
ers works independently, in the second configuration,
carriers work in Pairs. in 1 vs 3s, one carrier, the sam-
ple, is compared against 3 groups of 3 carriers. In 5
vs 5, 3 vs 7 and 1 vs 9 the 10 carriers are divided into
2 groups of differing sizes accordingly. In the final
configuration, All Cooperative, the 10 carriers work
together.
All Contracted
Pairs
1 vs 3s
5 vs 5
3 vs 7
1 vs 9
All Cooperative
75
80
85
90
95
Cost per request (£)
First group Second group
Figure 3: Cost per request for different carrier group con-
figurations.
Figure 3 confirms our earlier findings that work-
ing as a group can substantially reduce costs and ad-
ditionally shows that larger groups can attain bigger
cost reductions than smaller groups.
In each configuration, consignments are divided
equally between groups, not carriers, such that, for ex-
ample in the 1 vs 3s configuration each group of carri-
ers is assigned 100 consignments out of 400 but in the
1 vs 9 configuration, each group is assigned 200 con-
signments. Because of this, carrier 1 has more choice
in the 1 vs 9 configuration and can achieve slightly
better results than in the 1 vs 3s configuration, how-
ever the number of consignments that can actually be
served is dramatically reduced as can be seen in figure
4.
Figure 4 shows again the increase in network ca-
pacity made possible through cooperation. It is also
clear that the largest savings are made quickly: just
pairing with one other carrier can increase the num-
ber of scheduled deliveries from 72% to 80%.
5.2 Carrier Group Size
Extending our analysis, we seek to identify if there
are diminishing returns for increasing the number of
carriers in a cooperative group.
Figure 5 shows how both the cost per request and
the percentage of consignments scheduled improve
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
292
All Contracted
Pairs
1 vs 3s
5 vs 5
3 vs 7
1 vs 9
All Cooperative
75
80
85
Scheduled (%)
Across all carriers
Figure 4: Percentage of scheduled consignments for differ-
ent carrier group configurations.
1 2 3 4 5 6
7
8 9 10 15 20 25
75
80
85
Number of Carriers
Cost per request (£)
65
70
75
80
85
Scheduled (%)
Cost per request Consignments scheduled
Figure 5: Cost per request and number of consignments
successfully scheduled as the number of carriers working
together increases.
as the number of carriers in a cooperative group in-
creases.
Though there are linear savings evident above 10
carriers the majority of benefit is found between 1 and
5 carriers. These results must be qualified by stating
that our consignments cover the UK and our carriers
are randomly located across this area; since distance
costs are a dominant factor in real-world pricing; if
larger distances are involved, for instance across Eu-
rope, America or Asia a larger number of well dis-
tributed carriers would likely be necessary to produce
these savings. We also assume an infinite number
of vehicles at each carrier location; in practice there
will be a limited supply of vehicles at each carrier
and therefore multiple carriers in the same area would
need to work together. Our results can be thought of
more as suggesting that 10 major transport hubs is
sufficient for efficient vehicle routes in the UK assum-
ing an unlimited number of vehicles, clearly, given
that there are over 6000 carriers in the UK, more than
10 carriers would be required.
6 DISCUSSION AND
CONCLUSIONS
We have presented the VNDM hyper-heuristic, and
a set of LLHs optimised for the ordering constraints
of the problem, as an effective schedule optimisa-
tion for PMDP under dynamic consignment requests.
We use data from the RHA to explore pricing and
marginal costs of consignments, and show that cost
savings of 15% to 18% are possible when hauliers
cooperate. Cooperation also increases the capacity
of a group of hauliers, by as much as 21%. The
benefits of cooperation see diminishing returns above
10 separate carrier locations working together assum-
ing sufficient numbers of vehicles to meet demands.
Larger cooperatives will always have lower operating
costs than smaller ones as they are able to more effi-
ciently schedule their consignments to existing vehi-
cle routes.
In practice there would need to be some way to
distribute savings amongst all cooperating carriers
fairly, in order to encourage participation. Further
work is required to determine the best way to do this,
potential strategies involve allowing carriers to auc-
tion jobs to cooperating parties or having some central
control involved in deciding which carriers take what
consignments. We also do not consider issues of ve-
hicle reliability, for example, who pays the costs asso-
ciated with missed delivery slots and what effect this
has on customer perceptions. We intend to investigate
the impact of planning time and arrival time windows
on consignment cost and also to perform a compari-
son with alternative hyper-heuristic approaches.
We have not considered the fixed costs associated
with carrier owned vehicles in this research, only sav-
ings to marginal costs of delivery. Implementing the
strategies outlined in this paper may result in reduced
usage of carrier owned assets; we see future work as
being able to advise on appropriate fleet sizes for car-
riers and note that in general, cooperation allows for
an increase in capacity, allowing the same fixed cost
assets to be more productive, assuming there is suffi-
cient demand for service.
The Effect of Cooperation in Pickup and Multiple Delivery Problems
293
ACKNOWLEDGEMENTS
This work has been funded by the Large Scale Com-
plex IT Systems (LSCITS) EngD EPSRC initiative in
the Department of Computer Science at the Univer-
sity of York and by Transfaction Ltd.
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APPENDIX
Model Constraints
The constraints for the PMDP are laid out in tables 2
and 3. The constraints in table 2 have been adapted
and expanded from the formulation for the PDP pre-
sented by Desaulniers et al. (2002); Table 3 presents
the additional new constraints for the PMDP.
Table 2: Adapted constraints from Desaulniers et al. (2002),
here implies that this constraint is equivalent to a con-
straint presented by Desaulniers et al. and implies that
this constraint has been modified for the PMDP.
kK
uR
k
b
ruk
= 1 (3)
r R
uP
k
b
ruk
|D
j
|
wD
j
b
rwk
= 0 (4)
k K, r R
k
Removed (5)
Removed (6)
Removed (7)
b
ruk
t
rk
+tt
service
r
+tt
ru
t
uk
0 (8)
k K, (r, u) A
k
t
start
r
t
end
r
,t
start
r
t
rk
(9)
k K, r R
k
t
rk
+tt
service
r
+tt
ru
r
uk
(10)
k K, r P
k
, u D
r
b
ruk
(ll
rk
+ l
u
ll
uk
) = 0 (11)
k K, (r, u) A
k
0 < l
r
ll
rk
l
k
(12)
k K, r P
k
l
r
+
uD
r
l
u
= 0 (13)
r P
l
o
(k) = 0 (14)
k K
b
ruk
0 (15)
k K, (r, u) A
k
b
ruk
binary (16)
k K, (r, u) A
k
Constraints (3) and (4) ensure that each arc is only
included once and that a pickup and all its correspond-
ing deliveries are handled by the same truck. Here,
|D
u
| is the number of delivery-requests for pickup-
request u. Constraint (4) is non-standard for the PDP
and is necessary as there may be multiple delivery-
requests per pickup-request. It states that for each
pickup request there exists an b
ruk
= 1 and that this
multiplied by the number of deliveries is the same
Table 3: New constraints for the PMDP.
|P
c
| = 1 (17)
i I
|D
c
| 1 (18)
i I
t
rk
< t
uk
(19)
k K, r P
k
, u D
r
t
rk
< t
uk
t
vk
< t
wk
(20)
k K, r, u P
k
, v D
u
, w D
r
(r,u)A
k
b
ruk
tt
service
r
+tt
ru
tt
k
(21)
k K
as the number of arcs that end at each of the cor-
responding delivery requests. Constraints (5) to (7)
have been removed in comparison to the formulation
in Desaulniers et al. (2002) as we do not have a mul-
ticommodity flow. Constraint (8) remains unchanged
however, (9) and (10) have been modified, together
these allow for soft time windows. Constraints (11)
to (13) specify that a pickup node must have positive
load and that deliveries must have negative load, also
that the sum of pickup and delivery loads is zero. The
initial vehicle load, non-negativity and binary require-
ments are the same as Desaulniers et al. (2002). The
following constraints have been added for the PMDP:
(17) and (18) specify that a request has exactly one
pickup and may have arbitrarily many deliveries. (19)
specifies the precedence between a pickup and its de-
liveries while (20) expresses the LIFO constraint. Fi-
nally, (21) specifies that the length (in time) of any
vehicles route is less than a value E
k
which may be
set according to local conditions.
Minimising k, the number of vehicles used, is not
considered as part of this problem, though it is kept
low as a side effect of the heuristics used. For each
truck, requests may be nested within other requests if
LIFO and capacity constraints are not violated.
The Effect of Cooperation in Pickup and Multiple Delivery Problems
295