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ISSN 1750-9653, England, UKInternational Journal of Management Scienceand Engineering Management, 6(1): 3-13, 2011http://www.ijmsem.org/

A modified TOPSIS technique in presence of uncertainty

and its application to assessment of transportation

systems

Soheil Sadi-Nezhad , Kaveh Khalili Damghani

Industrial Engineering Department, Faculty of Engineering, Sciences & Research Branch, Islamic Azad University, Tehran1651893181, Iran

(Received 7 July 2010, Revised 19 November 2010, Accepted 12 December 2010)

Abstract. In this paper, a modified TOPSIS approach based on Preference Ratio (PR) and an efficient fuzzy distance measurementhas been proposed in an uncertain environment. As it is not proper to rank fuzzy numbers using crisp measurements, PR is supplied

to rank Generalized Fuzzy Numbers (GFNs) in a relative manner rather than absolute way. Moreover, as human logic says distancesbetween fuzzy numbers should not be a crisp value. So, an efficient measurement for calculating distance between fuzzy numbershas also been utilized in the core of proposed fuzzy TOPSIS procedure. The aforementioned segments of proposed procedure madeit well-posed and efficient for modeling complicated real life problems. The performance of proposed procedure has been comparewith an existing approach in selection of different transportation systems modes with conflicting subjective and qualitative. Theassociated expert system of proposed procedure has been developed through a linkage between MS-Excel 12.0 and Visual Basic 6.0.

Keywords: fuzzy sets, transportation, multiple criteria decision making, decision support systems, TOPSIS

1 Introduction

In the early decades Fuzzy Multiple Criteria DecisionMaking (FMCDM) techniques have attracted lots of re-searches efforts due to adaptability for real conditions de-

cision making problem. Decision making in complicatedreal world situations needs lots of considerations and of-ten multi-dimensional details (Gurumurthy and Kodali,2008 [14]). One of the well-known Multiple Attribute De-cision Making (MADM) approaches is the Technique forOrder of Preference by Similarity to Ideal Solution (TOP-SIS). TOPSIS was introduced by Hwang and Yoon (1981)[15]. TOPSIS ranks alternatives according to an algorithmicprocedure. Alternatives are sorted in decreasing order ofCloseness Coefficient (CC) which is calculated with respectto distance of a given alternative from both positive andnegative ideal solution concurrently in a multi-dimensionalspace.

Ambiguous data is a big challenge for Decision Makers(DMs). This fact motivated researchers to extend MCDMtechniques, including TOPSIS, in fuzzy environment. Ingeneral, fuzzy sets are assumed to be proper paradigm toplot some versions of uncertainty in which qualitative andsubjective criteria are considered. An impressive variety offuzzy TOPSIS algorithms with different applications aredeveloped in recent years.

Some of those can be outlined as Application of TOPSISin evaluating initial training aircraft under a fuzzy environ-ment by Wang and Chang (2007) [36], A note on groupdecision-making based on concepts of ideal and anti-ideal

points in a fuzzy environment by Wang and Hua (2007) [38],Compromise ratio method for fuzzy multi-attribute groupdecision making by Li (2007) [23], Generalizing TOPSISfor fuzzy multiple-criteria group decision-making by Wangand Lee (2007) [39], Fuzzy multi-criteria evaluation of in-

dustrial robotic systems by Kahraman et al. (2007a) [19],A two phase multi-attribute decision-making approach fornew product introduction by Kahraman et al. (2007b) [20],Group decision-making based on concepts of ideal and anti-ideal points in a fuzzy environment by Kuo et al. (2007)[22], Using fuzzy number for measuring quality of servicein the hotel industry by Bentez et al. (2007) [7], Multiple-attribute decision making methods for plant layout designproblem by Yang and Hung (2007) [43], Extensions of TOP-SIS for large scale multi-objective non-linear programmingproblems with block angular structure by Abo-Sinna et al.(2008) [3], Extension of the TOPSIS method for decision-making problems with fuzzy data by Jahanshahloo et al.(2006) [18], Multiple attribute decision-making methods forthe dynamic operator allocation problem by Yang et al.(2007) [36], A fuzzy approach for supplier evaluation andselection in supply chain management by Chen et al. (2006)[10], Fuzzy TOPSIS method based on alpha level sets withan application to bridge risk assessment by Wang and Elhag(2006) [37], An interactive algorithm for large scale multi-ple ob jective programming problems with fuzzy parame-ters through TOPSIS approach by Abo-Sinna and Abou-El-Enien (2006) [2], The Method of Grey Related Analysisto Multiple Attribute Decision Making Problems with In-terval Numbers by Desheng and Olson (2006) [41], Exten-

The authors would like to acknowledge the collaborating manner of experts in Research Committee of Iranian Traffic Center.

Correspondence to: E-mail address: [email protected].

International Society of Management Science

And Engineering Management

Published by World Academic Press,

World Academic Union

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4 S. Sadi-Nezhad & K. Damghani: A modified TOPSIS technique in presence of uncertainty

sions of the TOPSIS for group decision-making under fuzzyenvironment by Chen (2000) [9], The evaluation of airlineservice quality by fuzzy MCDM by Tsaura et al. (2002) [35],Combining Grey Relation and TOPSIS Concepts for Select-ing an Expatriate Host Country by Chen and Tzeng (2004)[11], Extensions of TOPSIS for multi-objective large-scalenonlinear programming problems by Abo-Sinna and Amer(2005) [3], and Transshipment site selection using the AHP

and TOPSIS approaches under fuzzy environment by Onutand Soner (2008) [30], Sadi-Nezhad and Damghani (2010)[32] proposed a fuzzy TOPSIS method based on preferenceratio and fuzzy distance measurement in order to assess per-formance of traffic police centers, Jadidi et al. (2009) [17]also proposed a TOPSIS extension for multi-objective sup-plier selection problem under price breaks based on Abo-Sinna and Amer (2005) [3] and Abo-Sinna et al. (2008) [4]approaches. Jadidi et al. (2008) [16] proposed an optimalgrey based approach based on TOPSIS concepts for sup-plier selection problem.

Chu and Lin (2009) [12] proposed interval arithmetic

based fuzzy TOPSIS model. Onut et al. (2010) [28] modeledshopping center site selection problem in Istanbul. Theyproposed a combined MCDM methodology using fuzzy

AHP for assigning weights of the criteria and fuzzy TOP-SIS to determine the most suitable alternative. Onut et al.(2009) [30] developed a supplier evaluation approach basedon fuzzy analytic network process (ANP) and fuzzy TOP-SIS methods to help a telecommunication company in theGSM sector in Turkey. Athanasopoulos et al. (2009) [6] pre-sented an expert system based on Max-Min set methodand fuzzy TOPSIS for coating selection of engineering partwhich could handle both qualitative and quantitative vari-ables. Malekly et al. (2010) [25] proposed a two-phased de-cision process based on QFD and TOPSIS for selecting themost suitable superstructure of a small to medium-spanhighway bridge design. Wu et al. (2009) [40] proposed afuzzy MCDM procedure to evaluate banking performance.They gathered associated criteria of banking performance

according to BSC perspectives. Then, the relative impor-tance of the aforementioned criteria was calculated throughfuzzy AHP. Finally, three well-known MCDM approachesincluding a fuzzy TOPSIS were used to evaluate three bankperformances as an illustrative example. Secme et al. (2009)[33] proposed a fuzzy MCDM model to evaluate the perfor-mances of banks. The proposed method used several finan-cial and non-financial indicators. Fuzzy AHP and TOP-SIS integrated in the proposed model. Dursun and Karsak(2010) [13] developed a fuzzy MCDM procedure based onfusion of fuzzy information principles, 2-tuple linguisticrepresentation model, and TOPSIS to personnel selectionproblem with different criteria such as organizing ability,creativity, personality, and leadership which exhibit vague-ness and imprecision. Kelemenis and Askounis (2010) [21]

also presented a fuzzy TOPSIS approach to select qualifiedpersonnel incorporating a new concept for the ranking ofthe alternatives. They applied a veto threshold. So the ul-timate decision criterion is not the similarity to the idealsolution but the distance of the alternatives from the vetoset by the decision makers.

As mentioned, in real cases some subjective and quali-tative criteria may arise that can properly be representedthrough fuzzy sets. Moreover, in many cases there is noenough information about the problem this vagueness canalso be modeled through fuzzy paradigm. Beside aforemen-tioned situations, exact data are not required or even noteasy/inexpensive to achieve in many real cases. Anyway,

fuzzy sets fuzzy sets have been proved to represent all afore-mentioned situations properly. Most to the all aforemen-tioned fuzzy TOPSIS procedures try to overcome the pre-vious points through some kind of fuzzy set theory. Dif-ferent extensions have been reported in literature of fuzzyTOPSIS.

In fuzzy TOPSIS, the relative importance of alternativesor rating of an alternative with respect to attributes can be

a fuzzy number. Therefore the calculations are done in fuzzyenvironment and fuzzy operators are used. Main differencesbetween fuzzy TOPSIS approaches can be summarized inchoosing a decision matrix normalization method, deter-mining Fuzzy Positive Ideal solutions (FPIS) and FuzzyNegative Ideal solutions (FNIS), distance calculation be-tween fuzzy numbers, and applied defuzzification method.Most to the all of aforementioned research works presenteda procedure which uses a defuzzification method at early ormiddle steps of the fuzzy TOPSIS algorithm. These meth-ods convert fuzzy numbers to an associated crisp valuewhich will cause some rounding error as well as probableand meaningful disturbance in final ranking of alternatives.

Moreover, a most likely used technique to determineFPIS & FNIS is to introduce (0 , 0 , 0 , 0) as FNIS, and (1,1,

1, 1) as FPIS for TrFNs as in Wang and Chang (2007) [36].Determination ideal and anti-ideal point in this way mayresult in following problems. These ideal and anti-ideal vec-tors are determined without considering the real data ofthe problem. Comparison of alternative with such ideal andanti-ideal vectors may not be reasonable in real case situ-ations. Moreover, this kind of ideal and anti-ideal determi-nation diminishes the possibility of periodical comparisonof best and worst alternatives in order to make an overalltrend for alternative in a predefined planning horizon.

Aforementioned shortages made us to modify the clas-sic TOPSIS procedure in an uncertain environment whichhas been modeled using fuzzy sets. The proposed modifiedfuzzy TOPSIS procedure is assumed to have the followingadvantages over existing fuzzy TOPSIS procedures in liter-

ature. All distances between alternatives and FPIS and FNIS arecalculated using an efficient fuzzy distance measurement.This yields more realistic properties in favor of representingall distances between fuzzy numbers in fuzzy environment.Although most to the all of existing fuzzy TOPSIS proce-dures serve crisp distance between fuzzy numbers. Comparisons of fuzzy numbers (i.e. closeness coefficients)have been accomplished using PR which has been known asan efficient ranking method for GFNs. This makes the rank-ing procedure more realistic in a fuzzy environment. UsingPR method for ranking, fuzzy numbers are ranked rela-tively in an interval rather than absolutely which is usualin existing fuzzy TOPSIS procedures. The proposed modified fuzzy TOPSIS procedure can

properly model subjective and qualitative criteria as wellas crisp one. The fuzzy environment is held till last step of proposedfuzzy TOPSIS procedure. This results in reducing roundingerror and plotting uncertainty in a proper manner. Determining the ideal and anti-ideal vectors are accom-plished according to real data of cases.

In this paper we have presented a modified fuzzy TOP-SIS approach based on Preference Ratio (PR) and fuzzydistance measurement. PR was first introduced by Modar-res and Sadi-Nezhad (2001) [26] for fuzzy multiple crite-ria group decision making. We used PR method for rank-ing Fuzzy closeness coefficients and ranking fuzzy numbers

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International Journal of Management Science and Engineering Management, 6(1): 3-13, 2011 5

to find a proper relative order of alternatives. PR woulddetermine the preference of fuzzy numbers in an intervalthrough a relative manner rather than absolute way. It isnotable that developing methods for ranking fuzzy num-bers has been attracted lots of research efforts. For in-stance the efforts of Asady and Zendehnam (2007) [5] orAbbasbandy and Asady (2006) [1] have worth to be men-tioned. Moreover, distances between fuzzy numbers should

be a fuzzy measure. So we used an efficient fuzzy distancemeasurement which was introduced by Chakraborty andChakraborty (2007) [8]. The calculations of distances be-tween fuzzy numbers and ranking CCs are mainly differentfrom what are well-known in reported fuzzy TOPSIS ap-proaches in literature.

The following sections of this paper are arranged as be-low. Section 2 is allocated to extend some main arithmeticoperators in fuzzy environment and notations are intro-duced. Concepts of PR are discussed in section 3. An ef-ficient fuzzy distance measure is also introduced in section4. Due to our best knowledge this type of fuzzy distancemeasure and concept of Preference Ratio are rarely used inmain core of proposed fuzzy TOPSIS algorithms. In section5 modified fuzzy TOPSIS approach is introduced. 6th sec-

tion talks about application of recommended algorithm forassessment of transportation systems performance in differ-ent modes and finally in section 7, the paper will be endedwith a brief conclusion.

2 Notations and fuzzy operators

Let X be the universe of discourse, X = {x1, x2, , xn}.A fuzzy set X of A is a set of order pairs {(x1,A(x1 )),

(x2,A(x2 )), , (xn,A(xn))} where A : X [0, 1] is the

membership function of A, and A(x1) stands for the mem-

bership degree ofxi in A. Some definitions are presented asfollow (Zimmermann, 1991 [44]).

Definition 1. A trapezoidal fuzzy number (TrFN) can be

defined as a m = (a, b, c, d), where the membership functionm is given by

m(x) =

xaba , a x b

1, b x cdxdc , c x d

where [b, c] is called a mode interval of m, and a and d arecalled lower and upper limits of m, respectively.

For sake of simplicity and without loss of generality, weconsider TrFNs throughout the paper and calculation areextended by this assumption.

Definition 2. m = (a, b, c, d) is called a positive trapezoidfuzzy number if a 0. a, b, c, and d are not identical.

Further detail about basic fuzzy arithmetic operators canbe found in (Zimmermann, 1991 [44]).

Lemma 1. In real cases something unusual occurs whenusing arithmetic calculation for some especial negative fuzzynumbers. For a convenient description note the followingexample.

Example 1. Let A = (0.305, 0.145, 0.894, 1.884) and B= {0.616, 0.491, 1.932, 3.483} as two negative TrFNs.A()B is calculated as below.

A()B = (a1, a2, a3, a4)()(b1, b2, b3, b4)

=

a4b1

,a3b2

,a2b3

,a1b4

, if A < 0, B < 0,

A = (0.305, 0.145, 0.894,1.884),

B =(0.616, 0.491, 1.932, 3.483),

A()B = (3.057, 1.820, 0.075,0.088). (1)

It is clear that the result expression is not a TrFN and thisis not a coincidence at all. This result is likely when TrFNshave the following properties:

TrFNs are negative (See definition 2.2 for positiveTrFNs).

TrFNs are asymmetric where plotting the TrFN will notcause a symmetric trapezoid shape in 2-D surface.

There are lots of such TrFNs which hold these propertiesand cause similar results.

Lemma 2. We modify such TrFNs in following mannerin order to prevent above problem in our algorithm. Letm = (a, b, c, d) be a negative TrFN in which a is per-manently negative and b, c and d are not equal. Allarguments are ordered ascending. A transition toward pos-

itive axis is used to prevent the aforementioned case. Thetransited TrFN is named

m = (0, |a| + b, |a|+ c, |a|+ d) = (0, b, c, d).

Lemma 3. If two or more than two fuzzy numbers meetthe aforementioned conditions in Lemma 1, it is more rea-sonable to transit the fuzzy numbers in a ways that theirrelative situations to each other are constant after transi-tion procedure. So we define a unique transition value forall of them as follow. Let A1, A2, A3, , An be n TrFNparameterized by (ai , bi , ci , di), i = 1,2, , n respectively.They meet aforementioned conditions in Lemma 1. Calcu-late |min (ai)|, i = 1,2, , n and name it unique transi-tion value (UTV). Transit all TrFNs toward positive axisby UTV. More formally a supposed TrFN in this string

will be (ai + |min (ai)|, b + |min (ai)|, c + |min (ai)|, d +|min (ai)|), i = 1,2, , n. It is clear that by this transi-tion method all TrFNs will be transited in a fairly mannerand their relative positions will be constant after transition.A direct result of this transition is that left spreads of alltransited TrFNs will be a positive or at least zero value forthose TrFNs which their left spreads are numerically equalto UTV. Since zero value for left spreads will cause a divi-sion by zero error during the fuzzy arithmetic, an arbitraryvery small positive value like > 0 is added to UTV. SoUTV= |min (ai)|+, i = 1,2, , n is the final UTV. Thetransited TrFNs are (ai + |min (ai)| + , b + |min (ai)| +, c + |min (ai)| +, d + |min (ai)| +), i = 1,2, , n.

3 Preference ratio

Our proposed approach is developed on the basis of Pref-erence Ratio concept. Preference Ratio is a ranking methodintroduced by Modarres and Sadi-Nezhad (2001) [26]. Theyevaluate fuzzy numbers point by point and rank them ateach point in PR method. Then, the overall preference overall points is calculated. So the preference in this way is rel-ative rather than absolute. Suppose the objective is to rankI fuzzy numbers. Let Ni be the ith one defined over a realdomain Si R, and it is identified by a membership func-tion (Ni (x)), x Si, with Ni (x) [0, 1]. Let Si be thesupport of Ni, one more precisely Si = {x,Ni (x) > 0},

and =I

i=1 Si then is the union of the support of all

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6 S. Sadi-Nezhad & K. Damghani: A modified TOPSIS technique in presence of uncertainty

a b bd a c d

)(x

xc

a0

a0

a0

a b bc , da c d

)(x

x

a0

a0

Fig. 1 Transition of a negative TrFN

fuzzy numbers. In other word, fuzzy numbers are rankedover To rank fuzzy numbers, we assume their spans arenot disjoint, because in that case the ranking is clear.

A fuzzy number is evaluated by a function called Pref-erence Function. At each point, this function is defined asfollows:

G() = U (x)dxUL (x)dx

, (2)

where (x) is the membership function of the fuzzy num-ber, L = min {x : x } and U = max{x : x }.This function has the same definition as 1 F() in prob-ability theory, where F() = P[x ] is the distributionfunction. At , let p() = i denote the ith fuzzy num-ber, which is the most preferred one. Therefore p() = i,if Gi() = max {G, j I}. where Gi() is the prefer-ence function of the jth fuzzy number. Let be the setof point at which the ith number is ranked number one.Then = { , p() = i}.

Definition 3. For the ith fuzzy number, R(i), the Prefer-ence Ratio, is by definition, the percentage of that theith fuzzy number is the most preferred one. Then

R(i) =|i|

||, (3)

where |i| and || are the lengths of the real set i and respectively. Modarres and Sadi-Nezhad (2001) [26] de-veloped an algorithm for determining preference ratio inTFN cases. An algorithm is also proposed for any contin-uous cases by Modarres and Sadi-Nezhad (2005) [27]. Alsoto make two fuzzy numbers A and B preference ratio equiv-alent, they define this concept as below.

Definition 4. (a) We define that two fuzzy numbers A andB are preference ratio equivalent if R(A) = R(B) = 0.5,where R(A) and R(B) are preference ratio of A and B, re-

spectively. Preference ratio equivalence is shown as A PR B.If

kAPR B,

then we say k is the Equivalence multiplier of A withrespect to B.

3.1 Ranking n fuzzy numbers

In all fuzzy TOPSIS approaches, a fuzzy dominance re-lation is needed to be defined in order to obtain a rankof fuzzy numbers. Modarres and Sadi-Nezhad (2001) [26]have proposed a ranking method for fuzzy numbers that

compares them relatively, rather than preferring one num-ber absolutely to the others. For two fuzzy number N1 andN2 they developed an algorithm to find scalar K such as

N1PR K N2.

Modarres and Sadi-Nezhad (2005) [27]. An algorithm for

ranking n TFNs was proposed by Sadi-Nezhad and Ghaleh-Assadi (2008) [31]. They also have developed an algorithmfor Max operator of two TFNs by preference ratio. Theyconsidered 3 initial estimation methods for Max operatorand five cases of two TFN positions in relation to eachother. They have proven that the final results are depen-dent to initial estimation method.

4 Fuzzy distance measure

Distance measures have become important due to thesignificant applications in diverse fields like remote sensing,data mining, pattern recognition, multivariate data anal-ysis, MADM methods and etc. Most of proposed meth-ods measure the distances of fuzzy numbers as a crispvalue, although human intuitions offer a fuzzy measureis more acceptable and reasonable for measuring the dis-tances of fuzzy numbers. Answering to this logical ques-tion that if the numbers themselves are not known ex-actly, how can the distance between them be an exactvalue? made researchers eager to develop several fuzzy dis-tance measures. Tran and Duckstein (2002) [34] developeda method for comparison of fuzzy numbers using a fuzzydistance measure. Voxman introduced a fuzzy distance forgeneralized fuzzy numbers (GFN) using-cut. The Voxmannew distances measurement method has been improved byChakraborty and Chakraborty (2007) [8]. They have intro-duced a novel fuzzy distance measure and have proven thequality of their method is over the Voxman method throughambiguity and fuzziness which are two main attributes offuzzy numbers.

We have used theirs as the main core of distance cal-culation step in our modified TOPSIS method as a val-idated method. In the next paragraph this method hasbeen presented briefly to make a comfort sense. Details canbe founded in Chakraborty and Chakraborty (2007) [8] orSadi-Nezhad and Damghani (2010) [32].

4.1 Defining fuzzy distance for GFNs

Let us consider two GFNs as A1 = (1 ,2 ;1 ,1) andA2 = (3,4;2 ,2). Therefore the -cut of A1 and A2 rep-

resents two intervals, respectively [ A1] = [AL1

(), AR1

()]

and [ A2] = [AL2

(), AR2

()], for all [0, 1]. Since it

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International Journal of Management Science and Engineering Management, 6(1): 3-13, 2011 7

may be possible to obtain the distance between two inter-val numbers by means of their difference, they employ theinterval-distance operation for interval [AL

1(), AR

1()] and

[AL2

(), AR2

()] to formulate the fuzzy distance between A1and A2. So the distance between [ A1] and [ A2] for every [0, 1] can be one of the following:

d(A1,

A2)=

[ A1] [ A2], ifAL1 (1)+A

R1 (1)

2

AL2 (1)+AR2 (1)

2

[ A2] [ A1], ifAL1 (1)+A

R1 (1)

2