The Effect of Cooperation in Pickup and Multiple Delivery Problems

Philip Mourdjis

, Fiona Polack

, Peter Cowling

, Yujie Chen

and Martin Robinson

YCCSA, University of York, York, U.K.

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

A’s 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

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

, l

start

, t

end

service

) where n

is the location, l

is the load (where

the summation of pickup load and delivery loads for

a consignment is equal to zero). t

start

, t

end

, represent

the start and end time of the arrival window respec-

tively where the service time tt

service

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

, n

)) is the arc (r, u). A

consignment c is identiﬁed by (p

, D

, t

) where p

is the pickup-request and D

= d

, . . ., d

is the se-

quence of delivery-requests. Each consignment has a

received time t

, which is the time at which the order

is entered in the system. C is the set of consignments.

⊂ A represents the feasible arcs for vehicle k. The

binary ﬂow variable b

ruk

is set to one if arc (r, u) ∈ A

is used by the vehicle k, and to zero otherwise. ll

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

∑

k∈K

∑

(r,u)∈A

ruk

∗ b

ruk

(1)

where:

ruk

= nc(nn

, ll

) +tc(ruk) + dc(tt

delay

) (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

, ll

), the cost of travel-

ling distance nn

(the length of arc (r, u)) with load

; tc(ruk), the cost of the time taken by vehicle k to

travel arc (r, u), and dc(tt

delay

), 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

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

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

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

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

Scheduled (%)

Across all carriers

Figure 4: Percentage of scheduled consignments for differ-

ent carrier group conﬁgurations.

1 2 3 4 5 6

8 9 10 15 20 25

Number of Carriers

Cost per request (£)

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.

≡

∑

k∈K

∑

u∈R

ruk

= 1 (3)

∀r ∈ R

∗

∑

u∈P

ruk

∗ |D

| −

∑

w∈D

rwk

= 0 (4)

∀k ∈ K, r ∈ R

∗ Removed (5)

∗ Removed (6)

∗ Removed (7)

≡ b

ruk



+tt

service

+tt

−t



≤ 0 (8)

∀k ∈ K, (r, u) ∈ A

∗ t

start

≤ t

end

start

≤ t

(9)

∀k ∈ K, r ∈ R

∗ t

+tt

service

+tt

≤ r

(10)

∀k ∈ K, r ∈ P

, u ∈ D

≡ b

ruk

(ll

+ l

− ll

) = 0 (11)

∀k ∈ K, (r, u) ∈ A

∗ 0 < l

≤ ll

≤ l

(12)

∀k ∈ K, r ∈ P

∗ l

∑

u∈D

= 0 (13)

∀r ∈ P

≡ l

(k) = 0 (14)

∀k ∈ K

≡ b

ruk

≥ 0 (15)

∀k ∈ K, (r, u) ∈ A

≡ b

ruk

binary (16)

∀k ∈ K, (r, u) ∈ A

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,

| 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.

| = 1 (17)

∀i ∈ I

| ≥ 1 (18)

∀i ∈ I

< t

(19)

∀k ∈ K, r ∈ P

, u ∈ D

< t

⇒ t

< t

(20)

∀k ∈ K, ∀r, u ∈ P

, ∀v ∈ D

, ∀w ∈ D

∑

(r,u)∈A

ruk



service

+tt



≤ tt

(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

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