Research ArticleStock Selection into Portfolio by Fuzzy Quantitative Analysisand Fuzzy Multicriteria Decision Making

Satit Yodmun and Wichai Witayakiattilerd

Department of Mathematics Faculty of Science King Mongkutrsquos Institute of Technology Ladkrabang Bangkok 10520 Thailand

Correspondence should be addressed to Wichai Witayakiattilerd wichaiwikmitlacth

Received 28 November 2015 Revised 26 February 2016 Accepted 7 April 2016

Academic Editor Igor L Averbakh

Copyright copy 2016 S Yodmun and W Witayakiattilerd This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper presents a stock selection approach assisted by fuzzy procedures In this approach stocks are classified into groupsaccording to business types Within each group the stocks are screened and then ranked according to their investment weightobtained from fuzzy quantitative analysis Groups were also ranked according to their group weight obtained from fuzzy analytichierarchy process (FAHP) and technique for order preference by similarity to ideal solution method (TOPSIS) The overall weightfor each stock was then derived from both of these weights and used for selecting a stock into the portfolio As a demonstrationour analysis procedures were applied to a test set of data

1 Introduction

Presently investors are more interested in investing in stocksand bonds than keeping their money in the bank because ityields a higher returnHowever this higher return also comeswith higher risk investors may lose some of their investmentget a lower-than-expected return or get a lower return thanthat from another type of investmentTherefore they have toanalyze a stock carefully before investing in it

In addition to several established approaches to stockanalysismdashsuch as quantitative fundamental analysis tech-nical analysis and stochastic analysismdashnew analytical toolshave been developed and widely used including ones that arebased on Brownian movement fuzzy logic and the analytichierarchy process

The analytic hierarchy process (AHP) is a multicriteriadecision-making approach and is a structured techniquefor organizing and analyzing complex decisions based onmathematics and psychology It was developed by Saaty inthe 1970s to help one make decision when one is facedwith the mixture of qualitative quantitative and sometimesconflicting factors that are taken into consideration AHP hasbeen very effective in making complicated often irreversible

decisions It has been extensively studied and refined sincethen (eg [1ndash11] and references therein)

Fuzzy sets and fuzzy logic especially are of wide interesttoday They are effective tools for modeling in the absenceof complete and precise information complex businessfinance and management systemsThe subjective judgementof experts who have used fuzzy logic techniques producesbetter results than the objective manipulation of inexactdata The concept of a fuzzy set is a reflection of realityreflection which serves as a point of departure for thedevelopment of theories which have the capability to modelthe pervasive imprecision and uncertainty of the real worldAs applied to stock analysis (eg [12ndash15] and referencestherein) fuzzy logic uses integrated experiential knowledgeof human experts to make better quantitative estimates notpossible with classical logic based on robust mathematicalprinciples

By reason of vagueness of boundaries of stock datain future and the attendant imprecision uncertainty andpreference of decision makers therefore fuzzy logic andAHP seem suitable for this problem This paper proposes anapproach to stock analysis based on calculated weights fromfuzzy quantitative analysis and fuzzy multicriteria decision

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016 Article ID 9530425 14 pageshttpdxdoiorg10115520169530425

2 Advances in Operations Research

making The idea of using fuzzy quantitative analysis andfuzzymulticriteria decisionmaking to imply final investmentweights for the stock selection into portfolio is different fromthe previous works The practicality of the approach wasdemonstrated by an application to a test set of data

2 Preliminaries

21 Fuzzy Logic Application and Definitions Fuzzy logicwas introduced by Zadeh [16] and has been widely appliedto problems in various fields of study Many researchersused fuzzy logic in stock market analysis (eg [12ndash15]) anddecision making (eg [1ndash4 6 7 9ndash15 17]) In this study weuse fuzzy logic in both stock market analysis and decisionmaking

In this subsection definitions of the fuzzy logic terms andconcepts used in this study are described below

Definition 1 Given a crisp set 119860 of a universe U a fuzzy set on 119860 is defined as

= (119909 119906 (119909)) | 119909 isin 119860 where 119906 (119909) isin [0 1] (1)

and 119906 is a membership function

Definition 2 Given a fuzzy set an 120572-cut set denoted by[]120572 for all 120572 isin [0 1] is defined as

[]120572

=

119909 isin 119860 | 119906 (119909) ge 120572 0 lt 120572 le 1

119909 isin 119860 | 119906 (119909) gt 0 120572 = 0

(2)

Definition 3 Let be a fuzzy set under the membership 119906 Rrarr [0 1] and is a fuzzy number if it satisfies the followingconditions

(1) is a normal fuzzy set that is exist119909 isin R 119906(119909) = 1

(2) is a convex fuzzy set that isforall120582 isin [0 1]forall1199091 1199092isin R

119906(1205821199091+ (1 minus 120582)119909

2) ge min119906(119909

1) 119906(1199092)

(3) For every 120572 isin [0 1] []120572 = [119886 119887] for some closedinterval [119886 119887]

Given an RF fuzzy number space condition (3) ofDefinition 3 ensures that every isin RF can be represented bya closed interval []120572 = [119906(120572) 119906(120572)] where 119906 119906 [0 1] rarr R

are functions that satisfy the following conditions

(1) 119906 is a bounded left continuous and nondecreasingfunction on [0 1]

(2) 119906 is a bounded right continuous and no-increasingfunction on [0 1]

(3) 119906(120572) le 119906(120572) for all 120572 isin [0 1]

Definition 4 = [119906(120572) 119906(120572)] is a positive fuzzy number thatcan be represented by the expression gt 0 if 119906(0) gt 0

Definition 5 Given 119886119871 le 1198861198721 le 1198861198722 le 119886119880 a trapezoidal fuzzynumber is a fuzzy number whose membership function119911(119909) is defined by

119911 (119909) =

119909 minus 119886119871

1198861198721 minus 119886119871 119886119871

le 119909 le 1198861198721

1 1198861198721 le 119909 le 119886

1198722

119909 minus 119886119880

1198861198722 minus 119886119880 1198861198722 le 119909 le 119886

119880

0 otherwise

(3)

and represented by the expression = ⟨119886119871 1198861198721 1198861198722 119886119880⟩

Definition 6 A trapezoidal fuzzy number =

⟨119886119871

119886119872

119886119872

119886119880

⟩ is called a triangular fuzzy numberand expressed as = ⟨119886119871 119886119872 119886119880⟩

Note For any real number 119886 119886 = ⟨119886 119886 119886⟩ = ⟨119886 119886 119886 119886⟩

Definition 7 Given any two positive fuzzy numbers =⟨119886119871

1198861198721 1198861198722 119886119880

⟩ and = ⟨119887119871

1198871198721 1198871198722 119887119880

⟩ and a realpositive number 119901 isin R+ operations oplus ⊖ otimes and ⊘ between and 119887 and an operation ⊙ between and 119901 are defined asfollows

oplus = ⟨119886119871

+ 119887119871

1198861198721 + 1198871198721 1198861198722 + 1198871198722 119886119880

+ 119887119880

⟩

⊖ = ⟨119886119871

minus 119887119880

1198861198721 minus 1198871198722 1198861198722 minus 1198871198721 119886119880

minus 119887119871

⟩

otimes = ⟨119886119871

119887119871

11988611987211198871198721 11988611987221198871198722 119886119880

119887119880

⟩

119901 ⊙ = ⟨119901119886119871

1199011198861198721 1199011198861198722 119901119886119880

⟩

⊘ = ⟨119886119871

1198871198801198861198721

11988711987221198861198722

1198871198721119886119880

119887119871⟩

(4)

Definition 8 Given two trapezoidal fuzzy numbers =

⟨119886119871

1198861198721 1198861198722 119886119880

⟩ and = ⟨119887119871 1198871198721 1198871198722 119887119880⟩ the distancebetween and represented by the symbol 119889( ) is definedas119889 ( )

= radic1

4[(119886119871 minus 119887119871)

2

+ (1198861198721 minus 1198871198721)2

+ (1198861198722 minus 1198871198722)2

+ (119886119880 minus 119887119880)2

]

(5)

For convenience 119868119899= 1 2 119899 is defined for further

use in this paper

Definition 9 = (119894119895)119898times119899

is a fuzzy matrix if 119894119895are fuzzy

numbers for all 119894 isin 119868119898and 119895 isin 119868

119899

Definition 10 = (119894119895)119899times1

is a fuzzy vector when all 119894=

⟨119898119871

119894 1198981198721

119894 1198981198722

119894 119898119880

119894⟩ 119894 isin 119868

119899 are trapezoidal fuzzy numbers

The aggregation of represented by agg is defined as

agg

= ⟨

119899

min119894=1

119898119871

119894 1

119899

119899

sum

119894=1

1198981198721

1198941

119899

119899

sum

119894=1

1198981198722

119894119899max119894=1

119898119880

119894⟩

(6)

Advances in Operations Research 3

22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]

Definition 11 Let 119860 = (119886119894119895)119899times119899

be an 119899 times 119899matrix where 119886119894119895gt

0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886

119895119894= 1119886119894119895for

all 119894 119895 isin 119868119899

Definition 12 Let119860 = (119886119894119895)119899times119899

be an 119899times119899matrix where 119886119894119895gt 0

for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist

weight vectors 119908 = (119908119894)119899times1

119908119894gt 0 for all 119894 isin 119868

119899 where

sum119899

119894=1119908119894= 1 and 119886

119894119895= 119908119894119908119895for all 119894 119895 isin 119868

119899

Definition 13 Let = (119894119895)119899times119899

be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868

119899and is a reciprocal

fuzzy matrix if 119895119894= 1 ⊘

119894119895for all 119894 119895 isin 119868

119899

In particular if every member of = (119894119895)119899times119899

is atriangular fuzzy number

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ is a reciprocal

fuzzy matrix if 119895119894= ⟨1119886

119880

119894119895 1119886119872

119894119895 1119886119871

119894119895⟩ for all 119894 119895 isin 119868

119899

Definition 14 Let = (119894119895)119899times119899

be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868

119899and is a consistency

fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868

119899

and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899

is a consistencymatrix that is there exist119908120572 = (119908120572

119894)119899times1

119908120572119894gt 0 for all 119894 isin 119868

119899

where sum119899119894=1119908120572

119894= 1 and 119886120572

119894119895= 119908120572

119894119908120572

119895for all 119894 119895 isin 119868

119899

According to Definition 14 since 119908120572119894gt 0 for all 119894 isin

119868119899 there exist fuzzy vectors = (

119894)119899times1

where 119908120572119894isin

[119908119894(120572) 119908

119894(120572)] gt 0 for all 119894 isin 119868

119899 These vectors are called fuzzy

weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a

fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (

119894)119899times1

for a fuzzy reciprocal matrix = (

119894119895)119899times119899

where 119894119895=⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ for all 119894 119895 isin 119868

119899by

using the method of geometric mean 119896= ⟨119908

119871

119896 119908119872

119896 119908119880

119896⟩

are defined for all 119896 isin 119868119899 where

119908119871

119896= 119862119871sdot

(prod119899

119895=1119886119871

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119872

119896=

(prod119899

119895=1119886119872

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119880

119896= 119862119880sdot

(prod119899

119895=1119886119880

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

(7)

119862119871= min119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119871

119894119895)1119899

119862119880= max119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119880

119894119895)1119899

(8)

In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows

Definition 15 Let = (119894119895)119899times119899

be a fuzzy reciprocal matrixof which

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ are triangular fuzzy numbers

evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590

119899() is defined as

119868120590

119899() = 119862

120590

119899sdotmax119894119895

max1003816100381610038161003816100381610038161003816100381610038161003816

119908119871

119894

119908119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119872

119894

119908119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119880

119894

119908119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(9)

where = (119894)119899times1

are fuzzy weight vectors and 119894=

⟨119908119871

119894 119908119872

119894 119908119880

119894⟩ for all 119894 isin 119868

119899as expressed in (7) and

119862120590

119899=

1

max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (

119899

2)

119899(119899minus2)

1

max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (

119899

2)

119899(119899minus2)

(10)

If the consistency index 119868120590119899() = 0 the fuzzy reciprocal

fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally

an acceptable value is 119868120590119899() lt 01 or 10

Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590

119899() le 1

23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management

The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock

4 Advances in Operations Research

The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875

119899Ratio which are defined as

follows

Definition 17 Let 1198991 1198992 and 119899

3be the number of common

stock preferred stock and treasury stock respectively 119875119905

current price per share and 119864119903119903th-quarter net profit price to

earnings ratio or 119875119864 is defined as

119875

119864=119875119905(1198991+ 1198992minus 1198993)

119864119903

(11)

119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for

Definition 18 Let 119899 be the number of be the number ofregistered share 119860

119905and 119877

119905the asset and liability of the

company respectively and 119875119905current price per share price to

book value ratio or 119875119861119881 is defined as

119875

BV=119875119905

119861119905

(12)

where 119861119905= (119860119905minus 119877119905)119899

119875BV denotes how many times the current stock price iscompared to its account value

Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th

year-end dividend per share 119896 isin 119868119899 and 119875

0the 119899th-quarter

historical price the current target price 119875119899is defined as

119875119899= 1198750(1 + 119903)

119899

minus

119899

sum

119896=1

119863119896(1 + 119903)

119899minus119896

(13)

Definition 20 Let 119875119899be the current target price and 119875 the

current stock price 119875119875119899is called price per target price ratio

represented by the symbol 119875119875119899

119875119875119899denotes how many times the current stock price is

compared to the current target price

3 Stock Selection Procedure

This section presents the proposed stock selection procedurewhich is done in the following 3 main steps

Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock

Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group

Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market

31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for

each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions

(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group

(2) More recent data reflect current trend better thanearlier ones

(3) Fuzzy rules are flexible and depend on expert infor-mation

The specific steps of the fuzzy analysis are as follows

Step 11 This step involves screening in only 119898 individualstocks (119878

1 1198782 119878

119898) in the same industrial group of which

sufficient financial data are provided for calculating 119875119864119875BV and 119875119875

119899of 119899 earlier years up to the present

Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894

119896)

and (119875119875119899)(119878119894

119896) for all 119894 isin 119868

119899and 119896 isin 119868

119898 where 119878119894

119896denotes the

119896th stock in the 119894th year

Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878

119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896)

119896 isin 119868119898 from the following equations

(119864

119875)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119864

119875) (119878119894

119896)

(119875

BV)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

BV) (119878119894

119896)

(119875

119875119899

)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

119875119899

) (119878119894

119896)

where 119908119894=

2119894

119899 (119899 + 1) 119894 isin 119868

119898

(14)

Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875

119899and a fuzzy set119882 of the investment weights from

(119864119875)119908

(119878119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896) 119896 isin 119868

119898

Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows

Advances in Operations Research 5

Rule-1 if 1199091is 11and 119909

2is 12and 119909

3is 13then 119910 is

1

Rule-2 if 1199091is 21and 119909

2is 22and 119909

3is 23then 119910 is

2

Rule-119902 if 1199091is 1199021and 119909

2is 1199022and 119909

3is 1199023then 119910 is

119902

1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875

119899

and 1198821 respectively and

1198961 1198962 and

1198963 119896 isin 119868

119902 are

linguistic terms of 119864119875 119875BV 119875119875119899 and 119882

1 respectively

that is 119864119875 = 11 21

1199021 119875BV =

12 22

1199022

119875119875119899= 13 23

1199023 and119882 =

1 2

119902

Step 16This step involves importing 119864119875 119875BV and 119875119875119899of

the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861

Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908

119911= 119906B(119911) int

119861

119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int

119861

119911119908119911119889119911 =

int119861

119911119906B(119911) 119889119911 int119861

119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group

32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)

AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω

119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus

1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω

119899is replaced by

a set of fuzzy preference values Ω120575119899= 1

120575 1(119899 minus 1)

13120575 12120575 1 2120575 3120575 (119899 minus 1)

120575 120575 where

120575= ⟨119896 minus 120575 119896 119896 +

120575⟩ and 1120575= 1⊘

120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868

119899

and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]

and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers

Applying this technique to 1198991decision makers 119899

2deci-

sion criteria and 1198993industrial groups as alternate choices the

analysis steps are as follows

Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899

1 is compared to another

decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as

120593 (119894 119895) =

119894119895 exist119894119895isin Ω119899 119895 gt 119894

1 119895 = 119894

1 ⊘ 120593 (119895 119894) 119895 lt 119894

(15)

The decision makerrsquos preference matrix = (119894119895)1198991times1198991

is areciprocal matrix where

119894119895=

120593 (119894 119895) 119894 lt 119895

1 119894 = 119895

1 ⊘ 120593 (119895 119894) 119894 gt 119895

(16)

Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =

(119894119895)1198991times1198991

) 119889119896= ⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ is a fuzzyweight vector for

all 119896 isin 1198681198991

where

119908119871

119889119896= 119862119871sdot

(prod1198991

119895=1119886119871

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119872

119889119896=

(prod1198991

119895=1119886119872

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119880

119889119896= 119862119880sdot

(prod1198991

119895=1119886119880

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

(17)

with

119862119871= min119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119871

119894119895)11198991

119862119880= max119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119880

119894119895)11198991

(18)

If its consistency index 1198681205901198991

() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21

Step 23 This step involves decision makers 1198891 1198892

1198891198991

constructing decision criteria 1198881 1198882 119888

1198992

for evaluatingindustrial groups 119866

1 1198662 119866

1198993

where 119888119894 119894 = 1 119899

2 is

constructed from investment weight of 1198993individual groups

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Operations Research

making The idea of using fuzzy quantitative analysis andfuzzymulticriteria decisionmaking to imply final investmentweights for the stock selection into portfolio is different fromthe previous works The practicality of the approach wasdemonstrated by an application to a test set of data

2 Preliminaries

21 Fuzzy Logic Application and Definitions Fuzzy logicwas introduced by Zadeh [16] and has been widely appliedto problems in various fields of study Many researchersused fuzzy logic in stock market analysis (eg [12ndash15]) anddecision making (eg [1ndash4 6 7 9ndash15 17]) In this study weuse fuzzy logic in both stock market analysis and decisionmaking

In this subsection definitions of the fuzzy logic terms andconcepts used in this study are described below

Definition 1 Given a crisp set 119860 of a universe U a fuzzy set on 119860 is defined as

= (119909 119906 (119909)) | 119909 isin 119860 where 119906 (119909) isin [0 1] (1)

and 119906 is a membership function

Definition 2 Given a fuzzy set an 120572-cut set denoted by[]120572 for all 120572 isin [0 1] is defined as

[]120572

=

119909 isin 119860 | 119906 (119909) ge 120572 0 lt 120572 le 1

119909 isin 119860 | 119906 (119909) gt 0 120572 = 0

(2)

Definition 3 Let be a fuzzy set under the membership 119906 Rrarr [0 1] and is a fuzzy number if it satisfies the followingconditions

(1) is a normal fuzzy set that is exist119909 isin R 119906(119909) = 1

(2) is a convex fuzzy set that isforall120582 isin [0 1]forall1199091 1199092isin R

119906(1205821199091+ (1 minus 120582)119909

2) ge min119906(119909

1) 119906(1199092)

(3) For every 120572 isin [0 1] []120572 = [119886 119887] for some closedinterval [119886 119887]

Given an RF fuzzy number space condition (3) ofDefinition 3 ensures that every isin RF can be represented bya closed interval []120572 = [119906(120572) 119906(120572)] where 119906 119906 [0 1] rarr R

are functions that satisfy the following conditions

(1) 119906 is a bounded left continuous and nondecreasingfunction on [0 1]

(2) 119906 is a bounded right continuous and no-increasingfunction on [0 1]

(3) 119906(120572) le 119906(120572) for all 120572 isin [0 1]

Definition 4 = [119906(120572) 119906(120572)] is a positive fuzzy number thatcan be represented by the expression gt 0 if 119906(0) gt 0

Definition 5 Given 119886119871 le 1198861198721 le 1198861198722 le 119886119880 a trapezoidal fuzzynumber is a fuzzy number whose membership function119911(119909) is defined by

119911 (119909) =

119909 minus 119886119871

1198861198721 minus 119886119871 119886119871

le 119909 le 1198861198721

1 1198861198721 le 119909 le 119886

1198722

119909 minus 119886119880

1198861198722 minus 119886119880 1198861198722 le 119909 le 119886

119880

0 otherwise

(3)

and represented by the expression = ⟨119886119871 1198861198721 1198861198722 119886119880⟩

Definition 6 A trapezoidal fuzzy number =

⟨119886119871

119886119872

119886119872

119886119880

⟩ is called a triangular fuzzy numberand expressed as = ⟨119886119871 119886119872 119886119880⟩

Note For any real number 119886 119886 = ⟨119886 119886 119886⟩ = ⟨119886 119886 119886 119886⟩

Definition 7 Given any two positive fuzzy numbers =⟨119886119871

1198861198721 1198861198722 119886119880

⟩ and = ⟨119887119871

1198871198721 1198871198722 119887119880

⟩ and a realpositive number 119901 isin R+ operations oplus ⊖ otimes and ⊘ between and 119887 and an operation ⊙ between and 119901 are defined asfollows

oplus = ⟨119886119871

+ 119887119871

1198861198721 + 1198871198721 1198861198722 + 1198871198722 119886119880

+ 119887119880

⟩

⊖ = ⟨119886119871

minus 119887119880

1198861198721 minus 1198871198722 1198861198722 minus 1198871198721 119886119880

minus 119887119871

⟩

otimes = ⟨119886119871

119887119871

11988611987211198871198721 11988611987221198871198722 119886119880

119887119880

⟩

119901 ⊙ = ⟨119901119886119871

1199011198861198721 1199011198861198722 119901119886119880

⟩

⊘ = ⟨119886119871

1198871198801198861198721

11988711987221198861198722

1198871198721119886119880

119887119871⟩

(4)

Definition 8 Given two trapezoidal fuzzy numbers =

⟨119886119871

1198861198721 1198861198722 119886119880

⟩ and = ⟨119887119871 1198871198721 1198871198722 119887119880⟩ the distancebetween and represented by the symbol 119889( ) is definedas119889 ( )

= radic1

4[(119886119871 minus 119887119871)

2

+ (1198861198721 minus 1198871198721)2

+ (1198861198722 minus 1198871198722)2

+ (119886119880 minus 119887119880)2

]

(5)

For convenience 119868119899= 1 2 119899 is defined for further

use in this paper

Definition 9 = (119894119895)119898times119899

is a fuzzy matrix if 119894119895are fuzzy

numbers for all 119894 isin 119868119898and 119895 isin 119868

119899

Definition 10 = (119894119895)119899times1

is a fuzzy vector when all 119894=

⟨119898119871

119894 1198981198721

119894 1198981198722

119894 119898119880

119894⟩ 119894 isin 119868

119899 are trapezoidal fuzzy numbers

The aggregation of represented by agg is defined as

agg

= ⟨

119899

min119894=1

119898119871

119894 1

119899

119899

sum

119894=1

1198981198721

1198941

119899

119899

sum

119894=1

1198981198722

119894119899max119894=1

119898119880

119894⟩

(6)

Advances in Operations Research 3

22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]

Definition 11 Let 119860 = (119886119894119895)119899times119899

be an 119899 times 119899matrix where 119886119894119895gt

0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886

119895119894= 1119886119894119895for

all 119894 119895 isin 119868119899

Definition 12 Let119860 = (119886119894119895)119899times119899

be an 119899times119899matrix where 119886119894119895gt 0

for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist

weight vectors 119908 = (119908119894)119899times1

119908119894gt 0 for all 119894 isin 119868

119899 where

sum119899

119894=1119908119894= 1 and 119886

119894119895= 119908119894119908119895for all 119894 119895 isin 119868

119899

Definition 13 Let = (119894119895)119899times119899

be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868

119899and is a reciprocal

fuzzy matrix if 119895119894= 1 ⊘

119894119895for all 119894 119895 isin 119868

119899

In particular if every member of = (119894119895)119899times119899

is atriangular fuzzy number

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ is a reciprocal

fuzzy matrix if 119895119894= ⟨1119886

119880

119894119895 1119886119872

119894119895 1119886119871

119894119895⟩ for all 119894 119895 isin 119868

119899

Definition 14 Let = (119894119895)119899times119899

be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868

119899and is a consistency

fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868

119899

and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899

is a consistencymatrix that is there exist119908120572 = (119908120572

119894)119899times1

119908120572119894gt 0 for all 119894 isin 119868

119899

where sum119899119894=1119908120572

119894= 1 and 119886120572

119894119895= 119908120572

119894119908120572

119895for all 119894 119895 isin 119868

119899

According to Definition 14 since 119908120572119894gt 0 for all 119894 isin

119868119899 there exist fuzzy vectors = (

119894)119899times1

where 119908120572119894isin

[119908119894(120572) 119908

119894(120572)] gt 0 for all 119894 isin 119868

119899 These vectors are called fuzzy

weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a

fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (

119894)119899times1

for a fuzzy reciprocal matrix = (

119894119895)119899times119899

where 119894119895=⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ for all 119894 119895 isin 119868

119899by

using the method of geometric mean 119896= ⟨119908

119871

119896 119908119872

119896 119908119880

119896⟩

are defined for all 119896 isin 119868119899 where

119908119871

119896= 119862119871sdot

(prod119899

119895=1119886119871

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119872

119896=

(prod119899

119895=1119886119872

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119880

119896= 119862119880sdot

(prod119899

119895=1119886119880

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

(7)

119862119871= min119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119871

119894119895)1119899

119862119880= max119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119880

119894119895)1119899

(8)

In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows

Definition 15 Let = (119894119895)119899times119899

be a fuzzy reciprocal matrixof which

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ are triangular fuzzy numbers

evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590

119899() is defined as

119868120590

119899() = 119862

120590

119899sdotmax119894119895

max1003816100381610038161003816100381610038161003816100381610038161003816

119908119871

119894

119908119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119872

119894

119908119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119880

119894

119908119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(9)

where = (119894)119899times1

are fuzzy weight vectors and 119894=

⟨119908119871

119894 119908119872

119894 119908119880

119894⟩ for all 119894 isin 119868

119899as expressed in (7) and

119862120590

119899=

1

max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (

119899

2)

119899(119899minus2)

1

max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (

119899

2)

119899(119899minus2)

(10)

If the consistency index 119868120590119899() = 0 the fuzzy reciprocal

fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally

an acceptable value is 119868120590119899() lt 01 or 10

Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590

119899() le 1

23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management

The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock

4 Advances in Operations Research

The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875

119899Ratio which are defined as

follows

Definition 17 Let 1198991 1198992 and 119899

3be the number of common

stock preferred stock and treasury stock respectively 119875119905

current price per share and 119864119903119903th-quarter net profit price to

earnings ratio or 119875119864 is defined as

119875

119864=119875119905(1198991+ 1198992minus 1198993)

119864119903

(11)

119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for

Definition 18 Let 119899 be the number of be the number ofregistered share 119860

119905and 119877

119905the asset and liability of the

company respectively and 119875119905current price per share price to

book value ratio or 119875119861119881 is defined as

119875

BV=119875119905

119861119905

(12)

where 119861119905= (119860119905minus 119877119905)119899

119875BV denotes how many times the current stock price iscompared to its account value

Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th

year-end dividend per share 119896 isin 119868119899 and 119875

0the 119899th-quarter

historical price the current target price 119875119899is defined as

119875119899= 1198750(1 + 119903)

119899

minus

119899

sum

119896=1

119863119896(1 + 119903)

119899minus119896

(13)

Definition 20 Let 119875119899be the current target price and 119875 the

current stock price 119875119875119899is called price per target price ratio

represented by the symbol 119875119875119899

119875119875119899denotes how many times the current stock price is

compared to the current target price

3 Stock Selection Procedure

This section presents the proposed stock selection procedurewhich is done in the following 3 main steps

Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock

Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group

Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market

31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for

each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions

(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group

(2) More recent data reflect current trend better thanearlier ones

(3) Fuzzy rules are flexible and depend on expert infor-mation

The specific steps of the fuzzy analysis are as follows

Step 11 This step involves screening in only 119898 individualstocks (119878

1 1198782 119878

119898) in the same industrial group of which

sufficient financial data are provided for calculating 119875119864119875BV and 119875119875

119899of 119899 earlier years up to the present

Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894

119896)

and (119875119875119899)(119878119894

119896) for all 119894 isin 119868

119899and 119896 isin 119868

119898 where 119878119894

119896denotes the

119896th stock in the 119894th year

Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878

119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896)

119896 isin 119868119898 from the following equations

(119864

119875)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119864

119875) (119878119894

119896)

(119875

BV)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

BV) (119878119894

119896)

(119875

119875119899

)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

119875119899

) (119878119894

119896)

where 119908119894=

2119894

119899 (119899 + 1) 119894 isin 119868

119898

(14)

Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875

119899and a fuzzy set119882 of the investment weights from

(119864119875)119908

(119878119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896) 119896 isin 119868

119898

Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows

Advances in Operations Research 5

Rule-1 if 1199091is 11and 119909

2is 12and 119909

3is 13then 119910 is

1

Rule-2 if 1199091is 21and 119909

2is 22and 119909

3is 23then 119910 is

2

Rule-119902 if 1199091is 1199021and 119909

2is 1199022and 119909

3is 1199023then 119910 is

119902

1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875

119899

and 1198821 respectively and

1198961 1198962 and

1198963 119896 isin 119868

119902 are

linguistic terms of 119864119875 119875BV 119875119875119899 and 119882

1 respectively

that is 119864119875 = 11 21

1199021 119875BV =

12 22

1199022

119875119875119899= 13 23

1199023 and119882 =

1 2

119902

Step 16This step involves importing 119864119875 119875BV and 119875119875119899of

the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861

Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908

119911= 119906B(119911) int

119861

119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int

119861

119911119908119911119889119911 =

int119861

119911119906B(119911) 119889119911 int119861

119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group

32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)

AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω

119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus

1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω

119899is replaced by

a set of fuzzy preference values Ω120575119899= 1

120575 1(119899 minus 1)

13120575 12120575 1 2120575 3120575 (119899 minus 1)

120575 120575 where

120575= ⟨119896 minus 120575 119896 119896 +

120575⟩ and 1120575= 1⊘

120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868

119899

and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]

and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers

Applying this technique to 1198991decision makers 119899

2deci-

sion criteria and 1198993industrial groups as alternate choices the

analysis steps are as follows

Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899

1 is compared to another

decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as

120593 (119894 119895) =

119894119895 exist119894119895isin Ω119899 119895 gt 119894

1 119895 = 119894

1 ⊘ 120593 (119895 119894) 119895 lt 119894

(15)

The decision makerrsquos preference matrix = (119894119895)1198991times1198991

is areciprocal matrix where

119894119895=

120593 (119894 119895) 119894 lt 119895

1 119894 = 119895

1 ⊘ 120593 (119895 119894) 119894 gt 119895

(16)

Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =

(119894119895)1198991times1198991

) 119889119896= ⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ is a fuzzyweight vector for

all 119896 isin 1198681198991

where

119908119871

119889119896= 119862119871sdot

(prod1198991

119895=1119886119871

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119872

119889119896=

(prod1198991

119895=1119886119872

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119880

119889119896= 119862119880sdot

(prod1198991

119895=1119886119880

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

(17)

with

119862119871= min119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119871

119894119895)11198991

119862119880= max119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119880

119894119895)11198991

(18)

If its consistency index 1198681205901198991

() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21

Step 23 This step involves decision makers 1198891 1198892

1198891198991

constructing decision criteria 1198881 1198882 119888

1198992

for evaluatingindustrial groups 119866

1 1198662 119866

1198993

where 119888119894 119894 = 1 119899

2 is

constructed from investment weight of 1198993individual groups

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 3

22 Consistency Fuzzy Matrix In this subsection we intro-duce the definition of consistency fuzzy matrix and consis-tency index which was developed by Ramik [3 4]

Definition 11 Let 119860 = (119886119894119895)119899times119899

be an 119899 times 119899matrix where 119886119894119895gt

0 for all 119894 119895 isin 119868119899and 119860 is a reciprocal matrix if 119886

119895119894= 1119886119894119895for

all 119894 119895 isin 119868119899

Definition 12 Let119860 = (119886119894119895)119899times119899

be an 119899times119899matrix where 119886119894119895gt 0

for all 119894 119895 isin 119868119899and 119860 is a consistency matrix if there exist

weight vectors 119908 = (119908119894)119899times1

119908119894gt 0 for all 119894 isin 119868

119899 where

sum119899

119894=1119908119894= 1 and 119886

119894119895= 119908119894119908119895for all 119894 119895 isin 119868

119899

Definition 13 Let = (119894119895)119899times119899

be an 119899times119899 fuzzy matrix where119894119895gt 0 are fuzzy numbers for all 119894 119895 isin 119868

119899and is a reciprocal

fuzzy matrix if 119895119894= 1 ⊘

119894119895for all 119894 119895 isin 119868

119899

In particular if every member of = (119894119895)119899times119899

is atriangular fuzzy number

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ is a reciprocal

fuzzy matrix if 119895119894= ⟨1119886

119880

119894119895 1119886119872

119894119895 1119886119871

119894119895⟩ for all 119894 119895 isin 119868

119899

Definition 14 Let = (119894119895)119899times119899

be an 119899times119899 fuzzymatrix where119894119895= [119886119894119895(120572) 119886119894119895(120572)] gt 0 for all 119894 119895 isin 119868

119899and is a consistency

fuzzy matrix if there exist 119886120572119894119895isin [119886119894119895(120572) 119886119894119895(120572)] for all 119894 119895 isin 119868

119899

and some 120572 isin [0 1] with which 119860 = (119886120572119894119895)119899times119899

is a consistencymatrix that is there exist119908120572 = (119908120572

119894)119899times1

119908120572119894gt 0 for all 119894 isin 119868

119899

where sum119899119894=1119908120572

119894= 1 and 119886120572

119894119895= 119908120572

119894119908120572

119895for all 119894 119895 isin 119868

119899

According to Definition 14 since 119908120572119894gt 0 for all 119894 isin

119868119899 there exist fuzzy vectors = (

119894)119899times1

where 119908120572119894isin

[119908119894(120572) 119908

119894(120572)] gt 0 for all 119894 isin 119868

119899 These vectors are called fuzzy

weight vectorsIt is clear that if is a fuzzy consistency matrix then it is a

fuzzy reciprocal fuzzymatrix and is not a fuzzy consistencymatrix if it is not a fuzzy reciprocal fuzzy matrix Becauseof these reasons construction of a fuzzy consistency matrixusually starts by first constructing a reciprocal fuzzy matrix Ramik and Korviny [4] proposed a method for calculatingfuzzy weight vector = (

119894)119899times1

for a fuzzy reciprocal matrix = (

119894119895)119899times119899

where 119894119895=⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ for all 119894 119895 isin 119868

119899by

using the method of geometric mean 119896= ⟨119908

119871

119896 119908119872

119896 119908119880

119896⟩

are defined for all 119896 isin 119868119899 where

119908119871

119896= 119862119871sdot

(prod119899

119895=1119886119871

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119872

119896=

(prod119899

119895=1119886119872

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

119908119880

119896= 119862119880sdot

(prod119899

119895=1119886119880

119896119895)1119899

sum119899

119894=1(prod119899

119895=1119886119872

119894119895)1119899

(7)

119862119871= min119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119871

119894119895)1119899

119862119880= max119894isin119868119899

(prod119899

119895=1119886119872

119894119895)1119899

(prod119899

119895=1119886119880

119894119895)1119899

(8)

In addition Ramik and Korviny [4] defined a consistencyindex for measuring the nearness of a fuzzy reciprocal matrixto the corresponding fuzzy consistency matrix as follows

Definition 15 Let = (119894119895)119899times119899

be a fuzzy reciprocal matrixof which

119894119895= ⟨119886119871

119894119895 119886119872

119894119895 119886119880

119894119895⟩ are triangular fuzzy numbers

evaluated from a scale 119878 = [1120590 120590] for some real number120590 gt 1 the consistency index of represented by the symbol119868120590

119899() is defined as

119868120590

119899() = 119862

120590

119899sdotmax119894119895

max1003816100381610038161003816100381610038161003816100381610038161003816

119908119871

119894

119908119880

119895

minus 119886119871

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119872

119894

119908119872

119895

minus 119886119872

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816

119908119880

119894

119908119871

119895

minus 119886119880

119894119895

1003816100381610038161003816100381610038161003816100381610038161003816

(9)

where = (119894)119899times1

are fuzzy weight vectors and 119894=

⟨119908119871

119894 119908119872

119894 119908119880

119894⟩ for all 119894 isin 119868

119899as expressed in (7) and

119862120590

119899=

1

max 120590 minus 120590(2minus2119899)119899 1205902 ((2119899)2(119899minus2) minus (2119899)(119899minus2)2) 120590 lt (

119899

2)

119899(119899minus2)

1

max 120590 minus 120590(2minus2119899)119899 120590(2minus2119899)119899 minus 120590 120590 ge (

119899

2)

119899(119899minus2)

(10)

If the consistency index 119868120590119899() = 0 the fuzzy reciprocal

fuzzy matrix is absolutely consistent The closer the valueof 119868120590119899() to 0 is the more consistent the matrix is Generally

an acceptable value is 119868120590119899() lt 01 or 10

Theorem 16 (see [4]) If is an 119899 times 119899 fuzzy reciprocal matrixwith triangular fuzzy elements evaluatedwith the scale [1120590 120590]for some 120590 gt 1 then 0 le 119868120590

119899() le 1

23 Financial Ratios A sustainable investment and missionrequires effective planning and financial management

The quantitative stock analysis is a useful tool that willimprove investmentrsquos understanding of financial results andtrends over time and provide key indicators of organizationalperformance Investormay use the quantitative stock analysisto pinpoint strengths and weaknesses of each company thatimpact to its stock

4 Advances in Operations Research

The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875

119899Ratio which are defined as

follows

Definition 17 Let 1198991 1198992 and 119899

3be the number of common

stock preferred stock and treasury stock respectively 119875119905

current price per share and 119864119903119903th-quarter net profit price to

earnings ratio or 119875119864 is defined as

119875

119864=119875119905(1198991+ 1198992minus 1198993)

119864119903

(11)

119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for

Definition 18 Let 119899 be the number of be the number ofregistered share 119860

119905and 119877

119905the asset and liability of the

company respectively and 119875119905current price per share price to

book value ratio or 119875119861119881 is defined as

119875

BV=119875119905

119861119905

(12)

where 119861119905= (119860119905minus 119877119905)119899

119875BV denotes how many times the current stock price iscompared to its account value

Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th

year-end dividend per share 119896 isin 119868119899 and 119875

0the 119899th-quarter

historical price the current target price 119875119899is defined as

119875119899= 1198750(1 + 119903)

119899

minus

119899

sum

119896=1

119863119896(1 + 119903)

119899minus119896

(13)

Definition 20 Let 119875119899be the current target price and 119875 the

current stock price 119875119875119899is called price per target price ratio

represented by the symbol 119875119875119899

119875119875119899denotes how many times the current stock price is

compared to the current target price

3 Stock Selection Procedure

This section presents the proposed stock selection procedurewhich is done in the following 3 main steps

Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock

Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group

Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market

31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for

each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions

(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group

(2) More recent data reflect current trend better thanearlier ones

(3) Fuzzy rules are flexible and depend on expert infor-mation

The specific steps of the fuzzy analysis are as follows

Step 11 This step involves screening in only 119898 individualstocks (119878

1 1198782 119878

119898) in the same industrial group of which

sufficient financial data are provided for calculating 119875119864119875BV and 119875119875

119899of 119899 earlier years up to the present

Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894

119896)

and (119875119875119899)(119878119894

119896) for all 119894 isin 119868

119899and 119896 isin 119868

119898 where 119878119894

119896denotes the

119896th stock in the 119894th year

Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878

119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896)

119896 isin 119868119898 from the following equations

(119864

119875)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119864

119875) (119878119894

119896)

(119875

BV)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

BV) (119878119894

119896)

(119875

119875119899

)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

119875119899

) (119878119894

119896)

where 119908119894=

2119894

119899 (119899 + 1) 119894 isin 119868

119898

(14)

Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875

119899and a fuzzy set119882 of the investment weights from

(119864119875)119908

(119878119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896) 119896 isin 119868

119898

Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows

Advances in Operations Research 5

Rule-1 if 1199091is 11and 119909

2is 12and 119909

3is 13then 119910 is

1

Rule-2 if 1199091is 21and 119909

2is 22and 119909

3is 23then 119910 is

2

Rule-119902 if 1199091is 1199021and 119909

2is 1199022and 119909

3is 1199023then 119910 is

119902

1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875

119899

and 1198821 respectively and

1198961 1198962 and

1198963 119896 isin 119868

119902 are

linguistic terms of 119864119875 119875BV 119875119875119899 and 119882

1 respectively

that is 119864119875 = 11 21

1199021 119875BV =

12 22

1199022

119875119875119899= 13 23

1199023 and119882 =

1 2

119902

Step 16This step involves importing 119864119875 119875BV and 119875119875119899of

the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861

Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908

119911= 119906B(119911) int

119861

119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int

119861

119911119908119911119889119911 =

int119861

119911119906B(119911) 119889119911 int119861

119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group

32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)

AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω

119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus

1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω

119899is replaced by

a set of fuzzy preference values Ω120575119899= 1

120575 1(119899 minus 1)

13120575 12120575 1 2120575 3120575 (119899 minus 1)

120575 120575 where

120575= ⟨119896 minus 120575 119896 119896 +

120575⟩ and 1120575= 1⊘

120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868

119899

and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]

and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers

Applying this technique to 1198991decision makers 119899

2deci-

sion criteria and 1198993industrial groups as alternate choices the

analysis steps are as follows

Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899

1 is compared to another

decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as

120593 (119894 119895) =

119894119895 exist119894119895isin Ω119899 119895 gt 119894

1 119895 = 119894

1 ⊘ 120593 (119895 119894) 119895 lt 119894

(15)

The decision makerrsquos preference matrix = (119894119895)1198991times1198991

is areciprocal matrix where

119894119895=

120593 (119894 119895) 119894 lt 119895

1 119894 = 119895

1 ⊘ 120593 (119895 119894) 119894 gt 119895

(16)

Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =

(119894119895)1198991times1198991

) 119889119896= ⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ is a fuzzyweight vector for

all 119896 isin 1198681198991

where

119908119871

119889119896= 119862119871sdot

(prod1198991

119895=1119886119871

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119872

119889119896=

(prod1198991

119895=1119886119872

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119880

119889119896= 119862119880sdot

(prod1198991

119895=1119886119880

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

(17)

with

119862119871= min119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119871

119894119895)11198991

119862119880= max119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119880

119894119895)11198991

(18)

If its consistency index 1198681205901198991

() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21

Step 23 This step involves decision makers 1198891 1198892

1198891198991

constructing decision criteria 1198881 1198882 119888

1198992

for evaluatingindustrial groups 119866

1 1198662 119866

1198993

where 119888119894 119894 = 1 119899

2 is

constructed from investment weight of 1198993individual groups

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Operations Research

The quantitative stock analysis presented in this study isbased on the following financial ratios price to earnings ratioor 119875119864 Ratio price to book value ratio or 119875BV Ratio andprice to intrinsic ratio or 119875119875

119899Ratio which are defined as

follows

Definition 17 Let 1198991 1198992 and 119899

3be the number of common

stock preferred stock and treasury stock respectively 119875119905

current price per share and 119864119903119903th-quarter net profit price to

earnings ratio or 119875119864 is defined as

119875

119864=119875119905(1198991+ 1198992minus 1198993)

119864119903

(11)

119875119864 denotes the stock price per 1 baht of net profit that theinvestor is willing to pay for

Definition 18 Let 119899 be the number of be the number ofregistered share 119860

119905and 119877

119905the asset and liability of the

company respectively and 119875119905current price per share price to

book value ratio or 119875119861119881 is defined as

119875

BV=119875119905

119861119905

(12)

where 119861119905= (119860119905minus 119877119905)119899

119875BV denotes how many times the current stock price iscompared to its account value

Definition 19 Let 119903 be the reference interest rate 119863119896the 119896th

year-end dividend per share 119896 isin 119868119899 and 119875

0the 119899th-quarter

historical price the current target price 119875119899is defined as

119875119899= 1198750(1 + 119903)

119899

minus

119899

sum

119896=1

119863119896(1 + 119903)

119899minus119896

(13)

Definition 20 Let 119875119899be the current target price and 119875 the

current stock price 119875119875119899is called price per target price ratio

represented by the symbol 119875119875119899

119875119875119899denotes how many times the current stock price is

compared to the current target price

3 Stock Selection Procedure

This section presents the proposed stock selection procedurewhich is done in the following 3 main steps

Step 1 The first step is analysis of individual stocks withineach industrial group from their financial ratios using fuzzylogic principles to calculate the investment weight for eachindividual stock

Step 2 The second step is analysis of industrial groups (egfinance communication technology and property) usingfuzzy multicriteria decision-making principles to calculatethe investment weight for each industrial group

Step 3 The third step is analysis of individual stocks acrossall industrial groups using the 2 types of weights from Steps1 and 2 to calculate the final weight for ranking all individualstocks in the market

31 Step 1 Analysis of Individual Stocks within Each IndustrialGroup In this step we apply the method of Bumlungponget al [15] to analyze individual stocks within each industrialgroup Price to earnings ratio (119875119864 ratio) price to bookvalue ratio (119875BV ratio) and price to intrinsic value ratio(119875119875119899ratio) are used to calculate the investment weight for

each individual stock within an industrial group based onquantitative fuzzy analysis under these assumptions

(1) A calculated investment weight of an individual stockcan be compared only to another one in the sameindustrial group

(2) More recent data reflect current trend better thanearlier ones

(3) Fuzzy rules are flexible and depend on expert infor-mation

The specific steps of the fuzzy analysis are as follows

Step 11 This step involves screening in only 119898 individualstocks (119878

1 1198782 119878

119898) in the same industrial group of which

sufficient financial data are provided for calculating 119875119864119875BV and 119875119875

119899of 119899 earlier years up to the present

Step 12 This step involves calculating (119864119875)(119878119894119896) (119875BV)(119878119894

119896)

and (119875119875119899)(119878119894

119896) for all 119894 isin 119868

119899and 119896 isin 119868

119898 where 119878119894

119896denotes the

119896th stock in the 119894th year

Step 13 This step involves calculating the following weightedarithmetic mean (119864119875)119908(119878

119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896)

119896 isin 119868119898 from the following equations

(119864

119875)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119864

119875) (119878119894

119896)

(119875

BV)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

BV) (119878119894

119896)

(119875

119875119899

)

119908

(119878119896) =

119899

sum

119894=1

119908119894(119875

119875119899

) (119878119894

119896)

where 119908119894=

2119894

119899 (119899 + 1) 119894 isin 119868

119898

(14)

Step 14 This step involves an expert constructing fuzzy setsin linguistic terms of the ranked financial ratios 119864119875 119875BVand 119875119875

119899and a fuzzy set119882 of the investment weights from

(119864119875)119908

(119878119896) (119875BV)119908(119878

119896) and (119875119875

119899)119908

(119878119896) 119896 isin 119868

119898

Step 15 This step involves an expert constructing fuzzy rulesfor estimation based on the fuzzy sets constructed in Step 14These fuzzy rules are in the form of an ldquoif-thenrdquo rule asfollows

Advances in Operations Research 5

Rule-1 if 1199091is 11and 119909

2is 12and 119909

3is 13then 119910 is

1

Rule-2 if 1199091is 21and 119909

2is 22and 119909

3is 23then 119910 is

2

Rule-119902 if 1199091is 1199021and 119909

2is 1199022and 119909

3is 1199023then 119910 is

119902

1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875

119899

and 1198821 respectively and

1198961 1198962 and

1198963 119896 isin 119868

119902 are

linguistic terms of 119864119875 119875BV 119875119875119899 and 119882

1 respectively

that is 119864119875 = 11 21

1199021 119875BV =

12 22

1199022

119875119875119899= 13 23

1199023 and119882 =

1 2

119902

Step 16This step involves importing 119864119875 119875BV and 119875119875119899of

the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861

Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908

119911= 119906B(119911) int

119861

119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int

119861

119911119908119911119889119911 =

int119861

119911119906B(119911) 119889119911 int119861

119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group

32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)

AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω

119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus

1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω

119899is replaced by

a set of fuzzy preference values Ω120575119899= 1

120575 1(119899 minus 1)

13120575 12120575 1 2120575 3120575 (119899 minus 1)

120575 120575 where

120575= ⟨119896 minus 120575 119896 119896 +

120575⟩ and 1120575= 1⊘

120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868

119899

and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]

and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers

Applying this technique to 1198991decision makers 119899

2deci-

sion criteria and 1198993industrial groups as alternate choices the

analysis steps are as follows

Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899

1 is compared to another

decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as

120593 (119894 119895) =

119894119895 exist119894119895isin Ω119899 119895 gt 119894

1 119895 = 119894

1 ⊘ 120593 (119895 119894) 119895 lt 119894

(15)

The decision makerrsquos preference matrix = (119894119895)1198991times1198991

is areciprocal matrix where

119894119895=

120593 (119894 119895) 119894 lt 119895

1 119894 = 119895

1 ⊘ 120593 (119895 119894) 119894 gt 119895

(16)

Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =

(119894119895)1198991times1198991

) 119889119896= ⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ is a fuzzyweight vector for

all 119896 isin 1198681198991

where

119908119871

119889119896= 119862119871sdot

(prod1198991

119895=1119886119871

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119872

119889119896=

(prod1198991

119895=1119886119872

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119880

119889119896= 119862119880sdot

(prod1198991

119895=1119886119880

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

(17)

with

119862119871= min119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119871

119894119895)11198991

119862119880= max119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119880

119894119895)11198991

(18)

If its consistency index 1198681205901198991

() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21

Step 23 This step involves decision makers 1198891 1198892

1198891198991

constructing decision criteria 1198881 1198882 119888

1198992

for evaluatingindustrial groups 119866

1 1198662 119866

1198993

where 119888119894 119894 = 1 119899

2 is

constructed from investment weight of 1198993individual groups

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 5

Rule-1 if 1199091is 11and 119909

2is 12and 119909

3is 13then 119910 is

1

Rule-2 if 1199091is 21and 119909

2is 22and 119909

3is 23then 119910 is

2

Rule-119902 if 1199091is 1199021and 119909

2is 1199022and 119909

3is 1199023then 119910 is

119902

1199091 1199092 1199093 and 119910 are fuzzy variables of 119864119875 119875BV 119875119875

119899

and 1198821 respectively and

1198961 1198962 and

1198963 119896 isin 119868

119902 are

linguistic terms of 119864119875 119875BV 119875119875119899 and 119882

1 respectively

that is 119864119875 = 11 21

1199021 119875BV =

12 22

1199022

119875119875119899= 13 23

1199023 and119882 =

1 2

119902

Step 16This step involves importing 119864119875 119875BV and 119875119875119899of

the latest day and making estimation with Mamdani methodusing the fuzzy rules constructed in Step 15 hence obtainingan output of a fuzzy setB under the membership 119906B on 119861

Step 17 This step involves performing defuzzification of thefuzzy output to a crisp output by a centroid method Acrisp 119911119888119892 is the average weight of the weight at each point119911 on domain 119861 where 119908

119911= 119906B(119911) int

119861

119906B(119911) 119889119911 for all119911 isin 119861 that is the crisp output is 119911119888119892 = int

119861

119911119908119911119889119911 =

int119861

119911119906B(119911) 119889119911 int119861

119906B(119911) 119889119911 It is the investment weight ofeach individual stock in a particular industrial group Theseweights are then used to rank stocks in an industrial group

32 Step 2 Analysis of Industrial Groups Industrial groupsare ranked by weights calculated by themethod of fuzzymul-ticriteria decision-making consisting of AHP fuzzy analytichierarchy process and Fuzzy Technique for Order Preferenceby Similarity to Ideal Solution Method (FTOPSIS)

AHP is a method for calculating decision weights devel-oped by Saaty [11] and Paul Yoon and Hwang [5] It com-pares paired data that are metrics of real quantities suchas price weight and preference Here these quantities arepreferences Levels of preferences are represented by numbersin a set Ω

119899= 1119899 1(119899 minus 1) 13 12 1 2 3 119899 minus

1 119899 expressed as a reciprocal matrix Generalizing thisidea the set of crisp preference values Ω

119899is replaced by

a set of fuzzy preference values Ω120575119899= 1

120575 1(119899 minus 1)

13120575 12120575 1 2120575 3120575 (119899 minus 1)

120575 120575 where

120575= ⟨119896 minus 120575 119896 119896 +

120575⟩ and 1120575= 1⊘

120575= ⟨1(119896+120575) 1119896 1(119896minus120575)⟩ for all 119896 isin 119868

119899

and 0 le 120575 le 1The other technique FTOPSIS developed by Chan [17]

and Balli and Korukoglu [10] is a fuzzy technique for rankingpreference levels by comparing the similarity of alternatechoice to the ideal choice in order to find the best alternativeIt covers diverse alternate choices decision criteria anddecision makers

Applying this technique to 1198991decision makers 119899

2deci-

sion criteria and 1198993industrial groups as alternate choices the

analysis steps are as follows

Step 21 (finding weights for decision makers) In this stepa decision maker 119894 119894 = 1 119899

1 is compared to another

decision maker 119895 in terms of their preference level based on apreference function 120593(119894 119895) defined as

120593 (119894 119895) =

119894119895 exist119894119895isin Ω119899 119895 gt 119894

1 119895 = 119894

1 ⊘ 120593 (119895 119894) 119895 lt 119894

(15)

The decision makerrsquos preference matrix = (119894119895)1198991times1198991

is areciprocal matrix where

119894119895=

120593 (119894 119895) 119894 lt 119895

1 119894 = 119895

1 ⊘ 120593 (119895 119894) 119894 gt 119895

(16)

Step 22 (finding a fuzzyweight vector 119889= (119889119896)1198991times1for =

(119894119895)1198991times1198991

) 119889119896= ⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ is a fuzzyweight vector for

all 119896 isin 1198681198991

where

119908119871

119889119896= 119862119871sdot

(prod1198991

119895=1119886119871

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119872

119889119896=

(prod1198991

119895=1119886119872

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

119908119880

119889119896= 119862119880sdot

(prod1198991

119895=1119886119880

119896119895)11198991

sum1198991

119894=1(prod1198991

119895=1119886119872

119894119895)11198991

(17)

with

119862119871= min119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119871

119894119895)11198991

119862119880= max119894isin1198681198991

(prod1198991

119895=1119886119872

119894119895)11198991

(prod1198991

119895=1119886119880

119894119895)11198991

(18)

If its consistency index 1198681205901198991

() as defined in Definition 15is less than 01 it is accepted as being valid Otherwise thedecision makerrsquos weight is reevaluated by repeating Step 21

Step 23 This step involves decision makers 1198891 1198892

1198891198991

constructing decision criteria 1198881 1198882 119888

1198992

for evaluatingindustrial groups 119866

1 1198662 119866

1198993

where 119888119894 119894 = 1 119899

2 is

constructed from investment weight of 1198993individual groups

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Operations Research

given by decision makers in the term of linguistic terms (seeTable 1)

Thedecision criteria constructed are in the formof a fuzzymatrix with members 119887

119895119894119896= ⟨119887119871

119895119894119896 1198871198721

119895119894119896 1198871198722

119895119894119896 119887119880

119895119894119896⟩ 119895 isin 119868

1198993

119894 isin 1198681198992

and 119896 isin 1198681198991

which are trapezoidal fuzzy numbersrepresenting the linguistic terms of 119888

1 1198882 119888

1198992

shown in(19)

Decision Criteria for Evaluating Industrial Groups 1198661

1198662 119866

1198993

Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111

211

119899311

112

212

119887119899312

119887111198991

119887211198991

119887119899311198991

1198882

1198661

1198662

1198661198993

121

221

119899321

122

222

119887119899322

119887121198991

119887221198991

119887119899321198991

1198881198992

1198661

1198662

1198661198993

111989921

211989921

119899311989921

119887111989922

119887211989922

119887119899311989922

119887111989921198991

119887211989921198991

119899311989921198991

= (19)

Step 24 This step involves decision makers 1198891 1198892 119889

1198991

evaluating decision criteria 1198881 1198882 119888

1198992

constructing fromthe linguistic terms VL LMLMMHHVH as in Step 23A fuzzy matrix = (

119894119895)1198992times1198991

for evaluation is then obtainedwhere

119894119895isin VL LMLMMHHVH for all 119894 isin 119868

1198992

and119895 isin 1198681198991

as shown in (20)

Evaluation of Decision Criteria 1198881 1198882 119888

1198992

Consider

11988911198892sdot sdot sdot 119889

1198991

11988811112sdot sdot sdot 11198991

11988822122sdot sdot sdot 21198991

=

1198881198992

1198992111989922sdot sdot sdot 11989921198991

(20)

Step 25 This step involves calculating decision criteria basedon decisionmakersrsquo weights bymultiplying the decision crite-rion of a decisionmaker in each column in Step 24 (depictedin (20))with the corresponding decisionmakerrsquos fuzzyweightvector

119889= (

119889119896)119899times1

where 119889119896= ⟨119908

119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ =

⟨119908119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from Step 22 Equation (21)

shows these multiplication results

Decision Criteria Based on Weights of Decision MakersConsider

1198891

1198892

sdot sdot sdot 1198891198991

119888111otimes 1198891

12otimes 1198892sdot sdot sdot 11198991

otimes 1198891198991

119888221otimes 1198891

22otimes 1198892sdot sdot sdot 21198991

otimes 1198891198991

= 119908

1198881198992

11989921otimes 119889111989922otimes 1198892sdot sdot sdot 11989921198991

otimes 1198891198991

(21)

Next we multiply the decision criterion for evaluatingindustrial groups in the column representing each decisionmaker constructed in Step 23 with the corresponding deci-sion makerrsquos fuzzy weight vector = (

119889119896)119899times1

where 119889119896=

⟨119908119871

119889119896 119908119872

119889119896 119908119880

119889119896⟩ = ⟨119908

119871

119889119896 119908119872

119889119896 119908119872

119889119896 119908119880

119889119896⟩ calculated from

Step 22 The multiplication results are in (22)

Decision Criteria for Evaluating Industrial Groups Based onWeights of Decision Makers Consider

1198891

1198892

1198891198991

1198881

1198661

1198662

1198661198993

111otimes 1198891

211otimes 1198891

119899311otimes 1198891

112otimes 1198892

212otimes 1198892

119887119899312otimes 1198892

111198991

otimes 1198891198991

211198991

otimes 1198891198991

119887119899311198991

otimes 1198891198991

1198882

1198661

1198662

1198661198993

121otimes 1198891

221otimes 1198891

119899321otimes 1198891

122otimes 1198892

222otimes 1198892

119887119899322otimes 1198892

121198991

otimes 1198891198991

221198991

otimes 1198891198991

119887119899321198991

otimes 1198891198991

1198881198992

1198661

1198662

1198661198993

111989921otimes 1198891

211989921otimes 1198891

119899311989921otimes 1198891

119887111989922otimes 1198892

119887211989922otimes 1198892

119887119899311989922otimes 1198892

119887111989921198991

otimes 1198891198991

119887211989921198991

otimes 1198891198991

119887119899311989921198991

otimes 1198891198991

= 119908

(22)

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 7

Table 1

Linguistic term Fuzzy numberVery low (VL) ⟨0 0 01 02⟩

Low (L) ⟨01 02 03⟩

Medium low (ML) ⟨02 03 04⟩

Medium (M) ⟨03 04 06 07⟩

Medium high (MH) ⟨06 07 08⟩

High (H) ⟨07 08 09⟩

Very high (VH) ⟨08 09 1 1⟩

Step 26 This step involves aggregating weights of decisioncriteria based on the decision makersrsquo weights as follows

119888119894= ⟨119908119871

119888119894 1199081198721

119888119894 1199081198722

119888119894 119908119880

119888119894⟩ (23)

where 119908119871119888119894= min1198991

119896=1119888119871

119908119894119896 1199081198721119888119894= (1119899

1) sum1198991

119896=11198881198721

119908119894119896 1199081198722119888119894=

(11198991) sum1198991

119896=11198881198722

119908119894119896 119908119880119888119894= max1198991

119896=1119888119880

119908119894119896 for all 119894 isin 119868

1198992

119908= (119908119895119896)1198992times1198991

and 1198991is the number of decision makers

Equation (24) shows these aggregation results

Weights of Decision Criteria 1198881 1198882 119888

1198992

Consider

1198881

1198882sdot sdot sdot 119888

1198992

119882211988811198882sdot sdot sdot

1198881198992

(24)

Next we aggregate industrial groups based on the deci-sion makersrsquo weights (see (22)) by the following equations

119895119894= ⟨119909119871

119895119894 1199091198721

119895119894 1199091198722

119895119894 119909119880

119895119894⟩ (25)

where 119909119871119895119894= min1198991

119896=1119887119871

119908119895119894119896 1199091198721119895119894= (1119899

1) sum1198991

119896=11198871198721

119908119895119894119896 1199091198722119895119894=

(11198991) sum1198991

119896=11198871198722

119908119895119894119896 119909119880119895119894= max1198992

119896=1119887119880

119908119895119894119896 for all 119895 isin 119899

3 119894 isin 119899

2

119908= (119908119895119894119896)11989931198992times1198991

and 1198991is the number of decision makers

These results are shown in (26)

Evaluation Matrix of Industrial Groups 1198661 1198662 119866

1198993

Con-sider

1198881

1198882sdot sdot sdot 119888

1198992

11986611112sdot sdot sdot

11198992

11986622122sdot sdot sdot

21198992

=

1198661198993

1198993111989932sdot sdot sdot 11989931198992

(26)

Step 27 This step involves constructing a decision matrix bynormalizing the industrial groupsrsquo evaluation matrix (see(26)) as follows

= (119895119894)1198993times1198992

119895119894= ⟨

119909119871

119895119894

119909lowast

119894

1199091198721

119895119894

119909lowast

119894

1199091198722

119895119894

119909lowast

119894

119909119880

119895119894

119909lowast

119894

⟩ where 119909lowasti =1198993max119895

119909119880

119895119894

(27)

Then multiplying the normalized matrix with the decisionweights from Step 26 = (V

119895119894)1198993times1198992

where V119895119894=

⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894when 119895 isin 119868

1198993

119894 isin 1198681198992

Industrial Groupsrsquo Evaluation Matrix Consider

11988811198882sdot sdot sdot 119888

1198992

1198661

V11

V12sdot sdot sdot V11198992

1198662

V21

V22sdot sdot sdot V21198992

=

1198661198993

V11989931V11989932sdot sdot sdot V11989931198992

(28)

Step 28 This step involves defining positive ideal solution(119866lowast

) and negative ideal solution (119866minus) from (28) as 119866lowast =(Vlowast1 Vlowast2 Vlowast

1198992

) and119866minus = (Vminus1 Vminus2 Vminus

1198992

) respectively whereVlowast119894= max1198993

119895V119880119895119894 and Vminus

119894= min1198993

119895V119871119895119894 119895 isin 119868

1198993

119894 isin 1198681198992

= (V

119895119894)1198993times1198992

Step 29 This step involves calculating the distances betweenthe industrial groupsrsquo evaluation results with the positive andnegative ideal solutions as defined by the following

119889lowast

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowast

119894) 119895 isin 119868

1198993

119889minus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vminus

119894) 119895 isin 119868

1198993

(29)

where 119889V(V119895119894 Vlowastminus

119894) are calculated in the same way as fuzzy

numbers are calculated according to Definition 8 (depictedin (30))

Distances between the Industrial Groupsrsquo Evaluation Resultsand Positive andNegative Ideal Solutions119866lowast and119866minus Consider

1198881

1198882

sdot sdot sdot 1198881198992

119889lowastminus

119895=

1198992

sum

119894=1

119889V (V119895119894 Vlowastminus

119895)

1198661119889V (V11 V

lowastminus

1) 119889V (V12 V

lowastminus

2) sdot sdot sdot 119889V (V1119899

2

Vlowastminus1198992

) 119889lowastminus

1

1198662119889V (V21 V

lowastminus

1) 119889V (V22 V

lowastminus

2) sdot sdot sdot 119889V (V2119899

2

Vlowastminus21198992

) 119889lowastminus

2

1198661198993

119889V (V11989931 Vlowastminus1) 119889V (V119899

32 Vlowastminus2) sdot sdot sdot 119889V (V119899

31198992

Vlowastminus1198992

) 119889lowastminus

1198993

(30)

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Operations Research

Table 2 119864119875 of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 369360995 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Number of treasury stocks 0 0 0 0Latest 12-month profit 1908520000 1089760000 399510000 2021430000119875119864 148500 30385 212183 264733 49093119864119875 00673 03291 00471 00378 02037119864119875 (weighted average) 01383119864119875 ( weighted average) 1383

Step 210 This step involves calculating the nearness coeffi-cients to the positive ideal solution 119862119862

119895 and ranking the

industrial groups according to them 119862119862119895are defined as

follows

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

119895 isin 1198681198993

(31)

From the calculation a set of investment weightsfor industrial groups 119882

3= (119908

1 1199082 119908

1198993

) where1199081 1199082 119908

1198993

are weights of individual groups is obtainedThe industrial group of which investment weight value isnearest to one (the closest to the positive ideal solution) isthe best industrial group

33 Step 3 Analysis of All Stocks from Different IndustrialGroups In this step the Correlation-Product Implication isused the two investment weights from Steps 1 and 2 areused to calculate the integrated final investment weights forall of the stocks in the market denoted as 119882

119874119860(119904119894119895) where

119882119874119860(119904119894119895) = 119882

1(119904119894119895) sdot 1198822(119866119895) and119882

1(119904119894119895) are the weight of the

119894th stock from the 119895th group from Step 1 and 1198822(119866119895) is the

weight of the 119895th group from Step 2 These weights are thenused to rank the stocks for making decisions and planningout strategies

4 Application of the Analysis Procedures toa Demonstration Case

As a demonstration of the applicability of our analysisprocedures a simulated case of stock selection into a portfoliofor a given period of time was conducted Suppose that the 6industrial groups of investment interest were the followingagricultural and food industry (119866

1) consumer product and

service industry (1198662) financial industry (119866

3) industrial

product and technology industry (1198664) property and con-

struction industry (1198665) and resource industry (119866

6) Stocks

from each individual industry were analyzed as follows

Step 1 (analysis of stocks in an industrial group) As an exam-ple the analysis of the property and construction industry1198665 is shown below

In this group1198665 we use the past 5-year financial fact data

of the companies from Stock Exchange of Thailand 2010ndash2014 httpwwwsettradecom

Step 11 This step involves gathering the past 5-year financialdata of the companies in this group and screening in stockswith complete data from 12 companies CK CNT ITDNWRPREB SEAFCO STEC STPI SYNTEC TRC TTCL andUNIQ

Step 12 This step involves calculating the 119864119875 119875BV and119875119875119899values of each individual stock

Step 13 This step involves calculating the following weightedarithmetic mean of 119864119875 119875BV and 119875119875

119899 Tables 2 3 and

4 show data of some stock (STPI) and Table 5 shows theweighted arithmetic mean of each individual stock in 119866

5

Step 14 This step involves an expert constructing a fuzzy setbased on the latest 5-year financial data of which linguisticterms are represented by trapezoidal and triangular fuzzynumbers

Values of119864119875119875BV and119875119875119899were grouped into 3 levels

low (119871) medium (119872) and high (119867) and so the fuzzy setsrepresenting these levels were

119871 = ⟨119897119871

1198971198721 1198971198722 119897119880

⟩

119872 = ⟨119898119871

1198981198721 1198981198722 119898119880

⟩

119867 = ⟨ℎ119871

ℎ1198721 ℎ1198722 ℎ119880

⟩

(32)

The fuzzy sets of linguistic terms were as follows

119864119875 rArr 119871119883 = ⟨0 0 1 3⟩119872119883 = ⟨1 3 8 10⟩ 119867119883 =⟨8 10 100 100⟩119875BV rArr 119871119884 = ⟨0 0 5 7⟩119872119884 = ⟨5 7 10 16⟩119867119884 =⟨10 16 100 100⟩119875119875119899rArr 119871119885 = ⟨0 0 1 11⟩ 119872119885 = ⟨1 11 19 23⟩

119867119885 = ⟨19 23 100 100⟩

Step 15 This step involves an expert constructing fuzzy rulesfrom the fuzzy sets constructed from Step 14 as follows

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 9

Table 3 119875BV of STPI

STPI stock 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 157 6275 2875 27

Number of common stocks 1477443980 368492092 367873233 367546097Number of preferred stocks 0 0 0 0Total assets 10867008638 7347262706 3522893354 4259624240Total liabilities 4956210154 2922198628 423972604 1021904292Accounting value per share 4000692117 1200857271 842388212 8809017357119875BV 3924320978 5225433658 3412915754 3065041072119875BV of 2014 (2nd quarter) 48119875BV (weighted average) 4350963831119875BV (highest) 2518861616119875BV 1727353263

Table 4 119875119875119899of STPI

STPI stock 14102014 27122013 28122012 30122011 30122010Closing price of commonstock (baht) 208 157 6275 2875 27

Dividend interest rate () 163 159 05 1216 786Dividend amount (baht) 0339 02496 03138 3496 21222Expected interest (119903) 00703 00707 00728 00750 00641Baht gained from 1 bahtinvestment (1 + 119903) 10703 10707 10728 10750 10641

Target price in 2014 293056Closing price to target priceratio 07098

Table 5 119864119875 119875BV and 119875119875119899of stocks in 119866

5

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 1086 786 089 614 105 638 559 1383 326 798 42 766119875BV 871 91 735 473 819 719 1606 1727 38 899 1619 83119875119875119899

243 112 094 238 294 097 167 071 186 083 287 24

Rule 1 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was 119871119885 then119908 was 119877119867119882

Rule 2 if 119909 was 119871119883 and 119910 was 119871119884 and 119911 was119872119885 then119908 was119872119882

Rule 27 if 119909 was 119867119883 and 119910 was 119867119884 and 119911 was 119867119885then 119908 was 119877119871119882

Step 16 This step involves importing the values of current119875119864 (inversing to 119864119875) 119875BV and 119875119875

119899 which in this study

were the values of the 22nd of January 2015 shown in Table 6

Note The 119864119875s of CNT and NWR were not applicablemeaning that they suffered a loss so they were not includedin further calculation

Step 17 This step involves performing defuzzification of thefuzzy output values to crisp values with the centroid methodobtaining the investment weights shown in Table 7

For the purpose of easy demonstration the investmentweights of the stocks from the other 5 industrial groups weremade up All of the weights are tabulated in Table 8

Step 2 (analysis of industrial groups) Stocks from 6 industrialgroups119866

1 1198662 119866

6 were analyzedThree decisionmakers

1198891 1198892 1198893constructed 4 decision criteria 119888

1 1198882 1198883 1198884

calculated in the following steps

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Operations Research

Table 6 Financial ratios of the 22nd January 2015 httpwwwsettradecom

Financial ratio CK CNT ITD NWR PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQ119864119875 () 476 NA 214 NA 576 529 471 823 484 547 358 351119875BV 254 236 366 169 341 388 483 409 182 346 304 373119875119875119899

255 091 134 234 429 179 155 066 237 108 283 312

Table 7 Investment weights from the analysis procedures

Stock CK ITD PREB SEAFCO STEC STPI SYNTEC TRC TTCL UNIQInvestment weights 0084 0105 0084 01091 01091 01435 0084 0113 0084 0084

Table 8 Investment weights of all stocks the ones for 1198665were actually calculated while the rest were made up

1198661

1198662

1198663

1198664

1198665

1198666

11990411

00418 11990412

026 11990413

01276 11990414

00518 11990415(CK) 0084 119904

1600261

11990421

0024 11990422

0169 11990423

01528 11990424

01077 11990425(ITD) 0105 119904

2601258

11990431

01148 11990432

01359 11990433

00282 11990434

01745 11990435(PREB) 0084 119904

3600667

11990441

01704 11990442

01006 11990443

00843 11990444

00528 11990445(SEAFCO) 01091 119904

4602034

11990451

01003 11990452

0004 11990453

00822 11990454

01108 11990455(STEC) 01091 119904

5600315

11990461

0097 11990462

01376 11990463

00841 11990464

01399 11990465( STPI) 01435 119904

6601576

11990471

00764 11990472

01825 11990473

00335 11990474

00916 11990475(SYNTEC) 0084 119904

7602068

11990481

00705 11990482

00104 11990483

00421 11990484

01099 11990485(TRC) 0113 119904

8600638

11990491

01484 11990493

0211 11990494

00825 11990495(TTCL) 0084 119904

9601215

119904101

01565 119904103

02517 119904104

00986 119904105

(UNIQ) 0084

Step 21This step involves calculating theweights for decisionmakers The preference level of the 119894th decision maker wascompared to that of the 119895th decision maker with a scale[19 9] obtaining

= (

(1 1 1) (1 2 3) (2 3 4)

(1

31

2 1) (1 1 1) (1 2 3)

(1

41

31

2) (

1

31

2 1) (1 1 1)

) (33)

Step 22 This step involves calculating the fuzzy weightvectors

119889= (

119889119896)3times1

for = (119894119895)3times3

and obtain-ing the following respective vectors for decision mak-ers 1198891 1198892 1198893 1198891= ⟨047165 053991 053991 053991⟩

1198892

= ⟨025869 029712 029712 034012⟩ and 1198893

=

⟨016296 016296 016296 018717⟩ and a consistency index119868120590

3() = 009403

Step 23 This step involves the 3 decision makers 1198891 1198892

1198893evaluating 6 industrial groups 119866

1 1198662 119866

6 according

to the decision criteria 1198881 1198882 1198883 1198884utilizing linguistic terms

VL LMLMMHHVH represented by trapezoidal fuzzynumbers as in Table 9

Step 24 This step involves decision makers 1198891 1198892 1198893

evaluating the decision criteria 1198881 1198882 1198883 1198884utilizing the

linguistic terms VL LMLMMHHVH represented by thementioned trapezoidal fuzzy numbers as in Table 10

Step 25 This step involves calculating fuzzy decision criteriaand the evaluation criteria for industrial groups based on theweights of decisionmakers as in Tables 11 and 12 respectively

Step 26 This step involves aggregating the decision criteriaand the fuzzy evaluation of industrial groups based on theweights of decisionmakersThe aggregation results are shownin Tables 13 and 14

Step 27 This step involves normalizing the weights ofindustrial groups for each decision criteria shown in Table 13and thenmultiplying the normalizedmatrix with the weightsof decision criteria from Step 26 defined by = (V

119895119894)6times4

where V

119895119894= ⟨V119871119895119894 V1198721119895119894 V1198722119895119894 V119880119895119894⟩ and V

119895119894= 119895119894otimes 119888119894 when

119895 isin 1 2 6 119894 isin 1 2 4 to obtain a decision matrixshown in Table 15

Step 28This step involves stipulating a positive ideal solution(119878lowast

) and a negative ideal solution (119878minus) to be

119878lowast

= [(054 054 054 054)

(0486 0486 0486 0486) (054 054 054 054)

(0432 0432 0432 0432)]

119878minus

= [(0021 0021 0021 0021)

(0021 0021 0021 0021)

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 11

Table 9 Trapezoidal fuzzy numbers representing linguistic terms used for fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

06 07 07 08 06 07 07 08 08 09 1 11198662

08 09 1 1 07 08 08 09 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

06 07 07 08 06 07 07 08 06 07 07 081198666

06 07 07 08 07 08 08 09 06 07 07 08

1198882

1198661

06 07 07 08 07 08 08 09 07 08 08 091198662

07 08 08 09 06 07 07 08 06 07 07 081198663

08 09 1 1 08 09 1 1 08 09 1 11198664

06 07 07 08 07 08 08 09 07 08 08 091198665

06 07 07 08 06 07 07 08 07 08 08 091198666

07 08 08 09 07 08 08 09 07 08 08 09

1198883

1198661

07 08 08 09 07 08 08 09 07 08 08 091198662

08 09 1 1 07 08 08 09 07 08 08 091198663

08 09 1 1 08 09 1 1 07 08 08 091198664

07 08 08 09 06 07 07 08 06 07 07 081198665

07 08 08 09 06 07 07 08 07 08 08 091198666

06 07 07 08 06 07 07 08 07 08 08 09

1198884

1198661

06 07 07 08 06 07 07 08 06 07 07 081198662

06 07 07 08 08 09 1 1 07 08 08 091198663

07 08 08 09 07 08 08 09 07 08 08 091198664

08 09 1 1 08 09 1 1 08 09 1 11198665

07 08 08 09 07 08 08 09 07 08 08 091198666

07 08 08 09 06 07 07 08 07 08 08 09

Table 10 Evaluation of fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

08 09 1 1 08 09 1 1 08 09 1 11198882

07 08 08 09 07 08 08 09 07 08 08 091198883

08 09 1 1 08 09 1 1 08 09 1 11198884

05 06 07 08 08 09 1 1 07 08 08 09

(0024 0024 0024 0024)

(0021 0021 0021 0021)]

(34)

Step 29 This step involves calculating the distances fromthe results of industrial groups evaluation in Table 14 to the(119878lowast

) and the (119878minus) ideal solutions shown in Tables 16 and 17respectively

Step 210 This step involves obtaining the nearness coeffi-cients 119862119862

119895 119895 = 1 6 to the positive ideal solution and the

investment weights shown in Table 18

Step 3 (analysis of all stocks from different industrial groups)The two kinds of investment weights obtained from Steps 1and 2 were used to calculate the final investment weights forall of the stocks in the market 119882

119874119860(119904119894119895) where 119894 represents

the 119894th company and 119895 the 119895th industrial group and the finalweights were ranked as shown in Table 19

From Table 19 investors can use the calculated weightsto help with their decision-making and strategy-planningThe better stocks to invest in show higher final investmentweights

5 Conclusions

The innovation appearing in this paper is to present thetactic of conveying the stock selection to portfolio by usingtwo tactics fuzzy quantitative analysis and fuzzy hierarchicalanalysis The two tactics imply the final investment weightInvestors can determine their strategies by using the finalinvestment weights The final investment weights may be

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Advances in Operations Research

Table 11 Fuzzy decision criteria

Criteria Decision maker1198891

1198892

1198893

1198881

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198882

03302 04319 04319 04859 01811 02377 02377 03061 01141 01304 01304 016851198883

03773 04859 05399 05399 0207 02674 02971 03401 01304 01467 0163 018721198884

02358 03239 03779 04319 0207 02674 02971 03401 01141 01304 01304 01685

Table 12 Fuzzy evaluation of industrial groups

Criteria Industrial group Decision maker1198891

1198892

1198893

1198881

1198661

0283 0378 0378 0432 0155 0208 0208 0272 013 0147 0163 01871198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198666

0283 0378 0378 0432 0181 0238 0238 0306 0098 0114 0114 015

1198882

1198661

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198662

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198663

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198664

0283 0378 0378 0432 0181 0238 0238 0306 0114 013 013 01681198665

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 01681198666

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 0168

1198883

1198661

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198662

0377 0486 054 054 0181 0238 0238 0306 0114 013 013 01681198663

0377 0486 054 054 0207 0267 0297 034 0114 013 013 01681198664

033 0432 0432 0486 0155 0208 0208 0272 0098 0114 0114 0151198665

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 01681198666

0283 0378 0378 0432 0155 0208 0208 0272 0114 013 013 0168

1198884

1198661

0283 0378 0378 0432 0155 0208 0208 0272 0098 0114 0114 0151198662

0283 0378 0378 0432 0207 0267 0297 034 0114 013 013 01681198663

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198664

0377 0486 054 054 0207 0267 0297 034 013 0147 0163 01871198665

033 0432 0432 0486 0181 0238 0238 0306 0114 013 013 01681198666

033 0432 0432 0486 0155 0208 0208 0272 0114 013 013 0168

Table 13 Aggregation of decision criteria

Criteria1198881

1198882

1198883

1198884

Weight 013 03 0333 054 0114 0267 0267 0486 013 03 0333 054 0114 0241 0268 0432

Table 14 Aggregation of evaluation of industrial groups

Group Criteria1198881

1198882

1198883

1198884

1198661

013 0244 025 0432 0114 0249 0249 0432 0114 0267 0267 0486 0098 0233 0233 04321198662

0114 0285 0303 054 0098 0251 0251 0486 0114 0285 0303 054 0114 0259 0268 04321198663

0114 0267 0267 0486 013 03 0333 054 0114 0295 0322 054 0114 0267 0267 04861198664

013 03 0333 054 0114 0249 0249 0432 0098 0251 0251 0486 013 03 0333 0541198665

0098 0233 0233 0432 0114 0239 0239 0432 0114 0257 0257 0486 0114 0267 0267 04861198666

0098 0243 0243 0432 0114 0267 0267 0486 0114 0239 0239 0432 0114 0257 0257 0486

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Operations Research 13

Table 15 Decision matrix

Group Criteria1198881

1198882

1198883

1198884

1198661

0031 0136 0154 0432 0024 0123 0123 0389 0028 0148 0148 0486 0021 0104 0116 03461198662

0028 0158 0187 054 0021 0124 0124 0437 0028 0158 0168 054 0024 0115 0134 03461198663

0028 0148 0165 0486 0028 0148 0165 0486 0028 0164 0179 054 0024 0119 0133 03891198664

0031 0167 0206 054 0024 0123 0123 0389 0024 014 014 0486 0028 0134 0166 04321198665

0024 013 0144 0432 0024 0118 0118 0389 0028 0143 0143 0486 0024 0119 0133 03891198666

0024 0135 015 0432 0024 0132 0132 0437 0028 0133 0133 0432 0024 0114 0128 0389

Table 16 Distances between 119866119895 119895 = 1 6 and 119878lowast for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889lowast

1= 119889V (1198661 119878

lowast

) 0381572 0348712 0378268 0309819 1418371119889lowast

2= 119889V (1198662 119878

lowast

) 0364995 0346628 0369604 0301308 1382533119889lowast

3= 119889V (1198663 119878

lowast

) 0374072 0326879 0365443 0298251 1364645119889lowast

4= 119889V (1198664 119878

lowast

) 0356869 0348712 0384022 0284309 1373911119889lowast

5= 119889V (1198665 119878

lowast

) 0388341 0351266 0381127 0298251 1418985119889lowast

6= 119889V (1198666 119878

lowast

) 038534 0341527 0389187 0300654 1416708

Table 17 Distances between 119866119895 119895 = 1 6 and 119878minus for each decision criterion

Distance Criteria Sum1198881

1198882

1198883

1198884

119889minus

1= 119889V (1198661 119878

minus

) 0223775 0197715 0247374 0174345 0843208119889minus

2= 119889V (1198662 119878

minus

) 0281158 0220808 0276399 017835 0956716119889minus

3= 119889V (1198663 119878

minus

) 0251745 0251745 0278567 0198531 0980589119889minus

4= 119889V (1198664 119878

minus

) 0285192 0197715 0245285 0225285 0953478119889minus

5= 119889V (1198665 119878

minus

) 0221504 0196478 0246015 0198531 0862528119889minus

6= 119889V (1198666 119878

minus

) 0223065 0222647 0218246 0197315 0861274

Table 18 Nearness coefficients to the positive ideal solution

Industrial group 1198661

1198662

1198663

1198664

1198665

1198666

119862119862119895=

119889minus

119895

119889minus

119895+ 119889lowast

119895

0304297 038056 0392965 0380328 0318558 0315015

Weights 0157599 0172877 0176738 0173169 0159816 0159816

Table 19 The final investment weights of all of the stocks in the market

119904119894119895

11990412

119904102

11990493

11990476

11990446

11990472

11990434

11990422

11990423

11990441

11990466

119904101

11990464

119882119874119860(119904119894119895) 00473 00472 00396 00332 00317 003114 00307 003063 00287 002543 002503 002478 00247

119904119894119895

11990462

11990432

11990491

11990465

11990413

11990426

11990496

11990454

11990484

11990424

11990431

11990485

11990455

119882119874119860(119904119894119895) 00237 00227 00218 00215 00201 001998 00195 001894 00183 001829 001792 001720 002373

119904119894119895

11990445

11990442

119904104

11990425

11990474

11990461

11990451

11990443

11990463

11990453

11990494

11990415

11990435

119882119874119860(119904119894119895) 00167 00166 00166 00166 00159 001583 00157 001544 0015 001459 001411 001279 001279

119904119894119895

11990475

11990495

119904105

11990471

11990481

11990436

11990486

11990444

11990414

11990483

11990411

11990473

11990456

119882119874119860(119904119894119895) 00127 00127 00127 00111 00102 001004 00094 000960 00096 00079 000608 000608 000529

119904119894119895

11990433

11990416

11990421

11990482

11990452

119882119874119860(119904119894119895) 00047 00039 00034 00018 00007

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Advances in Operations Research

used to select stocks and allocate asset into portfolio Acase study presented in Table 19 shows that if we use thefinal investment weights as decision criteria to select stocksinto portfolio stock that has the highest weight is the mostinteresting and is chosen first In contrast stock that hasthe lowest weight is the least interesting and is chosen lastHowever decision-making and strategy-planning of eachinvestor may be different and depend on their financial risktolerance For example some investors whose financial risktolerance is high level maybe invest in only one stock with thehighest final investment weights while some investors reducerisk by investing in many stocks with high final investmentweights You should keep in your mind that there is no besttool in the world for financial analysis but you can alter toolsthat fit for each situation The purpose of this research isto construct the tool for financial analysis that may be analternative for investors At least we hope that this researchwill help investors to make an appropriate decision

For future work we will improve our model and compareresults with others in each situation Moreover the softwareof this model will also be provided

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

Thefinancial support for this study was fromKingMongkutrsquosInstitute of Technology Ladkrabang Bangkok Thailand

References

[1] G Kabir and M Ahsan Akhtar Hasin ldquoComparative analysisof AHP and fuzzy AHP models for multi-criteria inventoryclassificationrdquo International Journal of Fuzzy Logic Systems vol1 no 1 pp 1ndash16 2011

[2] J J Buckley T Feuring and Y Hayashi ldquoFuzzy hierarchicalanalysis revisitedrdquo European Journal of Operational Researchvol 129 no 1 pp 48ndash64 2001

[3] J Ramik Consistency of Pair-Wise Comparison Matrix withFuzzy Elements School of Business Administration in KarvinaFSA-EUSFLAT 2009

[4] J Ramik and P Korviny ldquoInconsistency of pair-wise compar-ison matrix with fuzzy elements based on geometric meanrdquoFuzzy Sets and Systems vol 161 no 11 pp 1604ndash1613 2010

[5] K Paul Yoon and C-L Hwang Multiple Attribute DecisionMaking An Introduction 1995

[6] G F Milanka and Z S Dragan ldquoMulticriteria optimizationin a fuzzy environment the fuzzy analytic hierarchy processrdquoYugoslav Journal of Operations Research vol 20 no 1 pp 71ndash85 2010

[7] M B Ayhan ldquoA fuzzy AHP approach for supplier selectionproblem a case study in a gearmotor companyrdquo InternationalJournal of Managing Value and Supply Chains vol 4 no 3 pp11ndash23 2013

[8] M Gavalec J Ramık and K Zimmermann Decision Makingand Optimization vol 677 of Lecture Notes in Economics andMathematical Systems Springer 2015

[9] P Srichetta andWThurachon ldquoApplying fuzzy analytic hierar-chy process to evaluate and select product of notebook comput-ersrdquo International Journal of Modeling and Optimization vol 2no 2 pp 168ndash173 2012

[10] S Balli and S Korukoglu ldquoOperating system selection usingfuzzy AHP and topsis methodsrdquo Mathematical and Computa-tional Applications vol 14 no 2 pp 119ndash130 2009

[11] T L Saaty The Analytic Hierarchy Process Planning PrioritySetting Resource Allocation Decision Making Series Mcgraw-Hill New York NY USA 1980

[12] A Escobar J Moreno and S Munera ldquoA technical analysisindicator based on fuzzy logicrdquo Electronic Notes in TheoreticalComputer Science vol 292 pp 27ndash37 2013

[13] A A Gamil R S El-Fouly and N M Darwish ldquoEgyptstock technical analysis using multi agent and fuzzy logicrdquo inProceedings of theWorldCongress onEngineering (WCE rsquo07) volI London UK July 2007

[14] RDC T Raposo andA J DOCruz ldquoStockmarket predictionbased on fundamentalist analysis with fuzzy neural networksrdquoin Proceedings of the 3rd WSEAS International Conference onNeural Networks and Applications 2002

[15] P Bumlungpong R Chinarak AThaimai andWWitayakiatil-erd Fuzzy Quantitative Analysis of the Property and Construc-tion Industrial Group in the Stock Exchange of Thailand SpecialProblem King Mongkutrsquos Institute of Technology LadkrabangBangkok Thailand 2015

[16] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[17] C-T Chen ldquoExtensions of the TOPSIS for group decision-making under fuzzy environmentrdquo Fuzzy Sets and Systems vol114 no 1 pp 1ndash9 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of