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Handbook of Research
on Applied Optimization
Methodologies in
Manufacturing Systems
Ömer Faruk Yılmaz
Istanbul Technical University, Turkey & Yalova University, Turkey
Süleyman Tüfekçí
University of Florida, USA
A volume in the Advances in Logistics,
Operations, and Management Science (ALOMS)
Book Series
Published in the United States of America by
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Names: Yilmaz, Omer Faruk, 1989- editor. | Tufekci, Suleyman, editor.
Title: Handbook of research on applied optimization methodologies in
manufacturing systems / Omer Faruk Yilmaz and Suleyman Tufekci, editors.
Description: Hershey, PA : Business Science Reference, [2018] | Includes
bibliographical references.
Identifiers: LCCN 2017012416| ISBN 9781522529446 (hardcover) | ISBN
9781522529453 (ebook)
Subjects: LCSH: Manufacturing processes--Mathematical models--Handbooks,
manuals, etc. | Mathematical optimization--Handbooks, manuals, etc. |
Heuristic algorithms--Handbooks, manuals, etc.
Classification: LCC TS183 .H3595 2018 | DDC 670--dc23 LC record available at https://lccn.loc.gov/2017012416
This book is published in the IGI Global book series Advances in Logistics, Operations, and Management Science
(ALOMS) (ISSN: 2327-350X; eISSN: 2327-3518)
77
Copyright © 2018, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
Chapter 5
DOI: 10.4018/978-1-5225-2944-6.ch005
ABSTRACT
The Hub Location-Allocation Problem is one of the most important topics in industrial engineering and
operations research, which aims to find a form of distribution strategy for goods, services, and information.
There are plenty of applications for hub location problem, such as Transportation Management, Urban
Management, locating service centers, Instrumentation Engineering, design of sensor networks, Computer
Engineering, design of computer networks, Communication Networks Design, Power Engineering,
localization of repair centers, maintenance and monitoring power lines, and Design of Manufacturing
Systems. In order to define the hub location problem, the present chapter offers two different metaheuris-
tic algorithms, namely Particle Swarm Optimization or PSO and Differential Evolution. The presented
algorithms, then, are applied to one of the hub location problems. Finally, the performances of the given
algorithms are compared in term of benchmarking.
INTRODUCTION
In this chapter, we discuss some services, such as database transaction, movements of people, commodi-
ties, information or unfinished parts that take place between an origin-destination pair of nodes. Such
pairs of nodes can be found in the domain of a manufacturing site or spread along continents, as each
origin-destination pair needs a service different from the other pairs.
Hub Location Allocation
Problems and Solution
Algorithms
Peiman A. Sarvari
Istanbul Technical University, Turkey
Fatma Betül Yeni
Istanbul Technical University, Turkey
Emre Çevikcan
Istanbul Technical University, Turkey
78
Hub Location Allocation Problems and Solution Algorithms
Hub location problem is one of the most important topics of location problems. The facility location
problem, also known as location analysis or k-center problem is a branch of operations research and
computational geometry. Facility location problems try to reduce the costs of operations considering
some set of constraints and relevant demands with locating different ranges of facilities. Making deci-
sions for facility location are critically challenging regarding strategic planning for all types of business
entities. Property acquisition and establishment are naturally costly so that one can consider facility
location and relocation operations as long-term investments. Decision makers are challenging with dif-
ferent geographical, demographical and trending factors for selecting profitable sites. Thus, selection
of robust facility locations is an important task, as far as future events are uncertain and unpredictable.
Hub location problem is an extension of the classical facility location problems. Hubs are facilities
that operate as consolidating, connecting, and switching points for flows between the stipulated origins
and destinations (Farahani et al., 2013). Hubs are also defined as special facilities that serve as switch-
ing, transshipping and sorting points in many-to-many distribution systems. The hub location problem is
concerned with locating hub facilities and allocating demand nodes to hubs in order to route the traffic
between origin–destination pairs (Alumur & Kara, 2008). Many applications are available for the hub
location problem, and this section is primarily dedicated to introducing this problem to readers.
In this chapter, we have tried to fit what moves between an origin-destination pair of nodes, like
information, people and commodities into the concept of HLP. Basically, each different pair of origin-
destination node has to be serviced exclusively. For instance, people traveling from i to j are not inter-
changeable with those traveling from j to i. In order to have a fully connected network (a network in
which all nodes are connected) with N nodes, in which each node can be either an origin or a destination,
the number of pairs (i-j pairs which are different from j-i pairs) should be N (N-1). Fig.1 illustrates a
network composed of nodes and connections.
Assuming that we have different traffic services in this network and that each vehicle can service
five origin-destination pairs every day, with 18 vehicles, we will be able to service ten nodes every day.
If we set one of the nodes as a hub node and connect it to all the other nodes, which are introduced as
spokes, we will have 2(n-1) connections to service all origin-destination node pairs. This network is
presented in Figure 2 (Daskin, 1995).
Assume, if there are different traffic services, and if each vehicle can provide service for origin-
destination pairs every day, with 18 vehicles, we will be able to service 46 nodes every day. Thus, with
fixed traffic resources, we can service more cities with a hub network than with a completely connected
network. Multi-hub network is another type of hub and spoke network that is a formation of two or
several hubs and spoke networks in which all hubs are fully connected to each other.
This chapter is organized as follows. Section two, presents a technical and comprehensive literature
review. In section three, the taxonomy of HLP is given, and section four, introduces some of the basic and
fundamental models developed for the hub problem. In section five, an application with two metaheuristic
solution algorithms is suggested together with its application in terms of performance evaluation of the
proposed metaheuristics, and finally, the conclusion of the study is given in section six.
LITERATURE REVIEW
The hub location problem has been studied for many years. With a glance at the related literature, one can
find out the importance of the issue for the researchers. A panoramic view of its applications, research
79
Hub Location Allocation Problems and Solution Algorithms
area, location and time, as illustrated in Figure 3, shows a boom of hub location subjected documents
between the years 2000 and 2015. The U.S.A and China were pioneering regarding the number of the
published documents more than half of which are mostly articles in engineering, mathematics, and
computer sciences based areas.
Ostresh (1975) introduced a procedure to solve the two-center location-allocation problems. Likewise,
O’Kelly (1987) formulated the P-hub median problem (P-HLMP) as a quadratic integer programming
problem, which is a particular case of the None Linear Programming, when the objective function is
in quadratic form, and the constraints are linear. Showing that the problem is NP-hard, he proposed
two enumeration-based heuristics to solve it. Besides some exchange clustering methods presented by
Klincewicz (1991), new heuristic approaches, such as Tabu search, genetic algorithm, and greedy search
have also been used recently (Farahani et al., 2009).
Figure 1. A regular full-connected network with connections between nodes
Figure 2. A hub and spoke network
80
Hub Location Allocation Problems and Solution Algorithms
Sung and Jin (2001) proposed a dual-based solution algorithm which minimizes the total cost for hub
location problems. They tested the effectiveness of the model with numerical examples among which the
cost minimization objective is the most common one. For the same purpose, Martín and Román (2003)
suggested a two-stage spatial competition game. Moreover, Rodríguez-Martín and Salazar-González
(2008) and Contreras et al. (2010) have proposed a mixed integer programming (MIP) where the for-
mulation is strengthened with valid inequalities. He et al. (2015) have also proposed an improved mixed
integer programming (MIP) heuristic, which includes branch-and-bound, Lagrangian relaxation and
linear programming relaxation. Thomadsen and Larsen (2007) represented branch-and-price algorithm
or IP column generation (the combination of column generation and branch-and-bound algorithm) to
solve a two-layered network (a hierarchical network) consisting of clusters of nodes, each defining an
access network and a backbone network. Yaman et al. (2007) presented a generic mathematical model
to solve the latest arrival hub location problem for cargo delivery systems. In their model, they aimed
to minimize the longest delivery time. Sasaki et al. (2014) presented a general discrete Stackelberg hub
location problem by adopting a multiple allocation hub-arc location models. They examined how the
optimal solutions are affected by different customer allocation functions, different revenue sets, the
number of hub arcs, and the degree of discount for hub arc travel. Rothenbächer et al. (2016) proposed
branch-and-price-and cut algorithm for the solution of service network design hub location problems
(SDNHLP). They conducted a computational experiment based on the combined road-rail transporta-
tion data. In addition to the mentioned studies, Marianov and Serra (2003) suggested a heuristic Tabu
search algorithm for airline location in which each hub node is assumed to be an M/D/C queue. In this
method, only a part of the feasible solution is surveyed, and the best neighborhood is selected as a new
Figure 3. Research area and affiliation based systematic literature review
81
Hub Location Allocation Problems and Solution Algorithms
hub node with respect to continuous iterations on neighborhood nodes of the former hub, even the target
function gains a worse solution. Furthermore et al. (2013) presented a hybrid heuristic approach based on
simulated annealing and Tabu search algorithms to solve fully interconnected network design problem
(FINDP) which is a specific application of hub location to network design.
Unlike the studies mentioned above, Berman et al. (2007) have analyzed three different kinds of the
transfer point location problem in their study. They formulated each one of them based on two objective
functions named Mini-Sum and Mini-Max.
P-hub location problems have also been studied in the literature. Besides some mathematical models
aimed to minimize the total cost, maximum distance and total transportation time (Aversa et al., 2005;
Campbell, 2009; Puerto et al., 2016), some heuristic algorithms have also been developed. Pamuk and
Sepil (2001), for example, represented a single relocating heuristic by Tabu search to solve P-hub center
problems (P-HLCP). They used two single-allocation schemes for the evaluation of the algorithm and
employed a greedy local search to improve the resulting allocations.
Yaman (2008) represented a heuristic algorithm based on Lagrangian relaxation and local search to
solve P-hub location median single allocation problems (P-HLMP-S). Mohammadi et al. (2016) have
developed a bi-objective mixed integer non-linear model to study the bi-objective single allocation
p-hub center median problems. They used a fuzzy queuing approach to model the uncertainties in the
network and did several experiments besides a real transformation case to show the applicability of the
proposed method.
Table 1. Literature review matrix of Hub Location problem
Article Objective Type Solution Algorithm
Ebery et al. (2000) Cost minimization CHLP-M Linear programming, Heuristic
approach
Sung and Jin (2001) Cost minimization HLP Dual-based solution approach
Pamuk and Sepil (2001) Maximum distance minimization p-HLPC Tabu search
Ebery (20019 Cost minimization USApHMP and
pHAP Mixed integer linear programming
Mayer and Wagner (2002) Cost minimization UHLP-M Branch-and-bound method
Marianov and Serra (2003) Cost minimization HLP Tabu search
Martin and Roman (2003) Cost minimization HLP Spatial competition game
Bollapragada et al. (2005) Maximizing expected total demand
covered CHMCLP-M Greedy algorithm, Network-planning
model
Aversa et al. (2005) Cost minimization p-HUB median Mixed integer programming model
Marin (2005) Cost minimization SCHLP-M Integer linear programming
Labbe et al. (2005) Cost minimization UHLP-S Polyhedral analysis – Branch and cut
algorithm
Topcouglu et al. (2005) Cost minimization UHLP-S Genetic algorithm
Rodriguez et al. (2007) Cost minimization CHLP Simulated annealing algorithm
Wagner (2007) Cost minimization CHLP Tabu search
Berman et al. (2007) Distance minimization HLP MiniSum model, MiniMax model
Thomadsen and Larsen
(2007) Cost minimization HLP Branch-and-price algorithm
82
Hub Location Allocation Problems and Solution Algorithms
Bollapragada et al. (2005) represented a new network planning model and an efficient greedy solution
heuristic to solve a model that is most closely related to the capacitated hub maximum-covering loca-
tion problem with multi allocations (CHMCLP-M). The quality of the heuristic algorithms is evaluated
by comparing its coverage with the optimal (for small problems) or with an upper bound obtained by
solving a linear programming relaxation. Marn (2005) presented an integer linear programming formu-
lation for splittable capacitated multiple allocation hub location problems and evaluated the model with
a well-known data from the literature.
The uncapacitated hub location problems are the most common problem type at literature. Ebery
(2001) proposed a new mixed integer linear programming to solve the uncapacitated single allocation
p-hub median problems (USApHMP). Later, Mayer and Wagner (2002) used an aggregated branch and
bound model. Labbe et al. (2005) represented a solution method based on branch and cut algorithm to
solve uncapacitated single allocation hub location problem (UHLP-S). In this approach, the network
connecting the hub nodes is called Backbone Network and the connected network of the terminal nodes
is called access network. Cánovas et al. (2007) have also studied uncapacitated multiple allocation hub
location problems (UHLP-M).
Focusing on the dual problem of a four-indexed formulation, they proposed a heuristic approach
based on a dual ascent technique. They evaluated the obtained results based on two well-known data
sets. Hsu and Chen (2007) have developed a hybrid heuristic approach based on Simulated Annealing
Table 2. Literature review matrix of Hub Location problem (Continues)
Article Objective Type Solution Algorithm
Yaman, Kara and Tansel (2007) Longest delivery time
minimization LAHLP Generic mathematical model
Canovas, Garcia and Marin (2007) Cost minimization UHLP-M Dual-ascent technique, Integer programming
Chen (2007) Cost minimization UHLP-S Simulated annealing, Tabu list
Cunha and Silva (2007) Cost minimization UHLP-S Hybrid genetic algorithm
Kratica et al. (2007) Cost minimization USApHMP Genetic algorithm
Rodriguez and Salazar (2008) Cost minimization CHLP-M Benders decomposition, Branch and cut
algorithm
Costa, Captivo and Climaco (2008) Service time
minimization CHLP-S Bi-criteria model, Interactive methods
Alamur and Kara (2008) Cost minimization HLP Survey
Yaman (2008) Cost minimization p-HLMP-S Lagrangian relaxation based heuristic
Camargo, Miranda and Luna (2008) Cost minimization UHLP-S Benders decomposition algorithm
Campbell (2009) Cost minimization p-HLMP-M Mathematical model
Contreras, Fernandez and Marin
(2010) Cost minimization THLP Mixed integer programming (MIP)
Contreas, Cordeau and Laporte
(2011) Cost minimization UHLP Monte-carlo sampling, Benders decomposition
Alamur, Kara and Karasan (2012) Cost minimization MHLP Linear mixed integer programming
Saboury et al. (2013) Cost minimization HLP Tabu search and simulated annealing
Martin de Sa, Camargo and Miranda
(2013) Cost minimization HLP-S Benders decomposition method
83
Hub Location Allocation Problems and Solution Algorithms
and Tabu Search. They tried to find a solution for uncapacitated single allocation hub location problems
(USAHLP) with the hub-and-spoke network structure. Table 1, 2 and 3 indicate a summary of the related
works, containing problem types and solution approaches, of different researchers. The examined studies
are between 2000 and the first half of 2016.
TAXONOMY OF HLPs
There are many applications of hub problems in real world. Here, we are going to give four major prac-
tices of hub location-allocation problems as following:
• Energy Transfer: Energy generator sites or drilling areas need to access to the best-located stor-
ages or transmission sites as hubs via pipelines or cables to transfer energy efficiently to the cus-
tomers. As an example, we can name Waha Hub near Midland, Texas, the Katy Hub near Houston,
Texas, and the Carthage Hub in East Texas.
• Airlines and Airports: As they aim to avoid empty direct flights and unreasonable flight fares, all
airlines need to find the best-located airports as hubs for performing better operations and flight
services. Frankfort airport, for example, is a hub in Europe that makes connection flights more
reasonable than some direct flights.
• Environmental Design: It aims to handle the problems related to the transportation. For instance;
it is concerned with finding the optimized land dump locations, with respect to garbage transpor-
tation stations.
• Postal Logistics Network: The strategic decisions for a hub based mail system include the fol-
lowing: the selection of suitable locations for consolidation, the assignment of customers to send-
Table 3. Literature review matrix of Hub Location problem (Continues)
Article Objective Type Solution Algorithm
Rodriguez, Salazar and Yaman
(2014) Cost minimization HLP Mixed integer programming (MIP), Brunch-
and-cut algorithm
Sasaki et al. (2014) Cost minimization HLP-M Stackelberg competition model
He et al. (2015) Cost minimization HLP Improved mixed integer programming
(IMMIP) heuristic
Damgacıoğlu et al. (2015) Cost minimization UPHLP-S Genetic algorithm
Mohammadi et al (2016)
Total transportation
time minimization,
Total system costs
minimization
BpHCMP-S
Bi-objective mixed-integer non-linear
programming (BMINLP), Fuzzy Queuing
Approach, Game theory, Invasive weed
optimization
Zhalechian et al. (2016)
Total transportation
time minimization,
Total system costs
minimization
Multimodal HLP Multi-objective mixed-integer non-linear
mathematical model (MOMINLP)
Puerto et al. (2016) Cost minimization OMHLP-S İnteger programming
Rothenbacher, Drexl and Irnich
(2016) Cost minimization SNDHLP Branch-and-price and Cut algorithm
84
Hub Location Allocation Problems and Solution Algorithms
ing and receiving depots, the determination of line-haul routes, and the choices of the types of
transportation facilities. Operational decisions, which are based on strategic decisions, include the
disposition of the number of vehicles for line-haul, and the planning of pick-up and delivery tours
for parcels or part-loads to the customers from each depot (Zäpfel & Wasner, 2002).
Table 4 is giving a quick summary of the related works and the applications for location-allocation
problem. The most common formulations, which have been widely applied in the literature, are intro-
duced in the next section.
FUNDAMENTAL HUB LOCATION MODELS
The problem of hub location has attracted many researchers who have worked on a variety of hub model-
ing problems. Since most of the applications of hub problems in the real world are discrete, the models
developed so far are mostly discrete models. Sections 3.1-3.12 are introducing some most commonly
used hub location problems in relevant literature and Tables 5, 6, 7 and 8 summarize the proposed math-
ematical models and their notations, model inputs and outputs (decision variables).
Single Hub Location Problem
O’Kelly (1987) represented this problem with the following specifications:
• The total cost incurred by the location of hub nodes and allocation of non-hub nodes to hub nodes
is minimized (Mini-Sum).
• The solution domain is all of the network nodes (network).
• The non-hub nodes are connected to the hub node.
• The number of hub nodes to locate is primarily specified (exogenous) and is equal to one.
• There is no cost for establishing the hub facility.
• The hub facility to locate is uncapacitated (capacity is not limited).
• The problem is the allocation of a non-hub node to just one hub (single allocation).
Considering the characteristics of this problem, its decision variables are binary (0 or 1). The math-
ematical formulation of single-HLP is depicted in Table 1.
P-Hub Location Problem
In this problem, each non-hub node must be allocated to just one hub node. It is basically considered as
a single allocation p-hub location problem. In this model:
• The total cost incurred by the location of hub nodes and allocation of non-hub nodes to hub nodes
is minimized (criterion is Mini-Sum).
• The solution domain is all of the network nodes.
• The hub nodes are completely linked together.
• Every non-hub node is linked to a single hub node.
85
Hub Location Allocation Problems and Solution Algorithms
Table 4. Applications of hub location problem and the related works
No Article
Application
No Article
Application
Yes No Field Yes No Field
1 Wagner (2007) x Transportation and
Handling Problems 21 Campbell (2009) x truck transportation
2 Ebery et al. (2000) x
Airline Passenger
Transportation, Postal
Logistic
22 Yaman (2008) x Turkish network
3Rodriguez and
Salazar (2008) x Telecommunications 23 Pamuk and Sepil
(2001) xairline passenger
transportation
4
Costa, Captivo
and Climaco
(2008)
x Postal Logistic 24 Aversa et al.
(2005) x Ports
5Bollapragada et al.
(2005) x Telecommunications 25 Contreas, et al
(2011) x -----
6Sung and Jin
(2001) x Transportation 26 Mayer and
Wagner (2002) x
airline passenger
transportation, postal
logistic
7Marianov and
Serra (2003) x Airlines and Airports 27 Canovas et al.
(2007) x
airline passenger
transportation, postal
logistic
8Martin and Roman
(2003) x Airlines and Airports 28 Labbe et al.
(2005) x Telecommunications
9 Beran et al. (2007) x 29 Topcouglu et al.
(2005) x
airline passenger
transportation, postal
logistic
10 Thomadsen and
Larsen (2007) x ------ 30 Chen (2007) x
airline passenger
transportation, postal
logistic
11 Alamur and Kara
(2008) x 31 Cunha and Silva
(2007) xtransportation and
handling problems
12 Saboury et al.
(2013) x 32 Camargo et al.
(2008) x
airline passenger
transportation, postal
logistic
13
Rodriguez,
Salazar and
Yaman (2014)
x
Airline Passenger
Transportation, Postal
Logistic
33 Damgacıoğlu et
al. (2015) x
airline passenger
transportation, postal
logistic
14 He et al. (2015) x ------- 34 Kratica et al.
(2007) x
airline passenger
transportation, postal
logistic
15 Sasaki et al.
(2014) xAirline Passenger
Transportation 35 Ebery (20019 x postal logistic
16 Martin de Sa et al.
(2013) x Postal Logistic 36 Marin (2005) x postal logistic
17 Alamur, Kara and
Karasan (2012) x Turkish Network 37 Rothenbacher et
al. (2016) x road-rail transportation
18 Zhalechian et al.
(2016) x 38 Contreras et al.
(2010) x
airline passenger
transportation, postal
logistic
19 Puerto et al.
(2016) x Postal Logistic 39 Yaman et al.
(2007) x cargo logistic
20 Mohammadi et al
(2016) x Passenger Transportation
86
Hub Location Allocation Problems and Solution Algorithms
• The number of hub nodes to locate is primarily specified and is denoted by p, and at least one or
at most two hub nodes have to be traversed for traveling between two non-hub nodes.
• The other features of p-Hub Location Problem are the same of the Single HLP.
P-Hub Median Location Problem (Multiple Allocation p-HLP)
Because every non-hub node could be allocated to one hub node or more in p-hub median location
problems, this model is named multiple allocation p-HLP. Most of the characteristics of this model are
similar to those of the p-Hub LP except the following specifications:
• The problem tries to minimize the total transportation cost based on a nonlinear objective function.
• Its formulation is similar to the p-median formulation and is named p-hub median location
problem.
• Non-hub nodes can be allocated to several hub nodes.
Flow rate between two nodes that has to be determined as part of the solution is a relaxed variable
≥ ≥
( )
0 0 .
P-Hub Median Location Problem With Fixed Costs
Basically, the models mentioned above could be extended with fixed-link costs for connecting non-hub
nodes to hub nodes. The following factors are the only differences between Multiple Allocation p-HLP
and p-Hub Median LP with Fixed Costs:
• As the number of hubs to locate is not pre-specified so it must be considered both as a decision
variable and as a part of the solution.
• A fixed cost related with links is incorporated into the model.
Single Allocation p-Hub Location Problem
Unlike p-Hub Median Location model that allows the assignment of spokes to multiple hubs, sometimes
we need to have each of the spoke nodes assigned to a single hub. Most of the assumptions of this model
are similar to those of the median P-hub model except the following two features:
• Each non-hub node is assigned to only one hub.
• All of the outputs are binary variables (0–1).
Minimum Value Flow on Any Spoke/Hub Connection Problem
Instead of arguing that each non-hub node should be allocated to a single hub node, we may contend
that the flow between connections must be greater than or equal to some minimum flow threshold value.
The assumptions of this model are similar to those of the median P-hub model except that there is a
minimum flow for each spoke/hub connection.
87
Hub Location Allocation Problems and Solution Algorithms
Table 5. Mathematical models of the most commonly used hub models
Problem Type Model
Single
HLP hij : Amount of flow
between the ith node and
jth node.
Cij : Cost amount
between the ith node and
jth node.
Y
node i is allocated
to hub j
otherwise
ij =
1
0
min h C C Y Y
ik ij jk ij kj
kji
+
( )
∑∑∑
Subject to
Yjj
j
∑=1
Y Y
ij jj
− ≤ 0 ∀i j,
Yij ∈
{ }
0 1, ∀i j,
p-Hub LP hij : Amount of flow
between the ith node and
jth node.
Cij : Cost amount
between the ith node and
jth node.
α
: Discount factor
denoting economies of
scale for transferring
between hub nodes
0 1≤ <
( )
α
Y
node i is allocated
to hub j
otherwise
ij =
1
0
min C Y h
C Y h
ik ik ij
jki
kj ik ji
Jik
∑∑∑
∑∑∑
+
+αα h C Y Y
ij km ik jm
mkji ∑∑∑∑
Subject to
Yij
j
∑=1 ∀i
Y P
jj
j
∑
=
Y Y
ij jj
− ≤ 0
∀
i j,
yij ∈
{ }
0 1,
∀
i j,
p-Hub
Median
LP
Cij
km : The
transportation cost
between start node i, end
node j, the kth hub nodes
and the mth node.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥;The non-hub nodes are
allowed to be allocated to several hub nodes.
min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
Subject to
X P
k
k
=
∑ Zij
km
mk ∑∑ =1
∀i j,
Z X
ij
km
m
≤
∀
i j k m, , ,
Z X
ij
km
k
≤
∀
i j k m, , ,
Zij
km ≥0
∀
i j k m, , ,
Xk∈
{ }
0 1,
∀
k
continued on following page
88
Hub Location Allocation Problems and Solution Algorithms
Single
HLP hij : Amount of flow
between the ith node and
jth node.
Cij : Cost amount
between the ith node and
jth node.
Y
node i is allocated
to hub j
otherwise
ij =
1
0
min h C C Y Y
ik ij jk ij kj
kji
+
( )
∑∑∑
Subject to
Yjj
j
∑
=1
Y Y
ij jj
− ≤ 0 ∀i j,
Yij ∈
{ }
0 1,
∀
i j,
p-Hub LP hij : Amount of flow
between the ith node and
jth node.
Cij : Cost amount
between the ith node and
jth node.
α
: Discount factor
denoting economies of
scale for transferring
between hub nodes
0 1≤ <
( )
α
Y
node i is allocated
to hub j
otherwise
ij =
1
0
min C Y h
C Y h
ik ik ij
jki
kj ik ji
Jik
∑∑∑
∑∑∑
+
+αα h C Y Y
ij km ik jm
mkji ∑∑∑∑
Subject to
Yij
j
∑=1
∀i
Y P
jj
j
∑=
Y Y
ij jj
− ≤ 0 ∀i j,
yij ∈
{ }
0 1,
∀
i j,
p-Hub
Median
LP
Cij
km
: The
transportation cost
between start node i, end
node j, the kth hub nodes
and the mth node.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥;The non-hub nodes are
allowed to be allocated to several hub nodes.
min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
Subject to
X P
k
k
=
∑
Zij
km
mk ∑∑ =1
∀
i j,
Z X
ij
km
m
≤ ∀i j k m, , ,
Z X
ij
km
k
≤
∀
i j k m, , ,
Zij
km ≥0
∀
i j k m, , ,
Xk∈
{ }
0 1, ∀k
Table 5. Continued
89
Hub Location Allocation Problems and Solution Algorithms
P-HLP With Limited Capacity
If the incoming or outgoing flows in a network are limited to a fixed and certain value that is considered
as hub capacity, we are faced with a p-HLP with Limited Capacity. The problem is formulated in a similar
way to a general p-hub median location problem plus an extra capacity constraint.
Table 6. Mathematical models of the most commonly used hub models (continues)
Problem
Type
Model
Inputs
Model
Outputs
Mathematical
Model
p-Hub Median LP with
Fixed Costs gik : The fixed cost of
connecting non-hub node i
to a hub facility located at
node k.
Wik : The binary variable
denoting selection of link
(i, k) if it is equal to one.
min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
Subject to
X P
k
k
=
∑
Zij
km
mk ∑∑ =1
∀
i j,
g W
ik ik
ki ∑∑
Z X
ij
km
m
≤ ∀i j k m, , ,
Z X
ij
km
k
≤ ∀i j k m, , ,
Zij
km ≥0 ∀i j k m, , ,
Xk∈
{ }
0 1, ∀k
Single Allocation
p-Hub LP Cij
km : The transportation
cost between start node i,
end node j, the kth hub
nodes and the mth node.
Yik : The ith none-hub
node is assigned to the kth
hub node.
Min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
X P
k
k
∑=
Zij
km
mk ∑∑ =1 ∀i j,
Y X
ik k
≤ ∀i k,
Yik
k
∑=1 ∀i
Y Y Z
ik jm ij
km
+ + ≥2 0 ∀i j k m, , ,
Xk=0 1, ∀k
Yik =0 1,
∀
i k,
Zij
km =0 1,
∀
i j k m, , ,
90
Hub Location Allocation Problems and Solution Algorithms
P-Hub Center Location Problem
Aiming to minimize the maximum travel time (or cost) between any origin–destination pair, the p-hub
center problem is to locate p hubs and to allocate non-hub nodes to hub nodes. One may use this approach
for decomposable or sensitive goods in a hub system. The characteristics of this model are similar to
those of the median p-hub model except that some decision variables are relaxed (not necessary binary)
and the objective function is MiniMax.
Hub Set Covering Location Problem
The number of hubs in hub set covering problem is not determined. Hence, the objective function in
this problem minimizes the establish cost of hubs. This issue is defined when all origin–destination are
fully covered. Origin–destination can be allocated to one hub or more than one hub (Karimi & Bashiri,
2011). The assumptions of this model are similar to median P-hub model except that:
• The number of hubs are as decision variables and are not known before solving.
• A fixed cost of hub location is incorporated in the model.
Hub Maximal Covering Location Problem
If the time (cost or distance) to cover all origin–destination pairs is greater than the available time (budget
or distance), we can solve it by using a hub maximal covering problem, i.e., maximize the demand covered
with a given number of hub facilities. The hub maximal covering objective function is maximizing the
total flow between all origin–destination nodes which are allocated to the structured network (Karimi &
Bashiri, 2011). The assumptions of this model are similar to those of the median P-hub model except that:
The number of hubs is known.
The fixed cost of hub location does not matter of consideration in the model.
Multi-Objective p-Hub Location Problem
Costa et al. (2008) proposed a multi-objective HLP in which the first objective minimizes the total trans-
portation cost, while the second one minimizes the maximum time that the hub nodes take to process
the flow (i.e., minimizes the maximum service time of the hub nodes) (Farahani et al., 2013):
• In a similar manner to the p-HLP, each non-hub node in this problem is assigned to only one hub
node.
• In this model, the criteria are Mini-Sum and Mini-Max.
• The solution domain is the nodes of the network.
• There is a full connection between hubs.
• Every non-hub node is linked to a single hub ultimately.
• The number of hubs to locate is pre-defined, and one or two hub nodes have to be traversed for
traveling between two non-hub nodes.
91
Hub Location Allocation Problems and Solution Algorithms
Table 7. Mathematical models of the most commonly used hub models (continues)
Problem
Type
Model
Inputs
Model
Outputs
Mathematical
Model
Minimum Value
Flow on any
Spoke/Hub
Connection
Problem
Lik : The
minimum flow
between spoke i
and hub k
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥;The non-hub nodes are
allowed to be allocated to several hub
nodes.
min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
Subject to
X P
k
k
=
∑
Zij
km
mk ∑∑ =1
∀
i j,
Z X
ij
km
m
≤ ∀i j k m, , ,
Z X
ij
km
k
≤ ∀i j k m, , ,
Zij
km ≥0 ∀i j k m, , ,
Xk∈
{ }
0 1, ∀k
Y Y Z
ik jm ij
km
+ − ≥2 0 ∀i j k m, , ,
h Z
h Z L Y i k
ij ij
km
jm
pi pi
sk
sp
ik ik
∑∑
∑∑
+ ≥ ∀ ,
Capacity
Limitation of
HLP
θk: The
capacity of a hub
at the kth
candidate.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥; The non-hub nodes are
allowed to be allocated to several hub
nodes.
min C h Z
ij
km
ij ij
km
mkji ∑∑∑∑
Subject to
X P
k
k
=
∑
Zij
km
mk ∑∑ =1 ∀i j,
Z X
ij
km
m
≤ ∀i j k m, , ,
Z X
ij
km
k
≤ ∀i j k m, , ,
Zij
km ≥0
∀
i j k m, , ,
Xk∈
{ }
0 1,
∀
k
h Z
h Z X k
ij ij
km
jim
ij ij
sk
jis
k k
∑∑∑
∑∑∑
+ ≤ ∀θ
continued on following page
92
Hub Location Allocation Problems and Solution Algorithms
• There are not any fixed costs for hub nodes.
• The capacities of hubs are not limited.
• The decision variables are binary.
Continuous p-HLP
In a hub location problem, sometimes we have to consider a continuous domain of solution that is not
like a discrete set of nodes on a graph, yet like a plane or a sphere. The specifications of this model are:
• The criterion is Mini-Sum.
• The solution domain is a plane and is continuous.
• The hub nodes are completely linked, and every non-hub node is linked to only one hub facility.
• For traveling between two non-hub nodes, the number of hub nodes to locate is primarily specified
as one or two.
• The fixed cost of opening hub facilities is not considered.
• The capacities of hubs are not limited.
• The decision variables are binary.
Uncapacitated Single Allocation p-Hub Median Problem (USApHMP)
USApHMP belongs to the class of NP-hard problems. Even when the set of hubs is given, the assignment
sub-problem of optimal allocation of non-hub nodes to hubs is also NP-hard (R.F. Love, J.G. Moris,
1988). The objective is to minimize the overall flow cost in a network under the following assumptions:
Problem
Type
Model
Inputs
Model
Outputs
Mathematical
Model
p-Hub Center
LP Cij
km
: The
transportation
cost between
start node i, end
node j, the kth
hub nodes and
the mth node.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥;The non-hub nodes are
allowed to be allocated to several hub
nodes.
Xk∈
{ }
0 1, ∀k
min max
, , ,i j k m ij
km
ij ij
km
C h Z
{ }
Subject to
X p
k
k
∑=
Zij
km
mk ∑∑ =1
∀
i j,
Z X
ij
km
k
≤ ∀i j k m, , ,
Z X
ij
km
m
≤
∀
i j k m, , ,
Zij
km ≥0 ∀i j k m, , ,
Table 7. Continued
93
Hub Location Allocation Problems and Solution Algorithms
Table 8. Mathematical models of the most commonly used hub models (continues)
Problem
Type
Model
Inputs
Model
Outputs
Mathematical
Model
Hub Set
Covering
LP
FK: The
fixed cost in
the kth
candidate
node.
Vij
km : The
node hubs of
m and k
cover the
origin-
destination
of I and j.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥;The non-hub nodes are
allowed to be allocated to several hub
nodes.
min F X
K K
k
∑
Subject to
V Z
ij
km
ij
km
mk ∑∑ ≥1
∀
i j,
Z X
ij
km
k
≤
∀
i j k m, , ,
Z X
ij
km
m
≤ ∀i j k m, , ,
Zij
km ≥0 ∀k
Xk∈
{ }
0 1, ∀k
Hub
Maximal
Covering
LP
hij : The
demand flow
from origin i
to
destination j.
X
a hub is located
at node j
otherwise
j=
1
0
Z o
ij
km ≥; The non-hub nodes are
allowed to be allocated to several hub
nodes.
max h V Z
ij ij
km
ij
km
mkji ∑∑∑∑
Subject to
X p
k
k
∑=
Zij
km
mk ∑∑ =1 ∀i j,
Z X
ij
km
k
≤
∀
i j k m, , ,
Z X
ij
km
m
≤
∀
i j k m, , ,
Zij
km ≥0
∀
k
Xk∈
{ }
0 1,
∀
k
Multi
Objective
p-HLP
Tk: The
time unites
that the hub
node k takes
to process
one unit of
flow.
Y
node i is allocated
to hub j
otherwise
ij =
1
0
min h Y Y C C C
ij ik jm ik km jm
mkji
+ +
( )
∑∑∑∑ α
min max
kk ij ik
ji
ji jm ik
ji
T h Y h Y Y
∑∑ ∑∑
+
Subject to
Yik
k
∑=1 ∀i
Y p
kk
k
∑=
Y Y
ik kk
− ≤ 0 ∀i k,
Y
ik ∈
{ }
0 1,
∀
i k,
94
Hub Location Allocation Problems and Solution Algorithms
• The number of hubs to be located is predetermined (p).
• There are no capacities or fixed costs involved.
• Each origin/destination node is assigned to a single hub.
• Direct transportation between non-hub nodes is not allowed.
The p-hub median formulation can sometimes lead to unsatisfactory results, for example, when the
worst origin–destination distance (cost) is important. Difficulties of this kind can be avoided by using
the p-hub center formulation, which minimizes the maximum distance between origin–destination pairs
(Stanimirović, 2010).
As presented in Tables 1, 2 and 3 based on the relevant literature, there have been proposed a wide
range of different solution algorithms to solve various types of HLPs. Even though integer program-
ming optimization approaches are applied to solve small hub problems, larger instances of HLPs need
to be solved by heuristic or meta-heuristic procedures. As a matter of fact, while large-size instances can
be dealt with specialized exact methods, development of meta-heuristics has helped many real-world
applications, in which optimal/near-optimal solutions can even be obtained in less computational time
(Gelareh & Nickel, 2011).
In the past, few solving methods were proposed for hub location problems in which the number of
hubs is a decision variable, and the fixed cost of establishing a hub is considered. Nevertheless, with
the growth of meta-heuristic methods, the number of ways to solve such problems has been increased
(Farahani et al., 2009). Although literature review of this episode is including most of the hub location-
allocation problems and solution methods, the subsequent cases are, however, the major ones: a mixed
method, Simulated Annealing and Tabu Search provided by Chen (2007); Genetic Algorithm method
presented by Topcouglu et al. (2005); a Bi-criteria Integer Linear Programming to solve the capacitated
single allocation hub location problem proposed by Costa et al. (2008); a heuristic algorithm based on
Lagrangian Relaxation and Local Search to solve P-hub location median single allocation problems
by Yaman (2008); and a Genetic Algorithm to solve UPHLP-S problem by Damgacioglu et al. (2015).
PROPOSED HEURISTICS ALGORITHMS
In order to solve a general hub set covering location problem; that is fully introduced in Table 8, an
application of hub location problem in manufacturing is addressed in this section by using two distinct
metaheuristic algorithms, namely Differential Evolution (DE) and Particle Swarm Optimization (PSO)
with different number of nodes, transportation charges, and fixed costs.
Representation of Solution
The aim of this problem is to find the hub locations and the allocation of demand nodes to hubs. For
presenting the given network, some discrete nodes are used here. The solutions are presented as a ma-
trix. Each column shows a node in the network, in which its elements value explains the number of the
hub and the nodes, which are allocated to them. Furthermore, when the value of each element on the
entire column is equal to zero, the node is considered as a demand node. If the value of the item that is
in the same row and the same column is equal to one, that node is considered as a hub and the rest of
95
Hub Location Allocation Problems and Solution Algorithms
the elements show the demand nodes allocated to it. For example, a sample solution is obtained as in
the following matrix (EghbaliZarch et al. 2013):
S=
00000010
00000010
00000001
00000001
00000100
00000100
00000010
000000001
In this solution, the matrix S is giving the set of connections between eight nodes. As it is clear,
nodes 6, 7 and 8 are hubs. Also, node five is allocated to hub 6, nodes 1 and two are allocated to hub 7
and nodes 3 and four are assigned to hub 8.
The example mentioned above shows how a hub location problem should be solved. This solution
is, however, inadequate for solving a very large set of nodes where some advanced solution techniques
are needed. In such cases, metaheuristic algorithms should be used to receive an acceptable result in the
reasonable amount of time. For that purpose, here we are going to introduce two popular metaheuristic
algorithms namely Differential Evolution Algorithm and Particle Swarm Optimization Algorithm.
Differential Evolution Algorithm
Differential Evolution is a Stochastic Direct Search as well as a Global Optimization algorithm that is
an instance of an Evolutionary Algorithm from the field of Evolutionary Computation. It is related to
sibling Evolutionary Algorithms such as the Genetic Algorithm, Evolutionary Programming, and Evo-
lution Strategies. Differential Evolution (DE) was introduced by Ken Price and Rainer Storn in a series
of papers that followed in quick succession (Storn, 1996; 2008, Storn & Price, 1996; 1997; 1995). DE
is a population-based stochastic method for global optimization.
There are three kinds of vector in the literature of DE algorithm: a parent vector from the current
generation that is called target vector; a mutant vector obtained through the differential mutation operation
that is known as donor vector; and finally, an offspring formed by recombining the donor with the target
vector namely trial vector (Das & Suganthan, 2011). The primary stages and algorithm steps of DE is
illustrated in Flowchart 1. We are also giving details and descriptions of DE stages regarding algorithm
installation. Moreover, the main stages and pseudocodes of DE algorithm are shown in Flowchart 1.
The original version of DE can be defined by the following descriptions:
Stage 1: The population.
96
Hub Location Allocation Problems and Solution Algorithms
Pop X i N g y
X x j
x y i y Pop
i y j i y
, , max
, , ,
, , , , , , , , ,
,
=
( )
= − =
=
( )
=
0 1 1 0 1… …
00 1 1, , , .…Dim − (1)
where NPop denotes the number of population vectors, y defines the generation counter, and Dim
the dimensionality, i.e. the number of parameters.
Stage 2: The initialization of the population via x.
x U L L
j i j j j, , .
0= −
( )
+α (2)
For creating the initial population, the j numbers of hubs are firstly located randomly.
The dimensional initialization vectors, L and U, indicate the lower and upper bounds of the param-
eter vectors xi y,. The random number generator α returns a uniformly distributed random number from
within the range [0,1), i.e., 0 1≤ <α. The subscript, j, indicates that a new random hub is generated
for each parameter.
Stage 3: The perturbation of a base vector by using a difference vector based mutation to generate a
mutation vector mi y,.
M g X X
i y i y r y r y, , , ,
.= + −
( )
β
1 2
(3)
The difference vector indices, and r
1, r2 are randomly selected once per base vector. The setting
g x
i y r,=
0
, g defines what is often called classic DE where the base vector is also a randomly chosen
population vector. The random indexes r0, r
1 and r2 should be mutually exclusive. There are also vari-
ants of perturbations that are different to Eq. (3) and some of them will be described later. For example,
setting the base vector to the current best vector or a linear combination of various vectors is also popu-
lar.
Stage 4: Diversity Enhancement.
The classic variant of diversity enhancement is a crossover, which mixes parameters of the mutation
vector mi y, and the so-called target vector xi y, in order to generate the trial vector vi y,. The most com-
mon form of crossover is uniform and is defined as:
V v
m if Cr
x otherwise
i y j i y
j i y
j i y
, , ,
, ,
, ,
= = ≤
α (4)
97
Hub Location Allocation Problems and Solution Algorithms
Stage 5: Selection.
DE uses simple one-to-one survivor selection where the trial vector vi y, competes against the target
vector xi y,. The vector with the lowest objective function value survives into the next generation y+1.
XV if f V f X
X otherwise
i y
i y i y i y
i y
,
, , ,
,
+=
( )
≤
( )
1 (5)
Flowchart 1: DE Algorithm
1. Begin
2. Iteration=0
3. Create a random initial population
4. for each node i = 1 to Npop-1 do
5. for j=1 to Dim do
6. Equation 2.
7. end for
8. end for
9. Evaluate Objective Function for each node of population
10. for i=1 to Npop-1 do
11. Fitness function
12. end for
13. Test vector generation
14. for Iteration=1 to MaxIteration do
15. for i=1 to Npop do
16. Select randomly r1, r2, r3
∈
[1,Npop], r1
≠
r2
≠
r3
≠
i
17. Mutation and Crossover Process
18. jrand=
α
19. for j=1 to Dim do
20. if(
α
<Cr or j==jrand) then
21. Equation 3.
22. else
23. m(i,j)=x(i,j)
24. end if
25. end for
26. end for
27. Selection
28. if f(m) ≤f(x(i)) then
29. x(i)=m(i,j)
30. else
31. x(i)= x(i) of prior Iteration
32. end if
98
Hub Location Allocation Problems and Solution Algorithms
33. end for
34. end for
35. End
In order to prevent the case v x
i y i y, ,
= at least one component is taken from the mutation vector vi y,, a
detail that is not expressed in Eq. (4). Other variants of crossover are described by Storn (2008).
Particle Swarm Optimization
Inspired by the studies in neurosciences, cognitive psychology, social ethology and behavioral sciences,
the concept of swarm intelligence (SI) was introduced in the domain of computing and artificial intel-
ligence in 1989 as an innovative collective and distributed intelligent paradigm for solving problems,
mostly in the domain of optimization, without centralized control or the provision of a global model
(Marini & Walczak, 2015).
Particle Swarm Optimization (PSO) algorithm is a bio-inspired metaheuristic that is inspired by
the swarm behavior in nature such as fish schooling or flocking of birds. A PSO consists of a pool of
particles where a particle position in n-space represents a solution to a given problem. As each particle
moves around in n-space, it remembers its best position so far, and it is also aware of the global best
position found by all particles (Bailey et al., 2013). For the sake of simplification, the basic stages and
algorithm steps of PSO are depicted in Flowchart 2.
Flowchart 2: PSO Algorithm
1. for each particle i = 1, ..., N do
2. Initialize the particle’s position with a uniformly distributed
random vector: xi ~ U(L, U)
3. Initialize the particle’s best known position to its initial po-
sition: b(i) ← x(i)
4. if f(b[i]) < f(g) then
5. update the swarm’s best-known position: g ← b(i)
6. Initialize the particle’s velocity: v(i) ~ U(-|U-L|, |U-L|)
7. while a termination criterion is not met do:
8. for each particle i = 1, ..., N do
9. for each dimension dim = 1, ..., n do
10. Pick random numbers: r(i), r(g) ~ U(0,1)
11. Update the particle’s velocity: v(id) ← ω v(id) + φ(p)
r(p) [p(id) –x(id)] + φ(g) r(g)
12. [g(d)-
x(id))
13. Update the particle’s position: x(i) ← x(i) + v(i)
14. if f(x(i)) < f(p(i)) then
15. Update the particle’s best known position: p(i) ← x(i)
99
Hub Location Allocation Problems and Solution Algorithms
16. if f(p(i)) < f(g) then
17. Update the swarm’s best-known position: g ← p(i)
To give more detailed information about stages of PSO, let f x x
( )
∀ ∈
{ }
| be the objective function
that must be minimized. The function takes a candidate solution as an argument in the form of a vector
of real numbers and produces a real number as output which indicates the objective function value of
the given candidate solution. The goal is to find a solution g for which, f (g) ≤ f (d) for all d in the search-
space, which would mean g is the global minimum. Let N be the number of particles in the swarm, each
having a position x i N
i∈ ∈| in the search space and a velocity v i N
i∈ ∈|. Let bi be the best-
known position of particle i and let g be the best-known position of the entire swarm. The values L and
U are respectively the lower and upper boundaries of the search-space. The termination criterion can be
a number of iterations performed, or a solution with adequate objective function value is found. The
parameters ω, φp, and φg are selected by the practitioner and control the behavior and efficacy of the
PSO method (Clerc, 2012).
FINDINGS AND DISCUSSION
The DE and PSO were run on a predesigned data sets with fixed model parameters and different numbers
of nodes, and the solution qualities were compared with each other. Before designing the DE and PSO,
we first report the results with preset parameters for both DE and PSO algorithms. Table 9, describes
empirically established parameter settings.
Regarding obtaining optimal solution, we applied an integer linear programming (ILP) algorithm
for discrete space that was introduced and used by Marin (2005). We observed that the ILP algorithm
was unable to manage more than 50 nodes as it just found optimum values of an objective function for
the data sets with 10, 20, 30, 40 and 50 nodes. For the sake of comparison, we have used these optimum
values for determining the best gaps of both DE and PSO using the formulation (5). The best result is
obtained after the accomplishment of each metaheuristic at the end of 500th iterations or in 3600 seconds.
The results summary of the analyses of a classic hub set covering location problem is given in Table 10.
All analyses are fulfilled by a computer with 4770 (i7) CPU using MATLAB software.
Table 9. Established parameter settings for DE and PSO in Matla
%% DE Parameters
MaxIt=500; % Maximum Number of Iterations
NPop=200; % Population Size
beta_min=0.8; % Lower Bound of Scaling Factor
beta_max=1.5; % Upper Bound of Scaling Factor
pCR=0.2; % Crossover Probability
%% PSO Parameters
MaxIt=500; % Maximum Number of Iterations
NPop=200; % Population Size (Swarm Size)
w=0.4; % Inertia Weight
wdamp=1; % Inertia Weight Damping Ratio
c1=0.3; % Personal(Cognitive) Learning Coefficient
c2=0.9; % Global Learning Coefficient
% Velocity Limits
VelMax=0.1*(VarMax-VarMin);
VelMin=-VelMax;
100
Hub Location Allocation Problems and Solution Algorithms
best cost of obtained by the metaheurestic - optimum value
bbest result
×100 (5)
Figure 4 and Figure 5 are graphically showing performances of both algorithms aiming to solve hub
location-allocation problem for instance with 50 nodes. Proposed algorithms tried to minimize objec-
tive functions by assigning nodes to hubs. It is clear that both algorithms have found 11 hubs but with
extremely different best costs. Moreover, the operation elapsed time for PSO and DE algorithms are
behaving completely different.
In this certain example, the best cost and total CPU time to get the best cost by DE are respectively,
75663767 and 593 seconds, whereas these values for PSO are 71354969 and 537. Regarding preciseness
and accuracy, gaps of PSO algorithm are lesser than DE.
In general, as it is depicted in Table 10, considering the first 10, 20 and 30 nodes, performances of
DE are considerably better in terms of the best cost and CPU time.
With booming the numbers of nodes, the performance of PSO is getting better and better. As it is
clear, the obtained best costs by PSO are very close to the optimum values obtained by an ILP tool. As
illustrated in Table 10, ILP could not perform to catch optimum values of the experiments with more
than 50 nodes and the results are not available (N.A). One might also consider the comparisons between
the best costs and the total elapsed times for each experiment by using DE and PSO. Since minimization
is the objective function of hub set covering location problem, so the best costs of PSO are significantly
better than those of DE.
Gaps are metaheuristics variations that result from the difference between the best cost of metaheuristic
algorithm and the optimum value. There are many possible reasons that cause variations, yet one of the
most important factors is distribution type in the mutation phase for DE and the initialization stage for
PSO. We used Uniform distribution for both metaheuristic algorithms as one can get better results with
lower variations using different probabilistic distributions.
Figure 4. Hub location allocation for an instance with 50 nodes using DE (left) and PSO (right) meta-
heuristics
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Figure 5. Best cost (total cost) of objective function using DE (up) and PSO (down) metaheuristics
Table 10. Analysis results for 10 experiments with different numbers of sets
Hub Best Cost CPU Total (Sec) Optimum
Value
Lower Bound (%)
or Gap (%)
No. Node DE PSO DE PSO DE PSO DE PSO
110 1 1 2106312 2106312 65 116 2106312 0 0
220 3 3 10813414 10813414 105 161 10813414 0 0
330 4 4 20981139 20981139 208 235 20981139 0 0
440 5 8 49605297 48284833 453 331 48165245 2.9 0.24
550 11 11 75663767 71354969 593 537 71168452 5.9 0.26
660 14 15 95287163 92245522 856 616 N/A N/A N/A
770 14 16 167840678 144327597 845 763 N/A N/A N/A
880 14 16 204174556 182860891 1250 925 N/A N/A N/A
990 15 15 254737053 226993504 1531 1126 N/A N/A N/A
10 100 27 27 343083222 303949181 2709 1382 N/A N/A N/A
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FUTURE RESEARCH DIRECTIONS
With the development of technology, the logistics sector has found faster and more cost-effective ways
of shipping freight. The hub-and-spoke model was born from industry’s efforts to develop more efficient
networks. The functionality of the hubs and spokes differ according to the industry. Today, using solution
results of hub location-allocation problems can lead managers and researchers to new horizons. We faced
with the concept of hub location allocation not just as a transportation problem optimizer, but also as a
new paradigm in industry. As managerial applications to move into the future, one can consider some
challenges, as dependent variables or constraints, for hub location-allocation problem:
• Differences in Local Regulatory Environment, Culture and Time Zones: With centers in vari-
ous parts of the world, one company may face differences in regulatory environment, cultural pref-
erences, time zone, etc. which may pose a challenge. Due to differences in time zones and cultures,
communication between centers can be impeded. It is important to understand the ‘softer’ aspects
of the location and plan accordingly to ensure smooth program management and open bilateral
communication.
• Integration of Resource Pools to Provide Seamless Services: The solution results of a hub
location problem can help to synchronize operations between the Hubs and Spokes and tightly
integrate their resource pools to provide a seamless service offering to clients. Failure to do so may
defeat the entire purpose of running this business model.
• Addressing Tax Issues: Different countries have different tax structures; understanding and com-
plying with them can be an arduous task for companies. Thus, firms need to formulate a plan and
seek expert advice to optimize tax treatment, minimize uncontrolled tax risks, and ensure ongoing
compliance with laws.
CONCLUSION
Even though integer programming optimization approaches are applied to solve small hub problems,
larger instances of HLPs need to be solved by heuristic procedures or meta-heuristic procedures as one
can clearly observe that it is tough to solve such problems effectively with the conventional approaches.
While large-size instances can be dealt with specific exact methods, development of meta-heuristics
has, in fact, helped many real-world applications, in which optimal/near-optimal solutions can even be
obtained in less computational time. With a glance at the literature of HLP, we can see that the trend of
heuristic and metaheuristic algorithms in HLPs is similar to the exact solution algorithms. Two points
are clear in the related literature: first, the majority of studies have dealt with the uncapacitated cases of
HLPs. Second, most of the capacitated HLPs have been investigated in recent years.
In this chapter, we tried to introduce a hub location problem based on the already existing models
proposed by the studies. Thus, we reviewed over 40 papers between the years 2000 and the first half of
2016 dealing with or related to hub location problem. In this regard, we mentioned applications, apprehen-
sions and the definitions of the hub location problem. Moreover, we explained the basic classifications
and fundamental mathematical models and formulations for different variants of hub location problem.
Then, we delivered a categorization of solution approaches and algorithms including exact methods as
well as heuristics and meta-heuristics. Afterward, we reviewed an application of HLP and analyzed the
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shapes of different experimental designs with a different number of nodes. We finally introduced, in
Table 8, DE and PSO metaheuristics for a general hub set covering location problem that is one of the
popular hub location-allocation problems. We tested their performance comparing them with an exact
solution technique and also with each other. The comparisons showed that for booming the engineering
competence and control, we need to handle more advanced metaheuristics. We have concluded that we
need to use up-to-date and valid parameters of algorithms in order to improve the efficacy and influence
of metaheuristics.
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KEY TERMS AND DEFINITIONS
Database Transaction: A unit of work performed within a database management system (Sarvari
et al., 2016).
Hub: The facilities that are servicing many origin-destination pairs as transformation and tradeoff
nodes.
Hybrid Manufacturing System: A system is one in which functional layout (generally job-shop
type) and cells (manufacturing or assembly) coexist.
Metaheuristics: In computer science and mathematical optimization, a metaheuristic is a higher-
level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm)
that may provide a sufficiently good solution to an optimization problem, especially with incomplete or
imperfect information or limited computation capacity (Bianchi et al., 2009).
Quadratic Integer Programming: A special case of the None Linear Programming, when the ob-
jective function is in quadratic form and the constraints are linear.
Spoke: A none hub node that is connecting to a hub.
Uncapacitated Single Allocation p-Hub Median Problems (USApHMP): In the classical USApHMP,
transportation costs are modeled as linear functions of the transport volume, where a fixed discount fac-
tor on hub-hub connections is introduced to simulate economies of scale.
