Multi-trip Pickup and Delivery Problem, with Split Loads, Proﬁts and

Multiple Time Windows to Model a Real Case Problem in the

Construction Industry

Atef Jaballah

and Wahiba Ramdane Cherif-Khettaf

LORIA, UMR 7503, Lorraine University, Nancy, France

Keywords:

Vehicle Routing Problem, Construction Industry, Pickup and Delivery, Split Loads, Proﬁt, Multi-trips,

Multiple Time Windows, Heuristic.

Abstract:

This paper presents the ﬁrst optimization study of multi-site transportation in the construction industry, which

allows mutualizing building material delivery and construction waste removal. This study is inspired by a real-

world problem encountered in the framework of the French R&D project DILC, in which a pooling platform

must centralize the delivery of building materials to the construction sites and the pickup of their waste, using

a limited and heterogeneous ﬂeet that is allowed to perform multiple trips, under time and capacity limitation

constraints. The problem under study, called the Multi-Trip Pickup and Delivery Problem, with Split loads,

Proﬁts and Multiple Time Windows is a new extension of the vehicle routing problem with pickup and delivery,

that considers new realistic constraints speciﬁc to the construction industry such as each construction site may

have a priority on its delivery request or its pickup request or both, with a higher priority level for delivery

request, and each construction site may have several time windows. To solve this problem, we propose new

insertion criteria that takes into consideration several aspects of our problem, which we have embedded in a

construction heuristic. Experiments performed on new instances have shown the efﬁciency of our method.

1 INTRODUCTION

The problem addressed in this paper is a multi-site si-

multaneous optimization of building material delivery

and waste pickup through a pooling platform in the

construction sector. This issue is the result of a col-

laboration that we are conducting within the French

framework of the R&D project DILC, whose aim is

to design an innovative platform for optimizing con-

struction site logistics, that is adapted to multi-site

eco-city construction projects. The optimization lever

studied in the DILC project is the consolidation of the

transportation ﬂows and human resources through a

physical platform that is modular, removable and mo-

bile, and the development of decision support tools to

help the platform managers to optimize their logistics.

Unlike direct transportation from suppliers to con-

struction sites, the pooling platform aims grouping

many delivery building materials from different sup-

pliers, receiving them in pallets according to a sched-

ule corresponding to the progress of construction ac-

https://orcid.org/0000-0002-7199-4415

https://orcid.org/0000-0002-2822-0262

tivities on the construction sites. From the received

building materials, ready-to-use kits are prepared on

the platform, stored and delivered in pallets to the

construction sites. The kit represents the site supply

unit, that is, a kit must be delivered in full. It is not

possible to split the kit into several deliveries. Each

kit is characterized by its ID, the number of pallets it

contains, and its weight. The delivery request from

the construction sites may involve different kits, and

the quantities of material delivery demands are known

to sometimes exceed the truck’s capacity, which re-

quires to supply the construction sites several times,

so splitting the delivery demand is allowed in our

case. The quantity of waste is relatively inferior to

the quantity of material to be delivered, but can also

be split.

The platform must also manage the removal of

waste from construction sites to the platform. It

should be noted that there are two types of waste:

Big-bag waste and tipper waste. Big-bag wastes are

packed on pallets and concern wastes that are pro-

duced with small and medium quantities such as soft

plastic, hard plastic, and cardboard. Tipper waste con-

cerned the wastes that are produced with high quan-

200

Jaballah, A. and Cherif-Khettaf, W.

Multi-trip Pickup and Delivery Problem, with Split Loads, Proﬁts and Multiple Time Windows to Model a Real Case Problem in the Construction Industry.

DOI: 10.5220/0010253002000207

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 200-207

ISBN: 978-989-758-485-5

tities like wood and metals. In this study, we focus

only on Big-bag wastes because their removal can

be pooled with the delivery of building materials us-

ing a limited and heterogeneous tail-lift truck ﬂeet,

whose capacity is given in pallets. The vehicles per-

form multiple trips between the platform and the con-

struction sites to load the kits at the platform, deliver

them to the construction sites, collect waste from the

construction sites and unload these wastes in the re-

cycling center located just next to the platform.

It should also be noted that the planning of opera-

tions within the platform does not concern this study.

The problem addressed here is the routing optimiza-

tion between the platform and the construction sites

to mutualize the materiel delivery and the waste re-

moval, and more speciﬁcally we present a new vari-

ant of the well-known pickup and delivery problem,

named MTPDPSPMTW for the Multi-Trip Pickup

and Delivery Problem, with Split loads, Proﬁts and

Multiple Time Windows.

The MTPDPSPMTW belongs to the class of Ve-

hicle Routing Problem with Pickup and Delivery

(VRPPD), which has been studied for more than 30

years (Parragh et al., 2007; Parragh et al., 2008), and

consists of transporting objects or people between ori-

gins and destinations. More precisely, the problem

studied in this paper is included in the class VRP with

Backhauls (VRPB) where objects or individuals are

transported from a depot to linehaul customers and

from backhaul customers to a depot (Parragh et al.,

2008), and the most adequate VRPB subclass with

our problem is the VRP with Divisible Deliveries

and Pickups (VRPDDP), which is a special case of

the VRPPD, where each customer may have deliv-

ery and/or pickup requests that must be served with

capacitated vehicles, and the pickup and the delivery

quantities can be served, if helpful, in two separate

visits(Nagy et al., 2015). Many variants of VRPPD

with several constraints have been proposed in the

literature to model real transportation problems. In

some cases, the available resources are insufﬁcient to

service all customers. Thus, a known proﬁt is associ-

ated with each customer, the goal is to ﬁnd the subset

of customers to be serviced and to determine vehi-

cle routes that: maximize the total acquired proﬁt,

minimize the total traveling cost, and satisfy tem-

poral and capacity constraints (Chentli et al., 2018).

Several variants of VRPPD introduced the concept

of time windows where the service at each customer

must start within one given time windows (Sun et al.,

2020). The most relevant extension of the VRPPD is

the VRPPD with split loads (VRPDPSL) where cus-

tomer demands can be split between several vehicles

(Nowak et al., 2008). Other variants of VRPPD which

combines split and multi-trip are available in the lit-

erature such as (Haddad et al., 2018) and (Yin et al.,

2013).

Despite the abundant literature on the pickup and

delivery problem, the problem that we present here is

a novel one and allows us to model constraints that are

speciﬁc to the construction sector. Our contribution

can be summarized in the two following issues:

- The MTPDPSPMTW allows to simultaneously

consider constraints that have never been previously

combined in the pickup and delivery variants stud-

ied in the literature. More speciﬁcally, these are the

constraints on heterogeneous ﬂeet of vehicles, multi-

trips, splitting demands, proﬁt and time windows.

Note that in our problem, proﬁt is associated with

a pickup request and/or a delivery request, while in

most studies in the literature proﬁt is associated with

a customer and includes both pickup and delivery re-

quests.

- The MTPDPSPMTW is an NP-hard problem,

and to resolve it we propose new score criteria to deal

with the complexity of our problem, and we embed

these criteria in a constructive polynomial heuristic,

named SBH (Score Based Heuristic). Experiments on

new instances show the effectiveness of this approach.

The remainder of the paper is organized as fol-

lows. Section 2 describes the problem. A construc-

tive heuristic is proposed in Section 3. Experimental

results with the deﬁnition of benchmarks are given in

Section 4. Finally, concluding remarks and guidelines

for future research are given in Section 5.

2 PROBLEM DEFINITION AND

NOTATION

The Multi-Trip Pickup and Delivery Problem, with

Split loads, Proﬁts and Multiple Time Windows,

named MTPDPSPMTW can be deﬁned on a com-

plete, undirected graph G = (E,V ), where V =

{0, . . . , n} is the set of vertices and E = {(i, j ) : i, j ∈

V, i 6= j} is the set of edges. Vertex 0 is the pooling

platform while the other vertices are the construction

sites. A travel time t

i j

and cost c

i, j

are assigned to

each edge (i, j ). A ﬂeet of heterogeneous tail lift ve-

hicles is located on the platform. The vehicle ﬂeet is

composed by m vehicles with different capacities and

time availability. We noted Q

the capacity in pallets

of the k

vehicle k ∈ {1, . . . , m},W

its volume capac-

ity in tons, and by D

its maximum working time.

Each site i ∈ V has a pickup demand

−→

a deliv-

ery demand

−→

. Note that, all demands are integer

vectors.

−→

is in this form (big − bag

, . . . , big − bag

)

which expresses the pickup demand of each type of

Multi-trip Pickup and Delivery Problem, with Split Loads, Proﬁts and Multiple Time Windows to Model a Real Case Problem in the

Construction Industry

201

Big-bag waste.

−→

is represented as (kit

, . . . , kit

)

that describes the delivery demand of each type of

kits. A kit can contain one or more pallets and the

Big-bag waste unit is the pallet. Thus, all demands of

site delivery and pickup are expressed in pallets. In

the rest of the paper we denote by qt(

−−→

vect) the size

of demand in pallets (

−−→

vect can be

−→

). Some-

times, the demands of sites (delivery and/ or pickup)

are greater than the vehicle capacity (for example

qt(

−→

) > Q

), then the site can be served by the same

vehicle with several trips or by several vehicles. Each

site can have a priority on its delivery demands or its

pickup demands or both. To satisfy these require-

ments, two real values pp

and pd

are associated

with each site i and correspond to the pickup proﬁt

and delivery proﬁt, respectively. Unlike the litera-

ture approaches where the proﬁt is associated with

customers, in our model, the proﬁt is associated with

each demand.

Each vertex i ∈ V \ {0} has a service time s

which corresponds to the loading/unloading time

on site, and a set of time windows TW

{[e

, l

], [e

, l

], . . . , [e

, l

]} where e

p ∈ {1, . . . ,t} is

the earliest time to begin service at the vertex i and

is the latest time to ﬁnish service at the vertex

i . Some time windows are ﬂexible, so, time de-

lays allow arrival before e

and departure after l

These time delays are noted me

(ml

respectively)

for the upstream delay (the downstream delay respec-

tively),so each time window of a site i can be enlarged

to [e

− me

, l

+ ml

] = [e

∗

, l

∗

]. If a delay time of a

time window is null, the time window is called hard

time window, otherwise the time window is ﬂexible,

the ﬂexibility of a given time window is more impor-

tant if the delay time is larger. In fact, this time delay

can be used mainly by the platform managers to help

them to negotiate more effectively the time window

constraints with the construction sites.

Furthermore, we deﬁned [e

, l

] as the single time

window of the platform that designates the earliest

possible departure from the platform and the latest

possible arrival at the platform. The service time s

the platform is given by the sum of the loading time

of kits and unloading time of Big-bag waste. This

service time is not considered for the ﬁrst trip of each

vehicle since the ﬁrst vehicle loading can be done in-

dependently of its tour. If a vehicle travels directly

from site i to site j. The service of site j starts at b

max{e

, b

+ s

+ t

i, j

} where e

= min

1≤k≤|TW

−

| l

+ ml

− (b

+ s

+ t

i, j

+ s

) ≥ 0} designated

the lower bound of the most adequate time window.

If the vehicle arrives too early at j, the service can

start at b

= max{e

− me

, b

+ s

+ t

i, j

} if me

6= 0.

Note that waiting is not allowed because construction

site activities do not allow access to the sites beyond

the imposed time constraints.

A feasible solution to our problem is composed of

a set of feasible trips assigned to adequate vehicles.

A feasible trip is a sequence of nodes that satisﬁes the

following set of constraints:

• Each trip must start and end at the pooling plat-

form.

• Each kit must be delivered in full, no possibility

to split the kit into several trips.

• The overall amount of materials delivered and

wastes picked along the route must not exceed the

vehicle capacity (Q

, W

• The total duration of each trip calculated as the

sum of all travel duration required to visit all the

construction sites of the trip sequence, and service

time needed for each visit to a construction site

during the tour could not exceed D

;

• Each site can be visited at most once during the

trip while respecting one of its time windows.

We seek to construct a feasible solution of a minimum

number of trips, and affecting one or several trips to

the available vehicles such that:

• The total duration of each vehicle’s route, calcu-

lated as the sum of all its trips duration, and the

sum of the platform’s service times don’t exceed

• Each vehicle must start at the pooling platform no

earlier than e

and ﬁnish at the pooling platform

no later than l

• No more than m vehicles are used;

• Each construction site may be visited several

times with the same or different vehicles, so split-

ting is allowed for delivery requests and pickup

requests, and some sites may not be visited at all.

• The sum of the quantities delivered to a given con-

struction site must be less than or equal to its de-

livery request and the sum of the quantities col-

lected from a given construction site must be less

than or equal to its pickup request. This means

that the customer can be delivered and/or col-

lected partially.

The objective function is a weighting of three objec-

tives: time, proﬁt, and the number of priority cus-

tomers fully served. Our goal is to minimize the ﬁrst

objective and maximize others.

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

202

3 SCORE BASED HEURISTIC

The MTPDPSPMTW is an NP-hard problem. Hence,

the use of exact optimization methods is not able

to solve this problem in polynomial time, when the

problem concerns the very large real-world data sets.

However, heuristics are more suitable for this prob-

lem. Accordingly, we proposed a new construction

heuristic called SBH (Score Based Heuristic) and new

score criteria adapted to our problem such as dis-

tance, time, customer service urgency, the deadline to

serve a customer, priority (proﬁt delivery and proﬁle

pickup).In this section, we describe this heuristic.

Initially, the heuristic selects a vehicle available

from a heterogeneous ﬂeet. Throughout this work, the

vehicle selection has undergone the following rules:

If the total demand of the sites not yet served is greater

than the maximum capacity in this ﬂeet, then the vehi-

cle available with the maximum capacity is selected.

Otherwise, the vehicle with the minimum capacity

that meets all demands is selected. In the next step,

the heuristic creates an empty trip and associate it

with the selected vehicle. This trip is extended by

appending the feasible site j that has the minimum

score based on Cr

i, j

Eq(7) to the sites i (the latest

routed site). A free client j is said feasible if it can

be added to the current route without violating time

window constraint TW

, vehicle capacity constraint

(pallet capacity), W

(tons capacity), and vehicle

time availability D

. During the loading of a vehi-

cle at the platform, two cases are possible: either that

the site demand d

is less than the vehicle’s capacity,

in this case, all customer delivery will be loaded into

this vehicle (the customer is fully delivered). other-

wise, the customer is partially delivered on the cur-

rent trip. Two sites or more will probably be deliv-

ered partially on the same trip due to atomic deliver-

ies of kits. For example, if we consider a vehicle k

with Q

= 16 and two sites where

−→

= (4 kit

, 3 kit

)

and

−→

= (2 kit

, 1 kit

). Knowing that kit

, kit

and

kit

are composed by 3, 2 and 5 palettes, respectively.

Thus 4 kit

will be delivered to site 1 and 2 kit

to site

2. As soon as the limit vehicle capacity is reached or

no site can be inserted to the current trip, the vehicle

returns to the platform and a new route is initialized.

If the vehicle availability time has expired a new ve-

hicle is selected. The algorithm converges when all

customers are satisﬁed or all resources are used. The

main structure of this heuristic is given by algorithm

In this part of our paper, we describe the selec-

tion criteria of the SBH heuristic proposed to the MT-

PDPSPMTW. Firstly, we consider the distance d

i, j

Eq(1) which allows to select the closest sites to the

current vehicle location. The second criterion is the

travel time T

i, j

Eq(2). In addition, we consider the

customer service urgency using all time windows and

their margins Ur

i j

Eq(3) which favors the selection of

sites where their remaining time service is very short.

The fourth criterion is the deadline to serve the site

Eq(5) this promotes the selection of sites whose

deadlines will expire as soon as possible.

The last are the proﬁt related to delivery Pd

and

pickup Pp

which are represented by Eq(5) and Eq(6),

respectively. The values of Pd

and Pp

are ﬁxed ac-

cording to real cases. The goal of these criteria is to

foster the serving of the sites with priority. To ensure

continuity of deliveries service for a site, we multiply

the delivery proﬁle Pd

by progress rate delivery as

shown in Eq(7). For example, if a site i has a demand

such as qt(

−→

)=15 palettes and only 10 palettes

are delivered on the current trip.It will have a more

chance than the other sites where the service has not

yet started to be selected on the next trip because the

progress rate delivery of i is

∑

qt(

−→

)

= 0.75, but,

the progress rate delivery of other sites is null. This to

avoid multiplying partial services.

•

i j

(1)

the distance between the last site visited i and a

site j not yet satisﬁed either in delivery or in col-

lection

•

i j

= b

− (b

+ s

) (2)

the time difference between the end of service at i

and the start of service at j

•

i j

= l

− (b

+ s

i j

) +

∑

1≤k≤|TW

k6=c

≥b

i j

− e







∑

1≤k≤|TW

k6=c

≥b

i j

+ ml







(3)

the site service urgency when we consider all time

windows and their margins. β = 1 if we allow the

use of time windows margins 0 otherwise.

•

= max

1≤k≤|TW

) − (b

+ s

i j

) (4)

deadline to serve the site j.

•







5 if site i has priority in deliveries

2 if site i has non-priority in deliveries

0 otherwise

(5)

Multi-trip Pickup and Delivery Problem, with Split Loads, Proﬁts and Multiple Time Windows to Model a Real Case Problem in the

Construction Industry

203

•







2 if site i has priority in pickup

1 if site i has non-priority in pickup

0 otherwise

(6)

i j

= γ

i j

max

i j

)

+ γ

i j

max

i j

)

+ γ

i j

max

(Ur

i, j

)

+ γ

max

(Ds

)

− γ

∑

qt(

−→

)

max

(Pd

)

− γ

max

(Pp

)

(7)

where:

is the number of pallets delivered to site j so far.

∑

i=1

= 1 γ

≥ 0 ∀i ∈ 1, . . . , 6

4 COMPUTATIONAL STUDY

In this section, we describe our experimental results.

Section 4.1 presents the characteristics of the MT-

PDPSPMTW test instances. Section 4.2 the conﬁg-

urations for the values of the parameters of our al-

gorithm using the irace package. The results of de-

tailed and comprehensive computational studies are

summarized in Section 4.3.

4.1 Instance Generation

To evaluate the performance of our heuristic, we

designed two groups of MTPDPSPMTW instances

based on DILC real scenarios. The distance between

two sites or between a site and the platform is cho-

sen randomly between 1 and 150 km. For each in-

stance, 20% of sites have a load of a delivery request

in [Q, 3Q], 60% in [

Q, Q] and 20% lower than

The load of a pickup request is lower than

Q for

all sites, where Q = 16 is the pallet capacity of the

largest vehicles. The percentage of priority requests

waste collection is always ﬁxed at 50%. Each site

can have one, two, or three time windows. The set

of site time windows can be ﬁxed chosen from a list

{[06 : 00 − 08 : 00 0 − 30], [11 : 00 − 14 : 00 30 −

0], [17 : 00−20 : 00 30−0]} of predeﬁned or chosen

randomly respecting platform time windows, where

the notation [e

–l

− ml

] correspond to the time

window [e

, l

] of site i with upstream delay me

and

downstream delay ml

To evaluate the effectiveness of our heuristics on

several real cases, we create the ﬁrst group of in-

stances named G1 which contains three basic in-

stances DILC10, DILC20, and DILC100 or 10,20

Algorithm 1: Score Based Heuristic.

Input: A MTPDPSPMTW instance;

A list L

uns

of unsatisﬁed sites ;

uns

←

A list L

f eas

of feasible sites ;

f eas

← V;

Output: A feasible solution

1 Total D ← Calculate Total delivery();

2 Total P ← Calculate Total pick up();

3 k ← Select vehicle() ;

4 vehicleLoad ← 0;

5 Availability k ← D

;

6 Create an empty route r for this vehicle and

initialize it with the platform ;

7 vehCount ← 1 ;

8 while (vehCount ≤ m) AND (L

uns

0 ) do

9 while (vehicleLoad < Q

) AND

(Availability k < D

) do

10 Update L

f eas

;

11 if L

f eas

0 then

12 Let j ∈ L

f eas

such that

i, j

= min

j∈L

f eas

(Cr

i, j

) referring to Eq.

(7) ;

13 Update(

−→

);

14 Update(Total

D, Total P);

15 Connect j to last visited site i ;

16 Update(vehicleLoad);

17 Update(Availability k);

18 Update L

uns

;

19 else

20 break;

21 end

22 end

23 Connect the last site of the route r to the

platform ;

24 if Availability k < D

then

25 Create an empty route and initialize it

with the platform;

26 Associate this route to the current vehicle;

27 else

28 k ← Select vehicle() ;

29 vehicleLoad ← 0;

30 Availability k ← D

;

31 Create an empty route for this vehicle ;

32 vehCount ← vehCount + 1;

33 end

34 end

and 100 designating the number of sites. The num-

ber of vehicles is 3, 5, and 10 for instances DILC10,

DILC20, and DILC100, respectively. In all instances,

70% of vehicles have a large capacity, Q = 16 and the

leftovers have a small capacity of Q = 4. In each in-

stance, we vary the percentage of priority requests de-

liveries from 0%, 30%, 60%, and 100%. Then we ob-

tain 4 subgroups. For each subgroup described above,

we vary the percentage of random time windows from

0%, 30%, 70%, and 100%. So we have 16 instances

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

204

per group, resulting in a total of 48 instances in our

test group G1.

The most representative cases are the instances

DILC100. Consequently, we create a second group of

instances noted G2-100-70-50 where 100 is the num-

ber of clients of which 70 are priority and 50 is the

percentage of sites who have random time windows.

10 instances are generated in this group to assess the

impact of each criterion studied in section 3 on the

behavior of SBH.

4.2 Training SBH Parameters

Our heuristics use a score to select the most suitable

site. This score utilizes a set of criteria. To weigh

these criteria in the most effective way, we have cho-

sen automatic tuning using Iterated Racing for Auto-

matic Algorithm Conﬁguration (IRACE). IRACE is a

tool based on machine learning methods to tune opti-

mization algorithms, i.e. automatically ﬁnding good

conﬁgurations for the parameter values of a target al-

gorithm. (L

opez-Ib

nez et al., 2016). We set to 2000

the maximum number of algorithms runs during the

tuning for the algorithm runs for all instances. The

chosen parameters space for this racing are γ

∈ [0, 1]

and

∑

i=1

= 1. Table 1 shows the best setting pa-

rameters found by irace. These results show that all

parameters are not null, hence the importance of all

proposed criteria to calculate the score.

Table 1: Best conﬁgurations found by irace for the SBH

heuristic.

Parameter γ

Value 0.14 0.32 0.1 0.16 0.25 0.012

4.3 Resultats Analysis

The SBH heuristic was coded in python. All tests

were carried out on a personal computer with Intel

Core(TM) i7-7920HQ processor at 3.10 GHz and the

Microsoft Windows 10 operating system using 32.00

GB RAM.

Table 2 shows the results obtained by our heuris-

tic to solve the instance DILC 10, 20, and 100 de-

scribed above. These results reveal that the heuris-

tic can completely satisfy a high percentage of prior-

ity requests (for example, for the DILC20 group of

instances, we have 93.49% of priority delivery re-

quests completely satisﬁed, and 98.12% of priority

pickup requests). The heuristic adds a lower percent-

age of partially satisfying requests to complete routes.

It can be observed that the rate of partially satisfy-

ing requests is higher for priority delivery requests

compared to non-priority delivery requests (18.84%

for partially fulﬁlled priority delivery requests ver-

sus 6.9% for partially fulﬁlled non-priority pickup re-

quests for DILC10 instances). The quantities of pick-

up requests are much lower than the delivery quan-

tities, which explains the low percentage of partially

fulﬁlled requests.

For instances DILC10 we observe that 78.2% of

priority delivery requests are fully satisﬁed and 18.8%

are partially satisﬁed. The pallets delivered corre-

spond to 66.6% of the total number of pallets re-

quested. For non-priority delivery requests, 77.1%

are totally satisﬁed, and 6.9% partially satisﬁed. The

percentage of totally satisﬁed pickup requests is also

high 77.1% for the priority and 98.4% for the non-

priority. The collected pallets represent 66.6% of the

total.

For instances DILC20, we note that the percentage

of priority and non-priority delivery requests com-

pletely satisﬁed is very high 93.4% and 94.3%, re-

spectively. The percentages of pickup requests totally

satisﬁed are very lofty 98.12% for both priority and

non-priority requests. This explains the high rate of

pallets delivered 94.9% and collected 98.5%.

Finally, for large instances DILC100, the results

indicate that 53.4% of priority delivery requests are

entirely satisﬁed and 7.9% are incompletely satisﬁed.

The percentage of non-priority delivery requests com-

pletely satisﬁed is 25.6% and 6.3 for the requests par-

tially satisﬁed. The percentage of pickup requests

fully satisﬁed is 46.2% for the priority requests and

41.2% for non-priority requests. The rates of pallets

delivered and collected are proportional to the per-

centage of requests served. According to these re-

sults, we conclude that the SBH is effective for all

instances in terms of all metrics. Consequently, our

heuristic algorithm can be used to solve real industrial

cases.

4.4 Impacts of the Criteria on SBH’s

Behavior

Referring to table 2 we show that instances DILC100

are the most difﬁcult, so we decided to evaluate the

impact of each criterion discussed in Section 3 on the

behavior of our heuristic using instances of G2-100-

70-50 group. To reach these goals, we used the fol-

lowing metrics: The priority deliveries (Figure 1a)

and non-priority deliveries (Figure 1b) metrics are

statistics on the average number of priority and non-

priority delivery sites, respectively. Priority pickup

(Figure 1c) and non-priority pickup (Figure 1d) rep-

resent statistics on the average number of priority and

non-priority sites in pickup. The total distance and to-

tal time are exposed by Figure 2a and 2b, respectively.

Multi-trip Pickup and Delivery Problem, with Split Loads, Proﬁts and Multiple Time Windows to Model a Real Case Problem in the

Construction Industry

205

The number of material pallets delivered (Figure 2c)

represents the average of the pallets delivered on all

sites priority or non-priority. The number of Big-bag

pallets collected (Figure 2d) designates the average of

pallets collected from all sites.

Figures1 illustrates these statistics which repre-

sent the result of the evaluation of each criterion sep-

arately, i.e. γ

= 1 for the i

criterion, and the other

cities are not taken into account. If we consider the

distance criterion only d

i, j

Eq(1), we notice that the

average of priority delivery requests totally satisﬁed

is quite important 26.5 among 70 (see Figure 1a), as

well as the average of non-priority delivery requests

totally satisﬁed which is equal to 12 (see Figure 1b).

The same goes for the average of priority and non-

priority pickup requests. We also note that the average

of delivery requests (priority or not) partially satisﬁed

is high, which contradicts our objective, which aims

to satisfy completely all sites. This can be explained

by the fact that when we use this criterion, the heuris-

tic aims to minimize the total distance traveled by the

ﬂeets of vehicles 5059 km which increases the num-

ber of tours and consequently the number of requests

served in terms of deliveries and pickup without tak-

ing into account priority and complete request fulﬁll-

ment. Thus, the average of the amount of pallet deliv-

ery and pallet pickup is important that corresponding

to 575.8 and 289.6, respectively. As time is propor-

tional at the distance, we obtain the same results for

both criteria.

70 70 70 70 70 70 70

41.3 41.3

21.2

31.9

29.1

40.6

26.5 26.5

19.3

12.5

28.8

24.7

34.2

#priority deliveries

30 30 30 30 30 30 30

17.2 17.2

7.5

9.5

2.1

7.6

12 12

6.1

0.3

5.8

#non-priority deliveries

completely delivered partially delivered undelivered

(a) Priority deliveries (b) Non-priority deliveries

50 50 50 50 50 50 50

30.1 30.1

16.1

15.6

10.9

28.8

24.3

27.2 27.2

14.6

12.8

9.4

27.8

22.9

#priority pickup

50 50 50 50 50 50

27.2 27.2

15.3

14.2

19.4

2.4

22.6

25.6 25.6

13.8

12.3

18.9

1.6

21.3

#Non-priority pickup

completely collected partially collected not collected

Figure 1: Comparisons between the impact of each criterion

on deliveries and pickup.

If we consider only the site service urgency Ur

i j

Eq(3), we observe that the average of delivery re-

quests totally satisﬁed (priority or not) and the av-

erage of pickup requests totally satisﬁed is very low

compared to the other criteria. This is explained by

the fact that our heuristic selects the site which has the

minimum service time such a site can be far from the

current vehicle position which increases routes and

reduces the number of trips. Consequently, the num-

ber of sites served and the average of pallets delivered

and collected decreases as shown in Figure 1.

5,000

5,500

6,000

6,500

5,0595,059

6,290

6,397

6,084

6,082

5,348

Total distance

110

112

114

111.71111.71

111.82

112.4

113.45

110.19

114.01

Total time

(a) Total distance (b) Total time

300

400

500

600

575.8575.8

304.7

274

492

399

557.8

# of materials pallets delivered

150

200

250

300

289.6289.6

152.9

138

158

162.9

235.2

#ofBig-bag pallets collected

Figure 2: Comparisons between the impact of each criterion

on distance, time, and pallets delivered and collected.

When we assess the deadline to serve the site Ds

Eq(4). We note that the average of priority delivery

requests totally satisﬁed and the average of material

delivered pallets are the smallest 12.5 and 274, re-

spectively. However, the distance traveled by the ﬂeet

of vehicles is the longest 6397 km . This is due to

the behavior of our heuristic that chose the site with

the time service that will expire the earliest. These

sites are often distributed in distant geographical ar-

eas, which proves the results obtained.

Table 2: Results of the score based heuristic for solving

instance DILC 10, 20, and 100.

DILC10 DILC20 DILC100

% of sites 100% satisﬁed 78.2 93.4 53.4

Priority Delivery

% of partially satisﬁed sites 18.8 5.2 7.9

% of sites 100% satisﬁed 77.5 98.1 46.2

Pickup

% of partially satisﬁed sites 98.9 0 3.3

% of sites 100% satisﬁed 77.1 94.3 25.6

Non-Priority Delivery

% of partially satisﬁed sites 6.9 4.8 6.3

% of sites 100% satisﬁed 98.4 98.1 41.2

Pickup

% of partially satisﬁed sites 0 0 2.3

Distance 1236.4 2522.7 5188.6

Number of hours 31.2 54.3 111.1

Number of trucks 3 4.9 10

% pallet served in deliveries 66.6 94.9 42.2

% pallet served in pickup 85.1 98.5 45.6

When we test the inﬂuence of delivery proﬁt Pd

Eq(5) on heuristic behavior, we notice that the max-

imum of the average of priority delivery requests to-

tally satisﬁed 28.8 is reached for this criterion and the

average of priority delivery requests partially satisﬁed

is the smallest 3.1 thanks to the progress rate deliv-

ery which encourages the continuity of site service.

The average of non-priority delivery requests totally

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

206

or partially satisﬁed tends towards zero. The average

of non-priority pickup requests is greater than the av-

erage of priority pickup requests since the majority of

sites served are the priority in the delivery and not in

the collection. The total time is the biggest despite the

distance is no longer this is due to the bigger service

time.

We ﬁnd that the pickup proﬁt Pp

Eq(6) allows

us to serve a large number of pickup priority sites as

well as delivery priority sites, since, most of the sites

served, in this case, are the priority in pickup and de-

liveries. For all the criteria we notice that the average

number of partially satisfying pickup requests is very

small since the quantity of Big-bag waste generated

by construction sites is low below 3/4 vehicle capac-

ity, which allows them to be collected easily.

Our heuristic SBH combines all the criteria de-

scribed previously in a weighted manner according

to Table 1. This combination noted H in Figures 1

and 2 allows obtaining good results on all compara-

tive metrics. Indeed, our heuristic satisﬁes the highest

number of priority delivery requests 34.2, and a sig-

niﬁcant number of non-priority delivery requests 5.8.

The average pickup requests totally satisﬁed is also

important compared to the other criteria. The average

amount of pallet delivery and pallet pickup are com-

parable to the ones obtained by d

i, j

and T

i, j

. The total

time duration taken by our heuristic is highest because

the priority sites are not necessarily nearby in addition

to the time of loading and unloading of vehicles.

5 CONCLUSION

In the present paper, we described a new variant of

the vehicle routing problem with pickup and deliv-

ery, named MTPDSPMTW for multi-trip pickup and

delivery problem, with split loads, proﬁts, and multi-

ple time windows. This new variant allows modelling

a real-world problem encountered in the construction

industry and combines for the ﬁrst time characteristics

that have been studied separately in the literature on

vehicle routing problems. These are multi-trip, split

loads, proﬁt, time window and the use of heteroge-

neous vehicles. The issue is that not all customers can

be served because the number of vehicles is limited,

and partial service is allowed, so the difﬁculty arises

in selecting the customers to be included in the tour

while prioritizing service satisfaction for customers

with a high priority (maximizing proﬁt), and allow-

ing non-priority customers to be included to to ﬁll the

residual capacity and/or time of the vehicles. We pro-

posed a new insertion score based on six criterai, that

is embedded in a construction heuristic (SBH). Nu-

merical results on new instances show that all pro-

posed criteria have a non-zero weight in the score and

that the combination of all criteria gives the most sig-

niﬁcant results in relation to the objective of maxi-

mizing the number of priority customers completely

satisﬁed. The effectiveness of these results has been

validated by our industrial partners, and have been re-

tained for further experimentation. The next step is

to develop metaheuristic approaches to improve ob-

tained results.

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