Similarity based TOPSIS applied to equity portfolio

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LAPPEENRANTA UNIVERSITY OF TECHNOLOGY Faculty of Technology

M.Sc (Computational Engineering)

Oundo Herbert Masinde

SIMILARITY BASED TOPSIS APPLIED TO EQUITY PORTFOLIO

Examiners: Assoc Prof Pasi Luukka.

Prof. Mikael Collan.

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i

ABSTRACT

Lappeenranta University of Technology Faculty of Technology

M.Sc(Computational Engineering) Oundo Herbert Masinde

Similarity based TOPSIS applied to equity portfolio Master's thesis for the degree of Master of Science in Technology 2016

57 pages, 17 gures, 23 tables, 1 appendix Examiners: Assoc Prof Pasi Luukka

Prof. Mikael Collan.

Keywords: Similarity, TOPSIS, Ranking, Equity, Portfolio, Returns.

The aim of this thesis is to study applicability of similarity based TOPSIS to equity portfolios. Average annual returns are a basis to analyse portfolios. They are annually computed at same time for all nancial ratios by applying similarity based TOPSIS. To form the ratios, nancial values P,B,Ev, Ebit, E,S and Ebitda are selected from data set quoted on main list of Helsinki stock exchange (HEX) for period 1996-2012. They are chosen because from those values one can form widely used nancial ratios. The following nancial ratios;EV /Ebit, P/B, P/S, P/E and EV /Ebitdaare formed out of the values. Any two of these ratios are joined to form a combination. Since similarity based TOPSIS is multiple criteria decision making, by combining two of the ratios, we examine whether combinations of these ratios bring added value compared to using a single nancial ratio. Ranking is done according to closeness coecient computed from similarity based TOPSIS and ve equal size portfolios simultaneously. Portfolios are formed by dividing ranked companies into about equal size sets.

Results obtained generally indicate that 1st portfolios have highest average annual returns while 5th portfolios have the lowest. Similarly best performing portfolios occur whenp-values are less than1, implying that reducingp-values greatly improved performance of portfolios. Specically combination (EV /Ebit, P/E) has highest average annual returns of 15.71 and corresponding p−value of 0.75. This average annual return is higher than 15.29 for (EV /Ebit) which was the best single ratio result.

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ii Another interesting comparison is that combination (EV /Ebit, EV /Ebitda) has highest dierence between 1st portfolio returns and 5th portfolio returns of 14.49 which is higher than 13.73 for EV /Ebit the highest single ratio dierence. Hence we can note that there is added value to use similarity based TOPSIS.

The above results are in conformity with critical objectives of the study that; using two criteria instead of one brings added value since higher average returns have been gained this way and also that dierence between 1st portfolio returns and 5th portfolio returns is higher when we use combinations of ratios as compared to single nancial ratios.

Therefore, similarity based TOPSIS approach is practically robust and ecient in analysing portfolios.

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iii

Acknowledgements

I would like to thank department of Mathematics at Lappeenranta University of Technology for the scholarship given to me during my studies.

My sincere gratitude goes to my supervisor Associate professor Pasi Luukka for sparing your valuable time to guide, encourage and support me. You truely mentored me through your constructive ideas.

Special thanks goes to Prof. Mikael Collan for examining my thesis and all sta at Lappeenranta University of Technology who taught me various courses.

I would also like to thank all my family members; my parents Mr. and Mrs. Masinde for bringing me to this world and gaving all the necessary support i needed. My Sisters Lonah and Lovisa, brother Fred, i am so grateful for all the family ideas we have shared. Special thanks Lovisa and your husband John for taking care of my education at a critical time. My wife Pamelah and children Larry and Helsa, you have been patient and understanding during my absence from home, i thank you for keeping the home going.

I also thank my friends; Constance, Margaret and Simon, we have shared alot aca- demically and not forgeting Dr. Isambi and Idrissa for their assistance. Thanks to all my friends both in Uganda and Finland.

Lappeenranta, November 30, 2016.

Oundo Herbert Masinde

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CONTENTS iv

List of Tables

1 Ten combinations formed from ve nancial ratios . . . 6

2 Valid records from data source . . . 7

3 p-values with corresponding similarities . . . 29

4 Sample data . . . 30

5 Decision matrix . . . 31

6 Positive and negative ideal solutions . . . 32

7 Relative closeness to ideal solutions . . . 32

8 Returns for single rankings . . . 36

9 Ten applied combinations for the ve variables using two value ratios 38 10 Average annual returns using similarity based TOPSIS with p= 1 . . 38

11 Highest returns for similarity based TOPSIS with p = 1 compared with highest individual returns . . . 40

12 Best returns, Volatility and Sharpe with respective p-values . . . 42

13 Highest deviations . . . 42

14 Returns w.r.t p-parameter with combination (EV/Ebit,P/E) . . . 43

15 Returns w.r.t p-parameter with comb. (P/B, P/E) . . . 44

16 Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda) . . . 45

17 Returns w.r.t p-parameter with comb. (EV /Ebit, EV /Ebitda) . . . . 46

18 Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda) . . . 47

19 Returns w.r.t p-parameter with combination (EV/Ebit, P/B) . . . 54

20 Portfolio returns w.r.t p-parameter with comb. (P/E, P/S) . . . 55

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LIST OF TABLES viii

21 Returns w.r.t p-parameter with combination(P/S, EV/Ebitda) . . . 56 22 Returns w.r.t p-parameter with combination (EV/Ebit,P/S) . . . 57 23 Portfolio returns w.r.t p-parameter with comb. (P/B, P/S) . . . 58

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LIST OF FIGURES ix

List of Figures

1 Fuzzy numberA= [a₁, a₂, a₃] . . . 14

2 α−cut of fuzzy number . . . 15

3 Similarity between x and y . . . 29

4 Flow chart . . . 35

5 Returns for individual rankings . . . 37

6 Portfolio returns with TOPSIS (p=1) . . . 39

7 structure of combinations . . . 41

8 Returns w.r.t p-parameter with combination (EV/Ebit, P/E) . . . . 43

9 Returns w.r.t p-parameter with comb. (P/B, P/E) . . . 44

10 Returns w.r.t p-parameter with comb. (P/E, EV/Ebitda) . . . 45

11 Returns w.r.t p-parameter with comb. (EV/Ebit, EV/Ebitda) . . . . 46

12 Returns w.r.t p-parameter with comb. (P/B, EV/Ebitda) . . . 47

13 Returns w.r.t p-parameter with combination (EV/Ebit, P/B) . . . . 54

14 Returns w.r.t p-parameter with comb. (P/E, P/S) . . . 55

15 Returns w.r.t p-parameter with comb. (P/S, EV/Ebitda) . . . 56

16 Returns w.r.t p-parameter with combination (EV/Ebit, P/S) . . . . 57

17 Returns w.r.t p-parameter with comb. (P/B, P/S) . . . 58

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1 INTRODUCTION 1

1 INTRODUCTION

The TOPSIS method presented by Hwang and Yoon in 1981 [17] is one of the Mul- tiple Criteria Decision Making (MCDM) methods and has the basic principle that chosen alternatives should have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS). Ideally, positive ideal solution aims to maximize the benets and minimize the costs whereas the negative ideal solution aims to maximize the costs and minimize the benets.

Decision making involves nding feasible alternatives (see i.e. Jahanshahloo et al., [21]). The criteria used in selection of feasible alternatives usually conict with each other (see i.e. Ling et al., [25]). For example in design of a car, the criteria of higher fuel economy might mean a reduced confort rating due to the smaller passen- ger space. So there may be no solution satisfying all criteria simultaneously . Since its discovery, TOPSIS has been applied in wide range of elds with a great deal of interesting results such as decision making and support systems, negotiation systems, logistics management, wireless networks, project management, ecology, building and construction and feature selection.

Examination of decision approaches for portfolio selection by value ratios can be derived from some researchers. Nguyen et al., [46], initiated a new risk measure; the so called fuzzy sharpe ratio in the modeling context for assessing portfolio performance.

Research done by Yue et al., [52] using mean variance eciency and diversication on Chinese stocks joint construct portfolio constraints of upper bounds market values, P/E ratios, turn over ratios and industries found that upper bounds are eective in alleviating the contradiction, while market values, P/E ratios, turn over ratios and industries have much dampened inuence when applied separately or joints. In the same way Wang et al., [43], used TOPSIS method to measure the relative performance index of each project to select for a portfolio the rms which demonstrates the closeness of their overall nancial performance by listing companies in Vietnam stock market using inventory Turn over, Net Income Ratio, Earnings per share and current ratio, Return on total assets (ROA) and Return on common Equity (ROE) as estimation standard. Yuzi et al., [2], evaluated the returns performance of Is- lamic mutual funds in Malaysia based on four asset portfolios i.e. Equity, Debt, Money market and asset allocation using Sharpe and adjusted Sharpe Ratio. Simi- larly Kadri et al., [32], developed an improved equity valuation model that predicts rm's market value using rm's balanced score card (BSC) metrics by associatiing market value, book value and earnings.

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1 INTRODUCTION 2 Studies on equity portfolios by MCDM methods particularly brought interesting results. Panagiotis et al., [48] presented a methodology for supporting decisions that concern the selection of equities, on the basis of nancial analysis for Athens stock exchange, in which ELECTRE Tri outranking classication method was employed for selection of attractive equities.

Promethee V. Multiple criteria method in the second round of the participatory bud- geting (PB) Fontana et al., [31] was used in nding feasible alternative compatible with the city's goal. Similarly the holistic approach for nding an eective allocation of available research and development (R & D) resources by Gackstatter [40]

involved puting all potential portfolios one of them selected using (MCDM) methods.

Studies by Mendoza et al., [34], used four multiple criteria methods; ELECTRE, PROMOTHEE, TOPSIS and also a new and simple method called FUCA to select the best alternative among three criterias; NPV, risk and makespan for a new product Development (NPD) problem in the Pharmacentical Industry. The fuzzy decision theory was employed by Pai et al., [13], to tackle the uncertainty arising out of possible market scenarios in the fund manager's view point. The performance eciencies of the optimal fuzzy portfolios were measured using Sharpe and Treynor ratios and compared with those of the crisp counterparts.

Wachowicz et al., [45] designed a TOPSIS based approach to scoring negotiating oers in Negotiation support systems (NSS), in which a Simple Additive Weigh- ing (SAW) model was used in negotiations preference analysis. Raia et al., [22], had similarly used SAW by applying formal models, which allow for analyzing negotiators preferences and determining a scoring system for negotiation oers. This system was indeed real in building negotiators own proposals and analyzing partner's counter oers. Raia et al., [37] later modied this into a new scientic displine called negotiation analysis which was implemented as a software solution in form of negotiation support system (NSS). Hordijk L [18] developed a system using RAINS model which supports real world negotiation problems for instance to resolve the dispute between the European countries negotiating air pollution limits. Recently, the basic supportive ideas derived from SmartSettle system (see i.e Thiessen et al., [44]) have been used for supporting First Nations Negotiation in Canada. Other models are also applied into NSSs, based on dierent analytical approach,like the AHP (see i.e. Mustajoki et al., [35]) or ELECTRE (see i.e. Wachowicz [47]). On the other hand, the Internet expansion and e-commerce development cause that the vast majority of the business processes, including the negotiations, are conducted by means of computers and the web, using both simple communication software such as electronic mail clients and instant messaging systems, and more sophisticated ne-

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1 INTRODUCTION 3 gotiation support systems (NSS) or electronic negotiation systems [45] . The Deep Ocean Mining Model used in the United Nations UNCLOS III negotiations (see i.e.

Sebenius [38]) on the rights to exploit the natural resources from beneath of the sea bed and sharing the prots yielded from the exploitation was also developed.

Several studies have still emerged bringing newer techniques involving TOPSIS, for instance Chamodrakas et al., [6], developed a model for aggregating function of TOPSIS based on Fuzzy set representation of the closeness to ideal and negative ideal solution. Further Chamodrakas et al., [20], presented a method that takes into account user preferences, network conditions, QoS and energy consumption re- quirements in order to select the optimal network which achieves the best balance between performance and energy consumption. The proposed network selection method incorporates the use of parameterized utility functions in order to model diverse QoS elasticities of dierent applications, and adopts dierent energy consumption metrics for real time and non- real- time applications. Maryam et al., [29], used graph theory and matrix methods as decision analysis tools for contractor selection as decision support system for identifying eligible contractor to be awarded a contract. Zavadskas et al., [53], used grey theory technique for performing pre- dictive, relation analysis and decision making for assesing contractors competitive ability. Krohling et al., [23], presented fuzzy TOPSIS to handle uncertain data and proposed a fuzzy TOPSIS for group decision making which was applied to evaluate the ratings of response alternatives to a simulated oil spill. Hence combat responses in case of accidents with oil spill in the sea. Chen [7], extended TOPSIS to fuzzy environment in which the ratings of each alternative and weight of each criterion are described by linguistic terms which can be expressed in triangular fuzzy numbers.

He then proposed a vertex method to calculate the distance between two triangular fuzzy numbers. Milani et al., [30], employed entropy method and TOPSIS to weigh selected failure criteria and to rank the selected material IDs, respectively. This was applied specically to the gear material for selection of power transmission.

There are also lots of theoretical works on TOPSIS extensions showing how the method may be modied to solve problems of a particular formal structure with additional assumptions (see i.e. Jahanshahloo et al., [21] and Shih et al., [41]).

In this study, similarity based TOPSIS [26] is applied to equity portfolios for Finnish non-nancial stocks. This will involve computation of average annual returns for each combination of nancial ratios. Results are obtained simultaneously for ve portfolios on annual basis so that we can examine performance of similarity based TOPSIS.

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1 INTRODUCTION 4 This thesis is organized as follows. The rst chapter is an introductory part which highlights the background, In chapter two, research problem is introduced and we also look at objectives of the study, data and methodology used. Chapter three contains some mathematical concepts . In chapter four, the Similarity based TOPSIS is introduced. Chapter ve shows the results from our computation and discussions about the results. In chapter six, we conclude the study and give prospects for future studies.

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2 RESEARCH PROBLEM 5

2 RESEARCH PROBLEM

Similarity based TOPSIS has been introduced (see i.e Luukka et al. [26]), in which histogram ranking is as well introduced to relax parameter dependency for problems where suitable parameter values are exactly unknown. In this study, similarity based TOPSIS will be applied to criteria which are gained from nancial ratios.

To examine performance of similarity based TOPSIS, average annual returns for ve portfolios are computed. The ve portfolios are formed by ranking companies based on their closeness coecient value and forming ve sets based on these values.

The portfolios employed in testing the applicability of adjusted valuation measures as a basis of stock selection criterion are composed of Finnish non-nancial stocks quoted on main list of the Helsinki stock market (HEX) for the period 1996- 2012.

Financial values, P,B,Ev, Ebit, E,S and Ebitda are selected because they are easily measurable. They are explained below;

• Ebit- Earnings Before Interest and Taxes

• EV(Enterprise value) Market value of Equity (ME) plus Short term Debt plus Long term Debt Plus preferred stock value Minus Cash and Short term investments. ME is stock price multiplied by shares outstanding from the CRSP monthly le & obtained as end of April of yeart throughout the paper.

• B- Book value of equity is the stock holders equity plus deferred taxes minus preferred stock.

• P- Price

• E-(Earnings) is income before extra ordinary items minus preferred Dividends plus income statement deferred taxes.

• S(Sales)

• Ebitda- Earnings Before Interest, Taxes, Depreciation and Amortization.

From the above values, we derive ve most widely used nancial ratios; EV /Ebit, P/B,P/S,P/E and EV /Ebitda through division. Then we form ten combinations out of ve given nancial ratios by joining one ratio with atleast each of the other four remaining ratios. Basically this should lead to unique combinations seen in table 1 below.

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2 RESEARCH PROBLEM 6

(EV/Ebit,P/B)

(EV/Ebit,P/E) (P/B, P/E)

(EV/Ebit,P/S) (P/B,P/S) (P/E,P/S)

(EV/Ebit, EV/Ebitda) (P/B, EV/Ebitda) (P/E,EV/Ebitda) (P/S,EV/Ebitda)

Table 1: Ten combinations formed from ve nancial ratios

Ranking is done according to closeness coecient computed from similarity based TOPSIS and ve equal size portfolios. The returns of portfolios are examined with respect to parameter changes in similarity measure and with respect to dierent value ratio combinations. The results for dierent combinations and respective p- values are obtained to determine best performing and also the eect of changing p−values.

The objectives of this study are to;

• Study the Similarity based TOPSIS

• Apply similarity based TOPSIS to equity portfolios and compute returns from the available data basing on ve valuation ratios;

• Perform ranking to determine a portfolio which gives better percentage returns and the corresponding p- values.

• Analyse eect of changing p- values with similarity based TOPSIS on equity portfolios.

• Compare results for single criteria and two criteria to nd if we can ascertain which gives higher average returns and hence see if there is added value in using combinations of ratios.

• Compare dierence between 1st portfolio average annual returns and 5th portfolio average annual returns.

To achieve the above objectives, data was collected, consisting of Finnish non- nancial stocks quoted on the main list of the Helsinki Stock exchange (HEX) during the period 1996-2012. This sample comprehensively includes all Finnish non-nancial companies that have been quoted on the main list of the OMX HEX and have met all the criteria for inclusion. The stocks in sample are rst ranked based on conventional individual valuation ratios.

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2 RESEARCH PROBLEM 7 Normalization has been done to near minimum by reversing the valuation ratios.

The valuation ratios and respective parameter(p)-values are the inputs while the% returns are the outputs. However, two important considerations are made on the ratios;

• Missing values: This kind of scenario could lead to 0 or undened, hence non-representable.

• Very small numbers: This was leading to penny stocks i.e. stocks with price less than 1 euro. It is a common practice not to include such and hence they were also removed.

Therefore, particular companies were removed if for one nancial ratio one of these conditions was valid. All together, in the 17 years, total amount of companies was 160 records but as a result of above eects, data set is now less than original 160 records in each excel worksheet (each year). New sample of 1279 valid records is displayed in table 2 below:

Years No. of valid records

1996 49

1997 50

1998 55

1999 73

2000 83

2001 81

2002 80

2003 75

2004 68

2005 75

2006 75

2007 80

2008 95

2009 84

2010 85

2011 87

2012 84

Total No. of valid records 1279

Table 2: Valid records from data source

Ranking is done according to closeness coecient computed from similarity based TOPSIS and ve equal size portfolios on yearly basis. Portfolios are examined with

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2 RESEARCH PROBLEM 8 respect to parameter (p) changes in similarity measure and with respect to dierent value ratio combinations. Specically we;

• Form portfolios based on ranking companies with respect to value ratios;

EV /Ebit, P/B, P/E, P/S, and EV /Ebitda

• Apply similarity based TOPSIS to compute average annual returns for the ten combinations of valuation ratios with varying parameter which are formed out of ve given nancial ratios by joining any two of the ratios as seen in table 1.

• Determine best returns with respectivep−values and also corresponding volatility and sharpe.

• Compare returns from single criteria and two criteria to see which one gives highest average annual returns.

• Find the dierence between rst portfolio average annual returns and 5th portfolio average annual returns.

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3 MATHEMATICAL CONCEPTS 9

3 MATHEMATICAL CONCEPTS

Professor L.A Zadeh [28] introduced the concept of a fuzzy set which has played a major part in many science models. We examine three main aspects of fuzzy sets discussed by Dubois [11] to understand the set concept.

Uncertainty: it is the ability to judge whether a proposition is true or false for example we can describe the weather today as sunny if we dene any cloud cover of 20%or less sunny, that implies that cloud cover of20.5% is not sunny (see Klir [14]).

Impreciseness:It is a characteristic of language and pertains to measurable concepts and particularly metric properties. In traditional theories, world representations are forced to comply with extremely precise models, avoiding and rejecting imprecise as a perturbation fact [5]. However, impreciseness plays an important role in information representations where increase in precision would otherwise become unmanagable.

Vagueness: A notion is said to be vague when its meaning is not xed by sharp boundaries. Example of vague information; data quality is 'good' or transparency of optical element is acceptable. Dubois [11] generally observe that impreciseness and vagueness refer to the contents of a piece of information expressed in some language, while uncertainty refers to ability of an agent to claim whether a proposition holds or not.

3.1 Fuzzy sets and crisp sets

Let us consider that we have elements of a set A with membership values in the range0≤x≤1. For a crisp set, an element is either a member of the set A or not, while for fuzzy sets, elements can be partially in a set with a degree of membership such that for value 0, x /∈A and for value 1, x ∈A. On the other hand if only the extreme membership values of 0and 1are allowed, then it is a crisp set.

Crisp sets

A crisp set is dened in such a way as to classify the individuals in some given universe of discourse X into two groups: members and nonmembers (see i.e. [10]

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3 MATHEMATICAL CONCEPTS 10 and [14]). Klir [14], further outlines three basic methods by which sets can be dened within a given universal set X:

The list method: A set is dened by naming all its members. This method can only be used for nite sets. Set A, whose members are {a₁, a₂, . . . , a_n} is usually written as A={a₁, a₂, . . . , a_n}.

The rule method: A set is dened by a property satised by its members. A common notation expressing this method is

A={x|P(x)},

where ⁰|⁰ denotes the phrase "such that" and P(x) destinates a proposition of the form "x has the property P". That is, A is dened by this notation as the set of all elements ofxfor which the propositionP(x)is true. It is required that the property P be such that for any given x∈X, the proposition P(x) is either true or false.

Characteristic function: A set A is dened by its characteristic function that declares which elements of X are members of the set and which are not. Set A is dened by its characteristic function as follows;

λ_A(x) =







1, for x∈A 0, for x /∈A

That is, the characteristic function maps elements of X to the elements of the set {0,1}, which is formally expressed by

λ_A:X → {0,1} (1)

For eachx∈X,whenλ_A(x) = 1,xis declared to be a member of A; whenλ_A(x) = 0, x is declared as a nonmember of A.

Fuzzy sets

According to Klir [14], a fuzzy set can be dened mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. Dubois [10] presents a discussion for concept of a fuzzy set as follows;

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3 MATHEMATICAL CONCEPTS 11 Let X be a classical set of objects called the universe, whose generic elements are denotedx, membership in a classical subset A of X is often viewed as a characteristic function; µ_A from X to valuation set {0,1}such that pairs

µ_A(x) =







1, i x∈A 0, i x /∈A

If the valuation set is allowed to be real interval [0,1], A is called a fuzzy set (see i.e Zadeh [28]). µ_A(x) denotes the grade of membership of x in A. The closer the value ofµ_A(x)is to 1,the higher the certainty is thatxbelongs to A. A is completely characterized by the set of pair

A={(x, µ_A(x), x∈X)} (2) Letα, β, γ and δ be real numbers, some commonly used fuzzy sets [27] are dened below:

• Γ-Shaped fuzzy set: A function with one variable and two parametersΓ :x→ [0,1]is dened by

Γ(x;α, β) =











0, if x < α

x−α

β−α if α≤x≤β 1, if x > β

• S-shaped fuzzy set is dened by

S(x;α, β, γ) =











0, if x < α 2_(x−α)

(β−α)

if α≤x≤β 1−2_x−γ)

γ−α

if β ≤x≤γ 1, if x > γ

where β = α+γ 2

• L-Shaped fuzzy set: is decreasing piecewise continuous function L:x→[0,1]

dened by

L(x;α, β) =











1, if x < α

β−x

β−α if α≤x≤β 0, if x > β

• Λ-shaped fuzzy set: is dened as;

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Λ(x;α, β, γ) =











0, if x < α

x−α

β−α if α≤x≤β

γ−x

γ−β if β ≤x≤γ 0, if x > γ

• Bell-shaped fuzzy set: is dened by;

π(x;β, γ) =







S(x;γ−β,^γ−β_2,γ ), if x≤γ 1−S(x;γ,^γ+B₂ , γ+β), if x > γ

• Π-shaped (Trapezoidal fuzzy set):

Π(x;α, β, γ, δ) =











0, if x < α

x−α

β−α if α≤x≤β 1, if β <≤γ

δ−x

δ−γ if γ ≤x≤δ 0, if x > δ

3.2 Properties of fuzzy sets

Consider the universe of discourse X; as crisp sets, Klir [14] denes the following main properties of fuzzy sets;

• Given two fuzzy sets A and B, ifA⊆B and alsoB ⊆A, then A and B contain the same members and are called equal sets. If A and B are not equal, we write A6=B

• The support of a fuzzy set A within a universal set X is the crisp set that contains all the elements of x that have nonzero membership grades in A.

Supp(A) ={x∈X|A(x)>0}. (3)

• The core of a fuzzy set A is a crisp set

Core(A) = {x∈X|A(x) = 1}. (4)

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• The height, h(A) of a fuzzy set A is the largest membership grade obtained by any element in that set.

hgt(A) =Supx∈XA(x) (5)

A fuzzy set A is called normal when h(A)=1; it is called subnormal when h(A)<1.

• Given a fuzzy set A dened on x and any numberα∈[0,1], theα-cut^αA and the strong α-cut , ^α+A, are the crisp sets

αA = {x∈X|A(x)≥α}. and

α+A = {x∈X|A(x)> α} (6)

That is, the α-cut (or the strong α−cut) of a fuzzy set A is the crisp set ^αA (or the crisp set^α+) that contains all the elements of the universal set X whose membership grades in A are greater than or equal to (or only greater than ) the specied value ofα.

3.3 Operations on fuzzy sets

We present three special operations of fuzzy sets often called standard fuzzy operations discussed by Klir [14]. Consider two fuzzy subsets A and B of the universe X, and A(x), B(x) their respective membership values for allx∈X.

1. The intersection of two fuzzy sets A and B, A(x)∧B(x), is dened as:

(A∩B)(x) = A(x)∧B(x) = min{A(x), B(x)}. (7) 2. The union of two fuzzy sets A and B,A(x)∧B(x), is dened as:

(A∪B)(x) =A(x)∨B(x) = max{A(x), B(x)}. (8) 3. The complement of a fuzzy set A, isA¯is dened as:

A(x) =¯ A(x) = 1−A(x). (9)

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3.4 Fuzzy numbers

Shang et al., [39] dened a fuzzy number as an ordinary number whose precise value is somewhat uncertain. Klir [14], further explains fuzzy numbers as special types of fuzzy sets that are dened by the set of real numbers R, with membership functions of the formA:R→[0,1], i.e. they are close to a given real number or numbers that are around a given interval of real numbers. Therefore, a fuzzy set A on R must possess atleast the following properties for a fuzzy number.

(i) A must be a normal fuzzy set;

(ii) ^αA must be a closed interval for every α ∈(0,1]; (iii) the support of A, ⁰⁺A, must be bounded .

Operations of fuzzy numbers

If a fuzzy set is convex and normalized, and its membership function is dened in R and piecewise continuous, it is called a fuzzy number. Hence a fuzzy number /set represents a real number interval whose boundary is fuzzy.

Figure 1: Fuzzy number A= [a1, a2, a3]

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3 MATHEMATICAL CONCEPTS 15 Further, a fuzzy number can be expressed as a fuzzy set dening a fuzzy interval in the real number R. Since the boundary of this interval is ambiguous, the interval is also a fuzzy set. Generally a fuzzy interval can be represented by two end points and a peak point. Let use consider end points a₁ and a₃ with a peak point a₂ as shown in Figure 1. We also consider that the fuzzy number is normalized and convex, i.e.

∃x₀ ∈R, µA¯(x₀) = 1. Theα-cut operations can be also applied to the fuzzy number.

If we denote α-cut interval for fuzzy number A as A_α, the obtained interval A_α is dened as Aα =

h

a^(α)₁ , a^(α)₃

i. We can also know that it is an ordinary crisp interval as shown in Figure 2

Figure 2: α−cut of fuzzy number

The convex condition is that the line by α-cut is continuous and α-cut interval satises the following relationA_α = [a^(α)₁ , a^(α)₃ ]

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3.5 Fuzzy relations and equivalence

The basic ideas of fuzzy relations and concepts of fuzzy equivalence, compatibility and fuzzy orderings were rst introduced by Zadeh(1965) [28].

3.5.1 Crisp and Fuzzy relations

A crisp relation represents the presence or absence of association, interaction or interconnectedness between the elements of two or more sets (see i.e. Klir et al., [14]). A relation among crisp sets X₁, X₂, . . . , X_n is subset of the cartesian product X_i∈_N_nX_i.It can be denoted by either R(X₁, X₂, . . . , X_n)or by the abbreviated form R(X_i|i∈Nn). Thus

R(X₁, X₂, . . . , X_n)⊂X₁×X₂×. . .×X_n, (10) Each crisp relation R can be dened by a characteristic function which assigns a value of 1to every n tuple if the universal set belongs to the relation and0to every tuple not belonging to it.

⇒ R(x₁, x₂, . . . , x_n) =







1, if(x₁, x₂, . . . , x_n)∈R

0, otherwise (11)

3.5.2 Binary relations on a single set

Types of relations R(X, X) can be distinguished basing on three dierent characteristic properties [14]:

1. Reexivity: A crisp relation R(X, X) is reexive i (x, x) ∈ R for each x∈X, that is, if every element ofxis related to itself otherwise it is irreexive.

If (x, x)6=R for every x∈X, the relation is called antireexive.

2. Symmetric: A crisp relation R(X, X) is symmetric i for every (x, y) ∈ R and (y, x) ∈ R where x, y,∈ X. Thus, whenever an element x is related to an element y through a symmetric relation, y is also related to x. Otherwise it is asymmetric. If both < x, y >∈ R and < y, x >∈ R implies x = y then

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3 MATHEMATICAL CONCEPTS 17 the relation is called antisymmetric. If either < y, x >∈ R or < y, x >∈ R, whenever x6=y, then the relation is called strictly antisymmetric.

3. Transitive: For a crisp relationR(X, X)to be transitive(x, z)∈R whenever

< x, y >∈ R and < y, z >∈ R for at least one y ∈ X. In other words the relation of x toy and y to z implies the relationx to z. A relation that does not satisfy this property is called non transitive. However, if < x, z > /∈ R whenever both < x, y >∈ R and < y, z >∈ R, then the relation is called antitransitive.

3.5.3 Fuzzy equivalence relations

A crisp binary relation R(x, x) that is reexive, symmetric, and transitive is called an equivalence relation. We can dene a crisp set A_x containing all elements of x that are related to x by the equivalence relation.

A_x ={y|< x, y >∈R(x, x)}. (12) A fuzzy binary relation that is reexive, symmetric, and transitive is known as a fuzzy equivalence relation or similarity relation. (see i.e. Klir et al., (1995) [14]).

3.6 Fuzzy Decision Making

The concept of decision making has been applied in various elds such as logistics management, wireless networks, project management, building and construction and ecology. Klir [14] denes decision making as nding the best option among the available alternatives. Decision problems are further categorized into four main classes; individual decision making, multiperson decision making, multiple criteria decision making and multistage decision making (see i.e. [14] and [3]).

3.6.1 Individual decision making

This is a model of decision making in which one decision maker is involved in nding the best alternatives. Relevant goals and constraints are expressed in terms of fuzzy sets and a decision is determined by an appropriate aggregation of these fuzzy sets.

It is made up of the following components.

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• a set X of possible actions.

• a set of goals Pi(i∈Nn), each expressed in terms of a fuzzy set dened on X;

• a set of constraints Q_j(j ∈Nn), each of which is also expressed by a fuzzy set dened on X.

If we let P⁰_i and Q⁰_i to be fuzzy sets dened on sets Ai and Bi, respectively, where i∈Nn andj ∈Nm and assume that these fuzzy sets represent goals and constraints expressed by the decision maker. Then, for eachi∈Nnandj ∈Nm, we can describe the meanings of actions in set X in terms of sets Ai and Bj by functions

pi :X →Ai, (13)

q_j :X →B_j, (14)

If we express goals Pi and constraints Qj by the compositions of pi with P⁰i and the compositions of qj and Q⁰_j ; then,

P_i(a) = P_i⁰(p_i(a)), (15)

Q_j(a) = Q⁰_i(q_i(a)), (16)

for each a ∈ X. Now given a decision situation characterized by fuzzy sets X, Pi(i ∈Nⁿ), and Qj(j ∈ N^m), a fuzzy decision, D, is represented in form of a fuzzy set on X. That is,

D(a) = min

i∈infNⁿ

P_i(a), inf

j∈N^m

Q_j(a)

for all a∈X (17)

This simultaneously satises the given goals Pi and constraints Qj. We can now choose the best single crisp alternative from this fuzzy set by selecting an alternative that attains maximum membership grade in D.

3.6.2 Multiperson decision making

This arises when decisions made by more than one person are modeled. There are two dierences to consider from the case of single decision making;

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• the goals of the individual decision makers may dier such that each places a dierent ordering on the alternatives.

• the individual decision makers may have access to dierent information upon which to base their decision.

Each member of a group of n-individual decision makers is assumed to have a reexive, antisymmetric, and transitive preference ordering Pi, i∈Nn which totally or partially orders a set X of alternatives. A social choice function must then be found which produces the most acceptable overall group preference ordering from the individual preference orderings. The model allows an individual decision maker to have dierent aims and values while assuming that the overall purpose is to reach a common, acceptable decision. Let the social preference S be represented by a binary relation with membership grade function to deal with the multiplicity of opinions.

S :T ×T →[0,1] (18)

which assigns the membership grade S(t_i, t_j) to show the degree of reference of alternatives t_i over t_j. We then use the method of popularity of alternatives t_i over t_j which involves dividing the number of persons prefering t_i tot_j, denoted by N(t_i, t_j), by the total number of decision makers, n

S(t_i, t_j) = N(t_i, t_j)

n (19)

S is then converted into its resolution form to determine the trial non fuzzy group preference.

S =Uα∈[0,1]α^αS (20)

which is the union of the crisp relation^αScomprising theα−cuts of the fuzzy relation S, each scaled by α. α represents the level of agreement between the individual concerning the particular crisp ordering ^αS. The largest value of α for which the unique compatible ordering on T ×T is found represents the maximum level of agreement of the group while the crisp ordering represents the group decision.

3.6.3 Multiple criteria decision making

Each object is assigned several numerical evaluations which refer to dierent criteria of the objects. (see i.e Dubois et al., [10]). Hence relevant alternatives are evaluated

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3 MATHEMATICAL CONCEPTS 20 according to a number of criteria, each inducing a particular ordering of alternatives.

We therefore need a procedure by which to construct one overall preference ordering.

The number of criteria and alternatives are assumed to be nite.

LetX ={x₁, x₂, . . . , x_n}be a set of alternatives to be evaluated andC ={c₁, c₂, . . . , c_m} be a set of criteria to be followed for a decision problem. We can represent this as a matrix.

R =







X1 X2, . . . , Xn

C₁ r₁₁ r₁₂ . . . r_1n C₂ r₂₁ r₂₂ . . . r_2n ... ... ... ... ...

C_n r_m1 r_m2 . . . r_mn







It may happen that instead of matrix R with entries [0,1], an alternative matrix R⁰ = [r_ij⁰ ], whose entries are arbitrary real numbers is initially given. R⁰ can then be converted to a desired matrix R by the formula

r_ij =

r_ij⁰ −min

j∈Nn

r⁰_ij maxj∈Nn

r_ij⁰ −min

j∈Nn

r⁰_ij ∀i∈Nm and j ∈Nn (21) One approach is by converting to single criterion decision problems, whereby we nd a global criterion, r_j =h(r_ij, r_2j, . . . r_mj), i.e. for each x_j ∈X is an adequate aggre- gate of values r_1j, r_2j, r_mj to which the individual criteria c₁, c₂, . . . c_n are satised.

3.6.4 Multistage decision making

In this case, a required goal is achieved by solving a sequence of decision- making problems. The decision making problems, which represent stages in overall multistage decision making are dependent on one another in the dynamic sense. Gen- erally, multistage decision making may be viewed as part of the theory of general dynamic systems. The most important being that of dynamic programming, which can be fuzzied (see i.e. Bellman et al.,[3]). A fuzzication of dynamic programming extends its practical utility since it allows decision makers to express their goals, constraints, decisions in appropriate fuzzy terms. The basic ideas of fuzzy dynamic programing (see i.e. Bellman et al.,[3] ) are formulated as follows;

A decision problem concieved in terms of fuzzy dynamic programming is viewed as a decision problem regarding a fuzzy nite- state automaton with two restrictions

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• the state-transition relation is crisp and hence, characterized by the usual state transition function of classical automata.

• n special output is needed i.e. next internal state is also utilized as output and; consequently the two need to be distinguished.

From the above restrictions, we dene

A=< X, Z, f >, (22)

where X and Z are respectively the sets of input states and output states of A, and

f :Z×X →Z (23)

is the state-transition function of A whose meaning is to dene, for each discrete time t(t∈N),the next internal state, z^t+1 of the automaton in terms of its present internal state, z^t, and its present input state x^t, i.e.

Z^t+1 =f(z^t, x^t). (24)

3.7 Fuzzy ranking methods

The nal scores of alternatives can be represented in terms of fuzzy numbers, to try to resolve the ambiquity of concepts that are associated with human beings judgements. We need to construct crisp total ordering from fuzzy numbers in order to express crisp preferences of alternatives.

Fuzzy ranking methods are common in establishing an ordering relation on F. In comparing with previously studied methods, they are divided into two main types, (see i.e., Matteo et al., [33]).

3.7.1 First type ranking methods

These map fuzzy numbers directly into real line. The transformation is of the form M:F →R. Implying that they associate each fuzzy number with a real number and then use the ordering ≥ on the real line. Hence a higher associated value indicates a higher rank.

M(A_i)≥M(A_j)⇒A_i _M A_j (25)

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where _M is the dominance relation induced by M.

Several examples of rst type ranking methods have been proposed which include;

Hamming distance on the set R of all fuzzy numbers.

This is a method for ranking fuzzy numbers which is based on distance. For any given fuzzy numbers A and B, the hamming distance d(A, B)is dened as;

d(A, B) = Z

R

|A(x)−B(x)|dx (26)

We therefore determine the least upper bound,M AX(A, B)for the numbers A and B which we want to compare. Then calculate the Hamming distancesd(M AX(A, B), A) andd(M AX(A, B), B)and deneA≤Bifd(M AX(A, B), A)≥d(M AX(A, B), B). If A≤B, then M AX(A, B) =B and hence A≤B.

Other rst type ranking methods have been compared [33]. These are briey discussed below.

Adamo

When using this method, (see i.e. Adamo [1]), we simply evaluate the fuzzy numbers based on the right most point of the α- cut for a given α.

AD_α(A) =a⁺_α. (27)

Center of maxima

This is calculated, (see i.e. Klir [14]), as the average value of the end points of the modal values interval by the formula

CoM(A) = a⁻₁ +a⁺₁

2 . (28)

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3 MATHEMATICAL CONCEPTS 23 Center of gravity

The center of gravity of a fuzzy number is obtained [54] using CoG(A) =

R∞

−∞xA(x)dx R∞

−∞A(x)dx , (29)

Median

The median value of a fuzzy number [4] and [9], generalizes the denition of median to fuzzy numbers by minimizing the following expression.

Z med(A)

−∞

A(x)dx− Z ∞

M ed(A)

A(x)dx

(30) Hence the median can be interpreted as the center of area (CoA) of a fuzzy number A as it divides the area under the membership function into two equal parts.

Credibilistic mean

This is based on four axiomatic properties [24] and is proved that the original denition is equivalent to the following formulation

Cr(B) = P os(B) +N ec(B)

2 , (31)

where B ⊂ R i.e., the credibility measure is the arithmetic mean of the possibility and necessity measures. By this concept, the credibility expectation of a fuzzy variable is dened as

CrM ean(A) = Z 0

−∞

Cr(A≥x)dx− Z ∞

0

Cr(A≤x)dx. (32)

Chang's method

This ranking method is based on the index C(A) =

Z

x∈suppA

xA(x)dx.

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3 MATHEMATICAL CONCEPTS 24 From above [42], it can be observed that

CoG(A) = C(A) R∞

−∞A(x)dx. (33)

Possibilistic mean

The possibilisitic mean value [19] of a fuzzy number A∈F is the weighted average of the middle points of the α-cuts of a fuzzy number A;

E_p(A) = Z 1

0

α(a⁻_∞+a⁺_α)dα. (34)

Yager's approaches

Four dierent ranking methods for fuzzy quantities in the unit interval are proposed by Yager[49], [50], [51]. These methods are represented by equations (35), (36), (37) and (38) below.

-

Y₁(A) = R1

0 g(x)A(x)dx R1

0 A(x)dx (35)

where g(x) measures the importance of x, can be seen as a generalization of the ranking based on the center of gravity.

-

Y₂(A) =

Z hgt(A) 0

M(A_α)dα, (36)

where hgt(A) = sup_x∈sup_AA(x) is the height of A and M is the mean value operator. This can be used for ranking fuzzy numbers with arbitrary support.

In this case, hgt(A) = 1 and M(Aα) = ^a⁻^α^+a₂ ⁺^α -

Y₃(A) = Z 1

0

|x−A(x)|dx, (37)

-

Y₄(A) = sup

x∈[0,1]

min(x, A(x)) (38)

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3 MATHEMATICAL CONCEPTS 25 Chen's method

This is dened (see i.e. Chen [8] ) using the concepts of fuzzy maximizing and minimizing sets:

A_max(x) =

x−x_min x_max−x_min

k

, A_min(x) =

x_max−x x_max−x_min

k

wherex_max = sup∪ⁿ_i=1supA_i andx_min = inf∪ⁿ_i=1supA_i and k >0is a real number.

The left and right utility of a fuzzy numberA_i are dened as follows:

L(A_i) = sup

x∈R

min(A_min(x), A_i(x)), R(A_i) = sup

x∈R

min(A_max(x), A_i(x)),

Hence the nal ranking index is obtained as CH^k(A_i) = 1

2(R(A_i) + 1−L(A_i)) (39)

Kerre's method

The ranking index [16] is based on the Hamming-distance of fuzzy numbers by determing the distance between A_i and max(A₁, . . . , A_n) :

K(Ai) = Z

x∈S

|Ai(x)−max(A1, . . . An)|dx, (40) where S =∪ⁿ_i=1supA_i.

3.7.2 Second type ranking methods

They generate fuzzy binary relations where by the methods are functions M :F × F → [0,1] where the value of the relation M(A_i, A_j)∈[0,1] is the degree to which A_i is greater than A_j. The fuzzy numbers for this type are ranked according to the following rule;

M(A_i, A_j)≥M(A_j, A_i)⇒A_i _M A_j (41) Examples of second type ranking methods compared by Matteo et al., [33] are discussed below.

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3 MATHEMATICAL CONCEPTS 26 Baas and Kwakernaak's method

With this method, the value of the relation P_BK(A_i, A_j) quanties the degree to which A_i is greater than A_j as follows:

P_BK(A_i, A_j) = sup

xi≥xj

min(A_i(x_i), A_j(x_j))

which leads to the ranking of the fuzzy number as BK(A_i) = min

j6=i P_BK(A_i, A_j). (42)

It is worth noting that P_BK coincides with the fuzzy relation PD introduced by Dubois and Prade [12]. It is important to mention that the rankings produced by the two methods can be dierent: Baas and Kwakernaak's approach is based on the minimum value of P_BK and Dubois and Prade's PD relation can be used according to the ordering to the ordering procedure described in [15],

Nakamura's method.

Here the parametric method, (see i.e. Nakamura [36]), is based on the fuzzy relation

P_Nλ(A_i, A_j) = λd_H(A_i,min(A_i, A_j)) + (1−λ)(d_H( ¯A_i,min( ¯A_i,A¯_j)))

λd_H(A_i, A_j) + (1−λ)(d_H( ¯A_i,A¯_j)) (43) with λ ∈ [0,1] and where d_H(A_i, A_j) = R

R|A_i(x)−A_j(x)|dx is the Hamming distance between two fuzzy numbers, A_i(x) = sup_y≤xA_i(y) and A¯_i(x) = sup_y≥xA_i(y). When λd_H(A_i, A_j) + (1−λ)(d_H( ¯A_i, A_j)) = 0, the value of the relation is dened as P_Nλ(A_i, A_j) = 0.5.

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4 SIMILARITY BASED TOPSIS

The original TOPSIS is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative ideal solution [28]. The aim is to maximize the benets and minimize the costs. Therefore TOPSIS is a muticriteria decision making technique which aim to nd feasible alternatives. In this study we will apply similarity based TOPSIS. The underlying idea in this method is that comparison is done by computing similarity between alternatives and ideal solutions. These alternatives should have highest similarity to positive ideal solution and lowest to negative ideal solutions.

Similarity for two elements x1 ∈ [0,1] and x2 ∈ [0,1] can be computed using the formular:

S(x₁, x₂) = p^p

1− |x^p₁−x^p₂| (44) For the case of two vectors,x₁ ∈[0,1]ⁿ and x₂ ∈[0,1]ⁿ similarity can be calculated as

S(x₁,x₂) = 1 n

n

X

i=1

w_i ^p q

1− |(x₁(i))^p−(x₂(i))^p| (45)

The procedure of similarity based TOPSIS starts from the construction of an eval- uation matrix X = [x_ij], where x_ij denotes the score of the ith alternative, with respect to the jth criterion, and can be summarized in the following steps;

Step I: Calculation of normalized, decision matrix R r_ij =

x_ij − |min

i (x_ij)|

maxi (x_ij)−min

i (x_ij) (46)

i= 1, . . . m, j = 1, . . . n

Step II: Calculation of weighted normalized decision matrix V = [v_ij]

v_ij =r_ij(.)w_j j = 1, . . . , m, i= 1, . . . , n. (47) Step III: Determine positive and negative ideal solutions A⁺ and A⁻

A⁺={v₁⁺, . . . , v⁺_m}={(max

j v_ij|j ∈B),(min

j v_ij|j ∈C)}

A⁻ ={v⁻₁, . . . , v⁻_n}={(min

j v_ij|j ∈B),(max

j v_ij|j ∈C)} (48) Where B is for benet criteria, and C is for cost/non-benet criteria.

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4 SIMILARITY BASED TOPSIS 28 Step IV: Calculation of the similarities of each alternative from positive ideal

solution and negative ideal solution.

S_i⁺= 1 n

m

X

j=1

p

q

1− |(v_ij)^p −(v⁺_j ) ^p| i= 1, . . . , n

S_i⁻= 1 n

m

X

j=1

p

q

1− |(v_ij)^p −(v⁻_j ) ^p| i= 1, . . . , n (49) Step V: Calculation of the relative closeness to Ideal solutions.

CCi = S_i⁺

S_i⁺+S_i⁻, i= 1, . . . , n (50) From the above steps, we are able to achieve important aspects of the similarity based approach and they are discribed as follows;

• The closer the CCi is to 1 implies the higher priority of the ith alternative.

Hence the alternative with highest value in interval [0,1] will be selected.

• From equation (49), the parameter p is used to set the strength of the similarity. The higher, the parameter p value, the higher the similarity degrees.

• In step 1, normalization is done to ensure all elementsrij are between0and 1 which is required in step 4.

• We apply closeness coecient designed for similarity measure in step 5.

Example 4.1

Let x = [0.3,0.8,0.9,0.3,0.5] and y = [0.35,0.9,1,0.5,0.65]. Calculate similarity S(x, y)using p values p= [1,2,3].

Solution:

We calculateS(x, y) using the formular S(x, y) = 1

n

X

i=1

pp

1− |(x_i)^p−(y_i)^p| i= 1, . . . , n (51) The results are displayed in table

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P−values S(x, y)

1 0.88

2 0.9276

3 0.9460

Table 3: p-values with corresponding similarities

From above results, we conclude that there is similarity between x and y since results for all p = 1, p = 2 and p = 3 are all very close to 1. Indeed vectors x and y are highly similar.

If we apply higher values of ofpi.e. p= [1,2,3, . . .10] to vectorsxand yto observe similarity between them for the Example 4.1, we nd similar trend as seen in gure 3 below.

Figure 3: Similarity between x and y

Generally, we can see that values ofS(x, y) are closer to1 as we increasep−values from 1upto 10. Therefore the vectors xand y are similar with high values of p.

Example 4.2

Five companies A1, A2, A3, A4 and A5 are to be evaluated, using four criteria C1, C2, C3 and C4. Let us consider data in table 4 below:

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Valuations A1 A2 A3 A4 A5

Sales 1059 1109,7 215,7 218,3 83,5

EPS 19,1 12,1 3,76 2,57 0,13

B 122,3 118,08 24,05 29,37 15,62

P 154,00 240,00 45,58 46,50 13,20

EBIT 22,94 14,42 34,46 34,48 87,11

EV 189,59 200.28 174,92 192,57 551,33

Table 4: Sample data The criteria are calculated as follows;

C1 = B P C₂ = EP S

P C₃ = Sales

P C₄ = Ebit

EV

Assume C1, C2, C3 and C4 are all benets then form decision matrix as shown in table 5 below.

We note that if we had calculated them as ;

C1 = P B

C₂ = P

EPS

C₃ = P

Sales C₄ = EV

Ebit Then they would be handled as cost criterias.

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C1 C2 C3 C4

A1 0.7942 0.1240 6.8766 0.1210 A2 0.4920 0.0504 4.6238 0.0720 A3 0.5344 0.0836 4.7933 0.1970 A4 0.6316 0.0551 4.6946 0.1790 A5 1.1833 0.0098 6.3258 0.1580

Table 5: Decision matrix

We now assume the weight vector for decision makers to be W = [1 1 1 1]

Step I Calculation of normalized, decision matrix R

rij =

x_ij − |min

i (x_ij)|

maxi (x_ij)−min

i (x_ij) i= 1, . . . m, j = 1, . . . n From the formula, we obtain in matrix form the following results:

D=







0.4371 1.0000 1.0000 0.3920 0.0000 0.3555 0.0000 0.0000 0.0613 0.6462 0.0752 1.0000 0.2019 0.3967 0.0314 0.8560 1.0000 0.0000 0.7555 0.6880







Step II Calculation of weighted normalized decision matrix V = [v_ij] v_ij =r_ij(.)w_j j = 1, . . . , m, i= 1, . . . , n.

and given that W = [1,1,1,1], we obtain same decision matrix as in step I.

Step III Determine positive and negative ideal solutions A⁺ and A⁻ A⁺ = {v₁⁺, . . . , v_m⁺}={(max

j v_ij|j ∈B),(min

j v_ij|j ∈C)}

A⁻ = {v₁⁻, . . . , v_n⁻}={(min

j v_ij|j ∈B),(max

j v_ij|j ∈C)}

Where B is for benet criteria, and C is for cost/non-benet criteria.

The positive and negative ideal solutions are in table 6 in which all criteria are assumed to be benets.

Similarity based TOPSIS applied to equity portfolio

Acknowledgements

Contents

List of Tables

List of Figures

1 INTRODUCTION

2 RESEARCH PROBLEM

3 MATHEMATICAL CONCEPTS

3.1 Fuzzy sets and crisp sets

3.2 Properties of fuzzy sets

3.3 Operations on fuzzy sets

3.4 Fuzzy numbers

Operations of fuzzy numbers

3.5 Fuzzy relations and equivalence

3.6 Fuzzy Decision Making

3.7 Fuzzy ranking methods

Hamming distance on the set R of all fuzzy numbers.

4 SIMILARITY BASED TOPSIS