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ABSTRACT

This study focuses on a distribution problem involving

incompatible products which cannot be stored in a com-

partment of a vehicle. To satisfy different types of customer

demand at minimum logistics cost, the products are stored

in different compartments of eet vehicles, which requires

the problem to be modeled as a multiple-compartment ve-

hicle routing problem (MCVRP). While there is an extensive

literature on the vehicle routing problem (VRP) and its nu-

merous variants, there are fewer research papers on the

MCVRP. Firstly, a novel taxonomic framework for the VRP

literature is proposed in this study. Secondly, new mathe-

matical models are proposed for the basic MCVRP, together

with its multiple-trip and split-delivery extensions, for obtain-

ing exact solutions for small-size instances. Finally, heuristic

algorithms are developed for larger instances of the three

problem variants. To test the performance of our heuris-

tics against optimum solutions for larger instances, a lower

bounding scheme is also proposed. The results of the com-

putational experiments are reported, indicating validity and

a promising performance of an approach.

KEY WORDS

multiple-compartment vehicle routing problem; incompa-

tible products; split delivery; multiple trips; mathematical

model; heuristic algorithms;

1. INTRODUCTION

The vehicle routing problem (VRP) arises from the

logistics eld and deals with the distribution of goods

to customers [1]. It is a generalization of the traveling

salesman problem (TSP), which is an NP-hard prob-

lem [2]. The concept of a multiple-compartment VRP

(MCVRP) includes separated compartments carrying

incompatible products requested by one or more cus-

tomers. The problem aims to meet different types of

customer demands in the same vehicle while seeking

minimum distribution cost. There is an extensive liter-

ature on routing problems which includes taxonomic

studies [3-7]. Figure 1 proposes a new taxonomy for the

VRP.

The general VRP can be represented using six

main classes with respect to constraints describing

the problem structure, those being operational policy,

objective, vehicle, product, depot and period. The op-

erational policy dimension represents problem char-

acteristics and constraints regarding system congu-

ration and operating principles. In the VRP, soft time

windows (STW) [8] or hard time windows (HTW) [9]

can be dened to determine operational constraints

while servicing each node. Another operational classi-

cation identies the problem as pickup and delivery

(PD) [10], backhaul (BH) [11] or backhaul and linehaul

(BH&LH) [12], as opposed to the classical linehaul

problems. If the demand of a customer can be satis-

ed by more than one vehicle, the VRP with split deliv-

ery (SLD) [13] is valid; otherwise, no split deliveries are

allowed. The last operational classication category di-

vides the problem as a single-trip (ST) or multi-trip (MT)

VRP [14], allowing a vehicle to perform one or several

trips within the planning period. The VRPs also differ

according to their desired objectives. Literature wide-

ly uses minisum objectives, which minimize the total

distribution cost (MSTC) [15], total distribution time

(MSTT) [16] or total length of routes (MSTL). Minimizing

the total waiting time of the customers (MSWT) [17] is

BAHAR TAŞAR, M.Eng.1

E-mail: bahar.turantasar@gmail.com

DENİZ TÜRSEL ELİİYİ, Ph.D.2

(Corresponding author)

E-mail: deniz.eliiyi@bakircay.edu.tr

LEVENT KANDİLLER, Ph.D.3

E-mail: levent.kandiller@yasar.edu.tr

1 Ege University

Faculty of Economics and Administrative Sciences

Department of Business Administration

35100, Bornova, İzmir, Turkey

2 Izmir Bakircay University

Faculty of Engineering and Architecture

Department of Industrial Engineering

35665, Menemen, İzmir, Turkey

3 Yaşar University, Faculty of Engineering

Department of Industrial Engineering

35100, Bornova, İzmir, Turkey

Tasar B, Türsel Eliiyi D, Kandiller L. Vehicle Routing with Compartments Under Product Incompatibility Constraints

Trafc Management

Original Scientic Paper

Submitted: 24 July 2017

Accepted: 21 Jan. 2019

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 25

VEHICLE ROUTING WITH COMPARTMENTS UNDER PRODUCT

INCOMPATIBILITY CONSTRAINTS

Tasar B, Türsel Eliiyi D, Kandiller L. Vehicle Routing with Compartments Under Product Incompatibility Constraints

26 Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36

planning horizon of the problem may involve a single

time period (STP), which is mostly the case, or multiple

time periods (MTP) [27]. Our taxonomy is inspired by

Kendall’s notation [28]. The notation, including one

segment for each dimension and subsegments sepa-

rated by commas, is illustrated in Figure 2.

Recent MCVRP studies are listed in Table 1 using

our taxonomy. Lahyani et al. [15] studied the collec-

tion of olive oil using heterogeneous multiple-com-

partment vehicles with split delivery for different

types of products. They solved the problem using a

branch-and-cut algorithm. Coelho and Laporte [37]

described the MCVRP for fuel distribution, which dis-

tinguishes between split and unsplit compartments

and tanks. They proposed two mixed integer linear

programming formulations and a branch-and-cut al-

gorithm for all problem variants. Since the MCVRP

is an NP-Hard problem, most studies use heuristic

another objective considered for representing custom-

er satisfaction. Minimax objectives, such as minimizing

the maximum travel time (MMTT) [18] or travel length

(MMTL) [19], can be useful in balancing the workload

of drivers, and explicit load balancing objectives (LBTT

and LBTL) [20, 21] can be used for the same purpose.

Depending on the problem structure, other objectives

can also be considered, such as non-monetary objec-

tives in humanitarian logistics problems [22].

In vehicle dimension, depending on the type of ve-

hicle, the eet can be identical (IF) or heterogeneous

(HF) [23]. In terms of capacity restrictions of the ve-

hicles in the eet, the most studied problem is xed

capacity (FC), where vehicle number and capacity are

assumed to be predetermined and cannot be altered.

In contrast, the number of vehicles or their capacities

can be increased in the exible eet size (FLC) catego-

ry [24]. In terms of compartments, the vehicles may

have a single compartment (SC), xed-size multiple

compartments (FMC) or exible-size multiple compart-

ments (FLMC) [25]. It should be noted that compart-

ment size exibility does not necessarily bring exibility

to vehicle capacity; total vehicle capacity may be xed

while allowing exible compartments. For other dimen-

sions, the distribution problem may involve a single

product (SP) or multiple compatible (MCP) or incom-

patible (MICP) products. The depots in the problem

may be single (SD) or multiple (MD) [26]. Finally, the

OPERATIONAL POLICY

VEHICLE

Time window

Carrying

Split delivery

Trip

Fleet type

Capacity

Compartment

OBJECTIVE

Minisum

Minimax

Other

Load balancing

PRODUCT

Product type

DEPOT

PERIOD

Number

of depots

Number

of periods

STW│HTW MSTC│MSTT│MSTL│MSV

│MSWT

MMTT│MMTL│MSWT

LBTT│LBTL

SP│MCP│MICP

SD│MD

STP│MTP

PD│BH│BH&LH

SLD

ST│MT

IF│HF

FC│FLC

SC│FMC│FLMC

Figure 1 – VRP taxonomy

Period

Depot

Product

Vehicle

Objective

Operational policy

X X X X X X

Figure 2 – Proposed notation

Tasar B, Türsel Eliiyi D, Kandiller L. Vehicle Routing with Compartments Under Product Incompatibility Constraints

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 27

stressed the need for heuristic approaches for larger

practical instances, although no such approach was

presented.

In the next section, we adapt the basic MCVRP

model [41] with a modication in the subtour elimina-

tion constraints for ST | MSTC | IF,FC,FMC | MICP |

SD | STP. We then propose multiple-trip (MT | MSTC

| IF,FC,FMC | MICP | SD|STP) and split-delivery (SLD

| MSTC | IF,FC,FMC | MICP | SD| STP) extensions

for constructing other acceptable routing alternatives

for a feed distributer. Since the problem increases in

practical applications, we propose heuristic approach-

es and lower bounds for all three variants in Section

3. Comprehensive computational results, including

both exact and heuristic performances, are presented

in Section 4. Section 5 includes our conclusions and

future research directions.

2. FORMULATION

In this section, mathematical formulations of the

problem are presented. The basic model, adapted

from [41], is presented in Section 2.1, while multi-

ple-trip and split-delivery extensions are presented in

Section 2.2.

2.1 The basic model

The basic problem modeled in this section does

not allow a split delivery or multiple trips, represented

as ST | MSTC | IF,FC,FMC | MICP | SD | STP. Identi-

cal vehicles are available and fully loaded at the single

depot at the start of the planning period. An 8-hour

shift is assumed for all vehicles. Each compartment of

a vehicle is dedicated to a single type of product due to

algorithms to solve the problem and its variants. For

instance, El Fallahi et al. [32] proposed a memetic

algorithm as a genetic algorithm hybridized with a lo-

cal search procedure for constructive heuristics and

a tabu search [45] for path relinking. Abdulkader et

al. [38] introduced a distance-constrained variant of

the problem by applying a hybrid ant colony optimiza-

tion algorithm and local search procedures. Koch et al.

[39] worked on the MCVRP with exible compartment

sizes and developed a genetic algorithm where each

gene represents a customer location and their supply

of a specic product type. The initial population was

generated by a completely randomized procedure and

savings method [43]. They used swap and insertion

algorithms for mutations. Rabbani et al. [26] consid-

ered a distance-constrained variant of the MCVRP with

heterogeneous vehicles, multiple depots and mixed

open and closed tours, proposing a mathematical for-

mulation based on genetic algorithms as metaheuris-

tics for the solution. Genetic algorithms were used for

initial solutions and the iterative swap procedure [44]

was used for the improvement phase. Alinaghian and

Shokouhi [40] introduced the MCVRP under multiple

depot and split delivery assumptions. The authors de-

veloped a mathematical model and a hybrid algorithm,

composed of an adaptive large neighborhood search

and a variable neighborhood search for large scale in-

stances.

Our study derives its roots from a real-life VRP of

a local feed distributor. The basic VRP and its multi-

trip extension was modeled in an earlier study [41]. As

the problem is clearly NP-hard, pilot experimentation

revealed that only small-size instances could be solved

to optimality within acceptable CPU times. The authors

Table 1 – Recent MCVRP literature

Authors Year Problem

Van der Bruggen et al. [29] 1995 ST | MMTC | HF, FC, FMC | MICP | SD | STP

Avella et al. [30] 2004 ST | MMTL | IF, FC, FMC | MICP | SD | STP

Suprayogi et al. [31] 2006 SLD, MT | MSV, MMTT, LBTT | IF, FC, FMC | MICP | SD | STP

El Fallahi et al. [32] 2008 SLD, ST | MMTL | IF, FLC, FMC | MICP | SD | STP

Mendoza et al. [33] 2010 ST | MMTL | IF, FLC, FMC | MICP | SD | STP

Derigs et al. [25] 2011 ST | MMTL | IF, FC, FMC | MICP | SD | STP

Surjandari et al. [34] 2011 SLD, ST | MMTL | HF, FC, FMC | MICP | MD | MTP

Benantar and Ou [35] 2012 TW, ST | MMTL | HF, FC, FMC | MICP | SD | STP

Asawarungsaengkul et al. [36] 2013 SLD, ST | MMTL | HF, FC, FMC | MICP | SD | STP

Lahyani et al. [15] 2015 SLD, ST | MSV, MMTL | HF, FC, FMC | MICP | SD | MTP

Coelho and Laporte [37] 2015 SLD, ST | MSTC | IF, FC, FMC | MICP | SD | MTP

Abdulkader et al. [38] 2015 ST | MSTC | IF, FC, FMC | MICP | SD | STP

Koch et al. [39] 2016 ST | MSTC | IF, FC, FLMC | MICP | SD | STP

Rabbani et al. [26] 2016 ST | MSTC | HF, FC, FLMC | MICP | MD | STP

Alinaghian and Shokouhi [40] 2017 SLD, ST | MSV, MSTL | IF, FC, FMC | MICP | MD | STP

28 Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36

,,;vu Ni ji11

iij

i

N

1

6f+=

==

=

Y

/ (7)

zy

i

i

N

k

k

K

11

=

==

//

(8)

yv

ik

k

K

i

N

11

=

==

// (9)

,,;, ,,;xuxk KijNij111

ki ij kj

6f f#

-- ===

Y

^h

(10)

,,;, ,,;xuxk KijNij111

kj ij ki

6f f#

-- ===

Y

^h

(11)

,,;, ,,;

tTTxu

kKij Ni j

2

11

ij

k

ij ij ki ij

6f f

$

---

===

Y

^h

(12)

,,;,

,,tTTxzk KijN211

j

k

jj ki j

0

00 6f f$---= =

^h

(13)

,,;, ,,tTTxvk KijN211

i

k

ii ki i

000

6f f$

---= =

^h

(14)

,,,;tTui

jNij

1

ij

kij ij

k

K

1

6f#

==

=

Y

/ (15)

,,

tTzj N1

j

kjj

k

K

00

1

6f

#=

=

/ (16)

,,

tTvi N1

i

kii

k

K

00

1

6f

#=

=

/ (17)

,,

tt tT

kK

1

j

ki

kij

kk

j

N

i

N

i

N

j

N

00

1111

6f

#++ =

====

//// (18)

,,,;uu ij Ni j11

ij ji 6f#+==

Y

(19)

,, ,,;

uuuu uu

ijmNijm

2

1

ij ji im jm mi mj

6f

#++ +++

===

YY (20)

,,,, ,,,;,,,;xyzvukKijNij01 11

ki kiiij

6f f!

===

Y

"

,

(21)

,,,; ,,;,,alLk Kp P01 11 1

lkp 6f

ff

!

== =

"

,

(22)

,, ,,;, ,,;tt tkKijNij01 1

ij

kj

ki

k

00

6f f$

===

Y (23)

The Objective Function 1 minimizes the sum of the

xed vehicle cost and the fuel cost. A vehicle can serve

customers only if it is in use (2). Capacity restrictions

are enforced through Constraint 3, while Constraint 4

ensures one type of product in each compartment/

silo. Each customer can be served by only one vehi-

cle; hence, no split deliveries are allowed, which is

described by Constraint 5. Constraints 6 and 7 satisfy

the ow balance of vehicles’ routes, while 8 and 9 en-

sure that the entrance and exits to/from the depot are

equal to the total number of vehicles used. Continuity

of routes is guaranteed through Constraints 10 and 11.

Constraints 12–14 calculate travel times for feasible

routes. Time and route variables are linked through

Constraints 15–17. Constraint 18 guarantees time-feasi-

ble routes in terms of the 8-hour vehicle shifts. Con-

straints 21–23 dene the values of decision variables

as binary or real non-negative.

We replace the all-subtour elimination constraint

that was used in the model presented in the paper

[41] with Constraints 19 and 20, which are subtour

product incompatibility. Loading and unloading times

are included in the traveling time. Each customer or-

ders one type of product and, since there is no split

delivery, the customer demand cannot exceed the ve-

hicle capacity. The parameters and decision variables

are dened below:

fk

- xed cost for vehicle k (USD), k=1,…,K

Clk - capacity of compartment l of vehicle k [tons],

l=1,…,L, k=1,…,K

Dip - demand of customer i for product type p [tons],

i=1,…,N, p=1,…,P

sij - distance from customer i to customer j [km],

i, j=1,…,N

\

- fuel cost [USD/km]

Tk - time capacity of vehicle k [min], k=1,…,K

Tij - time from customer i to customer j [min],

i, j =1,…,N

,

,

,,,,,

x

i

kKiN

k1

0

11

if vehicleservescustomer

otherwise

ki

ff

=

==

)

,

,,,

,

yk

K

k1

0

1

if vehicleisused

otherwise

kf

==

)

,

,

,,,

u

i

ij N

j1

0

1

if routefromcustomertocustomer is used

otherwise

ij

f

=

=

)

,

,

,,

z

i

iN

1

0

1

if routefromdepot customer is used

otherwise

to

i

f

=

=

)

,

,

,,

v

i

iN

1

0

1

if routefromcustomer to depotisused

otherwise

i

f

=

=

)

,

,

,,,,,, ,,

a

lk

p

lLkKpP

1

0

11 1

if compartmentofvehicle is used for

producttype

otherwise

lkp

ff f

=

== =

Z

[

\

]

]

]

]

]

]

]

]

tij

k

- time taken by vehicle k from customer i to

customer j, k=1,…,K, i,j=1,…,N

The mathematical model formulation is as follows:

Min fy sv sz su

kk ii ii ij ij

j

N

i

N

k

K

00

111

a+++

===

^^ ^hh h

=

G

*

4

/// (1)

Subject to:

,,

xNyk K1

ki k

i

N

1

6f

#=

=

/ (2)

,,

;,

.Dx Ca kK

pp11

ip ki lk lkp

i

N

1

6f f#

==

=

// (3)

,,

;,

,aylL

kK11

lkpk

p

P

1

6f f#

==

=

/ (4)

,,

xiN

11

ki

k

K

1

6f

==

=

/ (5)

,,;zu

jNij

11

jij

i

N

1

6f+=

==

=

Y

/ (6)

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 29

,,;,,;vuxk Ki Ni j11

ki kijki

i

N

1

6f f+=

===

=

Y

/ (7’)

,,;,,;

uv uz

kKjNij11

kjikikij kj

i

N

i

N

11

6f f

+= +

===

==

Y

// (25)

,,

;,

,Dq Ca kK

pp11

ip ki lk lkp

l

L

i

N

11

6f f#

==

==

// (3’)

,,;,.,qx kKiN11

ki ki 6f f#==

(26)

,,

qiN

11

ki

k

K

1

6f

==

=

/ (27)

,,;,.,qkKi N011

ki

6f f$==

(28)

,,,;tTuijNij1

ij

kij kij

6f#

==

Y (15’)

,,tTzj N1

j

k

jkj

0

0

6f

#= (16’)

,,tTvi N1

i

k

iki

006f

#

=

(17’)

3. SOLUTION APPROACH

The MCVRP model and its two extensions are

solved to optimality for small-size test problems, using

IBM ILOG CPLEX 12.6. While obtaining optimal solu-

tions, subtours including four or more customers are

checked after obtaining the initial optimal solution,

and further necessary subtour elimination constraints

are added as cuts to the initial model. For the basic

model, optimal solutions could not be obtained with-

in reasonable computation times for instances of 20

or more customers. For multiple-trip and split-delivery

extensions, even a group of 15 customers was large

enough for obtaining exact solutions. Therefore, the

heuristic approaches for the basic, multiple-trip and

split-delivery variants are developed and explained in

the following subsections.

3.1 Construction and improvement heuristics

In the construction heuristics for the basic prob-

lem, the customers are selected in a random manner

and assigned to the same vehicle, as long as there

is enough capacity. If no more customers can be as-

signed to the same vehicle, a new vehicle is opened

to be used. The construction procedure continues un-

til all customers’ demands are assigned to vehicles.

The vehicles visit the customers in the order they are

assigned, hence, an initial route for each vehicle is

formed automatically as the assignments are being

made. Our improvement phase starts with a modi-

ed variable neighborhood search (mVNS) algorithm,

which consists of swap and insertion moves to im-

prove the initial route of each vehicle shown in Figure 3.

A VNS is based on the systematic change of the

neighborhood during the search [42]. A greedy re-

move/insert type heuristic with an mVNS substage is

elimination constraints that include only 2-node and

3-node subtours. Our initial numerical experiments

show that these constraints eliminate the majority of

the resulting subtours at a small computational cost.

Therefore, we included only these constraints in the

model, as opposed to the highly time-consuming one

used in [41]. Further subtour constraints for four or

more customers are added to the model whenever

subtours appear in the optimal solution, which will be

explained in Section 4.

2.2 Multiple trips and split delivery

The multiple-trip extension of the problem is denot-

ed as MT | MSTC | IF,FC,FMC | MICP | SD|STP based

on our taxonomy. This extension allows a maximum of R

trips per vehicle during the working day. A trip index is de-

ned as r= 1,…,R, and the number of vehicles in the mod-

el is replaced by k=1,…,K, K+1,…,2K,…,2K+1,…,RK.

With this representation, indices K+1,…,2K denote

the 2nd trip of the K vehicles, 2K+1,…,3K denote the

3rd trip, etc. Constraint 24 then ensures that a vehicle

can make its next tour only if it performs its previous

one. To satisfy the time constraint for a vehicle per-

forming multiple trips, Constraint 18 is replaced by 18’

in the model.

,,;,,yy kKrR11

rK krKk1

6f f#==

+-+

^

h

(24)

,,

tt

tT

kK1

j

rKk

i

rKk

i

N

r

R

j

N

r

R

ij

rKkk

j

N

i

N

r

R

0

1

0

1

1111

1

111

6f

#

++

+

=

-+ -+

====

-+

===

^

^

^

h

h

h

////

/// (18’)

When split delivery is allowed, the problem be-

comes SLD | MSTC | IF,FC,FMC | MICP | SD| STP. A

customer’s demand can be met through a delivery by

several vehicles, hence, Constraint 5 needs to be omit-

ted from the model. A vehicle index is added to deci-

sion variables u, z and v, and Constraints 6 and 7 are

modied as stated below. Constraint 25 is added to en-

sure ow balance. Another change handles fulllment

of a customer’s demand through a new continuous

non-negative decision variable qki between 0 and 1,

representing the percentage of customer i’s demand

met by vehicle k. Constraint 3 is then replaced by 3’ to

accommodate this change. Constraint 26 ensures that

a vehicle can satisfy a demand percentage of a cus-

tomer only if it visits that customer. Demand of each

customer is met via the additional Constraint 27, and

the new decision variable is dened by Constraint 28.

Constraints 15–17 are also updated to handle relations

between vehicle visits and time used. The modied

constraints are listed below.

,,;,,;zuxk Ki Ni j11

kj kijkj

i

N

1

6f f+= ===

=

Y

/ (6’)

30 Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36

The solution approach has similar construction and

improvement steps, but the algorithm also considers

possible ways of demand splitting and assigning each

portion to a different vehicle. This assumption consid-

erably increases the complexity of the algorithm.

3.2 Lower bounds

We were able to show the efciency of our heuris-

tic results for small-size instances by comparing them

with the optimal values. Since the problem is NP-hard,

in order to test our heuristic model for large-size prob-

lem instances, we need to compute some lower bound

values of the objective function. Three lower bounds

are developed for the problem. For the base model, a

lower bound on total xed cost is computed by multi-

plying the total number of vehicles with the xed cost

of a vehicle. To nd a bound for the number of vehi-

cles, the total amount of demands for each type of

product

,:;:Dicustomerspproducttype

ip

i

bl

/

is divided by

the compartment capacity (C) and the upper bound

integer value is taken as the number of compartments

needed for a certain product type. The total number of

compartments needed for all products C

D

ip

i

p

fp

fp

/

/ is

then divided by the number of compartments that can

t on vehicles (four in practice), giving a bound on the

total number of required vehicles:

#

#

/

Vehicles

Compartments

DC

ip

ip

=

f

bl

p

//

For the multiple-trip extension, a maximum of two

trips is assumed per vehicle per day as a reasonable

and practical assumption. Hence, a lower bound on

the number of needed vehicles is found by dividing

the number of vehicles found above by two and taking

the integer upper. For the split-delivery model, lower

bound is the same as the base model. There are three

different lower bounding methods for the routing cost,

as explained below.

LB1: The rst bound tries to nd the minimum dis-

tribution distance by seeking customers’ neighbors

and by forming clusters equal to the lower bound on

the number of vehicles (computed as above). The pro-

cedure starts with node 1 and selects two of its near-

est customers. This is applied to each node, so that

each customer is connected to its two closest neigh-

bors. Repetitions are removed; if a customer appears

in more than two clusters, only the minimum two dis-

tances for that customer are kept while the rest are

erased. The next step is to enforce depot connection

arcs. The number of entrances/exits from/to depot

should be equal to the bound of the number of vehi-

cles. For each vehicle, the nearest customers from the

depot are selected as tour start nodes. Selected cus-

tomers’ previous connections are removed and they

used subsequently. For this purpose, the vehicles are

sorted in an increasing order according to their total

capacities. The rst customer’s demand is removed

from the rst vehicle. Then, starting with the last vehi-

cle and moving towards the rst, the algorithm checks

whether the remaining capacity is sufcient (for the

same type of product) for inserting the removed de-

mand. If there is such a vehicle, the initial and the

changed route costs are calculated for both vehicles.

The switch is performed if there is a saving in cost and

the remaining capacities of the vehicles are updated.

The routes of both vehicles are then improved through

the mVNS algorithm as the visited nodes are changed.

The displacement starts from the rst customer of the

rst vehicle, and the procedure tries to nd room in

the last vehicle. Since it is not easy to empty a vehi-

cle loaded to its maximum capacity, the aim of the al-

gorithm is to maximize the total amount of demands

served by that vehicle, with the hope of reducing the

number of vehicles used. The algorithm continues

with the next customer of the rst vehicle in the same

manner. After all the customers that could be removed

are processed, total savings are computed. It should

be noted that the xed cost of an emptied vehicle is

also included in the savings if all its customers can be

moved to other vehicles. The improvement procedure

described above is outlined in Figure 4.

The solution approach for the multiple-trip problem

is the same as for the basic model, with the additional

consideration of multiple trips per day. Hence, to com-

pletely remove a vehicle from the schedule and elimi-

nate its xed cost, all trips of the same vehicle should

be empty. In the case of a split delivery, one or two

vehicles can meet a customer’s demand. Due to prac-

tical considerations, the percentage of a customer’s

demand served by any vehicle should be greater than

a threshold, which is called a minimum splitting ratio.

Figure 3 – Procedure for mVNS

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 31

Number of customers (N) directly affects the

problem size and its levels are determined as 10,

15, 20, 50 and 100. Each customer orders a single

product among 4 types of products. The demand of

each customer is generated from three predete-

rmined demand levels according to vehicle capacity,

which is 16 tons. U[1,8] indicates that demands are

generated between 1 ton and 8 tons from a discrete

uniform distribution. U[a,b] denotes that the random

variable U has the discrete uniform distribution with a

nite number of equally spaced and equally probable

are connected to depot. The sum of all arc distances

is multiplied by the fuel cost, providing a lower bound

for total route cost.

LB2: A second lower bound is obtained by solving

an assignment problem without depot and subtour

elimination constraints to optimality using CPLEX. This

bound provides an optimal customer-vehicle assign-

ment while ignoring feasible tours. To add the depot

connections on the assignment solution, we follow the

same procedure as in LB1.

LB3: The last lower bound is obtained by solving

the original model with a 5-minute computational time

limit.

4. NUMERICAL RESULTS

A computational study is carried out to evaluate

the performance of our mathematical formulation and

heuristic algorithms. In Table 2, the parameters and

their levels used in numerical tests are listed.

Sorting vehicles in increasing order

according to their total capacities

Selection of i-th customer

in k-th vehicle

Selection of (K-s)-th

vehicle

Dip≤R(K-s)p

Remove customer i to the vehicle (K-s)-th

and calculate Ck and CK-s

CK=CK+CK-s-Ck

Remove customers to initial

position for negative changes

of CK-s-Ck

All customers are

checked in vehicle k

Ck=0

f≥CK

Close the vehicle k

k=k+1

k=k+1

s=s+1

k≠(K-s)

i=i+1

K - number of vehicles

Dip - demand of customer i for product type p

R(K-s)p - remaining capacity for product type p in vehicle (K-s)

Ck - changed route costs of vehicle k

CK-s - changed route costs of vehicle (K-s)

CK - total change of route cost

f - fixed cost of a vehicle

Figure 4 – Flowchart of the improvement phase

Table 2 – Factors and Levels

Factors Levels

# of customers (N)10, 15, 20, 50, 100

Demand U[1,8], U[1,16], U[4,12]

Fixed cost USD 200, USD 400

32 Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36

solved to optimality using the base model, whereas

10-customer instances could be solved for other ex-

tensions.

Table 3 presents computational results for the base

model, where each row represents ten instances of

the same setting. Due to dominant xed costs of ve-

hicles as compared to fuel cost, optimal solutions try

to keep the number of vehicles used at the minimum

feasible number. The lowest average objective values

belong to the U[1,8] setting, since the highest number

of orders can be loaded onto the same vehicle. The

numbers of vehicles for U[1,16] and U[4,12] settings

are similar. U[1,16] has a higher variance; some vehi-

cles can carry orders of many customers while some

visit only one customer, increasing the routing cost.

Hence, U[1,16] instances yield higher objective values.

As expected, low-demand-low-variance setting (U[1,8])

has signicantly higher computation times. Since ve-

hicles can carry many orders, an increased number of

assignment combinations increases CPU times in this

setting. Following the reverse reasoning, U[4,12] yields

the fastest solutions. The combinatorial nature of the

problem prevails as the number of customers increas-

es, and the instances cannot be solved in reasonable

times for more than 20 customers. The suggested

heuristic algorithms, coded in C++ integrated with

outcomes with integer parameters a and b, where a<b.

One of the options which may occur is that the val-

ue of the demand generated from one of the demand

levels is equal to the half capacity of a vehicle and

the distribution range is low. This option is called low-

demand-low-variance setting. Considering demand

value and distribution range, we can have two more

options, high-demand-high-variance setting (U[1,16])

and high-demand-low-variance setting (U[4,12]). The

fuel cost is assumed to be a=1.4 USD/km in all set-

tings, while the xed cost of using a vehicle is set at

one of two levels, either USD 200 or USD 400. For the

number of vehicles, the total demand is divided by the

vehicle capacity to yield the rst level of the parame-

ter V. The second and third levels are taken as V+1

and V+2. For the multiple-trip extension, a maximum

of two trips is assumed per vehicle per day. In the

case of the split delivery, a customer’s demand can

be split among a maximum of two vehicles, and the

splitting ratio is not less than 25%. The coordinates

of customers are generated over a grid (-200000m,

+200000m) divided into 16 equal zones. To reect a

realistic scatter considering the regional demand, 12

low-density (HD) and 4 high-density (LD) zones are

dened, where a single high-density zone can con-

tain up to 4 times more customers than a low-density

one. The number of customers in a low-density zone

is obtained through

()

N

28 44 12

$

=+

bl

and customer

locations are assigned to zones according to their ca-

pacities. The depot is at the origin (0, 0), as shown in

Figure 5. Euclidian distances are used.

Ten instances are generated from each setting, to-

taling to 300 instances. Hence, a total of 900 runs was

made for three models. All runs are performed on a

PC with Intel® Core™ i5-3360M CPU @ 2.80GHz and

8.00 GB RAM. Up to 15-customer problems could be

Table 3 – Base model results for N=10, 15

Avg. obj. Avg. CPU [s] Max CPU [s] Min CPU [s] Avg KMax KMin K

N10_D1-16_FC400 4203.27 0.79 1.34 0.56 7 9 5

N10_D1-16_FC200 2903.27 0.65 1.31 0.30 7 9 5

N15_D1-16_FC400 6064.13 254.15 691.38 26.92 10 11 7

N15_D1-16_FC200 4224.13 493.73 2456.03 14.28 10 11 7

N10_D1-8_FC400 2609.85 1.16 3.52 0.28 443

N10_D1-8_FC200 1889.85 1.07 2.24 0.28 443

N15_D1-8_FC400 3551.98 330.34 686.00 9.28 5 6 4

N15_D1-8_FC200 2591.98 524.34 1187.32 12.11 56 4

N10_D4-12_FC400 4128.67 1.44 2.73 0.79 7 8 5

N10_D4-12_FC200 2868.67 1.62 2.05 0.95 7 8 5

N15_D4-12_FC400 5826.20 316.57 982.94 10.13 911 8

N15_D4-12_FC200 4099.75 356.22 934.25 8.63 9 11 8

(-200000, 200000) (200000, 200000)

(-200000, -200000) (200000, -200000)

LD

LD

LD

LD

LD

LD HD

HD

LD

LD

LD

LD

LD

LD

HD

HD

(0,0)

Figure 5 – Zones and customers on the grid

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 33

CPLEX within hours was not possible, let alone within

5 minutes. For this reason, the gaps in Best LB Gap %

column are computed for larger instances using only

the best of LB1 and LB2 as the lower bound. It should

be noted that much higher gaps are observed immedi-

ately after this threshold. The increased gaps between

the heuristic and Best LB for the larger instances can

therefore be attributed not to the decreased perfor-

mance of our heuristics for these instances but to the

evident weakness of LB1 and LB2 as compared to

LB3.

To better reveal this effect, a third column, LB Gap

%, is added for each variant, which presents percent-

age gaps between heuristic solutions and the best

of LB1 and LB2. In other words, this column lists the

Visual Studio Express, obtained solutions much faster.

Solution quality performance of heuristics is shown in

Table 4 for all 300 instances for each model.

Best UB gaps are computed as percentage gaps

between optimal and heuristic solutions when optimal

solutions are obtained. For the rest, lower bounds are

used for comparison. Best LB gap % columns present

the gaps between optimal solution and the best low-

er bound. In our computations, we observed that LB3

consistently dominated the other two lower bounds for

all three models when it could be computed. There-

fore, Best LB Gap % column lists the percentage gaps

between the heuristic solutions and LB3 for 10-node

and 15-node instances for the base model. However,

for larger instances, obtaining a feasible solution from

Table 4 – Performances of heuristics

Basic model Multiple-trip model Split-delivery model

BEST LB

GAP [%] BEST UB

GAP [%] LB GAP

[%] BEST LB

GAP [%] BEST UB

GAP [%] LB GAP

[%] BEST LB

GAP [%] BEST UB

GAP [%] LB GAP

[%]

N10_D1-16_FC400 0.00 0.00 15.98 0.00 0.00 23.99 2.19 1.21 13.97

N10_D1-16_FC200 0.00 0.00 23.09 0.00 0.00 31.17 3.06 1.01 21.46

N10_D1-8_FC400 0.00 0.00 20.65 0.00 0.32 37.96 0.00 0.22 22.41

N10_D1-8_FC200 0.00 0.00 31.31 0.00 0.40 40.86 0.00 0.31 34.01

N10_D4-12_FC400 0.00 0.00 21.14 0.00 0.62 27.88 6.59 1.50 13.82

N10_D4-12_FC200 0.00 0.00 28.99 0.00 0.80 35.27 4.77 1.45 21.23

N15_D1-16_FC400 0.00 0.00 22.58 8.26 31.66 17.14 19.94

N15_D1-16_FC200 0.27 0.00 31.00 8.98 40.93 23.34 29.25

N15_D1-8_FC400 0.08 0.14 20.39 13.47 38.41 14.02 23.44

N15_D1-8_FC200 0.18 0.52 30.66 11.02 28.45 19.24 36.73

N15_D4-12_FC400 0.07 0.00 25.84 14.41 33.39 14.03 17.87

N15_D4-12_FC200 0.00 0.00 35.10 20.69 44.48 23.42 29.31

N20_D1-16_FC400 10.40 21.84 9.80 13.86 17.66 18.66

N20_D1-16_FC200 12.36 29.39 15.66 23.68 25.95 27.60

N20_D1-8_FC400 16.95 24.95 25.07 27.15 19.86 24.16

N20_D1-8_FC200 24.91 36.86 34.97 39.59 30.37 36.45

N20_D4-12_FC400 14.42 27.17 19.46 19.84 19.38 19.63

N20_D4-12_FC200 17.86 35.66 27.01 31.87 29.31 29.81

N50_D1-16_FC400 25.73 37.67 24.33

N50_D1-16_FC200 36.41 42.58 34.41

N50_D1-8_FC400 38.14 71.98 38.19

N50_D1-8_FC200 56.00 70.47 50.80

N50_D4-12_FC400 30.71 36.80 25.62

N50_D4-12_FC200 43.02 48.40 35.95

N100_D1-16_FC400 29.13 40.74 28.28

N100_D1-16_FC200 42.07 52.16 39.99

N100_D1-8_FC400 45.59 76.72 44.08

N100_D1-8_FC200 66.84 86.63 62.26

N100_D4-12_FC400 29.56 43.56 30.48

N100_D4-12_FC200 47.94 56.89 44.46

34 Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36

To illustrate the cost advantage of heuristic solu-

tions, Table 5 presents the percentage of improvement

of multiple-trip and split-delivery objective values over

the basic problem.

It seems that the multiple-trip solutions can de-

crease the number of used vehicles considerably with-

out violating the 8-hour shift limit; the average savings

over the basic problem reach up to 38% when the xed

cost is high. The split-delivery approach, on the other

hand, tries to decrease routing cost by splitting cus-

tomer orders among vehicles, thereby resulting in rela-

tively small, but still valuable savings.

5. CONCLUSION

In this study, we consider a vehicle routing problem

with compartments under product incompatibility. A

taxonomic framework was proposed for many variants

of the VRP as a result of the extensive literature sur-

vey. In addition to the basic problem, the multiple-trip

and split-delivery extensions are formulated. Since

exact solutions could only be obtained for small in-

stances of the three variants, heuristic approaches are

developed for solving practical larger instances of the

problem. Simplistic lower bounds are also developed

for comparison purposes. While the solutions with

heuristics for small-size instances could be obtained

in milliseconds, CPU time increased as the number of

customers increased.

Our initial numerical tests show that the extensions

bring considerable savings when compared with the

basic model. The operational decision of performing

multiple trips per vehicle substantially decreases the

number of vehicle usage, and in turn, the total xed

cost. Although the split-delivery approach decreased

the overall cost of the basic model, it was not found

to be as effective as the multiple-trip approach. The

results indicate that the heuristic approaches devel-

oped in this study may be adapted and used by various

companies that are dealing with these types of prob-

lems and want good solutions computed effectively in

reasonable computing times.

percentage gaps when LB3 is not used. It can be clear-

ly seen that, in the absence of LB3, the gaps would

get much worse, including for the smallest problem

instances.

The number of customers directly affects CPU time

for the heuristics. Figure 6 shows the relation between

the problem size and CPU times for the basic prob-

lem. U[1-8] setting for demand distribution has the

highest CPU times, increasing parabolically after 50

customers. The reason is that the improvement phase

of the heuristic algorithm tries to combine customer

demands while minimizing the number of vehicles

used. Since many customer demands can t into a ve-

hicle with low demand settings, nding a better route

combination among the many customer locations in-

creases. As expected, a lower xed cost yields higher

computation times due to the trade-off between rout-

ing and vehicle costs. Multiple-trip and split-delivery

variants have similar patterns in solution times. When

it comes to multiple trips, the maximum solution time

is observed to be 2292 s, while the average time (over

all instances) is 1981. As might be expected, split

delivery has the highest CPU times, going up to 3345 s

for the largest instances, while averaging 2539 s over

all instances.

Fixed cost = USD 200 Fixed cost = USD 400

CPU time [s]

CPU time [s]

2,000

1,500

1,000

500

0

2,000

1,500

1,000

500

0

0 20 40 60 80 100 0 20 40 60 80 100

# of customers # of customers

D1-16 D1-8 D4-12

Figure 6 – Average time performance of heuristics for the base problem

Table 5 – Savings over the basic model

Fixed cost

[USD]

# of

customers

Multiple-trip

model Split-delivery

model

400

10 36.56% 2.61%

15 36.76% 3.74%

20 36.95% 3.96%

50 37.84% 4.54%

100 37.23% 4.79%

200

10 23.88% 2.41%

15 23.45% 2.61%

20 24.93% 2.80%

50 24.98% 3.43%

100 25.90% 3.83%

Promet – Trafc & Transportation, Vol. 31, 2019, No. 1, 25-36 35

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ment and the vehicle routing problem with time-de-

pendent travel speeds. Computers & Industrial Engi-

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[17] Çetin S, Gencer C. Heterojen Araç Filolu Zaman Pencer-

eli Eş Zamanlı Dağıtım-Toplamalı Araç Rotalama Prob-

lemleri: Matematiksel Model. International Journal of

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lem with split delivery and heterogeneous demand.

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System to the Vehicle Routing Problem. In: Voss S, Mar-

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[20] Lee T, Ueng J. A Study of Vehicle Routing Problems

Developing more sophisticated algorithms that can

yield better results, including metaheuristics, is one of

the future research directions. In such a case, it is im-

portant to keep in mind the quality-time trade-off for

practical applications, as well as the applicability and

ease-of-use of the solution approach by industry pro-

fessionals.

BAHAR TAŞAR, M.S.1

E-mail: bahar.turantasar@gmail.com

DENİZ TÜRSEL ELİİYİ, Prof. Dr.2

(Yazışmadan sorumlu yazar)

E-mail: deniz.eliiyi@bakircay.edu.tr

LEVENT KANDİLLER, Prof. Dr.3

E-mail: levent.kandiller@yasar.edu.tr

1 Ege Üniversitesi

İktisadi ve İdari Bilimler Fakültesi

İşletme Bölümü

35100, Bornova, İzmir, Türkiye

2 İzmir Bakırçay Üniversitesi

Mühendislik ve Mimarlık Fakültesi

Endüstri Mühendisliği Bölümü

35665, Menemen, İzmir, Türkiye

3 Yaşar Üniversitesi Mühendislik Fakültesi

Endüstri Mühendisliği Bölümü

35100, Bornova, İzmir, Türkiye

ÜRÜN KARIŞMAMA KISITLARI ALTINDA ÇOK

KOMPARTIMANLI ARAÇ ROTALAMA

ÖZET

Bu çalışma, bir aracın kompartmanlarında birbiriyle

karışmaması gereken ürünleri içeren bir dağıtım problemine

odaklanmıştır. Farklı müşteri taleplerini minimum lojistik

maliyetiyle karşılamak için ürünler lodaki araçların farklı

kompartmanlarında depolanır. Bu durum problemin Çok

Kompartımanlı Araç Rotalama Problemi (ÇKARP) olarak

modellenmesini gerektirir. Araç Rotalama Problemi (ARP)

ve çeşitli varyantları hakkında geniş bir literatür olmasına

rağmen, ÇKARP üzerine çok daha az araştırma bulunmak-

tadır. İlk olarak bu çalışmada ARP literatürü için yeni bir

taksonomik çerçeve önerilmiştir. Buna ek olarak, temel

ÇKARP ile birlikte problemin çoklu sefer ve bölünebilir si-

pariş uzantıları için kesin çözümler elde etmek amacıyla

yeni matematiksel modeller önerilmiştir. Son olarak, üç

problem varyantının daha büyük örnekleri için sezgisel al-

goritmalar geliştirilmiştir. Geliştirilen sezgisel yöntemlerin

performansını daha büyük problem örnekleri için optimum

çözümlere karşı test etmek amacıyla bir alt sınırlama şeması

da önerilmiştir. Hesaplamalı deneylerin sonuçları modellerin

geçerliliğini ve algoritmaların gelecek vaat eden performan-

slarını göstererek raporlanmıştır.

ANAHTAR SÖZCÜKLER

Çok kompartımanlı araç rotalama problemi; karışamayan

ürünler; bölünmüş dağıtım; çoklu sefer; matematiksel mo-

delleme; sezgisel algoritmalar;

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