ARISTOTLE UNIVERSITY OF THESSALONIKI SCHOOL OF ECONOMIC SCIENCES MSc “LOGISTICS AND SUPPLY CHAIN MANAGEMENT”
THE FACILITY LOCATION PROBLEM
MASTER THESIS: Submitted for the degree of Master of Science by:
NTOUFA MARGARITA
(AEM: 13)
EXAMINATION COMMITTEE:
SUPERVISION: DIAMANTIDIS ALEXANDROS
MEMBER: BOUTSOUKI CHRISTINA, ZIKOPOULOS CHRISTOS
Thessaloniki, February 2017
1
2 DECLARATION OF OWN WORK
I declare that this master thesis with the following issue ‘The Facility Location Problems’ is my own work and any material could be considered as the work of others, it is fully referenced.
I agree that the Library can make its copies freely available for inspection. Also, as prescribed in the Copyright Law, the extensive copying of this thesis is allowable only for scholarly purposes, consistent with “fair use”, for any other reproduction or any purposes is not be allowed without my written permission.
Signature:
Name of student: NTOUFA MARGARITA
Name of supervisor: DIAMANTIDIS ALEXANDROS
DATE…………………………….
Copyright © Ntoufa Margarita, 2017
3
Dedicated To My Precious Family
4 ABSTRACT This master thesis investigates Facility Location Problem, Known as the Location theory or Location analysis. The first optimization problem which is studied from many researchers for many centuries was the location problems. Various disciplines have studied this problem, such as economy, computer science, mathematics, geography, marketing, operation research and many more.
There are four elements which describe location problems:
A. Customers (who are already located at points or on route). B. Facilities where will be located yet. C. A space in which located customers and facilities. D. Finally, standard metrics that show distances or time between customers and facilities.
The target of the location problem has to do with service of facilities with minimum speed and the minimal cost, in order to satisfy the customer under some set of constraints. In addition, each facility location problems is unique and these problems are very easy to be understood but not easy to be solved.
On one hand, the selection of a facility location is an important decision for the following: companies, plants, warehouses, distribution centers, shops, hospitals, schools, fire station, courier service, police station, post office, ambulance facilities etc., and for supply chain generally. But on the other hand, is a difficult decision because it would be costly, and after an installation it is difficult to change the location. Furthermore, many of the operations that are become in the company rely on the location of its facilities.
In order to understand this topic better, the following project is divided into three parts and seven chapters. Figure 1, shows the way of study from the first chapter to the last chapter, from introduction to discussion and conclusion. Also, we quote literature on recent developments of facility location problem for the selection suitable sites not only to locate their firms but also to give profitability for the facility’s lifetime, consequently the optimum location of the company’s facilities can lead to growth rapidly and it has a target to satisfy the customers as good as possible.
5 Chapter 1 Chapter 2
Introduction Literature Review
Chapter 3 Chapter 4
Methodology Results and Analysis I
Chapter 5 Chapter 6
Discussion Conclusion
Chapter 7
Direction for future research
Figure 1: a brief review of chapters
Chapter 1: In this chapter, a brief review of Facility Location and mainly in Facility Location Problems. Chapter 2: In this chapter, we will go into details of facility location problems, such as definition, classifications, and factors which affect the location decision and the international facility location-decision making factors.
Chapter 3: In this chapter, we will present the methodology of location facility. We will refer AHP and fuzzy method. In addition, we will mention some location analysis techniques, the classification of models, the basic models of facility location and the measurement types for distances.
Chapter 4: In this chapter, we will mention the analysis and the results of a real case study of a Greek company COCOLAK.
Chapter 5: In this chapter, we will discuss about the result and we will suggest an optimal solution for a rent or build a new warehouse in order to store all amusement games. Moreover, we will apply some of techniques and models which we mentioned in Chapter 3.
Chapter 6: In this chapter, we will quote our conclusion of this research.
Chapter 7: Last but not least chapter, we will give some information about future research.
6 ACKNOWLEDGMENTS After, one and half year of my studying, we have come to the end of our journey at Aristotle University of Thessaloniki, specialized in Logistics and Supply Chain Management. Since the start of my Master’s program i knew that i would follow hard work and endless hours of study. This project has been conducted in order to obtain an MSc degree in Logistics and was carried out from October 2015 to February 2017. This project would not have been possible without the support of many people.
First of all, I would like to express my deepest gratitude to my supervision Diamantidi Alexandro who encouraged, advised and guided me during of writing the master thesis and led me in the right direction.
In addition, grateful acknowledgement to my parents, George and Anastasia for providing me the opportunity to be here, and their love and support during this hard work. They were always there when I needed them. Also, I would like to thank my brother John, who helped me to maintain a sense of humor reminding me not to take myself too seriously. It is impossible for me not to recognize their full support this year.
I would like also to thank my friend Christiana where the company ‘’COCOLAK’’ is family company. We had an excellent communication and she responded quickly whenever I have had any questions about the data who gave me in order to finish my master thesis.
Also, thanks to my faculty for the knowledge offered me in order to improve myself.
Last, I would like to thank my numerous friends who offered support, love, encouragement, patience and above understood me during this project. Without them, this master thesis would never have been finished.
Yours sincerely,
Ntoufa G. Margarita
7 TABLE OF CONTENTS
ABSTRACT ...... 5
ACKNOWLEDGMENTS ...... 7
HISTORICAL EVOLUTION OF LOGISTICS ...... 13
CONCEPTS ...... 14
PART I ...... 15
CHAPTER 1 ...... 15
1. INTRODUCTION ...... 15
CHAPTER 2 ...... 16
2. LITERATURE REVIEW ...... 16
2.1 FACILITY LOCATION ...... 16
2.2 CLASSIFICATION OF LOCATION FACILITY PROBLEMS ...... 19
2.3 FACTORS AFFECTING THE LOCATION DECISION ...... 20 2.3.1 INTERNATIONAL FACILITY LOCATION – DECISION MAKING FACTORS ...... 22 PART II ...... 27
CHAPTER 3 ...... 27
3.1METHODOLOGY ...... 27
3.2 THE ANALYTIC HIERARCHY PROCESS (AHP)...... 28 3.2.1 THE CHARACTERISTICS OF AHP...... 29
3.2.2 THE STEPS OF AHP ...... 29
3.2.3 PROS AND CONS OF AHP ...... 30 3.3. FUZZY LOGIC ...... 30
3.4 LOCATION ANALYSIS TECHNIQUES ...... 31 3.4.1. LOCATION FACTOR RATING METHOD...... 31
3.4.2 CENTER OF GRAVITY TECHNIQUE ...... 32
3.4.3 LOAD –DISTANCE TECHNIQUE ...... 34 3.5 CLASSIFICATION OF MODELS OF LOCATION FACILITIES ...... 35 3.5.1 THE CLASSIFICATION OF FRANCIS AND WHITE (1974) ...... 36
3.5.2 THE CLASSIFICATION OF DASKIN (1995) ...... 40
3.5.3 THE CLASSIFICATION OF REVELLE ET AL (2008) ...... 41 3.6 DISTANCE MEASUREMENT TYPES ...... 44
8 3.6.1 DISTANCE OBJECTIVE ...... 45 3.7 APPLICATIONS OF FACILITY LOCATION MODELS ...... 46
3.8 BASIC MODELS OF LOCATION FACILITIES ...... 46 3.8.1 P-MEDIAN MODEL ...... 47
3.8.2 THE FIXED CHARGE MODEL ...... 49
3.8.3 HUB AND SPOKE LOCATION MODEL ...... 50
3.8.4 A) MAXISUM LOCATION MODEL ...... 53
3.8.4B) MAXISUM LOCATION MODEL ...... 54
3.8.5 SET COVERING LOCATION MODEL ...... 55
3.8.6 MAXIMAL COVERING MODEL ...... 56
3.8.7 THE P-CENTER MODEL ...... 57
3.8.8 P-DISPERSION MODEL ...... 59 3.9 UNCAPACITATED FACILITY LOCATION PROBLEM (UFLP) ...... 60
3.10 THE LOCATION FACILITY PROBLEMS – INVENTORY ...... 61
PART III ...... 64
CHAPTER 4 ...... 64
4. ANALYSIS AND RESYLTS ...... 64
4.1 THE COMPANY ‘’COCOLAK’’- HISTORICAL EVOLUTION ...... 64
4.2 THE SUPPLY CHAIN OF COCOLAK ...... 68
4.3 THE WAREHOUSE OF COCOLAK ...... 69
4.4 DEMAND- NUMBER OF CUSTOMERS- NUMBER OF STORES ...... 70
CHAPTER 5 ...... 76
5. 1 DISCUSSION ...... 76
5.2 EXAMPLES OF LOCATION ANALYSIS TECHNIQUES ...... 76 5.2.1 IMPLEMENTATION OF FACTOR LOCATION RATING METHOD IN COCOLAK ...... 76
5.2.3 IMPLEMENTATION OF THE CENTER-OF-GRAVITY TECHNIQUE IN COCOLAK ..... 78
5.2.4 IMPLEMENTATION OF THE LOAD-DISTANCE TECHNIQUE IN COCOLAK ...... 80 5.3 EXAMPLES OF LOCATION ANALYSIS MODELS-PROBLEMS ...... 82
5.4 APPLICATION OF THE P-MEDIAN MODEL-PROBLEM ...... 82
5.5 APPLICATION OF THE P-CENTER MODEL-PROBLEM ...... 84
5.6 APPLICATION OF THE UFLP MODEL-PROBLEM ...... 84
9 CHAPTER 6 ...... 87
6.1 CONCLUSION ...... 87
CHAPTER 7 ...... 88
7.1 DIRECTION FOR FUTURE RESEARCH ...... 88
CHAPTER 8 ...... 89
8.1 REFERENCES ...... 89
10 LIST OF TABLES
Table 1: shows the classification of the Models ...... 35 Table 2: Shows the separation of models...... 46 Table 3: shows the number of branches stores, the branches stores and the year of establishment...... 66 Table 4: shows the number of games that company has in county of Kavala (for own uses and exploitation). The 1-2-3 shows the number of stores which are in exploitation...... 70 Table 5: shows the number of games that company has in the country of Katerini (for own uses and exploitation). The 1-2-3 shows the number of stores which are in exploitation...... 71 Table 6: shows the number of shops of exploitation and the number of amusement games in different areas of county of Halkidikis and mainly in region of Kassandras...... 71 Table 7: shows the number of games that company has in Halkidiki and mainly in Sithonia area of Halkidiki...... 72 Table 8: a conclusive table of above four tables, COCOLAK has the greatest demand in Kassandria area of Halkidiki...... 72 Table 9: depict the demand values in percentage ...... 72 Table 10: shows the distances between the areas of Kassandria and Sithonia of Halkidiki with the four candidate sites in which we would be located the new warehouse of company or rented. Company suggested the following four candidate points, Nea Moudania, Potidaia...... 73 Table 11: shows the number of customers in shops of Kassandria and Sithonia areas of Halkidiki, the years: 2016, 2015and 2014 and the months June, July, August and September...... 74 Table 12: depicts the location factor, weight score and the candidate sites ...... 77 Table 13: depicts the weighted score for each candidate site and for each location factor the value of weight...... 77 Table 14: shows the data that we used in order to make a decision...... 79 Table 15: coordinates of the four candidate locations...... 80 Table 16: shows the distances between the four candidate’s sites with each existing facility...... 81 Table 17: depicts the Load distances of four candidates’ sites ...... 81 Table 18: shows the distances between five sites in which company decided to locate its warehouse...... 83 Table 19: shows the fixed and variable costs...... 85 Table 20: shows the candidate cities and their objective values ...... 85 Table 21: depict the total costs for each candidate location: ...... 85 Table 22: aggregated results of Techniques ...... 87 Table 23: aggregated results of Models...... 87 Table 24 ...... 96 Table 25 ...... 96 Table 26 ...... 96 Table 27 ...... 96
11 LIST OF FIGURES
Figure 1: a brief review of chapters ...... 6 Figure 2: depicts the movements to be done for sending product ...... 14 Figure 3: shows the facility location problems...... 18 Figure 4: shows that the location decision is a sequence of the three regions...... 20 Figure 5: classification of Francis and White ...... 36 Figure 6:Classification of Daskin ...... 40 Figure 7: Classification of ReVelle et al ...... 41 Figure 8: Continuous models ...... 41 Figure 9:Network models ...... 42 Figure 10: Discrete models ...... 42 Figure 11: Separation of discrete models ...... 43 Figure 12:Euclidean distance ...... 44 Figure 13:Rectilinear distance ...... 44 Figure 14:Great circle ...... 45 Figure 15: four stages of strategic planning...... 61 Figure 16: four candidate sites depending on co-ordinates x and y...... 79
LIST OF APPENDIX
A. P-MEDIAN MODEL……………………………………………………………96
B. P-CENTER MODEL…………………………………………………………….98
C. UFLP MODEL…………………………………………………………………..99
D. AMUSEMENT GAMES………………………………………………………..100
12 HISTORICAL EVOLUTION OF LOGISTICS Here we will present some information about the historical of Logistics, thus the terms Logistics, was used by Leo the Wise for the first time. Logistics come from a Greek word ‘’ACCOUNTING ’’. Logistics is the most important element of the modern economy and it is not new terms but it existed from antiquity for example:
. Around 2700 B.C, blocks of stone must be transported at the construction site for construction of pyramid. . Around 300 B.C, it was used by Greek rowing vessels and it helped the development of trade. . Around 700 A.D, was needed procurement for the construction of the Mezquita Mosque. . Around 1200 A.D, appeared the development of international network . Around 1500 A.D, appeared the development of mail shipping service. . Around 1800 A.D, appeared the development of new road i.e railroad. . Around 1940 A.D, the soldiers used this term, hence the logistics was involved in the movement and supply troops. . Around 1956 A.D, appeared the world trade. . Around 1970-1980: we have just in time goods. . Around 1990: appeared the development on distribution. . Today: Logistics is the supply chain management with easy words is the transport of products from suppliers to the end customers.
It should be emphasized that the definition of the logistics is mainly the flow of goods, acquisition, storage, movement and transportation of products. Also, The D.W.Engels, in his book titled “Alexander the Great and the Logistics of the Macedonia Army’’ [20], referred that the greatest logistician was Alexander the Great. Alexander in order to succeed the combat effectiveness of the Macedonian Army, he followed a strategy that was based entirely on proper organization of a logistics circuit. As we reported before, the logistics were used in military area and were spreaded in business, in management science and in operations research. The homeland of logistics is considered USA. Companies have to do with the competition and the profit, with a result to create <> between them, the war of profit, which it is faced with the same support with this of military operations. Consequence, the identity of the logistic was formally adopted in 1963 with the establishment of Council of Logistics Management [87], [88].
Finally, an important element of a logistics system is that in concludes a set of facilities which is linked by transportation services.
13 CONCEPTS What is logistics- What is Supply Chain Management? There are many concepts of logistics, but the simplest explanation about logistics and supply chain management can be summarized as below,
‘’Logistics’’ is the process of strategically managing the procurement, movement and storage of materials, parts and finished inventory(and the related information flows) through the organization and its marketing channels in such a way that current and future profitability are maximized through the cost-effective fulfillment of orders. (Logistics & supply chain management by Martin Christopher. [56]
Αnother simple meaning of logistics is the following: ‘’Logistics’’ is about getting the right product, to the right customer, in the right quantity, in the right condition, at the right place, at the right time, and at the right cost. (The seven Rs of Logistics)" - Supply Chain Management: A Logistics Perspective By John J. Coyle et al [43].
"Supply Chain’’ is the network of organizations that are involved, through upstream and downstream linkages, in the different processes and activities that produce value in the form of products and services in the hands of the ultimate consumer" - Logistics and Supply Chain Management by Martin Christopher [56].
In the following figure, we can observe a supply chain network of suppliers, factories, distribution centers and customers. The arrows depict the movements to be done for sending goods.
Figure 2: depicts the movements to be done for sending product
Supply Chain Management Knowledge Simplified by SupplyChainOp
14 PART I
CHAPTER 1
1. INTRODUCTION Facilities are defined as buildings in which people use machines and other materials in order to make a tangible product or to provide an intangible good such as a service. With other words facilities are sites in which materials are processed, e.g. manufactured, stored, sorted, sold or consumed. In addition, facility is a real property which is consisted of one or more of the below: accommodation, restaurant, industrial factories, company, shop, warehouse, bank, manufacturing center, distribution center, transshipment points, retail outlets, hospital, transportation terminals, fire station etc. As regards customers can be consumers, retailers, wholesalers etc.
During the first stage of designing and operation of a logistics system, you need to address some fundamental topics such as, where should new facilities be located? What is the best size? Where the materials are stored? Where spare parts are stocked? How should warehouse operate? If you want to open a facility, you can answer these above questions. In this master thesis we will examine only where to be located?
Many issues complicate the facility location decisions. The location of the facility affects the company in different ways and there are many different variables which you have to in your mind in order to reach in the optimal decision.
Moreover, in the decision making process takes part in people from different departments of the company, general it shows easy but it is not [28], [76] . All public and private organizations face issues of facility location and an optimal location gives an advantage against competitors.
As we mentioned previously, the location facility is a well-known phenomenon in operational research area. As quantitative as qualitative methods are applied for the location problems. Facilities must be located in a suitable place, targeting to minimize the annual total logistics cost subject to some constraints which are related to facility capacity and customer service level. Managers try to meet the needs of customers, minimizing fixed cost and transportation cost.
15 CHAPTER 2
2. LITERATURE REVIEW
2.1 FACILITY LOCATION In the following chapter, we will mention some important economists. They defined the theory of the location facility in order to help people to decide for an optimal location. Hence:
Thunen (1875) [79] was the first who applied a method for evaluating location decisions by the economic scope.
Launhardt (1885) [47] examined the differences between the cost and demand factors at alternative locations in the location decision process.
Alfred Weber (1909), [1] a German economist and sociologist defined and followed a systems theory, the ‘’Theory of the Location of Industries’’. This theory of location is the starting point of all analytical studies which were made about this topic, and mainly in location of plant and office. The important element of this theory is that includes factors which detect and choose the basic location of these industries.
Harold Hotelling (1929) [38], examined the competition between companies. Also, he mentioned that the companies tend to locate their facility close to market. On the one hand, many researches tried to improve his model. But on the other hand, some others researches questioned his theory, (Lerner and Singar 1937, Balvers and Szerb 1996, Katz 1995, Smithies 1941, Chamberlain 1946, Ohlin 1935, 1952).
Furthermore, Hoover (1937), (1948), [36], [37] made reference in his papers studies which rely on cost and demand.
After years, Hakimi (1964) [31] decided to develop the Weber’s problem. He used some applications in order to minimize the sum of the weighted distance from the points, so with this manner he found the P point on graph, but there was not limited far the locations of the points, so he supposed that the facilities could be located in any points of graph, and this is a solution. However, he considered that this problem is
16 candidate of Weber’s problem because the solution is given only through constraint. Furthermore, he suggested a way how to distribute the switching centers in the Tele- communication network.
According to Cohon (1978), [12] divided the location facility problems in private and public section. So, he considered that the main separation is the differentiation. For example, in private sector, the service units and the facilities like a warehouse or a customer service station, target to minimize cost or to maximize profit and it is estimated in money, distances or time.
Also, from the early 1980 and with the innovations in technology are created facilities such as chemical plants, wastewater etc. i.e. undesirable facilities, noxious and hazardous to humanity. In this case, the targets are to minimize the negative effects on the people living around them and are built far away from the people.
In addition, Epping (1982), [23] stressed that a decision to construct a new plant or to expand the existing facilities is result not only monetary resources but also the human resources. Furthermore, he considered that an optimal location today it is not the best one next year. For this reason, he suggested that companies need to relocate their facilities more often in order to maintain their competitiveness and maybe bring profit. So, he tried to change the opinion of them who believe that a facility is not necessary to relocate until the economic life of them comes to the end.
McCarthy (2003), [59] declared that the location decisions are very important for manufacturing firms. Moreover, the optimal locations offer not only competitive advantage but also the success of a company.
Later, the Ballou (2004) [5] considered that the location of fixed facilities is a decision problem that gives structure, form and shape in the supply chain system. These above three elements play an important role in order to operate this system. Hence, the first multi-period location problem was introduced by him. In this problem the parameters change over time and the target is to change the configuration of the facilities to these parameters.
17 According to Snyder (2010) [75], the location facility problems regards the optimal choice to locate facilities in order to manage the best customer service and achieve a harmony between cost and service. For instance, on one hand if you construct few facilities, some of customers who are away from the nearest facility do not serviced, whereas on the other hand if you build many facilities then all customers will be pleased. Also, the transportation services move materials between facilities. For moving, they use vehicles such as trucks, tractors, cars, trains, Lorries, trailers, and they put the equipment for transport in pallets, containers, boxes etc.
Facility Location Problem: is given a set of facility locations and a set of customers who are served from the facilities. Below we depict a graph that shows this problem:
Figure 3: shows the facility location problems
18 2.2 CLASSIFICATION OF LOCATION FACILITY PROBLEMS [28] Location problem can be classified as follow:
i. Time horizon: is divided in single-period problems and in multi-period problems. In single-period problems the location decisions have to be counted at the beginning of the planning horizon. Whereas in multi-period problems must make at the beginning of the planning horizon some changes.
ii. Facility typology: in single-type location problems, is located only one kind of facility like Distribution Center, whereas in multi-type problems are located many types of facility like warehouses and distribution center. iii. Material flows: in single-commodity problems there is a single homogeneous flow of materials whereas in multi-commodity problems there are many elements with various features. iv. Interaction among facilities: there are cases which a logistic system can be move material among facilities with similar type. In this case, the optimal facility location depends on the location of the facilities that interact.
v. Dominant material flows: in single-echelon location problem the material flow that comes is negligible, whereas in multi-echelon location problems the inbound and outbound commodities are relevant. vi. Demand divibility: divisible is defined when one facility or one customer be supplied by a single center, whereas when a customer or a facility is served by two or more center then we have an indivisible distribution system. vii. Influence of transportation on location decisions: many location models consider that transportation cost is estimated multiplied by the freight volume and the distance between the sites. It depends on the kilometers which the truck does, we cannot estimate the accurate cost because the truck carries goods in different positions.
19 2.3 FACTORS AFFECTING THE LOCATION DECISION
Many different factors may influence the decision of firms, where to locate production facilities across national boundaries (McCarthy 2003) [59]. These factors are many and different in the country, and they differ from country to country. Some of them have impact in the location decisions of a business and if you ignore them, then the results will be false. These factors cannot be changed. They show the efficiency and effectiveness of facilities. As we mentioned before these factors differ and the differentiation depends on the facilities, for example, other factors affect a manufacturing business where to locate their facilities and other factors affect where a retailing business will be set up. As we can observe in the following figure, the Location decision is a sequence of the three below regions: country, community and site. Barkley and McNamara (1994), [7] classified the locations factors depend on their plant size.
Figure 4: shows that the location decision is a sequence of the three regions.
Also, we emphasized that the factors which affect the location decision are not the same, consequently they will be different in the above three regions, Thus:
20 Country factors are the following:
a. Political risks, government rules, attitudes, incentives b. Cultural and economic issues c. Location of markets d. Exchange rates and current risks e. Availability of supplies, energy and communication f. Labor availability, productivity and costs. Community Factors are the following:
a. Corporate desires b. Attractiveness of region c. Costs and availability of utilities d. Labor availability, costs, attitudes towards unions e. Environmental regulations f. Government incentives and policies g. Proximity to raw materials and customers h. Land costs Site factors are the following:
a. Size and cost b. Air, rail, highway systems c. Zone restrict d. Nearness of services or suppliers needed e. Environmental regulations at local level
The target of facility location problems is to achieve a perfect balance between the above factors in order to locate business in the best possible region. With this manner it manages lower production cost, lower transportation cost, maximization of profit or minimization of environmental effects. Hence, we will examine these factors in order to reach in an optimal solution. So, we will use some optimization models and other operational research techniques such as linear programming and other that we will mention in the following chapter.
21 2.3.1 INTERNATIONAL FACILITY LOCATION – DECISION MAKING FACTORS Bass, McGragor and Walters (1977), [8] suggested that if you want to invest in a foreign country, you have to in your mind the following factors such as:
a. accessibility, b. basic services available, c. environment, d. site costs, e. industrialization, f. labor and staff availability, g. host taxes and incentives, h. area reputation, i. the nature of the host government and its policies
Moreover, about the multinationals Rummel and Heenan (1978), [65] quoted a list of factors which are important in making international industrial location decisions. So:
a. domestic b. instability c. foreign d. conflict e. political climate f. economic climate
Masood Badri, Donald Davis and Donna Davis (1995), [57] did a study and they concluded that the most important factors in international location decisions are the following:
1. Factors which are related about the transportation: . Availability of highway, railroad, pipeline, airway facilities . Availability of trucking services . Cost of raw material transportation
2. Factors which are related about the labor: . Availability of skilled labor . Wage rates . Educational level of labor
22 . Cost of living (housing) . Worker stability
3. Factors about the raw materials: . Availability of raw materials . Closeness to materials . Availability of storage facilities . Location of suppliers . Freight cost
4. Factors about the market: . Proximity to consumers and to producers . Forecasting of growth of markets . Shipping costs . Availability of marketing services . Consumer characteristics . Location of competitors . Future expansion opportunities and the size of market in industrial site
5. Factors which are related about the industrial site: . Cost of industrial land . Space required or for future expansion . Lending institutions like bank and insurance rates . Closeness to other industries
6. Factors which are related about the utilities: . Adequacy of water supply . Quality and cost of water . Availability of disposable facilities of industrial waste . Cost and availability of fuels and electronic power
23 7. Government factors: . Zoning codes . Compensation, insurance and safety inspection laws . Environment pollution laws
8. Tax factors: . Tax assessment basis . Industrial property and state corporate tax rates . Availability of tax free operations . State sales tax
9. Climate factors: . Living conditions . Humidity . Temperature . Air pollution
10. Community factors . Availability of universities or colleges or schools . Availability of religious like church and library facilities . Availability of medical and malls or hotels facilities . Availability of banks and financial institution
11. Factors which are related about the political situation of foreign country: . Stability of regime o Protection of expropriation . Type of treaties, pacts and military alliances (or with which countries) . Attitude towards foreign capital
12. Factors which are related about the global competition and survival: . Availability of material and of labor . Market opportunities . Availability of foreign capital . Proximity to other international markets
24 13. Factors of government regulations: . Clarity of corporate investment laws . Regulations of joint ventures, mergers and transfer out of country . Taxation and laws of foreign owned companies . Allowable percentage of employees who may be foreign . Prevalence bureaucratic red tape . Imposing price controls by government . Requirements for setting local corporations
14. Economic factors: . Standard of living . Size of per capital income . Strength of currency against US dollar . Balance of payment status . Availability and size of government aids
Canel and Khumawala (1996), [9] presented a mixed-integer programming approach for the international facilities location. Also, they emphasized in capacitated and uncapacitated multi-period international facilities location problems. The list of factors is depicted below:
Trade barriers and regulations International customers and international competition Low cost and incentives Proximity to market and easily access Quickly responsiveness of customers News or expand markets Taxes, economies of scale, Power and prestige
McCarthy (2003) [59] presented a set of factors that influence international location decisions from analysis of existing literature. He announced that each facility has specific factors and the importance of each factor is not equal for every case.
25 Badri (2007) [3] suggested that there are other factors. Some examples we will mention in the following research project. Some of those factors include:
Proximity to schools, colleges, universities (Audretsch and Stephen, 1996) [2]. Expected or percent of market share (Drezner and Drezner, 1996) [21]. Changes in the location of users (Hansen and Roberts, 1996) [34]. Amount of development in region (Wojan and Pulver, 1995) [81]. Level of wages (Manders, 1995; Ma, 2006) [53], [51]. Changes in transport rates (Mai and Hwang, 1994; Leitham et al., 2000; Mazzarol and Choo, 2003) [52], [49], [58]. The location of other competitors (Serra and ReVelle, 1994; Cieslik, 2005; Siebert, 2006) [69], [11], [74]. Changes in local demand (Justman, 1994; Figueiredo et al., 2002) [44], [26]. Hazardous waste and pollution laws (Groothuis and Miller, 1994) [29].
26 PART II
CHAPTER 3
3.1METHODOLOGY The location facility is a multidimensional problem. The methodology which we will follow in this master thesis regards the detection of an optimal location for the installation of a warehouse. This is succeeded with different tools, methods and techniques. All these tools we will help us in order to solve a problem and to reach in the best result. For example the AHP method is a tool for making decision. Also, the marginal analysis is useful for evaluating alternatives. In addition, break –even analysis is used for making a decision in order to evaluate the available alternatives based on fixed cost, variable cost and price. On the other hand, the ratio analysis is very useful tool, it help us to find the weaknesses of a firms and to improve them. Furthermore, there are many quantitative methods in the process of decision making such as simulation, decision tree analysis, center of gravity, etc. Some of the previous we will examine in the following subsections. The phases which are related about the decision making for location facilities are the following:
1. find the total number of candidate location 2. the optimal location in a region 3. estimate the efficient and effectiveness of the previous location 4. the improvement of the existing location
Furthermore, Rahman and Smith (2000), [70] suggested some steps about the methodology of facility location, below we quote these steps:
a. understanding the problems b. development of model( qualitative and quantitative) c. analysis of model d. valuation of results e. implementation of results
The important element in this process of location facility is found the suitable model which will optimize the objective function.
27 3.2 THE ANALYTIC HIERARCHY PROCESS (AHP) In this section, we will present one of the most notorious methods for making multi- criteria decisions. The AHP has been applied in many cases such as: Dyer (1990), [22] Perez (1995), [62] Satty (1980), [67] Sun et al. (1996), Tversky and Simonson (1993), and Yang and Lee (1997) [82].
The Analytic Hierarchy Process (AHP), was introduced by Thomas Satty (1980), [67] and is published in 1992 in Expert Choice publication. AHP is an effective tool which regards the decision making. Also, it gives set priorities in order to reach in the best solution among several alternatives. In addition, the AHP is a helpful technique for checking the stability of the decision.
Furthermore, all tools use inputs and outputs, hence, the AHP for inputs uses real measures, such as prices or personal opinions into a numerical grid whereas the outputs are ratio scales and indices which are result of eigenvalues and eigenvectors. Some applications which have used AHP are the following [60]:
. First of all, factors which influence software development and productivity. . Also, for improving safety characteristic in motor vehicles with the choice of the suitable strategy. . In addition, it used for estimating cost and scheduling options for material planning. . Another application is to select desired software elements from different software vendors. . Finally, it is very important for evaluating the quality of research proposals. Now, we will examine how the AHP works.
First of all, AHP has two sets, one of assessment criteria and other of alternative ideas which target to the best decision. Hence, when the different criteria succeed manage a balance, and then we consider that is the best option. Moreover, AHP creates a weight for each assessment criterion in which the highest is the best. Subsequently, allocates a score to each option. As higher is the score, so better is the performance of the option. Thereafter, AHP includes criteria of weights and a total score for every option. The total score is a sum of the scores which estimated from all criteria.
28 3.2.1 THE CHARACTERISTICS OF AHP Furthermore, we will report the characteristics of the AHP:
. First of all, AHP is very useful, helpful and flexible tool. . In addition, it is a tool which has the ability to translate the appreciation as qualitative as quantitative. . Moreover, AHP is a simple model relevant to decision making. . Furthermore, AHP is needed a large number of assessments by the user. . Finally, as we mentioned before, despite it is very simple, it requires comparison between two options or criteria for making a decision.
3.2.2 THE STEPS OF AHP The methodology of the AHP can be described in the below steps:
Step 1: In this step the problem is divided into a hierarchy of criteria, alternatives and sub-criteria. Hierarchy depicts a relationship between components of the level, with other words every component is connected to every other one.
Step 2: In this step, is emphasized in data which are collected from specific or decision-maker. It classifies the comparison as equal, marginally strong, strong, very strong and extremely strong.
Step 3: In this step, we classify in a matrix the result which we estimated in step 2. The elements of the matrix are: if the criterion of the row i, is better than criterion of the column j. This comparison is based on a value equal to 1, thus, for example if the value of element i, is greater than 1, and vice versa.
Step 4: In this step, we estimate the eigenvalue and the eigenvector and we put the number in the matrix. They show the importance of the different criteria. The elements of the eigenvector are called weights.
Step 5: In this step, we will calculate the index of consistency with the following formula. So, CR= CL (max n)/(n 1),where max is the maximum eigenvalue of the matrix. Also, Saaty considered that the value of the index of consistency (CR) should be less than 0.1.
Step 6: Last but not least step, here we will take the relative weights of the decision matrix in order to reach in a various assessments of alternative decisions.
29 3.2.3 PROS AND CONS OF AHP
On one hand, we will present the pros of AHP and on the other hand, we will quote the cons of AHP. Thus,
Pros
. AHP lets multi-criteria decision making . Sometimes, it is difficult to evaluate the criteria . It is applicable for group decision making environments. . The elements which effect a decision can be mixed. . It contains flexibility and ability in order to be adapted in different problems. . Also, AHP involves a software programs in order to help makers during the mathematical operations that they need eigenvalues and eigenvectors. Cons
. In AHP there are hidden assumptions. . Also, when the number of alternatives criteria is greater than seven it is difficult to be used. . General, in AHP is difficult to add a new criterion or alternative. . Finally, it is difficult to take an existing criterion or alternative.
3.3. FUZZY LOGIC [86]
Fuzzy theory was developed in order to solve problems which there are not fixed boundaries. A fuzzy set is not limited by binary ‘’yes or no’’ but it gives a value which expresses the degree of each element that belongs to each set. A well-known algorithm in fuzzy classification is Fuzzy C-Means (Hatzichristos 1999) [86].This algorithm analyzes the input elements, the relations between them and then classifies them.
30 3.4 LOCATION ANALYSIS TECHNIQUES
In this point, we will give more emphasis in Location Techniques. The Location Techniques are very important in location decision, for example, as we mentioned previously, there are many questions about the location facilities, such as where the suitable place is in order to locate the facilities or which the information is which we will need, etc. Thus, the role of these techniques is to organize this information, and to conclude a starting point for comparing different locations under some different criteria. Expect of the analytical techniques which we mentioned in previous subsection, there’s more like: the location rating factor, the center-of-gravity technique, the load- distance technique, the break-even analysis, the transportation model, simulation methods, crossover charts, geographic information systems, purchasing power analysis of area, traffic counts, demographic analysis of drawing area etc., but we will present and examine the first three techniques.
Also, the analysis must be followed by the below process, thus:
Step1: identify dominant location factors
Step2: develop alternatives locations
Step3: evaluate the locations alternatives
3.4.1. LOCATION FACTOR RATING METHOD This technique is most popular because in the analysis can be used a wide variety of factors. Also, it is very useful mainly for industrial location and service. In this method, the decision is based on quantitative (intangible like education quality) and qualitative (tangible such as raw materials) factors. Each factor is appropriated weight. This weight is from 0 to 1.00 which demonstrates the priority and shows the importance of this decision. Finally, the site with the highest weighted score is the best choice.
31 Six steps of factor rating method
Step 1: In step 1, we will develop a list of relevant factors which is called critical success factors. Step 2: In this step, we will assign a weight to each factor. This connection shows the importance of the company (1-0). Step 3: Here, we will develop a rating scale for the factors (0-100, etc). Step 4: Score each location using factor scale. Step 5: In this step, we will multiply the scores by the weights for each factor and total the weighted scores for each location. Step 6: In finally step, we will select location, so the optimal location in order to locate your facility is this with the maximum total score.
3.4.2 CENTER OF GRAVITY TECHNIQUE
The center-of-gravity is a quantitative technique for location a facility such as a distribution center or a warehouse, hence, the question in which the center of gravity answers is the following, where is located a distribution center or warehouse. Here, the decision is based on reduction cost and in increased service level in storing.
There are some characteristics which will help you in order to reduce costs in your warehouse, thus you have in your mind the following three elements:
a. It is necessary you know the warehousing processes. b. Moreover, what warehouse metrics do you use in order to see if your processes are going well or not? c. Also, your warehouse location must be optimal in order to meet service level and to satisfy all customers.
One common approach about where to be built the next new fulfillment center location is a method which is called Center of Gravity Method. Hence, we will use this method, we will find the geographic coordinates (X and Y) for a potential new facility aiming to minimize costs.
32
Three steps of Center of Gravity Method
Step1:
Step 1 is made up by three phases:
- First of all, you must place the locations on coordinate system. - Also, the origin of the coordinate system and the scale which is used and the relative distances are correctly. - Furthermore, this process can be done easily by placing a grid over an ordinary map.
Step2
The above coordinates for the location of the new facility are estimated with the following formulas:
n n xiWi yiWi x i1 , y i1 n n Wi Wi i1 i1
Where:
. x, y: are the coordinates of the new facility at center of gravity . xi, yi: are the coordinates of existing facility i . Wi: is the annual weight which is comprised from facility i
Step 3
In this step, we have obtained the X and Y coordinates, i.e. exactly the point which they are depicted on the map. In addition, we stress that these coordinates are based on straight-line distances, but in case of a real situation these coordinates maybe follow more circuitous routes.
33 3.4.3 LOAD –DISTANCE TECHNIQUE
In this method, a set of location coordinated is not identified. The different location uses a load-distance value which counts the weight and distance. Hence, a load- distance value is calculated as following:
n LD l i di i1
Where:
. LD: is the load-distance value . li: is the load which is expressed as a weight i.e. number of trips or units being shipped from the proposed site to location i . di: declare the distance between the proposed site and location i
As regards the distance di, is the travel distance or it can be calculated from a map. In addition, there is another way in order to estimate this distance di, it can be estimated by using the below formula for the straight-line distance between two points, which is the hypotenuse of a right triangle:
di=√(푥푖 − 푥)2 + (푦푖 − 푦)2
Where: . (x, y) : are the coordinates of proposed site
. (xi, yi) : are the coordinates of existing facility
The load-distance technique was applied by calculating a load-distance value for every possible facility location. The conclusion of this method is that the location with the lowest value has result to reduce the transportation cost and so it would be preferable.
34 3.5 CLASSIFICATION OF MODELS OF LOCATION FACILITIES
In this section, we will mention and will analyze the models of location facilities which are very important in making decision of location facility and they aim to reach in optimal solution. Before we present and analyze the models of location facilities, we would like to quote a matrix of the general classification of the problems. So, we will consider a matrix with four criteria. More details we can observe below, thus [27]:
Table 1: shows the classification of the Models
1) Number 2)Feasible 3)Distance 4)Objective of facilities region metric function to be located
Single Discrete Euclidean Maximin Facility Multiple Continuous Rectilinear Maxisum Facility
Network
The above table is divided by four columns. So, the number of facilities can be divided in single and multiple facilities. The second column depict that the feasible region can be divided in discrete, continuous and network. The next column shows the distance metric which are Euclidean or rectilinear, and finally the objective function is maximin or maxisum. We will analyze in detail below.
Αs previous mentioned by Snyder(2010), [75] a location facility problem can answer to the question where is the suitable location for facilities. According to Mozafari (2009) and Tafazzoli, [68] the location science in order to continue the researches should be a classification of location facilities models. So, we will present below the three classifications of models.
35 3.5.1 THE CLASSIFICATION OF FRANCIS AND WHITE (1974)
The first classification about the location facility problems and layout is the Francis and White (1974), [27] (figure 5). This classification consist six criteria: a) new facilities features, b) existing facility locations, c) solution space, d) objective, e) distance measure, f) New/existing facility interaction. In the following figures we can see more detail for each one, hence:
Figure 5: classification of Francis and White
a) New Facilities Characteristics
New Facilities Characteristics
Single Multiple
Point Area
Layout Problem Other
Parameter Decision Variable
Independent Dependent
36
b) Existing Facility Locations
Existing Facility Locations
Static Dynamic
Deterministic Probabilistic
Point Area
Parameter Decision
c) Solution Space
Solution Space
Single Dimension Multidimensional
Discrete Continuous
Constrained Unconstrained
37
d) Objective
Objective
Qualitative Quantitative
Minimize Total Cost Minimize Maximum
Other
e) Distance Measure
Distance measure
Rectilinear Euclidean Other
38
f) New/existing facility Interaction
New/existing Facility
Interaction
Qualitative Quantitative
Location Dependent Location
Independent
Static Dynamic
Deterministic Probabilistic
Parameter Decision Variable
(Source: Francis and white, 1974)
39 3.5.2 THE CLASSIFICATION OF DASKIN (1995)
The classification of Daskin(1995) [15] (figure6) is based in 14 below criteria: 1)Topological structure, 2)Network type, 3) Distance measure, 4)number of facilities to locate, 5)time dependence, 6)certainty, 7)product diversity, 8)public and private sector, 9)number of objectives, 10)demand elasticity, 11)capacity of facilities, 12)demand allocation type, 13) hierarchical structure, 14)desirability of facility, which we will represent in the following matrix, so we can observe that there are two columns, one with the 14 criteria and the other with the name of the classification in which corresponding every criterion:
Figure 6:Classification of Daskin Criteria Classification 1.Topological structure Planar Discrete Network 2. Network type Tree problems General graph 3. distance measure Manhattan Euclidean Lp 4.Number of facilities to Single Facility Multi-facility locate 5. Time dependency Static Dynamic 6.Certainty Deterministic Probabilistic 7. Product diversity Single product Multi product 8. Public/private sector Public Private 9. Number of objectives Single objective Multi objective 10.Demand elasticity Elastic Inelastic 11.Capacity of facilities Capacitated Uncapacitated 12.Demand allocation Nearest facility General allocation type demand 13.Hierarchical structure Single- level Hierarchical 14.Desirability of Desirable Undesirable facilities
Source: Daskin, 1995
40
3.5.3 THE CLASSIFICATION OF REVELLE ET AL (2008)
Last but not least, we will present the classification of the ReVelle et al (2008) [64], (figure7). They classify the facility location models in two branches of discrete location science, first in: location models such as plant and the median, and second in: center models and covering. Furthermore, as mentioned before the location models are separated in four categories. The following shape demonstrate this separation, hence we will analyze the analytic models, the continuous model, the network models and the discrete models.
Figure 7: Classification of ReVelle et al Location Models
Analytic Continuous Network Discrete
Models Models Models Models
(Source: ReVelle et al., 2008)
Thus,
Analytic models: Despite the fact that they are simple models of location facilities and they are based on simple assumption, they have a disadvantage which is that they cannot be used to in actual decision-making problems.
Continuous models (figure 8): These models assume that the facilities can be sited anywhere in the place. An example of continuous model is the problem of Weber, it includes the location of a facility having a target to minimize the total Euclidean distance between the facilities and a set of fixed customers. In the following figure we can see a continuous model.
Figure 8: Continuous models
41 Network models (figure 9): these models assume that the network is composed of nodes and lines. The customers can be located on the nodes, the facilities can be located in nodes or lines and the movement between facilities and customers becomes through the lines. The following figure shows a network model.
Figure 9:Network models
Discrete models (figure 10): these models assume that there are a differentiation between the demand and the candidate sites of facilities. Furthermore, these models are modeled as integer problems or mixed integer programming. The following figure shows a discrete model.
Figure 10: Discrete models
42
In the following figure (figure 11) we can see the separation of discrete models according to Daskin [16]:
Figure 11: Separation of discrete models
Discrete Location Models
`
Covering-based Median-based Other Models Models Models
Set Covering P-Median P-
Dispersion Min# sites needed to Min average distance
cover all demand distance between Max the demand and nearest of minimum Max Covering P sites distance between Max # covered any pair of
demand with P sites facilities Fixed Charge
Min fixed facility and P-Center transport costs Min coverage Distance needed to cover all demand with P sites
(Source: Daskin, 2008)
Moreover, we should note that there are many classifications about the location facilities models but we mentioned in this master thesis only the above three.
43 3.6 DISTANCE MEASUREMENT TYPES Before proceeding with the analysis and development of the different models of location facility problem, it is important to mention some types measuring distances which are used in the application of models. The models use different types of measurement in order to estimate the distances, but many of them are based on symmetry or triangle inequality. Some of the types which we use in order to calculate the distance between two samples or between two variables are the following:
Euclidean distance: the distance between two points is the length of the path which connecting them. We use Euclidean distance in order to estimate the distance between them. The Euclidean distance between two points P1(x1, y1) and P2(x2,y2) is given by following formula as we can see in the following figure (figure 12):
Figure 12:Euclidean distance
Rectilinear distance: we use rectilinear distance when the distance between two points or two facilities is measured along path which is orthogonal to each other. Also, the distance between the facility and the demand points is measured with the rectilinear metric. In addition, this type is known as Manhattan because it allows the movement into a city with Vertical streets. We consider that we have two facilities which are located at points A(X1,Y1) and B(X2, Y2), as we can see in the following figure (figure 13) the rectilinear distance between them is : d(A,B)=|X1-X2|+|Y1-Y2|
Figure 13:Rectilinear distance
44
Great circle: the Euclidian distance and the rectilinear distance are applied at the level. There are cases in which the positions are determined by longitude and width we use the great circle distance in which the movement is became along a great circle. In the following figure, we depict the great circle distance between two points p and q. Thus, the great circle distance is estimated by the following formula: d=Ra (figure 14).
Figure 14:Great circle Highway: these distances can be found on websites such as Google Maps or various geographical information systems. Highway declares the actual distance between two geographical points, having first is solved the problem of the shortest route in the road network where these two points are connected.
Distance matrix: it is used when the distance represents fares or rates airline tickets. In addition, these matrixes record the distance between each pair.
3.6.1 DISTANCE OBJECTIVE
Total distance: with this concept we declare the total distance between customers and their assigned facilities. Usually this distance is mentioned in demand-weighted.
Maximum distance: with this concept we declare the maximum distance between a customer and its assigned facility. Usually this distance is unweighted.
45 3.7 APPLICATIONS OF FACILITY LOCATION MODELS This section is composed of some applications of location facilities, and then we analyze these models. General the below nine models are applied not only in public sectors but also in private sectors, some applications are the following: a) Emergency medical services /fire stations, b) Airline hubs, c) Blood banks, d) Hazardous waste disposal sites, e) Fast food, f) Public swimming pools ,g) Schools, h) Vehicle inspection stations, i) Bus stops , etc.
On the other hand, these models also applied to ‘’virtual facilities’’, some examples of these cases are the following: a) Location of bank accounts, b) Product positioning, c) Flexible manufacturing system tool selection, d) Vehicle routing problems, etc.
3.8 BASIC MODELS OF LOCATION FACILITIES In this section, we will present and analyze eight classical models of location and the uncapacitated facility location problem. In all below models, it is important to emphasize that the network is given. General, the problems have an aim. The target of each problem is to decide where to locate new facilities in order to optimize some objective. Thus, there is a catalog of basic models as we will mention below, if we observe the following table, we will see that models are divided in two categories. The first four are based on the overall or average distance and the other four are based on the maximum distance, so:
Table 2: Shows the separation of models.
MODELS # Overall or average distance # Maximum distance 1 P-Median problem 5 Set covering location problem 2 Fixed charge location problem 6 Maximam covering location problem 3 Hub location problem 7 P-Center problem
4 Maxisum location problem 8 P-Dispersion problem
46 3.8.1 P-MEDIAN MODEL
P-median is a model which concerns the overall or average distance. It is one of the classic models. First of all, p-median problem was defined by Hakimi, in 1994 & 1995 [31]. He considered that the p-median model is a model which aims to estimate the locations of p- facilities in order to minimize the total distance between the demand nodes and facilities which have been allocated, with other words the p- median model was applied on a network of nodes and arcs. The nodes declare the possible facility locations whereas the arcs are linkages and they are very important for facility locations.
On the other hand, the Jamshidi (2009) [41] followed some assumptions that they are based on the p-median model:
Assumptions:
. First assumption is that there is a linear relationship between cost and distance. . Second assumption is that in the begging the product is in facilities and then it is transferred. . Third assumption is that there is infinite time horizon. . Fourth assumption is that p-median problem is exogenous. . Fifth assumption is that the problem is discrete and not continues. . Sixth assumption is that there is not an initial installation costs. . Seventh assumption is that the facilities for location have similar kind. In addition, ReVelle and Swain (1970) [63] expressed the p-median problem as a model of integer programming. So, p-median is formulated as follow, firstly we will define the I, J and then the inputs.
I: all customers (sites of demand)
J: all possible sites of location facilities
Inputs
hi = customer I demand
dij = distance between customer I and candidate facility J
47 P = the number of facilities to be located
Decision Variables
1, 푖푓 푤푒 푤푖푙푙 푙표푐푎푡푒 푎푡 푐푎푛푑푖푑푎푡푒 푠푖푡푒 푗 Xj= { 0 , 표푡ℎ푒푟푤푖푠푒
1, 푖푓 푐푢푠푡표푚푒푟 푖푠 푠푒푟푣푒푑 푏푦 푓푎푐푖푙푖푡푦 푗 Yij = { 0, 표푡ℎ푒푟푤푖푠푒
Now, we will present the objective function, so:
Objective Function
Minimize Z= 훴푖 훴푗 ℎ푖 푑푖푗 푌푖푗 (3.8.1.1)
Subject to
∑푗 푌 푖푗 = 1 ∀푖 (3.8.1.2)
∑푗 푋푗 = 푃 (3.8.1.3)
푌푖푗 − 푋푗 ≤ 0 ∀푖, 푗 (3.8.1.4)
푌푖푗 = 0,1 ∀푖, 푗 (3.8.1.5)
푋푗 = 0,1 ∀푖, 푗 (3.8.1.6)
Now, we will explain the above constraints and the objective function. First of all, the target of the objective function (3.8.1.1) is to minimize the total demand or distance between the customer and the nearest facility.
The constraint (3.8.1.2) shows that every customer is located to exactly one facility for each iEJ. The constraint (3.8.1.3) declares that there are exactly P-facilities which are located. The next constraint (3.8.1.4) declares the connection of two variables, the location variable (X) with allocation variable(Y). And the last two constraint (3.8.1.5) and (3.8.1.6) demonstrate that the two above variables (X) and (Y) of the previous constraint are binary. Because of the two variables are binary, the constraint of the problem cannot be solved with stable linear programming, but it is solve with Lagrangian Relaxation. The Lagrangian relaxation concludes five steps but we will not mention these in this master thesis.
48 3.8.2 THE FIXED CHARGE MODEL
According to Mask S.Daskin (2013) [55], the fixed change facility location problem is a typical problem on which are based many models and they have been used in supply chain design. In this problem, are given a set of customer locations with known demand and a set of candidate facility locations wherever they are located. So, we have a fixed location cost, with other words there is a known unit transportation cost between every candidate point and every customer location. The target of the fixed charge model is to find the minimum point between the facilities and the customers and minimize the total costs of location and transport. Thus, we defined the following inputs and sets:
I: set of customer locations
J: set of candidate facility locations
Inputs
hi: demand at customer location iEI
fj: fixed cost of facility location at candidate site jEJ
cij: is the unit cost of transport between candidate facility point jEJ and a customer location iEI
We defined the decision variables, so:
Decision Variable
1, 푖푓 푤푒 푙표푐푎푡푒 푎푡 푐푎푛푑푖푑푎푡푒 푠푖푡푒 푗퐸퐽 푋푗 = { 0, 표푡ℎ푒푟푤푖푠푒
Yij= the fraction of demand at customer location iEI which is served by a facility at site jEJ.
According to Balinski (1965) [4] the fixed charge facility location problem can be modeled as below:
Objective function
Minimize ΣjEJ fj Xj + ΣieI ΣjEJ hi cij Yij (3.8.2.1)
49 Subject to
ΣjEJ Yij=1 for every iEI (3.8.2.2)
Yij-Xj<=0 for every iEI and jEJ (3.8.2.3)
Xj E {0, 1} for every jEJ (3.8.2.4)
Yij>=0 for every iEI and jEJ (3.8.2.5)
Now, we will explain the objective function and the constraints. Hence, the objective function (3.8.2.1) minimizes the sum of the fixed facility location costs and the transportation costs for the demand which is needed to be served. The (3.8.2.2.) constraint shows that every demand is assigned fully. The next constraint (3.8.2.3) depicts that if the facility is closed then the demand cannot be assigned. The (3.8.2.4) constraint is a standard constraint and finally the constraint (3.8.2.5) is a non-negative. Finally, there are cases in which the fixed charge location problem is referred to as the incapacitated fixed charge location problem.
3.8.3 HUB AND SPOKE LOCATION MODEL [39]
A different model in location facility problems is the hub location problem. The Hub location problems regard transportation like air passenger travel, large trucking systems etc. as well as telecommunication such as telephone networks, computer communications etc. between many origins and destinations. In addition, hub location problems include locating of facilities and designing hub networks so as to minimize the total cost of transportation between hubs, facilities and demands. The hub location problems are separated in multiple and single allocation versions. For example, in case of multiple each site can communicate with more than one hub whereas in single, each site has to communicate with one hub. General the flows can have the similar origin and different destinations and vice versa. On the other hand, the hub location problem has a disadvantage, in order to become a movement from a demand point to the other, it required mandatory movement through another hub and it is not becoming immediately. Many researchers studied the hub problem such as Campbell, O’kelly et al, Mayer and Wagner, Boland, Marin, Alumur and Kara etc [84] but O’Kelly (1987) [61] played an important role in this problem. Thus, the modeling of hub problem is the following:
50 In order to be described and be analyzed the hub location problem, first of all, it is needed to mention some model assumptions. Thus, we quoted the model assumptions below:
Model assumptions:
. The first assumption is that the objective function is minisum. . The second assumption declares that in hub location problems there is only one node location. . The third assumption declares that the number of nodes is known in advance. . The fourth assumption shows that all decision variables are binary. . The fifth assumption stresses that there is unlimited capacity. . Final assumption declares that the initial location node cost in not taken into consideration. We defined the variables I, J and the parameters Ci, Oi, Di
I: all point demand
J: all possible point of hub-location
Parameters
Ci: is the unit cost of local moving between i and j( i.e from the point of non- hub yo hub).
Oi: is the total outgoing flow of point i.
Di: is the total incoming flow of point i.
Variable Decision
1, 푖푓 푡ℎ푒 푛표푑푒 푤푎푠 푙표푐푎푡푒푑 푎푡 푡ℎ푒 푝표푖푛푡 푗 Xj={ 0, 표푡ℎ푒푟푤푖푠푒
1, 푖푓 푡ℎ푒 푝표푖푛푡 푖푠 푙푖푛푘푒푑 푡표 푡ℎ푒 푛표푑푒 푙표푐푎푡푒푑 푖 푛 푡ℎ푒 푝표푖푛푡 푗 Yij={ 0, 표푡ℎ푒푟푤푖푠푒
51 Objective Function
Minimize ∑i ∑ j 퐶푖푗 푌푖푗 (푂푖 + 퐷푖) (3.8.3.1)
Subject to
∑Xj = 1 (3.8.3.2)
Yij ≤ Xj ∀ i є I, ∀ j є J (3.8.3.3)
Xj є {0, 1} ∀ j є J (3.8.3.4)
Yij є {0, 1} ∀ i є I, ∀ j є J (3.8.3.5)
In this stage, we will explain the definition of the above constraint and of the objective function, thus: The objective function (3.8.3.1) minimizes the total cost of transport by the node location. The constraint (3.8.3.2) shows that the location has only one node. The inequality i.e the (3.8.3.3) constraint defines that the point of demand cannot be connected with node, if it is not located first at the point j. Also, the other two constraints (3.8.3.4) and (3.8.3.5) declare that are integrity.
Pros and Cons of Hub Location Model
On one hand, the pros of Hub location problem are the following: 1. Because of the Spokes are simple, it helps to be created a new one with easy way. 2. When there are few routes with frequent routes, it is necessary the programming. 3. Complicated operations are become at the hub and not at every node. 4. Final, the small number of routes, declare the efficient and effectiveness use of transportation sources. On the other hand, the cons of Hub location problem are the following:
1. Hub is Inflexible, i.e changes at the hub, or even in a single route where is affected by the whole network. 2. The hub concludes a bottleneck in the network. 3. The delays at a spoke also affect the whole operation of the network. 4. The cargo must pass through the hub before reaching its destination.
52 3.8.4 A) MAXISUM LOCATION MODEL [46]
On the other hand, dispersal could be maximizing the average distance between facilities. This method is called maxisum problem and it is related to the p-dispersion problem (maximin). Maximization of the minimum distances leads to the fact that it is not necessary the two facilities will be located near to one another. In this problem, the target is to there is equal distribution of potential facility locations. The maxisum problem located p facilities among n nodes so as to maximize the sum of distances or the average of distances between open facilities. The disadvantage of maxisum problem is when the location of the facilities is located very close one to another, so the best location is far away. For the modeling we defined the variables such as previous problems. So, where: D= smallest separation distance between any pair of open facilities
1, 푖푓 푓푎푐푖푙푖푡푦 푙표푐푎푡푒푑 푎푡 푛표푑푒 푖 푋푖 = { 0, 표푡ℎ푒푟푤푖푠푒
n= number of potential facility sites p= number of facilities to be located N=set of potential facility sites M=a very large number dij= shortest path distance between node i and node j
1, 푖푓 푓푎푐푖푙푖푡푖푒푠 푎푟푒 푙표푐푎푡푒푑 푎푡 푏표푡ℎ 푝표푖푛푡 푖 푎푛푑 푗 푍푖푗 = { 0, 표푡ℎ푒푟푤푖푠푒 Hence, we have the following objective function and the constraints
Objective function
푛 푛 Maximize 훴푖=1 훴푗=푖+1 푍푖푗 푑푖푗 (3.8.4.1A) Subject to
푛 훴푖=1 Xi=p (3.8.4.2A)
푍푖푗 ≤ 푋푖 for all i, j |j>i (3.8.4.3A) 푍푖푗 ≤ 푋푗 for all i, j |j>I (3.8.4.4A) 푋푖 ∈ {0,1} For all iEN (3.8.4.5A)
53 Now, we explain the objective function and the above constraints. Thus, the objective function (3.8.4.1A) consists the distances between all pairs of open facilities. The constraints (3.8.4.3A) and (3.8.4.4A) allow Zij to take the value equal to 1 only if Xi and Xj are equal to 1. Despite of the Zij is a continuous variable, sometimes it takes integer value, and this depends on what value the Xi and Xj variables will take.
3.8.4B) MAXISUM LOCATION MODEL [85]
As we mentioned before, the maxisum location problem assumes that location of facilities is desirable when it is located as close as possible to demand, but there are cases where facilities are undesirable and where the location facilities are far from demand nodes. We will examine this situation where the maxisum location problem regards the locations of p facilities and the target of them is to maximize the total demand-weighted distance between demand nodes and the facilities. This model is formulated as below:
Objective Function
Maximize 훴푖 ∈ 퐼 훴 ∈ 푗 ℎ푖 푑푖푗 푌푖푗 (3.8.4.1.B)
Subject to
훴 푗 ∈ 퐽 푌푖푗 = 1 ∀푖 ∈ 퐼 (3.8.4.2.B)
훴 푗 ∈ 퐽 푋푗 = 푃 (3.8.4.3B)
푌푖푗 − 푋푗 ≤ 0 ∀푖 ∈ 퐼, 푗 ∈ 퐽 (3.8.4.4B)
푌푖푗 = {0,1 } ∀푖 ∈ 퐼 , ∀ 푗 ∈ 퐽 (3.8.4.5B)
푋푗 = {0,1 } ∀푗 ∈ 퐽 (3.8.4.6B)
푚 훴푘=1푦푖(푘)푖 − 푋(푚)푖 ≥ 0 ∀푖 ∈ 퐼, 푚 = 1, … , 푁 − 1 (3.8.4.7. 퐵)
Now, we will explain the objective function and the constraint, so the objective function (3.8.4.1B) is to maximize the demand-weighted total distance and not to minimize it. The constraint (3.8.4.7.B) considers that demand is belonged to the nearest facility, and the (k)I is the index of the kth farthest candidate location from demand node i.
54 3.8.5 SET COVERING LOCATION MODEL [30]
First of all, the set covering location problem was introduced by Hakimi (1965), but Toregas (1971) et al, [80] formulated the set covering location problem as an integer programming problem. We will discuss the case of Toregas et al, [80] we will emphasize in the aim of this model which is to minimize the total number of facilities in order to be satisfied fully the demand of each customer. Hence, we will quote the inputs in order to make a decision under some constraints.
Inputs
I= the total number of customers (demand sites)
J= the set of candidate sites location facilities.
dij: distance between the demand point i and the candidate point j
Dc: the distance coverage
Ni: set of candidate sites that meet the demand at point i, Ni=j E J :{ dij<=Dc}
Decision Variables:
1, if a facility is located at point j Xj = { 0, otherwise
In addition, the set covering location problem is formulated as follows:
Objective function Minimize 훴푗 ∈ 퐽 푋푗 (3.8.5.1)
Subject to
훴푗 ∈ 푁푖 푥푗 ≥ 1 ∀푖 ∈ 퐼 (3.8.5.2)
푋푗 ∈ {0,1} ∀푗 ∈ 퐽 (3.8.5.3)
Now, we will explain the objective function and above constraints. So, the objective function (3.8.5.1) minimizes the required number of facilities that are necessary in order to cover the total customer demand. The constraint (3.8.5.2) defines that every point must be satisfied and the (3.8.5.3) constraint defines that the variable decision Xj as binary. Furthermore, if we want to include the cost of installation, we must add the parameter fi in the objective, so: Minimize 훴 푗 ∈ 퐽 푓푖푋푗
55 3.8.6 MAXIMAL COVERING MODEL
Church and ReVelle (1974), [10] were introduced the maximal covering location problem. The maximal model is the opposite of the above model, the set covering model. In maximal covering location model the target is to maximize the total demand which cover a specific number of facilities that are opened. So, we have the following:
Inputs
I= the total number of customers (demand sites)
J= the set of candidate sites location facilities.
dij: distance between the demand point i and the candidate point j
Dc: the distance coverage
P: the number of location facilities
hi: demand of customer at node i
Ni: set of all candidate sites that can meet demand at the point i, Ni=j E J: {dij<=Dc}
Decision Variables:
1, if customer i is covered Zi = { 0, othervise
1, if a facility is located at point j Xj = { 0, otherwise
The maximal covering location problem is formulated as follows:
Objective function Maximize 훴푖 ∈ 퐼 ℎ푖 푍푖 (3.8.6.1)
Subject to
훴푗 ∈ 퐽 푥푗 = 푃 (3.8.6.2)
푍푖 − 훴푖 ∈ 푁푖 푥푗 ≤ 0 (3.8.6.3)
푋푗 ∈ {0,1} ∀푗 ∈ 퐽 (3.8.6.4)
Zi ∈ {0,1} ∀푗 ∈ 퐼 (3.8.6.5)
56 As in previous models, so here, we will explain the objective function and the constraints. Thus, the objective function (3.8.6.1) maximizes the total demand which is covered. The first constraint (3.8.6.2) declares that will be located p facilities. The constraint (3.8.6.3) shows that customer not be able to be located if facilities not be located inside the covering distance. Also, the following two constraints (3.8.6.4) and (3.8.6.5) define that the Xj and Zi decision variables are binary.
Moreover, We can mention some examples of these above two models which have been applied to public-sector facility location problems like as the location of emergency medical service and vehicles( Eaton, et al. 1985), [14] fire stations (Schilling, et al 1980), [13] bus stops(Gleason 1975) [40] etc.
3.8.7 THE P-CENTER MODEL [6], [35]
Sylvester (1857) [77] was the first who considered that all possible destinations must be covered by a circle with the smallest radius. A few years later, the p-Center problem was solved by Hakimi (1964-1965) [31]. The p-Center problem gives p facilities, aiming to minimize the maximum distance from any customer to the nearest facility. In P-center model a demand point is serviced by its nearest facility. In addition, we will mention some examples where the P-center model is applied in some facilities such as hospital, fire station and police station. Of course, there are still many other applications about this model. Mainly, The readers mention, Handle (1990) [33], Daskin (2000) [19] etc.
Assumptions:
. In first assumption facilities can be located only above the nodes of network. . The second assumption is that the capacity of facilities is unlimited. . The third assumption is that in p-center model there are p location facilities. . The fourth assumption is that the demand points are located on the network nodes and they have not weighting. Inputs
I: the set of customers (i.e demand points)
J: the set of possible points location of facilities.
P: the number of facilities
dij: is the distance of the shortest route between the demand and the node
57 Because, the capacity is unlimited, we can suppose that every demand node is serviced by a single facility. Thus, we define the following decision variable in order to solve this problem:
Decision Variables
1, if facility located at j ∈ J Xj = { 0, otherwise
1, if the demand of customer i satisfied by facility j Yij = { 0, otherwise
Z= is the maximum distance between a demand point and the nearest facilities to this.
Objective function
Min Z (3.8.7.1)
Subject to
훴 푗 ∈ 퐽 푌푖푗 = 1 ∀푖 ∈ 퐼 (3.8.7.2)
훴푗 ∈ 퐽 푋푗 = 푃 (3.8.7.3)
푌푖푗 ≤ 푋푗 ∀푖 ∈ 퐼, ∀푗 ∈ (3.8.7.4)
푍 ≥ 훴푗퐸퐽 푑푖푗 푌푖푗 ∀푖 ∈ 퐼 (3.8.7.5)
푋푗{0,1} ∀푗 ∈ 퐽 (3.8.7.6)
푌푖푗 ∈ {0,1} ∀푖 ∈ 퐼, ∀푗 ∈ 퐽 (3.8.7.7)
Now, we will explain the objective function and the above constraints. Hence, the objective function (3.8.7.1) with the fourth constraint (3.8.7.5) minimizes the maximum distance between a node demand and the nearest facilities. The first constraint (3.8.7.2) shows that whole demand is transferred in facility j. The second constraint (3.8.7.3) defines that would be located p facilities. The next constraint (3.8.7.4) demonstrates that demand of each customer can be transferred only in facilities that located yet. Finally, the remaining constraints (3.8.7.6) and (3.8.7.7) claim that Xj and Yij belong to {0, 1} and declare that both are binary.
58 3.8.8 P-DISPERSION MODEL [46], [66]
First of all, it is interesting to emphasize that the p-dispersion problem has a duality relationship with the previous model, the p-center problem. Shier (1997),[72] demonstrated that on a tree network of dispersion problem, the minimum distance between facilities for p facilities is exactly two times larger by the maximum distance between points and their nearest facility for p-1 facilities where it is in a center problem. He considered that facilities can be located at any node or along any arc of a tree network. Furthermore, according to p-dispersion problem, the center problem is defined by lower bound (p-1)-. Moreover, According to, Handler and Mirchandani (1979), [32] there are many problems in this topic. The location facility on a network, aims to minimize the function of the distances between facilities and the nodes of the network. These problems are characterized as desirable, like hospitals, fire stations and warehouses. On the other hand, Erkut and Neuman (1989), [24] said that the proximity of facilities is undesirable. These problems are called p-dispersion problems where p facilities are located in order to maximize the function of the distance between any pair of nodes on the network. General, There are two categories of examples of p-dispersion model. One application regards the retail or service franchises, and the other the nuclear power plants, ammunition dumps etc., with other words in second type the facilities must be located as far away from the other facilities because in case of an accident do not affect any of the others. Tansel, Francis, Lowe and Chen (1982) [78] created an algorithm in order to solve nonlinear p- center problems with maximum distance, constraints and the p-dispersion problems. Thus, the target of the p-dispersion problem is to maximize the minimum separation distance subject to constraints with a relevant upper bound on that distance. This upper bound is the distance between two locations and it can be applied only when these two facilities are opened. Also, in order to explain the constraints about the upper bound, we define from 1 to 0 the value of location variables for open facilities and from 0 to 1 the value of location for facilities that are not opened.
For modeling we defined the following, where:
D= smallest separation distance between any pair of open facilities
1, if facility locates at node i Xi = { 0, otherwise
n= number of potential facility sites
p= number of facilities to be located N=set of potential facility sites M=a very large number dij= shortest path distance between node i and node j.
59 Hence, we have the following objective function and the constraints
Objective function
Maximize D (3.8.8.1)
Subject to
푛 훴푖=1 Xi=p (3.8.8.2)
퐷 ≤ 푑푖푗[1 + 푀(1 − 푋푖) + 푀(1 − 푋푖)] For all i, jEN |i 푋푖 ∈ {0,1} For all iEN (3.8.8.4) Now, we will explain the objective function (3.8.8.1) and the above constraints. Thus, the objective function (3.8.8.1) depicts the minimum distance between two or more open facilities. The constraint (3.8.8.2) shows that p facilities must be open. The (3.8.8.3) constraint declares an upper bound on D, equal to dij only in case where the facilities are open and equal to 0, otherwise if there are not facilities the upper bound will have a large number and it will tend to infinity. Finally, the constraint (3.8.8.4) shows that is a typical constraint of contentment. 3.9 UNCAPACITATED FACILITY LOCATION PROBLEM (UFLP) This model is known as the ‘’simple’’ facility location problem and regards the location of an undetermined number of facilities. The target of this model is to minimize the sum of the fixed annual costs and the variable costs of facilities that serve the market demand. Furthermore, in this model, the alternative facility locations and the customer zones are considered discrete points. With other word, it is one of the most widely discrete location problems and it has a variety of settings, such as distribution system design (klose & Drexl, 2003), [45] computer vision (Li, 2007, Lazic et al, 2009) etc. [50], [48] .According to Erlenkotter (1978) [25] the UFLP has two sets of decision variables, thus: Decision variables xij: is the fraction of customer zone j which satisfy the demand by the facility at i 1, 푖푓 푎 푓푎푐푖푙푖푡푦 푖푠 푡표 푏푒 푒푠푡푎푏푙푖푠ℎ푒푑 푎푡 푙표푐푎푡푖표푛 푖 yi: { 0, 표푡ℎ푒푟푤푖푠푒 fi: the fixed cost of establishing a facility as location i cij: the total capacity 60 So, the Erlenkotter (1978) [25] presents the following formulation of UFLP Objective function Max ∑푖 ∑푗 푐푖푗 푥푖푗 + ∑푖 푓푖 푦푖 (3.9.1) Subject to ∑ ixij = 1 for all j (3.9.2) xij<=yi for all i, j (3.9.3) xij>=0 and yi E {0, 1} for all i,j (3.9.4) Now, we will explain the objective function and the above constraints. So, the objective function (3.9.1) declares the total fixed and variable costs. The first constraint (3.9.2) depicts that the demand at each customer zone is satisfied. The second constraint (3.9.3) demonstrates that customer demand can be produced and shipped only from the location in which a facility is located. 3.10 THE LOCATION FACILITY PROBLEMS – INVENTORY [73] In this subsection, the main question in logistics and the supply chain is the design of distribution networks and the identification of facility locations. Ballou (1993) and Masters mentioned four region of strategic planning in the design of distribution network systems. In the following figure we can see the four stages: Customer Responsiveness Responsiveness Facility Location Decision Efficiency Inventory Decision Distribution Decision Figure 15: four stages of strategic planning. 61 We will explain these above four region, thus: . The first regards the customer service level. . The second is related with the facilities and demand assignments. . The third is about the inventory decisions and policies that involve inventory control. . The fourth regards the transportation decisions i.e. how transport nodes are selected, utilized and controlled. The Daskin et al.(2002) [17] and Shen et al. (2003) [71] mentioned models of locations- inventory which include in the objective function the cost inventory management, the transport costs and the non-lienar cost [83] In addition, the model of Shen et al.(2003) [71] assumed that a set i given to retailers with independent and uncertain demand, aiming to determine the number and the place in which the location distribution centers have to be located, also, the identification of retailers, the frequency of orders in distribution centers and the level of safety stock which should minimize the total cost. Hence, the model formulated as folows: Parameters μi: mean(yearly) demand at retailer i for each i E I, σi2 :variance of (daily) demand at retailer i, for each iE l, fj: fixed cost of a regional distribution center at retailer j, for each jEl vj(x) :cost to buy x units from the main supplier to a regional distribution center located at retailer j, for each j E l, dij :cost per unit for shipping from retailer j to retailer I, for each iEI and jEJ α :desired percentage of retailers orders satisfied β: weight factor associated with the transportation cost θ: weight factor associated with the inventory cost za: standard normal deviate such that P(z<=zα) =α h: inventory holding cost per unit of product per year 62 wj(x): total annual cost of working inventory held at distribution center j if the expected daily demand at j is x for each jEI Fj: fixed cost of placing an order at distribution center j, for each Jel L: lead time in days Decision Variables 1, if retailer j is selected as a distribution center location Xj = { 0, otherwise 1, if retailer i is served by a distribution center based at retailer j 푌ij = { 0, otherwise Objective function 퐌퐢퐧퐢퐦퐢퐳퐞 훴푗퐸퐼 { 푓푗 푋푗 + 훽 훴푖퐸퐼 휇푖푑푖푌푖푗 + 푤푗(훴푖퐸퐼 휇푖 푌푖푗) + 휃ℎ푧훼√훴푖퐸퐼휎푖^2푌푖푗 (3.10.1) Subject to 훴 푗퐸퐼 푌푖푗 = 1 ∀푖퐸퐼 (3.10.2) 푌푖푗 − 푋푗 ≤ 0 ∀푖, 푗퐸퐼 (3.10.3) 푌푖푗 ∈ {푂, 1} ∀ 푖, 푗 ∈ 퐼 (3.10.4) Χj ∈ {0,1} ∀j ∈ I (3.10.5) Now, we will explain the objective function and the constraints: The objective function minimizes the weighted sum of the following four costs: a)the fixed cost of location facilities , b)The shipping cost from the distribution centers to the non-DC retailers, c)The expected working inventory cost, d)The safety stock costs The constraint(3.10.2) defines that every retailer was assigned exactly one distribution center. The constraint(3.10.3) defines that retailers can only be assigned only in candidate sites that are selected as distribution centers. The remaining constraints (3.10.4) and (3.10.5) are integrality constraints. Finally, Daskin et al(2005), [18] supported that the combination of inventory management with facility location decisions is difficult to examine. 63 PART III CHAPTER 4 4. ANALYSIS AND RESYLTS In this section, we will study a real case study of a Greek company ‘’COCOLAK S.A’’, in the context of the issues in which we mentioned, discussed and analyzed in previous parts and chapters. Initially, we will present in few words the history of the company, and then we will make reference to the functions of supply chain because they are very important for the correct operation of companies. After that, we will mention the characteristics and the properties of its facilities. Furthermore, we will apply some of techniques and models of location facility which we mentioned in chapter 3 in order to be solved the problems. Finally, we will register the results of the research. 4.1 THE COMPANY ‘’COCOLAK’’- HISTORICAL EVOLUTION COCOLAK S.A is a Greek company with object the import, marketing- trading and exploitation of children’s fun games (amusement machines). The company was founded in 1994 by two brothers. The current form is the result of the path followed since then. 64 In 1994, the firm started trading games from a Greek merchant from Athens, who was supplied the amusement games from Italy. In 1995, the company took the decision to rent a small warehouse about 120 m2, in order to storage the equipment. Also, the company rent facilities approximately 10 m2 in order to use them as offices. These facilities are located in St. Paul area. In 1996, company decided to buy a truck in order to transport the amusement machines. In the same year, it began cooperation with the Italian supplier ‘’COGAN’’ and so started imports of rocking toys. In 1997, company started to gives its games under exploitation in different regions of Halkidiki such as Pefkohori. In 2003-2004, COCOLAK even made a step, started to go in exhibitions to abroad. Thus, the company was imported games from other countries such as China, Taiwan, and Poland. These imports had the result to increase the diversity of the amusement games. However, added skill games for example basket, cranes with dolls-teddy bears-watches-small balls, and hokey. In 2004, Company was transferred in a larger warehouse approximately 350 m2. In addition, it bought one more truck in order to transport the games, because it decided to spread in the other regions of Halkidiki. Furthermore, in the same year, it decided to rent the first branch in Platamonas. In 2006, COCOLAK decided to move for second time its facilities (warehouse and office) in which it is until today. Consequently, the company’s headquarters are located in Ionia Thessaloniki and specifically, near Vegetable Market. In 2008, COCOLAK rent extra two warehouses next to the old facilities. One warehouse in order to store games and one warehouse is used as a workshop for repairs of games. In 2010, Company started imports from other country like England, and is supplied game such as video games. Today, Company is located in facilities on a plot of 3000-3500 m2 and includes warehouses and offices. As imports grow, the games become more and more, this means increase in demand. Moreover, the secret of COCOLAK is that the Boss visit many exhibitions not only in Greece but mainly in abroad in order to develop public relations with foreign suppliers. 65 Nowadays, COCOLAK has nine branches in which work the company’s employees. Below we report the night branches, so: Table 3: shows the number of branches stores, the branches stores and the year of establishment. NUMBER OF YEAR OF BRANCHES BRANCHES STORES ESTABLISMENT STORES 1ο CENTRAL WAREHOUSES 1995 2ο PLATAMONAS 2004 3ο PEFKOHORI 2008 4ο POLICHRONO 2009 5ο ΚΑLLITHEA 2010 6ο VERGIA 2012 7ο LEPTOKARIA 2016 8ο PLAYGROUND-ΟΝΕ SALONIKA 2016 9ο ONE SALONIKA 2016 Moreover, COCOLAK AE is consisted of a network of customers which extends almost throughout the country's territory and includes customers even from islands. Hence, it has games to 66 points in Greece (such Halkidiki, Vrasna, Kavala, Tuzla, Nea Peramos, Iraklitsa, Nikiti, Marmara, Sarti, Toroni, Katerini, Katerinoskala, Karditsa, Lefkada, Preveza etc.), Most of them are seasonal. In the next section, we will mention more details about the customers of COCOLAK. Regarding the network of suppliers is composed of partners who are located in foreign countries such as China, England, Poland, and Italy. 66 CO-WORKERS The co-workers are illustrated in the below pictures, hence: WIK With above co-workers, company has long-term excellent cooperation and has been shown that their products are durable to long-term use. Moreover, COCOLAK provides security specification, high quality and competitive prices, making it in the first position in Greece. Finally, an important part of the Company which must be emphasized is that provides technical support (after sales service) in customers and this is an advantage which classifies the most reliable company in this sector. Furthermore, a first insight into the philosophy, operation, products, suppliers and activity of the company can get someone on the official website: www.cocolak.gr. 67 4.2 THE SUPPLY CHAIN OF COCOLAK As regards the supply chain of COCOLAK, we can consider that the flow of different functions between the supply chain links in which belongs, it begins with the suppliers from whom it is supplied amusement games and leads to customers who either buy or rent the amusement games. Suppliers are constantly multiplied. Furthermore, the process of market made through various exhibitions. In these exhibitions take part in companies with amusement games, retailers and wholesalers, who have the aim of presenting their products. The company through special representatives is visited these exhibitions in order to choose or perhaps to order innovative games which they will attract market, customer or children attention. These exhibitions are conducted mainly in April. Now, we will mention the time that elapses from order to delivery of games. The time varies depending on country, season and availability. For example, when the amusement games come from Taiwan make about one and half month, but when they come from Europe takes time two or three weeks. Moreover, some of them come with car (from countries such as Italy, Poland, English, and Spanish) while other with ships in container (from countries such as China, Taiwan). Some of which arrive at the port of Piraeus and after are coming to the port of Thessaloniki. The transport companies transport the containers in warehouse, and then the forklift unloads the container and places them in warehouse in such a way that all games are separated according to type and sometimes according to color. After, the employee check the orders, if all games are right i.e correct color, correct shape. For instance, they control if the orders are correct or if they have some damages. In addition, COCOLAK, some games of the same type places one above the other in order to save space. As regards, the transportation costs of amusement games, the company has contact with transport companies not only in Greece but also in abroad, so made arrangements between the suppliers and COCOLAK, COCOLAK and buyers. Furthermore, according to transport with car, COCOLAK sometimes receives the transfer of games with its own truck to the buyer but at buyer’s expense. On the other hand, there are many buyers who go to warehouse and transfer the amusement games with their own trucks. 68 Pertaining, with inventory, if a customer requests a game and has the money, and COCOLAK has it in stock, company send it in the same time, otherwise, if company has not in stock, it will sell the game which was for its own use and replaces it with a new order. Company aims to satisfy their customers, it wants to serve their customer immediately. 4.3 THE WAREHOUSE OF COCOLAK The establishment of the company, which is receiving, preservation and distribution of amusement games both for its own use as well as exploitation, as mentioned before. The extent of surface covers approximately 3500 m2 and the permanent staff which works in year is 10 employees. Additionally, as reported earlier, most stores are seasonal, it means that COCOLAK hires employees, at least one in each branch. Previous, we mentioned that COCOLAK has warehouse which is separated about in six smallest warehouses in order to store the amusement games. This separation is done according to kind of amusement game or shape, or if the game is for selling or servicing. The warehouse with spare parts is in shelving. Moreover, the boxes with crazy balls, dolls, bears, watches, are in pallets and the boxes one above the other in order to save space. 69 4.4 DEMAND- NUMBER OF CUSTOMERS- NUMBER OF STORES First of all, in order to be more interesting the application of location facility models and techniques in COCOLAK, in our research, we must quote the geographical information of company’s customers, distances and demands i.e. the number of amusement games and the number of branches exploitation or branches. In the following tables are depicted these information per counties where company has store. In addition, the matrix of demand regards 3years (2014-2016) for four months (June, July, August, and September for region of Halkidiki) i.e. in these months the number of customers who were visited the stores in order to play with amusement games. We mentioned only these four months because as we emphasized before it is seasonal job. However, we will present the tables: Table 4: shows the number of games that company has in county of Kavala (for own uses and exploitation). The 1-2-3 shows the number of stores which are in exploitation. NUMBER OF STORES TOTAL NUMBER OF NUMBER GAMES EXPLOITATION AND COUNTY AMUSEMENT FOR OWN USE THE NUMBER OF GAMES IN COUNTY GAMES 1 2 3 Ν.Κavalas Kavala 10 10 Iraklista 12 12 Peramos 20 20 Touzla 15 15 Vrasna 22 22 Asprovalta 6 6 SUM 85 Explanation: for the county of Kavala, COCOLAK has total games 85. 70 Table 5: shows the number of games that company has in the country of Katerini (for own uses and exploitation). The 1-2-3 shows the number of stores which are in exploitation. NUMBER OF STORES TOTAL NUMBER OF NUMBER GAMES EXPLOITATION AND COUNTY AMUSEMENT FOR OWN USE THE NUMBER OF GAMES IN COUNTY GAMES 1 2 3 Ν.Katerini Katerini Beach 68 28 96 Οlympiakh akth 89 89 Neoi Poroi 22 22 Leptokaria 35 8 8 Platamonas 55 SUM 90 305 Explanation: for the county of Katerini, COCOLAK has total games 305. Table 6: shows the number of shops of exploitation and the number of amusement games in different areas of county of Halkidikis and mainly in region of Kassandras. NUMBER GAMES NUMBER OF STORES COUNTY NUMBER OF GAMES FOR OWN USE EXPLOITATION 1 2 3 Kassandras Area Halkidikis Flogita 44 44 Kallikratia 7 7 Vergia 50 Sozopoli 6 6 Potidaia 8 13 17 38 Afitos 6 6 Κallithea 73 24 24 Siviri 14 14 Fourka 6 6 Poseidi 5 5 Mola kaliva 7 7 Skioni 24 26 50 Paliouri 5 5 Pefkohori 57 12 12 Chanioti 13 13 Polixrono 52 15 28 43 SUM 232 512 Explanation: for Kassandras area of Halkidikis, COCOLAK has total game 512. 71 Table 7: shows the number of games that company has in Halkidiki and mainly in Sithonia area of Halkidiki. TOTAL NUMBER OF NUMBER GAMES NUMBER OF STORES COUNTY AMUSEMENT FOR OWN USE EXPLOITATION GAMES IN COUNTY 1 2 3 Sithonia Area Halkidikis Poligiros 4 4 Nikiti 17 17 Marmaras 14 14 Toroni 25 25 Sarti 19 34 53 SUM 113 Explanation: for Sithonias region of Halkidikis, COCOLAK has total game 113. Table 8: a conclusive table of above four tables, COCOLAK has the greatest demand in Kassandria area of Halkidiki. COUNTY DEMAND N.KAVALAS 85 N.KATERINIS 305 KASSANDRIA AREA 512 HALKIDIKIS SITHONIA AREA 113 HALKIDIKIS Table 9: depict the demand values in percentage SITHONIA AREAS DEMAND OF HALKIDIKIS N.KAVALAS 11% 8% N.KATERINIS 30% KASSANDRAS AREA OF HALKIDIKIS 51% 72 We depict the demand of four counties with percentage. So, according to above shape, we can give the following explanation: the demand in Kassandria area of Halkidiki is 51%, in county of Katerini is 30%, in Sithonia area of Halkidiki is 11%, and the other 8% is in county of Kavalas. Because of the high demand in region of Halkidiki, COCOLAK Company decided to locate a new warehouse near there in order to store all the amusement games from the different cities of Halkidiki. Thus, in following table we quote some important information about the distances and the cities. Table 10: shows the distances between the areas of Kassandria and Sithonia of Halkidiki with the four candidate sites in which we would be located the new warehouse of company or rented. Company suggested the following four candidate points, Nea Moudania, Potidaia. DINSTANCES FOUR CANDIDATE POINTS FOR A NEW WAREHOUSE CITIES OF ΝΕΑ MOUDANIA POTIDAIA ΚΑLUVES FLOGITA KASSANDRIA AND SITHONIA AREAS FLOGITA 8,19 14,68 18,73 0 ΚALLIKRATIA 28,75 35,23 39,28 23,2 SOZOPOLI 15,16 21,64 25,69 7,03 POTIDAIA 8,91 0 16,83 16,12 AFITOS 22,63 15,77 31,5 30 KALLITHEA 25,38 18,51 34,24 32,74 SIBIRI 33,33 26,47 42,4 40,7 FOURKA 34,41 27,55 42,28 41,78 POSEIDI 43,65 36,78 52,51 51,02 ΜOLA KALIVA 45,89 39,03 54,76 53,26 SKIONI 52,97 46,11 61,84 60,34 PALIOURI 53,66 46,8 62,53 61,03 PEFKOHORI 44,03 37,17 52,9 51,4 CHANIOTI 40,52 33,66 49,39 47,89 POLICHRONO 36,04 29,18 44,91 43,41 POLIGUROS 28,1 33,21 17,45 35,61 ΝIKITI 38,01 43,11 27,36 45,51 ΜARMARAS 58,24 63,34 47,58 65,74 ΤORONI 80,36 85,47 69,71 87,86 SARTI 94,44 99,55 83,79 101,95 73 Table 11: shows the number of customers in shops of Kassandria and Sithonia areas of Halkidiki, the years: 2016, 2015and 2014 and the months June, July, August and September. NUMBER OF CUSTOMERS 2016 (in all stores) NUMBER OF CUSTOMERS 2015 (in all stores) NUMBER OF CUSTOMERS 2014 (in all stores) SITHONIA AREA OF HALKIDIKI JUNE JULY AYGUST SEPTEMBER JUNE JULY AYGUST SEPTEMBER JUNE JULY AYGUST SEPTEMBER SUM MO 2014-2016 POLIGIROS 1200 4500 7500 1500 1100 4300 7000 1300 900 4900 8900 1600 44700 3725 NIKITI 6000 13500 18000 8000 5000 12000 16000 8400 4000 11000 16010 7790 125700 10475 ΜARMARAS 8000 17000 21000 7500 7000 15000 16000 7000 7000 13000 24000 7500 150000 12500 ΤORONI 5000 12000 18000 6500 6000 10000 16000 7000 5000 7000 19000 6400 117900 9825 SARTI 12000 30000 38000 15000 10000 29000 37500 14000 9000 22000 36000 13000 265500 22125 SUM 32200 77000 102500 38500 29100 70300 92500 37700 25900 57900 103910 36290 KASSANDRIA AREA OF HALKIDIKI FLOGITA 7000 10000 13000 5000 6000 9000 11000 6000 3000 5000 13000 5000 93000 7750 KALLIKRATIA 7500 11000 13500 6800 6500 13000 11500 6500 4800 4800 13000 6700 105600 8800 VERGIA 1200 4500 7500 1500 1100 4300 7000 1300 900 4900 8900 1600 44700 3725 SOZOPOLI 1900 4000 6000 1200 2000 5000 6000 1000 1200 4000 7000 1200 40500 3375 POTIDAIA 8000 13500 16000 7000 8100 8000 17000 6000 7700 6000 15000 5000 117300 9775 AFITOS 7500 13000 18000 8700 7300 12000 16000 8500 6000 7000 17000 8900 129900 10825 KALLITHEA 24000 60000 90000 26500 20000 53000 80000 26000 23000 24000 83000 26000 535500 44625 SIVIRI 3500 5000 7000 4000 3700 6000 9000 3000 4000 6700 5000 3700 60600 5050 FOURKA 4000 5500 8000 3200 4000 5300 9000 3000 3000 5800 6000 500 57300 4775 POSEIDI 600 800 1200 450 700 400 1000 500 500 3700 2200 1030 13080 1090 MOLA KALIVA 1000 1500 2000 1200 900 1300 1800 2000 600 1700 2370 1750 18120 1510 SKIONI 9000 12000 14000 8100 9200 11000 12000 8000 8450 12100 13000 6510 123360 10280 PALIOURI 500 850 1000 550 800 490 500 300 300 600 1500 1010 8400 700 PEFKOHORI 1200 1600 2200 1500 3500 1500 2000 1900 1800 2000 2200 3800 25200 2100 CHANIOTI 3500 4000 4700 3100 530 3000 4500 3000 460 1500 4500 3150 35940 2995 POLICHRONO 600 1200 2000 7000 450 2000 2000 6000 550 1200 3110 5570 31680 2640 SUM 81000 148450 206100 85800 74780 135290 190300 83000 66260 91000 196780 81420 SUM OF SITHONIA AND KASSANDRIA AREA 113200 225450 308600 124300 103880 205590 282800 120700 92160 148900 300690 117710 Explanation: from the above table we can observe that despite the economic crisis, in 2016 the number of customer compared to the other two year is greater. Also, great number of customers means high demand and big profit. 74 In above table, as we can observe, first of all, we calculated the sum of customers per month for these three years in the two areas of Halkidiki (Kassandria and Sithonia). Also, we estimated the sum of customers per city for these three years as well as the average of customers. So, we depicted the average of customer in the following shape. Average Customer for 2014-2016 VERGIA 2% FLOGITA KALLIKRATIA 4% 5% SOZOPOLI ΤORONI 2% SARTI POTIDAIA ΜARMARAS 5% 12% 5% 7% AFITOS 6% NIKITI 6% KALLITHEA SKIONI 25% POLIGIROS 6% 2% POLICHRONO 1% CHANIOTI 2% MOLA KALIVA FOURKA SIVIRI PEFKOHORI 1% 3% 3% 1% PALIOURI POSEIDI 0% 1% In above figure, the explanation is that the 4% of number of customers are represented in Flogita, the 5% in Kallikratia, the 2% in Vergia, the 2% in Sozopoli, the 5% in Potidaia, the 6% in Afitos, the 25% in Kallithea, the 3% Siviri, the 3% in Fourka, the 1% in Poseidi, the 1% in Mola Kaliva, the 6% in Skioni, the 0% in Paliouri(because it is very small number comparatively with other cities, it shows 0,but in reality it is not 0, but very close to 0), the 1% in Pefkohori, the 2% in Chanioti, the 1% in Polichrono, the 2% in Poligiros, the 6% in Nikiti, the 7% in Marmara, the 6% in Toroni and the remaining 12% in Sarti. 75 CHAPTER 5 5. 1 DISCUSSION In this chapter, we will apply some techniques and models in order to COCOLAK take an optimal solution for where to locate, install, build or rent a new warehouse. Company is going to build or to rent a new warehouse in order to store the amusement games of different regions of Halkidiki. While on one hand, firm suggested four candidate sites (Nea Moudania, Potidaia, Kalives, Flogita) in order to locate the new warehouse, on the other hand it considered any part in region of Halkidiki, having low transportation cost and high profitability. For the analysis of the techniques we will study the case in which it will install in one of the four candidate sites, but in analysis of models we will try to find the optimal solution according to where it presented high demand. In addition, we should stress that each model and technique will give us a different solution. 5.2 EXAMPLES OF LOCATION ANALYSIS TECHNIQUES 5.2.1 IMPLEMENTATION OF FACTOR LOCATION RATING METHOD IN COCOLAK As regards the factor location rating method, the site selection team is the evaluating of four candidate sites, and the score that they have scored the important elements for each as below. We will use these results and then we will compare the locations in order to find the optimal solution. As we can observe in below matrix we have the following five location factor, (proximity to customer, roads and transport, space area, immediacy of space and land rental cost), weighted scores which for each site are estimated by multiplying the factor weights by the score for factor. For instance, weighted score for the proximity to customer in site1 (Nea Moudania) is formulated by multiplying the (5x90), with the same manner we will fill the table. Moreover, we can see the four different candidate sites (Nea Moudania, Potidaia, Kalives, Flogita), so we created the two following tables. 76 Table 12: depicts the location factor, weight score and the candidate sites site 1 Weight Site 2 Site 3 Site 4 Location Factor (Nea (score100) (Potidaia) (Kalives) (Flogita) Moudania) Proximity to 5 90 80 65 50 customer Roads and 15 75 100 91 70 transport Space area 15 75 60 63 55 Immediacy of 30 80 74 82 73 space Land rental cost 35 95 55 65 59 100 Table 13: depicts the weighted score for each candidate site and for each location factor the value of weight. Weight site 1 Site 2 Site 3 Site 4 Location Factor (Nea (score 100) (Potidaia) (Kaluves) (Flogita) Moudania) Proximity to 5 450 400 325 250 customer Roads and 15 1125 1500 1365 1050 transport Space area 15 1125 900 945 825 Immediacy of 30 2400 2220 2460 2190 space Land rental cost 35 3325 1925 2275 2065 100 8425 6945 7370 6380 Comparing the values of candidate sites, we conclude that the site1 (Nea Moudania) has the highest factor rating. So, according to this technique the optimal location for installing the warehouse is in Nea Moudania. 77 5.2.3 IMPLEMENTATION OF THE CENTER-OF-GRAVITY TECHNIQUE IN COCOLAK On one hand the Center of Gravity is a method used by for locating a single facility on the other hand, is presented as a refined version of the load-distance method. COCOLAK Company wants to reduce costs and mainly transportation costs, i.e. it wants to reduce the unnecessarily routes. In addition, it wants to ensure consistency of the firm’s products across all locations, hence, it is going to build or to rent a new warehouse in order to store the amusement games of the different regions of Halkidiki that distributed to the four candidate sites (Nea Moudania, Potidaia, Kalives, Flogita). In this Technique, we will suppose that the warehouse wants to expand its logistics network of four existing facilities. Hence, the question here is where to locate, install, build or rend this warehouse in order to be increased the profitability and simultaneous to serve all the following different cities of Halkidikis. Center of Gravity use coordinates of a facility in order to find the optimal solution so, We define the following: - x: is the longitude - y: is the latitude - w: is the number of amusement games where company sent. The locations of four candidate sites and the number of products that company transports are depicted in the following figure (figure16). Furthermore, we used Google maps in order to find the co-ordinates x (longitude) and y (latitude). Thus, - For site 1(Nea Moudania), we found in Google maps that x is equal to 40, 23 whereas y is equal to 23, 27. - For site 2(Potidaia), we found in Google maps that x is equal to 40, 19 whereas y is equal to 23, 32. - For site 3(Kaluves), we found in Google maps that x is equal to 40, 30 whereas y is equal to 23, 39. - For site 4 (Flogita), we found in Google maps that x is equal to 40, 25 whereas y is equal to 23, 22. Hence, we depicted the following co-ordinated x and y of the four candidates sites in figure 16. 78 CANDIDATE SITES 23.4 SITE 3 23.35 SITE 2 23.3 y SITE 1 23.25 SITE 4 23.2 40.18 40.2 40.22 40.24 40.26 40.28 40.3 40.32 Figure 16: four candidate sites depending on co-ordinates x and y. In this point, we use the Center-of-gravity method, in order to determine an optimal location for the warehouse. Table 14: shows the data that we used in order to make a decision. REGIONS DEMAND X Y xnew ynew A Poligiros 4 40,38 23,44 6883,858 4292,71 B Nikiti 17 40,21 23,66 C Marmaras 14 40,09 23,78 D Toroni 25 39,98 23,9 E Sarthi 53 40,09 23,97 F Flogita 44 40,25 23,21 G Kallikratia 7 40,31 23,06 H Vergia 50 40,29 23,11 I Sozopoli 6 40,27 23,14 J Potidaia 38 40,19 23,32 K Afitos 6 40,09 23,43 L Κallithea 97 40,07 23,44 M Siviri 14 40,03 23,35 N Fourka 6 40 23,41 O Poseidi 5 39,96 23,38 P Mola Kaliva 7 39,97 23,45 Q Skioni 50 39,94 23,52 R Paliouri 5 39,94 23,66 S Pefkohori 69 39,99 23,61 T Chanioti 13 40 23,57 U Polixrono 95 40,01 23,52 SUM 625 79 We will estimate the new co-ordinates x and y for the new warehouse, by using the following formulas: n n xiWi yiWi x i1 , y i1 n n Wi Wi i1 i1 Therefore, the coordinates x equal to 6883,858 and y equal to 4292, 71 are the coordinates for the new warehouse. In addition, we emphasize that these coordinates are based on straight-line distances and not in real distances, this means that the spherical shape of the Earth affects the distance, consequently, the result is a point in three-dimensional space and it is not a relevant results. Furthermore, we must quote that the location obtained by this methods is not generally the optimal one. 5.2.4 IMPLEMENTATION OF THE LOAD-DISTANCE TECHNIQUE IN COCOLAK Company ‘’COCOLAK’’ wants to appreciate four different sites, it has develop its supply chain network and it wants to examine the situations where to locate, construct or to rent a new warehouse in order to serve its customer in different regions of Khalkidhiki. So, as mentioned before the coordinates x and y of the four candidate sites under are the following, by using Google maps. Table 15: coordinates of the four candidate locations. CANDIDATE LOCATIONS x y SITE 1(NEA MOYDANIA) 40,23 23,27 SITE 2(POTIDAIA) 40,19 23,32 SITE 3(KALUVES) 40,3 23,39 SITE 4(FLOGITA) 40,25 23,22 Furthermore, we will calculate the distances between the four candidates’ sites with each existing facility. We will use the straight-line formula di, in order to estimate these distances, hence: di=√(푥푖 − 푥)2 + (푦푖 − 푦)2 But in our analysis, we want the real distances between the four candidates’ sites and the customers. So, by using Google maps and apostaseis.gr we will take the real distances. In addition, these distances are measured in kilometers. For instance, by using Google maps, we found that the distance from Flogita to Nea moudania is 8, 19 km. So, with this manner we filled the entire table (table16) as we quoted as below. 80 Table 16: shows the distances between the four candidate’s sites with each existing facility. DINSTANCES FOUR CANDIDATE POINTS FOR NEW ΝΕΑ MOUDANIA POTIDAIA ΚΑLIVES FLOGITA DEMAND WAREHOUSE A FLOGITA 8,19 14,68 18,73 0 44 B ΚALLIKRATIA 28,75 35,23 39,28 23,2 7 C VERGIA 18 24,7 27,1 11,8 50 D SOZOPOLI 15,16 21,64 25,69 7,03 6 E POTIDAIA 8,91 0 16,83 16,12 38 F AFITOS 22,63 15,77 31,5 30 6 G KALLITHEA 25,38 18,51 34,24 32,74 97 H SIBIRI 33,33 26,47 42,4 40,7 14 I FOURKA 34,41 27,55 42,28 41,78 6 G POSEIDI 43,65 36,78 52,51 51,02 5 K ΜOLA KALIVA 45,89 39,03 54,76 53,26 7 L SKIONI 52,97 46,11 61,84 60,34 50 M PALIOURI 53,66 46,8 62,53 61,03 5 N PEFKOHORI 44,03 37,17 52,9 51,4 69 O CHANIOTI 40,52 33,66 49,39 47,89 13 P POLICHRONO 36,04 29,18 44,91 43,41 95 Q POLIGIROS 28,1 33,21 17,45 35,61 4 R ΝIKITI 38,01 43,11 27,36 45,51 17 S ΜARMARAS 58,24 63,34 47,58 65,74 14 T ΤORONI 80,36 85,47 69,71 87,86 25 U ΣARTI 94,44 99,55 83,79 101,95 53 Next, we will calculate the load distance for each candidate site by using the following formula: 퐧 퐋퐃 = ∑ 퐥퐢퐝퐢 퐢=ퟏ Table 17: depicts the Load distances of four candidates’ sites SITE 1 SITE 2 SITE 3 SITE 4 24195,03 22619,73 27599,12 27262,91 Since, we observe that the site2 (Potidaia) has the lowest load-distance value. Thus, according to this technique, the optimal solution is the site 2(Potidaia). Also, it would be considered that this location would minimize not only the transportation costs but it will bring profit in company. 81 5.3 EXAMPLES OF LOCATION ANALYSIS MODELS- PROBLEMS In this section, we will apply three models (P-Median, P-CENTER, and UFLP) of location facility in real case study of COCOLAK. We will analyze the seasonal demand for three years 2016-2014 for months June, July, August and September because as we mentioned before its seasonal job. Because of the higher demand and the higher percentage of customers in Kassandra region and Sithonia region of Halkidiki, company has expressed thought creation of a warehouse in the region, so as to avoid unnecessary transportation of amusement games. After a discussion with company, the analysis models-problems will not mentioned in the four candidate sites as we mentioned before for the examination of techniques, but now COCOLAK decided to rent or to construct anywhere in Halkidiki aiming to serve all customers. So, it suggested the following sites, Kallithea, Potidaia, Polichrono, Flogita, Pechkohori, (which some of them have higher demand and other are candidates’ sites). We note that symbols and models are similar having the same interpretation of these which we mentioned in chapter 3. In addition, the assumptions of models remain the same. The tables in which we will rely on and we will use are mentioned in previous section. Finally, for solving integer problems and mixed integer programming we will use LINDO, a software package for linear programming, integer programming, nonlinear programming, and general optimization. 5.4 APPLICATION OF THE P-MEDIAN MODEL-PROBLEM Clarifications: a. The seasonal demand will be the average of the total demand of the company for the years 2014, 2015 and 2016. b. Company has two trucks for the transshipments, the capacity of trucks is about 10 items of the amusement games. As we mentioned before the fleet of the company is consisted by two diesel truck, with transportation cost 0, 30 euros/km when truck is full and 0, 25 euros/km when the truck is empty. c. The distances that we will use in our analysis are depicted in table 16. Also, the measuring type is highway and the unit is the kilometer. Moreover, in order to find these distances, we used the following web site: www.apostaseis.gr and then we rounding the results in order to find the optimal solution. 82 Table 18: shows the distances between five sites in which company decided to locate its warehouse. DISTANCES A/A CITIES KALLITHEA POTIDAIA POLYCHRONO FLOGITA PEYKOCHORI 1 KALLITHEA 0 22 12 36 20 2 POTIDAIA 22 0 29 16 37 3 POLYCHRONO 12 29 0 47 8 4 FLOGITA 36 16 47 0 51 5 PEYKOCHORI 20 37 8 51 0 Objective function 5 5 Minimize Z= 훴푖=1 ∑푗=1 ℎ푖 푑푖푗 푌푖푗 (5.4.1) Subject to 5 훴 푗=1푌푖푗 = 1 ∀푖 ∈ 퐼 (5.4.2) 푌푖푗 − 푋푗 ≤ 0 ∀푖 ∈ 퐼, ∀푗 ∈ 푗 (5.4.3) 5 Σj=1Χj = p (5.4.4) 푋푖 ∈ {0,1} ∀푖 ∈ 푗 (5.4.5) 푌푖푗{0,1} ∀푖 ∈ 퐼, 푗 ∈ 퐽 (5.4.6) Results: Objective Function Value: 17: is the total transportation cost which needed in order to transfer the required quantity to customers. According to P-Median Model, the optimal solution in order to locate the warehouse is in Polichrono. More details and tables that we used in modeling of this method, we can see in Appendix A. 83 5.5 APPLICATION OF THE P-CENTER MODEL-PROBLEM Modeling Objective Function Minimize z (5.5.1) Subject to 5 훴 푗=1푌푖푗 = 1 ∀푖 ∈ 퐼 (5.5.2) 5 Σj=1Χj = p (5.5.3) 푌푖푗 − 푋푗 ≤ 0 ∀푖 ∈ 퐼, ∀푗 ∈ 푗 (5.5.4) 5 z>= ∑푗=1 푑푖푌푖푗 ∀푖 ∈ 퐼 (5.5.5) 푋푗 ∈ {0,1} ∀푗 ∈ 퐽 (5.5.6) 푌푖푗{0,1} ∀푖 ∈ 퐼, 푗 ∈ 퐽 (5.5.7) Results: Objective function value: 8000, this means that the maximum distance between a demand node and installation in Polichrono and is minimized for a price equal to 8. According to p-center model, the optimal solution to locate the warehouse is in Polichrono. More details and tables about modeling of this problem are shown in appendix B and the distances of the shortest paths used are shown in Table 18. 5.6 APPLICATION OF THE UFLP MODEL-PROBLEM Clarifications: a. fi (location facility cost) : is the sum of the fixed cost and variable costs of each facility. b. The fixed costs include time costs, rent, land purchase costs, registration, road tax, construction costs and equipment costs. c. The variable costs include the purchase price, the cost of building repairs, electricity value and water, transport fares, fuel, insurances, loan capital interest. 84 In addition, for the application of this model, we will need the following tables, Hence: Table 19: shows the fixed and variable costs. FIXED COST VARIABLE COST Constructed cost 1.000.000 Labor cost 50.000 Equipment cost 500.000 Others 30.000 TOTAL 1.500.000 TOTAL 80.000 Table 20: shows the candidate cities and their objective values CITIES KALLITHEA POTIDAIA POLUCHRONO FLOGITA PEUKOCHORI Objective value 40.000 30.000 25.000 20.000 27.000 1,5 acre 60.000 45.000 37.500 30.000 40.500 The company wants to rent or to purchase or to construct 1,5 acre, so the objective value will be expressed in E/acre, so we will estimate the land purchase costs of five candidates cities. Thus, the land purchase costs for Kallithea is 40000 x1, 5=60000, and so on for the other cities, as we can observe in the above table. Furthermore, the total location facility cost is the sum of the fixed and variable costs and land acquisition costs. Table 21: depict the total costs for each candidate location: CITIES KALLITHEA POTIDAIA POLUCHRONO FLOGITA PEUKOCHORI Total costs 1.640.000 1.625.000 1.617.500 1.610.000 1.620.500 Modeling Objective function Minimize 5 5 5 z=∑j=1 fi Xj 훴푖=1 ∑푗=1 ℎ푖 푑푖푗 푌푖푗 (5.6.1) Subject to 5 훴 푗=1푌푖푗 = 1 ∀푖 ∈ 퐼 (5.6.2) 푌푖푗 − 푋푗 ≤ 0 ∀푖 ∈ 퐼, ∀푗 ∈ 푗 (5.6.3) 푋푗 ∈ {0,1} ∀푗 ∈ 퐽 (5.6.4) 푌푖푗{0,1} ∀푖 ∈ 퐼, 푗 ∈ 퐽 (5.6.5) 85 Results: Objective function value: 402593.8. This means that the total cost is minimized in the 402593.8 value when the warehouse is located in Flogita. Thus, according to this model, the optimal solution in order to locate the warehouse and in order to serve all customers is in Flogita. The modeling of problem is depicted in Appendix C. 86 CHAPTER 6 6.1 CONCLUSION Table 22: aggregated results of Techniques FACTOR LOCATION LOAD- TECHNIQUE RATING METHOD CENTER OF GRAVITY DISTANCE no direct effect, regard three- RESULTS Nea Moudania dimensional space Potidaia Table 23: aggregated results of Models MODELS P-MEDIAN P-CENTER UFLP RESULTS Polichrono Polichrono Flogita In our analysis from the above two tables 22 and table 23, we can observe that the techniques suggested us different results. Whereas on the other hand, as regards the problem-models of facility location, the P –median model and P-Center lead to the same result, i.e. in Polichrono. Unlike the UFLP model is end up in different solution, where the most optimal solution it suggested the region of Flogita. After that, was performed another meeting with the COCOLAK in order to discuss the results of research. Suggested the company not to deal with the case of UFLP model, because is needed data such as total cost, and they cannot be estimated so precisely. Furthermore, COCOLAK has expressed its desire to rent warehouse rather than build a new one. Now, between the P-Median and P-Center models in which the results are same i.e. the region of Polichrono, COCOLAK expressed the idea that they are not very happy with results because they wanted to rent a warehouse near to Nea Moudania. But on the other hand, they expressed the opinion that if they find the suitable space where it is needed in order to stock all its equipment, then it will be rented a warehouse there. 87 CHAPTER 7 7.1 DIRECTION FOR FUTURE RESEARCH In this Master Thesis, we presented and analyzed various models about the location facility problems. We concluded that every model gives a different solution, because every model regards different facilities. Moreover, the decision relates the optimal location of facilities and this is perhaps the most critical and most difficult decision to be taken for the realization of an efficient and effectiveness supply chain. Moreover, this decision has a long time horizon and is very difficult or extremely expensive to change. In addition, this decision plays an important role in the functions of logistics. Also, there are cases where a possible failure in the location of a facility may lead to unbearable cost no matter how well it is organized. The models were developed in this thesis also, demonstrate how crucial are the assumptions, parameters and sets, and they show that the more variables and constraints so more clear results. On the other hand, there are areas that have not been examined yet, one of them is the uncertainty. For example, the decision-making about the location facility, taking into account external factors and possible risks that would have an important impact on growth and viability business. Also, modeling and solving techniques would be a great value for problems with economies scale. Finally, it is important to mention the cases of location facilities, in which the customer demand becomes with time delay. 88 CHAPTER 8 8.1 REFERENCES 1. Alfred Weber (1909), Theory of the Location of Industries. Article: shared by Smriti Chand Theory of Alfred Weber: Definition, Features and Criticism 2. Audretsch, David B., and Paula E. Stephan (1996): "Company-scientist locational links: The case of biotechnology." The American Economic Review 86.3:641-652. 3. Badri, Masood A. (2007) "Dimensions of industrial location factors: review and exploration." Journal of Business and Public Affairs 1. , 1-26. 4. Balinski, M. L., (1965), Integer Programming: Methods, Uses, Computation, î Management Science, 12, 253-313. 5. Ballou R. (2004) Business logistics/supply chain management 5th edition. Pearson Prentice Hall. 6. Barbaros C. Tansel, Richard L. Francis and Timothy J. Low (1983) – Location on Networks: A Survey Management Science, (The p-Center and p-Median Problem) pp. 482-497. 7. Barkley, David L., and Kevin T. McNamara (1994): "Manufacturers' location decisions: do surveys provide helpful insights?" International Regional Science Review 17.1: 23-47. 8. Bass, Bernard M., Donald W. McGregor, and James L. Walters (1977) "Selecting Foreign Plant Sites: Economic, Social and Political Considerations." Academy of Management Journal 20.4 : 535-551 9. Canel, Cem, and Basheer M. Khumawala (1996): "A mixed-integer programming approach for the international facilities location problem." International Journal of Operations & Production Management 16.4:49-68. 10. Church, R. and ReVelle, C. (1974): The maximal covering location problem. Papers in Regional Science Association, 32(1), p. 101-18. 11. Cieślik, Andrzej (2005): "Location of foreign firms and national border effects: the case of Poland." Tijdschrift voor Economische en Sociale Geografie 96.3: 287-297. 12. Cohon, J.L (1978) Multiobjecive Programming and Planning, Academic Press New York, N.Y. 89 13. D. A. Schilling, C. ReVelle, J. Cohon, and D. J. Elzinga (1980): Some models for fire protection locational decisions. European Journal of Operational Research, 5(1):1–7. 14. D. J. Eaton, M. S. Daskin, D. Simmons, B. Bulloch, and G. Jansma (1985): Determining emergency medical service vehicle deployment in Austin, Texas. Interfaces, 15(1):96–108. 15. Daskin, M. (1995) Network and discrete location: models, algorithms, and application. New York, Wiley interscience. 16. Daskin, M. (2008) what you should know about location modeling. Naval Research Logistics, 55(4), p.283-94. 17. Daskin, M. et al. (2002) an inventory-location model: formulation, solution algorithm and computational results. Annals of Operations Research, 110(1), p. 83-106. 18. Daskin, M. et al. (2005): Facility location in supply chain design. In: Langevin, A. and Riopel, D. (eds) Logistics systems: design and optimization. New York, Springer, p. 39-65. 19. Daskin, M. S. (2000) “A new approach to solving the vertex p-center problem to optimality: Algorithm and computational results,” Communications of the Japanese OR Society, 9, 428-436. 20. Donald W. Engels (1980) - Alexander the Great and the Logistics of the Macedonian Army. 21. Drezner, Tammy and Zvi Drezner (1996) " Competitive facilities: market share and location with random utility. "Journal of regional science 36.1:1-15. 22. Dyer JS (1990) Remarks on the analytic hierarchy process. Management Science 36(3). 23. Epping, G. Micheal. (1982): "Important factors in plant location in 1980." Growth and Change 13.2: 47-51. 24. Erkut, E. and Neuman, S. (1989): “Analytical models for locating undesirable facilities”, European Journal of Operational Research, 40, 275-291. 25. Erlenkotter D (1978) A dual-based procedure for uncapacitated facility location. Oper Res 26:992– 1009. 26. Figueiredo, Octávio, Paulo Guimaraes, and Douglas Woodward (2002): "Home-field advantage: location decisions of Portuguese entrepreneurs." Journal of Urban Economics 52.2: 341-361. 90 27. Francis, R. and White, J. (1974) Facility layout and location: an analytical approach. Englewood Cliffs, Prentice Hall. 28. Gianpaolo Ghiani, Gilbert Laporte, Roberto Musmanno (2004) - Introduction to Logistics Systems Planning and Control, John Willey & Sons. 29. Groothuis, Peter A., and Gail Miller (1994): "Locating hazardous waste facilities: the influence of NIMBY beliefs." American journal of economics and sociology 53.3: 335-346. 30. H. A. Eiselt and V. Marianov (2011): Covering Problems-University Forthcoming in Foundations of Location Analysis, Springer Link. 31. Hakimi, S. (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Operations Research, 12(3), p. 450-9. 32. Hander, E, Y and Mirchandani, P.B (1979) Location Networks, M.I.T press Cambridge, MA. 33. Handler, G. Y. (1990). P-center problems. In: Mirchandani, P.B. and Francis R. L. (ed.), Discrete Location Theory, John Wiley, New York. 34. Hansen, Pierre, and Fred S. Roberts (1996): "An impossibility result in axiomatic location theory." Mathematics of Operations Research 21.1: 195- 208. 35. Hongzhong Jia, Fernando Ordóñez, and Maged Dessouky* Daniel J. (2005): A Modeling Framework for Facility Location of Medical Services for Large- Scale Emergencies-University of Southern California Los Angeles. 36. Hoover, E. (1937): Location Theory and the Shoe and Leather Industries. Cambridge, Mass: Harvard University. 37. Hoover, Edgar Malone (1948): The location of economic activity. New York: McGraw-Hill. 38. Hotelling, Harold (1929). Stability in competition. Economic Journal, 39, 41- 57. 39. Ivan Contreras (2015) – Location Science (pp 311-344)- Springer Link. 40. J. M. Gleason (1975): A set covering approach to bus stop location. Omega, 3(5):605–608. 41. Jamshidi (2009) - Rena Zanjirani Farahani, Masoud Hekmatfar- Facility Location: Concepts, Models, Algorithms and Case Studies–Springer Link) 91 42. Jamshidi, Masoomeh (2009) Median location problem. In: Farahani, Reza Z. and Hekmatfar, Masoud (eds) Facility location: concepts, models, algorithms and case studies. Heidelberg, Physica-Verlag, p. 177-91 43. John J.Coyle et al (2009)-Supply Chain Management: A Logistics Perspective. 44. Justman, Moshe (1994): "The effect of local demand on industry location." The Review of Economics and Statistics: 742-753. 45. Klose A, Drexl A (2003) Facility location models for distribution system design. Eur J Oper Res 162:4–29. 46. Kuby, M. (1987) Programming models for facility dispersion: the p-dispersion and maxisum dispersion problems. Geographical Analysis, 19(4), p. 315-29. 47. Launhardt, Wilhelm (1885). Mathematische Begründung der Volkswirthschaftslehre. W. Engelmann. 48. Lazic, N., Givoni, I, Frey, B, & Aarabi, P (2009). Floss: Facility location for subspace segmentation. In International conference on computer vision. 49. Leitham, Scott, Ronald W. McQuaid, and John D Nelson (2000) "The influence of transport on industrial location choice: a stated preference experiment." Transportation Research Part A: Policy and Practice 34.7: 515- 535 50. Li, H. (2007). Two-view motion segmentation from linear programming relaxation. In Computer vision and pattern recognition. 51. Ma, Alyson C (2006) "Geographical location of foreign direct investment and wage inequality in China." The World Economy 29.8: 1031-1055. 52. Mai, Chao-cheng, and Hong Hwang (1994). "On a location theory under duopoly." Regional Science and Urban Economics 24.6: 773-784. 53. Manders, A. J. C. (1995): "Fact and fiction: wage levels and the (re) location of production." International Journal of Social Economics 22.5: 15-26. 54. Mark S. Daskin(2003) :Facility Location in Supply Chain Design 55. Mark S.Daskin (2013) Network and Discrete Location, Models, Algorithms and Application, Second Edition by John Wiley & Sons. 56. Martin Christopher (2011) - Logistics & Supply Chain Management. 57. Masood A.Badri Donald L. Davis, Donna Davis, (1995) "Decision support models for the location of firms in industrial sites", International Journal of Operations & Production Management, Vol. 15 Iss: 1, pp.50 – 62. 92 58. Mazzarol, Tim, and Stephen Choo (2003) "A study of the factors influencing the operating location decisions of small firms." Property Management 21.2: 190-208. 59. McCarthy, Bart L, and Walailak Atthirawong (2003): "Factors affecting location decisions in international operations-a Delphi study." International Journal of Operations & Production Management 23.7: 794-818. 60. Melvin Alexander (2012): Decision-Making Using the Analytic Hierarchy Process (AHP)-paper, University of Maryland Medical Center, Baltimore, MD. 61. O’Kelly, M (1987) A quadratic integer program for the location of interacting hub facilities European Journals of Operational Research, 32(3), p.393-404. 62. Perez J. (1995) some comments on Saaty’s AHP. Management Science 41(6):1091-1095. 63. ReVelle, C. and Swain, R. (1970) Central facilities location. Geographical Analysis, 2(1), p. 30-42. 64. Revelle. C. et al (2008). A bibliography for some fundamental problem categories in discrete location science. European journal of operational research, 184(3), p.817- 25) 65. Rummel, Rudolph J., and David A. Heenan (1978) - How multinationals analyze political risk. 66. S. S. Ravi, D. J. Rosenkrantz and G. K. Tayi (1994): Heuristic and Special Case Algorithms for Dispersion Problems Source: Operations Research, pp. 299-310 67. Saaty, TL, (1980). “The Analytic Hierarchy Process.” McGraw-Hill, NY. 68. Sahedeh Tafazzoli and Marzieh Mozafari (2009) –classification of location models and location stoftwares. Contributions to Managemene Science pp505- 521. – Springer Link. 69. Serra, Daniel, and Charles ReVelle (1994):" Market capture by two competitors: the preemptive location problem." Journal of Regional Science 34.4: 549-561. 70. Shams-ur Rahman a , David K. Smith(2000) -Use of location-allocation models in health service development planning in developing nations- European Journal of Operational Research 123 437±452. 93 71. Shen, Z. et al. (2003) A joint location-inventory model. Transportation Science, 37(1), p. 40-55. 72. Shier, D. R. (1977). ‘‘A Min–Max Theorem for p-Center Problems on a Tree.’’ Transportation Science 11(3), 243–52. 73. Shu-Hsien Liao, Chia-Lin Hsieh b, Peng-Jen Lai c, journal (2011): An evolutionary approach for multi-objective optimization of the integrated location–inventory distribution network problem in vendor-managed inventory. 74. Siebert, H. (2006): Locational competition: a neglected paradigm in the international division of labor. The World Economy, 29.2, 137-159 75. Snyder L. (2010) Introduction to facility location. Cited in: Cochran, J. (2010) Wiley encyclopedia of operations Research and Management Science. John Wiley & Sons). 76. Sunderesh S. Heragu (2016) -Facilities Design Fourth Edition. 77. Sylvester, James Joseph, (1857): ‘A question in the geometry of situation’, Quarterly Journal of Pure and Applied Mathematics, 1, 79. 78. Tansel BC, Francis R.L, LOW E T.J, Chen ML (1992): Duality and distance constraints for the nonlinear p-center problem and covering problem on a tree network. Op Res 30: 725-744. 79. Thünen, J. H. "v. (1875): Der isolierte Staat in Beziehung auf Landwirtschaft und Nationalökonomie." Aufl., Teil I. Berlin. 80. Toregas, c. et al (1971): The location of emergency service facilities. Operations Research, 19(6), p.1363-73. 81. Wojan, Timothy R., and Glen C. Pulver (1995): "Location patterns of high growth industries in rural counties." Growth and Change 26.1: 3-22. 82. Yang, J. and Lee, H. (1997), ``An AHP decision model for facility location selection’’, Facilities, Vol. 15 No, 9/10, pp. 241-54 83. Zuo-Jun Max Shen, Collette Coullard, Mark S. Daskin (2003): A Joint Location-Inventory Model pp. 40-55. 84. Zvi Drezner Horst W.Hamacher (2001). Facility Location Applications and Theory. 85. Zvi Drezner, Horst W. Hamacher (2002) Facility Location: Applications and Theory. 94 a. GREEK BOOKS 86. Καρκαζή A., Χατζηχρήστος Θ., Μαυρόπουλος Α., Εμμανουηλίδη Β. και Αhmed Elseoud, 2001, Landfill Sitting. 87. Κωνσταντίνος Σιφνιώτης (1997)- Logistics Management- Θεωρία και Πράξη. 88. Στράτος Παπαδημητρίου &Ορέστης Σχινάς(2004)- Εισαγωγή στα logistics WEBSITES: WWW.COCOLAK.GR www.slideshare.net www.scribd.com www.springer.com www.hyuan.com/java/how.html http://mathworld.wolfram.com/Distance.html http://egon.cheme.cmu.edu http://www.prenhall.com 95 APPENDIX APPENDIX A: P-Median Table 24 DISTANCES NUMBER OF AVERAGE OF TRUCKS TRUCKS A/A CITIES KALLITHEA POTIDAIA POLYCHRONO FLOGITA PEYKOCHORI DEMAND REQUIRED (ROUNDING) COST-FULL 1 KALLITHEA 0 22 12 36 20 94 10,4 10 0,3 2 POTIDAIA 22 0 29 16 37 36 4 4 3 POLYCHRONO 12 29 0 47 8 92 10,2 10 4 FLOGITA 36 16 47 0 51 41 4,6 5 5 PEYKOCHORI 20 37 8 51 0 67 7,4 7 Table 25 COST OF KILOMETERS A/A CITIES KALLITHEA POTIDAIA POLHRONO FLOGITA PEFKOHORI 1 KALLITHEA 0 6,6 3,6 10,8 6 2 POTIDAIA 6,6 0 8,7 4,8 11,1 3 POLICHRONO 3,6 8,7 0 14,1 2,4 4 FLOGITA 10,8 4,8 14,1 0 15,3 5 PEFKOHORI 6 11,1 2,4 15,3 0 Table 26 COST OF TRUCKS ( REQUERED) A/A CITIES KALLITHEA POTIDAIA POLHRONO FLOGITA PEFKOHORI 1 KALLITHEA 0 66 36 108 60 2 POTIDAIA 26,4 0 34,8 19,2 44,4 3 POLICHRONO 36 87 0 141 24 4 FLOGITA 54 24 70,5 0 76,5 5 PEFKOHORI 42 77,7 16,8 107,1 0 Table 27 COST OF TRUCKS ( ROUNDED) A/A CITIES KALLITHEA POTIDAIA POLHRONO FLOGITA PEFKOHORI 1 KALLITHEA 0 66 36 108 60 2 POTIDAIA 26 0 35 19 44 3 POLICHRONO 36 87 0 141 24 4 FLOGITA 54 24 71 0 77 5 PEFKOHORI 42 78 17 107 0 96 Min 66Y12+36Y13+108Y14+60Y15+26Y21+35Y23+19Y24+44Y25+36Y31+87Y32+141Y34+24Y35+54 Y41+24Y42+71Y43+77Y45+42Y51+78Y52+17Y53+107Y54 Subject to Y12+Y13+Y14+Y15+Y21+Y23+Y24+Y25+Y31+Y32+Y34+Y35+Y41+Y42+Y43+Y45+Y51+Y52+ Y53+Y54=1 Y12-X2<=0 Y13-X3<=0 Y14-X4<=0 Y15-X5<=0 Y21-X1<=0 Y23-X3<=0 Y24-X4<=0 Y25-X5<=0 Y31-X1<=0 Y32-X2<=0 Y34-X4<=0 Y35-X5<=0 Y41-X1<=0 Y42-X2<=0 Y43-X3<=0 Y45-X5<=0 Y51-X1<=0 Y52-X2<=0 Y53-X3<=0 Y54-X4<=0 X1+X2+X3+X4+X5=1 End INTEGER X1 INTEGER X2 INTEGER X3 INTEGER X4 INTEGER X5 INTEGER Y12 INTEGER Y13 INTEGER Y14 INTEGER Y15 INTEGER Y21 INTEGER Y23 INTEGER Y24 INTEGER Y25 INTEGER Y31 INTEGER Y32 INTEGER Y34 INTEGER Y35 INTEGER Y41 INTEGER Y42 INTEGER Y43 INTEGER Y45 INTEGER Y51 INTEGER Y52 INTEGER Y53 INTEGER Y54 97 APPENDIX B: P-CENTER MODEL Min z Subject to Y12+Y13+Y14+Y15+Y21+Y23+Y24+Y25+Y31+Y32+Y34+Y35+Y41+Y42+Y43+Y45+Y51+Y52+ Y53+Y54=1 X1+X2+X3+X4+X5=1 Y12-X2<=0 Y13-X3<=0 Y14-X4<=0 Y15-X5<=0 Y21-X1<=0 Y23-X3<=0 Y24-X4<=0 Y25-X5<=0 Y31-X1<=0 Y32-X2<=0 Y34-X4<=0 Y35-X5<=0 Y41-X1<=0 Y42-X2<=0 Y43-X3<=0 Y45-X5<=0 Y51-X1<=0 Y52-X2<=0 Y53-X3<=0 Y54-X4<=0 22Y12+12Y13+36Y14+20Y15+22Y21+29Y23+16Y24+37Y25+12Y31+29Y32+47Y34+8Y35+ 36Y41+16Y42+47Y43+51Y45+20Y51+37Y52+8Y53+51Y54-z<=0 End INTEGER X1 INTEGER X 2 INTEGERS X 3 INTEGERS X 4 INTEGERS X5 INTEGER Y12 INTEGER Y13 INTEGER Y14 INTEGER Y15 INTEGER Y21 INTEGER Y23 INTEGER Y24 INTEGER Y25 INTEGER Y31 INTEGER Y32 INTEGER Y34 INTEGER Y35 INTEGER Y41 INTEGER Y42 INTEGER Y43 INTEGER Y45 INTEGER Y51 INTEGER Y52 INTEGER Y53 INTEGER Y54 98 APPENDIX C: UFLP MODEL Min 1640000X1+1625000X2+1617500X3+1610000X4+1620500X5+66Y12+36Y13+108Y14+60Y15+26 Y21+35Y23+19Y24+44Y25+36Y31+87Y32+141Y34+24Y35+54Y41+24Y42+71Y43+77Y45+42Y5 1+78Y52+17Y53+107Y54 Subject to Y12+Y13+Y14+Y15+Y21+Y23+Y24+Y25+Y31+Y32+Y34+Y35+Y41+Y42+Y43+Y45+Y51+Y52+ Y53+Y54=1 Y12-X2<=0 Y13-X3<=0 Y14-X4<=0 Y15-X5<=0 Y21-X1<=0 Y23-X3<=0 Y24-X4<=0 Y25-X5<=0 Y31-X1<=0 Y32-X2<=0 Y34-X4<=0 Y35-X5<=0 Y41-X1<=0 Y42-X2<=0 Y43-X3<=0 Y45-X5<=0 Y51-X1<=0 Y52-X2<=0 Y53-X3<=0 Y54-X4<=0 End INTEGER X1 INTEGER X 2 INTEGERS X 3 INTEGERS X 4 INTEGERS X5 INTEGER Y12 INTEGER Y13 INTEGER Y14 INTEGER Y15 INTEGER Y21 INTEGER Y23 INTEGER Y24 INTEGER Y25 INTEGER Y31 INTEGER Y32 INTEGER Y34 INTEGER Y35 INTEGER Y41 INTEGER Y42 INTEGER Y43 INTEGER Y45 INTEGER Y51 INTEGER Y52 INTEGER Y53 INTEGER Y54 99 APPENDIX D: SOME AMUSEMENT GAMES. Hello Kitty Car Royal Carousel 100 Sea adventure Magic Hokey Sonic Sports Basketball Cabinet 101