aristotle university of thessaloniki department of...
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ARISTOTLE UNIVERSITY OF THESSALONIKI
DEPARTMENT OF ECONOMICS
MASTER PROGRAM IN
LOGISTICS AND SUPPLY CHAIN MANAGEMENT
THE ROLE OF OPTIMAL SELECTION OF
FACILITIES IN A SUPPLY CHAIN NETWORK
By
Kanellas Konstantinos (R.N: 29)
Supervisor: Diamadidis Alexandros
Master Thesis submitted to the Department of Economics of Aristotle University of Thessaloniki
in partial of fulfillment of the requirements for the degree of Master of Science in Logistics and
Supply Chain Management
Thessaloniki, Greece, September 2018
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Acknowledgements
I would first like to thank my master thesis supervisor professor Mr. Diamadidis
Alexandros of the Department of Economics at Aristotle University of Thessaloniki.
Whenever I had questions or troubles about my research or writing, Mr. Diamadidis
was supporting me. Furthermore, he steered me in the right directions whenever he
thought I needed.
I would also like to thank the owner of Vlachodimos supermarket, Mr. Konstantinos
Vlachodimos who collaborated with me in order to conclude in results in the case study
part of this assignment.
Finally, I must express my gratitude to my parents for providing me support and
continuous encouragement throughout the year of study of this assignment. This
accomplishment would not have been possible without them. Thank you.
Author
Kanellas Konstantinos
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Abstract
The role of optimal selection of facilities in supply chain network
The current assignment investigates the role of facility location positioning in a
supply chain network. The location decisions are very important in the design of that
network because they require high expenses and cannot easily be changed. Particularly,
this assignment focuses on the retail location part of a supply network because in the
case study part the opening of new supermarket store in central Greece is investigated.
In the solution approach of the specific problem, different viewpoints are used. Such
viewpoints encompass simple but effective methods (Weighted Factor Rating Method,
Load Distance technique), a process that can be adjusted to multi-criteria problems
(Analytic Hierarchy Process ) as well as a prototype construction of a gravity model
(Huff model). The methods converge in two choices and it is up to decision maker’s
judgement which approach to follow in order to satisfy its requirements. In conclusion,
the results may propose different choices, but as far as gravity modeling is concerned,
it is worth of future research in order to be improved.
Keywords: facility location problem, retail location, Analytic Hierarchy Process, Huff
model
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Table of Contents
Abstract ..................................................................................................................................... iii
List of Tables ............................................................................................................................ vii
List of Figures ........................................................................................................................... ix
List of Images ............................................................................................................................ xi
Introduction ................................................................................................................................ 1
PART 1: LITERATURE REVIEW............................................................................................ 9
Chapter 1: Research of the facility location problem ................................................................. 9
1.1 Brief history of the major contributions to the facility location theory ............................ 9
1.2 Classification of the facility location models .................................................................13
1.2.1 Francis and White (1974) ........................................................................................13
1.2.2 Brandeau and Chiu (1989).......................................................................................15
1.2.3 Daskin (1995) ..........................................................................................................17
1.2.4 ReVelle, Eiselt and Daskin (2008) ..........................................................................17
1.2.5 Snyder (2010) ..........................................................................................................18
1.2.6 Eiselt and Marianov (2011) .....................................................................................20
1.3 The significant role of distance measurement in location theory ...................................22
1.3.1 Euclidean Distance ..................................................................................................23
1.3.2 Squared Euclidean distance .....................................................................................23
1.3.3 Rectilinear Distance ................................................................................................23
1.3.4 Aisle Distance .........................................................................................................24
1.3.5 Lp Norm Distance ...................................................................................................25
1.3.6 Shortest path ............................................................................................................25
1.3.7 Great Circle .............................................................................................................25
Chapter 2: Basic Facility Location Problems ...........................................................................27
2.1 Minisum Problem on the plane.......................................................................................27
2.2 Minisum Problems on the network.................................................................................29
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2.2.1 P- median problem ..................................................................................................29
2.2.2 Fixed charge facility location problem (UFLP) ......................................................33
2.2.3 Capacitated Facility Location Problem (CFLP) ......................................................37
2.3 Minimax Problem ...........................................................................................................38
2.4 Covering Problems .........................................................................................................41
2.4.1 Set Covering Location Problem (SCLP) .................................................................42
2.4.2 Maximal Covering Location Problem (MCLP).......................................................44
Chapter 3: Other Facility Location Problems ...........................................................................47
3.1 Competitive location problem (CLP) .............................................................................47
3.1.1 Major advances in the competitive location theory .................................................47
3.1.2 Maximum Capture - “Sphere Of Influence” Location Problem (MAXCAP) .........51
3.1.3 Other important concepts ........................................................................................53
3.1.4 Gravity Theory .........................................................................................................54
3.2 Hub Location Problem (HLP) ........................................................................................60
3.3 Undesirable Location Problem (ULP) ............................................................................61
3.4 Location Problem under Uncertainty (LPU) ..................................................................63
3.5 Location Routing Problem (LRP) ..................................................................................64
3.6 Location Inventory Problem (LIP) .................................................................................66
3.7 Generalizations – Extensions of the main concepts .......................................................67
Chapter 4: Solution techniques, methods and algorithms in Facility Location Problem .........69
4.1 Solution approaches in location problems ......................................................................69
4.2 Heuristics ........................................................................................................................70
4.3 Metaheuristics ................................................................................................................71
4.3.1 Genetic Algorithm ...................................................................................................71
4.3.2 Tabu Search .............................................................................................................72
4.3.3 Simulated Annealing ...............................................................................................73
4.4 Exact Methods ................................................................................................................74
4.5 Other Approaches ...........................................................................................................75
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4.5.1 Weighted Factor Rating Method .............................................................................75
4.5.2 Load Distance Technique ........................................................................................76
4.5.3 Center of Gravity .....................................................................................................76
4.5.4 Analytic Hierarchy Process (AHP) .........................................................................77
4.5.5 Geographic Information Systems (GIS) contribution to Location Analysis ...........86
PART 2: CASE STUDY ..........................................................................................................89
Chapter 5: Presentation of Vlachodimos company ..................................................................89
5.1 Vlachodimos supermarket image ...................................................................................89
5.2 Vlachodimos supermarket supply chain .........................................................................91
5.2.1 Vlachodimos Warehouse and Transportation of merchandise ................................91
5.2.2 Vlachodimos Supermarket Store .............................................................................93
Chapter 6: Implementation of models and techniques .............................................................95
6.1 Weighted Factor Rating Method and Facility Location Problem (FLP) ........................95
6.2 Load Distance Technique and Facility Location Problem (FLP) ...................................96
6.3 Analytic Hierarchy Process (AHP) and Facility Location Problem (FLP) ....................99
6.4 Huff model and Facility Location Problem (FLP) .......................................................107
Chapter 7: Results and Future Reasearch ...............................................................................125
Conclusion ..............................................................................................................................131
References ..............................................................................................................................133
APPENDICES ........................................................................................................................145
A) AHP ...............................................................................................................................145
B) Huff Model ....................................................................................................................155
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List of Tables
Table 1.1: Classification criteria..........................................................................17
Table 4.1: Comparison scale for the importance of factors..................................80
Table 6.1: Values of factors for the candidate locations.......................................96
Table 6.2: Total weight scores for the candidate locations...................................96
Table 6.3: Coordinates of candidate locations.....................................................98
Table 6.4: Estimated annual pallets.....................................................................98
Table 6.5: Calculated distances between candidate locations and warehouse......98
Table 6.6: Load-Distance resulted values............................................................98
Table 6.7: Abbreviations of the used factors......................................................101
Table 6.8: Prioritization of factors.....................................................................102
Table 6.9: Consistency Ratio (CR) result...........................................................103
Table 6.10: MPUR prioritization on the candidate locations.............................103
Table 6.11: EOFA prioritization on the candidate locations..............................103
Table 6.12: ATTC prioritization on the candidate locations..............................104
Table 6.13: ARPS prioritization on the candidate locations...............................104
Table 6.14: ASOH prioritization on the candidate locations..............................104
Table 6.15: PURP prioritization on the candidate locations...............................104
Table 6.16: INUN prioritization on the candidate locations...............................104
Table 6.17: COBE prioritization on the candidate locations..............................105
Table 6.18: COMN prioritization on the candidate locations.............................105
Table 6.19 : Consistency Ratio (CR’) result of MPUR......................................105
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Table 6.20 : Consistency Ratio (CR’’) result of EOFA......................................105
Table 6.21 : Consistency Ratio (CR’’’) result of ATTC....................................106
Table 6.22: Overall priority ranking for the candidate locations........................106
Table 6.23: Scenarios of different size of stores.................................................108
Table 6.24: Customers’ patronage possibility and Expected Consumers in
Trikala...............................................................................................................120
Table 6.25: Customers’ patronage possibility and Expected Consumers in
Karditsa.............................................................................................................121
Table 6.26: Customers’ patronage possibility and Expected Consumers in
Kalabaka...........................................................................................................121
Table 6.27: Customers’ patronage possibility and Expected Consumers in
Trikala...............................................................................................................122
Table 6.28: Customers’ patronage possibility and Expected Consumers in
Karditsa.............................................................................................................122
Table 6.29: Customers’ patronage possibility and Expected Consumers in
Kalabaka ...........................................................................................................122
Table 6.30: Overall final results.........................................................................125
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List of Figures
Figure A: Supply Chain Network..........................................................................2
Figure 1.1: Classification of facility location models..........................................15
Figure 1.2: Presentation of a continuous model structure....................................19
Figure 1.3: Presentation of a network model structure.........................................19
Figure 1.4: Presentation of a discrete model structure.........................................20
Figure 1.5: Depiction of Minisum problem.........................................................20
Figure 1.6: Depiction of Minimax problem.........................................................21
Figure 1.7: Depiction of Covering problem.........................................................22
Figure 1.8: Depiction of Euclidean distance........................................................23
Figure 1.9: Depiction of Rectilinear distance.......................................................24
Figure 1.10: Depiction of Aisle distance in plant layout......................................24
Figure 1.11: Depiction of Great Circle.................................................................26
Figure 3.1: (a) demand nodes (red circles) are assigned to one hub (blue squares),
(b) demand points are assigned to more than one hub..........................................61
Figure 4.1: Flow chart of general Tabu Search process........................................73
Figure 4.2: General graphical representation of an AHP structure.......................79
Figure 4.3: Pairwise comparisons of the selected factors example.......................80
Figure 4.4: Pairwise Comparison Matrix example...............................................81
Figure 4.5: Weighted Sum Vector.......................................................................82
Figure 4.6: Pairwise comparison matrix of each qualitative factor (example factor
1) for each alternative..........................................................................................83
Figure 4.7: Example of a matrix with overall depicted priorities.........................84
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Figure 4.8: AHP steps in the solution process of FLP..........................................85
Figure 4.9: Distinctive representation of GIS layers............................................87
Figure 6.1: Graphical representation of AHP-FLP............................................102
Figure 6.2: Diagrammatic representation of Expected Consumers in Walk Time
approach............................................................................................................121
Figure 6.3: Diagrammatic representation of Expected Consumers in Drive Time
approach............................................................................................................123
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List of Images
Image 1:Vlachodimos head offices.....................................................................90
Image 2:Vlachodimos warehouse........................................................................91
Image 3: Lifting forklift.......................................................................................92
Image 4: Supermarket store outside area.............................................................93
Image 5: Supermarket store greengrocer’s aisle..................................................94
Image 6: Supermarket store aisle.........................................................................94
Image 7: Input procedure of trade areas creation (GIS)......................................109
Image 8: Trikala trade area measured in walk time distance (GIS)....................110
Image 9: Trikala trade area measured in drive time distance (GIS)....................110
Image 10: Karditsa trade area measured in walk time distance (GIS)................111
Image 11: Karditsa trade area measured in drive time distance (GIS)................111
Image 12: Kalabaka trade area measured in walk time distance (GIS)...............112
Image 13: Kalabaka trade area measured in drive time distance (GIS)..............112
Image 14: Representation of candidate street-location (purple point),
Vlachodimos existing supermarket store (blue point), potential consumers (green
points) and existing competitors (red points) in the trade area of walk time
approach in city of Trikala.................................................................................114
Image 15: Representation of candidate street-location (purple point), potential
consumers (green points) and existing competitors (red points) in the trade area
of walk time approach in city of Karditsa..........................................................115
Image 16: Representation of candidate street-location (purple point), potential
consumers (green points) and existing competitors (red points) in the trade area
of walk time approach in city of Kalabaka.........................................................116
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Image 17: Representation of candidate street-location (purple point),
Vlachodimos existing supermarket store (blue point), potential consumers (green
points) and existing competitors (red points) in the trade area of drive time
approach in city of Trikala.................................................................................117
Image 18: Representation of candidate street-location (purple point), potential
consumers (green points) and existing competitors (red points) in the trade area
of drive time approach in city of Karditsa..........................................................118
Image 19: Representation of candidate street-location (purple point), potential
consumers (green points) and existing competitors (red points) in the trade area
of drive time approach in city of Kalabaka........................................................119
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Introduction
A supply chain is a network between a firm and its suppliers that produce and
distribute a product according to predefined specifications. The supply chain network
can be represented as a chain of Logistics functions. According to Council of Supply
Chain Management Professionals (2018) “Logistics is that part of supply chain
management that plans, implements, and controls the efficient, effective forward and
reverses flow and storage of goods, services and related information between the point
of origin and the point of consumption in order to meet customers' requirements”. That
functions occur in four main levels in such a network:
Level 1 - Production Center
At this level, vendors supply the plants with the appropriate raw materials in order to
produce goods.
Level 2 - Warehouse
When the production of goods is being finished, the final products are being transferred
to appropriate areas, the warehouses, where in accordance to strategic decisions for the
inventory management stay for long or short periods.
Level 3 - Distribution Center
At next level, the finished products are being carried to the distribution centers where
appropriate procedures occur in order to be delivered appropriately to the firm’s store
or warehouse before selling to the end-customer.
Level 4 – Store
In the final level, the products are being put to the selling areas inside the store in order
the end customer to buy them.
Across the whole chain, the distribution and delivery of products in the supply
chain occurs via different means of transportation or combination of them (i.e. train,
airplane, road)
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A typical representation where possible connections between collaborating parts
may occur is being depicted in the following Figure A.
Figure A: Supply Chain Network (Author, 2018)
According to the profile of the firm (e.g. supermarket, pharmacy, car seller), some
of the levels are being presented may not exist; some companies may outsource large
portions of their activities. (i.e. Level of outsourcing - collaborating firms that support
the supply procedure, 3PL companies). Rouse (2018) states that “A 3PL (third-party
logistics) is a provider of outsourced logistics services. Logistic services encompass
anything that involves management of the way resources are moved to the areas where
they are required”. For instance, in the case of the supermarket chain, the level of
production center probably does not exist while there are many external companies, the
3PL ones, that facilitate the flow of products from the start to the endpoint by providing
specialized services in order the chain to be optimized.
In general Supply Chain Management involves all the operations and decisions
need to be taken between the aforementioned levels in order firm to maximize its
profitability in terms of better-utilizing capacity of labor and equipment, time, facilities
-space, information, providing higher quality products and overall targeting to customer
satisfaction, retention and increasing its earnings.
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As it is mentioned before decisions play a significant role among counterparts.
The supply chain managers are being called to make these decisions that will affect the
company in daily basis or in a long-term horizon. These decisions fall into three
categories depending on their changeability (Riopel, Langevin and Campbell, 2005).
Additionally, it is necessary to mention that every decision that relies on these
categories are associated and affected to each other.
1. Strategic supply chain decisions
2. Planning network supply chain decisions
3. Operational supply chain decisions
1. Strategic supply chain decisions: Decisions pertaining to that category are
related to the high level of management, are long-term and address issues
concerning to interrelationship to overall firm’s mission, policy (e.g.
determination of vendors’ origin; which is company’s viewpoint to that topic;
local, national, international or mixed suppliers), ways to meeting customer’s
expectations and requirements, retrieving and management of resources (e.g.
human, capital), outsourcing of operations and competition activity.
2. Planning network supply chain decisions: Decisions referring to that category
have a direct correlation to the strategic ones. They have long-term impact on
the company, cannot be changed easily or can be changed at great expense and
are related to the optimization of the supply chain network in terms of facilities
and information. In detailed, decisions regarding facilities encompass subjects
about the number of new facilities, their ideal location, layout, capacity and role,
the potential use of existing ones, the distance to markets and vendors,
environmental issues for local communities and government incentives (i.e.
decrease of taxation in case of big investment that will create jobs).
Furthermore, decisions with respect to information involve issues about the flow
and sharing of information across the supply chain. Specifically, they are related
to the way that parties communicate the information, its centralized or
distributed character (i.e. some companies chose to keep data in local databases
unlike to other ones that chose to have fully integrated databases accessible to
responsible employees), the implemented information technology - enterprise
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resource planning systems (ERP), the importance of e-commerce in a globalized
environment and the telecommunication technologies (i.e. barcode, RFID) that
facilitate procedure of tracking cargo.
3. Operational supply chain decisions: Although decisions of that category are
in independence to the two previous ones have some significant differences too.
First of all, they have short-term impact on the firm’s activities. Secondly, they
can easily be changed in order to better respond to customer’s and cooperating
parties’ needs, requirements and meet their expectations; reflect agility in order
to incorporate new technologies and innovations, aiming to optimization of the
supply chain. Finally, they are related to the changeable logistics functions and
more specifically to the demand forecasting, inventory management (i.e. pull
versus push strategies or mixed; in push strategies companies produce and store
enough amount of goods in order to meet forecasted demand while in pull ones,
companies make products that fulfill only customer’s orders), production
scheduling, procurement policies (i.e. contracts with suppliers) and order
processing. Moreover, they are related to the appropriate transportation modes
and policies (e.g. vehicle routing problem - VRP; it is scheduled the assignment
of vehicles to the facilities and the number of them), warehouse role and layout
(i.e. storage versus cross docking; cargo in cross-docking stays in the warehouse
area for a very short period and is transmitted to the next location as soon as
possible).
The purpose of this assignment is to deal with an issue that plays a significant
role in the supply chain management. Especially, the current dissertation researches the
crucial subject of Facility Location Problem due to its importance in the optimization
of a supply chain network and its wide range of application.
Nowadays, it is necessary to be given special focus on the facility location
problem because of the globalization phenomenon and the rapidly evolving technology
that generate opportunities in terms of reducing costs and maximization of profits.
Specifically, the modern business environment makes it necessary, firms to investigate
new broader customer areas, new ways to respond to geographical shifts in demand, to
follow up technological changes, to take benefit of governments incentives
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internationally and examine relocation of their facilities, to give attention in workforce
diversity and examine more efficient ways of resources supply (Situmeang, 2015).
Areas of application of facility location theory are being met in private, public
sector and in virtual cases (Snyder, 2010; Bruno, Genovese and Improta, 2014).
Distinctive examples are the following:
Private Sector
Fast food restaurants, Supermarkets, Factories, Warehouses, Distribution
Centers, Banks, Gas Stations, Banks (or public), Airline hubs etc.
Public Sector
Hospital, Blood banks (or private), Schools, Fire Stations, Electricity Stations,
Airports, Train stations, Bus Stops, Hazardous Waste Disposal or Obnoxious
Storage Areas (landfills, incinerators) etc.
Virtual Facilities
Satellite orbits, Wildlife Reserves, Bank Account Location, Platforms of Political
Parties, Product Positioning etc.
The study of location problem is a field of controversy and there are diverse
perspectives that emphasize in different elements. In general, Farahani and Hekmatfar
(2009) claim that facility location problems locate a set of facilities (resources) to
minimize the cost of satisfying some set of demands (of the customers) with respect to
some set of constraints. Additionally, Eiselt and Marianov (2011) present different
points of view in the definition of facility location problem. From the viewpoint of
mathematicians, the facility location theory examines the determination of new points
in a space given some metric tools in order to optimize a distance function between the
new and existing ones while geographers try to find the ideal number of facilities that
will serve existing points on the map. People with a business-economic orientation
incorporate financial elements and associate the facilities with potential customer areas.
Lastly, the computer scientists focus on the minimum number of facilities that will be
capable to serve a specific area of points; these new facilities are called centroids.
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Although, there are various aspects considering different elements as the most
important, the procedure of facility location decision and the factors that affect it, are
being set in common ground. Regarding the general process of the selected location
decision, it consists of the following steps:
Step 1
Determination of the facility purpose
Step 2
Development of location alternatives
Step 3
Identification of the significant factors that will affect the final decision
Step 4
Decision for make or buy or rent
Step 5
Evaluation of the alternatives
Step 6
Selection of the optimal location in accordance to judgements and preferences of
l location decision maker
Regarding the factors, these can be divided in two main categories. The first
category encompasses factors related to the geographical determination of the site (e.g.
which location worth of investment) while the second category refers to factors that
affect and are affected by the operation of the site. Some of the factors that belong to
the previous categories are presented below (Joanmaines, 2010; Xatzigiannis, 2013;
Situmeang, 2015):
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1st Category
Country - Region Factors: These factors play a major role especially in the case
of global supply chains. They include the government tax policy, incentives,
stability, intervention, exchange rate, prospects of economy growth, market size,
penetration, expansion, supporting technology and industries, number and power
of existing competitive companies (i.e. oligopoly, monopoly), culture and
lifestyle differences, attractiveness of area (i.e. suitability of land and climate).
2nd Category
Site factors: Factors of that category refer to the proximity to resources and
customers, the site size and its construction cost, labor costs, labor, resource and
energy availability, transportation infrastructure (i.e. fast-moving roads),
environmental regulations and impact issues, local community attitudes.
At this point, it is necessary to present the framework of the current dissertation
which is as follows.
In the first part, a wide area of location theory will be presented, while special
focus will be given in the competitive-retail location theory. In the second part, models
and techniques that are met in the literature review will be implemented in a real case
scenario, the opening of a new supermarket store in central Greece. Specifically, the
selected models and techniques will try to reflect the previous viewpoints as well as
subjectivity and objectivity in the facility decision.
In detailed, in Chapter 1, which is the first chapter of Literature Review part, the
evolution of the study of location theory throughout years, the classification of facility
location models, as well as different approaches in the measurement of distance will be
presented.
In Chapter 2, the most major classic facility location models, their formulation,
some of their extensions and variants as well as some of their solution approaches it
will be presented.
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In Chapter 3, the competitive location model and the importance of Gravity
Theory in locating retail stores will be described. Furthermore, other distinctive location
models will be presented in short.
In Chapter 4, the general attributes of the most well-known solution approaches
in location theory will be briefly described while emphasis will be given to the
description of the Analytic Hierarchy Process. This process is implemented in order to
derive a proposal result for the opening of a new retail store in the case study section.
In Chapter 5, which is the first chapter of Case Study part, Vlachodimos
supermarket company will be presented because of its contribution to the current
dissertation. The author collaborated with the owner of that company in order to fulfil
its master thesis.
In Chapter 6, the implemented solution methods for the investigation of the
opening of new Vlachodimos supermarket store, the methodoly approach and the
derived final results will be presented.
In Chapter 7, the previous results will be analyzed while the limitations and
assumptions of the implemented methods will be pointed out. Future directions for
research in the field of location theory and more specifically to retail location theory
will further be provided.
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PART 1: LITERATURE REVIEW
Chapter 1: Research of the facility location problem
The current chapter presents the historical evolution in the study of facility
location theory as well as the general categorization of the problems. Furthermore, it
recognizes the importance of the distance measurement in the location decision and
presents the different types of it.
1.1 Brief history of the major contributions to the facility location
theory
The origin of the facility location problem is dated back to the sixteenth century.
Specifically, Eiselt and Marianov (2011) state the first who dealt with this issue in its
initial form was a French mathematician, Pierre de Fermat (1601-1665). Fermat posed
the following challenge:
“Given three points in a plane, find a fourth point such that the sum
of its distances to the three given points is as small as possible”
The first solution to this problem is attributed probably to Italian scientist
Evangelista Torricelli (1598-1647) while author Melzak (1967) claims that Jesuit
Bonaventura (1647) was the first who provided an initial solution as well as another
formulation to Fermat’s challenge as it is cited in the article of Bruno, Genovese and
Improta (2014). Furthermore, they state that Thomas Simpson (1750) extended and
generalized the original Fermat problem by assigning possible weights in the
aforementionted three points. In their article, they also present the contribution of the
Swiss Jakob Steiner who examine the location problem from different perspective.
Jakob Steiner’s viewpoint:
“Given n points, find the minimum tree that connects them”
The appliance and importance of the basis of this procedure is important even
nowadays. For instance, following the general direction of this viewpoint, train or
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railway stations, telecommunications newtork are constructed in order people to be
served in the most efficient way. Many years later, this viewpoint took its final name
which is the Minimum Cost Spanning Tree Problem.
According to Eiselt and Marianov (2011) and Bruno, Genovese and Improta
(2014), the farmer and amateur economist Von Thünen (1826) was the first who
introduced cost elements in the general location theory. His theory was based on the
query which is the best location for cultivation of land around cities. The transportation
and land use cost play a major role in his theory about the final decision of the
agricultural land. The closest to the city the higher the land use expenses. The best
choice involves the ideal combination and offset of the previous costs in terms of
minimizing both of them while the result is concentric rings around cities.
Another contribution in the nineteenth century is that of Sylvester in 1857 (Eiselt
and Marianov, 2011) who posed another question, that is presented below:
“It is required to find the least circle which shall contain a given system of
points in a plane”
Later on, Sylvester in 1860 and Chrystal in 1885 gave an answer to the previous
question while in present this subject is referred as the One-Center Problem in the plane.
In the twentieth century, a dramatic evolution happens in the location theory. The
pioneer of this development was Alferd Weber (1909) as it was referred by Bruno,
Genovese and Improta (2014), who incorporated the Fermat format in industrial
applications. Specifically, he wanted to minimize the sum of weighted Euclidean
distances between a region of customers and a warehouse. Although Weber couldn’t
propose a solution to the problem, its contribution to the development of the field was
so crucial and important that the initial Fermat problem renamed and referred nowadays
as Fermat-Weber problem or simply Weber Problem. The solution was proposed in
final by Weiszfeld in 1937 who used partial derivatives in its approach.
Another significant contribution to the extension of the theory was that of an
American economist, Harold Hotelling (1929) which was the first that incorporated the
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element of competition in the uprising study of location problems. Hotelling
investigated the issue of two competing suppliers that provide the same product and
“fight” each other in order to gain bigger share of the demand or in other words to obtain
larger number of customers.
Other contributions of that period was that of Reilly (1931) who integrated the
Gravity Theory to the location one. Its work is referred to the literature as the Reilly
Gravity Law. The Reilly Law is the base of the later on evolution of the
subcategorization of facility location theory, the retail location theory.
Furthermore, in 1933, the German geographer Walter Christaller
(planningtank.com, 2016) founded the central place theory who supported the general
location theory in its further evolution.
The present-day theory origin is dated back in 1964 when Hakimi showed a graph
version of Weber problem, the nowadays called P-Median problem which is suitable in
transportation and telecommunication design network systems. Hakimi described the
Weber problem as a graph [i.e. G = (V, A) where V is the set of nodes and A is the set
of arcs that connect the nodes] and proved that the optimal solution can be found if the
facilites are located in the nodes. Although Hakimi used Weber’s objective function in
its computations, he evolved the model by replacing the Euclidean distances with the
shortest route between the nodes and by presenting a weighted version in which he
incorporated different demand values in the nodes.
He further generalized his viewpoint on the location theory by introducing in
1964 the P-Center problem which is applicable in different areas. Specifically, this
problem refers to situations that the aim is to locate a number of facilities on a graph so
as to minimize the maximum distances between the facilities and the assigned to them
customers; it is necessary all the customers to be served. Applications of this problem
can be found in the case of construction of emergency services buildings (i.e. fire
stations, hospitals) and schools.
In the same period, another version of Weber’s problem was introduced by
Cooper (1963) who presented the Location-Allocation problem in which facilities need
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to be located in such a manner that total network costs can be minimized and the
assignment of customers (i.e. demand nodes) to the facilities take into account capacity
constraints.
Furthermore, professor Huff (1963) presented his theory about the investigation
of retail locations. The basis of his theory was based on the Reilly Gravity Law but he
overcame some of its limitations. His model is referred in the location theory as the
Huff model and targets on the delineation of trading areas of retail facilities. The aim
of the model is to define the possibility of customer patronage to facilities by taking
into account the presence of competition and the distances among customer and all
facilities (i.e. under examination facility and its competitive ones).
In the following years, Toregas et al. (1971) proposed another well-known
location model, the Set Covering model. The purpose of this model is to identify the
minimum number of facilities that need to be located in an area in order to satisfy its
customers. This area involves all the covered from the facility customers. An extension
of the aforementioned problem is the Maximal Covering Location problem which was
described by Church and Revelle (1974), aiming in the maximization of covered
customers in an area where the settlement of the number of the new facilities is standard.
Applications of that model can be found in the establishment of emergency buildings
(i.e. hospital, police and fire station etc.).
Another concept in location theory is presented by Hosseini and Esfahani, (2009)
the Undesirable one. While the most prevalent goal of location models is the
minimization of the distance between demand area and facility, this new type aims in
the maximization of that distance. In essence, the quality of people living in the previous
area is increasing as long as the distance away from the facility (i.e. landfill, nuclear
plant etc.) is increasing.
In the coming years, the research of location theory adjusted to the needs of the
era. Specifically, the new requirements in the global transportation, storage of cargo
provoked the rise of new location models like the Hub one by O’Kelly (1987) or the
combination of the studied facility location theory with theory referring to inventory,
routing policies.
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Although the impact of routing management to the facility location was
acknowledged by many professors [Boventer (1961), Maranzana (1965), Webb (1968),
Lawrence and Pengilly (1969), Higgins (1972), Christofides and Eilon (1969)] only
recently the two theories were combined and considered as one [ Jacobsen and Madsen
(1978), Or and Pierskalla (1979), Laporte and Norbert (1981) as cited by Hassanzadeh
et al. (2009)].
In the same attitude, Baumol and Wolfe (1958), Daskin, Snyder and Berger
(2005), perceived the role of inventory management as major one in the definition of
distribution costs associated with warehouse’s location but only recently it was
succeeded the incorporation of inventory policies in the facility location decisions
(Shen, Coullard and Daskin, 2003).
Challenges of our time stimulated professors to develop location models in order
to be agile in the case of intentional strike against a supply chain network and diminish
the negative impact to the facilities by fortifying them in the most appropriate way.
1.2 Classification of the facility location models
The number and the complexity of the location theory makes it necessary to
present a general framework of the literature; a categorization of the problems
according to their specific elements that make them suitable for the different situations.
The following classifications are the most well-known in the location science.
1.2.1 Francis and White (1974)
According to Francis and White (1974) the problems in the location theory are
classified according to the following elements: a) new facilities characteristics, b)
existing facility locations, c) solution space, d) objective, e) distance measure, f)
new/existing facility interaction.
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Page | 15
Figure 1.1: Classification of facility location models, Francis and White (1974)
1.2.2 Brandeau and Chiu (1989)
Furthermore, Brandeau and Chiu (1989) presented a classification proposal in
which emphasis is being given to the criteria of objective, decision variable(s) and
system parameters that are showed below.
I. Objective
• Optimizing:
➢ Minimize average travel time/average cost
➢ Maximize net income
➢ Minimize average response time
➢ Minimize maximum travel time/cost
➢ Maximize minimum travel time/cost
➢ Maximize average travel time/cost
➢ Minimize server cost subject to a minimum service constraint
➢ Optimize a distance-dependent utility function
➢ Other
• Non-optimizing
➢ Type of location dependence of objective function:
➢ Server-demand point distances
➢ Weighted vs. unweighted
➢ Some vs. all demand points
➢ Routed vs. closest
➢ Inter-server distances
➢ Absolute server location
➢ Server-distribution facility distances
➢ Distribution facility-demand point distances
➢ Other
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II. Decision variables
• Server/facility location
• Service area/dispatch priorities
• Number of servers and/or service facilities
• Server volume/capacity
• Type of goods produced by each server (in a multi-commodity
• situation)
• Routing/flows of server or goods to demand points
• Queue capacity
• Other
III. System parameters
• Topological structure:
➢ Link vs. tree vs. network vs. plane vs. n-dimensional space
➢ Directed vs. undirected
• Travel metric:
➢ Network-constrained vs. rectilinear vs. Euclidean vs. block norm vs. round
norm vs. L, vs. other
• Travel time/cost:
➢ Deterministic vs. probabilistic
➢ Constrained vs. unconstrained
➢ Volume-dependent vs. nonvolume-dependent
• Demand:
➢ Continuous vs. discrete
➢ Deterministic vs. probabilistic
➢ Cost-Independent vs. cost-dependent
➢ Time-invariant vs. time-varying
➢ Number of servers
➢ Number of service facilities
➢ Number of commodities
• Server location:
➢ Constrained vs. unconstrained
➢ Finite vs. infinite number of potential locations
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1.2.3 Daskin (1995)
Moreover, Daskin (1995) in his work described a classification scheme according
to fourteen criteria as presented in table 1.1.
Table 1.1: Classification criteria of Daskin (1995)
1.2.4 ReVelle, Eiselt and Daskin (2008)
In addition to the previous works, ReVelle, Eiselt and Daskin (2008) divided the
discrete location theory in two categories:
• the median and plant location models
Target: minimization of the average demand-distance between the facility and
its assigned node
• center and covering models
Target: provision of a service to a demand area completely or partially by a
number of facilities
Models are classified further to the analytical, continuous, network and discrete
ones.
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Analytical models: These models are based on large simplifying assumptions.
Specifically, they consider that demands are uniformly distributed and assume that the
establishment of a facility has the same fixed cost in any position of the service area.
Furthermore, the total cost is commonly expressed as a function of the number of the
facilities. Although models of such kind provide valuable information, they fall short
in realistic applications.
Continuous models: These models claim that facilities can be placed anywhere
in the service area while the demands are usually set into the same bracket with the
discrete location ones. One distinctive continuous problem is the Weber one. Their
application is limited.
Network models: These models are being expressed as a structure of lines and
nodes. The demand values are usually adopted in the nodes. In some occasions, the
demand values can be adapted to the lines too; for instance in the establishment of
emergency highway services.
Discrete models: These models state that both demand values and facilities’
locations are discrete. They are also formulated as integer or mixed integer
programming problems. Moreover, they are characterized as NP-hard on general
networks and they are widely applicable in real situations.
1.2.5 Snyder (2010)
In a similar way to the classification of ReVelle, Eiselt and Daskin, Snyder
(2010) presented a topological delineation of the location models. The following figures
depict its viewpoint.
Continuous models: As it was mentioned before facilities can be located
anywhere in space. In addition, they are optimized in a non-linear way.
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Figure 1.2: Presentation of a continuous model structure (Snyder, 2010)
Network models: Although facilities can be located anywhere on the network,
the traveling is occurring only through arcs. They are formulated as integer
programming problems and are characterized sometimes by Hakimi property in which
the optimal location can be found at the nodes.
Figure 1.3: Presentation of a network model structure (Snyder, 2010)
Discrete models: Facilities can be located in predifined spots and as it was stated
before they are formulate in inter programming.
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Figure 1.4: Presentation of a discrete model structure (Snyder, 2010)
1.2.6 Eiselt and Marianov (2011)
According to Eiselt and Marianov (2011), the location problems can be
categorized in order to reflect their different situations, purposes and set of constraints.
They identify three types of problems: Minisum, Minimax and Covering ones.
Minisum problems: Problems pertaining to that category aim to minimize the
sum of distances between customer (demand point) and its closest facility. Barbati
(2013) adds that because of the distribution of demand points, as they are depicted in
figure 1.5, there is a penalty for a customer that its distance is importantly bigger from
the assigned facility
Figure 1.5: Depiction of Minisum problem (Barbati, 2013)
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Minimax problems: Problems referring to that category aim to minimize the
maximum distance between customer and its assigned facility. From figure 1.6 it can
be resulted that the location of facility presented in figure 1.5 moved towards the
customer with the longest distance. Although this movement decreased the previous
longest penalized distance, in final it contributes to the increase of the average distance
of the total network and the decrease of its efficiency (Barbati, 2013).
Figure 1.6: Depiction of Minimax problem (Barbati, 2013)
Covering problems: While the first two categories target in the minimization of
distances, covering problems do not consider the distance as the most determinant
factor in the computations. Distance is considered as a constraint and is being taken into
account only if it exceeds a predefined value D¯ (Figure 1.7). Covering model can be
described as a circle in which the center is the location of the facility and the limits of
its covered area, as depicted with red line in figure 1.7, distinguish which customer will
be served from the facility and which will not. The purpose of these problems as
Barbati, 2013 claims, is the maximization of covered customers or the minimization of
costs in order to capture/cover all demand points, inside and outside of circle.
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Figure 1.7: Depiction of Covering problem (Barbati, 2013)
1.3 The significant role of distance measurement in location theory
One of the major components in the selection of the right location-decision is the
measurement of the distance. In most cases, like in the afformentioned Minisum and
Minimax problems, the distance metric determines the computations in the objective
function of the problem while in the Covering problems is part of its constraints.
Eiselt, Marianov and Bhadury (2015) add that while the most models have “pull”
goals (i.e. customers desire the minimum distance from the facilities; pull facilities
towards to them) or “push” goals (i.e. customers desire the maximum distance from the
facilities; push facilities away from them), there are some like dispersion or defender
ones in which according to Daskin (2008) (cited by Eiselt, Marianov and Bhadury 2015)
the element of distance is irrelevant.
As it is stated in the work of Zarinbal (2009) for defining the distance type it
must be considered the characteristics of the problem and distance as well as the way it
will be used.
Due to the distance’s importance in location theory, it is necessary to mention the
most used distance functions in the literature:
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1.3.1 Euclidean Distance
According to Snyder (2010) the Euclidean distance is calculated with the
following formula, d(A,B) = √(𝑥1 − 𝑥2)2 + (𝑦1 − 𝑦2)2. The A (x1, y1) and B (x2, y2)
are two points in the plane while the d is the distance between them. In other words, d
can be described as a straight line that connects the two points. This type of distance is
commonly used in the continuous location problems. Drezner and Wesolowsky (2001)
(Zarinbal, 2009) claim that traveling by air or water can be representative examples of
the Euclidean scheme. Moreover, Nickel, (2008/ 2009) states that this type of distance
is usually applied to the planning of power supply lines or pipeline-systems.
Figure 1.8 : Depiction of Euclidean distance (Xatzigiannis, 2013)
1.3.2 Squared Euclidean distance
The formula of Squared Euclidean distance is d2(A,B) = (𝑥1 − 𝑥2)2 +
(𝑦1 − 𝑦2)2. Nevertheless, there are similarities between the previous two types, the
Squared is useful in the estimations where the distances are long. Specifically, an
example is the establishment of a fire station that not only require to minimize the sum
of time distances but also to provide the service to the citizen with the longest distance
too (Nickel, 2008/ 2009).
1.3.3 Rectilinear Distance
Snyder (2010) further presents the Rectilinear distance between two points A (x1,
y1) and B (x2, y2). D is computed through the type of d = |𝑥1 − 𝑥2| + |𝑦1 − 𝑦2|.
Rectilinear is referred differently in the literature with the name Manhattan distance or
Page | 24
Taxicab Norm distance because it can be simulated as the distance that a car would
travel in a city formulated as a set of square blocks. The traveling is allowed only in
vertical or horizontal directions (i.e. North-South, East-West). Analytical are the kind
of problems that mainly use Manhattan distance, without excluding its use in other
problems like Discrete ones (Zarinbal, 2009). Applications of this distance can be found
in in-house location planning like warehouses where the layout is rectangular or in
locating facilities in cities (Nickel, 2008/ 2009).
Figure 1.9 : Depiction of Rectilinear distance (Xatzigiannis, 2013)
1.3.4 Aisle Distance
On the contrary Zarinbal (2009) judges as not realistic the rectilinear distance
application to in-house procedures like material handling in plants. Instead of
Manhattan, she proposes the Aisle distance as more appropriate for such operations.
Aisle distance can contribute in the estimation of the optimal route of a picker in which
he can traverses in a one-way direction the aisle and pick the items by giving priority
to the ones found in aisle nearest to Input/output station followed by the next nearest
aisle.
Figure 1.10 : Depiction of Aisle distance in plant layout (Zarinbal, 2009)
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1.3.5 Lp Norm Distance
Lp norm distance is a generalization of the previous two most common distances,
the Euclidean (i.e. p=2) and the Manhattan (i.e. p=1) one. Its formula is
(|𝑥1 − 𝑥2|𝑝 + |𝑦1 − 𝑦2|𝑝)1 ∕ 𝑝. When p takes the value ∞, Lp distance type gives the
Chebyshev distance (Zarinbal, 2009). Its existence for p except from the values 1,2 is
far more for mathematical than practical purposes (Xatzigiannis, 2013).
1.3.6 Shortest path
One common distance metric in the literature that is applied in real cases is the
finding of the shortest route between two points by using algorithms in order to be
achieved the minimization of all distances in a network. An indicative algorithm of such
kind is the Dijkstra or a different version of it, the algorithm of Mitchell et al. which is
applied in continuous problems (presented in the work of Aronov et al. 2005, cited by
Zarinbal, 2009).
The shortest path, in reality, can be mentioned as the estimation of distances
through the highways by taking into account all the possible obstacles (e.g. traffic). This
type of distance is usually estimated by using the Google Map tool or GIS (i.e.
Geographic Information Systems) programmes and it is applicable in the most real
problems.
1.3.7 Great Circle
One of the major weaknesses of the distances presented in the 1.3.1-1.3.2 sections
above is that they consider the location map as a plane. Because of that, it is very
difficult to proceed in the identification of an optimal location when the points are
represented by coordinates; except from GIS and Google map tools, there is another
distance metric which supports the process of optimality. This distance is called the
Great Circle. It takes into account the curvity of the earth and assumes that the routing
(represented as red line in the figure 1.11) from one point q to another p is following
the direction of a big circle, the Great Circle. Its formula is 𝑑 = 2 ∗ 𝑅 ∗
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𝑎𝑟𝑐 𝑠𝑖𝑛 (√𝑠𝑖𝑛2 (𝛥𝜑
2) + 𝑐𝑜𝑠 𝜑1 ∗ 𝑐𝑜𝑠 𝜑2 ∗ 𝑠𝑖𝑛2 (
𝛥𝜆
2)
) where (φ1,λ1) and (φ2,λ2) are the
geographical latitude and longitude in radians of two points 1 and 2, and Δφ, Δλ are
their absolute differences in the surface of a sphere. R is earth’s radius and has
approximately 6371 km value (Xatzigiannis, 2013).
Figure 1.11: Depiction of Great Circle (Xatzigiannis, 2013)
Although the aforementioned distances are the most widely applicable to the
different theoretical problems or real situations, there are other distance functions too.
For instance, an extension of norm distance is the Block distance developed by Witzgall
et al. in 1964 and Ward and Wendell in 1985 which overcome barriers and restrictions.
Other distinctive distances are the Matrix, Mahalanobis and Hausdorff (for full review
see Zarinbal, 2009).
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Chapter 2: Basic Facility Location Problems
The current assignment will adopt Eiselt and Marianov’s classification scheme.
Specifically, it will present the most well-known facility location problems belonging
to the minisum, minimax and covering categories.
2.1 Minisum Problem on the plane
Weber Problem
As it was presented before the facility location theory has its origins in the
formulation of the French mathematician, Pierre de Fermat who stated the question
about the existence of the three points in the plane and the locating of a new fourth point
in a spot that will minimize the total sum of distances to the three previous ones. In a
similar manner, Weber generalized Fermat’s initial formulation and assigned weights
to the aforementioned points. As Eiselt and Marianov (2011) state, Weber presented
Fermat’s approach in more realistic cases by identifying one new point in the map, that
represent one plant, in order to be minimized the sum of distances, representing the
transportation cost, from vendors to consumers. They represent the known points which
reflect different values of demand, the called assigned weights.
Due to the fact that Fermat’s formulation has many applications and had been
studied by different researchers in the literature it can be additionally referred as the
Fermat-Torricelli problem, the Steiner problem, the Weber problem, the Steiner-Weber
problem, the One median problem [Eiselt and Marianov (2011) add that the demand
points are located on the nodes of a network], the single facility Euclidean Minisum
problem, the Minimum aggregate travel point [from the perspective of geographers and
economists (Plastria, 2011)], the bivariate median, the spatial median (Xatzigiannis,
2013).
The Weber problem can be represented in the reality as the situation where a new
warehouse (with coordinates X, Y) is needed to be opened in an area in order to serve
different amounts of products (the weights) to existing demand points (with coordinates
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ai, bi) in such a manner that the total transportation cost will be minimized (represented
as the sum of distances in correlation to the number of products). Its mathematical
formulation is depicted in the following format.
𝑀𝑖𝑛 𝑧(𝑋) = ∑ 𝑤𝑖 ⅆ(𝑋, 𝑃𝑖)
𝑛
𝑖=1
(2.1)
where d (𝑋, 𝑃𝑖) is the distance between the warehouse and the demand points i.
The most commonly used distance metric is the Euclidean one, d(𝑋, 𝑃𝑖) =
√(𝑋 − 𝑎𝑖)2 + (𝛶 − 𝑏𝑖)2. Weiszfeld (1937) was the first who discovered the practical
solution to Weber’s problem. Its solution is an iterative algorithm that takes as an initial
solution a point that minimizes the sum of the squares of the distances. On the other
hand, Chen (2011) acknowledges the efficiency of iterative methods but judging their
solution procedure as quite long. As a result, he proposes a noniterative solution in his
research article.
There are many extensions and different approaches in the investigation of the
initial Weber problem. A distinctive one is the work of Cooper (1963, 1964) in which
there are more than three demand points and more than one new under-investigated
facility while a heuristic solution is proposed. It is referred in the literature as the Multi-
Weber or can be met as the Location-Allocation problem. In this kind of problem, it is
necessary to investigate which facility will serve which demand point.
One different approach to the aforementioned problem is the anti-Weber problem
presented by Hansel et al. (1981) in the work of Melachrinoudis (2011) refering to
Undesirable facility location problems. Specifically, they investigated the locational
patterns of nuclear power plants in France and provided a solution by using the brand
and bound technique.
Another approach is referring to the capacitated multi-facility Weber problem
examined by Aras et al (2007) as is cited by Xatzigiannis (2013) and took into account
different distance metrics. They used except from Euclidean distance, the Squared
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Euclidean distance and the Lp Norm Distance. In the same vein, Plastria (2011) noted
that different types of metrics are commonly used in the investigation of Weber
problem.
In addition, Kara and Taner (2011) concluded that the single-hub location
problem seems to behave in a same manner to that of the classical Weber problem.
According to Plastria (2011), further extensions include the assignment of negative
weights (Drezner and Wesolowsky 1991), or considering the initial problem into
buildings (Arriola et al. 2005) or taking into account price decisions (Fernández et al.
2007).
2.2 Minisum Problems on the network
2.2.1 P- median problem
One of the major contributions in location science was that of Hakimi (1964,
1965) who introduced the weighted version of Weber problem in the network for p
facilities, the so-called P-median problem. Its objective is to identify p facilities in order
the demand weighted distance between demand node and its closest assigned facility to
be minimized. It is necessary, except from the identification of the locating facilities,
the proper allocation of p facilities on the demand nodes to be found.
Hakimi stated three properties. Specifically, he showed that P-median problem
is NP-hard on general graph/ network and proved that at least one solution always can
be found locating only on the nodes of the network. Furthermore, he tracked that the
demand weighted total cost (relating to the distance function) decreases when a new
facility is added to the total network. Daskin and Maass (2015) add that it is better or at
least the same to locate p+1 facilities from the start comparing to p ones.
One important assumption, according to Melo, Nickel and Gama (2006) is that
there is the same setup fixed cost among all the candidate locations for establishing the
p facilities. On the other hand, Jamshidi (2009) claims that there is no initial setup cost
and adds the following assumptions:
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• Cost and distance are linearly related
• The goods are located in the facilities
• The facility has infinite capacity
• The time horizon is infinite
• The problem is exogenous
• Facilities are of the same kind and stationary
• The node’s demand is constant and steady
• The problem is discrete
According to ReVelle and Swain (1970) the model can be formulated as follows:
Notation about the parametres, variables and subscripts:
i: customers’ index
j: potential new facilities’ index
n: total number of customers
m: total number of candidate sites
p: total number of established facilities
hi: weighted demand of node i (demand in terms of number of products or customers)
di,j: distance from customer i to potential facility j
1, if customer i is assigned to facility located in position j
Xi, j =
0, otherwise
1, if there is an open facility in position j
Yj =
0, otherwise
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Objective function
𝑀𝑖𝑛 ∑ ∑ ℎ𝑖𝑑𝑖𝑗𝑋𝑖𝑗𝑗
𝑖
(2.2)
Constraints
Subject to
∑ 𝑋𝑖𝑗𝑗
= 1, i = 1, 2,…n (2.3)
Xi,j ≤ Yj , i = 1, 2,…n, j = 1, 2,…m (2.4)
∑ 𝑌𝑗𝑗
= 𝑝 (2.5)
Xi,j, Yj ϵ {0,1}, i = 1, 2,…n, j = 1, 2,...m (2.6)
The objective function (2.2) aims to the minimization of the total transportation
cost from the customers i to the established facilities j. The constraint (2.3) shows that
every customer is assigned exactly to one facility while the constraint (2.4) states that
the demand in nodes can be satisfied only by the established facilities. The (2.5)
constrain forces the locating of p facilities in the network and the (2.6) states that
decision variables are integer and binary.
A contribution to the problem is proposed in the work of Marianov and Serra
(2011) by replacing the so-called Balinski constraint (2.4) with the following one:
∑ 𝑋𝑖𝑗
𝑛
𝑖=1≤ 𝑚𝑌𝑗 (2.7)
By introducing the new constraint (2.7) instead of (2.4) it is prevented to assign
customers if there is no facility in node j and the size of the problem is reduced.
Although this may be beneficial, it may lead, by relaxing (2.7) to produce fractional Xi,j.
On the other hand, Balinski constraint may lead to a greater number of constraints but
finally in the solution computations (linearly relaxing p-median problem) will tend to
produce integer Xi,j.
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Marianov and Serra provided further extensions in their work. They presented the
generalization of the P-median, as it is nowadays known as the p-hub median location
problem which is developed by Goldman (1969) and extended by Hakimi and
Maheshwari (1972) for multiple goods and intermediate medians. Furthermore, they
referred new elements like elasticity in demand and probablistic behaviour that were
incorporated in their models by Holmes et al. (1972) and Frank (1966) correspondingly
(for complete review about p-median and its evolution see Marianov and Serra, 2011).
There are many procedures and techniques that support the solution of the NP-
hard P-median problem. Except from their formats presented before, Revelle and Swain
main contribution was the formulation of the problem as an integer programming one
and the solution they provided by using binary variables and an exact method, the
Branch and Bound one.
Additionally, in their work Daskin and Maass (2015) present some well-known
heuristic algorithms like the Greedy adding or Myopic (construction algorithm) that
was described by Kuehn and Hamburger (1963) (Mladenovic et al., 2007), the
Neighbourhood (improvement algorithm) that was proposed by Maranzana (1964) and
the Exchange (improvement algorithm) that was introduced by Teitz and Bart (1968).
In the same vein, Mladenovic et al. (2007) add other heuristic algorithms like the
Alternate that was proposed by Maranzana (1964), the Stingy or differently called Drop
or Greedy-Drop that was introduced by Feldman et al. (1966) and the well-known
DUALOC that was described by Erlenkotter (1978).
Morever, Daskin and Maass (2015) present a list of metaheuristic algorithms used
by researchers in the solution procedure of P-median problems. Distinctive examples
are the work of Murray and Church (1996) who are referring to the implementation of
Simulated Annealing, the application of Tabu search by Rolland et al. (1996) and the
proposal of Genetic by Alp et al. (2003).
In accordance to the heuristic algorithms, Mladenovic et al. (2007) add some
metaheuristic algorithms to the previous ones. Specifically, in their work they describe
the Ant Colony Optimization or briefly called AOC (Colorni et al. 1991), the Scatter
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search (Glover et al., 2000), the Neural Networks (Merino and Perez, 2002) and the
Heuristic concentration (Rosing and ReVelle, 1997) as well as other ones (for complete
review see Mladenovic et al., 2007).
Finally, Daskin and Maass (2015) introduce the Lagrangian relaxation algorithm
and judge that it has two major advantages comparing to the previous ones. First of all,
Lagrangian relaxation produces upper and lower limits in the objective function and
secondly, it can be incorporated in the Branch and Bound procedure in order optimality
to be achieved.
2.2.2 Fixed charge facility location problem (UFLP)
P- median is one problem that considers the setup cost as equivalent among all
candidate locations (Melo, Nickel and Gama, 2006). When this is not happening, a new
term must be added in the objective function that will reflect the different fixed costs.
Such an addition will lead to one of the most noticeable location models pertaining to
the category of minisum problems, the Fixed charge facility location problem or
differently known as Simple facility location problem (SFLP) (Verter, 2011) or
differently met in literature as Uncapacitated facility location problem (UFLP).
This problem and its extension the Capacitated Facility Location Problem
(CFLP), are considered as very suitable in the design of supply chain network due to
the fact that they can comprise different types of facilities and multiple flows of
commodities (Verter, 2011).
This problem examines the establishment of an unspecified number of facilities
in an area where it is known the location of the customers (customers are discrete points
on the plane or on the road network) and their demand values. Furthermore, the unit
shipment cost between the candidate locations and customer areas (variable costs) as
well as the fixed cost of locating a new facility in each candidate location are known.
The problem’s purpose is to identify the facilities’ locations in order to minimize the
total cost (i.e. variable and fixed costs) while all customers’ demand values must be
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satisfied. An important element of the UFLP is that facilities have infinite capacity to
serve all customers (Fernández and Landete, 2015).
Notation about the parametres, variables and subscripts:
i: customer index
I: set of customers
j: location index
J: set of candidate locations
hi: demand value at customer area i, i ϵ I
fj: fixed cost of establishing facility in candidate location j, j ϵ J
cj: variable cost (transportation cost) for the shipment of unit between candidate facility
j and customer area i, i ϵ I and j ϵ J
1, if facility located in candidate position j
Xj =
0, otherwise
Yi,j = fraction of demand value of customer i that is being satisfied by facility j
According to Balinski (1965) the model can be formulated as follows:
Objective function:
𝑀𝑖𝑛 ∑ 𝑓𝑗𝑋𝑗𝑗∈𝐽
+ ∑ ∑ ℎ𝑖𝑐𝑖𝑗𝑌𝑖𝑗𝑖∈𝐼
𝑗∈𝐽
(2.8)
Constraints:
Subject to
∑ 𝑌𝑖𝑗𝑗𝜖𝐽
= 1 ∀𝑖 ∈ 𝐼 (2.9)
𝑌𝑖𝑗 ≤ 𝑋𝑗 ∀𝑖 ∈ 𝐼; ∀𝑗 ∈ 𝐽 (2.10)
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𝑋𝑗 ∈ {0,1} ∀𝑗 ∈ 𝐽 (2.11)
𝑌𝑖𝑗 ≥ 0 ∀𝑖 ∈ 𝐼; ∀𝑗 ∈ 𝐽 (2.12)
The objective function (2.8) minimizes the total cost, as it was presented before.
Constraint (2.9) forces every demand point to be assigned and (2.10) ascertains that any
assignment cannot occur until a facility is being established in the candidate location.
The (2.11) states that variable 𝑋𝑗 is binary and conclude in the opening of a facility or
not and the (2.12) shows the non-negativity of variable 𝑌𝑖𝑗.
The research of UFLP has extended in many ways. Some of its distinctive
elements are the single commodity, the one type of facility and the single period
approach. Such elements are being formulated and examined by different scope as well
as some others are added in order to reflect realism in the results.
Specifically, Soland (1974) tried to use cost elements in a way to represent the
reality. He correlated the fixed establishment cost to the size of the facility. Another
approach is referring to the first attempts of examination of multi-echelon UFLP which
is introduced in the work of Kaufman et al. (1977) and examining the simultaneous
locating of facilities and warehouses in the network.
The assumption of single period is formulated and investigated by Van Roy and
Erlenkotter (1982) who gave a dynamic aspect in the calculations, by using multiple
time periods while Hodder and Jucker (1985) and Hodder and Dincer (1986) followed
the evolution in the global manufacturing system and attempted to incorporate the
uncertainty in the model so as to reflect the diversity in the worldwide operations.
Finally, another approach includes the effort of Klincewicz and Luss (1987) who were
the first that introduced the element of multi-commodity in the computations.
As P-median, UFLP is an NP-hard problem and many solutions have been
proposed over the years. The most common are heuristics and metaheuristics as it is
presented below.
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The solution procedure usually includes heuristics which start with a feasible
solution and proceed with adding or removing facilities from the map until this
procedure does not improve the solution anymore. In this vein, Maranzana (1964)
proposed an neighborhood algorithm which is an improvement algorithm as its was
stated before in the case of P-median.
In the following years, UFLP was solved through a branch and bound algorithm
by Efroymson and Ray (1966). The proposed algorithm by Teitz and Bart (1968) that
was used for the P-median solution can be extended to UFLP as Daskin, Snyder and
Berger (2005) claimed. Moreover, the enumeration algorithm proposed by Spielberg
(1969) was another solution approach.
In 1977, Bilde and Kraup developed one dual based algorithm for the solution of
the problem and in 1978 Erlenkotter stated his own dual based algorithm, the well-
known DUALOC which is quite similar to that of Bilde and Kraup (Vedat Verter, 2011)
and according to Daskin, Snyder and Berger (2005) it was characterized as one of the
most efficient solution techniques for the problem in this part.
More recent references about solution techniques are the incorporation of
Lagrangian relaxation in the branch and bound technique by Daskin (1995) and the
application of metaheuristic algorithm tabu search by Al-Sultan and Al-Fawzan (1999)
in small and moderate-sized problems.
As Snyder (2010) claims, most of the models have a capacitated version.
Embedding the element of capacity, the model becomes more realistic. In essence, the
capacitated versions of the corresponding problems are extensions of them.
As a result of the aforementioned, the UFLP has a capacitated version, the CFLP
(Capacitated Facility Location Problem).
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2.2.3 Capacitated Facility Location Problem (CFLP)
As it was mentioned in the UFLP, one distinctive characteristic is that the
limitless of capacity in the facilities. In essence, this means that every facility can serve
any customer independently to its amount of demand. This fact isn’t common in real
cases; most of the times, the facility cannot handle the overall demand. As a result, more
facilities must be assigned to the aforementioned customer in order its demand to be
fully satisfied. The CFLP is the suitable problem to reflect the situation described
before.
CFLP is an extension of the UFLP. The formats are the same except from the
addition of a new constraint that will indicate the maximum demand that can be
assigned to a facility, the bj. The added constraint (2.13) that restrict the assigned
demand at facility j to the maximum demand bj is the following:
∑ ℎ𝑖𝑌𝑖𝑗𝐼∈𝐼
≤ 𝑏𝑗𝑋𝑗 ∀𝑗 ∈ 𝐽 (2.13)
One of the most important extensions of CFLP is the work of Geoffrion and
Graves (1974). Their extension included plants, distributions centers, customers and
multiple commodities. The purpose of the model was to minimize the total cost of the
network (the fixed cost of distribution centers and the variable cost of the distribution
center as well as the transportation cost from the plants through distribution centers to
the customers) and not violate constraints pertaining to capacities and shipment of
products among facilities.
Another approach was that of Daskin, Snyder and Berger (2005), who
simplified the under examination model by considering as fixed values the variables Xj
and tracked that the optimality for the CFLP can be achieved through the solution of a
transportation problem and the suitable transformation of it. Moreover, in the next year
Melo, Nickel and Gama (2006) proposed a model that could be useful in supply chain
decisions by taking into account a dynamic approach combined with multiple
commodities.
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In the same manner as in the case of UFLP, the CFLP can be solved by heuristics
and metaheuristics algorithms. Some attempts are referring in 1963, when Khuen and
Hamburger presented a heuristic algorithm and in 1977, when Akinc and Khumawala
established a branch and bound procedure. Furthermore, in the following years, Van
Roy (1986) and Beasly (1988) developed two distinctive algorithms that were
considered as the most effective in the solution procedure of CFLP, the cross-
decomposition algorithm and the Lagrangian-based approach correspondingly. In
recent years, one noticeable work is that of Ahuja et al. (2004) who used the multi-
exchange neighborhood search algorithm for the single source capacitated facility
location problem (SSCFLP).
2.3 Minimax Problem
P-center
One of the most important problems in the literature of location science is the P-
center problem. The purpose of this problem is to minimize the maximum distance
between any demand point and its closest facility with the requirement all the demand
to be satisfied. This class of problem can be seen as the opposite of P-median one, with
the minisum objective.
Furthermore, because of its minimax target, P-center can be considered as
suitable in cases of establishment emergency service locations (i.e. hospitals, fire
stations, police stations), (Tansel, 2011) as well as other public service locations (i.e.
parks, post boxes, bus stops, military facilities) without excluding their use in
identification of facilities pertaining to supply chain network (e.g. warehouses,
distribution centers) (Biazaran and SeyediNezhad, 2009)
Hakimi was the first in 1964, 1965 who introduced and formulated the P-center
problem. According to Hakimi (1964) if the number of p facilities that will be
established is equal to one, the problem is called Absolute center problem while if the
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0, otherwise
p facilities are located only on the nodes of the network the problem is called Vertex P-
center problem and takes into account the following assumptions.
• Facilities can be located only on the nodes of the network
• Facilities have unlimited capacities
• The number of established facilities are p
• Demand points are on the nodes of the network
• Demand nodes are unweighted
Notation about the parameters, variables and subscripts:
i: demand point index
I: set of demand points
j: location index
J: set of candidate locations
p: number of locations that will be established
dij: shortest path route between demand point i and candidate facility j
1, if facility is located in candidate position j
Xj =
0, otherwise
1, if demand point i is assigned to facility located in candidate position j
Yi,j =
D: the maximum distance between a demand point and its closest facility
Objective function:
Min D (2.14)
Constraints:
Subject to
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∑ 𝑌𝑖𝑗𝑗
= 1 ∀𝑖 ∈ 𝐼 (2.15)
∑ 𝑋𝑗𝑗
= 𝑝 (2.16)
𝑌𝑖𝑗 ≤ 𝑋𝑗 ∀𝑖 ∈ 𝐼, ∀𝑗 ∈ 𝐽 (2.17)
𝐷 ≥ ∑ 𝑑𝑖𝐽𝑗∈𝑗𝑌𝑖𝑗 ∀𝑖 ∈ 𝐼 (2.18)
𝑌𝑖𝑗, 𝑋𝑗 ∈ {0,1} ∀𝑖 ∈ 𝐼,∀𝑗 ∈ 𝐽 (2.19)
The objective function (2.14) and the constraint (2.18) force the minimization of
the distance between one demand point and its closest facility. The (2.15) constraint
states that demand of point i is assigned to facility j while the (2.16) constraint show
that the number of located facilities will be p. Finally, the (2.17) constraint ensures that
the demand points will be assigned to only firstly established facilities and (2.19)
describes the variables 𝑌𝑖𝑗 , 𝑋𝑗 as binary.
Daskin (1995) added that each demand point can have weights such as time per
unit distance, cost per unit distance as well as loss per unit distance. These weights
would indicate that the target of the problem will be the minimization of maximum
time, cost or loss. The model is quite similar to the previous one, except from the new
input hi that indicates the demand value at node i and the constraint (2.18) that is being
replaced by the following constraint (2.20)
D ≥ hi ∑ ⅆijYijj∈J
∀ⅈ ∈ I (2.20)
As it was stated before by Snyder (2010) most of the facility location problems
have a capacitated version. In the same vein, there is the Capacitated Vertex P-center
problem that was introduced by Ozsoy and Pinar (2006). This model is almost the same
to the Vertex P-center except from the limits in the capacities that are represented with
the input Qj and the adding of a new constraint (2.21) that describes the incorporation
of capacities’ limitations in the form of the original problem.
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∑ hji
⋅ Yij ≤ 𝑄𝑗 ∀j ∈ J (2.21)
Other contributions to the research of the current problem encompass the
description of a problem that uses positive or negatives weights on the demand nodes
(Burkard and Dollani, 2003) that correspond to facilities with pull objectives and push
objectives (i.e. obnoxious) and the Anti P-center problem (Klein and Kincaid, 1994)
which aims to maximize the minimum weighted (negative) distance between demand
node and its closest facility (i.e obnoxious).
Finally, important extensions are the Continuous P-center problem where the
demand points are continuously distributed on the general graph as well the Conditional
P-center problem where there are established q facilities and the issue is to identify new
p facilities in order to be minimized the maximum sum of distances between demand
nodes and their nearest facilities (i.e q + p ).
As far as concern the solution procedure Kariv and Hakimi (1979) described the
NP hardness of the problem on the general graph. Furthermore, a well-structured
presentation of exact and approximate solutions for the understudied problem is
presented in the work of Biazaran and SeyediNezhad (2009) and Calik, Labbé and
Yaman (2015).
2.4 Covering Problems
The third class of problems as it was proposed by Eiselt and Vladimir Marianov,
(2011) encompasses the Covering ones. In contrast to the Minisum problem’s target,
the understudied category of the problem seeks the “coverage” of demand points by a
minimum number of facilities. In essence, customers or demand points are covered (i.e.
being serviced) by the facility if the distance between customer and facility is less or
equal to a threshold distance (or travel time) that is called coverage distance or coverage
radius (Fallah, NaimiSadigh, and Aslanzadeh, 2009) and is notated as Dc. Eiselt and
Marianov (2011) add that this distance value is being exculed by the formulation of the
objective function and is being now part of the set of constraints.
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The Covering location problems consists of two categories, the Set Covering
location (SCLP) problem and the Maximal Covering location problem (MCLP).
According to Snyder (2011) the Covering problems are quite similar to the P-center
model in terms of coverage distances and their application. Specifically, Covering
problems are mainly used in the establishment of public location services (e.g. fire
stations, hospitals, design of defend network at war etc) and in analysis of market
analysis of markets (Storbeck 1988), archaeology (Bell and Church 1985), wildfire
reserve selection (Church et al. 1996) and other areas as García and Marín (2015) state
in their work.
2.4.1 Set Covering Location Problem (SCLP)
Hakimi (1965) was the first who introduced the SCLP but Toregas et al. (1971)
formulated the problem mathematically and correlated to the location theory. The SCLP
seeks to identify the number of facilities that is necessary to be located in an area where
the total cost must be minimized and the set of customers must be covered. Facilities
have unlimited capacities and the same fixed costs.
Notation about the parameters, variables and subscripts:
i: demand point index
j: location index
I: set of demand points
J: set of candidate locations
Vi: the set of candidate locations that can cover customer i
Vi= {𝑗 ∈ 𝑉: 𝑑𝑖𝑗 ≤ D𝑐}, every node on the V is both customer/demand point and
candidate location
dij: distance between demand point i and candidate facility j
Dc: coverage distance
1, if facility will be established in candidate position j
Xj =
0, otherwise
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Objective function:
𝑧 = ∑ 𝑋𝑗𝑗∈𝑉
(2.22)
Constraints:
Subject to
∑ 𝑋𝑗𝑗∈𝑉𝑖≥ 1 ∀𝑖 ∈ 𝑉 (2.23)
𝑋𝑗 ∈ {0,1} ∀𝑗 ∈ 𝐽 (2.24)
The objective function (2.22) calculates the minimum number of opened facilities
in order the demand to be fully satisfied. The contsraint (2.23) indicates that every
demand node needs to be covered and (2.24) shows that variable Xj is a binary one.
One generalization of SCLP is the Weighted version of it where the fixed opening
costs fj are different among facilities j. The purpose of this generalized model is to
establish these facilities in order to satisfy the demand and minimize the costs. The only
element that needs to be added is the fj in the objective function. As a result, the (2.22)
is formulated as follows:
𝑧 = ∑ 𝑓𝑗𝑋𝑗𝑗∈𝑉
(2.25)
An extension of SCLP is the Redundant Covering Location Problem (RCLP)
which was described by Daskin and Stern (1981) and its purpose is to identify the best
optimal solution to SCLP that maximizes the number of demand points that were
covered at least twice. A greater extension is the Backup Set Covering Problem (BSCP)
that encompasses a wide range of problems. The purpose of this problem is to guarantee
that demand points will be covered by at least two facilities in order to surpass
malfunctions in the facilities.
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There are many other extensions of the original covering problem that are
formulated in a way to serve different purposes. The most well-known is the Maximal
Covering Location Problem that is presented in the section 2.4.2.
Distinctive exact solution technique pertaining to SCLP, encompasses a two-step
procedure proposed by Toregas et al. (1971) which solve the linear programming
relaxation of the problem. Furthermore, applications of variant brand and bound
techniques and cutting planes methods are presented by García and Marín, (2015). They
further describe other solution approaches, the heuristics and metaheuristics, which are
considered as more suitable for the SCLP due to its NP-hardness characteristic.
2.4.2 Maximal Covering Location Problem (MCLP)
In some cases, the whole demand cannot be covered because of budget constraints
or other restrictions. In that case, the covering problem is being formulated in order to
cover the maximum demand value of total network with a limited number of facilities.
In the literature, this problem is referred as the Maximal Covering Location Problem
(MCLP) and can be considered as reverse to the SCLP.
One major difference between SCLP and MCLP, is that in MCLP a priority is
being given to the coverage of customers with the highest demand value while the
adding of facilities that can serve customers results in the increase of the total covered
demand value.
MCLP was introduced by Church kαι ReVelle (1974) and is being formulated as
follows:
The notation is the same of the SCLP except from the addition of the two
parameters that are presented below:
αij: demand at node i per unit time
p: maximum limit of opened facilities
Furthemore, as far as the decision variable yi is concerned, it is stated the
following:
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1, if customer at demand point i is covered by some facility
yi =
0, otherwise
Objective function
Max 𝑧 = ∑ 𝑎𝑖𝑦𝑖𝑖𝜖𝑉 (2.26)
Constraints
Subject to
∑ 𝑥𝑗𝑗∈𝑉𝑖
≥ 𝑦𝑖 ∀𝑖 ∈ 𝑉 (2.27)
∑ 𝑥𝑗𝑗∈𝑉
= 𝑝 (2.28)
𝑥𝑗 , 𝑦𝑖 ∈ {0,1} ∀𝑖, 𝑗 ∈ 𝑉 (2.29)
The objective function (2.26) calculates the maximum covered demand. The
constraint (2.28) guarantee that a customer will be covered by a facility that has been
established. The (2.28) constraint forces that the number of the opened facilities will be
p and the (2.29) indicates that variables 𝑥𝑗 , 𝑦𝑖 as binary ones.
As it was presented before for the case of SCLP, fixed costs can be incorporated
in the MCLP. The difference is that the fixed cost value is embedded in the set of
constraints (2.30) on contrary to the appearance of this value, in the SCLP, in the
objective function.
∑ 𝑓𝑗𝑥𝑗𝑗∈𝑉
≤ 𝐵 (2.30)
Where B value is a constraint put on the total fixed costs (Snyder, 2011). A quite
different extension of MCLP is the MCLP with Mandatory Closeness Constraints
which is described by Church and ReVelle (Snyder, 2011). Purpose of this problem is
to investigate a secondary coverage distance s (s ≥ Dc) in which all the customers must
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be covered. It holds that 𝑈𝑖 = {𝑗𝜖𝑉: 𝑑𝑖𝑗 ≤ 𝑠}. The problem is formulated by adding the
(2.31) constraint.
∑ 𝑥𝑗𝑗∈𝑈𝑖
≥ 1 (2.31)
The aforementioned model can be considered as a combination of SCLP and
MCLP because it takes into account the optimal solutions provided by SCLP and uses
tools in order to choose the one that maximizes the covered demand (MCLP
requirement) for dij ≤ Dc.
According to Megiddo et al. (1983) MCLP is a NP-hard problem and heuristics
and metaheuristics are recommended for its solution. Distinctive proposals of such
solution approaches are investigated by Church and ReVelle and include the Greedy
Adding algorithm or Greedy Adding with Substitution, the use of linear programming
relaxation and branch and bound technique and the inspection method. Another solution
approach is the Lagrangian relaxation which is proposed by Galvão and ReVelle
(1996).
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Chapter 3: Other Facility Location Problems
In a globalized environment where the national and the international competition
plays a significant role in the profitability of companies, it is very important to be
investigated location theories that take into account the competition in their
computations. Because of that, the current assignment will present features of
Competitive Location model, will introduce the Gravity Theory and examine its role,
especially in the research of retail location facilities. Furthermore, in this chapter other
well - known location problems will be presented briefly as well.
3.1 Competitive location problem (CLP)
3.1.1 Major advances in the competitive location theory
A location problem can be described as a competitive one if in the investigation
procedure the decision maker takes into account the existence (or in future) of at least
two other firms and the new under examination facility must compete with these ones
in order to acquire a market share (Plastria, 2001). Hotelling (1929) is considered as the
pioneer in the introduction of that problem.
He described a market that is linearly represented (where customers are uniformly
distributed. This linear market is referred in literature as main street or “two ice cream
vendors on the beach”. Considerations of the model are the entering firms which
compete in terms of location at first stage and price at second stage while offering the
same product to the customers which have fixed and inelastic demand. Furthermore,
competing firms use mill price (i.e. mill price is defined above) and customers will
choose the facility with the lowest mill price in combination with a linear transportation
cost.
Hotelling’s major contribution is the well-known Hotelling Law or principle of
the minimum differentiation. According to that law, Hotelling stated that there is an
equilibrium (i.e. stable solution, where firms have no incentives to move; known in the
literature as Nash equilibrium) in the aforementioned linear market if the two
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competiting firms locate at the center of the market but not to close. Many approaches
were stated after the seminar work of Hotelling but the most distinctive is that of
Aspremont et al. (1979) who invalidated his results referring to the aforementioned
equilibrium.
Although there are different approaches across years with respect to the
competitive location theory that incorporate different elements, there is a concluded
common base that provides a framework for the categorization of that theory. Important
elements pertaining to competitive location problem are the following (Karakitsiou,
2015):
1. Spatial representation: As in every location problem, in the competitive
one it must be specified its space framework. This means that the
investigation problem must belong to one of the well-known Continuous,
Discrete or Network space as well as to incorporate the appropriate
distance metrics every time. On contrary to Karakitsiou, Eiselt, Marianov
and Drezner (2015) state that the used space in competitive models is
more simplified (e.g. Hotelling’s linear market)
2. Competition nature: This element is referring to the actions or reactions
or no actions at all that are being by competitive firms. In essence, there
is a three categorization of competition pertaining to those actions.
Specifically, there is the Static competition, the Dynamic competition and
the Sequential one or differently named as competition with Foresight.
The Static competition refers to a firm that enters into a market in order
to locate p facilities and gain the maximum market share whereas she
knows in advance features of the existing operating firms. The existing
competition is not expected to react to that new entrance. The Dynamic
competition refers to responsive actions that are being done by already
competitive firms in the case of an entrance of a new adversary company.
Their reactions are being stimulated by their lost profits. Sequential
competition refers to a two-stage action procedure with two players, the
leader and the follower. The leader is the company which enters in a virgin
market and must act with foresight that probably other competitors will
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follow. The following competitors are the followers and their actions are
being adapted to the leader’s actions. In the literature this situation is being
referred as the Stackelberg game while its solution is called Stackelberg
equilibrium.
3. Number of new facilities and their nature: This element refers to the p
facilities that are to be established as in the case of classical problems.
When p>1 facilities are going to be located, there must be done an
allocation of the customers as well. Facilities can be further distinguished
by the pull or push objective or the mixed one as it was mentioned in the
previous chapter.
4. Pricing policy: The pricing policy is one of most major elements of
competitive location models. Hotelling in is seminar work concerned
about the prices as it was referred before. In general, the most distinctive
pricing policy is considered the mill pricing. In this type of pricing, it is
being set a facility price which is the same for all customers and these
customers are going to be charged the transportation cost between them
and the facility that will patronize (the cost of the selling product is
separate to the transportation cost). The mill pricing is not necessarily the
same among the facilities of one seller. In the special case this price is the
same, the pricing policy is called uniform pricing (Eiselt, Marianov and
Drezner, 2015). They further describe the zone pricing where facility
creates zones of the market and charges for the delivery of the product to
the customer accordingly to which zone he belongs as well as the extreme
cases of it, the delivered policy and the spatial discrimination policy. In
the delivered policy, facility charges fixed prices for the delivery of its
products to customers independently to their location while in the case of
spatial discrimination the charge is fully depended on the customer’s
location.
5. Customer behavior: This element is probably the most major one that
distinguishes competitive location problems from the classic ones. Some
of its features are described below. First of all, in classic models, location
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planers decide which how the customers will be allocated in the facilities
while from the point of view of competitive firms, customers decide the
facility that will patronize. Most authors embrace the aspect that
customers patronize the closest facilities (Eiselt, Marianov and Drezner,
2015) but in reality, customer behavior is a far more complicated issue
and many features of her must be investigated. For instance, facilities that
provide products with high demand elasticity (i.e. demand of product is
highly dependent on the changes on their prices) will probably lose some
of their closest customers in the case of a permanent increase in prices. As
a result of this, customers will investigate other competing firms to
acquire products.
Moreover, other elements of this patronization refer to the deterministic
or probabilistic nature of customer behavior. In essence, in the case of
deterministic choices, customers are fully attracted by p=1 facility and
will continue to patronize her until something changes from the supply
side. This facility “gets it all” (i.e. acquire all of the customer demand).
On the other hand, probabilistic choices indicate that customers are
attracted by p>1 facilities. In that case, there is a proportional
patronization in more than one facility. Customers decide to patronize
facilities in respect to the attractiveness or utility of that facilities as well
as the distance from them. This issue was researched by the Huff (1964)
and finds the possibility of each facility to attract customers. The work of
Huff belongs to the literature of Gravity Theory that is described in the
next section and it is connected to the location of retailers. Retail facilities
are competitive facilities that want to gain as much as possible number of
customers. Their action is stimulated by the acquiring of bigger market
shares. As a result of this, they pursue to have the biggest possibility to be
patronized.
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0, otherwise
Yi =
3.1.2 Maximum Capture - “Sphere Of Influence” Location Problem (MAXCAP)
One of the most prominent models that set the foundations for evolutions in
competitive location theory is that of MAXCAP that was introduced by the ReVelle
(1986). MAXCAP model is mainly used in the identification of the possible market
share capture for an entering firm as she gets in a spatial market where there are existing
competitive facilities. The entering firm investigates to locate p facilities in a network
where there are existing competitive facilities. Firms are competing in terms of distance
and the markets (i.e. customers) patronize their closest facilities. The objective of
MAXCAP is to maximize the demand capture of the market areas. The formulation of
the problem is presented below.
Model assumptions:
• Existing competition is known and fixed
• Customers patronize the most attracted to them facility and the demand is fully
supplied from the facility (i.e. “winner gets is all”)
• MAXCAP leads to the combinatorial optimization models like the covering one
Notation about the parameters, variables and subscripts:
i: index of customers
I: sum of customers
S: sum of candidate location areas
Pi: set of candidate location areas s (sϵS) that customer i would patronize if a new
facility would be located there
Ti: set of candidate location areas that if a facility was located there, she would be the
same attractive to customer as competitor’s facility that satisfies now customer’s i
demand
wi: demand of customer i, ∀𝑖 ∈ 𝐼
p: number of facilities that will be established
1, if custoner i is completely captured by new firm
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1, if customer i that its demand has completely captured by an existing facility
that is attracted to, can change and patronize a new competitive facility that is
attracted too
0, otherwise
1, if a facility is located in location s
Xs =
0, otherwise
Objective function
𝑚𝑎𝑥 ∑ 𝑤𝑖𝑌𝑖𝑖𝜖1 + ∑ 𝑖𝜖1𝑤𝑖
2𝑧𝑖 (3.1.1)
Constraints
Subject to
𝑌𝑖 ≤ ∑ 𝑋𝑠𝑠∈𝑃𝑖 ∀𝑖 ∈ 𝐼 (3.1.2)
𝑍𝑖 ≤ ∑ 𝑋𝑠𝑠∈𝑇𝑖 ∀𝑖 ∈ 𝐼 (3.1.3)
𝑌𝑖+𝑍𝑖 ≤ 1 ∀𝑖 ∈ 𝐼 (3.1.4)
∑ 𝑋𝑠𝑠∈𝑆 = 𝑝 (3.1.5)
𝑌𝑖 , 𝑍𝑖 , 𝑋𝑠 ∈ {0,1} ∀𝑖 ∈ 𝐼 ∀𝑠 ∈ 𝑆 (3.1.6)
The objective function (3.1) maximizes entering firm’s market share. The
constraint (3.2) indicates that customer i can be captured completely if only a new
facility is located and is more attractive to that customer by the existing one. Constraint
(3.3) indicates that customer i is captured by an opening facility that is the same
attractive to an existing one if this new facility is located in a point s ∈ Ti whereas his
demand value is allocated to both facilities with the same percentage. The constraint
(3.4) shows that customer i can be captured only by the new entering firm or divided in
both two firms (i.e. new and existing) or by the existing one. It enforces that there cannot
be a simultaneous capture as is indicated in the three aforementioned cases. Constraint
(3.5) defines the establishment of p locations while the constraint (3.6) indicates that
variables 𝑌𝑖, 𝑍𝑖 , 𝑋𝑠 are binary.
Zi =
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Some of the most important extensions, formulations of MAXCAP comprise the
work of Eiselt and Laporte (1989 a,b) where is correlated the MAXCAP with the
Gravity Theory and the model of Revelle and Serra (1991) that considers an relocation
scheme for the existing facilities and location of new ones.
Furthermore, other extensions are the combination of MAXCAP with hierarchical
location problems where the competition is occurring in each level of hierarchy (Serra,
Marianov and ReVelle, 1992) and the PMAXCAP (Serra and ReVelle, 1999) where
there is a competition scheme among existing facilities and new ones in terms of
location and price strategies. In the same year, 1999, Serra, Revelle and Rosing also
adapted the original MAXCAP to incorporate a threshold demand level which is a
minimum level of demand that it is necessary to be captured in order a new entering
firm to survive in the market. In the same vein, Colomé, Lourenço and Serra (2003)
introduced the New Chance - Constrained maximum capture location problem that was
based on gravity modeling and on the introduction of stochastic threshold constraint
whereas they implemented the metaheuristic Max-Min Ant system and tabu search
procedure in order to solve it.
3.1.3 Other important concepts
Other approaches in competitive location theory include the rise of Gravity
Theory. According to that theory except from the distances firms are competing in terms
of attractiveness of their facilities in order to acquire the biggest market share. This
theory also is used to extend the binary customer choice (i.e. customers patronize the
closest facility, a facility that “gets it all”) and reflect the proportional customer’s
demand allocation among competitive facilities. It is a very appealing theory especially
to retailers due to the emphasis is given to the attractiveness of their retail stores.
A major contribution in the sequential problems is the work of Hakimi (1983) that
presents two models that follow the principles of leader, follower as it was mentioned
before. These problems are the medianoid (r|Xp) and the centroid (r|p). The product is
homogeneous and customers patronize the closest facility. According to centroid, the
leader firm wants to locate in an area p facilities in order to maximize its market share;
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the leader’s decision for the location of its facilities is based on the knowledge that a
follower will establish as well possible r facilities in the future.
Moreover in the case of medianoid it is examined the locating of r facilities from
the point of view of follower firm when he knows that a leader firm had already
established p ∈ Xp facilities. It can be concluded that centroid describes the problem of
opening a new facility from the point of view of the leader while the medianoid presents
the follower’s viewpoint.
Other contributions encompass models that are based on the classic formulations
and modified, extended or combined in order to provide dynamic approaches in terms
of pricing and quantity competition or incorporation of attractive elements.
3.1.4 Gravity Theory
The main difference between the classic competition approaches and the gravity
modeling is that in the estimation of customer’s capture apart from distance the
measurement of attractiveness elements play a major role too. The origins of Gravity
Theory is dated back to 1931 when Reilly introduced his famous Reilly Gravitation
Law that estimates influence trade areas between two cities.
The Reilly’s law takes into account two elements, the size of the cities’ population
and distances. Reilly’s theory encompasses two cities that attract customers from an
intermediate city and this attraction is proportional (i.e. linear increase) to the
population of these cities and inversely proportional to the square of the distances of
these two cities to an intermediate one. The mathematical format of the law is presented
below:
Notation:
a: index of city A
b: index of city B
Ba: the proportion of trade that city A attracts from intermediate city
Bb: the proportion of trade that city B attracts from intermediate city
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Pa: the population of city A
Pb: the population of city B
Da: the distance between city A and intermediate
Db: the distance between city B and intermediate
Ba
Bb= (
Pa
Pb) ∗ (
Db
Da)
2 (3.2.1)
Although Reilly recognized the impact of other factors (e.g. transportation,
topography, business attractions, population density, different type of customers) in the
estimation of influence trading areas, he judged that population and distance were
enough capable to provide a reliable result. Another limitation in Reilly’s work refers
to the assumption that the investigated area is flat. In essence, there are any kinds of
obstacles or detterent factors that could change customer’s opinion to travel to buy their
goods.
In 1949, Converse revised Reilly’s law and presented his viewpoint in the
estimation of the breaking point, a point where the trade influence of both cities would
be equal. Thus, if a customer was located in this breaking point, he would be attracted
at the same level from both cities and would have 50% probability to patronize each of
one. As Huff (1964) claims by applying the following formula (3.2.2) for a city and its
competitive ones this would result in the delineation of its influence trade area.
Notation:
a: index of city A
b: index of city B
Pa: the population of city A
Pb: the population of city B
Dab: distance separating city A from city B
Db: breaking point between A and B expressed in miles to city B
Db =Dab
1+√PaPb
(3.2.2)
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Although Converse identified the breaking point and extended the original model
of Reilly the aforementioned limitations of the original model still existed. These
limitations challenged professor Huff (1963, 1964) who acknowledged the previous
contributions and presented his viewpoint in the Gravity Theory.
Huff was the first to introduce the Luce axiom (Serra and Colomé, 2000) in
Gravity Theory. The used axiom in this model indicates that customers may visit more
than one store and estimates the probability of customer patronization of one store as a
ratio of the attractiveness or utility of that store to the sum of all stores’ attractivenesses
in an investigated or trade area where store trade area can be defined as the area around
the store that are located most of its captured customers.
Huff in his formulated model (3.2.3) used the size of stores in order to reflect their
attractiveness (i.e. the bigger the store the bigger the possibility to have larger and wider
assortment of goods) and the distances from the customers as well a parameter λ to
indicate the different customers travel trips for the different type of products. From the
(3.2.3) formula it is conducted that as λ increases, customer’s trip has a greater effect
on the possibility Pij. As a result, higher values should be used for traveling and buying
of convenient goods [i.e. a good that is widely available and frequently purchased with
minimum effort (investopedia.com, 2018)] rather than specialty goods.
The parameter λ is usually computed through empirical surveys and studies and
takes one to four values. According to the previous statement about the λ values and
different type of goods it can be concluded that Huff model is an agile model and can
be customized in order to propose different results as far as trade areas for different
types of products are concerned (Anderson,Volker and Phillips, 2016).
Notation:
i: index of customers
j:index of stores
n: number of all possible competing stores in the investigated area
Sj: size of store in square feet
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Dij: distance or travel time between customer i and store j
λ: parameter decay that reflects the different effects of customer trips for the different
types of products
Pij: the probability of a customer at an origin point i to travel and buy products from a
store located in point j
Pij =
sj
Dⅈjλ
∑sj
Dⅈjλ
n
j=1
(3.2.3)
Equation (3.2.3) can be further extended in order to compute the total number of
expected customers from origin point i (i.e point i is used to represent a set of customers
that located in that area) that would patronize the store in location j. Particularly, the
possible expected customers are calculated by multiplying the Pij value by the number
of customers at that point i (3.2.4).
Notation:
Eij: expected costumers in point i that is possible to travel and buy their products from
store j
Ni: number of customers at point i
Eij = Pij∗Ni (3.2.4)
Overall Huff presented a definition about the delineation of a store’s trade area
which is the following and can be represented symbolically by the equation (3.2.5):
“A geographically delineated region, containing potential customers for whom
there exists a probability greater than zero of their purchasing a given class of
products or services offered for sale by a particular firm or by a particular
agglomeration of firms”
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Notation:
Tj: trading area of a particular firm or agglomeration of firms j, that indicates the total
expected number of consumers within a given region who are likely to patronize store
j for a specific class of products or services
Pij: probability of individual consumer residing within a given gradient i shopping at j
Ni: number of consumers residing within a given gradient i
Tj = ∑ (Pij∗Ni)n
i=1 (3.2.5)
Huff’s model is an easily used model and can be applied to variant problems.
Therefore, many authors and researchers used that model and its modified or extended
forms in their computations for providing results referring to the delineation of trading
areas or differently met in literature, as catchment areas. Recent instances of such
different type of implementations are the work of Dolega, Pavlis and Singleton (2016)
and Lin et al. (2016).
Moreover, many researchers in order to provide validity in their theoretical results
gained from the implementation of Huff model they conducted a comperative analysis
between the aforementioned results and actual data derived from empirical surveys.
They concluded that even Huff model presents a probability framework for the
consumer behavior refering to patronization of stores it must be correlated to other data
(e.g. social, demographic, economic or other local community data) in order to have a
more compehensive analysis. Examples of such researches are the distinctive work of
Drezner and Drezner (2002) or more recently a work provided by (Mitríková, Šenková
and Antoliková, 2015).
Another impact of the Huff model in the retail location theory is its more recently
combination with the GIS (i.e. Geographical Information Systems) tools for the
estimation of consumer’s patterns. Particularly, an example is the ArcGIS system that
provide a systematic analysis for store selection based on a combination of the Huff
model with other specific area features.
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Despite the fact that Huff’s is one of the most important in Gravity Theory and
the consequent retail and in general competitive location theory, the original model has
a major drawback. This drawback is the oversimplification in the model to consider
that consumer behaviour can be reflected by only two parameters the size of the stores
and the distance between consumer and stores. Thus, in the following years, Nakanishi
and Cooper (1974) modified and improved the original model in order to incorporate
more consumer behaviour based attributes (e.g. store image, service level, accessibility
to the store etc.). Their work is refered as the Multiplicative Competitive Interaction
model or usually met with its abbreviation MCI model.
According to that model, the store floor area is replaced by a set of attributes that
each of them represent a component of the attractiveness and is being raised to a power
that reflects the sensitivity of Pij to the k attribute. The mathematical formulation is the
following (3.2.6).
Notation:
i: index of customers
j: index of stores
k: index of attributes of attractiveness
uijk: utility of customer i for attribute k in store j
a: sensitivity parameter of Pij with respect to attribute k
Pij: probability of customer located in point i to purchase products from store located at
point j
Pij =∏ u
ⅈjk
a𝑘
k
∑ ∏ uⅈj𝑘
ak
𝑘j
(3.2.6)
Throughout years many extensions, modifications and adjustments that combine
Gravity Theory with competitive location theory as well with the general location
science have been proposed. An example is the previously stated reference about
Colomé, Lourenço and Serra (2003) who used the gravity modeling in their approach
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and more specifically the MCI model in a combination with the introduction of a
threshold constraint to the MAXCAP model. Furthermore, another distinctive example
is the contribution of Drezner and Drezner (2006) who presented the gravity p-median
model in order to surpass the limitation of the original p-median that customers
patronize their closest facilities and introduce the concept of dividing the patronage in
more than one store indicated by a probability value.
3.2 Hub Location Problem (HLP)
More recently, the transportation of big amounts of cargo or a great number of
people as well the transactions of big amounts of data made it necessary the
development of a new model that could facilitate the flows of commodities, men and
information among counterparts.
O’Kelly (1987) investigated the aforementioned issue and introduced and
formulated a new problem called the Hub Location Problem or Hub-Spoke Location
Problem. He presented the single HLP and he further acknowledged the importance for
multiple hub nodes. As a result of this he proceeded to the formulation of p HLP, for
more than one hub nodes. A description of the Hub-Spoke network with multiple hub
nodes and spokes is presented in figure 3.1.
A Hub is a transfer point that aggregates commodities, people or data from
different origins (Spokes) and promotes them to another Hub in order to disaggregate
and deliver them to the suitable destinations (Spokes). The flow must always be done
through the Hub nodes in order to be achieved economies of scale. Economies of scale
are being achieved through the more efficient transportation modes between the hub
nodes. For example, the flows of cargo, people or information between point 6 and 4
must follow the path 6-11-13-4 and reversely (Figure 3.1).
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Figure 3.1: (a) demand nodes (red circles) are assigned to one hub (blue squares), (b) demand
points are assigned to more than one hubs (Ghaffarinasab et al. 2018; Author 2018)
Airline companies are taking advantage of such economies of scales by using big
airplanes with high capacity in the internal transshipments of cargo and smaller
airplanes with lower capacity in the external transshipments (Eiselt and Marianov,
2011). Except from the airline industry applications, the HLP is being applied in postal
and delivery package services, in transportation and handling services, in
telecommunication systems and for the design of a supply chain network for chain
stores (Hekmatfar and Pishvaee, 2009).
3.3 Undesirable Location Problem (ULP)
As far as concern the problems presented until now, they have one common
element, the pull objective. In essence, customers or demand points desire the facilities
to be located the closest to them. This situation cannot reflect all the cases in reality.
There are many cases, in which a facility must be as far as possible because it can be
harmful to the people or the environment (Melachrinoudis, 2011). As a result of this
the under-examined problem has a different objective, the push one. Problems
pertaining to that category are called Undesirable ones.
Except from these two objectives, in reality, a combination of them can be
appeared where a facility may have positive and negative impact on the lives of people
and enviroment. A distinctive example is a mall that can provide a wide range of goods
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and services (positive) as well as can create traffic and crowd congestion that bring
negative effects, like noise, to local society (Bruno and Giannikos, 2015).
Another approach is the aspect of Eiselt and Marianov (2011) who state that
although the Undesirable facilities must be located as far as possible, in reality, this
may lead to great expenses. As a result of this, a maximin or maxisum target in the
objective functions of the models must be proposed in order to tradeoff the unpleasant
consequences and the eminent expenses. The maxisum objective maximizes the
average distance between customers and facilities and the maximin objective
maximizes the distances between the facilities and their closest customers.
(Melachrinoudis, 2011).
In general, Undesirable facility location problems are characterized by two basic
principles the NIMBY (Not In My Backyard) and the NIABY (Not In Anyone’s
Backyard) (Hosseini and Esfahani, 2009). Furthermore, according to Daskin (1995)
who present the distinction provided by Erkut and Neuman in 1989 the Undesirable
FLP can be devided into subcategories: the establishment of the noxious facilities that
can be harmful for men and enviroment (e.g air and ground pollution from a power
plants, landfills etc.) and the establishment of obnoxious ones which may not be harmful
but can be annoying for the lifestyle of people in a wide area (e.g. noise of the operation
of an airport).
In final, there is a special problem closely related to the Undesirable one, the p-
dispersion one (Hosseini and Esfahani, 2009). This problem examines the issue of
establishing one facility in such a manner to reduce the negative effects from that
opening as much as possible to the existing facilities (i.e. avoid the cannibalization
effect). Purpose of this model is to identify p points for facilities in order to be
maximized the minimum distance to the already operating facilities. A distinctive
example is the establishment of a new store by a mother company as far as possible
from its representative (e.g. franchise) stores so as not affect their operation and to
capture a greater area of customers.
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3.4 Location Problem under Uncertainty (LPU)
An important approach in the research of location theory is the impact of the
element of uncertainty on the facility location decisions. On a continuously changeable
environment, one major drawback of the classic facility location problems is the
consideration that data are known and deterministic. In fact, they do not capture many
characteristics of the real world case scenarios. Since the strategic nature of location
decisions enforce facilities to operate for long time periods, the appropriate locations
must be selected in order the total supply chain network to operate optimally in present
and future times. As as result of this, data must be formulated in a way that will reflect
possible future changes.
Data and information that are being used in the location theory and can easily be
changed on a frequent basis are the following (Correia and Gama, 2015):
• Demand level
• Travel time or cost of servicing customers
• Location of customers
• Presence or Absence of customers
• Trends in the price of goods
When refering to such elements a location decision maker must incorporate the
uncertainty in its considerations. In general, there are two many subcategories of the
location problem under Uncertainity.
The first category, that is called stochastic programming includes problems
where probabilistic information is given and the uncertain parameters are represented
by random variables that are assosiated to discrete scenarios. Purpose of these problems
is the minimization of expected cost. In the second category, that is called robust
optimization encompasses problems that probabilistic information is not given and the
uncertain parameters are described by either discrete or continuous scenarios whereas
its target is the minimax cost or regret. According to Daskin, Snyder, Berger (2005)
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“The regret of a solution under a given scenario is the difference between the
objective function value of the solution under the scenario and the optimal
objective function value for that scenario.”
Correia and Gama (2015) define the term “scenario” as the complete realization
of all the uncertain parameters. Although both categories result in “good” results,
though they are not necessarily the optimal ones.
The first contribution to the correlation between stochastic location theory and
the general one is attributed to the Frank (1966) in his paper “Optimum Locations on
a Graph with Probabilistic Demands” (Berman, Krass and Wang, 2011). Another major
addition in this theory is the work of Sheppard (1974) who tried to minimize the
expected cost associated to the facility location considering an uncertain environment.
In final, as far concern the location theory combined with the element of
uncertainty it is important the presence of the risk in decision procedure to be
mentioned. Specifically, before choosing the location model the decision maker must
determine the level of risk in his approach. Specifically he must decide whether to be
risk-neutural or risk-averse. In essence, a risk neutral does not take into account the risk
in his decision while its target is the minimization of expected cost or maximization of
return. On the other hand, a risk-averse does not want any risk to be included in his
decision and its purpose is the minimization of future maximum costs (Correia and
Gama, 2015).
3.5 Location Routing Problem (LRP)
One extension of the basic problems presented in Chapter 2 is the Location
Routing problem. Many of the aforementioned problems have limitations. In essence,
when considering the establishment of facilities in a supply chain network these models
do not take into account important elements like inventory and routing policies in the
decision procedure.
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These elements are taken into account in the research of location science by a
more recent problem, the Location Routing one. This problem can be considered as a
combination of two smaller problems, the FLP and the VRP (Vehicle Routing
Problem). The current problem investigates the situation where there are LTL (Less
Than Truckload) routes in the service of customers, on contrary to the classical FLP
who consider FTL (Full Truck Load) routes in their computations. The delivery cost,
on LTL routes, depends directly on the multiple stops for the service of other customers
as well as the sequence that are being serviced these customers (Daskin, Snyder and
Berger, 2005).
According to Perl and Daskin (1985) the problem can be defined as follows: The
position and the demand values for a set of customers as well as the candidate locations
are known. Each customer is being assigned to one facility that will supply its demand.
For each candidate location its fixed and its linear variable cost are being given. The
transportation is linearly dependent on the delivered quantity and the routes encompass
multiple stops for the service of different customers. The maximum number of routes
is predefined and the total transportation cost is dependent on the total covered distance
from the fleet of vehicles.
As far as concerning the target of the combined problem, it must include three
optimization decisions (Perl and Daskin, 1985; Hassanzadeh et al., 2009):
• the establishment of facilities ( number, size, location)
• allocation of customers to facilities and routes
• design of the optimal route/s. (sequence of visiting customers) in order to be
minimized the total cost of the network
Although, Eilon, Watson-Gandy and Christofides (1971) tracked the error in the
calculation procedures by using FTL in the estimation of routes, only a decade later
approximately the LRP was proposed as combined one (Hassanzadeh et al., 2009).
Despite the fact that LRP has many applications (Hassanzadeh et al., 2009) in
real life like the distribution of consumer goods or newspapers, or healthcare industry
for blood bank locations, or in military the research of it, is in early stages. This is
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mainly attributed in its NP-hardness (i.e combination of two NP-hard problems) and its
consequently difficult solution procedure as well as in the difficulty of merging strategic
(i.e. FLP) with operational (i.e. VRP) decisions (Daskin, Snyder and Berger, 2005).
Such decisions have different impact on the location of facilities in terms of time
horizon, expenses, character of permanency etc.
3.6 Location Inventory Problem (LIP)
According to Shen, Coullard and Daskin (2003) inventory literature do not take
into account strategic location decisions and their associated costs and the facility
location science do not consider operational decisions like inventory management,
delivery costs and demand vulnerability as important ones.
Although operational decisions (i.e. inventory management) are not being
correlated directly to the strategic ones (i.e. facility location) in real circumstances, it is
necessary to combine them because of the huge impact of strategic on the operational
ones. Ignorance of the total costs, independently to the level of decision that are related
in, can result in suboptimality of the supply chain network (Shen and Qi, 2007)
An interesting approach is that of Daskin et al. (2002) who tries to track the
tradeoff between the opposing targets as far as concern the desires of customers and
optimal inventory policies. Specifically, he emphasizes the important role of
establishment of a distribution center in a location where the inventory costs will be
reduced through an implementation of a centralization policy and the proximity to that
location must be as smaller as possible in order final consumers get their goods.
As a consequence of the above, the need for a combined version of inventory and
location models gave birth to the Location Inventory Problem. The first reference to a
correlation between inventory and location theory attributed in Baumol and Wolfe
(1958) who tracked the interaction of inventory management with the number of
shipements through warehouses.
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Despite the fact that many researchers have acknwoledged the contribution of
inventory policies into the strategic facility location decisions only more recently a
feasible solution was presented from Shen, Coullard and Daskin (2003) who introduced
a location model with risk pooling (LMRP). According to that model, there is three-tier
supply chain which encompasses a plant, distribution centers and customers. Purpose
of the model is to minimize the sum of costs:
• Shipment cost from plant to distribution centers
• Fixed establishment facility location costs
• Inventory cost at the distribution centers
• Transportation cost from distribution centers to the customers
Allocation of customers to distribution centers has a direct impact on the first and
third type of costs.
In general, as it was stated for the LRP, the solution procedure for the LIP is a
difficult issue because of the incompatibility between the operational decisions (i.e.
inventory management) that require frequent scheduling and the strategic ones (i.e.
facility location) that have a more unchangeable character. As it is judged by Daskin,
Snyder and Berger (2005), priority in the decision procedure must be given to facility
location decisions compared to inventory or routing ones because of their high expenses
and their aforementioned permanent character.
3.7 Generalizations – Extensions of the main concepts
Most of the problems presented until now do not capture real location scenarios.
For instance, the majority of the aforementioned problems were investigating the issue
of locating one type of facility. In real circumstances, in the design of an optimal supply
chain network, it is required to be established more than one type of facility. This
concept is described in location science as the Hierarchical Location Problem (HLP)
and examines the simultaneously and dependently establishment of different types of
facilities.
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In a Hierarchical system, facilities belong to a set of levels (m) where level 1
represents the lowest level of service, level m the highest one and level 0 the customers.
An HLP is suitable in the investigation of locating facilities in health care, solid waste
disposal and education systems.
Another location problem referring to real scenarios that attracted much of
attention in recent years is the Location problem with Failure. The element of failure
may be attached in different aspects of supply chain network. For instance, locations
with an unstable climate, labor union actions, bad transportation network are some
features that must be incorporated in the location model search and consequent choice
in order failures in the network to be prevented or have the least impact in terms of any
cost related.
Following the previous approach, a decision maker must choose locations that
will be inexpensive and reliable (Daskin, Snyder and Berger, 2005). This problem can
be related to the Location problem under uncertanty as far as concern the decision
maker who seems to behave as a risk-averse because its purpose is the minimization of
possible future expenses.
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Chapter 4: Solution techniques, methods and algorithms
in Facility Location Problem
In the current chapter, the assignment will provide a reference to the most well-
known solution procedures in the FLP as well as some of their general features.
Emphasis will be given in the description of methods that were implemented (AHP,
Weighted Factory Rating Method, Load Distance Technique) in order results to be
acquired for the under investigation issue of the opening of a new supermarket (Chapter
6). Description of another solution procedure (Huff Model) to this issue will be
presented in the Chapter 6 too.
4.1 Solution approaches in location problems
Facility location models are being characterized as NP-hard problems.
Specifically, this means non-deterministic polynomial hard problems. The term non-
deterministic refers to the no particular rule that must be followed in order the problem
to be solved. The term polynomial refers to the time or steps an algorithm needs to solve
the problem. This algorithm is limited by a polynomial function of n where n
corresponds to the size of the input to the problem. An NP problem can easily and
quickly be solved by algorithms (Hosch, 2018).
On the other hand, when a problem is considered as NP-hard there is not any
known polynomial algorithm that can solve the problem and the time to find a solution
grows exponentially with problem size (combinatorial explosion phenomenon). As a
result, different approaches for the solution procedure of NP-hard problems have been
proposed in the literature. For large scale problems, they are used approximate methods
like the Heuristics and the Metaheuristics algorithms.
For small scale problems, it is more suitable for exact methods to be used (Nordin
et al., 2016). Some well know exacts methods in the concept of FLP include the Graph
Theory, the Pair-wise Exchange method, the Branch and Bound technique, the Cutting
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Planes method, the Benders Decomposition, the Column Generation algorithm and the
well-known and widely applicable Lagrangian Relaxation (Snyder, 2010).
Other approaches investigate the location problems from different scopes.
Specifically, they refer to useful simple techniques that provide exact results as well as
they incorporate qualitative elements in the configurations of the problem or try to
present hybrid models that encompass quantitative and qualitative data at the same time.
A well-known procedure in that category is the Analytic Hierarchy Process (AHP)
which is suitable for multi-decision problems. A very different approach, that is
continuously evolving nowadays is the support of Geographical Information Systems
(GIS) in the solution processes of facility location problems.
4.2 Heuristics
Heuristics’ name is dated back to the Greek word “heuriskein” which means
discover. According to BusinessDictionary.com (2018), heuristics are trial-error
procedures for solving problems through incremental exploration and by employing
known criteria to unknown factors. They can be considered as self-educating techniques
that provide improved solutions through the experimentation; solutions are
characterized as satisfactory rather than optimal, complete or precise. One simple
distinctive heuristic is described below:
Greedy Adding Algorithm
Greedy Adding is an algorithm which is commonly used in the solution of FLP.
It is a mathematical process that seeks for simple, fast, easy to implement solutions in
complex problems. Its name is attributed to its procedure. Specifically, it examines level
by level a problem and at any level it chooses the solution that seems to be the best at
this moment (hackerearth.com, 2018). It does not examine the whole problem but wants
to provide greedily a solution on each time level. The result can be considered as a local
optimal while it is anticipated to be the global optimal too.
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Rouse (2018) agrees with the previous statements and further claims that its major
advantage is that its solutions are straightforward whereas its major drawback is the
myopic view of the general problem. While it can result in optimal short-term solutions,
the same time it can lead to sub-optimal long-term ones.
Other examples of heuristics algorithms that are being met in the location science
are the Swap algorithm, the Neighborhood search algorithm, the Exchange
(Interchange) algorithm, the Vertex Substitution and many others (for a complete
review of implemented heuristics in the field of location science see Brimberg and
Hodgson, 2011).
4.3 Metaheuristics
Metaheuristics are high-level independent to the problem strategies that provide
guidelines to heuristic procedures in order the last ones to explore the search space and
give not only satisfactory results but as close as possible to the optimal ones (Glover F.
and Sörensen K. (2015). In contrast to heuristics which are problem - dependent,
metaheuristics are applicable to a wide range of problems. In general, metaheuristics
are designed to overcome the drawback of local optimal provided by heuristics and
broaden their solution area.
Furthermore, they are being differentiated by the exact methods which provide
exact results, in terms of providing an approximate solution (heuristic nature) that is
good enough and its computation time is small enough too. In that way, they are not
subject to the combinatorial explosion, a relevant to NP-hard problems phenomenon.
Some of the most applicable metaheuristics algorithms (general features) in
location analysis are described briefly below:
4.3.1 Genetic Algorithm
Genetic are metaheuristic algorithms that inspired by the natural evolution and
are used to provide high quality optimal solutions in problems with large and complex
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data (techopedia.com, 2018). Its structure is based on Charles Darwin theory of
"survival of the fittest.". In essence, this means that fittest individuals dominate over
the weak ones and they are combined in order to create offsprings.
Genetic algorithms use methods that are based on the evolutionary biology such
as selection, mutation, inheritance and recombination to solve a problem. Since Genetic
algorithms are designed to imitate the evolutionary biology procedure, much of the
terminology in attributed to the science of biology. Generally, a Genetic Algorithm
encompasses five steps (Mallawaarachchi, 2017):
1. Initial Population
2. Fitness Function
3. Selection
4. Crossover
5. Mutation
4.3.2 Tabu Search
Tabu search (Glover, 1986) is a metaheuristic algorithm. Like any other
metaheuristic, Tabu search provides the guidelines to heuristics to explore their solution
search area and bypass their local optimal results. Furthermore, it is iterative and local
search method which seeks at any iteration the “neighborhood” area of the until then
proposed solutions in order to investigate and provide an improved one.
Specifically, the procedure can be described as follows. At first, it must set an
initial solution. The value of the initial solution which is considered as the best solution
until the first iteration is stored in the long-term memory. Then the algorithm guides the
search, of the current solution, into its neighborhood area where there is the highest
improvement or the least deterioration in the solution value (Fera et al., 2013). These
accepted moves are marked as Tabu (forbidden) and stored at the short-term memory
of the algorithm in order to create a list, the Tabu one. This Tabu list is updated with
accepted moves that evaluation process provides and prevents the selection of recently
(i.e. short-term iterations) visited solutions. If the evaluation procedure leads to an
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improved solution, its value is replacing the previous one in the long-term memory.
Using these two memories, they are dodged cycling movements and procedure don’t
stick in suboptimal regions. The algorithm ends when a set of stopping criteria is being
satisfied (e.g. attempt limit, score threshold; Wikipedia). The flow chart of Tabu Search
is presented in the following figure 5.1.
Figure 4.1: Flow chart of general Tabu Search process (Fera el al., 2013)
4.3.3 Simulated Annealing
Simulated Annealing is a generic probabilistic metaheuristic method that like the
aforementioned ones is being used to provide a good approximation to global optimum
in the presence of many local optimums. It is commonly used when the search area is
discrete and it is based on the Metropolis–Hastings algorithm. Its name is attributed to
the physical process of repeated heating and cooling the temperature of metals in order
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to increase the dimensions of the crystals and decrease the defects. The method seeks
to minimize the system energy.
The method can be described as follows. An initial solution is being set
[Temperature (T): set of solutions}. At each step, the algorithm randomly selects a
solution close to the current one. If the new solution improves the energy of the system
(e.g. sum of distances in the computation of shortest path) by reducing it, this solution
is considered as a good move and it is accepted (Fera et al., 2013). Beyond that choice,
for certain probability value, the algorithm accepts solutions which are considered as
bad moves and do not improve the system energy. As algorithm reduces T and
converges to global optimum (i.e. minimum or maximum, depends on the problem’s
objective f) the acceptance of bad moves is decreasing and the research for new
solutions is limited. In essence, the algorithm exploits the local optimums and at the
same time explores possible solutions in order to find an approximate global optimal
one.
Other examples of Metaheuristics algorithms that are being met in the location
science are the well-known Grasp algorithm, the Ant-Colony algorithm, the Variable
Neighborhood, the Neural Networks and the Heuristic Concentration.
4.4 Exact Methods
Lagrangian Relaxation
Lagrangian Relaxation is a method used for solving combinatorial optimization
problems, like the under examination FLP. While its birth is dated back to 1970, 1971
when Held and Karp used a Lagrangian problem to plan a successful algorithm for the
traveling salesman problem, its current name is attributed to Geoffrion (1974) (Fisher,
2004). Lagrangian relaxation approaches the complex problems by decomposing the
initial problem through the set of difficult constraints into a simpler one. The final
proposed solution is an approximate and weaker compared to the optimal of the initial
problem. Methodology of Lagrangian Relaxation encompass the following steps
(Xatzigiannis, 2013):
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1. The procedure begins by putting non-negative prices, on one or more
constraints. Then, these values that are called Lagrangian multipliers are added
to the objective function of the problem. The new objective function is called
Lagrange function to (P) where P is the initial problem.
2. Then proceed to the solution of the relaxed problem and find optimal values for
the initial variables.
3. Then use the aforementioned variables in order to find one feasible solution
(optional step)
4. Use the solution of step 2 in order to create an upper limit for the optimal value
of the objective function.
5. For the solution of step 2, investigate the set of constraints that are violated.
Then, examine how these violated constraints should be modified in order to
reduce the probability of their violation in future iterations. After the
modification proceeds to another iteration from step 2.
4.5 Other Approaches
4.5.1 Weighted Factor Rating Method
One well-known evaluation method in location analysis is the Weighted Factory
Rating one (prenhall.com, 2018). This specific method proposes a different point of
view in the solution procedure of location problems. Despite the fact it is a simple
procedure, it is considered as an effective one too, because it can reflect the preferences
and the purposes of the decision maker. The decision maker can decide based on the
result of that procedure or more typically to use them in correlation to other elements
like costs, restrictions, loads in order to have a more objective decision. The process
concludes the following steps:
1. Identify all the important factors and make a list of them. The evaluation and
the subsequent choice of the most critical factors can be based on quantitative
data except from qualitative ones
2. Assign weights to the previous factors (0-1) based on the relative importance
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3. For every candidate location subjectively score the previous factors (0-100)
4. Calculate the weighted score for every factor for every candidate location by
multiplying its weight with the correspondent score
5. In accordance, calculate the sum of the weighted scores for each candidate
location and choose the location with the highest total score
4.5.2 Load Distance Technique
This technique, (prenhall.com, 2018) as it concluded from its name, emphasizes
on two basic elements, the transferred loads and the distances between a set of existing
facilities n (e.g. distribution centers, warehouses) and a set of potential new location
sites. Its purpose is to choose the candidate location with the lowest LD value in order
the transportation cost to be minimized. Its mathematical formulation is the following:
Notation:
LD: Load-Distance value
li: Loads expressed as weights or as a number of pallets or as other general shipped
elements that are being transmitted from the set of the existing facilities i to the potential
new location sites
di: distance is calculated by Euclidean or another type and differently can be expressed
as a time function
Euclidean di: √(𝑥𝑖 − 𝑥)2 + (𝑦𝑖 − 𝑦)2 where (x,y) the coordinates of candidate
locations and (xi, yi) the coordinates of the set of the origins i
Formulation:
𝐿𝐷 = ∑ 𝑙𝑖𝑑𝑖𝑛𝑖=1 (4.1)
4.5.3 Center of Gravity
Center of Gravity is a method (prenhall.com, 2018) that seeks to identify one
facility on central locations in the presence of existing ones. Hence, it is suitable in the
investigation of locating distribution centers. Its concept emphasizes on the
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minimization of distribution costs by considering the location of existing destinations
(e.g. markets, retail stores etc.) as well as the volume of commodities shipped to those
facilities. Procedure encompasses the following steps:
1. The existing facilities n must be placed on a coordinate grip map where the
origin of the grid and the scale are arbitrary and the distances relative
2. Then calculate the (Cx, Cy) coordinates of the new facility, the center of gravity
which minimize the transportation cost.
Mathematical formulation of (Cx, Cy):
Notation:
i: existing facilities n
xi, yi: coordinates of locations i
Wi: volume of shipped goods
𝐶𝑥 =∑ 𝑥𝑖𝑊𝑖
𝑛𝑖=1
∑ 𝑊𝑖𝑛𝑖=1
𝐶𝑦 =∑ 𝑦𝑖𝑊𝑖
𝑛𝑖=1
∑ 𝑊𝑖𝑛𝑖=1
(4.2)
Other simple techniques that help location managers to decide about new
facilities are the Break-Even or Cost-Volume Analysis that is used usually on the
locating of industrial facilities and compare costs and profits via exploitation of graphic
representations and the Transportation Model which is used for the minimizations of
costs on a network with multiple suppliers and multiple demand points to be served.
4.5.4 Analytic Hierarchy Process (AHP)
As it was stated before, the Analytic Hierarchy Process is a hybrid approach that
encompasses qualitative and quantitative data at the same time. Its special characteristic
is the agility to express personal preferences and subjective factors for various
multicriteria problems, like the facility location ones. Other applications can be found
in the sector of manufacturing, marketing, energy, healthcare etc. (Subramanian and
Ramanathan, 2012).
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AHP was developed in early 1970s by Saaty (1980) and its main concept is to
decompose complex problems into a hierarchy by evaluating multiple different factors
through a systematic and structured mathematical procedure (Saaty, 1982). One major
feature of this process is the emphasis given to the element of consistency in order the
results to be reliable, for problems that decision criteria are expressed subjectively
(Badri, 1999).
AHP’s results are a prioritized ranking of alternative choices and can be used by
decision maker solely or can be incorporated into other processes. Particularly, when
the problem seeks for selection of one decision, AHP provides good results (Badri,
1999) whereas when it is needed further prioritizing except from the first choice it is
preferable to combine AHP with other tools. In the literature, AHP has been combined
with Goal Programming, SWOT analysis, DEA analysis, metaheuristics, Delphi
method and many others (Subramanian and Ramanathan, 2012).
In gereral according to Saaty the development steps of the AHP solution approach
as they are presented in the work of Anderson et al. (2011), are the following:
1. Developing the hierarchy: In the first step, it must be decomposed the decision
problem. A graphical representation of the overall goal, the used factors or
criteria and the decision alternatives, supports the process (example: Figure 4.2).
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Figure 4.2: General graphical representation of an AHP structure (Author, 2018)
2. Comparative Analysis: The comparative analysis steps refers to the
computations for the prioritization of the criteria or the factors according to
preferences of the decision maker and the scores of each alternative for each
factor. The prioritization of the factors is being computed through pairwise
comparisons and subsequent mathematical validation procedure while in the
calculation of the alternative scores in the measurement of qualitative factors, a
scale of 1 to 10 is being used and for the quantitative ones their real values being
used (approch method on qualitative factors of Yang and Lee,1997)
3. The pairwise comparisons (Figure 4.3) (two a time) are conducted by decision
maker which rates the relative importance of each criterion to each other criterion
according to a scale from 1 to 9 where 1 means equally important and 9
extremely more important. The analytical scale is presented below (Table 4.1).
The number of pairwise comparisons must be (n-1*n)/2, where n the number of
factors.
Verbal
Judgement Numerical Rating
Extremely more
important 9
Overall Goal
Factor 1 Factor 2 Factor 3
Alternative 1
Alternative 2
Alternative 3
Alternative 1
Alternative 2
Alternative 3
Alternative 1
Alternative 2
Alternative 3
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Pairwise Comparison More Important Criterion How much more important Numerical rating
Very strongly to
Extremely more
important 8
Very strongly
more important 7
Strongly to Very
strongly more
important 6
Strongly more
important 5
Moderately to
Strongly more
important 4
Moderately more
important 3
Equally to
Moderately more
important 2
Equally
important 1
Table 4.1: Comparison scale for the importance of factors (Author, 2018)
Figure 4.3: Pairwise comparisons of the selected factors example (Author, 2018)
At next level, it is constructed the pairwise comparison matrix (Figure 4.4) for
the chosen factors as it was conducted in the previous pairwise comparisons. The
factors placed in the column More Important Criterion above indicate which row
of the pairwise comparison matrix the Numerical Rating must be placed in. The
diagonal elements are equal to 1 and the rest of the elements are filled out in the
way presented below.
Factor 1 - Factor 2
Factor 1 - Factor 3
Factor 2 - Factor 3
Factor 1
Factor 1
Factor 3
Moderately
Equally to moderately
Moderately to strongly
3
2
4
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Factor 1 Factor 2 Factor 3
Figure 4.4: Pairwise Comparison Matrix example (Author, 2018)
4. Establishment priorities between factors: This step refers to the calculation of
each factor priority in terms of contribution to the overall goal. The steps of this
synthesization process are the following:
• For every column proceed to the estimation of its sum
• Divide each value from the pairwise matrix by its column sum in order to
normalize them and make the normalized matrix
• In the normalized matrix, calculate the average of every row. These
average values indicate the priority ranking among factors to the overall
goal
5. Consistency evaluation: As it was mentioned before consistency is one feature
of AHP that provides validation in its procedure. When there are many pairwise
comparisons perfect consistency is difficult to be achieved. In order consistency
to be achieved AHP follows a suitable procedure. If the percentage of
consistency is unacceptable the pairwise comparisons should be revised.
Acceptable percentages of CR are considered values smaller or equal to the value
of 0,10.
Factor 1
Factor 2
Factor 3
1
1
1
3 2
1/3
1/2
1/4
4
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Procedure of consistency evaluation
• Firstly, calculate the consistency ratio (CR) in the following way. Multiply
the elements of the first column of pairwise matrix by the priority of the
first factor resulted in the synthetization step. Continue the same
procedure for the others columns and the corresponding priorities.
Continue by summing up the values of each row (i.e. Sum 1, Sum 2, Sum
3). This is the weighted sum vector (Figure 4.5).
Figure 4.5: Weighted Sum Vector (Author, 2018)
• Secondly, divide the elements of weighted sum vector (i.e. Sum 1, Sum
2, Sum 3) by the priority for each factor (e.g. Sum 1/ Priority factor 1 etc.)
• Thirdly, compute the λmax where λmax is calculated through the following
format:
λmax= Total sum (Sum 1, Sum 2, Sum 3)/ number of factors
• Then calculate the Consistency Index (CI) with the following format:
CI= (λmax – number of factors)/ (number of factors – 1)
• In final compute the Consistency Ratio (CR) with the following format:
CR=CI/ RI
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where RI is the consistency index of the randomly generated pairwise `
comparison matrix. The RI is depended on the number of compared factors
(N)
N 3 4 5 6 7 8 9 10
Ri 0,58 0,9 1,12 1,24 1,32 1,41 1,45 1,49
6. Establishment priorities of each alternative pertaining to each factor: This step
involves the process where the decision maker must express a judgment for its
preference among the alternatives choices for each factor. The prioritization for
each alternative, concerning the qualitative factors, is computed through the
pairwise comparison analysis (Figure 4.6) based on table 5.1 (same procedure as
step 3 in order to obtain the priorities values). On the other hand, as far as the
quantitative factors are concerned the process is the following (Yang, and Lee
(1997):
• Calculate the Wi of each alternative where Wi= 100/ Ti in order to
normalize the Ti values. Ti are the exact values of each factor for each
alternative. Then calculate the sum of Wi for each factor
• Find the priorities of each alternative relating to each factor by dividing
each Wi value with its related sum
Factor 1
Figure 4.6: Pairwise comparison matrix of each qualitative factor (example factor 1) for each
alternative (Author, 2018)
7. Overall priority ranking: The current step is the final one and refers to the
combination of the priorities of factors obtained in step 3 and 5 in order an overall
Alternative 1
Alternative 2
Alternative 3
Alternative 1 Alternative 2 Alternative 3
1 2 3
1/2 1 2
1/3 1/2 1
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priority ranking among the alternatives to be defined. Particularly the
computation procedure is to weight each alternative’s priority resulted in step 5
by the corresponding criterion of step 3.
Figure 4.7: Example of a matrix with overall depicted priorities (Author, 2018)
Values referring to priorities:
Step 3: X1, X2, X3
Step 5: Y1, Y2, Y3, R1, R2, R3, W1, W2, W3
Overall Prioritization Ranking for:
Alternative 1: (Y1*X1) + (R1*X2) + (W2*X3) = P1
Alternative 2: (Y2*X1) + (R3*X2) + (W1*X3) = P3
Alternative 3: (Y3*X1) + (R2*X2) + (W3*X3) = P2
P1, P2, P3 in the current representation are indicative prioritization values
because there are not any computed results from the above steps. An illustration of the
total computations is presented in the Appendix of AHP that is related to the Case Study
in Chapter 6.
Yang, and Lee (1997) in their paper describe an integrated AHP location decision
model. They mention that AHP can be characterized as an assistant tool for location
managers to analyze different location criteria, to evaluate the candidate locations based
on that criteria and to provide a final selection. Furthermore, they provide a wide range
of location factors and a schematic representation of AHP in the concept of FLP (Figure
4.8).
Priority 1 (Y1) Priority 1 (R1) Priority 2 (W2)
Priority 2 (Y2) Priority 3 (R3) Priority 1 (W1)
Priority 3 (Y3) Priority 2 (R2) Priority 3 (W3)
Alternative 1
Alternative 2
Alternative 3
Factor 1 (Priority: X1) Factor 2 (Priority: X2) Factor 3 (Priority: X3)
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Figure 4.8: AHP steps in the solution process of FLP (Yang and Lee, 1997)
One quite similar approach referring to an application of AHP in the solution
process of an FLP is the work of Erbıyık, Özcan and Karaboğa (2012). A range of
common and significant location factors or criteria is presented as well as the step by
step implementation of the AHP in the location problem.
Investigating the AHP, both strengths and weaknesses can be mentioned. On one
hand, as far as the strengths are concerned, it is very useful the combination of
qualitative and quantitative factors in the computations of complex problems with
different features. Through this combination, decision makers are appealed by the
ability to incorporate their personal judgments and preferences through a mathematical
validation procedure (i.e. consistency) for subjective choices and use them in
correlation to real numerical data.
Furthermore, AHP is considered as a flexible process because of its decomposed
structure. Specifically, it is very easy to identify areas where wrong judgments occurred
due to possible lack of availability of information and use new data sets that will support
the decision making and change or not the final result (Yang, and Lee, 1997). This issue
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is defined as sensitivity analysis and reflects the agility of the procedure to react in
possible changes and reevaluate its proposed solution.
Overall, AHP is considered as a simple and easy applicable process that does not
require time spending for its learning. This advantage is supported further by the
information technology through computer programs, like the Expert Choice. Expert
Choice is a program with a friendly to user interface that helps decision maker in its
comparisons and the subsequent mathematical computations as well as it provides the
tool of conducting sensitivity analysis.
On the other hand, some concerns are being raised for the arbitrary ranking
between alternatives with similar elements. However, these concerns in real cases stop
exist because it is too rare to encounter situations with similar characteristics (Yang,
and Lee, 1997).
In conclusion, when computer programs like the aforementioned Expert Choice
is not available the control of the consistency can be a very time consuming and
laborious procedure.
4.5.5 Geographic Information Systems (GIS) contribution to Location Analysis
One different approach concerning the solution of facility location problems is
the use of the GIS tool. According to Silva, Egami and Zerbini (2000) at it was cited in
the work of Mapa and Lima (2014), GIS can be defined as an organized collection of
hardware, software, skilled personnel, and geographic data, in order to manage a
database, making the insertion, storage, handling, removal, update, assessment and data
visualization of both spatial and non-spatial data. In the same vein, Murray (2010)
present the definition given in Church and Murray (2009) and describe the GIS as a tool
that combines hardware, software and procedures and helps decision maker in terms of
collecting, managing, analyzing and depicting referenced information.
One major feature of GIS is its database and their incorporation into vector maps
where they can be represented as different levels of layers (Figure 4.9). Many layers
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can be added in order to reflect the different elements that need to be examined in the
different cases problems. As Widaningrum (2015) cites the statement of Mallach (1994)
the GIS is a general term and can be described as an umbrella that encompasses any
data that can be recorded on maps.
Figure 4.9: Distinctive representation of GIS layers (usgs.gov, 2018)
Trubint, Ostojić and Bojović (2006) acknowledge the multidisciplinary nature of
location problems and investigate the use of GIS in the determination of an optimal
retail location. They further distinguish three main database sets, the Demographical,
the Business Demographic and the Database on existing retail outlets whereas they
notice that location decision must be taken under consideration of multiple parameteres
expressed as layers by GIS programmes.
In the same manner, Murray (2010) provides an integration of the GIS in the
concept of location theory. Specifically, he states that GIS supports location analysis in
terms of providing the models’ inputs, in the visualization of acquired data, in the
proposal of location’s problem solution as well as in the advances in the location
models’ theoretical frame.
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Another work that deals with the combination of GIS and location science is the
work of Turk, Kitapci and Dortyol (2014) that refers to the determination of
supermarket locations under the usage of a GIS program. Further references of
integrated GIS - location applications are presented in the work of Mapa and Lima
(2014).
Overall it can be concluded that GIS contributes in the location science by
providing to the decision maker large quantities of data about candidate locations as
well as the tools to analyze these data in order to evaluate better its alternatives and
provide an improved in quality solution. There is a number of GIS software packages
in the market with different and multiple capabilities. Distinctive well-known examples
are the ArcGIS, the MapInfo, the Maptitude and the TransCAD.
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PART 2: CASE STUDY
The current assignment investigates the real case scenario of the opening of a new
supermarket store in central Greece. This new store location is examined under the
collaboration between the author and the owner of the supermarket Mr. Vlachodimos.
At first stage Vlachodimos company’s image will be presented as well with some key
elements about it. At second stage it will be presented the methodology approach, the
conducted results and directions for future research as well.
Chapter 5: Presentation of Vlachodimos company
5.1 Vlachodimos supermarket image
Vlachodimos’ company is a classic supermarket firm that supplies consumers the
best products in terms of quality and price in the area of central Greece. The owner and
CEO of the company is Mr. Konstantinos Vlachodimos. Mr. Vlachodimos started his
activity in retailing by setting up a company in the 1990s that produced and traded
clothes while in 2000 this company was formulated and changed his activity.
Particularly, the main focus was given to the trade of small house devices as well as
some assortment of clothes too. Mr. Vlachodimos’ company took his nowadays form
as a supermarket firm in 2010.
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Image 1: Vlachodimos head offices
Mr. Vlachodimos has one vision, to give products to consumers with high quality
and low prices while at the same to support local communities by signing contracts with
Greek suppliers and more specifically with suppliers that are located in central Greece.
His target is to be one major player in the supermarket sector in central Greece. At
present Vlachodimos supermarket maintains an important market share of 17% in 2017
in his activity area and more specifically in the wide area of Larissa that encompasses
the cities of Larissa and Elassona.
Mr. Vlachodimos gives emphasis on his personnel and as consequence provides
many opportunities for young people to work in his company. Nowadays, firm’s
workforce is 170 people for the seven supermarket stores. Except from Mr.
Vlachodimos the management of the company consists of one Regional director that is
above him in the hierarchy. Furthermore, above the Regional Director the hierarchy
encompasses the Stores’ Directors and the Warehouse Director as well. The stores’
administrative personnel include the Heads of the aisles (e.g. Butchery aisle) and the
employees whereas the warehouse consists of the pickers and the truckers’ drivers.
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5.2 Vlachodimos supermarket supply chain
Mr. Vlachodimos invested the amount of 2.500.000€ in order to set up
Vlachodimos supermarket. That investment refers to the formulation of the warehouse
as an appropriate place for storage of the goods and all the consequent equipment, to
the formulation of stores and to the total merchandise.
5.2.1 Vlachodimos Warehouse and Transportation of merchandise
Vlachodimos supermarket has one central warehouse, outside of the city of
Larissa that supplies all the stores. Head offices of the company are located in that
location too. The size of the warehouse is 8000 square meters, can storage dry and cold
merchandise while its layout is I. In essence, this means that there is one entrance of
unloading the cargo and on the opposite side of the warehouse an exit where it occurs
the loading operation in order the trucks to supply the stores
Image 2: Vlachodimos warehouse
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Furthermore, warehouse consists of 40 corridors while each one of them can
storage 200 pallets. The height of that corridors is 6.5 meters. The employees of the
warehouse use the WMS Exelixis for the computerization of the warehouse whereas
they use six forklifts for their daily operations, one type of lifting usage, three type of
pedestrian usage and five type of manual usage.
Image 3: Lifting forklift
Vlachodimos supermarket supplies its stores in a combined way. It has
outsourced 50% of its transportation operation and 50% is being conducted by his own
truck fleet. That fleet includes threes types of trucks: two fridge types and one normal
type that can be modified in order to support transportation of fridge type cargo. The
company has signed contracts with 50 % international and 50% national suppliers while
the overall number of them is 300. Particularly, 50 suppliers are located in central
Greece, 100 are located in the wide area of Greece and the 150 international ones are
located in Belgium, Spain, Italy, Poland and England. Moreover, the international
supply is conducted through trucks and delivery of containers.
Vlachodimos inventory management comprises a mixed pull and push policy.
Specifically, Vlachodimos supermarket supplies its warehouse with merchandise on a
daily basis for stores’ needs for every day and sensitive products while at the same time
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it buys big amounts of seasonal products (e.g. lent products) in order to take advantage
of economies of scale for better prices.
5.2.2 Vlachodimos Supermarket Store
Mr. Vlachodimos is giving special concern to the customer, to the economy, to
the quality and to the service as it is stated in the website of the company
(http://www.vlachodimos.gr/). Therefore, he pursues to have well-structured and
attractive stores and well-trained personnel in order to facilitate and make pleasant the
shopping operation.
Image 4: Supermarket store outside area
Vlachodimos company consists of seven stores while the owner’s intention is to
open more in the future. Particularly, there is a store in the city of Elassona with a size
of 300 square meters, a new (i.e. 2017) established store in Trikala with a size of 900
square meters and five in the city of Larissa. More specifically, in the city of Larissa it
has the following stores:
• Area of Fillipoupoli: 700 square meters
• Area of Agios Konstantinos: 600 square meters
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• Area of Giannouli: 1400 square meters
• Street Mandilara: 300 square meters
• Fillelinon Street: 250 square meters
Image 5: Supermarket store greengrocer’s aisle
Image 6: Supermarket store aisle
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Chapter 6: Implementation of models and techniques
The models and techniques are selected in such a way to give decision maker the
opportunity to select among different choices for the particular case of a retail store
location. Particularly, the weighted factor rating method was selected in order to reflect
a subjectivity in results while the load distance technique to reflect simplicity and speed
in computations. Furthermore, the AHP method was selected in order to reflect
subjectivity and objectivity in the result by correlating decision maker’s preferences
and judgments with real spatial data. Finally, the Huff model was selected because of
its wide appliance in the investigation of retail stores in the literature, like in the case
of the current assignment, and its target is to reflect an objectivity in results.
At this point, it is needed to mention that according to Mr. Vlachodimos the
candidate locations for the opening of his new supermarket store are the city of Trikala,
the city of Karditsa and the city of Kalabaka, locations in central Greece. Moreover, all
the calculations were conducted in the Microsoft Excel.
6.1 Weighted Factor Rating Method and Facility Location Problem
(FLP)
The weighted factor rating method is a simple and fast in calculations procedure.
After face to face conversations between the author and Mr. Vlachodimos, the
following factors were resulted to be the most important ones. These factors reflect the
preferences and judgments of Mr. Vlachodimos when considering a new location
decision.
The steps of the method are these ones that presented in section 4.5.1. The priority
given by him is indicated as weights in parenthesis (the sum must be equal to 100%)
and presented as follows:
Factors:
• Position (55%)
• Size (25%)
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• Rent (15%)
• Competition (5%)
When considering these factors in the particular case of Trikala, Karditsa and
Kalabaka their values are depicted below.
Factors
Cities Position Size Rent Competitors
Trikala 80 60 55 15
Karditsa 95 80 40 20
Kalabaka 80 80 50 10
Table 6.1: Values of factors for the candidate locations
After the appropriate mathematical procedure, the result of the Weighted Factor Rating
indicates that the better proposal must be the city of Karditsa.
Factors
Cities Total Weighted Scores
Trikala 68
Karditsa 79,25
Kalabaka 72
Table 6.2: Total weight scores for the candidate locations
6.2 Load Distance Technique and Facility Location Problem (FLP)
The basic two elements of this technique are the loads and the distances between
the potential new supermarket stores in the candidate locations (Table 6.3) and the one
and central warehouse of Vlachodimos Company with coordinates Xi= 39.6901240 Yi=
22.3534510.
At the computations the estimated from Mr. Vlachodimos annual pallets (Table
6.4) were used as indicators of loads and three different types of distance metrics.
Moreover, Mr. Vlachodimos stated that the transportation cost of one pallet is 20€ and
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the same among three locations. Therefore, it is omitted in the calculations. In the case
of different transportation cost of one pallet, it should be multiplied by the
corresponding distances.
Referring to the distances, it was used the classic Euclidean distance measurement
(i.e. format 1.3.1), the Great Circle distance (or flying distance) depicted as straight line
in Google map measured in km and the drive time (min) distance in order to reflect a
more realistic investigation (Table 6.5). The drive time distance was calculated through
the estimation of the shortest path in the Google Maps tool and the used departure time
from the warehouse was between 6:00 a.m. and 7:00 a.m., times that the daily products
may be delivered to the stores. The Great Circle distance was computed in tool
https://www.distancefromto.net/ that is based on https://www.openstreetmap.org.
It is necessary to further mention that the candidate locations are represented by
specific streets inside the candidate cities. The selection of that streets was taken
arbitrarily by the author under the only restriction being set by Mr. Vlachodimos to be
central points of that cities. Particularly, the candidate city of Trikala represents the
street of Deligiorgi 20, 42100, the city of Karditsa represent the street of Dimitriou
Lappa 65, 43100 and the city of Kalabaka represent the street of Trikalon 40, 42200.
As far as the Deligiorgi 20 street, 42100 in Trikala is concerned, the particular
selection was taken under the assumption that there is an already operating
Vlachodimos supermarket in that city in the Ploutonos-28s Octomvriou street, 43100.
The target was to avoid the cannibalization effect. In essence, the new store must avoid
to capture customers of the operating store and in addition, it must extend company’s
overall customer population.
Purpose of this technique was to identify the smallest Load-Distance value in
order the transportation cost to be minimized. The resulted LD values were conducted
through format (4.1).
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Candidate locations Coordinate Xi' Coordinate Yi'
Trikala 39.5537280 21.7622158
Karditsa 39.3615680 21,9212430
Kalabaka 39.7053302 21.6264052
Table 6.3: Coordinates of candidate locations
Candidate locations Palletes
Trikala 42000
Karditsa 54000
Kalabaka 54000
Table 6.4: Estimated annual pallets
Distance from Vlachodimos Warehouse
Candidate locations Euclidean
Drive time
(min)
Great Circle
distance (km)
Trikala 0,606764312 60 52,93
Karditsa 0,542911411 60 52,10
Kalabaka 0,727204802 70 62,32
Table 6.5: Calculated distances between candidate locations and warehouse
Load distance computation
Candidate
locations Euclidean
Drive Time
(min)
Great Circle distance
(km)
Trikala 25484,10111 2520000 2223060
Karditsa 29317,2162 3240000 2813400
Kalabaka 39269,0593 3780000 3365280
Table 6.6: Load-Distance resulted values
According to table 6.6 values, the city of Trikala is proposed as the most efficient
choice in order the transportation cost of the shipped pallets be minimized. The result
is the same for the three different distance measurements.
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6.3 Analytic Hierarchy Process (AHP) and Facility Location Problem
(FLP)
In this section, the assignment implements the well-known AHP because of its
importance on multi-criteria problems. Most of the FLP include complex decisions
pertaining to locating single or multiple facilities while considering the existing ones
and evaluating the candidate cites based on multiple criteria.
As in the previous sections, there have been face to face conversations with Mr.
Vlachodimos in the central offices of the firm in order to define his preferences among
the selected criteria. Furthermore, demographic data (referring to drive time
measurement and are related to the year 2015) that are used in the following factors and
in the consequent mathematical calculations are derived through a GIS software, the
ArcGIS in a trial version of it. The computations of AHP were conducted in the
Microsoft Excel.
Methodology Approach
The structure of the current implemented AHP follows the procedure and the
elements described in section 4.5.4.
First of all, it must be chosen the evaluation criteria or factors. The author
investigated a wide range of factors that can an have impact on location decisions and
in consequence according to his proposals and the following judgments of the owner of
the supermarket, he resulted in the following 9 factors. The set of the factors created in
order to reflect objectivity rather than subjectivity (3 subjective and 6 objective):
1. ARPS (quantitative): The average rent is considered as a major factor in
operating firms of the retail sector. Its high expense plays a critical role in the
profitability of a supermarket. It was computed through author’s phone
conversations with real estate offices in the under-investigation areas
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2. ASOH (quantitative): The average size of household indicates the members
of the household and as consequent a possible consumption measurement. It
was computed through provided data from GIS
3. MPUR (qualitative): The multi-purpose shopping is a new element in the
consumer behavior and its contribution to the research of the consumer
patterns is judged as a serious one among the retailers. In essence most of the
competitive literature considers single purpose trips for the supply of the
goods, while in reality it is usual a customer to follow multi-stop or multi-
purpose trips, in order to perform comparison pricing for the former case or
to purchase more than one of type of good in the second case (Eiselt,
Marianov and Drezner, 2015). In the current factor, it was used in a manner
to reflect the market (i.e. number and variety of other stores) in the candidate
locations
4. EOFA (qualitative): The ease of accessibility is widely embedded as a
general factor scheme or as subcategories of it in the investigation of facility
location problem in the literature. In essence, it refers to the ability of
customers and personnel to approach the facilities. In the current assignment
it is considered as a qualitative factor but in other cases, it can be defined as
quantitative by conducting questionnaires on the appropriate parts and
measuring their judgments
5. PURP (quantitative): The purchasing power is one of the most important
indices when examining possible consumer consumption because it can
reflect financial strength or weakness and a standard of living. It was derived
from GIS data
6. INUN (quantitative): The index of unemployment is an important factor too
because it can indicate local society’s inability to do expenses for goods or
services. It was computed through provided data from GIS (in the
computation procedure the unemployment population for 2014 was used
because there were the only derived data from the GIS)
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7. COBE (quantitative): The consumer behavior was chosen in order to reflect
the specific consumption of supermarket products. It was computed through
estimations of provided data from GIS and express the supermarket products
as a percentage of total consumption for goods and services
8. ATTC: (qualitative): The attractiveness of the city was used in order to
propose if one city is suitable to do investments in terms of establishing retail
facilities on its location. One of the most important indicators used in the City
Attractiveness Model proposed by Redevco, a real estate agency that
expertise in the identification of investments in retail locations, shows that
bigger cities have better prospects.
9. COMN (quantitative): The number of competitors finally constitutes one
critical factor for the profitability of every firm dependless to the sector they
belong. They show the possibility of a new company to acquire a percentage
of market share. It was computed through provided data from three different
sources: The Panorama Ellinikon Supermarket, a magazine based in Athens
that deals with research in the supermarket sector of Greece, the Google Maps
and the Xrisos Odigos, a website that provides contacts and addresses about
companies and professionals
Table 6.7 presents the list of the abbreviations of the used factors in the current
AHP
LIST OF ABBREVIATIONS
ARPS Average rent per square
ASOH Average size of household
MPUR Multi - purpose shopping
EOFA Ease of accessibility
PURP Purchasing power per capita
INUN Index of unemployment
COBE Consumer behavior
ATTC Attractiveness of city
COMN Number of competitors
Table 6.7: Abbreviations of the used factors
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The figure 6.1 presents the graphical representation of the current problem as it
was stated in the section 4.5.4
Figure 6.1: Graphical representation of AHP-FLP (Author, 2018)
Thirdly the comparison of the factors must be conducted. The comparison scale
presented in the section 4.5.4 is used for the preferences of the supermarket owner. The
final results for the prioritization of the factors after the pairwise comparison and the
mathematical computations are presented in the following table 6.8. The pairwise
comparisons and the following mathematical computations are presented in the
Appendix of AHP.
Priority
ARPS 0,30328413
ASOH 0,091395829
MPUR 0,068910569
EOFA 0,071009985
PURP 0,147731262
INUN 0,170167159
COBE 0,064652569
ATTC 0,045201826
COMN 0,03764667
Table 6.8: Prioritization of factors
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Using this table, the factor ARPS has the highest priority in the preferences and
the judgments of the supermarket owner while the least priority is given to the factor
COMN.
In the next step, the level of consistency must be checked in the under
investigation problem for the pairwise comparisons. Particularly, the results in table 6.9
present an unacceptable level of consistency because the CR index is smaller than the
value of acceptance 0,10. The pairwise comparisons are considered as reasonable and
the computation procedure can be continued.
CR 0,058453572 ≤ 0,10
Table 6.9 : Consistency Ratio (CR) result
In the following step, the priorities related to qualitative and quantitative factors
for all candidate locations must be computed. As far as the qualitative factors are
concerned, priority is calculated through the comparison analysis and its subsequent
mathematical procedures. Whereas, as far as the quantitative factors are concerned, the
current assignment adopts the calculations procedure of Yang and Lee (1997) as it was
presented in 4.5.4 section. The computational results are the following:
Qualitative Factors
MPUR Priority
Trikala 0,557142857
Karditsa 0,320238095
Kalabaka 0,122619048
Table 6.10: MPUR prioritization on the candidate locations
EOFA Priority
Trikala 0,524675325
Karditsa 0,333766234
Kalabaka 0,141558442
Table 6.11: EOFA prioritization on the candidate locations
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ATTC Priority
Trikala 0,571428571
Karditsa 0,285714286
Kalabaka 0,142857143
Table 6.12: ATTC prioritization on the candidate locations
Quantitative Factors
ARPS Priority
Trikala 0,320922916
Karditsa 0,349239643
Kalabaka 0,329837441
Table 6.13: ARPS prioritization on the candidate locations
ASOH Priority
Trikala 0,325679859
Karditsa 0,327981484
Kalabaka 0,346338657
Table 6.14: ASOH prioritization on the candidate locations
PURP Priority
Trikala 0,310812899
Karditsa 0,32645282
Kalabaka 0,362734281
Table 6.15: PURP prioritization on the candidate locations
INUN Priority
Trikala 0,336180893
Karditsa 0,335579496
Kalabaka 0,328239612
Table 6.16: INUN prioritization on the candidate locations
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COBE Priority
Trikala 0,338096586
Karditsa 0,330540272
Kalabaka 0,331363142
Table 6.17: COBE prioritization on the candidate locations
COMN Priority
Trikala 0,109947644
Karditsa 0,157068063
Kalabaka 0,732984293
Table 6.18: COMN prioritization on the candidate locations
As far as the qualitative factors are concerned, their CR index in the cities’
prioritization (table 6.10 - 6.12) must be checked in order the pairwise comparisons (see
appendix of AHP) of MPUR, EOFA and ATTC to reflect consistency. The procedure
is the same as previously for the consistency check in the factor’s prioritization (table
6.8). The computation results of consistency check in the aforementioned factors for
the candidate cities are presented in the appendix of AHP. The final results presented
below show that there is an acceptable CR index in these factors. The pairwise
comparisons of MPUR, EOFA and ATTC are considered as reasonable and the
computation procedure can be continued.
MPUR
CR’ 0,015797236 ≤ 0,10
Table 6.19 : Consistency Ratio (CR’) result of MPUR
EOFA
CR’’ 0,046395303 ≤ 0,10
Table 6.20 : Consistency Ratio (CR’’) result of EOFA
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ATTC
CR’’’ 0,095785441 ≤ 0,10
Table 6.21 : Consistency Ratio (CR’’’) result of ATTC
In the final stage, the step 6 of 4.5.4 section is implemented. The result indicates
that the candidate location of Trikala seems to be better compared to the other ones.
CANDIDATE CITIES OVERALL PRIORITY RANKING
Trikala 0,357698146
Karditsa 0,327193577
Kalabaka 0,315108277
SUM 1
Table 6.22: Overall priority ranking for the candidate locations
Sensitivity analysis
As a final point to the implementation of AHP in the concept of facility location
problem, a sensitivity analysis was conducted (at the first table presented in the
appendix of AHP) in order to identify the sensitivity of the results to possible changes.
Specifically, three factors were chosen in order to conduct the aforementioned changes.
The second highest factor (INUN) in priority ranking was selected, a medium one
(EOFA) and the second lowest one (ATTC) were selected as well (see priority ranking
table 6.8).
As far as the INUN factor is concerned its strength was reduced by 2 in the
pairwise comparisons, the EOFA’s strength was reduced by 1 and increased by 2 and
factor ATTC’s strength was increased by 3. This approach was followed by the author
in order to reflect possible changes from the low-level factors to the highest ones. The
results that are presented in the tables of AHP appendix indicate that candidate location
of Trikala is superior in any case and therefore it must be selected as the optimal one.
Page | 107
6.4 Huff model and Facility Location Problem (FLP)
In the current section, the assignment will attempt to implement the well known
original Huff gravity model. Although Huff’s model considers only two factors that
indicate consumer behavior, size of stores and distances between customers and stores,
it is a model that can be characterized as a milestone in the Gravity Theory and generally
in retail location theory as well. Therefore, it is formulated in Microsoft Excel in order
to provide a proposal for the opening of a new Vlachodimos supermarket store in the
wide area of Thessaly, in central Greece and more specifically to the aforementioned
candidate cities of Trikala, Karditsa and Kalabaka. Its purpose is to identify the
possibility of customer patronage and expected consumers if Vlachodimos company
opens a supermarket store in the cities.
Methodology Approach
As it was stated before in section 6.2 the cities are represented by specific streets
selected arbitrarily by the author under the only restriction that was set by the owner of
the company, Mr. Vlachodimos, to be central places in these cities. Particularly these
streets are Deligiorgi 20, 42100 with coordinates Xi: 39.553685 and Yi: 21.762194 in
Trikala, Lappa 65, 43100 with coordinates Xi: 39.361568 and Yi: 21.921243 in
Karditsa and Trikalon 40, 42200 with coordinates Xi: 39.705435 and Yi: 21.626377.
Of course, in these streets proper establishments to locate a supermarket may not exist.
For this reason and for the competitors stores’ size measurement that is analysized
below, the current assignment’s results of the implementation of Huff model can be
considered as results derived from a simulation procedure.
The assignment for the previous candidate street-locations investigates the
opening of different size of stores. Particularly, the model will be implemented in order
to find one solution for three different scenarios as is depicted in the following table
6.23. The potential store’s size was selected by the author in order to reflect the current
biggest and smallest Vlachodimos stores’ size and one that is closely related to the rest
ones. According to referred data the size of the smallest Vlachodimos supermarket is
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250 square meters in Fillelinon street while the biggest one is 1400 square meters in the
area of Giannouli.
Vlachodimos New Store Square Meters
Scenarios
1 2 3
250 650 1400
Table 6.23: Scenarios of different size of stores
The assignment at first step identified the competitor’s stores in the previous cities
by conducting an analysis through three different sources (i.e. Panorama Ellinikon
Supermarket, Google Maps, Xrisos Odigos). Specifically, as it was mentioned in
section 6.3 of AHP for the computation of factor COMN (i.e. number of consumers) 20
competitor stores were identified in Trikala, 14 stores in Karditsa and 3 stores in
Kalabaka. Their street location, their coordinates as well as with their size are presented
in Appendix of Huff model.
As far as the competitors’ size of stores is concerned, it is necessary to mention
that there were not any provided data. Therefore, author proceeded with arbitrary
estimations of the sizes according to approaches provided by people that work in the
supermarket sector. As a consequence, the proposed results of Huff model can be
considered as a compilation of a simulation procedure in terms of measurement of
stores’ size and real distance measurement between potential consumers and stores as
it is explained below.
The author approached the establishment of potential consumers in the candidate
cities as follows. First of all, the trade areas that derived from the trial version of ArcGIS
were used. More specifically, according to author’s inputs (image 7), the GIS
programme created three different trade areas for each candidate street-location. The
departure time, towards facility when the origin is considered the home of customers is
Monday 7:00 pm which is a possible supermarket shopping time because many people
have left from work at that time.
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Image 7: Input procedure of trade areas creation (GIS)
Furthermore, two types of distance measurement were used, the walk time and
the drive distance measurement in order to capture different possible realistic scenarios.
As a result, six different trade area maps were derived, three for the walk time distance
measurement in the three candidate locations and three for distance measurement in the
same locations. The first trade area encompasses places that are located at most 5 min
(red line) far away from the candidate streets while the second and third trade area refers
to 10 min (green line) distance and 15 min distance (blue line) correspondingly (Images
8-13).
It is needed to be mention that in the computations of Huff model the derived
trade area of 10 min was used as the most possible real case scenario of consumer
shopping behavior. The 15 min trade is considered to be as a wide area and it is difficult
for a store to have customer patronage for too long distances. The 5 min trade area is
considered as too restricted because it does not capture consumers that can be located
for instance at 6 min far away from the store which is a reasonable trip for supermarket
shopping.
Page | 110
Image 8: Trikala trade area measured in walk time distance (GIS)
Image 9: Trikala trade area measured in drive time distance (GIS)
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Image 10: Karditsa trade area measured in walk time distance (GIS)
Image 11 : Karditsa trade area measured in drive time distance (GIS)
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Image 12: Kalabaka trade area measured in walk time distance (GIS)
Image 13: Kalabaka trade area measured in drive time distance (GIS)
Page | 113
In the previous trade areas of 10 min (green lines), author proceeded in the split
of that areas in four parts in order to locate potential consumers around the candidate
streets. In essence, author used conceivable lines that are crossing exactly in the
candidate streets in order to create a a southwest sublocation, a southeast sublocation,
a northwest sublocation and a northeast sublocation. In each of these locations,
according to supervisor professor’s guidelines, 20 potential customers were chosen,
represented as streets in computations. As a consquence, 80 potential customers were
choosen for each candidate location-street and for the two types of distance
measurement. The procedure was the same among the three streets in order to be
provided a statistical impartiality in the computation analysis.
This led to the overall locating of 480 potential customers points in maps. The
potential customers-streets with their coordinates are presented in the appendix of Huff
model. This procedure was followed by the author in order to provide different potential
customers that are located across all the investigated trade area. The potential
consumers as well as the candidate locations of Vlachodimos stores and the existing
competitor’s locations are represented further in the following maps-images 14-19.
The representation was conducted by the author through
www.mapcustomizer.com tool that uses OpenStreetMap data. The purple points
represent the candidate Vlachodimos stores, the blue point represents the existing
Vlachodimos supermarket in Trikala and red points represent the existing competitors
in the candidate cities.
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Image 14: Representation of candidate street-location (purple point), Vlachodimos
existing supermarket store (blue point), potential consumers (green points) and existing competitors
(red points) in the trade area of walk time approach in city of Trikala
Page | 115
Image 15: Representation of candidate street-location (purple point), potential consumers (green
points) and existing competitors (red points) in the trade area of walk time approach in city of Karditsa
Page | 116
Image 16: Representation of candidate street-location (purple point), potential consumers (green
points) and existing competitors (red points) in the trade area of walk time approach in city of Kalabaka
Page | 117
Image 17: Representation of candidate street-location (purple point), Vlachodimos
existing supermarket store (blue point), potential consumers (green points) and existing competitors
(red points) in the trade area of drive time approach in city of Trikala
Page | 118
Image 18: Representation of candidate street-location (purple point), potential consumers (green
points) and existing competitors (red points) in the trade area of drive time approach in city of Karditsa
Page | 119
Image 19: Representation of candidate street-location (purple point), potential consumers (green
points) and existing competitors (red points) in the trade area of drive time approach in city of
Kalabaka
Author then proceeded in the structure of the Huff model by calculating through
Google maps tool the walk/drive time distance between every aforementioned potential
customer and the potential new Vlachodimos store in the three different scenarios in the
three candidate locations. The same distance calculations are conducted for any
competitors’ store in that candidate locations (see results in the Appendix of Huff
model).
The derived distance data for Vlachodimos potential stores were used in the
computation of nominator of format 3.2.3 while the derived data from competitors
distance measurements were used in the denominator of the same format (the
denominator contains all the stores in the trade area, the potential new ones and the
existing ones). In the computations, it was further used the power λ=3 due to the fact it
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was used by the Huff (1966) for supermarket shopping as it was cited in the work of
Drezner and Drazner (2002). The value of λ is concluded by empirical researches. Of
course, Huff model can be adjusted in order to incorporate different values of λ that
reflect different types of products as it was mentioned before.
The resulted nominators, denominators as well with the probabilities referring to
each customer are presented in the appendix of Huff model. The procedure is ending
by using in the Microsoft Excel the AVERAGE function for each sublocation and
overall using this function in order to conclude in one final customer patronage
possibility for each candidate location-street.
As consequence, it can be stated that around every candidate street this set of
customers (i.e. 80 for every candidate city) is represented by one average possible
customer point. This approach is followed by the author in order to be use the format
3.2.4 that calculates the expected customers for a retail store. The provided population
data derived from the GIS programm for the used trade areas are referring to an overall
population of that areas (derived population data are presented in the appendix of Huff
model). The results as far as the customers’patronage possibility is concerned for the
three scenarios of a new Vlachodimos supermarket store as well with their expected
customers are presented in the following tables and figures. Analysis of the results and
the consequently proposed choice is provided in the chapter 7.
Walk Time Distance Measurement
Vlachodimos New Store Square Meters
Scenarios in Trikala
250 650 1400
City % Pij
5,846178852 12,08434758 19,96000673
City Expected Consumers
64 132 218
Table 6.24: Customers’ patronage possibility and Expected Consumers in Trikala
Page | 121
Vlachodimos New Store Square Meters
Scenarios in Karditsa
250 650 1400
City % Pij
5,242390296 11,1866232 19,28962997
City Expected Consumers
93 197 340
Table 6.25: Customers’ patronage possibility and Expected Consumers in Karditsa
Vlachodimos New Store Square Meters
Scenarios in Kalabaka
250 650 1400
City % Pij
11,0269182 22,22504735 35,83173452
34 68 109
Table 6.26: Customers’ patronage possibility and Expected Consumers in Kalabaka
Figure 6.2: Diagrammatic representation of Expected Consumers in Walk Time approach
64
132
218
93
197
340
34
68
109
0 50 100 150 200 250 300 350 400
250
650
1400
Expected Consumers
Sq
uare
Met
ers
Sel
lin
g A
rea
250 650 1400
KALABAKA 34 68 109
KARDITSA 93 197 340
TRIKALA 64 132 218
Walk Time
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Drive Time Distance Measurement
Vlachodimos New Store Square Meters
Scenarios in Trikala
250 650 1400
City % Pij
3,850319063 6,944919026 10,76103274
City Expected Consumers
529 953 1476
Table 6.27: Customers’ patronage possibility and Expected Consumers in Trikala
Vlachodimos New Store Square Meters
Scenarios in Karditsa
250 650 1400
City % Pij
1,641372876 3,947572591 7,586406597
City Expected Consumers
314 755 1451
Table 6.28: Customers’ patronage possibility and Expected Consumers in Karditsa
Vlachodimos New Store Square Meters
Scenarios in Kalabaka
250 650 1400
City % Pij
11,77254173 25,12118084 40,94852111
City Expected Consumers
239 508 828
Table 6.29: Customers’ patronage possibility and Expected Consumers in Kalabaka
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Figure 6.3: Diagrammatic representation of Expected Consumers in Drive Time approach
529
953
1476
314
755
1451
239
508
828
0 200 400 600 800 1000 1200 1400 1600
250
650
1400
Expected Consumers
Sq
uare
Met
ers
Sel
lin
g A
rea
250 650 1400
KALABAKA 239 508 828
KARDITSA 314 755 1451
TRIKALA 529 953 1476
Drive Time
Page | 124
Page | 125
Chapter 7: Results and Future Reasearch
Methods/ Techniques/ Models Results
Weighted Factor Rating Method Karditsa
Load Distance Technique Trikala
Analytic Hierarchy Process Trikala
Huff Model Trikala/ Drive Time – Karditsa/ Walk Time
Table 6.30: Overall final results
The concluded results from all the implemented methods are presented in table
6.30. According to that table the city of Trikala and the city of Karditsa are the proposed
ones. Purpose of the current assignment was to present different approaches in the
solution procedure of facility location problem in order the decision maker to have the
opportunity to select the best method according to his preferences or judgments or
available data or other affecting factors. Concerning the results and future research, the
following statements can be mentioned.
The Weighted Factor Rating Method is considered as effective and simple in
calculations method and can be adjusted in order to reflect completely the preferences
of decision maker. In the current assignment, factors were identified and classified by
Mr. Vlachodimos. It seems that this technique may omit critical objective factors that
can affect the proposed result but in fact, this can be changed by decision maker himself.
If one decision maker in future wants to present a more subjective result, he can
incorporate into the method objective and real factors that can be derived by empirical
studies, surveys or questionaries in order to identify consumer’s supermarket shopping
behavior. The method’s proposed result of Karditsa seems to be an optimal one in
terms of the agreement with another more objective method that follows.
As far as the Load Distance Technique is concerned, it can be stated that is a
simple but a more objective technique compared to the previous one. In the current
assignment, technique’s proposed result, the city of Trikala, is an informative result
about how the process works rather than a realistic comparative analysis. Particularly,
this is concluded by the fact that there is no differentiation in the transportation cost of
one pallet (i.e. 20€ among all candidate locations) and the differentiation in the distance
Page | 126
measurement is judged as little due to the fact the candidate cities are closely located.
Researchers in the future can conduct an analytical survey about the components of the
transportation cost (e.g. shipment cost, fuel consumption per km etc.) that affect the
result and are necessary to be minimized.
The Analytic Hierarchy Process is a procedure that many researchers have already
used in order to solve multi-criteria problems like the under investigation problem of
facility location. Its proposed result, the city of Trikala, is a combinational result that
encompasses quantitative and qualitative location factors. The selected factors were
chosen by the author and owner’s supermarket judgments in order to reflect already
used factors in location literature, demographical and economic factors that may affect
one location investment policy as well as some factors that reflect consumer behavior.
Study of consumer behavior is considered as a critical factor among retailers.
The result of the current AHP is validated by a mathematical procedure and reflect
in final objectivity rather than subjectivity. Future researchers are recommended to
formulate a fully objective AHP by incorporation of qualitative factors or on the
opposite a fully subjective AHP with only qualitative factors. Moreover it would be
interesting to incorporate subfactors for further analytical study. Although it always
exist the element of subjectivity in final result due to the fact of pairwise comparisons,
the validation of the procedure cannot be challenged or disputed.
Furthermore, reserchers are recommended to use Expert Choice programme in
order to approach the problem in a different way of the assignment’s use of Microsoft
Excel. They can further conclude in an aggregate result by conducting a sensitivity
analysis, in order to examine the result of different scenarios of pairwise comparisons.
According to the author, the most important contribution of the current
assignment is the use of the Huff model in order a solution to be proposed for the
opening of the new Vlachodimos supermarket store. Although Huff’s model is
criticized for its simplicity in terms of considering only two elements that are related to
consumer behavior, it is a landmark in retail location theory. Therefore, it was chosen
for a final solution.
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In the current assignment, the result of the implemented Huff model is being
limited by some challenging assumptions and approaches. First of all, there was a lack
of data concerning the size of the competitor’s stores. Therefore, author proceeded to
arbitrarily estimations of these sizes by conducting personal conversations with people
that work on supermarket sector as it was mentioned before in section 6.4. There was a
further lack of data concerning the population of the sublocations. In general,
demographic data that derived from the GIS programme were referring in an aggregate
manner to the overall trade area (10 min) population. As a consequence, it was
necessary to follow another approach in order to use Huff’s expected consumer format.
Furthermore, there was not any database referring to the exact locations of
supermarket stores. As a result of this, author proceeded in computation analysis in
order to identify the number and the locations of competitors’ stores. Last but not least,
it is necessary to mention that the arbitrary selection of the street inside the candidate
locations does not invalidate the model, because of its agility in future to move to other
location points that will meet decision maker’s preferences. The aforementioned
assumptions may not led, Huff model to propose an accurate result but once found the
exact data for the previous assumptions the model can propose exact solutions.
Despite the limitations of the current implement Huff model, its results are worth
of study. Particularly, according to the estimations of that model, it is concluded that
there is a differentiation between customer patronage possibility and the expected
consumers. For instance, for walk time distance measurement when locating a store
with a size of 1400 square meters, the best possibility for customers’ patronage is
allocated to the city of Kalabaka with a approximate 35,83% while at the same time the
expected consumers’ results indicate that the best proposal must be the city of Karditsa
with 340 population. The different results between the two estimations are the same for
the three store size scenarios. Consequently, the city of Karditsa must be chosen for
the walk time approach in the case of considering the expected consumers as the
decision criterion.
These results conclude to an important found of the current Huff model. More
specifically, the element of population density must be emphasized and how it can
differentiate the final result. The importance of population density is further derived in
Page | 128
the case of drive time distance measurement results. Particularly, the high population
density (see the population of trade areas in the appendix of Huff model) in the drive
time trade area of Karditsa seems to offset the differences in customer’s patronage
possibility among Karditsa and the other two locations (tables 6.27, 6.28, 6.29) when
examing the expected consumers. More precisely, Kalabaka and Trikala have higher
possibilities to be patronaged by customers compared to Karditsa for all the three
scenarios. Nevertheless, the aggregate population in the trade area of Karditsa leads to
a superior expected consumer result comparing to Kalabaka and a very close result
comparing to Trikala. Despite the fact that results of Trikala and Karditsa are quite
similar (in the estimations of expected consumers), the selection of Trikala seems to
be superior in the drive time approach and must be selected.
In conclusion, it’s up to decision maker’s preference, judgment, policy or
available data about what distance measurement will be used or which is the most
suitable criterion, the patronage possibility or the expected consumers.
Future researchers are encouraged to find accurate data that can be used in Huff
model in order to surpass the current limitations. Moreover, it is recommended the
investigation of λ power in the literature for further and more precise analysis in order
to reflect a different type of products. An interesting approach about different values of
λ are presented in the work of Drezner and Drezner (2002). Furthermore, future
researchers are recommended to increase the sample and examine how this change can
improve the final result as well which is the sample that could not lead to further
improvements.
One major drawback of gravity models and in general competitive location theory
is the assumption that the potential customers start their shopping trip from their home.
As a result, the origins i most times in the location literature reflect these homes. This
seems not to capture the general frame of consumer behavior. Many consumers are
going shopping after the end of their work or others are going on multistop shopping
trips. Future researchers are encouraged to investigate further this issue in order to
present a different viewpoint.
Page | 129
As a final point, it is recommended, authors to examine and implement the MCI
model. The MCI model is the model that in essence succeeded Huff model in gravity
theory. It surpassed its major limitation (i.e. the simplicity) and was formulated in a
way to encompass different attractive attributes that can reflect realistic scenarios.
Page | 130
Page | 131
Conclusion
The current assignment investigated an important issue pertaining to the design
of a supply chain network, the facility location problem. The location decisions play a
major role in the profitability of a company due to the fact that they necessitate high
expenses for the establishment of facilities and refer to a long-term horizon investment.
Purpose of the author was to describe the most important attributes in location
theory as well as to present a different viewpoint pertaining to retail location science.
Specifically, a brief history of the location theory was presented, different classification
schemes and the role of distance measurement were also emphasized. As far as the
facility location models are concerned, the most important contributions in location
science were described along with some of their extensions and their solution
approaches.
Furthermore, special focus was given by the author to the description of
competitve location problem and the role of gravity modeling in retail locations because
of the investigation of the opening of a retail store in the part of the case study.
Moreover, it was provided a brief description of other distinctive location problems.
The assignment further presented briefly the main solution approaches in facility
location problem. Its purpose was to provide a general and concise description of these
solutions and to emphasize only to the assignment’s implemented solution methods.
As final contribution to location science, the research of how a theoretical
problem can be applied to a real case scenario was undertaken. Therefore, author in
collaboration with the owner of supermarket Vlachodimos tried to identify which
should be the optimal location for its new supermarket store.
The implemented methods/ techniques excluded candidate city of Kalabaka of
their proposed results and conclude in that Vlachodimos supermarket should open his
new store either to the city of Trikala or to the city of Karditsa. The final choice is
depended on decision maker’s selection of the solution approach that reflects different
Page | 132
viewpoints. Finally, the limitations and assumptions of the methodology approaches
were presented along with some key points in order to surpass such constraints and
elements to examine for further future research.
Page | 133
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APPENDICES
A) AHP
Pairwise comparison table
Pairwise
comparison
More important
criterion How much more important
Numerical
Rating
MPUR - ATTC MPUR Moderately more important 3
MPUR - EOFA EOFA
Equally to Moderately more
important 2
MPUR - ARPS ARPS Strongly more important 5
MPUR -COMN MPUR
Equally to Moderately more
important 2
MPUR - ASOH ASOH
Equally to Moderately more
important 2
MPUR - INUN INUN Moderately more important 3
MPUR - PURP PURP Moderately more important 3
MPUR - COBE COBE
Equally to Moderately more
important 2
ATTC - EOFA EOFA Moderately more important 3
ATTC - ARPS ARPS Strongly more important 5
ATTC - COMN ATTC
Equally to Moderately more
important 2
ATTC - ASOH ASOH
Equally to Moderately more
important 2
ATTC - INUN INUN Moderately more important 3
ATTC - PURP PURP Moderately more important 3
ATTC - COBE COBE
Equally to Moderately more
important 2
EOFA - ARPS ARPS
Moderately to Strongly more
important 4
EOFA - COMN EOFA
Equally to Moderately more
important 2
EOFA - ASOH ASOH
Equally to Moderately more
important 2
EOFA - INUN INUN Moderately more important 3
EOFA - PURP PURP Moderately more important 3
EOFA - COBE COBE
Equally to Moderately more
important 2
ARPS - COMN ARPS Strongly more important 5
ARPS - ASOH ARPS
Moderately to Strongly more
important 4
ARPS - INUN ARPS
Moderately to Strongly more
important 4
ARPS - PURP ARPS Moderately more important 3
ARPS - COBE ARPS Strongly more important 5
COMN - ASOH ASOH Moderately more important 3
Page | 146
COMN - INUN INUN Moderately more important 3
COMN - PURP PURP Moderately more important 3
COMN -COBE COBE
Equally to Moderately more
important 2
ASOH - INUN INUN Moderately more important 3
ASOH - PURP PURP Moderately more important 3
ASOH - COBE ASOH
Equally to Moderately more
important 2
INUN - PURP PURP
Equally to Moderately more
important 2
INUN - COBE INUN Moderately more important 3
PURP - COBE PURP Moderately more important 3
Pairwise Comparison Matrix
Pairwise Comparison Matrix
AR
PS
AS
OH
MP
UR
EO
FA
PU
RP
INU
N
CO
BE
AT
TC
CO
MN
AR
PS
1
4
5
4
3
3
5
5
5
AS
OH
0,2
5
1
2
2
0,3
333
333
33
0,3
333
333
33
2
2
3
MP
UR
0,2
0,5
1
0,5
0,3
333
333
33
0,3
333
333
33
2
3
2
EO
FA
0,2
5
0,5
2
1
0,3
333
333
33
0,3
333
333
33
0,5
3
2
PU
RP
0,3
333
333
33
3
3
3
1
0,5
3
3
3
Page | 147
INU
N
0,3
333
333
33
3
3
3
2
1
3
3
3
CO
BE
0,2
0,5
0,5
2
0,3
333
333
33
0,3
333
333
33
1
2
2
AT
TC
0,2
0,5
0,5
0,3
333
333
33
0,3
333
333
33
0,3
333
333
33
0,5
1
2
CO
MN
0,2
0,3
333
333
33
0,5
0,5
0,3
333
333
33
0,3
333
333
33
0,5
0,5
1
SU
M
2,9
666
666
67
13
,333
333
33
17
,5
16
,333
333
33
8
6,5
17
,5
22
,5
23
Computation of factors’ prioritization
AR
PS
AS
OH
MP
UR
EO
FA
PU
RP
INU
N
CO
BE
AT
TC
CO
MN
Pri
ori
ty
AR
PS
0,3
370
786
52
0,3
0,2
857
142
86
0,2
448
979
59
0,3
75
0,4
615
384
62
0,2
857
142
86
0,2
222
222
2
0,2
173
913
04
0,3
032
841
3
AS
OH
0,0
842
696
63
0,0
75
0,1
142
857
14
0,1
224
489
8
0,0
416
666
67
0,0
512
820
51
0,1
142
857
14
0,0
888
888
9
0,1
304
347
83
0,0
913
958
29
MP
UR
0,0
674
157
3
0,0
375
0,0
571
428
57
0,0
306
122
45
0,0
416
666
67
0,0
512
820
51
0,1
142
857
14
0,1
333
333
3
0,0
869
565
22
0,0
689
105
69
Page | 148
EO
FA
0,0
842
696
63
0,0
375
0,1
142
857
14
0,0
612
244
9
0,0
416
666
67
0,0
512
820
51
0,0
285
714
29
0,1
333
333
3
0,0
869
565
22
0,0
710
099
85
PU
RP
0,1
123
595
51
0,2
25
0,1
714
285
71
0,1
836
734
69
0,1
25
0,0
769
230
77
0,1
714
285
71
0,1
333
333
3
0,1
304
347
83
0,1
477
312
62
INU
N
0,1
123
595
51
0,2
25
0,1
714
285
71
0,1
836
734
69
0,2
5
0,1
538
461
54
0,1
714
285
71
0,1
333
333
3
0,1
304
347
83
0,1
701
671
59
CO
BE
0,0
674
157
3
0,0
375
0,0
285
714
29
0,1
224
489
8
0,0
416
666
67
0,0
512
820
51
0,0
571
428
57
0,0
888
888
9
0,0
869
565
22
0,0
646
525
69
AT
TC
0,0
674
157
3
0,0
375
0,0
285
714
29
0,0
204
081
63
0,0
416
666
67
0,0
512
820
51
0,0
285
714
29
0,0
444
444
4
0,0
869
565
22
0,0
452
018
26
CO
MN
0,0
674
157
3
0,0
25
0,0
285
714
29
0,0
306
122
45
0,0
416
666
67
0,0
512
820
51
0,0
285
714
29
0,0
222
222
2
0,0
434
782
61
0,0
376
466
7
Consistency Computation
AR
PS
AS
OH
MP
UR
EO
FA
PU
RP
INU
N
CO
BE
AT
TC
CO
MN
Su
m
Div
isio
n
AR
PS
0,3
032
841
3
0,3
655
833
16
0,3
445
528
44
0,2
840
399
42
0,4
431
937
85
0,5
105
014
78
0,3
232
628
47
0,2
260
091
3
0,1
882
333
52
2,9
886
608
24
9,8
543
264
47
AS
OH
0,0
758
210
33
0,0
913
958
29
0,1
378
211
38
0,1
420
199
71
0,0
492
437
54
0,0
567
223
86
0,1
293
051
39
0,0
904
036
5
0,1
129
400
11
0,8
856
729
12
9,6
905
178
55
Page | 149
MP
UR
0,0
606
568
26
0,0
456
979
14
0,0
689
105
69
0,0
355
049
93
0,0
492
437
54
0,0
567
223
86
0,1
293
051
39
0,1
356
054
8
0,0
752
933
41
0,6
569
404
9,5
332
314
19
EO
FA
0,0
758
210
33
0,0
456
979
14
0,1
378
211
38
0,0
710
099
85
0,0
492
437
54
0,0
567
223
86
0,0
323
262
85
0,1
356
054
8
0,0
752
933
41
0,6
795
413
14
9,5
696
585
53
PU
RP
0,1
010
947
1
0,2
741
874
87
0,2
067
317
07
0,2
130
299
56
0,1
477
312
62
0,0
850
835
8
0,1
939
577
08
0,1
356
054
8
0,1
129
400
11
1,4
703
618
98
9,9
529
502
5
INU
N
0,1
010
947
1
0,2
741
874
87
0,2
067
317
07
0,2
130
299
56
0,2
954
625
23
0,1
701
671
59
0,1
939
577
08
0,1
356
054
8
0,1
129
400
11
1,7
031
767
4
10
,008
845
11
CO
BE
0,0
606
568
26
0,0
456
979
14
0,0
344
552
84
0,1
420
199
71
0,0
492
437
54
0,0
567
223
86
0,0
646
525
69
0,0
904
036
5
0,0
752
933
41
0,6
191
456
98
9,5
765
056
9
AT
TC
0,0
606
568
26
0,0
456
979
14
0,0
344
552
84
0,0
236
699
95
0,0
492
437
54
0,0
567
223
86
0,0
323
262
85
0,0
452
018
3
0,0
752
933
41
0,4
232
676
12
9,3
639
493
89
CO
MN
0,0
606
568
26
0,0
304
652
76
0,0
344
552
84
0,0
355
049
93
0,0
492
437
54
0,0
567
223
86
0,0
323
262
85
0,0
226
009
1
0,0
376
466
7
0,3
596
223
88
9,5
525
682
41
9,6
780
614
4
Ave
rag
e:λm
ax
9
Nu
mb
er o
f
fact
ors
: N
Page | 150
Consistency Index (Ci)- Consistency Ratio (CR)
Computation of Ci
Ci 0,08475768
Computation of CR
CR 0,058453572 ≤ 0,10
Estimations of the prioritization score of each factor to each location
Qualitative factors
Prefencerences of each factor to each location
1.MPUR Trikala Karditsa Kalabaka
Trikala 1 2 4
Karditsa 0,5 1 3
Kalabaka 0,25 0,333333333 1
SUM 1,75 3,333333333 8
2.EOFA Trikala Karditsa Kalabaka
Trikala 1 2 3
Karditsa 0,5 1 3
Kalabaka 0,333333333 0,333333333 1
SUM 1,833333333 3,333333333 7
3.ATTC Trikala Karditsa Kalabaka
Trikala 1 2 4
Karditsa 0,5 1 2
Kalabaka 0,25 0,5 1
SUM 1,75 3,5 7
Quantitave factors
Actual Data
Page | 151
Real data Ti
Trikala Karditsa Kalabaka
4.ARPS (€/m2) 9,25 8,5 9
5.ASOH (number of members) 2,85 2,83 2,68
6.PURP (€) 9912,61 9437,71 8493,73
7.INUN (%) 11,16 11,18 11,43
8.COBE (%) 51,18 52,35 52,22
9.COMN (number) 20 14 3
ASOH was computed through average estimations of GIS data. It was choosed not
integer numbers in order to reflect a slight difference between cities. Cities’
characteristics encompass many common elements
Mathematical Formulation for the computation of prioritization score of
each factor to each location
Qualitative factors
MPUR Trikala Karditsa Kalabaka Priority
Trikala 0,571428571 0,6 0,5 0,557142857
Karditsa 0,285714286 0,3 0,375 0,320238095
Kalabaka 0,142857143 0,1 0,125 0,122619048
EOFA Trikala Karditsa Kalabaka Priority
Trikala 0,545454545 0,6 0,428571429 0,524675325
Karditsa 0,272727273 0,3 0,428571429 0,333766234
Kalabaka 0,181818182 0,1 0,142857143 0,141558442
ATTC Trikala Karditsa Kalabaka Priority
Trikala 0,571428571 0,571428571 0,571428571 0,571428571
Karditsa 0,285714286 0,285714286 0,285714286 0,285714286
Kalabaka 0,142857143 0,142857143 0,142857143 0,142857143
Quantitave factors
Normilization and Prioritization
ARPS Wi = (100/Ti) Priority
Trikala 10,81081081 0,320922916
Karditsa 11,76470588 0,349239643
Kalabaka 11,11111111 0,329837441
Page | 152
SUM 33,6866278
ASOH Wi = (100/Ti) Priority
Trikala 35,0877193 0,325679859
Karditsa 35,33568905 0,327981484
Kalabaka 37,31343284 0,346338657
SUM 107,7368412
PURP Wi = (100/Ti) Priority
Trikala 0,01008816 0,310812899
Karditsa 0,010595791 0,32645282
Kalabaka 0,01177339 0,362734281
SUM 0,032457342
INUN Wi = (100/Ti) Priority
Trikala 8,960573477 0,336180893
Karditsa 8,944543828 0,335579496
Kalabaka 8,748906387 0,328239612
SUM 26,65402369
COBE Wi = (100/Ti) Priority
Trikala 1,953888238 0,338096586
Karditsa 1,910219675 0,330540272
Kalabaka 1,914975105 0,331363142
SUM 5,779083018
COMN Wi = (100/Ti) Priority
Trikala 5 0,109947644
Karditsa 7,142857143 0,157068063
Kalabaka 33,33333333 0,732984293
SUM 45,47619048
Consistency Computation of MPUR, EOFA and ATTC in the pairwise
comparisons of page 150
Page | 153
MPUR Trikala Karditsa Kalabaka Sum Division
Trikala
0,55714285
7 0,64047619 0,49047619
1,68809523
8 3,02991453
Karditsa
0,27857142
9
0,32023809
5
0,36785714
3
0,96666666
7
3,01858736
1
Kalabak
a
0,13928571
4
0,10674603
2
0,12261904
8
0,36865079
4
3,00647249
2
3,01832479
4
Average:
λ max
3
Number
of
factors: N
Consistency Index (Ci)- Consistency Ratio (CR’)
Ci CR’
0,009162397 0,015797236 ≤ 0,10
EOFA Trikala Karditsa Kalabaka Sum Division
Trikala
0,52467532
5
0,66753246
8
0,42467532
5
1,61688311
7
3,08168316
8
Karditsa
0,26233766
2
0,33376623
4
0,42467532
5
1,02077922
1
3,05836575
9
Kalabak
a
0,17489177
5
0,11125541
1
0,14155844
2
0,42770562
8
3,02140672
8
3,05381855
2
Average:
λ max
3
Number
of
factors: N
Consistency Index (Ci)- Consistency Ratio (CR’’)
Ci CR’’
0,026909276 0,046395303 ≤ 0,10
ATTC Trikala Karditsa Kalabaka Sum Division
Trikala
0,57142857
1
0,57142857
1
0,57142857
1
1,71428571
4 3
Karditsa
0,28571428
6
0,28571428
6
0,28571428
6
0,85714285
7 3
Kalabak
a 0,19047619
0,14285714
3
0,14285714
3
0,47619047
6
3,33333333
3
3,11111111
1
Average:
λ max
3
Number
of
factors: N
Page | 154
Consistency Index (Ci)- Consistency Ratio (CR’’’)
Ci CR’’’
0,055555556 0,095785441 ≤ 0,10
Sensitivity Analysis of factors’ pairwise comparisons
INUN (-2)
CR 0,090747446 ≤ 0,10
CANDIDATE CITIES OVERALL PRIORITY RANKING
Trikala 0,362774921
Karditsa 0,324435236
Kalabaka 0,312789843
SUM 1
EOFA (+2/-1)
(+2)
CR 0,0865507 ≤ 0,10
CANDIDATE CITIES OVERALL PRIORITY RANKING
Trikala 0,361951565
Karditsa 0,327759075
Kalabaka 0,310289359
SUM 1
(-1)
CR 0,048340429 ≤ 0,10
CANDIDATE CITIES OVERALL PRIORITY RANKING
Trikala 0,35531567
Karditsa 0,326458205
Page | 155
Kalabaka 0,318226125
SUM 1
ATTC (+3)
CR 0,079545684 ≤ 0,10
CANDIDATE CITIES OVERALL PRIORITY RANKING
Trikala 0,360640382
Karditsa 0,326980375
Kalabaka 0,312379243
SUM 1
B) Huff Model
Square Meters Selling area scenarios of new Vlachodimos store
Vlachodimos New Store Square Meters
Scenarios
1 2 3
250 650 1400
Existing competitor’s location adress, coordinates and size stores in candidate
cities of Trikala, Karditsa and Kalabaka
Trikala
Competitor Store
(C.S) j Adress
Latitude -
Longitude
Square Meters of
selling area
1 VLACHODIMOS
Ploutonos - 28s
Octovriou, 42100
39.556296,
21.7692640 900
2
ALFA BITA
BASILOPOULOS
S.A Pilis 39, 42100
39.542747,
21.758761 700
3
ALFA BITA
BASILOPOULOS
S.A Karditsis 65, 42100
39.544810,
21.776773 500
4 LIDL
1 km Trikalon -
Pilis, 42100
39.537908,
21.75438 600
5 LIDL
1 km E.O Trikalon
- Larisas, 42100
39.554968,
21.790405 600
6 MASOUTIS Kondili 73, 42100
39.563072,
21.770785 500
Page | 156
7 MASOUTIS
Kondili 15 -
Tiouson (Ermou),
42132
39.557187,
21.768358 400
8 MASOUTIS
Deligiorgi 2 -
Ziaka, 42131
39.553238,
21.764532 700
9 MASOUTIS
Eleftherias -
Meteoron, 42131
39.547741,
21.762780 500
10 MASOUTIS
Tsitsani - Averof ,
42132
39.554760,
21.772551 500
11 GALAXIAS
Kondili 50 -
Solomou, 42100
39.559943,
21.769865 800
12 GALAXIAS
Kokkinos Pirgos,
42100
39.558164,
21.763057 500
13 MY MARKET
Kalabakas 69,
42100
39.564704,
21.751932 900
14 MY MARKET
Kolokotroni 47 -
Theodosopoulou,
Bara, 42100
39.559197,
21.773124 2000
15 MY MARKET
Karditsis 58 -
Thiras, Saragia,
42100
39.546664,
21.773773 700
16 MY MARKET
Lakmonos 5 -
Psaron,
Trikaioglou, 42100 39.555584,
21.761483 900
17 SKLAVENITIS Kalabakas 6, 42131
39.559364,
21.757482 1000
18 SKLAVENITIS
Kolokotroni -
Ptolemaiou, 42132
39.553506,
21.770553 600
19 SKLAVENITIS Vironos 4, 42131
39.554915,
21.766576 300
20 SKLAVENITIS
Peraivou -
Kolokotroni, 42132
39.557047,
21.771594 300
Karditsa
Competitor Store
(C.S) j Adress
Latitude -
Longitude
Square Meters
of selling area
1
ALFA BITA
BASILOPOULOS
S.A
Koumoundourou
- Palaiologou,
43100
39.361937,
21.922644 700
2 LIDL
2 km Karditsas -
Trikalon, 43131
39.385181,
21.907983 700
3 LIDL
Leoforos
Dimokratias 156
39.364081,
21.937404 600
4 MY MARKET
Lappa - Iroon
Politexniou,
43100
39.361150,
21.923030 900
5 MY MARKET
Ipsilantou 21,
43100
39.363435,
21.928550 500
6
ELLINIKA
MARKET
Saradaporou -
Euzonon, 43100
39.357455,
21.911835 300
7 MASOUTIS
Karaiskaki 95 -
Thessaliotidos,
43132
39.364074,
21.921169 300
8 MASOUTIS Averof 22, 43132
39.362400,
21.921640 700
9 GALAXIAS
Periviou -
Trikalon -
Taliadourou,
43131
39.364921,
21.917025 1100
Page | 157
10
SUPER MARKET
GRIGORIADIS -
KRITIKOS Tzella 29, 43100
39.364776,
21.919242 900
11
KALIAKOUDAS
MARKET
Lappa - Agrafon,
43100
39.359807,
21.930790 300
12 SKLAVENITIS
Trikalon 185,
43100
39.370729,
21.917056 900
13 SKLAVENITIS
E.O Karditsas -
Volou, 43100
39.363928,
21.943852 1200
14 SKLAVENITIS
Garidikiou -
Amerikis, 43100
39.366394,
21.928706 600
Kalabaka
Competitor Store
(C.S) j Adress
Latitude -
Longitude
Square Meters
of selling area
1 LIDL
Trikalon 156,
42200
39.700208,
21.636243 600
2 GALAXIAS
Sidirodromon 17,
42200
39.704208,
21.624456 1000
3 SKLAVENITIS Ramidi 14, 42200
39.705278,
21.627681 1200
Potential consumers in the candidate cities of Trikala, Karditsa and Kalabaka,
represented as streets with the corresponding coordinates
Area of Consumers (walk time ≤ 10 min)
TRIKALA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - Northwest
1 Satovriandou, 42100 39.557903, 21.760639
2 Irakleous Retou 1 -11, 42100 39.555678, 21.754920
3 Thiseos 8, 42100 39.557083, 21.755703
4 Liakata 7, 42100 39.555474, 21.760675
5 Ntai 12 - 22, 42100 39.555964, 21.757291
6 Profiti Ilia, 42100 39.559047, 21.760909
7 Averof 429, 42100 39.559125, 21.758718
8 Ipirou 37, 42100 39.557941, 21.758183
9 Pindou 14, 42100 39.557480, 21.757194
10 Perikelous 9, 42100 39.556612, 21.756585
11 Perikleous 16, 42100 39.556508, 21.754790
12 Mavrokordatou 105, 42100 39.554934, 21.755682
13 Voulgari 7, 42100 39.554288, 21.761393
14 Dervenakion 4, 42100 39.556503, 21.758645
15 Lakmonos 34, 42100 39.557624, 21.759810
16 Amalias 58, 42100 39.556632, 21.761388
17 Xarilaou Trikoupi 30, 42100 39.555689, 21.759197
18 Deligiorgi 51, 42100 39.554871, 21.758435
Page | 158
19 Deligiorgi 39 - 43, 42100 39.554461, 21.759748
20 Mavrokordatou 66, 42100 39.554118, 21.757945
Group B - NorthEast
21 Agion Anargiron 7, 42100 39.558046, 21.764098
22 Kalamatas 17, 42100 39.554712, 21.763939
23 Othonos 4, 42100 39.554748, 21.768526
24 25th Martiou 47, 42100 39.556703, 21.766173
25 Ermoupoleos 12, 42100 39.557104, 21.767805
26 Koletti 8, 42100 39.553550, 21.763726
27 Xarilaou Trikoupi 19, 42100 39.554648, 21.762613
28 Amalias 46, 42100 39.555699, 21.763331
29 Adam 50, 42100 39.553922, 21.766195
30 Kanouta 3-7, 42100 39.555229, 21.765749
31 Stournara 7, 42100 39.556251, 21.767339
32 Zappa 6, 42100 39.554451, 21.769733
33
Iroon Alvanikou Metopou 12,
42100 39.556120, 21.769036
34 25th Martiou 31, 42100 39.557007, 21.764320
35 Agiou Ikonomiou 14, 42100 39.559246, 21.764121
36 Mataragiotou 5, 42100 39.558405, 21.766225
37 Ipsiladou 29 , 42100 39.557404, 21.766631
38 Diodopou 9, 42100 39.557905, 21.765093
39 Nelson 3, 42100 39.555822, 21.765056
40 Ptolemeou 1, 42100 39.553426, 21.770241
Group C - SouthWest
41 E.O Servon - Elatis, 42100 39.554090, 21.755697
42 Mavrokordatou 71 - 55, 42100 39.553353, 21.760192
43 Kakoplevriou 1, 42100 39.551615, 21.758752
44 Meteoron 20, 42100 39.549221, 21.760918
45 Xirokabou, 42100 39.550466, 21.758299
46 Valaoritou 92, 42100 39.552790, 21.757958
47 Mavrokordatou 26 - 28, 42100 39.552923, 21.761764
48 Voulgari 32, 42100 39.552212, 21.760261
49 Kanari 34 - 38, 42100 39.551975, 21.761460
50 Kakoplevriou 4, 42100 39.552161, 21.757921
51 Iasonos 40, 42100 39.551326, 21.760180
52 Orthovouniou, 42100 39.550876, 21.757513
53 Malakasiou 1, 42100 39.549867, 21.757499
54 Agiou Nikolaou 9 -11, 42100 39.549395, 21.758987
55 Koronidous 52, 42100 39.550550, 21.760532
56 Garidikiou 43, 42100 39.550647, 21.759102
57 Palaioxoriou 4, 42100 39.548293, 21.760638
58 Souliou, 42100 39.553500, 21.758930
59 Iasonos 35, 42100 39.550759, 21.761984
60 Meteoron 28, 42100 39.549943, 21.760239
Page | 159
Group D - SouthEast
61 Meteoron 8, 42100 39.547537, 21.762602
62 Garivaldi 14 -16, 42100 39.553419, 21.768360
63 Asklipiou 23, 42100 39.551022, 21.765750
64 Koronidous 7 - 3, 42100 39.549507, 21.764180
65 Fleming 18, 42100 39.550175, 21.768298
66 Kavrakou 26, 42100 39.548762, 21.767442
67 Asklipiou 72, 42100 39.548362, 21.764511
68 Erganis 5 - 7, 42100 39.550941, 21.764704
69 Sigrou 5, 42100 39.549023, 21.765720
70 Alexandras 44, 42100 39.549268, 21.762887
71 Omirou 11 - 17, 42100 39.550486, 21.767056
72 Alexandras 32, 42100 39.550405, 21.763339
73 Alexandras 20, 42100 39.551345, 21.763723
74 Mavrokordatou 14, 42100 39.552460, 21.763412
75 Korai 2 - 4, 42100 39.553529, 21.764971
76 Ippokratous 13, 42100 39.551899, 21.765492
77 Asklipiou 18, 42100 39.553215, 21.766717
78 Kapodistriou 20 , 42100 39.551978, 21.767538
79 Kapodistriou 47, 42100 39.550981, 21.769380
80 Apollonos 26, 42100 39.552629, 21.769167
KARDITSA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - NorthWest
1 Lappa 44, 43130 39.361931, 21.919044
2 Voriou Ipirou 10, 43100 39.364840, 21.915870
3 Vasiardani 89, 43100 39.364073, 21.914823
4 Vasiardani 70, 43100 39.363835, 21.916301
5 Blatsouka 26, 43100 39.365751, 21.919104
6 Palaiologou 20, 43100 39.362175, 21.920596
7 Laxana 17 - 19 , 43100 39.363039, 21.917293
8 Karaiskaki 47, 43100 39.362936, 21.920273
9 Tzella 32 - 34, 43100 39.363642, 21.919012
10 Karaiskaki 21, 43100 39.363770, 21.920932
11 Taliadourou 1, 43100 39.364595, 21.921349
12 Iezekil 42, 43100 39.366116, 21.920764
13 Tzella 45, 43100 39.366706, 21.919897
14 Riga Feraiou 6, 43100 39.366724, 21.921544
15 Trikalon 94, 43100 39.365879, 21.917754
16 Karamanli 85 - 95, 43100 39.364746, 21.917345
17 Tzella 44, 43100 39.364586, 21.919255
18 Karamanli 56, 43100 39.362077, 21.916638
Page | 160
19 Laxana 20 - 30, 43100 39.363085, 21.915284
20 Saradaporou 53 - 55, 43100 39.362304, 21.913617
Group B - NorthEast
21 Koummoundourou 31, 43100 39.362657, 21.922987
22 Ipsiladou 56, 43100 39.363916, 21.925699
23 Diakou 58, 43100 39.362028, 21.928955
24 Aza 19 - 21, 43100 39.366308, 21.924778
25 Palaiologou 23 , 43100 39.361936, 21.922142
26 Iezikil 32 - 34, 43100 39.365808, 21.922339
27 Plastira, 43100 39.363467, 21.922666
28 Xatzimitrou 65, 43100 39.361601, 21.923900
29 Diakou 23, 43100 39.362912, 21.924329
30 Palaiologou 88, 43100 39.360637, 21.929247
31 Nikitara 80, 43100 39.362778, 21.928322
32 Kaminadon 11, 43100 39.365743, 21.925948
33 Valtadorou, 43100 39.365275, 21.923721
34 Papandreou 20 - 22, 43100 39.366791, 21.923260
35 Ipsiladou 21, 43100 39.364378, 21.923480
36 Nikitara 21, 43100 39.363259, 21.925935
37 Palaiologou 76, 43100 39.361037, 21.927457
38 Palaiologou 41, 43100 39.361258, 21.925846
39 Diakou 41 - 45, 43100 39.362422, 21.926939
40 Averof 17, 43100 39.361840, 21.927842
Group C - SouthWest
41 Milou 1 -7, 43100 39.360541, 21.919105
42 Karaiskaki 115 - 123, 43100 39.358016, 21.916869
43 Thessaliotidos 96, 43100 39.358924, 21.918447
44 Kresnas 27, 43100 39.360388, 21.915041
45 Venizelou 17, 43100 39.361597, 21.917875
46 Giannitson 10, 43100 39.361700, 21.914508
47 Karamanli 51, 43100 39.361167, 21.916346
48 Karaiskaki 143-149, 43100 39.356678, 21.915831
49 Kedravrou 30 - 40, 43100 39.356661, 21.918613
50 Solomou 6, 43100 39.357761, 21.919622
51 Thessaliotidos 82, 43100 39.358490, 21.920847
52 Makedonias 11, 43100 39.360001, 21.920472
53 Venizelou 48, 43100 39.360996, 21.921156
54 Argitheas 9, 43100 39.360133, 21.917483
55 Saradaporou 67, 43100 39.361149, 21.913488
56 Navarinou 18, 43100 39.358148, 21.914945
57 Ploutarxou 1, 43100 39.359473, 21.913963
58 Karamanli 34, 43100 39.359857, 21.916003
59 Thessalonikis 3, 43100 39.359494, 21.918315
60 Karamanli 27 - 29, 43100 39.358952, 21.915714
Page | 161
Group D - SouthEast
61 Floraki 49, 43100 39.355173, 21.921798
62 Iroon Politexniou 60, 43100 39.360379, 21.922865
63 Androutsou 35, 43100 39.356564, 21.921189
64 Thessaliotidos 70, 43100 39.358140, 21.923073
65 Xarilaou Floraki 20, 43100 39.357622, 21.921703
66 Giannitson 119, 43100 39.359145, 21.928300
67 Emmanouil 30, 43100 39.360180, 21.927578
68 Evripidou 3, 43100 39.356311, 21.923677
69 Thessaliotidos 51, 43100 39.357591, 21.925941
70 Lappa 141 - 147, 43100 39.360169, 21.928927
71 Giannitson 104, 43100 39.359546, 21.926110
72 Skirou, 43100 39.358832, 21.924897
73 Politexniou 68-72, 43100 39.359429, 21.922548
74 Koumoundourou 51, 43100 39.361528, 21.922656
75 Lappa 80 - 82, 43100 39.360888, 21.924686
76 Isaiou 6, 43100 39.356404, 21.920456
77 Botsi 76, 43100 39.357605, 21.924366
78 Giannitson 100, 43100 39.359939, 21.924534
79 Plastira 57 - 61, 43100 39.360439, 21.921607
80 Skirou, 43100 39.358354, 21.927077
KALABAKA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - NorthWest
1 Ramou 1, 42200 39.706769, 21.625496
2 Plastira 13, 42200 39.708049, 21.621329
3 Liakata 14, 42200 39.709247, 21.624036
4 Ioanninon 26, 42200 39.705428, 21.619321
5
Patriarxou Dimitriou 39,
42200 39.709206, 21.618285
6 Masouta 2 - 8, 42200 39.706216, 21.624792
7 Ipirou 17, 42200 39.706908, 21.621663
8 Ipirou 2 - 4, 42200 39.706311, 21.623051
9 Perikleous 6, 42200 39.706973, 21.624480
10 Liakata 1, 42200 39.707371, 21.623794
11 Dimitriou 6, 4220 39.707473, 21.622137
12 Meteoron 18, 42200 39.706981, 21.619972
13 Plastira 27, 42200 39.708179, 21.619463
14 Ipirou 45, 42200 39.706054, 21.618811
15 Makedonias 16, 42200 39.707409, 21.618432
16 Kaiki 6, 42200 39.708495, 21.624218
17 Aggeli, 42200 39.708651, 21.626043
18 Vlaxava 26, 42200 39.707745, 21.625315
Page | 162
19 Kalostipi 22, 42200 39.709359, 21.625697
20 Metaxa 11, 42200 39.709004, 21.622717
Group B - NorthEast
21 Agias Triados 1, 42200 39.707628, 21.630248
22 Venizelou 3 - 9, 42220 39.705248, 21.628807
23 Zioga 17, 42200 39.705367, 21.631429
24 Agias Triados 17, 42200 39.709044, 21.630248
25 Koupi 29, 42200 39.706172, 21.634407
26 Iona, 42200 39.706733, 21.626673
27 Ramidi 15, 42200 39.705806, 21.627760
28 Ramidi 38, 42200 39.707128, 21.628573
29 Vlaxava 35, 42200 39.708375, 21.626951
30 Skarlatou 11, 42200 39.709576, 21.627134
31 Leukosias, 42200 39.710362, 21.628965
32 Leukosias, 42200 39.710681, 21.630217
33 Agias Triados 11, 42200 39.710053, 21.631076
34 Sopotou 22, 42200 39.709832, 21.629509
35 Vlaxava 8, 42200 39.709049, 21.628725
36 Katsika 2-8, 42200 39.708142, 21.628549
37 Klisthenous 15, 42200 39.706497, 21.630081
38 Kalostipi 11 - 19, 42200 39.706760, 21.632763
39 E.O Peristeras, 42200 39.704759, 21.634935
40 Katsika 17, 42200 39.706420, 21.631798
Group C - SouthWest
41 E.O Servion - Elatis, 42200 39.705038, 21.620732
42 Meteoron 23, 42200 39.707499, 21.619259
43 Meteoron 6, 42200 39.706054, 21.621110
44 Meteoron 2, 42200 39.704177, 21.622421
45 Kondili 28 , 42200 39.704767, 21.626409
46 Ikonomou 11, 42200 39.705391, 21.622883
47 Averof 13-17, 42200 39.704791, 21.624685
48 Averof 53, 42200 39.703838, 21.626148
49 E.O Servion - Elatis, 42200 39.703670, 21.624970
50 E.O Servion - Elatis, 42200 39.704683, 21.623326
51 Lesvou 25, 42200 39.702995, 21.621510
52 Alexiou, 42200 39.700717, 21.625942
53 Vitouma 8, 42200 39.702103, 21.626048
54 E.O Servion - Elatis, 42200 39.703161, 21.625967
55 Stavrodromi Takou, 42200 39.702770, 21.624819
56 Katsimitrou 6, 42200 39.703536, 21.623054
57 Lesvou 14, 42200 39.704291, 21.620780
58 E.O Servion - Elatis, 42200 39.704123, 21.624028
59 Xatzipetrou 15, 42200 39.705592, 21.624450
60 Xatzipetrou 1 - 7, 42200 39.706026, 21.623769
Page | 163
Group D - SouthEast
61 Trikalon 15 - 17, 42200 39.705048, 21.627311
62 Pindou 89, 42200 39.702198, 21.629121
63
E.O Trikalon - Ioanninon,
42200 39.702104, 21.633259
64 18th Octovriou, 42200 39.700259, 21.627369
65 Stavrodromi Takou, 42200 39.702095, 21.625241
66 18th Octovriou, 42200 39.701514, 21.626994
67 Dimoula 8, 42200 39.702567, 21.627514
68
Megalou Alexandrou 14,
42200 39.703468, 21.626911
69 Xazipetrou 40, 42200 39.703620, 21.628134
70 E.O Peristeras, 42200 39.703850, 21.634934
71 E.O Peristeras, 42200 39.703574, 21.633664
72 Ikonomou 7, 42200 39.704840, 21.630669
73 Koupi 11, 42200 39.704703, 21.632980
74 Rouvali 7, 42200 39.704059, 21.631848
75 Agiou Stefanou 1-5, 42200 39.703053, 21.632128
76 Pindou 40, 42200 39.701239, 21.629759
77 Deligianni 37, 42200 39.702126, 21.631508
78 Trikalon 87 - 91, 42200 39.703640, 21.630199
79 Ikonomou 2 - 12, 42200 39.704544, 21.628827
80 Deligianni 18, 42200 39.703019, 21.629366
Area of Consumers (drive time ≤ 10 min)
TRIKALA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - NorthWest
1 Kalabakas 128, 42100 39.571793, 21.747445
2 Bernadaki 7, 42100 39.563701, 21.755774
3 Periferiaki Trikalon, 42100 39.561693, 21.735917
4
E.O Trikalon - Peristeras,
42100 39.575230, 21.728892
5
E.O Trikalon - Rizomaton,
42100 39.580688, 21.755542
6
E.O Trikalon - Ioanninon,
42100 39.581078, 21.742011
7 Panourgia, 42100 39.574212, 21.738827
8
E.O Trikalon - Peristeras,
42100 39.568735, 21.735576
9 Mavrounioti, Pirgos, 42100 39.564409, 21.744121
10 Agamonioti, 42100 39.557169, 21.749160
11
E.O Trikalon - Kato Elatis,
42100 39.560637, 21.742479
12 Periferiki Trikalon, 42100 39.553587, 21.734138
13 Aliakmonos 13, Pirgos, 42100 39.563122, 21.749207
Page | 164
14 Loudia, Pirgos, 42100 39.559570, 21.750641
15
E.O Servion - Elatis 262,
42100 39.566308, 21.748596
16 Koraka, 42100 39.558708, 21.756982
17 Trikoupi 41, 42100 39.556045, 21.757940
18 Narkissou 32, 42100 39.568294, 21.753058
19 Megarxis, 42100 39.567596, 21.758584
20 Voulgari 5, 42100 39.554402, 21.761463
Group B - NorthEast
21 Kondili 40 - 50, 42100 39.559616, 21.769361
22 E.O Larisas - Trikalon, 42100 39.557745, 21.785197
23 Analipseos 4 - 14, 42100 39.570918, 21.771887
24 Balakouras 9, 42100 39.569462, 21.778859
25
E.O Trikalon - Rizomaton,
42100 39.573611, 21.762389
26 Stefanou Sarafi 21, 42100 39.556059, 21.765455
27 Tsakalof 13, 42100 39.556179, 21.774670
28 Agaristis 11, 42100 39.560800, 21.775685
29
E.O Larissas - Trikalon,
42100 39.564197, 21.780113
30 Ioustinianou 8, 42100 39.564597, 21.773569
31 Mela 20-22, 42100 39.560686, 21.765827
32
E.O Trikalon - Ioanninon,
42100 39.567523, 21.769290
33 Mesoxoras 26, 42100 39.568960, 21.761800
34 Dragoumi 18, 42100 39.562858, 21.770270
35 Vasili Tsitsani 81, 42100 39.555812, 21.780274
36 Ellis 41, 42100 39.558796, 21.777351
37 Sokratous 26, 42100 39.557710, 21.771237
38 Seferi 9, 42100 39.571146, 21.767156
39 Xasion 20, 42100 39.564865, 21.767902
40
Agion Anargiron 10 - 12,
42100 39.558509, 21.764017
Group C - SouthWest
41 Theotokou 9, 42100 39.553974, 21.753484
42 Agiou Georgiou 22, 42100 39.546248, 21.761173
43 Pilis, 42100 39.539109, 21.753682
44 Promitheos 10, 42100 39.541237, 21.754251
45 Periferiaki Trikalon, 42100 39.528738, 21.745650
46 Meteoron 62, 42100 39.552973, 21.757457
47 Koronidos 54, 42100 39.550785, 21.759889
48 Adioxias, Pirgetos, 42100 39.552242, 21.748430
49 Origeni, 42100 39.549389, 21.752300
50
Agias Paraskevis 26-34,
42100 39.547943, 21.758409
51 Dimitras 11, 42100 39.544724, 21.753721
52 Magira 362, 42100 39.537744, 21.757698
Page | 165
53 Periferiaki 28, 42100 39.534206, 21.748909
54 Periferiaki Trikalon, 42100 39.540616, 21.735874
55 Trapezoudos 8, 42100 39.551910, 21.744272
56 Malakasiou 2, 42100 39.550231, 21.757415
57 Kerasoudos 4, 42100 39.554746, 21.742698
58 Periferiaki Trikalon, 42100 39.546025, 21.747175
59 Mavrokordatou 39, 42100 39.552886, 21.761757
60 Flamouliou 27, 42100 39.540624, 21.758854
Group D - SouthEast
61 Kalipsous 14, 42100 39.549270, 21.773803
62 Agias Monis 5, 42100 39.528809, 21.766501
63 Karditsis, 42100 39.538881, 21.784119
64 Innouson 106, 42100 39.548039, 21.781003
65 Kefallinias 28, 42100 39.542793, 21.769288
66 Deligiorgi 8, 42100 39.553218, 21.763742
67 Aristotelous 10, 42100 39.548879, 21.763626
68 Katsimidou 38, 42100 39.551609, 21.775956
69 Patoulias 3, 42100 39.553403, 21.779247
70 Alexandras 27 - 29, 42100 39.550939, 21.763650
71 Fleming 9, 42100 39.551159, 21.767540
72 Thoukidou 5, 42100 39.546036, 21.766027
73 Garivaldi 5 - 7, 42100 39.553096, 21.767487
74 Karparthou 4, 42100 39.545519, 21.772917
75 Flamouliou 89, 42100 39.535013, 21.763155
76 Eleutherias, Karies, 42100 39.532538, 21.775657
77 Papamanou, 42100 39.538444, 21.769347
78 Rizariou 10, 42100 39.543609, 21.786752
79 Fleming 35, 42100 39.549512, 21.768897
80 Arianou 30 - 32, 42100 39.546769, 21.770786
KARDITSA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - NorthWest
1
E.O Karditsas - Argitheas,
43100 39.375590, 21.883622
2 Griva, 43100 39.374775, 21.907835
3 Fanariou 127, 43100 39.368969, 21.902779
4 Taliadourou 54, 43100 39.366025, 21.913246
5 Stratigou Papagou 72, 43100 39.372593, 21.915960
6 Averof 1 - 3, 43100 39.363150, 21.919088
7 Taliadorou 13, 43100 39.364828, 21.919571
8 Agiou Serafeim 26, 43100 39.368963, 21.915593
9 Dodekanisou 10 - 16 , 43100 39.371917, 21.921169
10 Dorieon 11, 43100 39.375909, 21.919598
Page | 166
11 Periferiaki Karditsas, 43100 39.384206, 21.914758
12 Trikalon 275 - 283, 43100 39.377320, 21.911653
13 Dodekanisou 92 - 96, 43100 39.370752, 21.910967
14
E.O Trikalon - Karditsas 2,
43100 39.389751, 21.904105
15 Mandilara 20 - 30, 43100 39.369859, 21.906057
16
E.O Karditsas - Argitheas,
43100 39.371949, 21.894351
17 Agiou Serafim 111, 43100 39.375103, 21.904213
18 Pasiali 7, 43100 39.374565, 21.911207
19 Fanariou 84, 43100 39.367505, 21.909940
20 Papapostolou 4 - 14, 43100 39.378944, 21.915753
Group B - NorthEast
21 Periferiaki Karditsas, 43100 39.382497, 21.934605
22 Dionisiou 15, 43100 39.370370, 21.933169
23 E.O Karditsas - Volou, 43100 39.369168, 21.973071
24 Gardikiou 49, 43100 39.366341, 21.931589
25 Panagouli 8 - 14, 43100 39.361714, 21.932812
26 Kapodistriou 82 , 43100 39.364465, 21.933139
27 Iezekiil 28, 43100 39.365661, 21.923290
28 Papandreou 81 ,43100 39.370061, 21.925550
29
Stratiogou Papagou 2 - 10,
43100 39.373276, 21.922451
30 Kapodistriou 27 , 43100 39.368153, 21.921327
31 Tertipi 159, 43100 39.375079, 21.927364
32
Karagianopoulou 17 - 25,
43100 39.370936, 21.928687
33 Sotiros 1 - 3, 43100 39.366808, 21.926872
34 Tertipi 2, 43100 39.363988, 21.929171
35 Titaniou 32- 36, 43100 39.368916, 21.934070
36 Periferiaki Karditsas, 43100 39.372949, 21.940367
37
Leoforos Dimokratias 50,
43100 39.364240, 21.938595
38 E.O Karditsas - Larisas, 43100 39.364614, 21.948157
39 Ipsiladou 40, 43100 39.364140, 21.924373
40 Kondili 18 ,43100 39.369078, 21.928674
Group C - SouthWest
41
E.O Karditsas - Kastanias,
43100 39.342960, 21.886221
42 Dragatsaniou, 43100 39.360029, 21.904007
43 Monis Petras, 43100 39.366171, 21.902400
44 Mavromixali 12 - 20, 43100 39.365908, 21.907809
45 Saradaporou 156 - 162, 43100 39.358171, 21.911977
46 Tamasiou, 43100 39.360089, 21.908577
47
E.O Killithirou - Neraidas,
43100 39.318813, 21.910940
48 Alevadon 19, 43100 39.354246, 21.915352
49 Thessaliotidos 100, 43100 39.359060, 21.917870
Page | 167
50 Niala, 43100 39.352758, 21.907468
51
E.O Karditsas - Rentina,
43100 39.339363, 21.914856
52 Kefalinias, 43100 39.350997, 21.911477
53
E.O Karditsas - Kastanias,
43100 39.355027, 21.909015
54
E.O Karditsas - Kastanias,
43100 39.335474, 21.859633
55
E.O Krias Vrisis - Agiou
Georgiou, 43100 39.325392, 21.873564
56
E.O Karditsas - Rentina,
43100 39.315978, 21.916740
57
E.O Karditsas - Rentina,
43100 39.345659, 21.919961
58 Ithakis 20, 43100 39.363887, 21.912185
59 Ploutonos 10, 43100 39.360965, 21.914884
60 Eptanisou, 43100 39.363623, 21.903854
Group D - SouthEast
61 Alevadon, 43100 39.354828, 21.933658
62
E.O Mouzakiou -
Palaioxoriou, 43100 39.350540, 21.940931
63
E.O Karditsas - Rentina,
43100 39.348828, 21.925494
64 Giannitson 100, 43100 39.360100, 21.923701
65 Koumoundrou 87, 43100 39.358836, 21.921757
66 Isaiou 6, 43100 39.356261, 21.920297
67 Alevadon, 4300 39.352994, 21.922329
68 Aristotelous 80, 43100 39.352587, 21.929401
69 Agrafon 62, 43100 39.355439, 21.926685
70 Thessaliotidos 46, 43100 39.357794, 21.925175
71 Lappa 58, 43100 39.361429, 21.921756
72 Lappa 129, 43100 39.360465, 21.927528
73 Xatzimitrou 32 - 36, 43100 39.362567, 21.924168
74 Athanasiou Diakou 58, 43100 39.362091, 21.928763
75 Fleming, 43100 39.351643, 21.936513
76 Leoforos Dimokratias, 43100 39.360065, 21.938147
77 Lappa 134 - 130, 43100 39.359606, 21.931791
78 Alkiviadou 20, 43100 39.355503, 21.924276
79 Makedonias 70 - 76, 43100 39.358635, 21.927476
80 Georgiou Souri, 43100 39.356420, 21.930387
KALABAKA
Potential Consumer (P.C) i
(address) Latitude - Longitude
Group A - NorthWest
1 Eparxiaki Odos, 42200 39.711551, 21.585834
2 E.O Kalabakas, 42200 39.715705, 21.618474
3
Patriarxou Dmitriou 15 - 7,
42200 39.707598, 21.621372
Page | 168
4 Kalostipi 17, 42200 39.709328, 21.625738
5 Aggeli, 42200 39.708648, 21.625734
6
E.O Trikalon - Ioanninon,
42200 39.717173, 21.597332
7 Diogenous 22-30, 42200 39.706679, 21.618309
8 Liakata 8, 42200 39.708048, 21.623783
9 Ioanninon 1, 42200 39.706950, 21.622709
10 Xatzipetrou 2-4, 42200 39.706152, 21.623539
11
E.O Trikalon - Ioanninon,
42200 39.710231, 21.609781
12
Patriarxou Dimitriou 39,
42200 39.709274, 21.616821
13 Ploutarxou, 42200 39.708973, 21.622428
14
Patriarxou Dimitriou 31,
42200 39.707831, 21.618267
15 Diogenous 40, 42200 39.705832, 21.615180
16
Patriarxou Dimitriou 39,
42200 39.711986, 21.616945
17
E.O Kalabakas - Agiou
Stefanou, 42200 39.719299, 21.615068
18
E.O Kalabakas,Meteora,
42200 39.722531, 21.629716
19
E.O Kalabakas,Meteora,
42200 39.724665, 21.619026
20 Lesvou 14, 42200 39.704264, 21.620686
Group B - NorthEast
21 Koupi 27, 42200 39.706342, 21.633218
22 E.O Kalabakas 39.706320, 21.642334
23 Kleisthenous 10, 42200 39.706488, 21.629661
24 Sokratous 6, 42200 39.708055, 21.629686
25 Agias Triados 11, 42220 39.708913, 21.630450
26 E.O Peristeras, 42220 39.705650, 21.633625
27 E.O Peristeras, 42220 39.705148, 21.637414
28
E.O Meteoron - Kallitheas,
42200 39.712704, 21.651262
29 E.O Peristeras, 42220 39.705547, 21.635264
30 Ramidi 18 - 20, 42200 39.706102, 21.627933
31 Gika, 42200 39.708231, 21.627506
32 Ramou 12, 42200 39.707157, 21.626369
33 Zioga 15, 42200 39.705542, 21.631341
34 Leukosias, 42200 39.710013, 21.627617
35 Vlaxava 9, 42200 39.709169, 21.629011
36 Sopotou 15, 42200 39.710344, 21.629554
37 Leukosias, 42200 39.710600, 21.631169
38 Leukosias, 42200 39.711744, 21.630458
39
E.O Meteoron - Kallitheas,
42200 39.717056, 21.643146
40 E.O Kalabakas, 42200 39.711858, 21.659204
Group C - SouthWest
Page | 169
41 Katsimitrou 6, 42100 39.700371, 21.622375
42 E.O Servion - Elatis,42200 39.703614, 21.625242
43 Athanasoula, 42200 39.702876, 21.623749
44 Terma Vitouma, 42200 39.699207, 21.619403
45 Alexiou, 42200 39.701779, 21.620993
46 E.O Servion - Elatis, 42200 39.705064, 21.622214
47 E.O Sarakinas - Diavas, 42200 39.695893, 21.590027
48 Ioanninon 49, 42200 39.705046, 21.618667
49 Terma Vitouma, 42200 39.694403, 21.622415
50 Terma Vitouma, 42200 39.697475, 21.617979
51 Diogenous 38, 42200 39.705815, 21.617412
52 Platonos 2 - 6, 42200 39.705837, 21.621278
53
E.O Kalabakas - Krias Vrisis,
42200 39.701638, 21.612934
54
E.O Trikalon - Kalabakas,
42200 39.705552, 21.611573
55 Diava, 42200 39.690988, 21.580635
56
E.O Kalabakas - Krias Vrisis,
42200 39.698685, 21.579359
57 E.O Kalabakas, 42200 39.705651, 21.603056
58 E.O Sarakinas - Diavas, 42200 39.692755, 21.602836
59 Vitouma Terma, 42200 39.701135, 21.619096
60 Averof 13, 42200 39.704980, 21.624192
Group D - SouthEast
61 Deligianni 2, 42200 39.703519, 21.628300
62 18th Octovriou, 42200 39.698896, 21.627886
63 E.O Peristeras, 42200 39.691374, 21.629641
64
E.O Trikalon - Ioanninon 50,
42200 39.698812, 21.639585
65 E.O Sarakinas - Diavas, 42200 39.662414, 21.641048
66 E.O Peristeras, 42200 39.703643, 21.636583
67 Rouvali 7, 42200 39.703998, 21.631957
68
E.O Trikalon - Ioanninon 50,
42200 39.695097, 21.641233
69
E.O Trikalon - Ioanninon 50,
42200 39.700241, 21.635264
70 E.O Peristeras, 42200 39.704162, 21.634450
71 Ikonomou 2 - 12, 42200 39.704556, 21.628892
72 E.O Peristeras, 42200 39.702200, 21.634151
73 Vitouma, 42200 39.697860, 21.624729
74 Alexiou, 42200 39.700711, 21.625811
75 Pindou 40 , 42200 39.701184, 21.630317
76 E.O Peristeras, 42220 39.697335, 21.631346
77
E.O Neas Zois, Theopetra
42200 39.673508, 21.681046
78
E.O Meteoron - Kallitheas,
42200 39.700503, 21.649362
79
E.O Trikalon - Ioanninon 50,
42200 39.701448, 21.639414
Page | 170
80
E.O Trikalon - Ioanninon ,
42220 39.676822, 21.659089
Computation of distance between potential consumers and candidate
locations-streets as well as all existing competitor’s stores in cities of Trikala,
Karditsa and Kalabaka
In the following tables P.T represent the aforementioned Potential Consumers and
the C.S represent the aforementioned Competitor’s Stores while the C.S.0 is the
proposed new stores. Values of the following tables are meausered in minutes.
Dij - Walk time ≤ 10 min
Tri
ka
la
New
Sto
re
Co
mp
etit
ors
j
Po
ten
tia
l
Co
nsu
mer
s i
C.S
.0
C.S
.1
C.S
.2
C.S
.3
C.S
.4
C.S
.5
C.S
.6
C.S
.7
C.S
.8
C.S
.9
C.S
.10
C.S
.11
C.S
.12
C.S
.13
C.S
.14
C.S
.15
C.S
.16
C.S
.17
C.S
.18
C.S
.19
C.S
.20
P.T
.1
8
12
25
26
33
35
17
11
9
17
15
15
6
5
18
21
5
6
15
9
15
P.T
.2
9
19
22
30
30
42
25
18
13
14
21
22
14
9
26
26
9
8
19
15
23
P.T
.3
10
18
25
30
33
41
24
17
12
17
21
21
12
8
24
25
8
6
19
14
21
P.T
.4
4
12
21
24
29
35
20
12
6
13
14
16
7
1
19
19
1
8
13
8
16
Page | 171
P.T
.5
8
15
23
27
31
38
22
15
10
15
18
19
10
5
22
23
5
7
16
12
19
P.T
.6
9
12
26
27
34
35
14
11
10
19
15
13
6
7
17
23
7
8
16
11
15
P.T
.7
10
16
27
29
35
39
21
14
12
19
18
18
10
7
22
24
7
2
18
12
19
P.T
.8
10
15
26
28
34
38
21
14
11
19
18
18
9
6
22
24
6
3
17
12
18
P.T
.9
10
16
27
29
35
39
22
15
12
19
18
19
10
7
22
24
7
4
18
12
19
P.T
.10
9
16
24
28
32
39
23
16
11
17
19
20
11
6
23
24
6
6
17
13
20
P.T
.11
10
18
23
30
31
41
25
18
13
16
21
22
13
8
25
26
8
7
19
14
22
P.T
.12
8
17
21
29
29
40
24
17
12
13
20
21
12
7
24
25
7
9
18
14
21
P.T
.13
2
11
18
23
26
34
19
11
5
11
14
15
7
2
18
18
2
10
12
7
15
P.T
.14
7
14
23
27
31
37
21
14
9
15
17
18
9
4
21
22
4
7
16
10
18
P.T
.15
8
13
24
26
33
36
19
12
9
17
16
16
7
4
19
22
4
4
15
10
16
P.T
.16
6
10
23
24
31
33
18
10
7
15
13
14
5
3
17
19
3
7
14
7
14
Page | 172
P.T
.17
5
13
22
25
30
36
21
13
8
14
16
17
9
3
20
21
3
8
14
9
17
P.T
.18
5
15
21
26
29
37
23
14
8
13
17
18
10
5
22
21
5
9
15
11
18
P.T
.19
3
13
19
24
27
36
21
13
7
12
16
17
9
4
20
20
4
10
13
9
17
P.T
.20
6
16
20
27
28
39
24
16
9
12
19
20
12
6
23
22
6
10
16
12
20
P.T
.21
8
7
25
26
33
31
12
5
8
17
12
9
1
5
12
21
5
11
13
8
10
P.T
.22
4
8
19
20
27
30
16
8
3
11
10
12
7
3
15
16
3
11
9
3
11
P.T
.23
9
3
22
18
30
25
13
5
7
14
5
9
10
8
10
14
8
16
3
3
7
P.T
.24
9
4
24
23
32
28
12
3
8
16
8
7
4
6
11
18
6
13
9
4
8
P.T
.25
10
3
24
22
32
27
9
1
8
16
8
5
5
9
8
18
9
15
8
4
5
P.T
.26
2
10
17
20
26
31
18
10
1
9
12
14
10
5
17
15
5
13
9
5
13
P.T
.27
2
9
19
21
27
32
18
9
4
12
12
14
7
3
17
17
3
10
10
5
13
P.T
.28
5
8
21
22
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16
14
11
18
16
9
27
27
16
P.T
.78
5
5
46
18
3
9
17
10
7
14
13
8
23
26
13
P.T
.79
2
3
42
22
2
12
14
7
3
11
9
11
20
29
14
P.T
.80
10
9
49
16
8
7
19
15
11
18
17
6
27
24
12
Kalabaka New store Competitors j
Potential Consumers i C.S.0 C.S.1 C.S.2 C.S.3
P.T.1 2 16 4 4
P.T.2 7 20 7 9
P.T.3 7 20 8 8
P.T.4 9 21 7 11
P.T.5 10 23 10 12
P.T.6 2 15 3 4
P.T.7 6 19 6 8
P.T.8 5 17 4 6
P.T.9 3 17 5 5
P.T.10 4 18 5 6
P.T.11 5 18 6 7
P.T.12 8 21 6 10
P.T.13 8 21 8 10
P.T.14 10 22 8 11
P.T.15 10 24 9 12
P.T.16 6 19 7 8
P.T.17 6 19 7 7
P.T.18 4 18 5 5
P.T.19 7 20 8 8
P.T.20 8 21 8 9
P.T.21 7 13 10 6
P.T.22 4 12 7 2
P.T.23 6 11 10 5
P.T.24 8 16 10 6
P.T.25 10 12 14 9
Page | 185
P.T.26 2 15 4 3
P.T.27 2 13 6 1
P.T.28 4 15 7 3
P.T.29 5 18 7 5
P.T.30 8 19 10 9
P.T.31 10 20 12 9
P.T.32 10 18 13 9
P.T.33 10 17 12 9
P.T.34 8 17 11 7
P.T.35 7 17 9 6
P.T.36 5 15 8 5
P.T.37 5 12 8 4
P.T.38 10 12 12 8
P.T.39 10 9 14 9
P.T.40 9 12 12 8
P.T.41 9 19 5 11
P.T.42 9 22 8 10
P.T.43 7 19 5 9
P.T.44 9 16 4 10
P.T.45 1 13 4 3
P.T.46 6 18 3 8
P.T.47 3 15 1 5
P.T.48 3 13 3 4
P.T.49 6 14 2 6
P.T.50 6 16 2 7
P.T.51 10 18 6 12
P.T.52 10 13 9 10
P.T.53 9 12 8 8
P.T.54 5 13 3 6
P.T.55 10 13 8 10
P.T.56 10 15 6 12
P.T.57 10 19 6 12
P.T.58 5 15 1 7
P.T.59 3 15 2 5
P.T.60 4 16 3 6
P.T.61 1 12 5 1
P.T.62 8 9 7 7
P.T.63 9 5 11 8
P.T.64 10 13 9 10
P.T.65 10 13 9 10
P.T.66 8 11 7 8
P.T.67 6 11 5 5
P.T.68 4 12 4 3
P.T.69 4 11 5 3
Page | 186
P.T.70 10 7 13 9
P.T.71 10 7 13 8
P.T.72 5 10 8 4
P.T.73 7 9 11 6
P.T.74 7 9 11 6
P.T.75 8 7 11 7
P.T.76 10 7 9 10
P.T.77 9 7 9 8
P.T.78 5 9 8 4
P.T.79 3 11 7 2
P.T.80 6 9 7 5
Dij – Drive time ≤ 10 min
Tri
ka
la
New
Sto
re
Co
mp
etit
ors
j
Po
ten
tia
l
Co
nsu
mer
s i
C.S
.0
C.S
.1
C.S
.2
C.S
.3
C.S
.4
C.S
.5
C.S
.6
C.S
.7
C.S
.8
C.S
.9
C.S
.10
C.S
.11
C.S
.12
C.S
.13
C.S
.14
C.S
.15
C.S
.16
C.S
.17
C.S
.18
C.S
.19
C.S
.20
P.T
.1
6 9 8 11 9 7 4 6 8 7 8 5 6 1 7 9 6 4 10 7 10
P.T
.2
7 9 9 12
10 9 6 8 8 9 9 7 6 2 10
11 5 3 9 7 10
P.T
.3
9 12 7 10 7 12
10
11
10
10
12
11 9 6 13
11 8 7 12
10
13
P.T
.4
9 13 9 13
10
10 8 10
11
10
12 9 10 5 11
12 8 7 13
11
14
Page | 187
P.T
.5
9
11
13
12
14
8
6
7
11
12
9
6
7
7
8
12
8
8
11
9
10
P.T
.6
10
13
11
14
12
10
8
10
11
11
11
9
10
5
11
13
9
7
13
11
14
P.T
.7
9
12
10
13
11
9
7
9
10
10
11
8
9
4
10
12
8
6
12
10
13
P.T
.8
8 12
8 11
9 9 7 9 9 9 10
8 9 4 10
11
7 6 12
10
13
P.T
.9
8 11
9 12
10
10
8 10
9 9 11
9 8 5 11
10
7 6 11
9 12
P.T
.10
5 9 6 9 7 12
9 8 6 6 8 9 5 5 10
8 4 5 9 6 9
P.T
.11
7 10
6 11
7 10
8 9 8 8 10
8 7 4 11
9 6 5 10
8 11
P.T
.12
8 12
7 11
8 10
8 9 10
9 11
8 9 5 11
11
7 6 12
10
13
P.T
.13
6 9 7 10 8 9 7 8 7 7 9 8 6 3 10 8 5 4 9 7 10
P.T
.14
5
9
6
9
7
10
8
7
6
6
8
8
5
3
10
8
4
4
9
6
9
P.T
.15
6 10 7 10 9 9 6 9 7 7 9 7 6 3 9 9 5 4 10 8 10
P.T
.16
4 7 7 10 8 10 8 6 6 7 7 7 4 3 8 8 3 1 7 5 7
Page | 188
P.T
.17
3
6
6
9
7
9
7
5
4
6
6
6
3
4
8
7
1
2
6
4
7
P.T
.18
8
10
9
12
11
7
5
7
9
9
9
6
7
2
8
11
6
4
10
8
11
P.T
.19
6
8
9
12
11
8
5
7
7
9
7
6
4
6
8
11
4
4
8
6
8
P.T
.20
1 6 5 7 6 9 7 5 3 4 6 6 5 5 8 6 2 4 6 3 7
P.T
.21
8 5 11
8 12
6 1 1 9 9 5 1 4 8 2 8 6 6 5 8 4
P.T
.22
9 5 11
6 12
2 3 5 8 8 3 4 7 7 6 6 8 8 5 7 6
P.T
.23
8 10
12
11
13
6 4 6 10
11
8 5 6 6 7 10
7 7 9 8 9
P.T
.24
9 9 13
9 14
5 4 6 11
12
6 5 7 7 7 9 8 8 8 10
9
P.T
.25
8 9 11
10
13 6 4 5 9 10 7 4 5 5 7 10 6 6 9 8 8
P.T
.26
5
3
8
8
10
6
4
2
6
8
3
3
2
5
5
6
3
3
3
5
4
P.T
.27
9 4 10 5 11 4 4 5 7 8 2 4 7 10 2 5 9 9 4 6 3
P.T
.28
10 6 11 6 13 5 2 3 9 9 3 2 5 8 5 6 8 8 5 7 6
Page | 189
P.T
.29
8
6
12
7
13
3
2
4
9
9
4
3
6
6
5
6
7
7
6
8
7
P.T
.30
9
7
12
8
14
4
2
3
10
11
5
2
5
7
5
8
7
7
7
9
6
P.T
.31
6
7
9
10
11
7
3
3
7
9
7
2
2
6
4
10
4
4
7
6
6
P.T
.32
7 8 11
8 12
4 2 4 8 10
5 3 4 5 5 8 5 5 7 7 7
P.T
.33
7 9 11
10
12
6 3 5 8 10
7 4 6 5 6 10
5 6 9 7 8
P.T
.34
9 6 12
8 14
5 1 2 10
11
5 1 4 7 3 8 7 7 6 9 5
P.T
.35
10
5 10
5 12
3 4 5 8 8 2 4 7 10
5 5 10
10
4 6 5
P.T
.36
10
5 11
6 12
4 2 3 8 9 3 2 5 9 5 6 8 8 5 7 6
P.T
.37
9 2 12 9 14 7 4 3 10
11 6 3 6 9 2 9 7 7 6 9 2
P.T
.38
8
9
12
10
13
6
4
5
9
11
7
4
5
6
7
10
7
7
9
8
8
P.T
.39
8 8 11 9 13 5 2 4 9 11 6 3 3 6 5 9 6 6 8 8 7
P.T
.40
5 5 9 10
10 8 4 4 7 8 5 3 1 6 6 8 4 4 5 6 6
Page | 190
P.T
.41
5
9
6
9
7
12
9
8
6
6
8
9
5
5
10
8
4
5
9
6
9
P.T
.42
4
8
2
6
4
9
10
9
6
3
6
10
8
7
9
5
6
7
6
6
9
P.T
.43
8
12
3
10
1
13
12
12
10
7
10
13
11
10
13
8
9
10
10
10
13
P.T
.44
8 12
3 10
2 13
13
12
10
7 10
13
10
10
13
9 9 10
10
10
13
P.T
.45
9 13
4 10
4 13
10
12
10
7 10
11
11
7 13
9 9 9 11
11
13
P.T
.46
3 7 4 7 5 10
8 6 4 4 7 7 4 3 9 6 3 3 7 5 8
P.T
.47
2 7 4 7 5 10
8 6 3 4 7 7 5 4 9 5 4 4 6 4 8
P.T
.48
5 9 4 10
5 12
10
8 7 6 9 9 6 5 11
8 5 5 9 7 10
P.T
.49
5 10 3 9 5 13
11 9 6 6 9 10 7 7 12 8 6 7 9 7 11
P.T
.50
4
9
4
7
5
10
10
8
5
4
7
9
7
6
10
6
6
6
7
6
10
P.T
.51
7 12 3 10 3 13
13
11 8 7 10
12 9 9 12 8 8 9 10 9 13
P.T
.52
8 11 2 9 6 12
13
12 9 6 9 13
11
10
12 8 10
10 9 10
12
Page | 191
P.T
.53
9
12
4
8
3
12
11
11
10
7
10
12
9
8
13
9
8
9
11
11
13
P.T
.54
9
13
6
9
6
11
9
10
11
9
12
9
10
6
12
10
8
7
13
11
14
P.T
.55
7
11
5
12
6
13
11
10
9
8
11
11
8
7
12
10
6
7
11
9
11
P.T
.56
3 9 4 8 6 11
9 8 5 5 8 9 6 6 10
6 5 6 8 5 9
P.T
.57
7 11
5 11
6 11
9 9 8 8 10
10
7 5 12
10
6 6 11
8 11
P.T
.58
8 11
5 11
5 14
12
10
9 9 11
11
8 8 13
10
7 8 11
9 12
P.T
.59
1 6 6 7 7 9 7 5 2 4 6 6 4 6 7 5 3 4 5 3 6
P.T
.60
7 10
1 8 3 11
13
11
8 5 8 12
10
9 11
7 9 9 8 9 11
P.T
.61
7 5 7 2 8 5 7 6 5 5 3 7 8 10 6 2 7 8 3 4 6
P.T
.62
8
12
5
7
5
12
13
12
10
7
10
13
12
11
12
7
10
11
10
10
12
P.T
.63
8 7 8 1 9 7 8 7 6 6 5 8 10
11 8 2 9 10 5 6 8
P.T
.64
10 6 9 3 11 4 6 7 8 8 4 6 9 12 6 4 11
11 5 7 6
Page | 192
P.T
.65
6
8
5
3
6
8
10
8
6
3
6
9
9
8
8
2
8
8
6
7
8
P.T
.66
1
6
6
7
7
9
7
5
2
4
6
6
5
6
8
5
3
4
5
3
7
P.T
.67
4
6
3
4
4
7
8
6
4
1
4
7
7
6
7
3
6
6
4
5
7
P.T
.68
8 4 8 3 9 4 5 5 7 6 2 5 8 11
5 3 9 9 4 6 5
P.T
.69
9 4 10
5 11
3 4 5 7 8 2 5 8 10
5 5 9 9 4 6 5
P.T
.70
2 7 6 8 7 10
8 6 3 5 7 7 6 6 9 7 4 5 6 4 8
P.T
.71
4 6 4 4 6 7 8 6 4 2 3 7 7 7 6 3 6 7 4 4 6
P.T
.72
5 7 4 4 5 8 9 7 5 2 5 8 8 7 8 3 7 7 5 6 8
P.T
.73
6 5 8 6 9 6 6 4 5 6 3 5 7 8 6 4 5 6 2 3 6
P.T
.74
7
7
6
2
8
8
8
7
6
4
4
8
9
10
7
1
8
9
5
5
7
P.T
.75
9 11 3 6 4 11
12
11
10 7 9 12
12
11
12 6 11
11 9 10
12
P.T
.76
10 9 8 4 9 9 9 9 8 7 6 10
12
12 9 3 10
11 7 7 9
Page | 193
P.T
.77
6
10
4
8
6
11
12
10
8
5
8
11
9
9
11
7
8
9
8
8
11
P.T
.78
10
7
9
3
11
5
7
7
8
8
4
7
10
12
7
4
10
11
6
8
7
P.T
.79
6
5
6
4
8
6
7
5
4
4
3
6
8
9
6
3
6
7
3
4
6
P.T
.80
6 7 5 2 6 8 9 7 6 3 5 8 9 8 7 1 8 8 5 5 7
Ka
rdit
sa
New
sto
re
Co
mp
etit
ors
j
Po
ten
tia
l
Co
nsu
mer
s i
C.S
.0
C.S
.1
C.S
.2
C.S
.3
C.S
.4
C.S
.5
C.S
.6
C.S
.7
C.S
.8
C.S
.9
C.S
.10
C.S
.11
C.S
.12
C.S
.13
C.S
.14
P.T
.1
9 10 5 12
10
10 6 7 9 6 6 12 6 10
11
P.T
.2
8 9 3 11 9 10 7 7 9 5 6 11 3 8 7
P.T
.3
6 7 3 9 7 7 4 5 7 3 4 9 5 8 8
P.T
.4
5 6 6 8 6 7 3 4 6 2 3 8 4 9 8
Page | 194
P.T
.5
7
8
3
9
8
8
7
6
8
4
5
9
2
8
5
P.T
.6
2
3
8
7
3
6
4
3
3
3
3
5
5
8
7
P.T
.7
3
4
8
6
4
4
5
1
3
4
5
6
5
7
5
P.T
.8
6 7 4 9 7 7 4 4 6 3 4 9 2 10
6
P.T
.9
7 8 4 8 7 8 7 5 7 4 4 8 2 8 5
P.T
.10
8 10
4 8 9 8 9 7 9 5 6 9 3 8 5
P.T
.11
9 10
1 11
10
10
7 7 9 6 7 12
4 10
8
P.T
.12
7 8 2 9 8 8 7 5 7 4 5 10
1 7 6
P.T
.13
8 9 4 10 8 9 6 6 8 5 5 10 2 10 7
P.T
.14
9
10
4
9
10
10
7
7
9
6
7
9
3
7
8
P.T
.15
8 9 4 10 8 9 5 6 8 4 5 10 4 9 8
P.T
.16
8 9 4 11 9 9 5 6 8 5 6 11 5 9 10
Page | 195
P.T
.17
8
9
5
11
9
9
4
6
8
5
5
11
5
9
9
P.T
.18
8
9
3
11
9
9
7
7
9
5
6
11
3
9
7
P.T
.19
5
6
4
8
6
6
4
4
6
2
3
8
4
9
7
P.T
.20
9 10
3 9 10
10
9 7 9 6 7 9 4 7 7
P.T
.21
9 8 3 5 8 6 8 9 9 8 9 5 6 3 5
P.T
.22
10
9 7 4 9 5 9 8 8 7 8 5 6 4 2
P.T
.23
10
9 8 6 9 7 8 10
10
10
11
6 9 4 6
P.T
.24
8 7 7 3 7 3 8 6 6 5 6 4 4 4 1
P.T
.25
7 6 7 1 5 3 6 6 6 8 8 1 7 2 3
P.T
.26
8
7
8
2
7
3
8
6
6
6
7
3
5
4
1
P.T
.27
8 8 7 7 8 7 8 7 8 5 6 7 4 8 4
P.T
.28
7 7 6 6 7 6 7 6 7 4 5 6 3 7 3
Page | 196
P.T
.29
7
8
5
8
8
7
7
6
8
4
5
8
3
7
4
P.T
.30
6
7
5
6
6
6
6
4
6
3
3
6
2
7
3
P.T
.31
9
9
4
7
8
7
9
7
8
6
6
8
4
6
4
P.T
.32
9 9 7 5 9 5 9 8 8 6 7 6 5 5 2
P.T
.33
8 6 6 4 6 4 8 6 6 5 5 4 4 5 1
P.T
.34
9 8 8 4 7 4 9 7 7 6 7 4 5 5 1
P.T
.35
9 8 9 4 8 4 8 8 8 7 7 4 6 4 2
P.T
.36
8 7 4 4 7 5 7 8 8 9 9 4 7 2 4
P.T
.37
8 6 6 2 6 6 6 8 8 10
10 3 8 1 4
P.T
.38
8
7
6
4
7
5
7
9
8
9
9
4
8
3
4
P.T
.39
8 7 9 3 7 2 9 7 7 8 9 4 7 5 3
P.T
.40
9 8 7 5 8 5 9 7 8 6 7 5 5 5 2
Page | 197
P.T
.41
7
8
9
10
8
11
4
7
8
8
9
9
9
9
11
P.T
.42
8
9
7
12
9
10
5
7
9
6
6
11
7
11
11
P.T
.43
7
8
5
10
8
8
4
6
8
4
5
10
6
10
9
P.T
.44
6 7 5 9 7 7 5 5 7 3 4 9 5 10
8
P.T
.45
5 6 6 8 5 8 1 5 5 4 5 7 6 7 9
P.T
.46
6 7 6 9 7 10
2 6 7 5 6 9 6 8 10
P.T
.47
8 7 11
8 7 9 6 8 9 11
10
7 12
7 9
P.T
.48
3 4 7 5 4 7 2 4 4 6 5 5 7 5 7
P.T
.49
2 3 7 7 3 6 2 3 3 5 4 5 6 6 7
P.T
.50
5
6
7
8
6
9
2
6
6
6
7
8
8
7
10
P.T
.51
6 5 9 6 5 8 5 6 7 9 8 6 10 5 7
P.T
.52
6 7 8 9 7 10 3 6 7 7 8 8 8 8 10
Page | 198
P.T
.53
4
5
5
7
5
8
1
5
5
5
6
6
6
6
8
P.T
.54
10
11
12
13
11
14
7
10
11
11
12
12
12
12
14
P.T
.55
9
10
12
12
10
13
6
9
10
10
11
11
11
11
13
P.T
.56
8 7 12
8 7 10
7 9 9 11
10
8 12
7 10
P.T
.57
5 4 9 6 4 7 4 6 6 8 8 5 9 5 7
P.T
.58
4 5 7 9 5 7 2 4 5 3 4 7 4 9 8
P.T
.59
3 4 8 8 4 7 2 4 4 4 4 6 5 8 8
P.T
.60
6 8 5 10
7 9 3 6 7 4 5 9 6 10
10
P.T
.61
9 7 7 3 7 5 8 8 8 9 10 3 8 2 4
P.T
.62
10
9
7
6
8
7
8
10
10
10
11
6
9
2
5
P.T
.63
6 5 9 6 5 8 5 6 7 9 8 6 10 5 7
P.T
.64
3 2 10 4 1 3 5 4 3 6 5 2 7 5 5
Page | 199
P.T
.65
3
1
10
5
1
5
5
4
3
6
5
3
7
7
6
P.T
.66
4
3
9
7
3
6
3
4
5
7
6
5
8
6
7
P.T
.67
5
3
8
4
3
6
3
5
5
7
7
4
9
3
6
P.T
.68
6 5 9 5 4 5 4 6 7 9 8 3 10
6 6
P.T
.69
5 4 9 4 4 4 5 6 6 8 8 2 9 5 5
P.T
.70
4 3 9 6 3 5 4 4 5 7 6 4 8 6 7
P.T
.71
4 1 10
5 1 4 5 4 2 7 6 3 8 6 5
P.T
.72
6 4 8 3 4 2 6 5 5 8 7 1 7 4 3
P.T
.73
5 4 11 4 4 3 7 3 3 6 7 4 8 6 4
P.T
.74
6
5
8
2
5
1
7
4
4
7
8
1
6
4
3
P.T
.75
8 7 8 5 7 6 7 9 9 11
10 4 10 4 6
P.T
.76
6 5 9 3 5 4 5 7 7 9 8 2 8 4 5
Page | 200
P.T
.77
6
5
7
2
4
3
6
6
6
9
8
1
8
3
4
P.T
.78
5
3
9
5
3
4
5
5
5
8
7
3
9
5
6
P.T
.79
5
3
9
4
3
3
5
5
5
8
7
1
8
5
4
P.T
.80
6 4 9 4 4 4 5 6 6 8 8 2 9 5 6
Kalabaka New store Competitors j
Potential Consumers i C.S.0 C.S.1 C.S.2 C.S.3
P.T.1 8 8 9 9
P.T.2 5 6 6 6
P.T.3 2 4 3 3
P.T.4 4 6 4 4
P.T.5 3 6 4 4
P.T.6 7 6 7 7
P.T.7 4 4 5 5
P.T.8 3 5 3 3
P.T.9 2 4 3 3
P.T.10 3 4 3 3
P.T.11 6 5 6 6
P.T.12 4 5 5 5
P.T.13 3 6 4 4
P.T.14 3 4 4 4
P.T.15 6 7 6 6
P.T.16 8 9 9 9
P.T.17 6 8 7 7
P.T.18 9 10 9 9
P.T.19 9 10 10 10
P.T.20 4 4 5 4
P.T.21 5 4 6 5
P.T.22 6 5 7 6
P.T.23 2 3 4 3
P.T.24 3 4 5 4
Page | 201
P.T.25 4 4 5 4
P.T.26 4 4 6 5
P.T.27 4 4 6 5
P.T.28 7 6 9 8
P.T.29 4 4 6 5
P.T.30 3 4 4 3
P.T.31 4 5 5 5
P.T.32 2 5 2 2
P.T.33 3 3 5 4
P.T.34 4 6 5 5
P.T.35 3 5 4 4
P.T.36 4 6 5 5
P.T.37 5 6 6 5
P.T.38 5 6 6 6
P.T.39 9 8 11 10
P.T.40 9 8 11 10
P.T.41 4 3 5 4
P.T.42 4 5 4 4
P.T.43 3 3 5 3
P.T.44 5 4 6 5
P.T.45 5 4 6 4
P.T.46 3 4 4 4
P.T.47 7 7 8 8
P.T.48 3 4 4 4
P.T.49 4 4 6 4
P.T.50 7 6 7 6
P.T.51 4 5 5 5
P.T.52 3 5 4 4
P.T.53 4 4 5 5
P.T.54 4 4 5 5
P.T.55 10 9 11 11
P.T.56 8 8 9 9
P.T.57 7 7 8 8
P.T.58 9 9 10 10
P.T.59 5 4 6 4
P.T.60 4 3 1 4
P.T.61 4 4 5 4
P.T.62 4 3 6 4
P.T.63 4 4 6 4
P.T.64 3 2 5 3
P.T.65 9 9 10 9
P.T.66 3 3 5 4
P.T.67 3 3 5 4
P.T.68 4 3 5 4
Page | 202
P.T.69 3 1 5 4
P.T.70 3 3 5 4
P.T.71 3 3 5 4
P.T.72 2 2 4 3
P.T.73 3 3 5 3
P.T.74 3 2 4 3
P.T.75 3 1 4 3
P.T.76 4 2 6 4
P.T.77 9 8 11 10
P.T.78 5 4 6 5
P.T.79 4 3 5 4
P.T.80 5 4 7 6
Computation of Numerators, Denominators, individual consumer patronage
possibities as well as consumer patronage possibities refering to sublocations of
Northwest (Group A), Northeast (Group B), Southwest (Group C) and Southeast
(Group D) in cities of Trikala, Karditsa and Kalabaka
Walk time ≤ 10 min
Trikala
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=20+1) Pij % Pij % Pij
P.T.1 0,48828125 25,40367979 0,019220
887
1,922088
666 Group A
P.T.2 0,342935528 6,278386901 0,054621
598
5,462159
843
4,280091
709
P.T.3 0,25 10,11114573 0,024725
19
2,472519
008
P.T.4 3,90625 1813,454041 0,002154
039
0,215403
86
P.T.5 0,48828125 20,50959778 0,023807
451
2,380745
129
P.T.6 0,342935528 13,16731481 0,026044
454
2,604445
424
P.T.7 0,25 132,7570689 0,001883
139
0,188313
889
P.T.8 0,25 48,26884177 0,005179
325
0,517932
461
P.T.9 0,25 23,32724304 0,010717
083
1,071708
301
P.T.10 0,342935528 15,4536509 0,022191
23
2,219123
042
Page | 203
P.T.11 0,25 8,231080209 0,030372
684
3,037268
422
P.T.12 0,48828125 9,043664169 0,053991
528
5,399152
831
P.T.13 31,25 268,1364772 0,116545
128
11,65451
278
P.T.14 0,728862974 35,22479442 0,020691
759
2,069175
948
P.T.15 0,48828125 48,80639636 0,010004
452
1,000445
201
P.T.16 1,157407407 80,67565328 0,014346
427
1,434642
746
P.T.17 2 74,93532756 0,026689
681
2,668968
116
P.T.18 2 21,417499 0,093381
585
9,338158
486
P.T.19 9,259259259 43,59566065 0,212389
47
21,23894
7
P.T.20 1,157407407 13,29417703 0,087061
23
8,706123
027
P.T.21 0,48828125 527,1106729 0,000926
335
0,092633
535 Group B
P.T.22 3,90625 115,8479394 0,033718
77
3,371876
978
2,226894
388
P.T.23 0,342935528 85,35921041 0,004017
557
0,401755
741
P.T.24 0,342935528 58,76529754 0,005835
681
0,583568
096
P.T.25 0,25 462,3324967 0,000540
736
0,054073
638
P.T.26 31,25 753,4910108 0,041473
62
4,147362
019
P.T.27 31,25 117,9420916 0,264960
538
26,49605
376
P.T.28 2 250,5915925 0,007981
114
0,798111
373
P.T.29 2 147,0139974 0,013604
147
1,360414
679
P.T.30 1,157407407 344,3117417 0,003361
51
0,336151
013
P.T.31 0,342935528 113,6368346 0,003017
82
0,301782
014
P.T.32 0,25 129,9044499 0,001924
491
0,192449
143
P.T.33 0,25 979,2210668 0,000255
305
0,025530
496
P.T.34 0,728862974 47,93577153 0,015204
991
1,520499
098
P.T.35 0,25 77,25249389 0,003236
141
0,323614
148
P.T.36 0,25 32,71716569 0,007641
249
0,764124
871
P.T.37 0,25 52,25192445 0,004784
513
0,478451
277
P.T.38 0,342935528 85,90931545 0,003991
832
0,399183
169
P.T.39 1,157407407 40,60535784 0,028503
81
2,850381
006
Page | 204
P.T.40 0,25 627,0111378 0,000398
717
0,039871
7
P.T.41 0,342935528 5,454088809 0,062876
777
6,287677
743 Group C
P.T.42 9,259259259 30,64730587 0,302123
107
30,21231
067
13,56472
417
P.T.43 0,728862974 8,334511324 0,087451
195
8,745119
485
P.T.44 0,728862974 24,08887311 0,030257
247
3,025724
659
P.T.45 0,342935528 4,971774822 0,068976
48
6,897648
031
P.T.46 0,728862974 7,488628566 0,097329
3
9,732929
966
P.T.47 31,25 61,40358756 0,508927
918
50,89279
184
P.T.48 3,90625 24,25420534 0,161054
545
16,10545
448
P.T.49 9,259259259 25,44253776 0,363928
29
36,39282
899
P.T.50 0,25 3,74577707 0,066741
826
6,674182
562
P.T.51 2 12,35883223 0,161827
587
16,18275
872
P.T.52 0,25 3,583675456 0,069760
781
6,976078
138
P.T.53 0,25 3,64739741 0,068542
024
6,854202
378
P.T.54 0,25 5,569561921 0,044886
834
4,488683
375
P.T.55 2 12,07999736 0,165562
95
16,55629
501
P.T.56 0,48828125 6,785686731 0,071957
529
7,195752
904
P.T.57 0,342935528 12,12815603 0,028275
983
2,827598
255
P.T.58 2 14,86001412 0,134589
374
13,45893
741
P.T.59 2 14,46111937 0,138301
88
13,83018
803
P.T.60 1,157407407 14,54518976 0,079573
208
7,957320
784
P.T.61 0,25 505,0863589 0,000494
965
0,049496
486 Group D
P.T.62 0,48828125 57,76223415 0,008453
296
0,845329
578
3,313005
14
P.T.63 0,728862974 24,01384266 0,030351
784
3,035178
434
P.T.64 0,48828125 14,83012518 0,032924
958
3,292495
808
P.T.65 0,25 12,46026641 0,020063
776
2,006377
646
P.T.66 0,25 7,999917616 0,031250
322
3,125032
182
P.T.67 0,25 68,47872859 0,003650
769
0,365076
872
P.T.68 1,157407407 22,6655519 0,051064
603
5,106460
289
Page | 205
P.T.69 0,342935528 15,38081907 0,022296
311
2,229631
117
P.T.70 0,48828125 26,15911191 0,018665
819
1,866581
907
P.T.71 0,342935528 12,73139648 0,026936
207
2,693620
678
P.T.72 1,157407407 19,90234746 0,058154
316
5,815431
619
P.T.73 2 37,15134024 0,053833
859
5,383385
867
P.T.74 9,259259259 46,57455643 0,198805
098
19,88050
981
P.T.75 3,90625 731,2159977 0,005342
129
0,534212
874
P.T.76 2 102,3343755 0,019543
775
1,954377
491
P.T.77 1,157407407 58,07809185 0,019928
468
1,992846
822
P.T.78 0,728862974 26,19831013 0,027820
992
2,782099
19
P.T.79 0,342935528 16,77155476 0,020447
45
2,044745
004
P.T.80 0,342935528 27,27743789 0,012572
131
1,257213
121
Karditsa
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=14+1) Pij % Pij % Pij
P.T.1 9,259259259 51,07259033
0,181296
057
18,12960
572 Group A
P.T.2 0,25 52,02072147
0,004805
777
0,480577
725
2,958154
836
P.T.3 0,25 25,472872
0,009814
363
0,981436
251
P.T.4 0,48828125 55,01831328
0,008874
886
0,887488
585
P.T.5 0,342935528 161,7381008
0,002120
314
0,212031
381
P.T.6 31,25 233,6236585
0,133762
138
13,37621
378
P.T.7 1,157407407 69,13334081
0,016741
668
1,674166
754
P.T.8 9,259259259 188,0579497
0,049236
202
4,923620
231
P.T.9 1,157407407 160,2134245
0,007224
16
0,722415
997
P.T.10 3,90625 379,9554553
0,010280
81
1,028081
041
P.T.11 2 357,9266844
0,005587
737
0,558773
65
P.T.12 0,48828125 30,71482054
0,015897
252
1,589725
225
P.T.13 0,25 49,55580121
0,005044
818
0,504481
804
P.T.14 0,342935528 18,83159841
0,018210
644
1,821064
366
Page | 206
P.T.15 0,25 177,8177864
0,001405
934
0,140593
36
P.T.16 0,342935528 1139,743866
0,000300
888
0,030088
824
P.T.17 0,728862974 1059,859834
0,000687
698
0,068769
751
P.T.18 2 32,16790304
0,062173
776
6,217377
606
P.T.19 0,48828125 25,47456652
0,019167
402
1,916740
171
P.T.20 0,25 6,410512021
0,038998
445
3,899844
493
P.T.21 9,259259259 920,3164073
0,010060
952
1,006095
207 Group B
P.T.22 0,48828125 36,35451566
0,013431
103
1,343110
315
1,866096
907
P.T.23 0,25 28,95825196
0,008633
118
0,863311
778
P.T.24 0,342935528 17,2714596
0,019855
619
1,985561
939
P.T.25 31,25 1551,861502
0,020137
106
2,013710
629
P.T.26 0,728862974 32,13193212
0,022683
447
2,268344
683
P.T.27 3,90625 151,0734146
0,025856
634
2,585663
407
P.T.28 3,90625 220,0189239
0,017754
155
1,775415
465
P.T.29 2 81,37540677
0,024577
45
2,457745
011
P.T.30 0,342935528 23,02042937
0,014897
008
1,489700
833
P.T.31 0,25 510,6703745
0,000489
553
0,048955
258
P.T.32 0,25 19,73165456
0,012669
997
1,266999
679
P.T.33 0,728862974 27,22594403
0,026770
898
2,677089
811
P.T.34 0,48828125 18,10233965
0,026973
378
2,697337
8
P.T.35 1,157407407 56,03570648
0,020654
82
2,065481
958
P.T.36 0,728862974 31,82596006
0,022901
524
2,290152
355
P.T.37 0,728862974 28,08242585
0,025954
416
2,595441
639
P.T.38 1,157407407 38,07504242
0,030398
06
3,039805
957
P.T.39 0,48828125 34,0410599
0,014343
891
1,434389
092
P.T.40 0,48828125 34,44360381
0,014176
253
1,417625
324
P.T.41 3,90625 21,59477708
0,180888
647
18,08886
466 Group C
P.T.42 0,48828125 6,616338033
0,073799
32
7,379932
034
11,16925
778
P.T.43 1,157407407 8,231425894
0,140608
374
14,06083
736
Page | 207
P.T.44 0,342935528 8,838764567
0,038799
034
3,879903
413
P.T.45 3,90625 29,2555517
0,133521
666
13,35216
659
P.T.46 0,48828125 8,872503308
0,055033
087
5,503308
74
P.T.47 0,728862974 18,23907182
0,039961
626
3,996162
64
P.T.48 0,25 5,491147328
0,045527
826
4,552782
598
P.T.49 0,25 4,336728688
0,057647
139
5,764713
866
P.T.50 0,342935528 4,598650836
0,074573
074
7,457307
378
P.T.51 1,157407407 12,82548598
0,090242
772
9,024277
203
P.T.52 9,259259259 31,45811617
0,294336
101
29,43361
011
P.T.53 250 421,1670036
0,593588
761
59,35887
614
P.T.54 1,157407407 12,80902541
0,090358
741
9,035874
081
P.T.55 0,25 5,550063931
0,045044
526
4,504452
617
P.T.56 0,25 8,397030666
0,029772
429
2,977242
908
P.T.57 0,25 8,972276538
0,027863
608
2,786360
841
P.T.58 0,48828125 9,161455186
0,053297
346
5,329734
633
P.T.59 1,157407407 9,470451474
0,122212
485
12,22124
849
P.T.60 0,342935528 7,331599754
0,046774
993
4,677499
313
P.T.61 0,25 3,484667873
0,071742
849
7,174284
871 Group D
P.T.62 9,259259259 944,4321769
0,009804
049
0,980404
891
4,976051
662
P.T.63 0,342935528 4,632165307
0,074033
525
7,403352
545
P.T.64 0,728862974 12,66191198
0,057563
421
5,756342
129
P.T.65 0,728862974 11,05361376
0,065938
886
6,593888
562
P.T.66 0,25 10,88260672
0,022972
437
2,297243
726
P.T.67 0,48828125 21,12124595
0,023118
014
2,311801
354
P.T.68 0,25 4,270559212
0,058540
343
5,854034
275
P.T.69 0,25 5,38510971
0,046424
31
4,642430
953
P.T.70 0,342935528 48,01189168
0,007142
721
0,714272
061
P.T.71 0,728862974 15,23484763
0,047841
829
4,784182
891
P.T.72 0,728862974 13,47187189
0,054102
576
5,410257
608
Page | 208
P.T.73 2 51,86157923
0,038564
194
3,856419
395
P.T.74 31,25 1724,368055
0,018122
581
1,812258
115
P.T.75 3,90625 152,2491643
0,025656
955
2,565695
528
P.T.76 0,48828125 5,30198044
0,092094
125
9,209412
511
P.T.77 0,25 5,205820538
0,048023
169
4,802316
91
P.T.78 2 45,94296228
0,043532
239
4,353223
869
P.T.79 31,25 199,6077356
0,156557
059
15,65570
588
P.T.80 0,25 7,477183013
0,033435
052
3,343505
162
Kalabaka
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=3+1) Pij % Pij % Pij
P.T.1 31,25 65,77148438
0,475129
918
47,51299
183 Group A
P.T.2 0,728862974 5,365405404
0,135844
902
13,58449
025
17,35032
051
P.T.3 0,728862974 5,100737974
0,142893
632
14,28936
318
P.T.4 0,342935528 4,224753004
0,081172
918
8,117291
775
P.T.5 0,25 1,993758162
0,125391
336
12,53913
362
P.T.6 31,25 87,21481481
0,358310
685
35,83106
846
P.T.7 1,157407407 8,218263346
0,140833
575
14,08335
75
P.T.8 2 23,30268053
0,085827
036
8,582703
597
P.T.9 9,259259259 26,98138423
0,343172
136
34,31721
36
P.T.10 3,90625 17,56468621
0,222392
245
22,23922
45
P.T.11 2 10,23105256
0,195483
308
19,54833
081
P.T.12 0,48828125 6,3826987
0,076500
752
7,650075
195
P.T.13 0,48828125 3,70619407
0,131747
351
13,17473
507
P.T.14 0,25 3,161051371
0,079087
611
7,908761
062
P.T.15 0,25 2,359589335
0,105950
64
10,59506
399
P.T.16 1,157407407 6,504085611
0,177950
826
17,79508
261
P.T.17 1,157407407 7,658877885
0,151119
71
15,11197
103
P.T.18 3,90625 21,60913066
0,180768
494
18,07684
937
P.T.19 0,728862974 5,100737974
0,142893
632
14,28936
318
Page | 209
P.T.20 0,48828125 4,152284605
0,117593
397
11,75933
965
P.T.21 0,728862974 7,557518211
0,096442
106
9,644210
618 Group B
P.T.22 3,90625 157,1689241
0,024853
832
2,485383
177
10,40403
016
P.T.23 1,157407407 12,20819629
0,094805
767
9,480576
656
P.T.24 0,48828125 7,190321181
0,067908
128
6,790812
785
P.T.25 0,25 2,607744244
0,095868
297
9,586829
712
P.T.26 31,25 91,49722222
0,341540
423
34,15404
232
P.T.27 31,25 1236,152729
0,025280
048
2,528004
773
P.T.28 3,90625 51,44392412
0,075932
193
7,593219
349
P.T.29 2 14,61833255
0,136814
51
13,68145
096
P.T.30 0,48828125 3,221848093
0,151553
157
15,15531
57
P.T.31 0,25 2,549794239
0,098047
127
9,804712
718
P.T.32 0,25 2,454137329
0,101868
79
10,18687
899
P.T.33 0,25 2,596919213
0,096267
916
9,626791
574
P.T.34 0,48828125 4,8602633
0,100463
95
10,04639
502
P.T.35 0,728862974 7,778285616
0,093704
835
9,370483
545
P.T.36 2 13,73090278
0,145656
847
14,56568
466
P.T.37 2 23,05034722
0,086766
589
8,676658
884
P.T.38 0,25 3,519675926
0,071029
267
7,102926
669
P.T.39 0,25 3,083567289
0,081074
929
8,107492
931
P.T.40 0,342935528 3,612611454
0,094927
321
9,492732
127
P.T.41 0,342935528 9,331989598
0,036748
383
3,674838
302 Group C
P.T.42 0,342935528 3,552409138
0,096536
045
9,653604
492
8,845639
663
P.T.43 0,728862974 10,46242982
0,069664
79
6,966478
978
P.T.44 0,342935528 17,3144199
0,019806
354
1,980635
39
P.T.45 250 310,3425441
0,805561
483
80,55614
827
P.T.46 1,157407407 40,6410751
0,028478
76
2,847875
959
P.T.47 9,259259259 1019,037037
0,009086
283
0,908628
335
P.T.48 9,259259259 65,31939598
0,141753
596
14,17535
959
Page | 210
P.T.49 1,157407407 131,9316219
0,008772
782
0,877278
238
P.T.50 1,157407407 129,8024341
0,008916
685
0,891668
493
P.T.51 0,25 5,676954733
0,044037
695
4,403769
482
P.T.52 0,25 3,094841794
0,080779
573
8,077957
345
P.T.53 0,342935528 4,98703275
0,068765
445
6,876544
536
P.T.54 2 44,86569227
0,044577
491
4,457749
114
P.T.55 0,25 3,676224681
0,068004
549
6,800454
86
P.T.56 0,25 5,751851852
0,043464
263
4,346426
272
P.T.57 0,25 5,661550383
0,044157
516
4,415751
572
P.T.58 2 1005,67632
0,001988
711
0,198871
144
P.T.59 9,259259259 144,037037
0,064283
878
6,428387
76
P.T.60 3,90625 46,64532697
0,083743
651
8,374365
138
P.T.61 250 1458,347222
0,171426
939
17,14269
388 Group D
P.T.62 0,48828125 7,725320687
0,063205
305
6,320530
497
7,507682
471
P.T.63 0,342935528 8,238000329
0,041628
492
4,162849
168
P.T.64 0,25 3,094841794
0,080779
573
8,077957
345
P.T.65 0,25 3,094841794
0,080779
573
8,077957
345
P.T.66 0,48828125 6,198272026
0,078776
996
7,877699
591
P.T.67 1,157407407 19,20819629
0,060255
913
6,025591
316
P.T.68 3,90625 64,32291667
0,060728
745
6,072874
494
P.T.69 3,90625 56,80148332
0,068770
211
6,877021
112
P.T.70 0,25 4,100527808
0,060967
761
6,096776
116
P.T.71 0,25 4,798187273
0,052103
01
5,210301
011
P.T.72 2 23,303125
0,085825
399
8,582539
895
P.T.73 0,728862974 7,858778598
0,092745
07
9,274507
033
P.T.74 0,728862974 7,858778598
0,092745
07
9,274507
033
P.T.75 0,48828125 6,487409462
0,075265
983
7,526598
296
P.T.76 0,25 4,57101325
0,054692
469
5,469246
89
P.T.77 0,342935528 5,807698778
0,059048
436
5,904843
575
Page | 211
P.T.78 2 23,52617027
0,085011
712
8,501171
152
P.T.79 9,259259259 162,6255
0,056936
085
5,693608
479
P.T.80 1,157407407 14,49590457
0,079843
752
7,984375
186
Drive time ≤ 10 min
Trikala
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=20+1) Pij % Pij % Pij
P.T.1 1,157407407 957,2452047
0,001209102
0,120910233
Group
A
P.T.2 0,728862974 176,3788565
0,004132372
0,413237158
3,945945
878
P.T.3 0,342935528 20,13783497
0,017029414
1,702941397
P.T.4 0,342935528 21,81842616
0,015717702
1,571770235
P.T.5 0,342935528 24,96968769
0,013734074
1,373407358
P.T.6 0,25 20,34528846
0,012287857
1,228785723
P.T.7 0,342935528 32,53444411
0,010540691
1,054069118
P.T.8 0,48828125 35,42466316
0,013783653
1,378365259
P.T.9 0,48828125 26,77839109
0,018234152
1,823415187
P.T.10 2 57,60745015
0,034717732
3,471773173
P.T.11 0,728862974 45,08711523
0,01616566
1,616565997
P.T.12 0,48828125 27,99848053
0,017439562
1,743956246
P.T.13 1,157407407 78,75161766
0,014696935
1,469693502
P.T.14 2 92,75928977
0,021561183
2,156118277
P.T.15 1,157407407 79,28739013
0,014597623
1,459762272
P.T.16 3,90625 1106,936327
0,003528884
0,35288841
P.T.17 9,259259259 1115,721412
0,008298899
0,829889896
P.T.18 0,48828125 155,6326536
0,003137396
0,313739591
P.T.19 1,157407407 68,08729133
0,016998876
1,699887578
P.T.20 250 470,4754012
0,53137741
53,13774096
P.T.21 0,48828125 1997,769088
0,000244413
0,024441326
Group B
Page | 212
P.T.22 0,342935528 168,1677746
0,002039246
0,20392464
0,390943
565
P.T.23 0,48828125 43,70872973
0,011171252
1,117125236
P.T.24 0,342935528 43,26217961
0,007926913
0,792691287
P.T.25 0,48828125 60,75221275
0,008037259
0,803725869
P.T.26 2 339,8536458
0,005884886
0,588488611
P.T.27 0,342935528 400,4001392
0,000856482
0,085648204
P.T.28 0,25 245,5829406
0,001017986
0,101798602
P.T.29 0,48828125 172,3512402
0,002833059
0,28330591
P.T.30 0,342935528 229,4127639
0,001494841
0,149484066
P.T.31 1,157407407 277,7905559
0,004166475
0,4166475
P.T.32 0,728862974 169,0364275
0,004311869
0,431186925
P.T.33 0,728862974 77,24510524
0,009435717
0,943571727
P.T.34 0,342935528 1462,991901
0,000234407
0,023440699
P.T.35 0,25 162,0973093
0,001542283
0,154228347
P.T.36 0,25 253,1491199
0,00098756
0,098756022
P.T.37 0,342935528 472,2875071
0,000726116
0,072611603
P.T.38 0,48828125 54,21677135
0,009006092
0,900609235
P.T.39 0,48828125 161,6354552
0,00302088
0,30208796
P.T.40 2 615,1999941
0,003250975
0,325097532
P.T.41 2 57,60745015
0,034717732
3,471773173
Group
C
P.T.42 3,90625 155,2039548
0,025168495
2,516849526
6,250440
148
P.T.43 0,48828125 638,0751273
0,000765241
0,0765241
P.T.44 0,48828125 112,7307331
0,004331394
0,433139426
P.T.45 0,342935528 34,11760303
0,010051572
1,005157155
P.T.46 9,259259259 177,282531
0,05222883
5,222883047
P.T.47 31,25 156,1707599
0,200101479
20,01014788
P.T.48 2 56,82174876
0,03519779
3,519779034
Page | 213
P.T.49 2 58,16209892
0,034386654
3,438665449
P.T.50 3,90625 63,28450316
0,061725222
6,172522189
P.T.51 0,728862974 63,20959049
0,011530892
1,15308922
P.T.52 0,48828125 104,2563471
0,004683468
0,468346785
P.T.53 0,342935528 47,40067112
0,007234824
0,723482432
P.T.54 0,342935528 23,39733356
0,014657035
1,465703463
P.T.55 0,728862974 27,66212536
0,02634877
2,634876982
P.T.56 9,259259259 66,31278305
0,139630081
13,96300809
P.T.57 0,728862974 36,06113873
0,020211868
2,021186794
P.T.58 0,48828125 25,42859409
0,019202055
1,920205451
P.T.59 250 457,094598
0,546932738
54,6932738
P.T.60 0,728862974 742,3063744
0,00098189
0,098188969
P.T.61 0,728862974 243,7134767
0,002990655
0,299065519
Group
D
P.T.62 0,48828125 23,58367098
0,020704209
2,070420888
4,813946
659
P.T.63 0,48828125 621,9043183
0,000785139
0,078513886
P.T.64 0,25 80,95662905
0,003088073
0,308807324
P.T.65 1,157407407 159,2353309
0,007268534
0,726853394
P.T.66 250 450,8431909
0,554516526
55,4516526
P.T.67 3,90625 635,6735438
0,006145057
0,614505675
P.T.68 0,48828125 184,4390343
0,002647386
0,264738563
P.T.69 0,342935528 163,9550869
0,002091643
0,209164311
P.T.70 31,25 118,3644061
0,26401518
26,40151801
P.T.71 3,90625 190,2670853
0,020530351
2,053035077
P.T.72 2 150,6071636
0,013279581
1,327958081
P.T.73 1,157407407 179,7803083
0,006437899
0,643789867
P.T.74 0,728862974 812,6292094
0,000896919
0,089691949
P.T.75 0,342935528 51,06992403
0,006715019
0,671501935
Page | 214
P.T.76 0,25 53,65098158
0,004659747
0,465974699
P.T.77 1,157407407 35,4108367
0,032685119
3,268511889
P.T.78 0,25 66,39485458
0,003765352
0,376535202
P.T.79 1,157407407 141,7672944
0,008164136
0,816413555
P.T.80 1,157407407 825,0649384
0,001402808
0,140280765
Karditsa
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=14+1) Pij % Pij % Pij
P.T.1 0,342935528 26,8642276
0,012765509
1,276550859
Group
A
P.T.2 0,48828125 82,88768963
0,005890878
0,589087779
1,405527
615
P.T.3 1,157407407 109,0875323
0,010609896
1,060989632
P.T.4 2 222,6168373
0,008984046
0,898404642
P.T.5 0,728862974 179,6528813
0,004057063
0,405706253
P.T.6 31,25 225,432165
0,138622632
13,86226318
P.T.7 9,259259259 415,8175816
0,022267599
2,226759923
P.T.8 1,157407407 203,3486835
0,005691738
0,569173789
P.T.9 0,728862974 174,6010341
0,004174448
0,417444821
P.T.10 0,48828125 71,61043697
0,006818577
0,68185766
P.T.11 0,342935528 729,9277004
0,000469821
0,046982123
P.T.12 0,728862974 1029,432743
0,000708024
0,070802389
P.T.13 0,48828125 151,3239207
0,003226729
0,322672878
P.T.14 0,342935528 63,04474395
0,005439558
0,543955779
P.T.15 0,48828125 62,15372621
0,007856025
0,785602537
P.T.16 0,48828125 42,5514566
0,011475077
1,147507721
P.T.17 0,48828125 42,7578352
0,011419691
1,141969063
P.T.18 0,48828125 82,37590122
0,005927477
0,592747688
P.T.19 2 225,3244705
0,008876089
0,887608876
P.T.20 0,342935528 58,87661975
0,005824647
0,582464703
Page | 215
P.T.21 0,342935528 97,66047239
0,003511508
0,351150798
Group B
P.T.22 0,25 125,5067276
0,001991925
0,199192509
0,358842
553
P.T.23 0,25 35,56083399
0,007030206
0,703020632
P.T.24 0,48828125 703,6167951
0,000693959
0,069395906
P.T.25 0,728862974 1116,499836
0,000652811
0,065281064
P.T.26 0,48828125 750,0318878
0,000651014
0,065101399
P.T.27 0,48828125 51,31140746
0,009516037
0,951603696
P.T.28 0,728862974 102,8617468
0,007085851
0,708585063
P.T.29 0,728862974 86,89448899
0,008387908
0,838790794
P.T.30 1,157407407 241,0583401
0,004801358
0,480135807
P.T.31 0,342935528 58,69706261
0,005842465
0,58424649
P.T.32 0,342935528 116,6485751
0,002939903
0,293990328
P.T.33 0,48828125 677,8894965
0,000720296
0,072029623
P.T.34 0,342935528 655,4196911
0,00052323
0,052323043
P.T.35 0,342935528 132,589788
0,00258644
0,258643998
P.T.36 0,48828125 201,7231558
0,002420551
0,242055132
P.T.37 0,48828125 1316,037182
0,000371024
0,03710239
P.T.38 0,48828125 87,43033189
0,005584804
0,55848038
P.T.39 0,48828125 136,6790548
0,003572466
0,357246581
P.T.40 0,342935528 118,8785969
0,002884754
0,288475417
P.T.41 0,728862974 19,84503827
0,036727718
3,672771822
Group
C
P.T.42 0,48828125 23,26688927
0,020986099
2,098609936
3,571845
988
P.T.43 0,728862974 49,3512138
0,014768897
1,47688966
P.T.44 1,157407407 85,33035636
0,013563841
1,356384125
P.T.45 2 359,5803088
0,00556204
0,556203983
P.T.46 1,157407407 71,80422763
0,016118931
1,611893123
P.T.47 0,48828125 17,91521929
0,027255109
2,725510875
Page | 216
P.T.48 9,259259259 124,348572
0,074462128
7,446212781
P.T.49 31,25 207,8851919
0,150323357
15,03233574
P.T.50 2 69,59439099
0,028737948
2,873794815
P.T.51 1,157407407 41,40656123
0,027952271
2,795227068
P.T.52 1,157407407 33,30548912
0,034751251
3,475125086
P.T.53 3,90625 358,2817364
0,010902733
1,090273269
P.T.54 0,25 6,967895023
0,035878841
3,587884134
P.T.55 0,342935528 9,275683458
0,036971457
3,697145657
P.T.56 0,48828125 16,40810894
0,029758533
2,975853291
P.T.57 2 60,40294187
0,03311097
3,311097006
P.T.58 3,90625 141,3736794
0,027630674
2,763067367
P.T.59 9,259259259 134,7355616
0,068721718
6,872171793
P.T.60 1,157407407 57,3408783
0,020184682
2,018468223
P.T.61 0,342935528 210,4625995
0,001629437
0,162943691
Group
D
P.T.62 0,25 170,0299296
0,001470329
0,147032937
1,229275
349
P.T.63 1,157407407 41,40656123
0,027952271
2,795227068
P.T.64 9,259259259 1125,182703
0,008229116
0,822911624
P.T.65 9,259259259 1684,076616
0,005498122
0,549812234
P.T.66 3,90625 108,4247287
0,036027298
3,602729788
P.T.67 2 152,4025666
0,013123139
1,312313857
P.T.68 1,157407407 62,30850537
0,018575432
1,857543205
P.T.69 2 109,218167
0,018311972
1,831197185
P.T.70 3,90625 107,0023095
0,036506221
3,650622141
P.T.71 3,90625 1742,404393
0,002241873
0,224187337
P.T.72 1,157407407 470,0041792
0,002462547
0,246254706
P.T.73 2 122,4234788
0,016336736
1,633673556
P.T.74 1,157407407 956,9279289
0,001209503
0,120950322
Page | 217
P.T.75 0,48828125 44,72310834
0,010917874
1,091787374
P.T.76 1,157407407 116,3423423
0,00994829
0,99482904
P.T.77 1,157407407 481,2417465
0,002405044
0,240504365
P.T.78 2 114,7277797
0,01743257
1,743256955
P.T.79 2 426,018154
0,004694636
0,469463562
P.T.80 1,157407407 106,3533522
0,01088266
1,088266033
Kalabaka
Potential
Consumer i
Numerator - Sj
(n=1, new)
Denominator - ΣSj
(n=3+1) Pij % Pij % Pij
P.T.1 0,48828125 4,677988897
0,104378454
10,43784542
Group A
P.T.2 2 14,96296296
0,133663366
13,36633663
13,27551
652
P.T.3 31,25 122,1064815
0,255924171
25,59241706
P.T.4 3,90625 41,05902778
0,095137421
9,513742072
P.T.5 9,259259259 46,41203704
0,199501247
19,95012469
P.T.6 0,728862974 9,920634921
0,073469388
7,346938776
P.T.7 3,90625 30,88125
0,126492613
12,64926128
P.T.8 9,259259259 95,54074074
0,09691425
9,691425027
P.T.9 31,25 122,1064815
0,255924171
25,59241706
P.T.10 9,259259259 100,1157407
0,092485549
9,248554913
P.T.11 1,157407407 16,14259259
0,071698979
7,169897901
P.T.12 3,90625 26,30625
0,148491328
14,84913281
P.T.13 9,259259259 46,41203704
0,199501247
19,95012469
P.T.14 9,259259259 53,00925926
0,174672489
17,46724891
P.T.15 1,157407407 13,09186373
0,088406619
8,84066189
P.T.16 0,48828125 4,329159165
0,112788935
11,27889346
P.T.17 1,157407407 8,743276577
0,132376849
13,23768495
P.T.18 0,342935528 3,960768176
0,086583085
8,658308513
P.T.19 0,342935528 3,142935528
0,109113128
10,91131285
Page | 218
P.T.20 3,90625 40,03125
0,097580016
9,758001561
P.T.21 2 25,60462963
0,078110874
7,811087405
Group B
P.T.22 1,157407407 14,42841486
0,080217225
8,021722544
13,68653
301
P.T.23 31,25 113,5416667
0,275229358
27,52293578
P.T.24 9,259259259 45,38425926
0,204019178
20,40191778
P.T.25 3,90625 40,03125
0,097580016
9,758001561
P.T.26 3,90625 27,51087963
0,14198928
14,19892803
P.T.27 3,90625 27,51087963
0,14198928
14,19892803
P.T.28 0,728862974 7,222132864
0,100920737
10,09207373
P.T.29 3,90625 27,51087963
0,14198928
14,19892803
P.T.30 9,259259259 78,7037037
0,117647059
11,76470588
P.T.31 3,90625 26,30625
0,148491328
14,84913281
P.T.32 31,25 311,05
0,100466163
10,0466163
P.T.33 9,259259259 58,23148148
0,159007791
15,90077914
P.T.34 3,90625 24,28402778
0,160856759
16,08567588
P.T.35 9,259259259 48,43425926
0,191171691
19,11716913
P.T.36 3,90625 24,28402778
0,160856759
16,08567588
P.T.37 2 19,00740741
0,105222136
10,52221356
P.T.38 2 14,96296296
0,133663366
13,36633663
P.T.39 0,342935528 3,466125329
0,09893916
9,893915989
P.T.40 0,342935528 3,466125329
0,09893916
9,893915989
P.T.41 3,90625 52,87847222
0,073872217
7,387221748
Group C
P.T.42 3,90625 43,08125
0,090671696
9,067169592
10,47171
741
P.T.43 9,259259259 83,92592593
0,110326567
11,03265666
P.T.44 2 25,60462963
0,078110874
7,811087405
P.T.45 2 34,75462963
0,05754629
5,754629013
P.T.46 9,259259259 53,00925926
0,174672489
17,46724891
Page | 219
P.T.47 0,728862974 6,775009111
0,107581106
10,75811061
P.T.48 9,259259259 53,00925926
0,174672489
17,46724891
P.T.49 3,90625 36,66087963
0,106550908
10,65509077
P.T.50 0,728862974 11,9776482
0,060851927
6,085192698
P.T.51 3,90625 26,30625
0,148491328
14,84913281
P.T.52 9,259259259 48,43425926
0,191171691
19,11716913
P.T.53 3,90625 30,88125
0,126492613
12,64926128
P.T.54 3,90625 30,88125
0,126492613
12,64926128
P.T.55 0,25 2,725937829
0,091711556
9,171155604
P.T.56 0,48828125 4,677988897
0,104378454
10,43784542
P.T.57 0,728862974 6,775009111
0,107581106
10,75811061
P.T.58 0,342935528 3,365980796
0,101882794
10,1882794
P.T.59 2 34,75462963
0,05754629
5,754629013
P.T.60 3,90625 1044,878472
0,003738473
0,373847304
P.T.61 3,90625 40,03125
0,097580016
9,758001561
Group D
P.T.62 3,90625 49,50810185
0,078901227
7,890122735
9,656399
968
P.T.63 3,90625 36,66087963
0,106550908
10,65509077
P.T.64 9,259259259 136,7037037
0,067732322
6,773232186
P.T.65 0,342935528 3,812071331
0,089960417
8,996041742
P.T.66 9,259259259 58,23148148
0,159007791
15,90077914
P.T.67 9,259259259 58,23148148
0,159007791
15,90077914
P.T.68 3,90625 52,87847222
0,073872217
7,387221748
P.T.69 9,259259259 636,0092593
0,014558372
1,455837179
P.T.70 9,259259259 58,23148148
0,159007791
15,90077914
P.T.71 9,259259259 58,23148148
0,159007791
15,90077914
P.T.72 31,25 166,3194444
0,187891441
18,78914405
P.T.73 9,259259259 83,92592593
0,110326567
11,03265666
Page | 220
P.T.74 9,259259259 144,3287037
0,06415397
6,415396953
P.T.75 9,259259259 669,3287037
0,01383365
1,383365035
P.T.76 3,90625 102,2858796
0,038189533
3,818953324
P.T.77 0,342935528 3,466125329
0,09893916
9,893915989
P.T.78 2 25,60462963
0,078110874
7,811087405
P.T.79 3,90625 52,87847222
0,073872217
7,387221748
P.T.80 2 19,84600745
0,100775937
10,07759372
Population data derived form ArcGIS for the investigated trade areas in
cities of Trikala, Karditsa and Kalabaka
Walk time ≤ 10 min
Trikala
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 52 140
Total Population Age 30-44 75 201
Total Population Age 45-59 71 190
Total Population Age 60+ 98 263
SUM 1090
Karditsa
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 97 237
Total Population Age 30-44 126 309
Total Population Age 45-59 123 301
Total Population Age 60+ 164 402
SUM 1759
Page | 221
Kalabaka
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 13 29
Total Population Age 30-44 19 43
Total Population Age 45-59 20 46
Total Population Age 60+ 41 93
SUM 304
Drive time ≤ 10 min
Trikala
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 472 1942
Total Population Age 30-44 681 2801
Total Population Age 45-59 642 2639
Total Population Age 60+ 888 3651
SUM 13716
Karditsa
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 1038 2591
Total Population Age 30-44 1351 3373
Total Population Age 45-59 1319 3293
Total Population Age 60+ 1760 4393
SUM 19118
Kalabaka
Number of Consumers 0-5 min 5-10 min
Total Population Age 15-29 106 173
Total Population Age 30-44 156 253
Total Population Age 45-59 167 272
Total Population Age 60+ 341 554
SUM 2022