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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

Page | 12

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

Page | 33

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

Page | 46

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

Page | 95

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

Page | 97

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

Page | 103

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

Page | 108

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.

Page | 109

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)

Page | 112

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.

Page | 114

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

Page | 120

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

Page | 122

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

Page | 123

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.

Page | 127

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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Page | 145

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

29

31

16

8

4

13

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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

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17

6

27

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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

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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C.S

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P.T

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6 9 8 11 9 7 4 6 8 7 8 5 6 1 7 9 6 4 10 7 10

P.T

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P.T

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13

P.T

.4

9 13 9 13

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12 9 10 5 11

12 8 7 13

11

14

Page | 187

P.T

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9

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P.T

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P.T

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9

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P.T

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6 9 7 10 8 9 7 8 7 7 9 8 6 3 10 8 5 4 9 7 10

P.T

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5

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6

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8

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6

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P.T

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6 10 7 10 9 9 6 9 7 7 9 7 6 3 9 9 5 4 10 8 10

P.T

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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

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3

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6

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5

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7

1

2

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P.T

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8

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P.T

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6

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P.T

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1 6 5 7 6 9 7 5 3 4 6 6 5 5 8 6 2 4 6 3 7

P.T

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8 5 11

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P.T

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P.T

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P.T

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9 9 13

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P.T

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P.T

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5

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8

8

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8

3

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5

5

6

3

3

3

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4

P.T

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9 4 10 5 11 4 4 5 7 8 2 4 7 10 2 5 9 9 4 6 3

P.T

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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

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9

7

12

8

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4

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3

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11

5

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7

5

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7

7

7

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6

P.T

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6

7

9

10

11

7

3

3

7

9

7

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4

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4

4

7

6

6

P.T

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7 8 11

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4 2 4 8 10

5 3 4 5 5 8 5 5 7 7 7

P.T

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7 9 11

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6 3 5 8 10

7 4 6 5 6 10

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P.T

.34

9 6 12

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5 1 2 10

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5 1 4 7 3 8 7 7 6 9 5

P.T

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10

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4 6 5

P.T

.36

10

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4 2 3 8 9 3 2 5 9 5 6 8 8 5 7 6

P.T

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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

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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

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7

11

5

12

6

13

11

10

9

8

11

11

8

7

12

10

6

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11

9

11

P.T

.56

3 9 4 8 6 11

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P.T

.57

7 11

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10

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10

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P.T

.58

8 11

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10

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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

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13

11

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10

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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

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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

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11

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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

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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

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6 7 5 2 6 8 9 7 6 3 5 8 9 8 7 1 8 8 5 5 7

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C.S

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C.S

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C.S

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C.S

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C.S

.8

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6 7 3 9 7 7 4 5 7 3 4 9 5 8 8

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.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

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5

P.T

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2

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8

7

3

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4

3

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5

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7

P.T

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3

4

8

6

4

4

5

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4

5

6

5

7

5

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6 7 4 9 7 7 4 4 6 3 4 9 2 10

6

P.T

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7 8 4 8 7 8 7 5 7 4 4 8 2 8 5

P.T

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P.T

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8

P.T

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7 8 2 9 8 8 7 5 7 4 5 10

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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

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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

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10

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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

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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

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P.T

.47

8 7 11

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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

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P.T

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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

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3 4 8 8 4 7 2 4 4 4 4 6 5 8 8

P.T

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6 8 5 10

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10

P.T

.61

9 7 7 3 7 5 8 8 8 9 10 3 8 2 4

P.T

.62

10

9

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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

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5

1

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5

4

3

6

5

3

7

7

6

P.T

.66

4

3

9

7

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7

6

5

8

6

7

P.T

.67

5

3

8

4

3

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7

7

4

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6

P.T

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6 5 9 5 4 5 4 6 7 9 8 3 10

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P.T

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5 4 9 4 4 4 5 6 6 8 8 2 9 5 5

P.T

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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