European Journal of Operational Research - Purdue …mmoshre/LLRP.pdfEuropean Journal of Operational...

European Journal of Operational Research 255 (2016) 604–619

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier.com/locate/ejor

Innovative Applications of O.R.

The Latency Location-Routing Problem

Mohammad Moshref-Javadi ∗, Seokcheon Lee

1

School of Industrial Engineering, Purdue University, 315 N. Grant St., West Lafayette, IN 47907, USA

a r t i c l e i n f o

Article history:

Received 10 September 2015

Accepted 26 May 2016

Available online 3 June 2016

Keywords:

Location routing

Minimum latency problem

Memetic Algorithm

Granular search

Humanitarian logistics

a b s t r a c t

This paper introduces the Latency Location-Routing Problem (LLRP) whose goal is to minimize waiting

time of recipients by optimally determining both the locations of depots and the routes of vehicles. The

LLRP is customer oriented by pursuing minimization of the latency instead of minimization of the length

of routes. One of the main applications of this problem is the distribution of supplies to affected areas

in post-disaster relief activities. It is also relevant in customer-oriented supply chain where latency at

demand locations plays a significant role in the satisfaction of the customers. The problem is formulated

mathematically and two heuristics, the Memetic Algorithm (MA) and the Recursive Granular Algorithm

(RGA), are proposed. An extensive experimental study shows that both algorithms are able to produce

promising results in reasonable time.

© 2016 Elsevier B.V. All rights reserved.

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1. Introduction

Transportation and delivery of products and services are inher-

ent in most manufacturing and service systems. The delivery is

generally performed by a fleet of vehicles from multiple depots to

customers. To design an efficient distribution system, whether for

emergency situations or commercial systems, several types of deci-

sions need to be carefully examined: locating the depots, allocating

vehicles to depots, and routing vehicles. While these decisions in

emergency and disaster situations seek to achieve minimum loss

and damage, profit maximization is the primary goal in commer-

cial delivery systems.

One of the main post-disaster activities is the distribution of

commodities from distribution centers to affected areas. Commodi-

ties may encompass a range of different supplies, such as food,

water, clothing, and medical supplies. Affected areas with their

associated demand size are examined and estimated after disas-

ter. To effectively distribute supplies to the victims despite lim-

ited resources, it is important to answer the three aforementioned

questions. The locations of temporary distribution centers (DCs)

must be appropriately selected among a set of potential locations

that are usually predetermined before a disaster happens. As time

passes after disaster, deaths and losses increase due to the lack of

supplies. Therefore, the focus of the disaster relief operations is on

minimizing waiting time of the recipients which eventually leads

∗ Corresponding author. Tel.: + 1 213 448 3241.

E-mail addresses: [email protected] , [email protected]

(M. Moshref-Javadi), [email protected] (S. Lee). 1 Tel.: + 1 765 494 5419.

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http://dx.doi.org/10.1016/j.ejor.2016.05.048

0377-2217/© 2016 Elsevier B.V. All rights reserved.

o enhanced safety and welfare of the victims. The limited capac-

ty of the vehicles and depots should also be taken into account to

andle the relief logistics operations realistically.

In commercial environments, product/service is delivered with

he goal of maximizing profit, which can be achieved by reduc-

ng customers’ waiting time and thereby improving customer sat-

sfaction. This satisfaction will eventually lead to more profit for

he company by receiving more orders from the existing and new

ustomers due to good reputation. The same decisions mentioned

bove, therefore, contribute to the commercial systems as well.

hese customer-oriented systems with focus on minimum waiting

ime are different from the server-oriented systems in which min-

mum travel distance is the primary objective.

According to the literature of disaster relief location-routing, no

rticle has assumed minimization of the total latency although it

lays an important role in reducing deaths and losses. The com-

on objective in the literature of disaster relief location-routing

roblems that involve the three types of decisions is maximizing

he amount of satisfied demands ( Ceselli, Righini, & Tresoldi, 2014;

ath & Gutjahr, 2014 ). However, this objective does not satisfy the

eed for disaster delivery systems as discussed above. For example,

t does not guarantee maximum survival of victims because it may

ead to the delivery of supplies to the victims who are no longer

live at the time of delivery. Hence, minimizing latency should be

aken into account as a primary goal of the LRP in disaster relief

ogistic.

The purpose of this paper is to introduce a problem called the

atency Location-Routing Problem (LLRP) whose objective is to min-

mize latency by optimally determining location, allocation, and

outing decisions at the same time. This problem can be viewed


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M. Moshref-Javadi, S. Lee / European Journal of Operational Research 255 (2016) 604–619 605

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s a hybrid of two optimization problems. The first problem is

he Facility Location Problem (FLP) ( Bramel & Simchi-Levi, 1997;

lose & Drexl, 2005 ) which determines the locations of depots, and

he second problem is the Cumulative Capacitated Vehicle Rout-

ng Problem (CCVRP) ( Lysgaard & Wøhlk, 2014; Ngueveu, Prins, &

olfler Calvo, 2010; Ribeiro & Laporte, 2012 ), which allocates ve-

icles to depots and obtains the sequences of visits to customers.

he FLP and CCVRP are closely interrelated and best results can

e obtained when both are solved simultaneously instead of se-

uential problem solving. Because of the complexity of the LLRP,

wo efficient algorithms are proposed to deal with the problem

ractically. The first approach is a Memetic Algorithm with elitism

hich executes several local search operators, while the second

pproach is a recursive granular search algorithm with different

eighborhood search strategies. The performances of the two pro-

osed algorithms are carefully examined in several ways. We com-

are them with a sequential approach that solves the decisions in

natural, sequential manner, as well as comparisons with two dif-

erent lower bounds. We also evaluate it by analyzing performance

nhancement over the conventional Location-Routing Problem set-

ings. The evaluation results indicate the importance of considering

atency in designing distribution systems and the effectiveness of

he two proposed algorithms in minimizing the latency.

The remainder of the paper is organized as follows. Section

presents the literature review of recent location-routing prob-

ems and solution approaches. Section 3 describes the problem and

ts assumptions, and presents a mathematical model of the prob-

em based on a network flow model. Section 4 describes the pro-

osed algorithms and the results of the implementation are pre-

ented in Section 5 . Finally, Section 6 concludes the paper.

. Related literature

Many authors have studied different variants of location rout-

ng problems. In the context of disaster relief operations, Wang,

u, and Ma (2014) presented a multi-objective formulation of the

pen location routing problem in post-disaster earthquake relief.

he model seeks to minimize the total depot locating and vehi-

les’ travel costs, minimize the maximum travel time of the ve-

icles, and maximize the minimum route reliability. Two heuris-

ic algorithms based on Non-dominating Sorting Genetic Algorithm

I (NSGA-II) and Non-dominated Sorting Differential Algorithm

NSDE) were developed and implemented in a case study on Great

ichuan Earthquake in China. Ceselli et al. (2014) designed an exact

lgorithm based on column generation with three different types

f columns and branch-price-cut algorithms. The algorithms were

sed for drug distribution problems to maximize the total satisfied

emands. Rath and Gutjahr (2014) formulated the LRP with three

bjective functions: minimizing the costs of opening facilities, min-

mizing the costs of transportation and warehousing, and maxi-

izing satisfied demands. The authors used a decomposition-based

pproach in which the location and routing problems were solved

teratively with a single objective in each step. A Variable Neigh-

orhood heuristic was also proposed and compared with NSGA-II

n a real problem in Manabi with 40 demand locations. Özdamar

nd Demir (2012) proposed a hierarchical clustering and rout-

ng (HOGCR) algorithm that obtains the delivery of supplies from

arehouses to recipients and pickup of victims to hospitals con-

idering the capacity of warehouses and hospitals. The algorithm

rst clusters demand nodes to form aggregate clusters and finally

nds the optimal routings of the vehicles in each cluster. It applies

ivide and conquer to cluster nodes recursively until the optimal

outing is found with the minimum total travel time. For a review

n emergency logistics the reader is referred to Caunhye, Nie, and

okharel (2012) and Luis, Dolinskaya, and Smilowitz (2012) .

The LRP has also been studied in commercial distribution

ystems. Boujelben, Gicquel, and Minoux (2014) presented a

lustering-based approach to deal with a three-level distribution

etwork design problem in automotive industry to minimize

rimary, secondary, and transit costs. The authors also proposed

ifferent heuristics by relaxing the MIP formulation of the problem

o solve large-sized problems with 500 customer nodes and 50

otential DCs. Averbakh and Berman (1994) presented a location-

outing problem to minimize the total latency of customers on

aths. The authors developed polynomial algorithms to solve the

roblem with one and multiple servers. Lin and Kwok (2006) pre-

ented a two-phased Tabu Search for costs minimizing and

ehicles’ workload balancing assuming homogeneous capacitated

eet with both time and load constraints. Chakrabarty and Swamy

2011) developed approximation algorithms for uncapacitated

acility location and minimum latency with objective function sum

f facility costs and customers’ latency. Using linear approximation

echniques, some improved constant factor approximations were

roposed to solve special cases of the problem. Contardo, Cordeau,

nd Gendron (2013) proposed a three-stage exact algorithm to

olve the capacitated location-routing problem. The first stage

olves the two-indexed flow formulation by branching on location

ariables. In the second and third stages, the gap is improved by

olving a column-and-cut generation of the linear relaxation of

he set-partitioning formulation and branch and bound on the

numerated columns. Rahmani, Ramdane Cherif-Khettaf, and Oula-

ara (2016) formulated the two-echelon location-routing problem

ssuming pickup and delivery, multi-product, and intermediate

acilities. Three heuristics based on nearest neighborhood, inser-

ion, and clustering were applied to the problems with up to 200

ustomers and 10 DCs to minimize the total travel costs, facility

pening costs, and vehicle fixed costs. Huang (2015) presented a

hree-stage solution approach to deal with the multi-compartment

apacitated location routing problem with pickup-delivery and

tochastic demands. The algorithm divides the problem to de-

ermine facility locations, assignment of customers to facilities,

nd routings by minimizing facility opening, vehicle, and travel

osts, and violation of the vehicle and depot capacity constraints.

adizadeh and Nasab (2014) formulated the capacitated-routing

roblem with fuzzy demands in a time horizon and developed

hybrid heuristic algorithm with four phases. The method also

stimates route failures by stochastic simulation of each route.

Many metaheuristics have been recently applied to the

ocation-routing problems. These approaches are popular since

hey are able to solve large scale problems with reasonable compu-

ation time. Karaoglan and Altiparmak (2015) proposed a Memetic

lgorithm for the LRP with backhauls to minimize transportation

osts, depot opening costs, and vehicle operating costs. Derbel,

arboui, Hanafi, and Chabchoub (2012) developed a hybrid Genetic

lgorithm with Iterated Local Search (ILS) to minimize costs in the

RP with capacitated depots and uncapacitated vehicles. Prodhon

2011) also proposed some hybrid evolutionary algorithms for

he periodic location routing problem. The algorithms combine

volutionary local search with the randomized extended Clarke

nd Wright algorithm. Hemmelmayr (2015) developed a large

eighborhood search (LNS) algorithm to solve the periodic location

outing problem with minimum costs. The author also presented

general parallelization strategy to reduce the computation time.

scobar, Linfati, and Toth (2013) presented a two-phased hybrid

lgorithm for the LRP with minimum costs. The approach first con-

tructs an initial solution and the solution is improved by granular

abu search using different diversification strategies. More algo-

ithms to tackle the LRP are: Genetic Algorithm ( Ardjmand, Weck-

an, Park, Taherkhani, & Singh, 2015 ), Tabu Search ( Martínez-

alazar, Molina, Ángel-Bello, Gómez, & Caballero, 2014 ), GRASP

nd path relinking ( Prins, Prodhon, & Calvo, 2006 ), Particle Swarm

606 M. Moshref-Javadi, S. Lee / European Journal of Operational Research 255 (2016) 604–619

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and Path Relinking ( Marinakis & Marinaki, 2008 ), Variable Neigh-

borhood Search ( Macedo et al., 2015 ), and Ant Colony ( Bouhafs

& Koukam, 2006; Ting & Chen, 2013 ). The reader is referred

to Prodhon and Prins (2014) for reviews on LRP and solution

approaches.

In this paper, we propose the LLRP, formulate the problem, in-

troduce two efficient algorithms, and finally evaluate on several

problem instances.

3. Problem description and formulation

This paper considers the combined location and routing prob-

lems which minimizes the total waiting time of customers. Sup-

pose J ( V , E ) is a graph where V is the set of vertices including

potential depots ( G ) and customers ( V’ ) and E is the set of arcs

{ ( v i , v j ) : v i , v j ∈ V, v i � = v j } . It is assumed that there are N f homo-

geneous candidate depots of which N g are opened. Regarding arcs,

c ij indicates the travel time of the arc between nodes v i and v j and

it is symmetric, i.e. , c ij =c ji . There are N v vehicles distributing com-

modities from depots and each vehicle is assigned to one route.

The route of each vehicle starts from an open depot and multi-

ple vehicles are allowed in each depot. Vehicles are capacitated

and depots are assumed to be uncapacitated. Numbers of avail-

able vehicles, candidate depots, and open depots are known. Each

customer is visited exactly once. The goal of the problem is mini-

mizing the total waiting time of the customers by finding the set

of open facilities and vehicle routings. Since the objective consid-

ers minimizing the sum of latencies at only customers, we do not

take into account the return travel to the depots. Thus, it can be

assumed that the routes are open. To formulate the problem, the

following notations are defined:

Indices

i,j,u Represent customers, totally N c customers

k Represents vehicle

g Represents candidate depots, totally N f

Sets

K Set of vehicles, | K| = N v G Set of candidate depots, | G | = N f

′ Set of customers, | V ′ | = N c

V Set of all customers and candidate depots | V | = N = N c +N f

Parameters

q j Demand quantity at customer j

Q k Capacity of vehicle k

c i j

Travel time between nodes i and j

N g Number of facilities to open

M Large positive constant

Variables

t k i

Arrival time of vehicle k at customer i

x k i j

1 if vehicle k traverses arc ( i , j ) from customer i to cus-

tomer j ; otherwise, 0

f gi

1 if customer i is supplied from depot g ; otherwise, 0

z g 1, if facility g is open; otherwise 0

The proposed mathematical model is based on the formulation

of the network flow problem ( Ahuja, Magnanti, & Orlin, 1988 ) and

is as follows:

Minimize : ∑

k ∈ K

∑

i ∈ V ′ t k i (1)

∑

j∈ V x k i j =

∑

j∈ V x k ji ∀ i ∈ V, ∀ k ∈ K (2)

d

k ∈ K

∑

j∈ V ′ i � = j

x k i j = 1 ∀ i ∈ V

′ (3)

g∈ G f g j = 1 ∀ j ∈ V

′ (4)

i ∈ V

∑

j∈ V ′ x k i j q j ≤ Q k ∀ k ∈ K (5)

g∈ G

∑

i ∈ V ′ x k gi = 1 ∀ k ∈ K (6)

∑

∈ V ′ x k gu +

∑

u ∈ V \{ i } x k ui ≤ 1 + f gi ∀ i ∈ V

′ , ∀ k ∈ K, ∀ g ∈ G (7)

i ∈ V ′ f gi ≤ M z g ∀ g ∈ G (8)

g∈ G z g ≤ N g (9)

k i + c i j − (1 − x k i j ) M ≤ t k j ,

∀ i ∈ V, ∀ j ∈ V

′ , ∀ i � = j, ∀ k ∈ K, ∀ g ∈ G, (10)

t k i

≥ 0 , ∀ i ∈ V, ∀ k ∈ K,

z g ∈ { 0 , 1 } ∀ g ∈ G

x k i j

∈ { 0 , 1 } ∀ i, j ∈ V, ∀ k ∈ K

(11)

In this model, the objective function is to minimize the total la-

ency, which is the sum of the arrival time of the vehicles at cus-

omers. Constraints ( 2 and 3 ) represent flow continuity in routes.

ince k ∈ K and | K | = N v , no more than N v vehicles are used. Con-

traints ( 4 ) ensure that each customer is supplied from exactly one

epot. Constraints ( 5 ) guarantee that the total load on each vehi-

le does not exceed the vehicle’s capacity. Constraints ( 6 ) ensure

hat each customer is assigned to exactly one route and each route

tarts from exactly one depot. Constraints ( 7 ) ensure that a cus-

omer is assigned to a depot ( f gi

= 1 ) if there is a route connecting

he depot to the customer. Therefore, constraints ( 6 ) and ( 7 ) also

epresent the relation vehicle-depot which determines the depot of

ach vehicle. Constraints ( 8 ) guarantee that a facility must be open

f a customer is assigned to that facility and constraint ( 9 ) ensures

he total number of open facilities. Latency at each node is calcu-

ated using constraints ( 10 ). If vehicle k arrives at node i at time t k i

nd traverses through arc i to j , the arrival time of the vehicle at

ode j is equal to the sum of t k i

and travel time on arc ( i , j ). Finally,

onstraints ( 11 ) specify the ranges and types of the variables.

. Solution algorithms

The LLRP is an extension of the facility location and vehicle

outing problems and both of the problems are NP-hard ( Sahni &

onzalez, 1976 ). Thus, the LLRP is NP-hard too. The mathemati-

al model is mixed integer linear and it is only applicable to find

he optimal solutions of problems with less than nine customers.

wo heuristic approaches, the Memetic Algorithm and the Recur-

ive Granular Algorithm are presented in this section and applied

o solve large-sized instances adopted from the literature.

.1. Memetic Algorithm (MA)

The Memetic Algorithm, proposed by Pablo Moscato ( Moscato,

989 ), is a population-based metaheuristic approach that combines

he classical genetic algorithm as a diversification strategy with

local search as an intensification method. Many authors have

tilized this approach to deal with several NP-hard problems,

specially routing problems ( Cattaruzza, Absi, Feillet, & Vidal,

014; Karaoglan & Altiparmak, 2015; Prins, Prodhon, Ruiz, Soriano,

Wolfler Calvo, 2007 ). The algorithm starts with an initial popula-

ion and different individuals are selected in each iteration to pro-

uce offspring using crossover. The new generated solutions un-

ergo mutation with a pre-defined probability and are exposed to


Fig. 1. Solution representation.

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ocal search for possible local improvements. The next generation

s formed using the current population and new generated solu-

ions. The algorithm stops after a prespecified number of iterations.

.1.1. Solution representation

Solution representation plays an important role in the efficiency

f the algorithm. Since MA is a population-based algorithm, the

epresentation should not consume much memory to be efficient.

ig. 1 shows the solution representation which has two sections;

he first section specifies the open depots while the second section

epresents vehicle assignments and routings. The numbers repre-

ent the customers from 1 to N c . Each “V ” signifies that the route

s terminated and another route from the same depot is started,

hile each “D ” terminates the assignments of routes to the current

epot and starts the assignment to the next depot. Note that every

D ” represents a new depot and a new vehicle. Thus, the length of

he chromosome is N g + N c + N v − 1 . In this example two depots,

and 5, are open. The three first vehicles are assigned to depot 2

ecause depot 2 appears first in section one of the representation,

hereas the last vehicle is assigned to depot 5. The first vehicle

isits customers 9, 7, and 8 and the sequence of visits is from left

o right.

.1.2. Fitness evaluation

The objective function value uses the total latency to eval-

ate the fitness of each chromosome; however, since infeasible

olutions are allowed in the population, the fitness value of

he chromosome is modified by the amount of total excess

oad on the vehicles. Therefore, the objective function value is

F V = total latency + P × (total excess load) where P is the penalty

oefficient.

.1.3. Initial population generation

Four different heuristics are used to generate a diverse initial

opulation. Heuristics #1 and #3 generate feasible solutions re-

arding the capacity of the vehicles, while heuristics #2 and #4

ay generate infeasible solutions.

Heuristic #1 is a greedy method that initially opens N g de-

ots randomly. An equal number of vehicles ( R g ) is assigned to

ach open depot. If the remainder of N v N g

is not zero, we assign

v − ( N g − 1) N v N g vehicles to a depot ( X represents the largest

nteger equal or smaller than X ). After assigning the vehicles to

epots, the first vehicle in the first depot is selected and uses the

earest neighborhood method to form a route. Customers are as-

igned to the current vehicle until the customer’s demand exceeds

he vehicle’s remaining capacity. The process is repeated for all of

he vehicles until every customer is assigned to a route.

Heuristic #1: Nearest neighborhood with capacity

1: Input N g , N v , N c , C ij 2: Open N g depots randomly

3: Assign vehicles to the open depots ( R g )

4: For depot g = 1 to N g

5: For k = 1 to R g 6: Current_node ← Depot g

7: Until vehicle k has enough capacity

8: New_node ← nearest node to Current_node

9: Add New_node to the end of the route

10: Update vehicle’s capacity

11: Calculate arrival time

12: Current_node ← New_node

13: End Until

14: End For

15: End For

16: Calculate the total latency

Heuristic #2: Nearest Neighborhood with equal visits

1: Input N g , N v , N c , C ij 2: Open Ng depots randomly

3: Assign vehicles to the open depots ( R g )

4: Calculate the number of customers to be visited by each vehicle ( S k )

5: For depot g = 1 to N g

6: For k = 1 to R g 7: Current_node ← Depot g

8: For s = 1 to S k 9: New_node ← nearest node to Current_node




13: End Until

14: End For

15: End For


Heuristic #2 is similar to heuristic 1; however, it does not con-

ider the capacity of the vehicles. This heuristic assumes that the

umber of customers visited by each vehicle is equal. N c N v cus-

omers are assigned to each vehicle. To ensure that all of the cus-

omers are visited, one vehicle visits N c − ( N v − 1) N c N v customers.

Heuristic #3 uses the centrality measure from the complex net-

orks science to open the depots and assign vehicles to the open

epots. In this method, a centrality measure is calculated using the

ollowing formula:

en t g =

∑

i ∈ V ′

1

1 + c gi

∀ g ∈ G (12)


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Heuristic #3: Centrality method

1: Input N g , N v , N c , C ij 2: For depot g = 1 to N g

3: Calculate centrality measure for all remaining candidate depots

4: Open a depot using Roulette Wheel on centrality values

5: Assign vehicles to the open depot g using formula ( 13 ) ( AV g )

6: For k = 1 to AV g 7: Current_node ← Depot g

8: Until vehicle has enough capacity

9: New_node ← nearest node to Current_node


11: Update vehicle’s capacity



14: End Until

15: End For

16: End For


In each loop, this centrality measure is calculated for all of

the remaining candidate depots considering only the remaining

customers. Having calculated the centrality measure, the roulette

wheel method is used to open the next depot. This method has

two advantages: first it gives a chance to the depots with lower

centrality values to be located. Second, it increases the diversifica-

tion of the initial population by opening different sets of depots.

Yet, depots with higher centrality values have higher chance to be

opened. The centrality measure is also used to assign vehicles to

the open depots. The number of vehicles that are assigned to open

depot g is calculated by the following formula in step 5 of the

algorithm:

A V g =

⌊

cen t g ∗(N v −

∑ g−1 e =1

A V e

)cent g

∗( N g − g + 1)

⌋

g = 2 , ..., N g (13)

where A V g is the number of assigned vehicles to open depot g .

cen t g is the centrality value of the opened depot in iteration g

of the loop and cent g denotes the average of centrality values of

the remaining candidate depots in iteration g of the loop. For the

first open depot ( g = 1), ∑ g−1

e =1 A V e = 0 . Using formula ( 13 ), depots

with higher centrality values receive more vehicles. In addition, the

number of vehicles is assigned dynamically in each iteration of the

algorithm according to the recalculated centrality value. After as-

signing the vehicles to the depots, the nearest neighborhood policy

is used to form the routes while keeping the capacity of the vehi-

cles into account. All of the solutions generated by this method are

feasible.

Heuristic #4: Subgrain growth

1: Input N g , N v , N c , C ij 2: Select N v customers randomly

3: For i = 1 to N v

4: j ← nearest node to node i

5: Connect nodes i and j to form a route

6: End For

7: For v = 1 to N c -2 ×N v

8: // Find the closest customer to every route’s ends

9: For k = 1 to N v

10: For e = 1 to 2

11: W(e, k) ← The closest customer to end e of route k

12: End For

13: End For

14: New_node ← Select the closest customer according to W(e, k)

15: Add New_node to the end of the corresponding route

16: Update route’s end

17: End For

18: Use Consensus Algorithm to open depots and connect routes to depots

c
Consensus Algorithm

1: Input N g , N v , N c , C ij , routes disconnected from depots

2: Depot_votes = (0, 0,…,0)//Depot_votes is a vector with size N g

3: // Count the number of votes for each depot

4: For k = 1 to N v

5: For e = 1 to 2

6: Q (e, k) ← The closest depot to end e of route k

7: End For

8: End For

9: For g = 1 to N g

10: For k = 1 to N v

11: If Q (1, k) = depot g and/or Q(2, k) = depot g

12: Depot_votes( g ) = Depot_votes( g ) + 1

13: End If

14: End For

15: End For

16: Open the first N g depots with the largest Depot_votes( g ) values

17: // Assign nodes to depots

18: For k = 1 to N v

19: Find the closest open depot to the nodes located in the ends of route k

20: Connect the route to the depot

21: End For

Heuristic #4 is based on the subgrain growth from the Mate-

ials Science. This algorithm selects N v customers randomly and

nds the nearest customer to each of them to form N v routes con-

aining two customers. To finalize the routes, each route grows

rom its end which has the closest customer to it. Having assigned

ll of the customers to the routes, the routes must be connected

o an open depot. First N g depots are opened by Consensus Algo-

ithm and each route is connected to the depot with the shortest

istance.

The idea behind the Consensus Algorithm is to open N g depots

hich have the highest number of requests to be opened among

ll depots. Each candidate depot receives one vote from each route

f at least one of the ends of the route selects that depot as the

losest depot to the route’s ends. Among N f depots, those N g de-

ots that receive the highest number of votes are opened. In case

f tie, one depot is opened randomly. The Consensus algorithm is

heuristic that solves the following optimization problem to select

he best set of open depots:

inimize : ∑

i ∈ V ′

∑

g∈ G ( c ig w ig + c ig v ig ) (14)

g∈ G w gi +

∑

g∈ G v gi = 1 ∀ i ∈ V

′ (15)

gi ≤ z g ∀ i ∈ V

′ , ∀ g ∈ G (16)

gi ≤ z g ∀ i ∈ V

′ , ∀ g ∈ G (17)

z∈ G z g = N g (18)

g , w gi , v gi ∈ { 0 , 1 } ∀ g ∈ G, ∀ i ∈ V

′ (19)

In this model, w gi and v gi are binary variables equal to 1 if

he first and second ends of route i is connected to depot g , re-

pectively. The objective function minimizes the total sum of arcs’

engths connected to the open depots. Constraints ( 16 ) and ( 17 ) en-

ure that a depot is open if a route is connected to the depot and

onstraint ( 18 ) guarantees that N g depots are open.


X X P1 1 5 2 D V 4 6 3 9 7 V 8 P2 7 2 6 3 8 1 V 4 D 5 9 V C1’ 7 2 2 D V 4 6 4 D 5 9 V C2’ 1 5 6 3 8 1 D 3 9 7 V 8 C1 7 3 2 D V 4 6 8 1 V 5 9 C2 5 2 6 3 8 1 V D 4 9 7 V

Fig. 2. Order Crossover (OX) maintaining feasibility.

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.1.4. Crossover and mutation operators

Crossover is used to generate offspring to explore the solution

pace. Using Order Crossover (OX), two parents are selected and

ombined to generate two new solutions. First, two points are se-

ected from the second section of the solution representation and

he alleles between these two points are moved to the offspring.

he remaining alleles in the offspring are filled using the corre-

ponding parent’s chromosome considering the feasibility of the

olution. In other words, all of the customers must appear once

n the solution representation and the numbers of “D ”s and “V ”s

ust be maintained. Fig. 2 illustrates Order Crossover. Parents P1

nd P2 generate a pair of new chromosomes C1’ and C2’ . These

hromosomes are then finalized maintaining feasibility. The first

ection of each offspring uses the parent’s chromosomes without

ny changes.

Mutation is a diversification strategy to explore new areas of

he solution space, prevent early convergence, and escape from lo-

al optimal. The mutation operator uses the Swap , 2 -Opt , and In-

ertion moves. Using the Swap operator, two segments from the

hromosome are selected and swapped. The Insertion move in-

erts one segment of the chromosome into another part of it. Us-

ng the 2 -Opt move, two points in the chromosome are selected

nd the order of the in-between alleles is reversed. Note that us-

ng the defined mutation operators, an inter-route or intra-route

ove may happen based on the selected alleles (customer, ‘ V ’,

r ‘ D ’). For example, two nodes that are swapped together can

e customers that belong to the same or different routes. It also

an be a ‘ V ’ or ‘ D ’ that may completely change the set of cus-

omers serviced by some vehicles or depots. The 2 -Opt move can

esult in more complicated changes, for example, reversing a route

nd a part of another route, or reversing a route and assign-

ng a part of a route to another route. These operators result in

ifferent changes in different chromosomes that help the explo-

ation of the solution space. Also, the open depots are randomly

hanged to mutate the first section. The described operators are

pplied according to a probability ( p m

) which increases if the best

olution does not improve after a specified number of iterations

NI m

).

.1.5. Improving solutions by local search

Local search is an intensification procedure that is added to

he algorithm to improve the quality of the resulting individ-

als. The Swap , Insertion , and 2 -Opt operators are utilized and

re performed on the best N nloc solutions in the pool of off-

pring and parents for N loc iterations using the first improving

trategy.

.1.6. Preparation of the next population

After local search, offspring solutions form a pool of solutions

ith the current population. All of the chromosomes in this pool

re sorted with respect to fitness value and N elite of the best solu-

ions in the pool are transferred to the next population. The rest of

he population is selected from the pool using the roulette wheel

ethod which assigns higher probability to those chromosomes

ith better fitness values.

Memetic Algorithm (MA)

1: Input N g , N v , N c , C ij , N Itr , p m , N loc , N nloc , N elite , NI m , αm , ps

2: Generate an initial population using Heuristics #1-4 ( pop )

3: Evaluate fitness value of each chromosome

4: Store Best Solution ( BS ) and Best Fitness Value (FV best )

5: For i = 1 to N Itr

6: //Offspring generation

7: Select parents from pop

8: Produce ps offspring using order crossover

9: Mutate offspring with probability p m 10: Evaluate fitness function of the offspring ( FV ofs )

11: If FV ofs < FV best and feasible

12: Update BS

13: FV best ← FV ofs

14: End If

15: //Local search

16: Create a pool of offspring and parents

17: Sort individuals in the pool according to FV

18: For e = 1 to N nloc

19: Select the local search operator according to probabilities

20: Perform local search on each individual N loc times

21: End For

22: Evaluate fitness function ( FV loc )

23: If FV loc < FV best and feasible

24: Update BS

25: FV best ← FV loc

26: End If

27: // Next generation preparation

28: Move N elite elite chromosomes from pool to new population

29: Select the rest of the new population from pool by Roulette Wheel ( new_pop

30: pop ← new_pop

31: Update p m = αm p m if FV best does not improve for NI m iterations

32: End For

.2. Recursive granular algorithm (RGA)

The proposed RGA is an individual-based search algorithm

hich combines various types of search operators to explore the

olution space. The idea of granularity ( Prins et al., 2007 ) is used

o reduce neighborhood search effectively and removes unneces-

ary and non-improving moves.

.2.1. Initial solution generation

RGA starts with the generation of an individual solution. After

everal experiments, we selected Heuristic #3 explained in Section

.1 due to the fact that it produces good solutions with respect to

uality and diversity.

.2.2. Neighborhood search

Three different operators are used to systematically change the

eighborhood of the search. The search process starts with per-

urbing the current best solution. To start the granular search, a


Fig. 3. Granular search.

Fig. 4. Search operators: (a) cross-move; (b) exchange; (c) relocation; (d) swap; (e) 2-Opt.

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neighborhood is defined for each customer and all of the neigh-

bor customers are stored. For a pre-specified number of customers

( N ng ), the selected customer is moved to one of its neighbor cus-

tomers’ routes if they are in different routes. If this move improves

the OFV compared to the best so far solution, the algorithm ac-

cepts the move and in case of feasibility, the best so far solution is

updated too. The granularity not only searches neighbor solutions

at random, but also limits the neighborhood of search to those

moves that have higher chance of finding better solutions. Fig. 3

shows the granular local search schematically. The granular size

increases if the best found solution does not improve after a pre-

specified number of iterations ( NI r ). To change the neighborhood

of open depots, they are randomly changed according to a small

probability ( p g ).

RGA continues with other search operators including Exchange

and Cross-move which are shown in Fig. 4 . The Exchange move

chooses two nodes from two different routes and exchanges them

together. The Cross-move selects some nodes from the end of two

routes and exchanges the nodes. To reduce the computation time

hen updating the total latency, the recalculation of latency is per-

ormed on only affected routes.

.2.3. Local search

After changing the search neighborhood, local search operators

re applied for local improvement in the neighborhood. Three dif-

erent moves Swap , Relocation , and 2-Opt are applied in this step.

he operator is selected with respect to a probability and applied

locg times on all routes of the solution.

.2.4. Post-optimization

This procedure aims at systematically changing the open de-

ots and connecting the routes to the possibly new open depots.

o do this, each route is disconnected from its depot and assumed

o form a super-customer. The Consensus Algorithm described in

ection 4.1.3 selects the best combination of N g depots and con-

ects each route to one of the open depots. Fig. 5 illustrates the

ost-optimization step that closes depot 5 and opens depot 1.


Fig. 5. Post-optimization step in RGA.

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Fig. 6. Illustration of lower bound 1.

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Recursive Granular Algorithm (RGA)

1: Input N g , N v , N c , C ij , N Itrg , p g , N locg , N ng , N ns , NI r 2: Generate an initial solution using Heuristics #3 ( Current_sol )

3: Calculate OFV

4: BS ← Current_sol, OFV best ← OFV

5: For i = 1 to N Itrg

6: //Neighborhood Search

7: For s = 1 to N ng

8: //Granular Search

9: Select a customer randomly (Cust1)

10: Select a customer from Cust1’s neighborhood (Cust2)

11: If Cust1 and Cust2 are in different routes

12: Insert Cust1 after Cust2 in the route (New_sol)

13: End If

14: Evaluate New_sol ( OFV new ) //Update Current_sol, BS, and OFV best

15: If OFV new < OFV best

16: Current_sol ← New_sol

17: If New_sol is feasible

18: BS ← New_sol

19: OFV best ← OFV new

20: End If

21: End If

22: Perform Exchange operator

23: Update Current_sol, BS , and OFV best

24: Perform Cross-move operator


26: Perturb open depots with probability p g 27: Update Current_sol, BS , and OFV best

28: //Local search

29: Current_sol ← BS

30: For all perturbed routes

31: Select the local search operator according to probability

32: For l = 1 to N locg

33: Perform local search


35: End For

36: End For

37: // Post optimization

38: Disconnect routes from depots

39: Execute Consensus Algorithm


41: Update granular size and nodes’ neighborhood lists

42: End For

43: End For

.3. Lower bounds

Two lower bounds were proposed by Ngueveu et al. (2010) for

he CCVRP. These LBs are modified here to be applied on the LLRP.

he lower bounds do not consider capacity constraints.

.3.1. Lower bound 1 (LB1)

Lower bound 1 relaxes the assumption of the number of ve-

icles; therefore, every customer is visited by exactly one vehi-

le and every vehicle visits only one customer ( | K| = N c ). Thus,

he lower bound locates N g depots and allocates customers to the

losest open depot without the consideration of the routing prob-

em. Fig. 6 shows the lower bound schematically. To obtain the LB1

alue, the following mixed-integer formulation is solved. The total

atency is calculated by the sum of all arcs’ travel times connect-

ng the customers to the open depots. Constraints ( 21 ) ensure that

f a node is connected to a depot, the depot must be open. Con-

traints ( 22 ) represent that each node must be assigned to exactly

ne open depot and constraint ( 23 ) guarantees that only N g depots

re located.

inimize : ∑

g∈ G

∑

i ∈ V ′ c gi f gi (20)

i ∈ V ′ f gi ≤ z g ∀ g ∈ G (21)

g∈ G f gi = 1 ∀ i ∈ V

′ (22)

g∈ G z g = N g (23)

f gi , z g ∈ { 0 , 1 } ∀ i ∈ V

′ , ∀ g ∈ G (24)

.3.2. Lower bound 2 (LB2)

Lower bound 2 assumes that each vehicle visits an equal num-

er of customers. This lower bound is composed of two parts. The

rst part is calculated based on the connections of customers to

epots and the second part is based on the connections of cus-

omers together. To calculate the first part, all of the arcs ( ( v g , v j ) : g ∈ G, v j ∈ V ′ ) are sorted ( w e indicates the eth shortest arc) and


Table 1

Algorithm’s parameters setting.

MA parameters Values RGA parameters Values

N Itr : Number of iterations 200 + N c N Itrg : Number of iterations 400 + N c p m : Mutation operator probability 0.2 p g : Depot neighborhood change probability 0.3

N elite : Number of elite individuals 0.3 N c N locg : Number of local searches 0.5 N c N loc : Number of local searches 0.2 N c N ns : Neighborhood size 5

N nloc : Number of individuals undergo local search 0.4 N c N ng : Number of nodes with neighborhood perturbation 0.4 N c NI m : Number of not improved iterations before p m increase 20 NI r : Number of not improved iterations before N ns increase 90

αm : Mutation probability increase rate 0.02

ps : Population size N/2

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the first N v arcs are selected. Since we are assuming an equal num-

ber of visits for all vehicles, the following formula is valid:

LB 2 _ a =

N v ∑

e =1

⌈N c + N v − e − ( N c mod N v )

N v

⌉w e (25)

(X mod Y) indicates the remainder of X divided by Y. Using this

formula, each arc is multiplied by a coefficient which is the num-

ber of times that the arc is considered in the calculation of the

total latency. Thus, N v nodes are connected by direct links to open

depots although the number of open depots may be greater than

N g .

To calculate the second part, all of the arcs connected to each

node v j ∈ V ′ are sorted and the shortest one is selected. This pro-

cess continues for all nodes v j ∈ V ′ until we have N c –N v arcs. All

of these arcs are sorted and the eth shortest arc is indicated as w

′ e .

Note that Ngueveu et al. (2010) selected the N c shortest arcs of the

network. However, our lower bound is tighter since we consider

the connected arcs to every node. The following formula calculates

the latency of the remaining nodes:

LB 2 _ b =

N c −N v ∑

e =1

⌈N c − e − ( N c mod N v )

N v

⌉w

′ e (26)

The LB2 value is the sum of LB2_a and LB2_b .

4.4. Sequential approach

The sequential approach decomposes the LLRP into sub-

problems to solve sequentially. To locate the depots and allocate

customers to them, the mathematical model ( 20 )–( 24 ) is solved.

Next, vehicles are assigned to the open depots. To allocate vehi-

cles, those depots with more assigned customers will have more

vehicles. Therefore, we assign one vehicle to each open depot and

the rest of the vehicles are assigned with respect to the ratio of

NC g / N g , where NC g is the number of customers serviced by de-

pot g . Finally, to obtain the routings of the vehicles, the Nearest

Neighborhood method is applied with equal number of visits for

vehicles.

5. Computational results

The proposed algorithms are evaluated on problem instances

adopted from the literature. Computation was performed on a

3.1 GHz computer with 4GB RAM. To tune the parameters of the

algorithms, we considered four values for each parameter and con-

ducted 10 experiments for each value. After preliminary experi-

ments and tuning, the algorithms’ parameters were set ( Table 1 ).

The Swap , Insertion , and 2- Opt probabilities for both algorithms are

0.4, 0.4, and 0.1, respectively and the penalty coefficient ( P ) for

vehicles’ excess capacity is 35. Twenty-five percent of the initial

population for the Memetic Algorithm is generated by each of the

Heuristics #1–4.

.1. Comparing initial solution generation heuristics

The Memetic Algorithm uses four different heuristics to con-

truct the initial population. Heuristics #1–4 are compared on

roblem instance #111112 in Tuzun and Burke (1999 ) with 100

odes, 10 candidate locations, and 11 vehicles. Fig. 7 illustrates the

ox plots after 100 runs by each algorithm. It shows that Heuris-

ic #3 which, uses the centrality measure, generates the best solu-

ions, whereas Heuristic #4 generates the lowest quality solutions.

e did not conduct any statistical test and leave it as a future re-

earch. However, we used all of the heuristics to form a diverse ini-

ial population in the proposed Memetic Algorithm. We also gen-

rated 100 solutions randomly. The average OFV of the randomly

enerated solutions was 68856.4, while this value is 5894.3 for

euristic #3. These values show the importance of using these

euristics for generating the initial solution.

.2. Comparison with LRP solutions

To show the importance of considering the right objective func-

ion (latency) in customer-oriented distribution systems and disas-

er relief, we compare our results with conventional LRP results.

e calculated the total latency of the LRP solutions obtained by

u, Lin, Lee, and Ting (2010) on 36 problem instances generated

y Tuzun and Burke (1999) . The objective function used in Yu et

l. (2010) was minimizing the total costs. The authors assumed

hat the numbers of open depots and vehicles were variables of

he problem. Thus, to have a fair comparison, we set the number

f open depots and vehicles to the values obtained by Yu et al.

2010) and ran the algorithms 10 times. The latency of the best ob-

ained solutions is illustrated in Fig. 8 . Assuming gap is defined as

LLRP-LRP)/LLRP , the average gap is 24.6 percent which shows the

ignificance of selecting the right objective function. The LRP can

lso serve as an upper bound for the LLRP. It can also be seen that

he gap value between LLRP and LRP is smaller for the problems

ith clustered networks.

.3. Evaluation of the proposed algorithms

The algorithms are compared on three sets of benchmarks

hich totally include 76 problems. These benchmarks are popular

or testing algorithms designed for different location routing prob-

ems ( Escobar et al., 2013; Huang, 2015; Jarboui, Derbel, Hanafi, &

ladenovi ́c, 2013; Karaoglan & Altiparmak, 2015; Prodhon, 2011;

u et al., 2010 ). Since our paper is the first paper that studies the

LRP, there is no other available algorithm to compare with. There-

ore, the performances of the algorithms are compared together

nd with the lower bounds and sequential approach.

Due to the fact that LLRP is greedy in using vehicles, it utilizes

ll of the available vehicles. Thus, the numbers of vehicles and

pen depots need to be set. We set them to the values obtained

n Yu et al. (2010) and the algorithms are run for 30 replications

n each problem and the best and average OFVs are reported.


Fig. 7. Comparison of solution generation heuristics.

Fig. 8. Comparing latency of LLRP solutions with LRP solutions in Yu et al. (2010 ) on Tuzun and Burke (1999 ) instances.

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roblem sizes and results are shown in Tables 2 –4 . The first six

olumns show the information about the sizes of the problems.

B1 , LB2 , and the sequential solution values are presented in

olumns 7–9. The Bold values in columns LB1 and LB2 indicate

he tighter LB value. The best and average obtained solutions

f 30 runs are reported for each algorithm. The %Gap LB value

s calculated by ( Best algorithm

− max ( LB1 , LB2 ))/ max ( LB1 , LB2 )) ∗100,

hich means smaller values are better. Also, the %Gap value of

eq. is calculated by ( Seq. − Best alg )/ Seq. ∗100 indicating that larger

alues are better. The Underlined value in each row indicates the

lgorithm that obtained the best solution for each problem. The

verage of %Gap LB and Seq. , and computation time (in seconds) of

ll benchmarks are reported in the last row.

The first set of problems was proposed by Tuzun and Burke

1999) . This set includes 36 problems with the number of nodes

anging from 100 to 200 and 10 to 20 candidate depots. The evalu-

tion results are shown in Table 2 . Both of the algorithms perform

ell according to the Avg. %Gap LB less than 40 percent. MA ob-

ains better results in 33 instances out of 36 with respect to %Gap

B . The Memetic Algorithm has %Gap LB less than 20 percent in

out of 36 problems (16.7 percent) and less than 30 percent in

7 out of 36 problems (47.2 percent). The Memetic Algorithm av-

ragely obtains better solutions with 4.75 percent better Gap LB .

omparing LB values together, LB1 has tighter gap values, whether

lustered or dispersed, except the first three problems. The av-

rage computation time of both algorithms is almost equal and

cceptable.

The second benchmark set was proposed by Prins et al.

2006) and contains 30 problems. The size of the problems ranges

rom 20 to 200 cities, 5 to 10 depot locations, and 3 to 49 vehicles.

ccording to the results in Table 3 , the Memetic Algorithm obtains

etter solutions with 3.01 percent better %Gap LB in 25 out of 30

nstances. The RGA was capable of obtaining the optimal solution

or the first problem since LB2 is equal to the best produced solu-

ion. Although in other problems the %Gap LB is not zero, it does

ot indicate that the solution is not optimal since the LB assumes

ncapacitated vehicles and can be loose. The Memetic Algorithm

lso has %Gap LB value less than 20 percent in 14 out of 30 prob-

ems (46.7 percent) and less than 30 percent in 21 problems (70

ercent). The gap value is lower in problems with more vehicles

ince the solution becomes very similar to LB1 and it is tighter

n larger sized networks and clustered networks. The computation

ime of RGA is averagely 23.1 seconds more than MA. The reason

s probably due to the local search in the RGA algorithm that is

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Table 2

Computational results on Tuzun and Burke (1999) problem instances.

Problem number Problem size LB1 LB2 Seq. RGA MA

Cities Vehicle

capacity

Depots Vehicles Open

depots

Best Avg. CPU %Gap LB %Gap Seq. Best Avg. CPU %Gap LB %Gap Seq.

111112 100 150 10 11 3 2265 .6 3189 .7 5933 .4 4208.7 4544 .2 21 31 .95 29 .07 4017.9 4270 .1 40 .0 25 .96 32 .28

111122 100 150 20 11 3 2462 .7 2530 .5 4731 .1 3980.5 4348 .2 21 .7 57 .30 15 .87 3719.1 4087 .5 40 .0 46 .97 21 .39

111212 100 150 10 10 3 2382 .3 2820 .4 4700 .4 4344.1 4769 .2 29 54 .02 7 .58 4264.4 4563 .5 41 .5 51 .20 9 .28

111222 100 150 20 11 2 2917 .4 2675 .2 5378 .0 4492.9 5025 .4 21 .4 54 .00 16 .46 4278.3 4557 .3 41 .4 46 .65 20 .45

112112 100 150 10 11 2 2208 .0 2059 .3 3656 .5 2908.5 3557 .3 27 .5 31 .73 20 .46 2795.6 3049 .7 41 .4 26 .61 23 .54

112122 100 150 20 11 3 1619 .0 1498 .7 3344 .0 2230.6 2524 .7 26 .4 37 .78 33 .30 2097.9 2702 .6 44 .3 29 .58 37 .26

112212 100 150 10 12 3 1086 .8 818 .4 1673 .0 1485.1 1615 .9 28 .2 36 .65 11 .23 1442.2 1604 .9 48 .5 32 .70 13 .80

112222 100 150 20 11 2 1346 .6 938 .9 2019 .8 1744.0 2470 .4 27 .4 29 .51 13 .65 1659.0 2084 .1 43 .8 23 .20 17 .86

113112 100 150 10 11 3 2186 .2 1973 .3 3720 .6 3001.8 3271 .5 26 .4 37 .31 19 .32 2897.7 3162 .6 39 .0 32 .55 22 .12

113122 100 150 20 11 3 2188 .4 1592 .7 3703 .6 3054.0 3322 .8 27 39 .55 17 .54 2912.0 3330 .0 41 .6 33 .07 21 .37

113212 100 150 10 12 3 1577 .0 1002 .4 2063 .9 1871.1 2095 .2 21 .8 18 .65 9 .34 1832.1 1901 .0 48 .6 16 .18 11 .23

113222 100 150 20 11 4 1477 .2 972 .7 2215 .1 2017.3 2239 .8 21 .8 36 .56 8 .93 2037.5 2360 .6 41 .3 37 .93 8 .02

131112 150 150 10 16 3 3899 .4 3760 .3 6860 .3 6135.7 6528 .5 65 .1 57 .35 10 .56 5863.8 6160 .7 59 .9 50 .38 14 .53

131122 150 150 20 16 3 3549 .2 3235 .5 7135 .3 5591.9 6114 .9 65 .9 57 .55 21 .63 5310.3 5649 .3 58 .2 49 .62 25 .58

131212 150 150 10 16 3 4176 .7 3435 .6 7215 .8 6257.7 6589 .8 64 49 .82 13 .28 5967.6 6234 .2 58 .2 42 .88 17 .30

131222 150 150 20 16 3 3469 .5 3347 .4 6309 .1 5634.9 6006 .8 65 62 .41 10 .69 5261.0 5610 .4 58 .9 51 .64 16 .61

132112 150 150 10 16 2 3545 .5 1861 .9 5263 .7 4164.6 4342 .8 63 .5 17 .46 20 .88 4026.7 4246 .2 58 .2 13 .57 23 .50

132122 150 150 20 16 2 3186 .1 1956 .6 4892 .7 3978.6 4348 .3 63 .6 24 .87 18 .68 3874.0 4185 .5 59 .5 21 .59 20 .82

132212 150 150 10 17 3 2563 .3 1195 .1 3342 .5 2994.3 3140 .4 65 .5 16 .81 10 .42 2906.0 3129 .9 59 .3 13 .37 13 .06

132222 150 150 20 17 3 1274 .9 1088 .2 2093 .0 1862.0 1956 .4 66 .4 46 .05 11 .04 1784.8 2152 .3 59 .1 40 .00 14 .73

133112 150 150 10 16 3 3894 .0 2074 .1 5843 .8 4918.9 5579 .3 66 .8 26 .32 15 .83 5034.0 5465 .9 57 .9 29 .28 13 .86

133122 150 150 20 16 3 2580 .0 1780 .8 4853 .0 3552.7 3921 .7 65 .8 37 .70 26 .79 3474.0 3849 .1 58 .9 34 .65 28 .42

133212 150 150 10 17 3 2530 .1 1290 .8 3351 .0 3066.1 3199 .3 66 21 .18 8 .50 3008.0 3284 .5 59 .0 18 .89 10 .24

133222 150 150 20 17 3 2144 .9 1374 .6 3074 .9 2732.7 3466 .2 64 .2 27 .40 11 .13 2617.4 3016 .9 58 .5 22 .03 14 .88

121112 200 150 10 21 3 5009 .8 4159 .1 8531 .9 7184.0 7753 .5 88 .4 43 .40 15 .80 7008.7 7371 .1 83 .4 39 .90 17 .85

121122 200 150 20 22 4 3882 .0 3803 .4 6942 .2 6181.5 6951 .9 89 .5 59 .23 10 .96 6039.6 6501 .6 83 .0 55 .58 13 .00

121212 200 150 10 21 3 4688 .1 4245 .8 8123 .2 7019.7 7515 .9 88 .7 49 .73 13 .58 6744.3 7318 .6 85 .3 43 .86 16 .97

121222 200 150 20 21 3 4840 .8 3940 .7 8756 .0 7330.3 7873 .2 89 .8 51 .43 16 .28 6828.5 7567 .2 85 .1 41 .06 22 .01

122112 200 150 10 21 3 5127 .7 2907 .1 7943 .7 6793.1 7106 .1 89 .2 32 .48 14 .48 6643.8 7106 .9 85 .7 29 .57 16 .36

122122 200 150 20 21 3 3228 .9 2188 .6 5514 .4 4160.9 4486 .4 90 .7 28 .86 24 .54 4012.9 4915 .7 84 .9 24 .28 27 .23

122212 200 150 10 21 2 3666 .0 1750 .1 5059 .2 4352.4 4 4 49 .8 95 .6 18 .72 13 .97 4227.5 4 4 48 .1 84 .5 15 .32 16 .44

122222 200 150 20 22 3 1606 .6 1582 .1 2568 .1 2251.7 2487 .1 91 .3 40 .15 12 .32 2127.9 2345 .5 83 .7 32 .45 17 .14

123112 200 150 10 22 4 3968 .2 3022 .4 6206 .8 5231.2 5537 .9 89 .8 31 .83 15 .72 5099.0 5527 .4 85 .3 28 .50 17 .85

123122 200 150 20 22 4 3762 .6 2720 .5 6380 .4 5046.4 5498 .2 87 .6 34 .12 20 .91 5188.7 5862 .9 84 .8 37 .90 18 .68

123212 200 150 10 22 3 4763 .7 2930 .1 5735 .6 5674.4 5865 .3 95 .9 19 .12 1 .07 5363.0 5678 .5 84 .8 12 .58 6 .50

123222 200 150 20 22 5 2071 .4 1725 .7 2972 .1 2726.8 3096 .7 88 .1 31 .64 8 .25 2657.5 3917 .6 85 .9 28 .29 10 .59

Avg . 59 .9 37 .52 15 .28 62 .63 32 .77 18 .13

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Table 3

Computational results on Prins et al. (2006) problem instances.


Cities Vehicle capacity Depots Vehicles Open depots Best Avg. CPU %Gap LB %Gap Seq. Best Avg. CPU %Gap LB %Gap Seq.

20-5-1 20 70 5 5 3 226 .6 331 .9 379 .3 331.9 366 .7 8 .7 0 .00 12 .50 337.3 378 .0 12 .9 1 .63 11 .07

20-5-1b 20 150 5 3 2 383 .3 390 .5 842 .6 620.0 662 .8 8 .5 58 .76 26 .42 608.1 636 .8 12 .5 55 .71 27 .84

20-5-2 20 70 5 5 3 198 .6 291 .7 349 .8 314.7 352 .9 8 .8 7 .88 10 .04 304.8 354 .4 12 .7 4 .48 12 .88

20-5-2b 20 150 5 3 2 272 .2 360 .4 622 .6 486.5 540 .1 8 .2 35 .00 21 .85 486.5 511 .2 12 .7 35 .00 21 .85

50-5-1 50 70 5 12 3 734 .4 679 .6 1020 .6 876.6 938 .4 24 .4 19 .36 14 .11 859.9 917 .3 18 .2 17 .08 15 .75

50-5-1b 50 150 5 6 2 977 .4 946 .5 1611 .9 1343.6 1477 .3 22 .6 37 .47 16 .64 1330.2 1379 .8 17 .4 36 .10 17 .48

50-5-2 50 70 5 12 3 601 .8 581 .4 756 .3 752.5 801 .0 24 .0 25 .04 0 .51 723.4 786 .2 18 .4 20 .21 4 .35

50-5-2b 50 150 5 6 3 601 .8 770 .1 1224 .8 990.8 1077 .6 22 .7 28 .66 19 .11 965.7 1009 .4 19 .1 25 .40 21 .15

50-5-2BIS 50 70 5 12 3 913 396 .1 1011 .0 957.3 974 .2 23 .7 4 .85 5 .31 955.2 981 .5 17 .9 4 .62 5 .52

50-5-2bBIS 50 150 5 6 2 655 .6 439 .2 882 .5 825.4 877 .8 23 .3 25 .90 6 .47 811.8 884 .9 17 .8 23 .83 8 .01

50-5-3 50 70 5 12 2 733 .1 648 .3 1012 .5 897.5 943 .2 23 .8 22 .42 11 .36 848.1 928 .9 20 .4 15 .69 16 .24

50-5-3b 50 150 5 6 2 733 .1 820 .3 1442 .2 1245.6 1331 .5 22 .7 51 .85 13 .63 1163.9 1198 .8 17 .7 41 .89 19 .30

100-5-1 100 70 5 24 3 1896 .3 1124 .6 2220 .4 2062.9 2096 .5 58 .4 8 .79 7 .09 2030.9 2044 .3 29 .7 7 .10 8 .53

100-5-1b 100 150 5 12 3 1896 .3 1344 .9 3192 .6 2437.9 2582 .5 53 .4 28 .56 23 .64 2374.9 2507 .8 24 .8 25 .24 25 .61

100-5-2 100 70 5 24 2 1032 .5 904 .5 1360 .6 1206.5 1243 .7 57 .5 16 .85 11 .33 1226.1 1500 .9 28 .4 18 .75 9 .89

100-5-2b 100 150 5 11 2 1032 .5 1282 .6 1995 .2 1706.6 1865 .2 52 .1 33 .06 14 .46 1622.9 1701 .0 28 .5 26 .53 18 .66

100-5-3 100 70 5 24 2 1465 .4 1021 .1 1837 .7 1641.2 1654 .1 63 .0 12 .00 10 .69 1710.4 1726 .2 30 .1 16 .72 6 .93

100-5-3b 100 150 5 11 2 1465 .4 1356 .6 2559 .0 2087.1 2201 .5 52 .3 42 .43 18 .44 2054.8 2190 .5 29 .0 40 .22 19 .70

100-10-1 100 70 10 26 3 1358 .0 746 .5 1689 .1 1573.0 1676 .0 70 .8 15 .83 6 .87 1524.1 1589 .9 40 .5 12 .23 9 .77

100-10-1b 100 150 10 12 3 1358 .0 1314 .8 2224 .7 2152.0 2285 .4 64 .5 58 .47 3 .27 1960.7 2138 .6 39 .5 44 .38 11 .87

100-10-2 100 70 10 24 3 1011 .5 875 .7 1363 .6 1211.9 1326 .1 73 .8 19 .81 11 .12 1175.0 1236 .3 44 .2 16 .16 13 .83

100-10-2b 100 150 10 11 3 1011 .5 1263 .2 1806 .7 1675.1 1857 .0 68 .1 32 .61 7 .28 1625.8 1724 .2 41 .0 28 .70 10 .01

100-10-3 100 70 10 25 3 1107 .2 825 1467 .8 1280.6 1308 .4 73 .8 15 .66 12 .75 1246.8 1288 .3 44 .4 12 .61 15 .06

100-10-3b 100 150 10 11 3 1107 .2 1135 .7 2091 .9 1860.8 2031 .9 61 .6 63 .85 11 .05 1799.0 1890 .2 42 .9 58 .40 14 .00

200-10-1 200 70 10 49 3 2688 .4 1446 .6 3209 .7 2953.2 3125 .1 191 .9 9 .85 7 .99 2920.7 3092 .4 113 .8 8 .64 9 .00

200-10-1b 200 150 10 22 3 2688 .4 2117 .9 4423 .1 3626.0 3992 .6 163 .2 34 .88 18 .02 3532.2 3809 .3 108 .0 31 .39 20 .14

200-10-2 200 70 10 49 3 1894 .1 1230 .3 2416 .2 2049.2 2095 .1 190 .1 8 .19 15 .19 2064.2 2153 .3 113 .7 8 .98 14 .57

200-10-2b 200 150 10 23 3 1894 .1 1361 .5 3063 .0 2598.9 2729 .9 179 .4 37 .21 15 .15 2516.4 2684 .5 104 .7 32 .85 17 .85

200-10-3 200 70 10 48 3 2620 .2 1344 .8 3156 .0 2820.1 2885 .4 177 .8 7 .63 10 .64 2805.9 3066 .1 102 .8 7 .09 11 .09

200-10-3b 200 150 10 22 3 2620 .2 1835 .3 3986 .2 3474.5 3651 .9 147 .9 32 .60 12 .84 3347.0 3454 .6 108 .9 27 .74 16 .04

Avg . 66 .02 26 .52 12 .53 42 .90 23 .51 14 .47

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Table 4

Computational results on Barreto (2004) problem instances.


Cities Vehicle capacity Depots Vehicles Open depots Best Avg. CPU %Gap LB %Gap Seq. Best Avg. CPU %Gap LB %Gap Seq.

Christ-50-5 50 160 5 6 2 1020 .5 1375 .7 2185 .5 1841 .7 1972 .3 24 .6 33 .87 15 .73 1690.8 1782 .4 19 .7 22 .90 22 .64

Christ-75-10 75 160 10 9 3 1604 .4 1916 .6 2929 .2 2703 .4 2893 .9 34 .1 41 .05 7 .71 2590.3 2689 .8 29 .1 35 .15 11 .57

Christ-100-10 100 200 10 8 2 2008 .0 3284 .0 5478 .6 4116 .9 4450 .8 49 .4 25 .36 24 .85 4058.2 4194 .9 34 .1 23 .57 25 .93

Gaskell-21-5 21 60 0 0 5 4 2 382 .4 612 .9 765 .7 699 .8 810 .2 10 .8 14 .18 8 .60 658.4 741 .1 14 .7 7 .43 14 .01

Gaskell-29-5 29 4500 5 4 2 768 .8 936 .2 1434 .1 1238 .4 1366 .1 12 .2 32 .28 13 .65 1224.5 1296 .3 15 .6 30 .79 14 .62

Gaskell-32-5b 32 110 0 0 5 3 2 773 .5 1189 .1 2517 .8 1622 .3 1786 .2 12 .4 36 .43 35 .57 1571.0 1668 .4 16 .1 32 .12 37 .60

Gaskell-36-5 36 250 5 4 2 644 .4 1575 .1 1730 .7 1646 .1 1694 .2 13 .6 4 .51 4 .89 1642.4 1647 .0 17 .4 4 .27 5 .10

Min-27-5 27 2500 5 4 2 4380 .6 2983 .1 7763 .1 5387 .5 6343 .8 10 22 .99 30 .60 5387.5 5697 .0 27 .8 22 .99 30 .60

Min-134-8 134 850 8 11 3 14791 .0 11468 .0 29755 .0 25496 .0 28872 .8 80 .2 72 .38 14 .31 23387.0 26012 .5 74 .0 58 .12 21 .40

Or-117-14 117 150 0 0 0 14 7 3 32896 .0 31965 .0 87974 .0 60580 .0 70626 .3 45 .8 84 .16 31 .14 56209.0 61396 .2 51 .5 70 .87 36 .11

Avg. 29 .31 36 .72 18 .71 30 30 .82 21 .96


Table 5

Summary of results on three problem sets.

Number of

customer nodes

Number of

instances RGA MA

CPU %Gap LB %Gap Seq. CPU %Gap LB %Gap Seq.

Tuzun and Burke (1999)

100 Dispersed 4 23 .28 49 .32 17 .24 40 .73 42 .70 20 .85

100 Clustered 8 25 .81 33 .47 16 .72 43 .89 28 .98 19 .40

Avg. 12 24 .97 38 .75 16 .89 42 .83 33 .55 19 .88

150 Dispersed 4 65 .00 56 .79 14 .04 58 .80 48 .63 18 .50

150 Clustered 8 65 .23 27 .23 15 .41 58 .80 24 .17 17 .44

Avg. 12 65 .15 37 .08 14 .95 58 .80 32 .32 17 .79

200 Dispersed 4 89 .10 50 .95 14 .16 84 .20 45 .10 17 .46

200 Clustered 8 91 .03 29 .62 13 .91 84 .95 26 .11 16 .35

Avg. 12 90 .38 36 .73 13 .99 84 .70 32 .44 16 .72

Prins et al. (2006)

20 Avg. 4 8 .55 25 .42 17 .70 12 .70 24 .21 18 .41

50 Avg. 8 23 .40 26 .94 10 .89 18 .36 23 .10 13 .47

100 Avg. 12 62 .43 28 .99 11 .50 35 .87 25 .59 13 .65

200 Avg. 6 168 .35 21 .73 13 .31 108 .65 19 .45 14 .78

Barreto (2004)

Avg. 10 29 .31 36 .72 18 .71 30 .00 30 .82 21 .96

Total average 58 .42 33 .07 14 .64 50 .21 28 .86 17 .19

Table 6

Hypothesis tests for comparing algorithms’ means.

Preferred algorithm Problem instances Percentage

Tuzun and Burke (1999) MA 111112, 111122, 111222, 112112, 112222, 113112, 113212, 121112, 121122, 121212, 121222, 122222,

123212, 131112, 131122, 131212, 131222, 132112, 132122, 133222

55 .56

RGA 112122, 113222, 122122, 123122, 123222, 132222, 133212 19 .44

Both 111212, 112212, 113122, 122112, 122212, 123112, 132212, 133112, 133122 25 .00

Prins et al. (2006) MA 20-5-1b, 20-5-2b, 50-5-1, 50-5-1b, 50-5-2, 50-5-2b, 50-5-3b, 100-5-1, 100-5-1b, 100-5-2b, 100-10-1,

100-10-1b, 100-10-2, 100-10-2b, 100-10-3, 100-10-3b,

20 0-10-1b, 20 0-10-2b, 20 0-10-3b 69 .44

RGA 20-5-1, 100-5-2, 100-5-3, 200-10-2, 200-10-3 13 .89

Both 20-5-2, 50-5-2BIS, 50-5-2bBIS, 50-5-3, 100-5-3b, 200-10-1 16 .67

Barreto (2004) MA Christ-50-5, Christ-75-10, Gaskell-21-5, Gaskell-29-5, Min-27-5, Min-134-8, Or-117-14 70 .00

RGA – 0 .00

Both Christ-100-10, Gaskell-32-5b, Gaskell-36-5 30 .00

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erformed on each route in each iteration. Thus, when the number

f vehicles increases, the local search must be executed on each

oute which increases the computation time.

The third problem set was proposed by Barreto (2004) . This set

as 10 problems from 21 cities to 117 and 5 to 14 depot locations

ith 2 or 3 vehicles. The results are given in Table 4 . The Memetic

lgorithm yields better solutions in all 10 problems with average

Gap LB 5.9 percent better than RGA. The Memetic Algorithm has

Gap LB less than 20 percent in 2 out of 10 problems (20 percent)

nd less 30 percent in 5 out of 10 problems (50 percent). The com-

utation time is almost equal for both algorithms.

Altogether, both MA and RGA algorithms perform well on

ll benchmarks while the Memetic Algorithm has slightly better

erformance. The results indicate that the MA obtained better so-

utions in 68 out of 76 problems (89.47 percent). The %Gap LB and

Gap Seq. of the Memetic Algorithm are averagely 4.21 percent and

.55 percent better than RGA as reported in Table 5 . Also, the %Gap

alues are small for the problems in which RGA yields better re-

ults; the average %Gap LB and average %Gap Seq. are 2.45 and 1.77

ercent respectively. According to Table 5 the algorithms perform

ell on dispersed and clustered networks while the Avg. %Gap LB is

ower for clustered networks because of tighter lower bound. The

omputation time of both algorithms is acceptable with maximum

08 and 191.9 seconds for the MA and RGA on the largest problem

ith 200 cities, 10 candidate locations, and 49 vehicles. The results

lso indicate that the RGA algorithm needs more computation time

or larger scale problems than MA.

r

To compare the algorithms’ means, we conducted hypothesis

ests using a one-way independent t -test after 30 runs of the al-

orithms on each problem. The results for the three problem sets

re shown in Table 6 assuming significance level 5 percent. Accord-

ng to this comparison, MA is selected as the preferred algorithm

n 46 out of 76 problems (60.53 percent), whereas RGA is preferred

n 12 problems (15.79 percent). In 23.68 percent of the problems,

ny algorithm can be used. Also, although RGA yields less mean

alue in problem 200-10-3, MA obtained the best value.

.4. Sensitivity analysis

This section aims to present the sensitivity of performance to

he number of vehicles or depots to provide decision makers with

nsights on resource planning, for example, how much latency will

hange if one more vehicle is available. The MA was run on prob-

em #111112 for 10 replications and the best obtained solutions are

eported in Fig. 9 , varying the number of vehicles and depots. It

an be seen that increasing the number of vehicles leads to the

eduction in latency. Likewise, increasing the number of depots

esults in decrease of latency due to the fact that vehicles now

an start their tours from more efficient locations. However, the

mount of reduction in latency from 2 depots to 3 depots is much

ore considerable than the change from 3 depots to 4 depots. Such

bservations from the sensitivity analysis can guide decision mak-

rs to make more economic decisions in utilizing and acquiring

esources.


Fig. 9. Results of the sensitivity analysis on the number of depots and vehicles.

Table 7

Evaluation of MA and RGA on CMT problem instances ( Christofides et al., 1979 ).

Problem Best known solution ( Ke & Feng, 2013) MA RGA

Best Avg. %Gap Best Avg. %Gap

CMT1 2230 .35 2346 .7 2528 .2 5 .22 2506 .9 2555.5 12 .40

CMT2 2391 .63 2613 .4 2758 .7 9 .27 2961 .2 2971.4 23 .82

CMT3 4045 .42 4333 .3 4511 .4 7 .12 4 4 48 .4 4642.4 9 .96

CMT4 4987 .52 5338 .1 5479 .8 7 .03 5473 5637.6 9 .73

CMT5 5809 .59 6120 .1 6241 .6 5 .34 6640 .9 6676.1 14 .31

CMT11 7314 .55 7986 .5 8323 .2 9 .19 8286 .9 8435.1 13 .29

CMT12 3558 .92 3642 .3 3671 .2 2 .34 3685 .7 3.7450 3 .56

6 .50 12 .44

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5.5. Comparison with CCVRP results

We also implemented our two algorithms on CCVRP problems

( Christofides, Mingozzi, & Toth, 1979 ) whose goal is to route multi-

ple capacitated vehicles from a single depot (i.e., the location prob-

lem is ruled out). The results are compared with the best known

solutions for these instances presented by Ke and Feng (2013) , as

shown in Table 7 . The Memetic Algorithm, which has been consis-

tently performing well and therefore is our recommendation, per-

forms better than RGA achieving average 6.5 percent gap with best

known solutions and limiting the gap to less than 10 percent. It

should be noted that our algorithms are designed for the LLRP

which is an extension of the CCVRP additionally involving location

problem and multiple depots, and some loss in performance is in-

evitable compared to those specifically designed for CCVRP.

6. Conclusions

In this paper, the location-routing problem in customer-

oriented systems was considered. The LLRP determines the loca-

tions of open depots, allocation of vehicles to the depots, and rout-

ings of the vehicles. Contrary to the conventional LRP, all of the

three decisions are made with the objective of minimizing the to-

tal latency at the customers. A Memetic Algorithm and a Recursive

Granular Algorithm were proposed and evaluated on 76 problems

adopted from the literature. We also proposed four heuristics to

generate a diverse initial population for the Memetic Algorithm,

among which the centrality-based heuristic generates promising

solutions. Comparing the results with two lower bounds and se-

quential approach indicates that both algorithms obtain promis-

ng results. The Memetic Algorithm performs averagely 4.21 per-

ent better than the RGA with almost equal average computation

ime. Comparing the results with the latency of the best known

olution for the conventional LRP demonstrated almost 25 percent

ap which indicates the importance of considering latency as the

bjective function in the LLRP. The sensitivity analysis shows that

ncreasing the number of open depots and vehicles reduces the to-

al latency at the customers. However, the amount of reduction in

atency varies for different values of open depots and vehicles.

This paper assumed LLRP with capacitated vehicles and unca-

acitated depots. However, in real-world depots cannot have un-

imited capacity and as a future research one can consider the LLRP

ith capacitated depots. The objective function of the LLRP was

nly the total waiting time of the customers. Although in disaster

elief logistics the cost of temporary DCs are negligible comparing

o humans’ lives, the cost of DCs in commercial distribution sys-

ems should be taken into account. We also assumed that all of the

emands are satisfied. However, due to limited resources, such as

epots and vehicles, all of the demands cannot be fulfilled in rea-

onable time. Therefore, another direction for future research is to

se multi-objective approaches to consider conflicting objectives.

eferences

huja, R.K., Magnanti, T.L., & Orlin, J.B. (1988). Network flows. DTIC document.

rdjmand, E. , Weckman, G. , Park, N. , Taherkhani, P. , & Singh, M. (2015). Applyinggenetic algorithm to a new location and routing model of hazardous materials.

International Journal of Production Research, 53 (3), 916–928 .

verbakh, I. , & Berman, O. (1994). Technical note—routing and location-routingp-delivery men problems on a path. Transportation Science, 28 (2), 162–166 .

Barreto, S. (2004). Análise e Modelização de Problemas de localização-distribuição[Analysis and modelling of location-routing problems] (Ph.D. thesis) . 3810-193:

University of Aveiro, campus Universitário de Santiago [in Portuguese] .

http://refhub.elsevier.com/S0377-2217(16)30385-X/sbref0001














B

B

B

C

C

C

C

C

C

D

E

H

H

J

K

K

K

L

L

L

M

M

M

M

N

N

Ö

P

P

P

P

R

R

R

S

T

T

W

Y

ouhafs, L. , & Koukam, A. (2006). A combination of simulated annealing and antcolony system for the capacitated location-routing problem. Lecture Notes in

Computer Science, 4251 , 409–416 . oujelben, M. K. , Gicquel, C. , & Minoux, M. (2014). A distribution network design

problem in the automotive industry: MIP formulation and heuristics. Computers& Operations Research, 52 , 16–28 .

ramel, J. , & Simchi-Levi, D. (1997). The logic of logistics: Theory, algorithms, and ap-plications for logistics management . New York: Springer-Verlag .

attaruzza, D. , Absi, N. , Feillet, D. , & Vidal, T. (2014). A memetic algorithm for the

multi trip vehicle routing problem. European Journal of Operational Research,236 (3), 833–848 .

aunhye, A. M. , Nie, X. , & Pokharel, S. (2012). Optimization models in emergencylogistics: A literature review. Socio-Economic Planning Sciences, 46 (1), 4–13 .

eselli, A. , Righini, G. , & Tresoldi, E. (2014). Combined location and routing problemsfor drug distribution. Discrete Applied Mathematics, 165 , 130–145 .

hakrabarty, D. , & Swamy, C. (2011). Facility location with client latencies: Linear

programming based techniques for minimum latency problems. In Oktay Gün-lük, & Gerhard J. Woeginger (Eds.), Integer programming and combinatoral opti-

mization (pp. 92–103). Springer . hristofides, N. , Mingozzi, A. , & Toth, P. (1979). The vehicle routing problem. In

N. Christofides, A. Mingozzi, P. Toth, & C. Sandi (Eds.), Combinatorial Optimization(pp. 315–338). Chichester: Wiley .

ontardo, C. , Cordeau, J.-F. , & Gendron, B. (2013). An exact algorithm based on

cut-and-column generation for the capacitated location-routing problem. IN-FORMS Journal on Computing, 26 (1), 88–102 .

erbel, H. , Jarboui, B. , Hanafi, S. , & Chabchoub, H. (2012). Genetic algorithm withiterated local search for solving a location-routing problem. Expert Systems with

Applications, 39 (3), 2865–2871 . scobar, J. W. , Linfati, R. , & Toth, P. (2013). A two-phase hybrid heuristic algorithm

for the capacitated location-routing problem. Computers & Operations Research,

40 (1), 70–79 . emmelmayr, V. C. (2015). Sequential and parallel large neighborhood search algo-

rithms for the periodic location routing problem. European Journal of OperationalResearch, 243 (1), 52–60 .

uang, S.-H. (2015). Solving the multi-compartment capacitated location routingproblem with pickup-delivery routes and stochastic demands. Computers & In-

dustrial Engineering, 87 , 104–113 .

arboui, B. , Derbel, H. , Hanafi, S. , & Mladenovi ́c, N. (2013). Variable neighborhoodsearch for location routing. Computers & Operations Research, 40 (1), 47–57 .

araoglan, I. , & Altiparmak, F. (2015). A memetic algorithm for the capacitated lo-cation-routing problem with mixed backhauls. Computers & Operations Research,

55 , 200–216 . e, L. , & Feng, Z. (2013). A two-phase metaheuristic for the cumulative capacitated

vehicle routing problem. Computers & Operations Research, 40 (2), 633–638 .

lose, A. , & Drexl, A. (2005). Facility location models for distribution system design.European Journal of Operational Research, 162 (1), 4–29 .

in, C. K. Y. , & Kwok, R. C. W. (2006). Multi-objective metaheuristics for a loca-tion-routing problem with multiple use of vehicles on real data and simulated

data. European Journal of Operational Research, 175 (3), 1833–1849 . uis, E. , Dolinskaya, I. S. , & Smilowitz, K. R. (2012). Disaster relief routing: Integrat-

ing research and practice. Socio-Economic Planning Sciences, 46 (1), 88–97 . ysgaard, J. , & Wøhlk, S. (2014). A branch-and-cut-and-price algorithm for the cu-

mulative capacitated vehicle routing problem. European Journal of Operational

Research, 236 (3), 800–810 .

acedo, R. , Alves, C. , Hanafi, S. , Jarboui, B. , Mladenovi ́c, N. , Ramos, B. , et al. (2015).Skewed general variable neighborhood search for the location routing schedul-

ing problem. Computers & Operations Research, 61 , 143–152 . arinakis, Y. , & Marinaki, M. (2008). A particle swarm optimization algorithm with

path relinking for the location routing problem. Journal of Mathematical Mod-elling and Algorithms, 7 (1), 59–78 .

artínez-Salazar, I. A. , Molina, J. , Ángel-Bello, F. , Gómez, T. , & Caballero, R. (2014).Solving a bi-objective transportation location routing problem by metaheuristic

algorithms. European Journal of Operational Research, 234 (1), 25–36 .

oscato, P. (1989). On evolution, search, optimization, genetic algorithms and martialarts: Towards memetic algorithms. Caltech concurrent computation program Tech-

nical Report, 826 . Pasadena, CA: California Institute of Technology . adizadeh, A. , & Nasab, H. H. (2014). Solving the dynamic capacitated location-rout-

ing problem with fuzzy demands by hybrid heuristic algorithm. European Jour-nal of Operational Research, 238 (2), 458–470 .

gueveu, S. U. , Prins, C. , & Wolfler Calvo, R. (2010). An effective memetic algorithm

for the cumulative capacitated vehicle routing problem. Computers & OperationsResearch, 37 (11), 1877–1885 .

zdamar, L. , & Demir, O. (2012). A hierarchical clustering and routing procedurefor large scale disaster relief logistics planning. Transportation Research Part E:

Logistics and Transportation Review, 48 (3), 591–602 . rins, C. , Prodhon, C. , & Calvo, R. W. (2006). Solving the capacitated location-routing

problem by a GRASP complemented by a learning process and a path relinking.

4OR, 4 (3), 221–238 . rins, C. , Prodhon, C. , Ruiz, A. , Soriano, P. , & Wolfler Calvo, R. (2007). Solving the

capacitated location-routing problem by a cooperative Lagrangean relaxation–granular tabu search heuristic. Transportation Science, 41 (4), 470–483 .

rodhon, C. (2011). A hybrid evolutionary algorithm for the periodic location-rout-ing problem. European Journal of Operational Research, 210 (2), 204–212 .

rodhon, C. , & Prins, C. (2014). A survey of recent research on location-routing prob-

lems. European Journal of Operational Research, 238 (1), 1–17 . ahmani, Y. , Ramdane Cherif-Khettaf, W. , & Oulamara, A. (2016). The two-echelon

multi-products location-routing problem with pickup and delivery: Formula-tion and heuristic approaches. International Journal of Production Research, 54 (4),

999–1019 . ath, S. , & Gutjahr, W. J. (2014). A math-heuristic for the warehouse location—rout-

ing problem in disaster relief. Computers & Operations Research, 42 , 25–39 .

ibeiro, G. M. , & Laporte, G. (2012). An adaptive large neighborhood search heuristicfor the cumulative capacitated vehicle routing problem. Computers & Operations

Research, 39 (3), 728–735 . ahni, S. , & Gonzalez, T. (1976). P-complete approximation problems. Journal of the

ACM, 23 (3), 555–565 . ing, C.-J. , & Chen, C.-H. (2013). A multiple ant colony optimization algorithm for

the capacitated location routing problem. International Journal of Production Eco-

nomics, 141 (1), 34–44 . uzun, D. , & Burke, L. I. (1999). A two-phase tabu search approach to the location

routing problem. European Journal of Operational Research, 116 (1), 87–99 . ang, H. , Du, L. , & Ma, S. (2014). Multi-objective open location-routing model with

split delivery for optimized relief distribution in post-earthquake. TransportationResearch Part E: Logistics and Transportation Review, 69 , 160–179 .

u, V. F. , Lin, S.-W. , Lee, W. , & Ting, C.-J. (2010). A simulated annealing heuristicfor the capacitated location routing problem. Computers & Industrial Engineering,

58 (2), 288–299 .


















































































































































































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