Research ArticleHamming Distance Method with Subjective andObjective Weights for Personnel Selection

R Md Saad1 M Z Ahmad1 M S Abu1 and M S Jusoh2

1 Institute of Engineering Mathematics Universiti Malaysia Perlis Pauh Putra Main Campus 02600 Arau Perlis Malaysia2 School of Business Innovation and Technopreneurship Universiti Malaysia Perlis Jalan Kangar-Alor Setar01000 Kangar Perlis Malaysia

Correspondence should be addressed to M Z Ahmad mzainiunimapedumy

Received 31 August 2013 Accepted 11 November 2013 Published 17 March 2014

Academic Editors R-M Chen and H Wu

Copyright copy 2014 R Md Saad et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Multicriteria decisionmaking (MCDM) is one of themethods that popularly has been used in solving personnel selection problemAlternatives criteria and weights are some of the fundamental aspects in MCDM that need to be defined clearly in order toachieve a good result Apart from these aspects fuzzy data has to take into consideration that it may arise from unobtainableand incomplete information In this paper we propose a new approach for personnel selection problem The proposed approachis based on Hamming distance method with subjective and objective weights (HDMSOWrsquos) In case of vagueness situation fuzzyset theory is then incorporated onto the HDMSOWrsquos To determine the objective weight for each attribute the fuzzy Shannonrsquosentropy is considered While for the subjective weight it is aggregated into a comparable scale A numerical example is presentedto illustrate the HDMSOWrsquos

1 Introduction

The rapid growth in globalization had created an intensecompetition between modern firms in global markets Thesesituations had urged the organization and firms to estab-lish a comprehensive procedure during personnel selectionprocess The personnel selection can be defined as a processof selecting the individuals who match the requirement andqualification to perform a particular job in an excellent way[1]Themain objective of this process is to assess the diversityamong the alternatives that could pave a way of predictingthe future performance [2] Knowing the fact that personnelselection is not an easy task to be solved has awakenedthe conscience of decision makers to make decisive actionto solve The decision makers have to consider all aspectsthat are needed in this process Hence some of the decisionmakers try to solve this problembyusing any kind ofmethodsthat are available and suitable for them to use

Despite restructuring and reorganizing the personnelselection process some of the firms had performed a so-called ldquostrategic decisionrdquo to choose the best candidate

during the selection process Some decision makers try toutilize rigorous and costly selection procedure and someeven used the traditional method which depends on onlyinformation stated on the application forms that turn outto be quickest and inexpensive methods [1] However thesemethods actually never bring satisfaction and their finalresults are sometimes deniable Thus when multicriteriadecisionmaking (MCDM)was introduced in the early 1970rsquosit had become one of favorable and importantmethods in thisarea Some of the decision makers took a chance and grabedthis opportunity to apply this method in solving personnelselection problem [3] MCDM is known for its capabilitiesin evaluating electing or ranking a finite set of availablealternatives with respect to multiple and conflicting criteria[4] A number of methods and theories had been introducedand extended based on the utilization of this approach andthe continuing study of this field had extended in a fixedrate Preference Ranking Organization Method for Enrich-ment Evaluation (PROMETHEE) [5] linear programmingtechniques [6] AnalyticHierarchy Process (AHP) [7] SimpleAdditive Weighting (SAW) [8] and Technique for Order

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 865495 9 pageshttpdxdoiorg1011552014865495

2 The Scientific World Journal

Preference by Similarity to Ideal Solution or TOPSIS [9] aresome of the numerous examples on MCDM methods thatparticularly have been used by the decision makers

Distance measure can be identified as one of the MCDMapproaches that can be used in personnel selection processThis approach holds an important key to solve many prob-lems related to biology science social and technology due toits capability of constructing some related distance measuresnotably similarity and proximity which always become anorm in various problems [10] In recent years the study ofthis method has been rapidly growing in which it resultedin proposing and improving the previous distance measuremethods Some of the well-known distancemeasuremethodsare Hamming Euclidean Hausdorff and Minkowski meth-ods Based on the existing literatures Hamming distance isone of the methods that can be used in personnel selectionprocess [11ndash13]This method was proposed by Hamming [14]in 1950 to count the number of flipping bits in a fixed-lengthbinary word as an estimate of error used in telecommunica-tion Hamming distance is known for its ability in calculatingthe difference between two sets or elements For example thedistance between interval-valued fuzzy sets Consequentlyapart from the decision making problem it also has beenapplied in various fields such as communication [15] irisrecognition [16] and engineering [17]

Literally evaluation of certain criteria or attributes toselect an appropriate alternative for specified position couldbecome tremendous and challenging task for the decisionmakers It is because some of the criteria such as leadershippersonality and creativity are referred to as qualitative criteriain which exhibits imprecise and vagueness data In generalthis uncertainty and subjective scene that occurs during theevaluation of the alternatives based on respective criteria andcriteria weight may come from various sources includingunquantifiable information incomplete information unob-tainable information and partial ignorance [18] For this situ-ation commonly classical MCDM will be put aside since thealternatives rating and criteria weights for classical MCDMare usually measured in crisp numbers Therefore one ofthe best resorts to solve this problem is by applying fuzzyset theory The fuzzy set theory is known for its flexibilityin handling imprecise and uncertainty in human judgmentsBellman and Zadeh [19] had introduced the use of fuzzy settheory inMCDM and it proved to be an effective approach indealing with uncertainty in human decision making processSince then it had become an important tool in constructinga decision making framework that incorporates subjectivejudgments that entails in the personnel selection process

Themain objective of this paper is to propose an approachto solve personnel selection process by using Hammingdistance method Inspired by algorithm proposed by Canoset al [11] we extend and improve Canosrsquos algorithm byadding weight in the classical Hamming distance In ourproposed method we suggest two types of weight whichare subjective and objective weights The linguistic termscorrespondence to triangular fuzzy numbers are used toevaluate the performance rating values as well as the weightof the criteria in which later will be expressed into intervalvalued fuzzy numbers In this approach we also identify the

changes in ranking of the alternatives when different valuesof 120572 are used The remaining of this paper is organized asfollows The next section we briefly explain the preliminaryconcerning fuzzy set and Hamming distance Section 3 willbriefly explain about the Hamming distance method andsubjective and objective weights In Section 4 we proposea new algorithm for personnel selection problem The newalgorithm is called HDMSOWrsquos Section 5 validates theHDMSOWrsquos by conducting a numerical example The lastsection concludes this paper

2 Preliminaries

A fuzzy set 119860 in 119883 is defined as a set of ordered pairs (see[20])

119860 = ⟨119909 120583119860 (119909)⟩ 119909 isin 119883 (1)

where 119883 is denoted as a universe of discourse and 120583119860

(119909) isthe membership function of 119860 defined as

120583119860

119883 997888rarr [0 1] (2)

A triangular fuzzy number is specified by three parametersand can be defined as triplet 119860 = (119886

1 1198862 1198863) where 119886

1lt 1198862

lt

1198863with the 119909 = 119886

2as the core of the triangle Its membership

function can be represented as [21]

120583119860 (119909) =

0 119909 lt 1198861

(119909 minus 1198861)

(1198862

minus 1198861)

1198861

le 119909 le 1198862

(1198863

minus 119909)

(1198863

minus 1198862)

1198862

le 119909 le 1198863

0 119909 gt 1198863

(3)

The 120572-cuts of this fuzzy number 119860 are denoted by

[119860]120572= [1198861

+ 120572 (1198862

minus 1198861) 1198863

minus 120572 (1198863

minus 1198862)] 120572 isin (0 1]

(4)

An interval-valued fuzzy set 119860 in universe discourse 119883 isdenoted by (see [22 23])

119860 = (119909 [120583119871

119860(119909) 120583

119880

119860(119909)]) | 119909 isin 119883 (5)

where 120583119871

119860(119909) 120583

119880

119860(119909) 119883 rarr [0 1] 120583

119871

119860(119909) is lower bound and

120583119880

119860(119909) is upper bound of membershipThemultiplication of two interval-valued fuzzy numbers

119860 = [119886119871 119886119880

] and 119861 = [119887119871 119887119880

] can be defined as (see [21])

119860 sdot 119861 = [119886119871 119886119880

] sdot [119887119871 119887119880

] = [119888119871 119888119880

] (6)

where

119888119871

= min 119886119871119887119871 119886119871119887119880

119886119880

119887119871 119886119880

119887119880

119888119880

= max 119886119871119887119871 119886119871119887119880

119886119880

119887119871 119886119880

119887119880

(7)

Hamming distance methods to be used in this paper arepresented as follows

The Scientific World Journal 3

Definition 1 (see [24]) Given two fuzzy subsets of 119860 and 119861

with a reference set 119883 = 1199091 1199092 119909

119899 and memberships

function 120583119860and 120583

119861

Then the Hamming distance is defined as

119889 (119860 119861) =

119899

sum

119895=1

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (8)

The normalized Hamming distance for two interval-valuedfuzzy numbers 119860 and 119861 whose membership functions are asfollows

120583119860

(119909119895) = [119886

119871

119909119895 119886119880

119909119895] 120583

119861(119909119895) = [119887

119871

119909119895 119887119880

119909119895]

119895 = 1 2 119899

(9)

is defined as

119889NHD (119860 119861) =

1

2119899

(

119899

sum

119895=1

(

100381610038161003816100381610038161003816

119886119871

119909119895minus 119887119871

119909119895

100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816

119886119880

119909119895minus 119887119880

119909119895

100381610038161003816100381610038161003816

)) (10)

Definition 2 (see [25]) The weighted Hamming distance ofdimension 119899 is a mapping 119889WHD [0 1]

119899times [0 1]

119899rarr [0 1]

that associated with weighting vector 119882 of dimension 119899 with119882 = sum

119899

119895=1 119908119895

= 1 and 119908119895

isin [0 1] Then the weightedHamming distance is defined as

119889WHD (119860 119861) =

119899

sum

119895=1

119908119895

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (11)

According to [12] the weighted Hamming distance can be thenormalized Hamming distance if 119908

119895= 1119899 for 119895 = 1 2 119899

3 Hamming Distance Method andSubjective and Objective Weights

31 Hamming Distance Method Hamming distance is oneof the distance measures that can be applied in personnelselection process This is due to its ability in calculatingthe distance between ideal alternative and alternative Theideal alternative is a virtual alternative in which the criteriavalues are expressed as close as possible to ideal valueswhich is rationale for human thinking to achieve There areseveral methods that focus on identifying and measuring theideal alternative However this measurement is beyond ourscope of research In this paper the evaluation on the idealalternative is made based on assumption of the optimumvalue of each criterion that alternatives should achieve forthe specified job We also disregard the usage of maximumvalue for example (1 1 1) in case of the triangular fuzzynumber of all criteria evaluations Rationally it is hard forthe alternatives to achieve a perfect score for some criteriaespecially when the evaluation of the criteria itself is madefrom human based judgment that mostly in subjective termscould be varied from one person to others The rankingof alternatives is made through the comparison betweenthe alternatives and the ideal alternative [26] such that thealternatives with the minimum distance values are likely to

be selected However when the distance values between thealternatives are the same the decision makers will face aproblem in ranking them Thus with the help of weights itwill help decision makers to distinguish between the criteriathat valued the most for the specified job than the othercriteria

32 Subjective and Objective Weights The decision makersare genuinely aware that they cannot assume that all criteriaare equally important as it holds its own meaning andneediness especially when its focus is only to one subjector position For example when recruiting the appropriateapplicant for position credit officer the criteria that might bevaluedmost are experienced in credit analysis andpersonalityassessment Generally the other criteria are also valuablebut they are not as important as these two criteria Plusit is a human nature to have diverse opinion in evaluatingprocessThus it is undeniable that the criteria weight plays animportant role in MCDM problem as it depicted the relativeweightiness of the criteria must be assigned [3] Alternativelynumerous approaches has been generalized and introducedto solve this problemThese methods can be categorized intotwo groups which are subjective and objective weights

The subjective weight are determined solely based on thepreference of the decision makers [27 28] These evaluationsare basically based on experience perception and knowledge[29] In a general view it is a process of assigning subjectivepreferences to the criteria [29] AHP method eigenvectormethod and weighted least square method can be used tocalculate this approach Beside objective weight measuredthe weight with the use of mathematical models such asentropy method [30] and multiple objective programming[31]This approach solves without any consideration from thedecision makers preference The use of objective weight canovercome some of the limitations in subjective weight suchas inconsistency problem in subjective weight Furthermoreit is useful when the reliable subjective weight is not available[29]

One of the objective weighting measure that vastly hasbeen used in MCDM field is Shannonrsquos entropy concept[32] Shannonrsquos entropy concept is a general measure ofuncertainty in information formulated in terms of probabilitytheory [30] This concept is appropriate for calculating therelative contrast intensities of criteria to represent the averageintrinsic information transmitted to the decision maker [33]It began when Shannon first introduced the application ofentropy in communication theory and since then he hadcontributed the most fundamental definition of the entropymeasure in the information theory [34] This concept hadbeen applied in wide range area exemplified mathematics[35] spectral analysis [36] and economics [37] Entropyweight is a parameter that describes how much diversealternatives approach one another with respect to a certaincriteria [3 28] This concept is also relatively known in themeasurement of fuzziness [38] Hence this method is suitableto be applied in our approach as we will deal with fuzzydata Apart from that the total weights for all criteria valueswill equal to one in which satisfy the condition that need inweighted Hamming distance method

4 The Scientific World Journal

4 Hamming Distance Method withSubjective and Objective Weights

In this section the description and algorithm for the HDM-SOWs is constructed To our best knowledge the studyof using a weighted Hamming distance method in solvingpersonnel selection problem has rarely been done Merigoand Gil-Lafuente [12] had presented a study involving theuse of weighted Hamming distance method integrated withOrdered Weighted Averaging (OWA) but without the useof fuzzy numbers Hence we would like to expand the useof weighted Hamming distance in personnel selection byusing fuzzy data and we propose two types of weights whichare subjective and objective weights The elements of thisHDMSOWs can be presented in the following descriptions

Let us assume that there is a set of 119898 possible alternatives119860 = 119860

1 1198602 119860

119898 to be evaluated based on a set of 119899

respective criteria 119862 = 1198621 1198622 119862

119899 These evaluations

are done by a set of 119898 decision makers 119864 = 1198641 1198642 119864

119898

by using linguistic variables To capture the linguistic termswe use triangular fuzzy numbers The linguistic variablesare divided into two categories which are the evaluationon criteria weight and the evaluation on criteria The givenalgorithm is unfolded as follows

Step 1 (construct a decision matrix for ideal alternative) Thedecision matrix for ideal alternative is given as follows

119868 = [V1 V2 V

119899] (12)

The ideal alternativematrix represents the optimum values of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 that the alternatives

should achieve These values are set up by decision makers

Step 2 (construct a decision matrix for alternatives) Thedecision matrix for performance alternatives is given asfollows

119863 =

1198621

1198622

sdot sdot sdot 119862119899

1198601

1198602

119860119898

[

[

[

[

[

11990911

11990912

sdot sdot sdot 1199091119899

11990921

11990922

sdot sdot sdot 1199092119899

sdot sdot sdot

1199091198981

1199091198982

sdot sdot sdot 119909119898119899

]

]

]

]

]

(13)

where 119909119894119895represent the linguistic assessment on the utility

ratings of alternative 119860119894

(119894 = 1 2 119898) with respect to119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers

Step 3 (construct a decision matrix for weight (criteriaimportance)) The weighting matrix for criteria weight 119908

119894119895

evaluated by the decision makers 119864119894

(119894 = 1 2 119898) is givenas follows

119882 =

1198641

1198642

119864119898

1198621

1198622

sdot sdot sdot 119862119899

[

[

[

[

[

11990811

11990812

sdot sdot sdot 1199081119899

11990821

11990822

sdot sdot sdot 1199082119899

sdot sdot sdot

1199081198981

1199081198982

sdot sdot sdot 119908119898119899

]

]

]

]

]

(14)

The weighting matrix represents the relative importance of119899 selection criteria 119862

119895(119895 = 1 2 119899) given by the decision

makers

Step 4 (construct an interval-valued fuzzy number) By using120572-cut of triangular fuzzy number the interval performancematrix for alternatives ideal alternatives and criteria weightare derived as follows respectively

(i) The interval decision matrix for the ideal alternative

119868120572

= [[(V1)119871

120572 (V1)119880

120572] [(V2)119871

120572 (V2)119880

120572] [(V

119899)119871

120572 (V119899)119880

120572]]

(15)

(ii) The interval decisionmatrix for performance alterna-tives

119863120572

=

[

[

[

[

[

[(11990911)119871

120572 (11990911)

119880

120572] [(11990912)

119871

120572 (11990912)

119880

120572] sdot sdot sdot [(1199091119899)

119871

120572 (1199091119899)

119880

120572]

[(11990921)119871

120572 (11990921)

119880

120572] [(11990922)

119871

120572 (11990922)

119880

120572] sdot sdot sdot [(1199092119899)

119871

120572 (1199092119899)

119880

120572]

sdot sdot sdot

[(1199091198981)

119871

120572 (1199091198981)

119880

120572] [(1199091198982)

119871

120572 (1199091198982)

119880

120572] sdot sdot sdot [(119909119898119899)

119871

120572 (119909119898119899)

119880

120572]

]

]

]

]

]

(16)

(iii) The interval decision matrix for criteria weight

119882120572

=

[

[

[

[

[

[(11990811)119871

120572 (11990811)

119880

120572] [(11990812)

119871

120572 (11990812)

119880

120572] sdot sdot sdot [(1199081119899)

119871

120572 (1199081119899)

119880

120572]

[(11990821)119871

120572 (11990821)

119880

120572] [(11990822)

119871

120572 (11990822)

119880

120572] sdot sdot sdot [(1199082119899)

119871

120572 (1199082119899)

119880

120572]

sdot sdot sdot

[(1199081198981)

119871

120572 (1199081198981)

119880

120572] [(1199081198982)

119871

120572 (1199081198982)

119880

120572] sdot sdot sdot [(119908119898119899)

119871

120572 (119908119898119899)

119880

120572]

]

]

]

]

]

(17)

where 0 le 120572 le 1 The value of 120572 represents the degreeof confidences in the decision makersrsquo assessmentwith respect to ideal alternative alternatives ratingand criteria weights

Step 5 (calculating of criteria weight) The criteria weight of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers will be calculated using twomethods whichare subjective and objective weights

(a) Subjective weightThe subjective weight of 119899 selectioncriteria 119862 = 119862

1 1198622 119862

119899 may be considered as the

average weights [9] and its calculation is [9 28]

119908119895

=

1

119898

(

119898

sum

119894=1

119908119894119895

) 119894 = 1 2 119898 119895 = 1 2 119899 (18)

(b) Objective weight The interval valued fuzzy numberis transformed into crisp number before using Shan-nonrsquos entropy concept

The crisp value of interval weight is given by [39]

119908119894119895

=

(119908119897

119894119895+ 119908119906

119894119895)

2

(19)

Then Shannonrsquos entropy concept is used to obtain the weight

The Scientific World Journal 5

The details of Shannonrsquos entropy concept are defined asfollows [27 39]

Step 51 Normalized each criterion weight to obtain theprojection value 119901

119894119895

119901119894119895

=

119908119894119895

sum119898

119894=1119908119894119895

119894 = 1 119898 119895 = 1 119899 (20)

Consequently a projection matrix representing a relativeweight of each criterion from the decision maker evaluationis expressed as

119875 =

[

[

[

[

[

11990111

11990112

sdot sdot sdot 1199011119899

11990121

11990122

sdot sdot sdot 1199012119899

sdot sdot sdot

1199011198981

1199011198982

sdot sdot sdot 119901119898119899

]

]

]

]

]

(21)

Step 52 Calculate entropy values 119890119895as

119890119895

= minus119896

119898

sum

119894=1

119901119894119895ln119901119894119895

119895 = 1 119899 (22)

where 119896 is constant and let 119896 = (ln119898)minus1 If 119901

119894119895= 0 then

119901119894119895ln119901119894119895is equal to 0

Step 53 Calculate the degree of diversification 119889119895

119889119895

= 1 minus 119890119895 119895 = 1 119899 (23)

Step 54 Calculate the criteria weight 119908119895

119908119895

=

119889119895

sum119899

119896=1119889119896

(24)

Step 6 (calculating the distance values) (a) Subjective weightBefore calculating the distance values calculate the overallperformance evaluation of ideal alternatives and alternativesby multiplying the aggregate weight with each criterion [21]

For the ideal alternative

119877119895

= [V119871119895 V119880119895

] sdot [119908119871

119895 119908119880

119895] = [119903

119871

119895 119903119880

119895] (25)

and for the alternatives119878119894119895

= [119909119871

119894119895 119909119880

119894119895] sdot [119908119871

119895 119908119880

119895]

= [119904119871

119894119895 119904119880

119894119895]

(26)

where119903119871

= min V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119903119880

= max V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119904119871

= min 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

119904119880

= max 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

(27)

Then calculate the distance values between the ideal alterna-tives with the alternatives by using Definition 1

119889NHD (119877 119878) =

1

2119899

(

119899

sum

119895=1

(

10038161003816100381610038161003816119903119871

119895minus 119904119871

119894119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816119903119880

119895minus 119904119880

119894119895

10038161003816100381610038161003816)) (28)

(b) Objective weight For the objective weight the dis-tance values are calculated by using Definition 2

119889WHD (119868 119863) =

119899

sum

119895=1

(119908119895

10038161003816100381610038161003816V119871119895

minus 119909119871

119894119895

10038161003816100381610038161003816

+ 119908119895

10038161003816100381610038161003816V119880119895

minus 119909119880

119894119895

10038161003816100381610038161003816) (29)

Step 7 (ranking the candidate) The alternatives are rankedin ascending order according to the distance values forrespective 120572 values The alternative with the less distancevalue is considered as the best choice

Step 8 (repeat Steps 4 5 and 6 for different values of 120572) Thealternatives are ranking according to the different values of 120572

Step 9 (selection of the appropriate alternative by the decisionmakers)

5 A Numerical Example

An example on the personnel selection in an academicinstitution is provided to validate the proposed algorithmSuppose that the academic institution intends to employ alecturer based on consideration of four main criteria whichare experienced in teaching areas (119862

1) proficiency in per-

forming research (1198622) personality assessment (119862

3) and past

contribution (1198624) Assume that after preliminary selection

phase four alternatives 1198601 1198602 1198603 and 119860

4are qualified for

final evaluation A committee of experts (decision makers)consisting of three persons is formed namely119863

11198632 and119863

3

The information of this study is given in Figures 1 and 2 andTables 1 2 3 4 and 5 while the results from the numericalexamples are shown in Tables 6 7 8 9 and 10 As mentionedbefore TOPSIS is one of the existing MCDM methods thatcan be used to solve personnel selection problemThus it canbe used to validate the proposed method and the results byusing this method that is shown in Table 11 More explanationon these figures and tables are explained in the discussionsection

51 Discussion Based on the results obtained the proposedHDMSOWs can be summarized as follows

Step 1 Ideal alternative matrix (12) is built from the evalua-tions of the criteria based on linguistic variables taken fromWang and Lee [28] as illustrated in Figure 2 and Table 2 Thelinguistic terms are represented by triangular fuzzy numberranging from ldquovery poorrdquo to ldquovery goodrdquo Table 4 showsthe decision makers evaluation on ideal alternative In thispaper we assume that the 119898 decision makers had come toan agreement in standardizing into one final value for eachcriterion

Step 2 Decision matrix for alternatives evaluation on eachcriterion (13) is obtained by using the same linguistic variablesadopted from Wang and Lee [28] as illustrated in Figure 2and Table 2 Similar to Step 1 these terms are captured inthe form of the triangular fuzzy number The alternativesperformance evaluations are ranging from ldquovery poorrdquo to

6 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

2 The Scientific World Journal

Preference by Similarity to Ideal Solution or TOPSIS [9] aresome of the numerous examples on MCDM methods thatparticularly have been used by the decision makers

Distance measure can be identified as one of the MCDMapproaches that can be used in personnel selection processThis approach holds an important key to solve many prob-lems related to biology science social and technology due toits capability of constructing some related distance measuresnotably similarity and proximity which always become anorm in various problems [10] In recent years the study ofthis method has been rapidly growing in which it resultedin proposing and improving the previous distance measuremethods Some of the well-known distancemeasuremethodsare Hamming Euclidean Hausdorff and Minkowski meth-ods Based on the existing literatures Hamming distance isone of the methods that can be used in personnel selectionprocess [11ndash13]This method was proposed by Hamming [14]in 1950 to count the number of flipping bits in a fixed-lengthbinary word as an estimate of error used in telecommunica-tion Hamming distance is known for its ability in calculatingthe difference between two sets or elements For example thedistance between interval-valued fuzzy sets Consequentlyapart from the decision making problem it also has beenapplied in various fields such as communication [15] irisrecognition [16] and engineering [17]

Literally evaluation of certain criteria or attributes toselect an appropriate alternative for specified position couldbecome tremendous and challenging task for the decisionmakers It is because some of the criteria such as leadershippersonality and creativity are referred to as qualitative criteriain which exhibits imprecise and vagueness data In generalthis uncertainty and subjective scene that occurs during theevaluation of the alternatives based on respective criteria andcriteria weight may come from various sources includingunquantifiable information incomplete information unob-tainable information and partial ignorance [18] For this situ-ation commonly classical MCDM will be put aside since thealternatives rating and criteria weights for classical MCDMare usually measured in crisp numbers Therefore one ofthe best resorts to solve this problem is by applying fuzzyset theory The fuzzy set theory is known for its flexibilityin handling imprecise and uncertainty in human judgmentsBellman and Zadeh [19] had introduced the use of fuzzy settheory inMCDM and it proved to be an effective approach indealing with uncertainty in human decision making processSince then it had become an important tool in constructinga decision making framework that incorporates subjectivejudgments that entails in the personnel selection process

Themain objective of this paper is to propose an approachto solve personnel selection process by using Hammingdistance method Inspired by algorithm proposed by Canoset al [11] we extend and improve Canosrsquos algorithm byadding weight in the classical Hamming distance In ourproposed method we suggest two types of weight whichare subjective and objective weights The linguistic termscorrespondence to triangular fuzzy numbers are used toevaluate the performance rating values as well as the weightof the criteria in which later will be expressed into intervalvalued fuzzy numbers In this approach we also identify the

changes in ranking of the alternatives when different valuesof 120572 are used The remaining of this paper is organized asfollows The next section we briefly explain the preliminaryconcerning fuzzy set and Hamming distance Section 3 willbriefly explain about the Hamming distance method andsubjective and objective weights In Section 4 we proposea new algorithm for personnel selection problem The newalgorithm is called HDMSOWrsquos Section 5 validates theHDMSOWrsquos by conducting a numerical example The lastsection concludes this paper

2 Preliminaries

A fuzzy set 119860 in 119883 is defined as a set of ordered pairs (see[20])

119860 = ⟨119909 120583119860 (119909)⟩ 119909 isin 119883 (1)

where 119883 is denoted as a universe of discourse and 120583119860

(119909) isthe membership function of 119860 defined as

120583119860

119883 997888rarr [0 1] (2)

A triangular fuzzy number is specified by three parametersand can be defined as triplet 119860 = (119886

1 1198862 1198863) where 119886

1lt 1198862

lt

1198863with the 119909 = 119886

2as the core of the triangle Its membership

function can be represented as [21]

120583119860 (119909) =

0 119909 lt 1198861

(119909 minus 1198861)

(1198862

minus 1198861)

1198861

le 119909 le 1198862

(1198863

minus 119909)

(1198863

minus 1198862)

1198862

le 119909 le 1198863

0 119909 gt 1198863

(3)

The 120572-cuts of this fuzzy number 119860 are denoted by

[119860]120572= [1198861

+ 120572 (1198862

minus 1198861) 1198863

minus 120572 (1198863

minus 1198862)] 120572 isin (0 1]

(4)

An interval-valued fuzzy set 119860 in universe discourse 119883 isdenoted by (see [22 23])

119860 = (119909 [120583119871

119860(119909) 120583

119880

119860(119909)]) | 119909 isin 119883 (5)

where 120583119871

119860(119909) 120583

119880

119860(119909) 119883 rarr [0 1] 120583

119871

119860(119909) is lower bound and

120583119880

119860(119909) is upper bound of membershipThemultiplication of two interval-valued fuzzy numbers

119860 = [119886119871 119886119880

] and 119861 = [119887119871 119887119880

] can be defined as (see [21])

119860 sdot 119861 = [119886119871 119886119880

] sdot [119887119871 119887119880

] = [119888119871 119888119880

] (6)

where

119888119871

= min 119886119871119887119871 119886119871119887119880

119886119880

119887119871 119886119880

119887119880

119888119880

= max 119886119871119887119871 119886119871119887119880

119886119880

119887119871 119886119880

119887119880

(7)

Hamming distance methods to be used in this paper arepresented as follows

The Scientific World Journal 3

Definition 1 (see [24]) Given two fuzzy subsets of 119860 and 119861

with a reference set 119883 = 1199091 1199092 119909

119899 and memberships

function 120583119860and 120583

119861

Then the Hamming distance is defined as

119889 (119860 119861) =

119899

sum

119895=1

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (8)

The normalized Hamming distance for two interval-valuedfuzzy numbers 119860 and 119861 whose membership functions are asfollows

120583119860

(119909119895) = [119886

119871

119909119895 119886119880

119909119895] 120583

119861(119909119895) = [119887

119871

119909119895 119887119880

119909119895]

119895 = 1 2 119899

(9)

is defined as

119889NHD (119860 119861) =

1

2119899

(

119899

sum

119895=1

(

100381610038161003816100381610038161003816

119886119871

119909119895minus 119887119871

119909119895

100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816

119886119880

119909119895minus 119887119880

119909119895

100381610038161003816100381610038161003816

)) (10)

Definition 2 (see [25]) The weighted Hamming distance ofdimension 119899 is a mapping 119889WHD [0 1]

119899times [0 1]

119899rarr [0 1]

that associated with weighting vector 119882 of dimension 119899 with119882 = sum

119899

119895=1 119908119895

= 1 and 119908119895

isin [0 1] Then the weightedHamming distance is defined as

119889WHD (119860 119861) =

119899

sum

119895=1

119908119895

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (11)

According to [12] the weighted Hamming distance can be thenormalized Hamming distance if 119908

119895= 1119899 for 119895 = 1 2 119899

3 Hamming Distance Method andSubjective and Objective Weights

31 Hamming Distance Method Hamming distance is oneof the distance measures that can be applied in personnelselection process This is due to its ability in calculatingthe distance between ideal alternative and alternative Theideal alternative is a virtual alternative in which the criteriavalues are expressed as close as possible to ideal valueswhich is rationale for human thinking to achieve There areseveral methods that focus on identifying and measuring theideal alternative However this measurement is beyond ourscope of research In this paper the evaluation on the idealalternative is made based on assumption of the optimumvalue of each criterion that alternatives should achieve forthe specified job We also disregard the usage of maximumvalue for example (1 1 1) in case of the triangular fuzzynumber of all criteria evaluations Rationally it is hard forthe alternatives to achieve a perfect score for some criteriaespecially when the evaluation of the criteria itself is madefrom human based judgment that mostly in subjective termscould be varied from one person to others The rankingof alternatives is made through the comparison betweenthe alternatives and the ideal alternative [26] such that thealternatives with the minimum distance values are likely to

be selected However when the distance values between thealternatives are the same the decision makers will face aproblem in ranking them Thus with the help of weights itwill help decision makers to distinguish between the criteriathat valued the most for the specified job than the othercriteria

32 Subjective and Objective Weights The decision makersare genuinely aware that they cannot assume that all criteriaare equally important as it holds its own meaning andneediness especially when its focus is only to one subjector position For example when recruiting the appropriateapplicant for position credit officer the criteria that might bevaluedmost are experienced in credit analysis andpersonalityassessment Generally the other criteria are also valuablebut they are not as important as these two criteria Plusit is a human nature to have diverse opinion in evaluatingprocessThus it is undeniable that the criteria weight plays animportant role in MCDM problem as it depicted the relativeweightiness of the criteria must be assigned [3] Alternativelynumerous approaches has been generalized and introducedto solve this problemThese methods can be categorized intotwo groups which are subjective and objective weights

The subjective weight are determined solely based on thepreference of the decision makers [27 28] These evaluationsare basically based on experience perception and knowledge[29] In a general view it is a process of assigning subjectivepreferences to the criteria [29] AHP method eigenvectormethod and weighted least square method can be used tocalculate this approach Beside objective weight measuredthe weight with the use of mathematical models such asentropy method [30] and multiple objective programming[31]This approach solves without any consideration from thedecision makers preference The use of objective weight canovercome some of the limitations in subjective weight suchas inconsistency problem in subjective weight Furthermoreit is useful when the reliable subjective weight is not available[29]

One of the objective weighting measure that vastly hasbeen used in MCDM field is Shannonrsquos entropy concept[32] Shannonrsquos entropy concept is a general measure ofuncertainty in information formulated in terms of probabilitytheory [30] This concept is appropriate for calculating therelative contrast intensities of criteria to represent the averageintrinsic information transmitted to the decision maker [33]It began when Shannon first introduced the application ofentropy in communication theory and since then he hadcontributed the most fundamental definition of the entropymeasure in the information theory [34] This concept hadbeen applied in wide range area exemplified mathematics[35] spectral analysis [36] and economics [37] Entropyweight is a parameter that describes how much diversealternatives approach one another with respect to a certaincriteria [3 28] This concept is also relatively known in themeasurement of fuzziness [38] Hence this method is suitableto be applied in our approach as we will deal with fuzzydata Apart from that the total weights for all criteria valueswill equal to one in which satisfy the condition that need inweighted Hamming distance method

4 The Scientific World Journal

4 Hamming Distance Method withSubjective and Objective Weights

In this section the description and algorithm for the HDM-SOWs is constructed To our best knowledge the studyof using a weighted Hamming distance method in solvingpersonnel selection problem has rarely been done Merigoand Gil-Lafuente [12] had presented a study involving theuse of weighted Hamming distance method integrated withOrdered Weighted Averaging (OWA) but without the useof fuzzy numbers Hence we would like to expand the useof weighted Hamming distance in personnel selection byusing fuzzy data and we propose two types of weights whichare subjective and objective weights The elements of thisHDMSOWs can be presented in the following descriptions

Let us assume that there is a set of 119898 possible alternatives119860 = 119860

1 1198602 119860

119898 to be evaluated based on a set of 119899

respective criteria 119862 = 1198621 1198622 119862

119899 These evaluations

are done by a set of 119898 decision makers 119864 = 1198641 1198642 119864

119898

by using linguistic variables To capture the linguistic termswe use triangular fuzzy numbers The linguistic variablesare divided into two categories which are the evaluationon criteria weight and the evaluation on criteria The givenalgorithm is unfolded as follows

Step 1 (construct a decision matrix for ideal alternative) Thedecision matrix for ideal alternative is given as follows

119868 = [V1 V2 V

119899] (12)

The ideal alternativematrix represents the optimum values of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 that the alternatives

should achieve These values are set up by decision makers

Step 2 (construct a decision matrix for alternatives) Thedecision matrix for performance alternatives is given asfollows

119863 =

1198621

1198622

sdot sdot sdot 119862119899

1198601

1198602

119860119898

[

[

[

[

[

11990911

11990912

sdot sdot sdot 1199091119899

11990921

11990922

sdot sdot sdot 1199092119899

sdot sdot sdot

1199091198981

1199091198982

sdot sdot sdot 119909119898119899

]

]

]

]

]

(13)

where 119909119894119895represent the linguistic assessment on the utility

ratings of alternative 119860119894

(119894 = 1 2 119898) with respect to119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers

Step 3 (construct a decision matrix for weight (criteriaimportance)) The weighting matrix for criteria weight 119908

119894119895

evaluated by the decision makers 119864119894

(119894 = 1 2 119898) is givenas follows

119882 =

1198641

1198642

119864119898

1198621

1198622

sdot sdot sdot 119862119899

[

[

[

[

[

11990811

11990812

sdot sdot sdot 1199081119899

11990821

11990822

sdot sdot sdot 1199082119899

sdot sdot sdot

1199081198981

1199081198982

sdot sdot sdot 119908119898119899

]

]

]

]

]

(14)

The weighting matrix represents the relative importance of119899 selection criteria 119862

119895(119895 = 1 2 119899) given by the decision

makers

Step 4 (construct an interval-valued fuzzy number) By using120572-cut of triangular fuzzy number the interval performancematrix for alternatives ideal alternatives and criteria weightare derived as follows respectively

(i) The interval decision matrix for the ideal alternative

119868120572

= [[(V1)119871

120572 (V1)119880

120572] [(V2)119871

120572 (V2)119880

120572] [(V

119899)119871

120572 (V119899)119880

120572]]

(15)

(ii) The interval decisionmatrix for performance alterna-tives

119863120572

=

[

[

[

[

[

[(11990911)119871

120572 (11990911)

119880

120572] [(11990912)

119871

120572 (11990912)

119880

120572] sdot sdot sdot [(1199091119899)

119871

120572 (1199091119899)

119880

120572]

[(11990921)119871

120572 (11990921)

119880

120572] [(11990922)

119871

120572 (11990922)

119880

120572] sdot sdot sdot [(1199092119899)

119871

120572 (1199092119899)

119880

120572]

sdot sdot sdot

[(1199091198981)

119871

120572 (1199091198981)

119880

120572] [(1199091198982)

119871

120572 (1199091198982)

119880

120572] sdot sdot sdot [(119909119898119899)

119871

120572 (119909119898119899)

119880

120572]

]

]

]

]

]

(16)

(iii) The interval decision matrix for criteria weight

119882120572

=

[

[

[

[

[

[(11990811)119871

120572 (11990811)

119880

120572] [(11990812)

119871

120572 (11990812)

119880

120572] sdot sdot sdot [(1199081119899)

119871

120572 (1199081119899)

119880

120572]

[(11990821)119871

120572 (11990821)

119880

120572] [(11990822)

119871

120572 (11990822)

119880

120572] sdot sdot sdot [(1199082119899)

119871

120572 (1199082119899)

119880

120572]

sdot sdot sdot

[(1199081198981)

119871

120572 (1199081198981)

119880

120572] [(1199081198982)

119871

120572 (1199081198982)

119880

120572] sdot sdot sdot [(119908119898119899)

119871

120572 (119908119898119899)

119880

120572]

]

]

]

]

]

(17)

where 0 le 120572 le 1 The value of 120572 represents the degreeof confidences in the decision makersrsquo assessmentwith respect to ideal alternative alternatives ratingand criteria weights

Step 5 (calculating of criteria weight) The criteria weight of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers will be calculated using twomethods whichare subjective and objective weights

(a) Subjective weightThe subjective weight of 119899 selectioncriteria 119862 = 119862

1 1198622 119862

119899 may be considered as the

average weights [9] and its calculation is [9 28]

119908119895

=

1

119898

(

119898

sum

119894=1

119908119894119895

) 119894 = 1 2 119898 119895 = 1 2 119899 (18)

(b) Objective weight The interval valued fuzzy numberis transformed into crisp number before using Shan-nonrsquos entropy concept

The crisp value of interval weight is given by [39]

119908119894119895

=

(119908119897

119894119895+ 119908119906

119894119895)

2

(19)

Then Shannonrsquos entropy concept is used to obtain the weight

The Scientific World Journal 5

The details of Shannonrsquos entropy concept are defined asfollows [27 39]

Step 51 Normalized each criterion weight to obtain theprojection value 119901

119894119895

119901119894119895

=

119908119894119895

sum119898

119894=1119908119894119895

119894 = 1 119898 119895 = 1 119899 (20)

Consequently a projection matrix representing a relativeweight of each criterion from the decision maker evaluationis expressed as

119875 =

[

[

[

[

[

11990111

11990112

sdot sdot sdot 1199011119899

11990121

11990122

sdot sdot sdot 1199012119899

sdot sdot sdot

1199011198981

1199011198982

sdot sdot sdot 119901119898119899

]

]

]

]

]

(21)

Step 52 Calculate entropy values 119890119895as

119890119895

= minus119896

119898

sum

119894=1

119901119894119895ln119901119894119895

119895 = 1 119899 (22)

where 119896 is constant and let 119896 = (ln119898)minus1 If 119901

119894119895= 0 then

119901119894119895ln119901119894119895is equal to 0

Step 53 Calculate the degree of diversification 119889119895

119889119895

= 1 minus 119890119895 119895 = 1 119899 (23)

Step 54 Calculate the criteria weight 119908119895

119908119895

=

119889119895

sum119899

119896=1119889119896

(24)

Step 6 (calculating the distance values) (a) Subjective weightBefore calculating the distance values calculate the overallperformance evaluation of ideal alternatives and alternativesby multiplying the aggregate weight with each criterion [21]

For the ideal alternative

119877119895

= [V119871119895 V119880119895

] sdot [119908119871

119895 119908119880

119895] = [119903

119871

119895 119903119880

119895] (25)

and for the alternatives119878119894119895

= [119909119871

119894119895 119909119880

119894119895] sdot [119908119871

119895 119908119880

119895]

= [119904119871

119894119895 119904119880

119894119895]

(26)

where119903119871

= min V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119903119880

= max V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119904119871

= min 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

119904119880

= max 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

(27)

Then calculate the distance values between the ideal alterna-tives with the alternatives by using Definition 1

119889NHD (119877 119878) =

1

2119899

(

119899

sum

119895=1

(

10038161003816100381610038161003816119903119871

119895minus 119904119871

119894119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816119903119880

119895minus 119904119880

119894119895

10038161003816100381610038161003816)) (28)

(b) Objective weight For the objective weight the dis-tance values are calculated by using Definition 2

119889WHD (119868 119863) =

119899

sum

119895=1

(119908119895

10038161003816100381610038161003816V119871119895

minus 119909119871

119894119895

10038161003816100381610038161003816

+ 119908119895

10038161003816100381610038161003816V119880119895

minus 119909119880

119894119895

10038161003816100381610038161003816) (29)

Step 7 (ranking the candidate) The alternatives are rankedin ascending order according to the distance values forrespective 120572 values The alternative with the less distancevalue is considered as the best choice

Step 8 (repeat Steps 4 5 and 6 for different values of 120572) Thealternatives are ranking according to the different values of 120572

Step 9 (selection of the appropriate alternative by the decisionmakers)

5 A Numerical Example

An example on the personnel selection in an academicinstitution is provided to validate the proposed algorithmSuppose that the academic institution intends to employ alecturer based on consideration of four main criteria whichare experienced in teaching areas (119862

1) proficiency in per-

forming research (1198622) personality assessment (119862

3) and past

contribution (1198624) Assume that after preliminary selection

phase four alternatives 1198601 1198602 1198603 and 119860

4are qualified for

final evaluation A committee of experts (decision makers)consisting of three persons is formed namely119863

11198632 and119863

3

The information of this study is given in Figures 1 and 2 andTables 1 2 3 4 and 5 while the results from the numericalexamples are shown in Tables 6 7 8 9 and 10 As mentionedbefore TOPSIS is one of the existing MCDM methods thatcan be used to solve personnel selection problemThus it canbe used to validate the proposed method and the results byusing this method that is shown in Table 11 More explanationon these figures and tables are explained in the discussionsection

51 Discussion Based on the results obtained the proposedHDMSOWs can be summarized as follows

Step 1 Ideal alternative matrix (12) is built from the evalua-tions of the criteria based on linguistic variables taken fromWang and Lee [28] as illustrated in Figure 2 and Table 2 Thelinguistic terms are represented by triangular fuzzy numberranging from ldquovery poorrdquo to ldquovery goodrdquo Table 4 showsthe decision makers evaluation on ideal alternative In thispaper we assume that the 119898 decision makers had come toan agreement in standardizing into one final value for eachcriterion

Step 2 Decision matrix for alternatives evaluation on eachcriterion (13) is obtained by using the same linguistic variablesadopted from Wang and Lee [28] as illustrated in Figure 2and Table 2 Similar to Step 1 these terms are captured inthe form of the triangular fuzzy number The alternativesperformance evaluations are ranging from ldquovery poorrdquo to

6 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

The Scientific World Journal 3

Definition 1 (see [24]) Given two fuzzy subsets of 119860 and 119861

with a reference set 119883 = 1199091 1199092 119909

119899 and memberships

function 120583119860and 120583

119861

Then the Hamming distance is defined as

119889 (119860 119861) =

119899

sum

119895=1

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (8)

The normalized Hamming distance for two interval-valuedfuzzy numbers 119860 and 119861 whose membership functions are asfollows

120583119860

(119909119895) = [119886

119871

119909119895 119886119880

119909119895] 120583

119861(119909119895) = [119887

119871

119909119895 119887119880

119909119895]

119895 = 1 2 119899

(9)

is defined as

119889NHD (119860 119861) =

1

2119899

(

119899

sum

119895=1

(

100381610038161003816100381610038161003816

119886119871

119909119895minus 119887119871

119909119895

100381610038161003816100381610038161003816

+

100381610038161003816100381610038161003816

119886119880

119909119895minus 119887119880

119909119895

100381610038161003816100381610038161003816

)) (10)

Definition 2 (see [25]) The weighted Hamming distance ofdimension 119899 is a mapping 119889WHD [0 1]

119899times [0 1]

119899rarr [0 1]

that associated with weighting vector 119882 of dimension 119899 with119882 = sum

119899

119895=1 119908119895

= 1 and 119908119895

isin [0 1] Then the weightedHamming distance is defined as

119889WHD (119860 119861) =

119899

sum

119895=1

119908119895

10038161003816100381610038161003816120583119860

(119909119895) minus 120583119861

(119909119895)

10038161003816100381610038161003816 (11)

According to [12] the weighted Hamming distance can be thenormalized Hamming distance if 119908

119895= 1119899 for 119895 = 1 2 119899

3 Hamming Distance Method andSubjective and Objective Weights

31 Hamming Distance Method Hamming distance is oneof the distance measures that can be applied in personnelselection process This is due to its ability in calculatingthe distance between ideal alternative and alternative Theideal alternative is a virtual alternative in which the criteriavalues are expressed as close as possible to ideal valueswhich is rationale for human thinking to achieve There areseveral methods that focus on identifying and measuring theideal alternative However this measurement is beyond ourscope of research In this paper the evaluation on the idealalternative is made based on assumption of the optimumvalue of each criterion that alternatives should achieve forthe specified job We also disregard the usage of maximumvalue for example (1 1 1) in case of the triangular fuzzynumber of all criteria evaluations Rationally it is hard forthe alternatives to achieve a perfect score for some criteriaespecially when the evaluation of the criteria itself is madefrom human based judgment that mostly in subjective termscould be varied from one person to others The rankingof alternatives is made through the comparison betweenthe alternatives and the ideal alternative [26] such that thealternatives with the minimum distance values are likely to

be selected However when the distance values between thealternatives are the same the decision makers will face aproblem in ranking them Thus with the help of weights itwill help decision makers to distinguish between the criteriathat valued the most for the specified job than the othercriteria

32 Subjective and Objective Weights The decision makersare genuinely aware that they cannot assume that all criteriaare equally important as it holds its own meaning andneediness especially when its focus is only to one subjector position For example when recruiting the appropriateapplicant for position credit officer the criteria that might bevaluedmost are experienced in credit analysis andpersonalityassessment Generally the other criteria are also valuablebut they are not as important as these two criteria Plusit is a human nature to have diverse opinion in evaluatingprocessThus it is undeniable that the criteria weight plays animportant role in MCDM problem as it depicted the relativeweightiness of the criteria must be assigned [3] Alternativelynumerous approaches has been generalized and introducedto solve this problemThese methods can be categorized intotwo groups which are subjective and objective weights

The subjective weight are determined solely based on thepreference of the decision makers [27 28] These evaluationsare basically based on experience perception and knowledge[29] In a general view it is a process of assigning subjectivepreferences to the criteria [29] AHP method eigenvectormethod and weighted least square method can be used tocalculate this approach Beside objective weight measuredthe weight with the use of mathematical models such asentropy method [30] and multiple objective programming[31]This approach solves without any consideration from thedecision makers preference The use of objective weight canovercome some of the limitations in subjective weight suchas inconsistency problem in subjective weight Furthermoreit is useful when the reliable subjective weight is not available[29]

One of the objective weighting measure that vastly hasbeen used in MCDM field is Shannonrsquos entropy concept[32] Shannonrsquos entropy concept is a general measure ofuncertainty in information formulated in terms of probabilitytheory [30] This concept is appropriate for calculating therelative contrast intensities of criteria to represent the averageintrinsic information transmitted to the decision maker [33]It began when Shannon first introduced the application ofentropy in communication theory and since then he hadcontributed the most fundamental definition of the entropymeasure in the information theory [34] This concept hadbeen applied in wide range area exemplified mathematics[35] spectral analysis [36] and economics [37] Entropyweight is a parameter that describes how much diversealternatives approach one another with respect to a certaincriteria [3 28] This concept is also relatively known in themeasurement of fuzziness [38] Hence this method is suitableto be applied in our approach as we will deal with fuzzydata Apart from that the total weights for all criteria valueswill equal to one in which satisfy the condition that need inweighted Hamming distance method

4 The Scientific World Journal

4 Hamming Distance Method withSubjective and Objective Weights

In this section the description and algorithm for the HDM-SOWs is constructed To our best knowledge the studyof using a weighted Hamming distance method in solvingpersonnel selection problem has rarely been done Merigoand Gil-Lafuente [12] had presented a study involving theuse of weighted Hamming distance method integrated withOrdered Weighted Averaging (OWA) but without the useof fuzzy numbers Hence we would like to expand the useof weighted Hamming distance in personnel selection byusing fuzzy data and we propose two types of weights whichare subjective and objective weights The elements of thisHDMSOWs can be presented in the following descriptions

Let us assume that there is a set of 119898 possible alternatives119860 = 119860

1 1198602 119860

119898 to be evaluated based on a set of 119899

respective criteria 119862 = 1198621 1198622 119862

119899 These evaluations

are done by a set of 119898 decision makers 119864 = 1198641 1198642 119864

119898

by using linguistic variables To capture the linguistic termswe use triangular fuzzy numbers The linguistic variablesare divided into two categories which are the evaluationon criteria weight and the evaluation on criteria The givenalgorithm is unfolded as follows

Step 1 (construct a decision matrix for ideal alternative) Thedecision matrix for ideal alternative is given as follows

119868 = [V1 V2 V

119899] (12)

The ideal alternativematrix represents the optimum values of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 that the alternatives

should achieve These values are set up by decision makers

Step 2 (construct a decision matrix for alternatives) Thedecision matrix for performance alternatives is given asfollows

119863 =

1198621

1198622

sdot sdot sdot 119862119899

1198601

1198602

119860119898

[

[

[

[

[

11990911

11990912

sdot sdot sdot 1199091119899

11990921

11990922

sdot sdot sdot 1199092119899

sdot sdot sdot

1199091198981

1199091198982

sdot sdot sdot 119909119898119899

]

]

]

]

]

(13)

where 119909119894119895represent the linguistic assessment on the utility

ratings of alternative 119860119894

(119894 = 1 2 119898) with respect to119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers

Step 3 (construct a decision matrix for weight (criteriaimportance)) The weighting matrix for criteria weight 119908

119894119895

evaluated by the decision makers 119864119894

(119894 = 1 2 119898) is givenas follows

119882 =

1198641

1198642

119864119898

1198621

1198622

sdot sdot sdot 119862119899

[

[

[

[

[

11990811

11990812

sdot sdot sdot 1199081119899

11990821

11990822

sdot sdot sdot 1199082119899

sdot sdot sdot

1199081198981

1199081198982

sdot sdot sdot 119908119898119899

]

]

]

]

]

(14)

The weighting matrix represents the relative importance of119899 selection criteria 119862

119895(119895 = 1 2 119899) given by the decision

makers

Step 4 (construct an interval-valued fuzzy number) By using120572-cut of triangular fuzzy number the interval performancematrix for alternatives ideal alternatives and criteria weightare derived as follows respectively

(i) The interval decision matrix for the ideal alternative

119868120572

= [[(V1)119871

120572 (V1)119880

120572] [(V2)119871

120572 (V2)119880

120572] [(V

119899)119871

120572 (V119899)119880

120572]]

(15)

(ii) The interval decisionmatrix for performance alterna-tives

119863120572

=

[

[

[

[

[

[(11990911)119871

120572 (11990911)

119880

120572] [(11990912)

119871

120572 (11990912)

119880

120572] sdot sdot sdot [(1199091119899)

119871

120572 (1199091119899)

119880

120572]

[(11990921)119871

120572 (11990921)

119880

120572] [(11990922)

119871

120572 (11990922)

119880

120572] sdot sdot sdot [(1199092119899)

119871

120572 (1199092119899)

119880

120572]

sdot sdot sdot

[(1199091198981)

119871

120572 (1199091198981)

119880

120572] [(1199091198982)

119871

120572 (1199091198982)

119880

120572] sdot sdot sdot [(119909119898119899)

119871

120572 (119909119898119899)

119880

120572]

]

]

]

]

]

(16)

(iii) The interval decision matrix for criteria weight

119882120572

=

[

[

[

[

[

[(11990811)119871

120572 (11990811)

119880

120572] [(11990812)

119871

120572 (11990812)

119880

120572] sdot sdot sdot [(1199081119899)

119871

120572 (1199081119899)

119880

120572]

[(11990821)119871

120572 (11990821)

119880

120572] [(11990822)

119871

120572 (11990822)

119880

120572] sdot sdot sdot [(1199082119899)

119871

120572 (1199082119899)

119880

120572]

sdot sdot sdot

[(1199081198981)

119871

120572 (1199081198981)

119880

120572] [(1199081198982)

119871

120572 (1199081198982)

119880

120572] sdot sdot sdot [(119908119898119899)

119871

120572 (119908119898119899)

119880

120572]

]

]

]

]

]

(17)

where 0 le 120572 le 1 The value of 120572 represents the degreeof confidences in the decision makersrsquo assessmentwith respect to ideal alternative alternatives ratingand criteria weights

Step 5 (calculating of criteria weight) The criteria weight of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers will be calculated using twomethods whichare subjective and objective weights

(a) Subjective weightThe subjective weight of 119899 selectioncriteria 119862 = 119862

1 1198622 119862

119899 may be considered as the

average weights [9] and its calculation is [9 28]

119908119895

=

1

119898

(

119898

sum

119894=1

119908119894119895

) 119894 = 1 2 119898 119895 = 1 2 119899 (18)

(b) Objective weight The interval valued fuzzy numberis transformed into crisp number before using Shan-nonrsquos entropy concept

The crisp value of interval weight is given by [39]

119908119894119895

=

(119908119897

119894119895+ 119908119906

119894119895)

2

(19)

Then Shannonrsquos entropy concept is used to obtain the weight

The Scientific World Journal 5

The details of Shannonrsquos entropy concept are defined asfollows [27 39]

Step 51 Normalized each criterion weight to obtain theprojection value 119901

119894119895

119901119894119895

=

119908119894119895

sum119898

119894=1119908119894119895

119894 = 1 119898 119895 = 1 119899 (20)

Consequently a projection matrix representing a relativeweight of each criterion from the decision maker evaluationis expressed as

119875 =

[

[

[

[

[

11990111

11990112

sdot sdot sdot 1199011119899

11990121

11990122

sdot sdot sdot 1199012119899

sdot sdot sdot

1199011198981

1199011198982

sdot sdot sdot 119901119898119899

]

]

]

]

]

(21)

Step 52 Calculate entropy values 119890119895as

119890119895

= minus119896

119898

sum

119894=1

119901119894119895ln119901119894119895

119895 = 1 119899 (22)

where 119896 is constant and let 119896 = (ln119898)minus1 If 119901

119894119895= 0 then

119901119894119895ln119901119894119895is equal to 0

Step 53 Calculate the degree of diversification 119889119895

119889119895

= 1 minus 119890119895 119895 = 1 119899 (23)

Step 54 Calculate the criteria weight 119908119895

119908119895

=

119889119895

sum119899

119896=1119889119896

(24)

Step 6 (calculating the distance values) (a) Subjective weightBefore calculating the distance values calculate the overallperformance evaluation of ideal alternatives and alternativesby multiplying the aggregate weight with each criterion [21]

For the ideal alternative

119877119895

= [V119871119895 V119880119895

] sdot [119908119871

119895 119908119880

119895] = [119903

119871

119895 119903119880

119895] (25)

and for the alternatives119878119894119895

= [119909119871

119894119895 119909119880

119894119895] sdot [119908119871

119895 119908119880

119895]

= [119904119871

119894119895 119904119880

119894119895]

(26)

where119903119871

= min V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119903119880

= max V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119904119871

= min 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

119904119880

= max 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

(27)

Then calculate the distance values between the ideal alterna-tives with the alternatives by using Definition 1

119889NHD (119877 119878) =

1

2119899

(

119899

sum

119895=1

(

10038161003816100381610038161003816119903119871

119895minus 119904119871

119894119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816119903119880

119895minus 119904119880

119894119895

10038161003816100381610038161003816)) (28)

(b) Objective weight For the objective weight the dis-tance values are calculated by using Definition 2

119889WHD (119868 119863) =

119899

sum

119895=1

(119908119895

10038161003816100381610038161003816V119871119895

minus 119909119871

119894119895

10038161003816100381610038161003816

+ 119908119895

10038161003816100381610038161003816V119880119895

minus 119909119880

119894119895

10038161003816100381610038161003816) (29)

Step 7 (ranking the candidate) The alternatives are rankedin ascending order according to the distance values forrespective 120572 values The alternative with the less distancevalue is considered as the best choice

Step 8 (repeat Steps 4 5 and 6 for different values of 120572) Thealternatives are ranking according to the different values of 120572

Step 9 (selection of the appropriate alternative by the decisionmakers)

5 A Numerical Example

An example on the personnel selection in an academicinstitution is provided to validate the proposed algorithmSuppose that the academic institution intends to employ alecturer based on consideration of four main criteria whichare experienced in teaching areas (119862

1) proficiency in per-

forming research (1198622) personality assessment (119862

3) and past

contribution (1198624) Assume that after preliminary selection

phase four alternatives 1198601 1198602 1198603 and 119860

4are qualified for

final evaluation A committee of experts (decision makers)consisting of three persons is formed namely119863

11198632 and119863

3

The information of this study is given in Figures 1 and 2 andTables 1 2 3 4 and 5 while the results from the numericalexamples are shown in Tables 6 7 8 9 and 10 As mentionedbefore TOPSIS is one of the existing MCDM methods thatcan be used to solve personnel selection problemThus it canbe used to validate the proposed method and the results byusing this method that is shown in Table 11 More explanationon these figures and tables are explained in the discussionsection

51 Discussion Based on the results obtained the proposedHDMSOWs can be summarized as follows

Step 1 Ideal alternative matrix (12) is built from the evalua-tions of the criteria based on linguistic variables taken fromWang and Lee [28] as illustrated in Figure 2 and Table 2 Thelinguistic terms are represented by triangular fuzzy numberranging from ldquovery poorrdquo to ldquovery goodrdquo Table 4 showsthe decision makers evaluation on ideal alternative In thispaper we assume that the 119898 decision makers had come toan agreement in standardizing into one final value for eachcriterion

Step 2 Decision matrix for alternatives evaluation on eachcriterion (13) is obtained by using the same linguistic variablesadopted from Wang and Lee [28] as illustrated in Figure 2and Table 2 Similar to Step 1 these terms are captured inthe form of the triangular fuzzy number The alternativesperformance evaluations are ranging from ldquovery poorrdquo to

6 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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 The Scientific World Journal

4 Hamming Distance Method withSubjective and Objective Weights

In this section the description and algorithm for the HDM-SOWs is constructed To our best knowledge the studyof using a weighted Hamming distance method in solvingpersonnel selection problem has rarely been done Merigoand Gil-Lafuente [12] had presented a study involving theuse of weighted Hamming distance method integrated withOrdered Weighted Averaging (OWA) but without the useof fuzzy numbers Hence we would like to expand the useof weighted Hamming distance in personnel selection byusing fuzzy data and we propose two types of weights whichare subjective and objective weights The elements of thisHDMSOWs can be presented in the following descriptions

Let us assume that there is a set of 119898 possible alternatives119860 = 119860

1 1198602 119860

119898 to be evaluated based on a set of 119899

respective criteria 119862 = 1198621 1198622 119862

119899 These evaluations

are done by a set of 119898 decision makers 119864 = 1198641 1198642 119864

119898

by using linguistic variables To capture the linguistic termswe use triangular fuzzy numbers The linguistic variablesare divided into two categories which are the evaluationon criteria weight and the evaluation on criteria The givenalgorithm is unfolded as follows

Step 1 (construct a decision matrix for ideal alternative) Thedecision matrix for ideal alternative is given as follows

119868 = [V1 V2 V

119899] (12)

The ideal alternativematrix represents the optimum values of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 that the alternatives

should achieve These values are set up by decision makers

Step 2 (construct a decision matrix for alternatives) Thedecision matrix for performance alternatives is given asfollows

119863 =

1198621

1198622

sdot sdot sdot 119862119899

1198601

1198602

119860119898

[

[

[

[

[

11990911

11990912

sdot sdot sdot 1199091119899

11990921

11990922

sdot sdot sdot 1199092119899

sdot sdot sdot

1199091198981

1199091198982

sdot sdot sdot 119909119898119899

]

]

]

]

]

(13)

where 119909119894119895represent the linguistic assessment on the utility

ratings of alternative 119860119894

(119894 = 1 2 119898) with respect to119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers

Step 3 (construct a decision matrix for weight (criteriaimportance)) The weighting matrix for criteria weight 119908

119894119895

evaluated by the decision makers 119864119894

(119894 = 1 2 119898) is givenas follows

119882 =

1198641

1198642

119864119898

1198621

1198622

sdot sdot sdot 119862119899

[

[

[

[

[

11990811

11990812

sdot sdot sdot 1199081119899

11990821

11990822

sdot sdot sdot 1199082119899

sdot sdot sdot

1199081198981

1199081198982

sdot sdot sdot 119908119898119899

]

]

]

]

]

(14)

The weighting matrix represents the relative importance of119899 selection criteria 119862

119895(119895 = 1 2 119899) given by the decision

makers

Step 4 (construct an interval-valued fuzzy number) By using120572-cut of triangular fuzzy number the interval performancematrix for alternatives ideal alternatives and criteria weightare derived as follows respectively

(i) The interval decision matrix for the ideal alternative

119868120572

= [[(V1)119871

120572 (V1)119880

120572] [(V2)119871

120572 (V2)119880

120572] [(V

119899)119871

120572 (V119899)119880

120572]]

(15)

(ii) The interval decisionmatrix for performance alterna-tives

119863120572

=

[

[

[

[

[

[(11990911)119871

120572 (11990911)

119880

120572] [(11990912)

119871

120572 (11990912)

119880

120572] sdot sdot sdot [(1199091119899)

119871

120572 (1199091119899)

119880

120572]

[(11990921)119871

120572 (11990921)

119880

120572] [(11990922)

119871

120572 (11990922)

119880

120572] sdot sdot sdot [(1199092119899)

119871

120572 (1199092119899)

119880

120572]

sdot sdot sdot

[(1199091198981)

119871

120572 (1199091198981)

119880

120572] [(1199091198982)

119871

120572 (1199091198982)

119880

120572] sdot sdot sdot [(119909119898119899)

119871

120572 (119909119898119899)

119880

120572]

]

]

]

]

]

(16)

(iii) The interval decision matrix for criteria weight

119882120572

=

[

[

[

[

[

[(11990811)119871

120572 (11990811)

119880

120572] [(11990812)

119871

120572 (11990812)

119880

120572] sdot sdot sdot [(1199081119899)

119871

120572 (1199081119899)

119880

120572]

[(11990821)119871

120572 (11990821)

119880

120572] [(11990822)

119871

120572 (11990822)

119880

120572] sdot sdot sdot [(1199082119899)

119871

120572 (1199082119899)

119880

120572]

sdot sdot sdot

[(1199081198981)

119871

120572 (1199081198981)

119880

120572] [(1199081198982)

119871

120572 (1199081198982)

119880

120572] sdot sdot sdot [(119908119898119899)

119871

120572 (119908119898119899)

119880

120572]

]

]

]

]

]

(17)

where 0 le 120572 le 1 The value of 120572 represents the degreeof confidences in the decision makersrsquo assessmentwith respect to ideal alternative alternatives ratingand criteria weights

Step 5 (calculating of criteria weight) The criteria weight of119899 selection criteria 119862 = 119862

1 1198622 119862

119899 evaluated by the

decision makers will be calculated using twomethods whichare subjective and objective weights

(a) Subjective weightThe subjective weight of 119899 selectioncriteria 119862 = 119862

1 1198622 119862

119899 may be considered as the

average weights [9] and its calculation is [9 28]

119908119895

=

1

119898

(

119898

sum

119894=1

119908119894119895

) 119894 = 1 2 119898 119895 = 1 2 119899 (18)

(b) Objective weight The interval valued fuzzy numberis transformed into crisp number before using Shan-nonrsquos entropy concept

The crisp value of interval weight is given by [39]

119908119894119895

=

(119908119897

119894119895+ 119908119906

119894119895)

2

(19)

Then Shannonrsquos entropy concept is used to obtain the weight

The Scientific World Journal 5

The details of Shannonrsquos entropy concept are defined asfollows [27 39]

Step 51 Normalized each criterion weight to obtain theprojection value 119901

119894119895

119901119894119895

=

119908119894119895

sum119898

119894=1119908119894119895

119894 = 1 119898 119895 = 1 119899 (20)

Consequently a projection matrix representing a relativeweight of each criterion from the decision maker evaluationis expressed as

119875 =

[

[

[

[

[

11990111

11990112

sdot sdot sdot 1199011119899

11990121

11990122

sdot sdot sdot 1199012119899

sdot sdot sdot

1199011198981

1199011198982

sdot sdot sdot 119901119898119899

]

]

]

]

]

(21)

Step 52 Calculate entropy values 119890119895as

119890119895

= minus119896

119898

sum

119894=1

119901119894119895ln119901119894119895

119895 = 1 119899 (22)

where 119896 is constant and let 119896 = (ln119898)minus1 If 119901

119894119895= 0 then

119901119894119895ln119901119894119895is equal to 0

Step 53 Calculate the degree of diversification 119889119895

119889119895

= 1 minus 119890119895 119895 = 1 119899 (23)

Step 54 Calculate the criteria weight 119908119895

119908119895

=

119889119895

sum119899

119896=1119889119896

(24)

Step 6 (calculating the distance values) (a) Subjective weightBefore calculating the distance values calculate the overallperformance evaluation of ideal alternatives and alternativesby multiplying the aggregate weight with each criterion [21]

For the ideal alternative

119877119895

= [V119871119895 V119880119895

] sdot [119908119871

119895 119908119880

119895] = [119903

119871

119895 119903119880

119895] (25)

and for the alternatives119878119894119895

= [119909119871

119894119895 119909119880

119894119895] sdot [119908119871

119895 119908119880

119895]

= [119904119871

119894119895 119904119880

119894119895]

(26)

where119903119871

= min V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119903119880

= max V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119904119871

= min 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

119904119880

= max 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

(27)

Then calculate the distance values between the ideal alterna-tives with the alternatives by using Definition 1

119889NHD (119877 119878) =

1

2119899

(

119899

sum

119895=1

(

10038161003816100381610038161003816119903119871

119895minus 119904119871

119894119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816119903119880

119895minus 119904119880

119894119895

10038161003816100381610038161003816)) (28)

(b) Objective weight For the objective weight the dis-tance values are calculated by using Definition 2

119889WHD (119868 119863) =

119899

sum

119895=1

(119908119895

10038161003816100381610038161003816V119871119895

minus 119909119871

119894119895

10038161003816100381610038161003816

+ 119908119895

10038161003816100381610038161003816V119880119895

minus 119909119880

119894119895

10038161003816100381610038161003816) (29)

Step 7 (ranking the candidate) The alternatives are rankedin ascending order according to the distance values forrespective 120572 values The alternative with the less distancevalue is considered as the best choice

Step 8 (repeat Steps 4 5 and 6 for different values of 120572) Thealternatives are ranking according to the different values of 120572

Step 9 (selection of the appropriate alternative by the decisionmakers)

5 A Numerical Example

An example on the personnel selection in an academicinstitution is provided to validate the proposed algorithmSuppose that the academic institution intends to employ alecturer based on consideration of four main criteria whichare experienced in teaching areas (119862

1) proficiency in per-

forming research (1198622) personality assessment (119862

3) and past

contribution (1198624) Assume that after preliminary selection

phase four alternatives 1198601 1198602 1198603 and 119860

4are qualified for

final evaluation A committee of experts (decision makers)consisting of three persons is formed namely119863

11198632 and119863

3

The information of this study is given in Figures 1 and 2 andTables 1 2 3 4 and 5 while the results from the numericalexamples are shown in Tables 6 7 8 9 and 10 As mentionedbefore TOPSIS is one of the existing MCDM methods thatcan be used to solve personnel selection problemThus it canbe used to validate the proposed method and the results byusing this method that is shown in Table 11 More explanationon these figures and tables are explained in the discussionsection

51 Discussion Based on the results obtained the proposedHDMSOWs can be summarized as follows

Step 1 Ideal alternative matrix (12) is built from the evalua-tions of the criteria based on linguistic variables taken fromWang and Lee [28] as illustrated in Figure 2 and Table 2 Thelinguistic terms are represented by triangular fuzzy numberranging from ldquovery poorrdquo to ldquovery goodrdquo Table 4 showsthe decision makers evaluation on ideal alternative In thispaper we assume that the 119898 decision makers had come toan agreement in standardizing into one final value for eachcriterion

Step 2 Decision matrix for alternatives evaluation on eachcriterion (13) is obtained by using the same linguistic variablesadopted from Wang and Lee [28] as illustrated in Figure 2and Table 2 Similar to Step 1 these terms are captured inthe form of the triangular fuzzy number The alternativesperformance evaluations are ranging from ldquovery poorrdquo to

6 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

The Scientific World Journal 5

The details of Shannonrsquos entropy concept are defined asfollows [27 39]

Step 51 Normalized each criterion weight to obtain theprojection value 119901

119894119895

119901119894119895

=

119908119894119895

sum119898

119894=1119908119894119895

119894 = 1 119898 119895 = 1 119899 (20)

Consequently a projection matrix representing a relativeweight of each criterion from the decision maker evaluationis expressed as

119875 =

[

[

[

[

[

11990111

11990112

sdot sdot sdot 1199011119899

11990121

11990122

sdot sdot sdot 1199012119899

sdot sdot sdot

1199011198981

1199011198982

sdot sdot sdot 119901119898119899

]

]

]

]

]

(21)

Step 52 Calculate entropy values 119890119895as

119890119895

= minus119896

119898

sum

119894=1

119901119894119895ln119901119894119895

119895 = 1 119899 (22)

where 119896 is constant and let 119896 = (ln119898)minus1 If 119901

119894119895= 0 then

119901119894119895ln119901119894119895is equal to 0

Step 53 Calculate the degree of diversification 119889119895

119889119895

= 1 minus 119890119895 119895 = 1 119899 (23)

Step 54 Calculate the criteria weight 119908119895

119908119895

=

119889119895

sum119899

119896=1119889119896

(24)

Step 6 (calculating the distance values) (a) Subjective weightBefore calculating the distance values calculate the overallperformance evaluation of ideal alternatives and alternativesby multiplying the aggregate weight with each criterion [21]

For the ideal alternative

119877119895

= [V119871119895 V119880119895

] sdot [119908119871

119895 119908119880

119895] = [119903

119871

119895 119903119880

119895] (25)

and for the alternatives119878119894119895

= [119909119871

119894119895 119909119880

119894119895] sdot [119908119871

119895 119908119880

119895]

= [119904119871

119894119895 119904119880

119894119895]

(26)

where119903119871

= min V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119903119880

= max V119871119908119871 V119871119908119880 V119880119908119871 V119880119908119880

119904119871

= min 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

119904119880

= max 119909119871119908119871 119909119871119908119880

119909119880

119908119871 119909119880

119908119880

(27)

Then calculate the distance values between the ideal alterna-tives with the alternatives by using Definition 1

119889NHD (119877 119878) =

1

2119899

(

119899

sum

119895=1

(

10038161003816100381610038161003816119903119871

119895minus 119904119871

119894119895

10038161003816100381610038161003816

+

10038161003816100381610038161003816119903119880

119895minus 119904119880

119894119895

10038161003816100381610038161003816)) (28)

(b) Objective weight For the objective weight the dis-tance values are calculated by using Definition 2

119889WHD (119868 119863) =

119899

sum

119895=1

(119908119895

10038161003816100381610038161003816V119871119895

minus 119909119871

119894119895

10038161003816100381610038161003816

+ 119908119895

10038161003816100381610038161003816V119880119895

minus 119909119880

119894119895

10038161003816100381610038161003816) (29)

Step 7 (ranking the candidate) The alternatives are rankedin ascending order according to the distance values forrespective 120572 values The alternative with the less distancevalue is considered as the best choice

Step 8 (repeat Steps 4 5 and 6 for different values of 120572) Thealternatives are ranking according to the different values of 120572

Step 9 (selection of the appropriate alternative by the decisionmakers)

5 A Numerical Example

An example on the personnel selection in an academicinstitution is provided to validate the proposed algorithmSuppose that the academic institution intends to employ alecturer based on consideration of four main criteria whichare experienced in teaching areas (119862

1) proficiency in per-

forming research (1198622) personality assessment (119862

3) and past

contribution (1198624) Assume that after preliminary selection

phase four alternatives 1198601 1198602 1198603 and 119860

4are qualified for

final evaluation A committee of experts (decision makers)consisting of three persons is formed namely119863

11198632 and119863

3

The information of this study is given in Figures 1 and 2 andTables 1 2 3 4 and 5 while the results from the numericalexamples are shown in Tables 6 7 8 9 and 10 As mentionedbefore TOPSIS is one of the existing MCDM methods thatcan be used to solve personnel selection problemThus it canbe used to validate the proposed method and the results byusing this method that is shown in Table 11 More explanationon these figures and tables are explained in the discussionsection

51 Discussion Based on the results obtained the proposedHDMSOWs can be summarized as follows

Step 1 Ideal alternative matrix (12) is built from the evalua-tions of the criteria based on linguistic variables taken fromWang and Lee [28] as illustrated in Figure 2 and Table 2 Thelinguistic terms are represented by triangular fuzzy numberranging from ldquovery poorrdquo to ldquovery goodrdquo Table 4 showsthe decision makers evaluation on ideal alternative In thispaper we assume that the 119898 decision makers had come toan agreement in standardizing into one final value for eachcriterion

Step 2 Decision matrix for alternatives evaluation on eachcriterion (13) is obtained by using the same linguistic variablesadopted from Wang and Lee [28] as illustrated in Figure 2and Table 2 Similar to Step 1 these terms are captured inthe form of the triangular fuzzy number The alternativesperformance evaluations are ranging from ldquovery poorrdquo to

6 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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 The Scientific World Journal

Table 1 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion weight

Linguistic terms Fuzzy numbersVery low (VL) (0 0 02)Low (L) (005 02 035)Medium low (ML) (02 035 05)Medium (M) (035 05 065)Medium high (MH) (05 065 08)High (H) (065 08 095)Very high (VH) (08 1 1)

Table 2 Fuzzy linguistic terms and respective fuzzy numbers foreach criterion

Linguistic terms Fuzzy numbersVery poor (VP) (0 0 02)Poor (P) (005 02 035)Medium poor (MP) (02 035 05)Fair (F) (035 05 065)Medium good (MG) (05 065 08)Good (G) (065 08 095)Very good (VG) (08 1 1)

Table 3 Decision makersrsquo evaluation on each criterion weight

Criteria 1198621

1198622

1198623

1198624

1198631

VH H H MH1198632

H VH MH H1198633

VH VH H H

Table 4 Decision makersrsquo evaluation on ideal alternative

Criteria 1198621

1198622

1198623

1198624

119868 VG G VG MG

ldquovery goodrdquo Table 5 illustrates each fuzzy linguistic term toits corresponding fuzzy number for each alternative

Step 3 The weighting matrix (14) for each criterion isevaluated and determined by the decision makers based onlinguistic variables pictured in Figure 1 and Table 1 Likethe previous step these linguistic terms are expressed in theform of triangular fuzzy numbers and are ranging from ldquoverylowrdquo to ldquovery highrdquo Table 3 marks the evaluation of thecriteriaweights by the decisionmakers according to their ownjudgment in evaluating criteriarsquos importance for the specifiedjob

Step 4 By using the 120572-cuts of fuzzy numbers the intervalvalue of the fuzzy number of the performance matrics for theideal alternative (15) the alternatives (16) and criteria weight(17) are built The values of 120572 show the degree of confidencesfor the decisionmakers in evaluating the criteria performanceof each alternative

VL L ML M MH H VH

0

02

04

06

08

1

0 02 04 06 08 1

Figure 1 The fuzzy linguistic variables for each criterion weight

0

02

04

06

08

1

0 02 04 06 08 1

VP P MP M MG G VG

Figure 2 The fuzzy linguistic variables for each alternative

Step 5 The objective and subjective weights are identifiedThe subjective weight is measured based on (18) Table 6shows the subjective weight for each criterion at 120572 = 0

and 120572 = 05 While for objective weight Shannonrsquos entropyconcept (19)ndash(24) is used to obtain the weightThe projectionvalues are shown in Table 7 Table 8 consist of entropy values(119890119895) degree of diversifications (119889

119895) and the objective weight

(119908119895) The use of objective weight will give an insight to

the decision maker in determining which criteria is neededthe most in which 119862

3and 119862

4are considered as the most

important criteria based on Shannonrsquos entropy concept Itis known that objective weight can be obtained withoutconsideration of decisionmakerrsquos preferences however sincethe evaluation of criteria weight exists the objective weight isobtained based on the evaluation of criteria weight

Step 6 The distance values between the ideal alternative andthe alternatives are calculated by using theHamming distancemethod For the subjective weight the overall performanceevaluation for the ideal alternative (25) and the alternatives(26) are determined beforehand the use of the normalizedHamming distance method (28) For the objective weightthe distance values are obtained from the use of the weightedHamming distance method (29) The distance values showhow much is the similarity between the alternatives and theideal alternative

Step 7 The ranking of the alternatives is made based on thedistance values obtained before The alternative with the lessdistance value is considered as a preferable alternative to beselected Table 9 shows the distance value for each alternativeat 120572 = 0 and 120572 = 05 Table 10 shows the ranking ofthe alternatives based on the distance values with the use ofsubjective and objective weights

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

The Scientific World Journal 7

Table 5 Decision makers rating on alternative performance

Alternatives 1198621

1198622

1198623

1198624

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198631

1198632

1198633

1198601

G G F F MG F G VG VG G VG MG1198602

F G G F F F G MG G MG G G1198603

F VG F MG VP G VG G MG VG G G1198604

G G G MG G G VG VG VG G G MG

Table 6 Subjective weight for each criterion at 120572 = 0 and 120572 = 05

Criteria 120572 = 0 120572 = 05

1198621

(075 09833) (084166 095833)1198622

(075 09833) (084166 095833)1198623

(060 090) (0675 0825)1198624

(060 090) (0675 0825)

Table 7 Each criterion projection value at 120572 = 0 and 120572 = 05

Criteria 1198631

1198632

1198633

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

(034615) (035185) (030769) (029630) (034615) (035185)1198622

(030769) (029630) (034615) (035185) (034615) (035185)1198623

(035556) (035556) (028889) (028889) (035556) (035556)1198624

(028889) (028889) (035556) (035556) (035556) (035556)

Table 8 Shannonrsquos entropy based weight

Criteria 119890119895

119889119895

119908119895

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

1198621

099864 099713 000136 000287 012385 0204391198622

099864 099713 000136 000287 012385 0204391198623

099585 099585 000415 000415 037615 0295611198624

099585 099585 000415 000415 037615 029561

Table 9 Distance value of subjective and objective weights at 120572 = 0 and 120572 = 05

Distance 1198601

1198602

1198603

1198604

120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05 120572 = 0 120572 = 05

Subjective 012604 013993 015181 017915 017305 020001 004979 006338Objective 023685 032263 031193 040220 036193 045220 011239 014087

Table 10 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingHDMSOWrsquos

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198602

1198602

1198602

1198602

4 1198603

1198603

1198603

1198603

Step 8 The steps are repeated by using different values of120572 120572 isin [0 1] Under different values of 120572 the decisionmakers may expect the different outcome in the ranking ofthe alternatives If there exist two or more alternatives on

Table 11 Ranking of alternatives at 120572 = 0 and 120572 = 05 by usingTOPSIS

Ranking Subjective weight Objective weight120572 = 0 120572 = 05 120572 = 0 120572 = 05

1 1198604

1198604

1198604

1198604

2 1198601

1198601

1198601

1198601

3 1198603

1198603

1198603

1198603

4 1198602

1198602

1198602

1198602

the same ranking which indicate that they having the samedistance values the decision makers may refer to the criteriaweight which mean the alternative that perform well in thecriteria that is needed the most is likely to be selected

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

8 The Scientific World Journal

Step 9 The decision makers then will select the suitablealternative to fill the vacancy based on the ranking of thealternatives The decision makers also can make the decisionbased on the preferable 120572 levels since the ranking may bechanged at the different values of 120572 Apparently the mostsuitable alternative for the post by using both subjectiveand objective weights is the alternative with the minimumdistance values From Table 10 119860

4is likely to be selected

by the decision makers regarding hisher distance valuesHere we also present the results by using TOPSIS methodto validate the proposed approach Consequently the sameresults are recorded by using TOPSIS method in which 119860

4

is the possible alternative to be selected The ranking for theother alternatives also can be clarified as almost similar to theresults by using the proposed method

6 Conclusions

In this paper we have presented a novel approach of handlingpersonnel selection process by using the Hamming distancemethod Based on the fact that most of criteria assessment isin qualitative or in subjective measurement fuzzy set theoryhas been applied to overcome this limitation Furthermorerealizing the importance of weighting the criteria in deter-mining which criteria are valued the most two types ofweights have been applied in this paper which are objectiveand subjective weights The objective weight is determinedby the application of Shannonrsquos entropy concept and thesubjective weight is obtained based on the preference of thedecision maker With the use of the weighted Hammingdistance the distance values between the ideal alternativeand the alternatives are identified and the ranking of thealternatives based on the overall evaluation of the criteria ismade The final results showed that the criteria 119862

3and 119862

4

are considered as the important criteria and 1198604is considered

as the best alternative to choose based on the use of sub-jective and objective weights With emphasis on finding thedistance measure between ideal alternative and alternativeswith the use of subjective and objective weights our methodprovides an effective way to be used In addition we are alsoincorporating fuzzy linguistic terms to express the subjectiveassessment that the decision makers often exhibit whileevaluating the alternatives performance in certain criteriaWe also provided the numerical example to prove the validityof this approach To verify the proposedmethod the TOPSISmethod is used to compare the result and we can justifythat the final results are almost the same for both methodsThe proposed method also can overcome some limitation inthe existing methods of MCDM that are involved with theinconsistency of judgement when there are the addition ofalternatives and criteria For further research we are going tostudy the appropriatemethods in evaluating ideal alternativeshence improving the HDMSOWs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research was funded by the Ministry of Educationof Malaysia under Research Acculturation Grant Scheme(RAGS) 9018-00004

References

[1] M Dursun and E E Karsak ldquoA fuzzy MCDM approach forpersonnel selectionrdquo Expert Systems with Applications vol 37no 6 pp 4324ndash4330 2010

[2] Z Gungor G Serhadlıoglu and S E Kesen ldquoA fuzzy AHPapproach to personnel selection problemrdquoApplied SoftComput-ing vol 9 pp 641ndash646 2009

[3] H Niakan M Zowghi and A Bakhshandeh-Fard ldquoA fuzzyobjective and subjective decisionmakingmethod by non-linearnormalizing and weighting operationsrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[4] Y H Chang C H Yeh and YW Chang ldquoA newmethod selec-tion approach for fuzzy group multicriteria decision makingrdquoApplied Soft Computing vol 13 no 4 pp 2179ndash2187 2013

[5] T Dereli A Durmusoglu S U Seckiner and N Avlanmaz ldquoAfuzzy approach for personnel selection processrdquoTurkish Journalof Fuzzy Systems vol 1 no 2 pp 126ndash140 2010

[6] I S Fagoyinbo and I A Ajibode ldquoApplication of linear pro-gramming techniques in the effective use of resources for stafftrainingrdquo Journal of Emerging Trends in Engineering andAppliedSciences pp 127ndash132 2010

[7] C K VoonAnalytic hierarchy process in academic staff selectionat Faculty of Science in University Technology Malaysia [MSthesis] Faculty of Science Universiti TeknologiMalaysia JohorMalaysia 2009

[8] A Afshari M Mojahed and R M Yusuff ldquoSimple additiveweighting approach to personnel selection problemrdquo Interna-tional Journal of Innovation Management and Technology vol1 no 5 pp 511ndash515 2010

[9] P V Polychroniou and I Giannikos ldquoA fuzzy multicrite-ria decision-making methodology for selection of humanresources in a Greek private bankrdquo Career Development Inter-national vol 14 no 4 pp 372ndash387 2009

[10] EMarinov E Szmidt J Kacprzyk andR Tcvetkov ldquoAmodifiedweighted Hausdorff distance between intuitionistic fuzzy setsrdquoin Proceedings of the 6th IEEE International Conference onIntelligent System pp 138ndash141 Sofia Bulgaria September 2012

[11] L Canos T Casasus E Crespo T Lara and J C Perez ldquoPerson-nel selection based on fuzzy methodsrdquo Revista de MatematicaTeorıa y Aplicaciones vol 18 no 1 pp 177ndash192 2011

[12] J M Merigo and A M Gil-Lafuente ldquoDecision-making tech-niques with similarity measures and OWA operatorsrdquo Statisticsand Operations Research Transactions vol 36 no 1 pp 81ndash1022012

[13] L Canos and V Liern ldquoSome fuzzy models for human resourcemanagementrdquo International Journal of Technology Policy andManagement vol 4 no 4 pp 291ndash308 2004

[14] R W Hamming ldquoError detecting and error correcting codesrdquoBell System Technical Journal vol 29 no 2 pp 147ndash160 1950

[15] W Huang Y Shi S Zhang and Y Zhu ldquoThe communicationcomplexity of the Hamming distance problemrdquo InformationProcessing Letters vol 99 no 4 pp 149ndash153 2006

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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

The Scientific World Journal 9

[16] S Ziauddin and M N Dailey ldquoIris recognition performanceenhancement using weighted majority votingrdquo in Proceedingsof the 15th IEEE International Conference on Image Processing(ICIP rsquo08) pp 277ndash280 San Diego Calif USA October 2008

[17] T Morie T Matsuura S Miyata T Yamanaka M Nagata andA Iwata ldquoQuantum-dot structures measuring Hamming dis-tance for associative memoriesrdquo Superlattices and Microstruc-tures vol 27 no 5-6 pp 613ndash616 2000

[18] C H Yeh and H Deng ldquoAlgorithm for fuzzy multi-criteriadecisionmakingrdquo in Proceedings of the IEEE International Con-ference on Intelligent Processing Systems pp 1564ndash1568 October1997

[19] R E Bellman and L A Zadeh ldquoDecision-making in a fuzzyenvironmentrdquo Management Science vol 17 no 4 pp 141ndash1641970

[20] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[21] M Z AhmadM K Hasan and B De Baets ldquoA newmethod forcomputing continuous functions with fuzzy variablerdquo Journal ofApplied Sciences vol 11 no 7 pp 1143ndash1149 2011

[22] T Y Chen and C Y Tsao ldquoThe interval-valued fuzzy TOPSISmethod and experimental analysisrdquo Fuzzy Sets and Systems vol159 no 11 pp 1410ndash1428 2008

[23] B Ashtiani F Haghighirad A Makui and G A MontazerldquoExtension of fuzzy TOPSIS method based on interval-valuedfuzzy setsrdquoApplied SoftComputing Journal vol 9 no 2 pp 457ndash461 2009

[24] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[25] J M Merigo and A M Gil-Lafuente ldquoA method for decisionmaking with the OWA operatorrdquo Computer Science and Infor-mation Systems vol 9 no 1 pp 357ndash380 2012

[26] L Canos and V Liern ldquoSoft computing-based aggregationmethods for human resource managementrdquo European Journalof Operational Research vol 189 no 3 pp 669ndash681 2008

[27] FH Lotfi andR Fallahnejad ldquoImprecise shannonrsquos entropy andmulti attribute decision makingrdquo Entropy vol 12 no 1 pp 53ndash62 2010

[28] T C Wang and H D Lee ldquoDeveloping a fuzzy TOPSISapproach based on subjective weights and objective weightsrdquoExpert Systems with Applications vol 36 no 5 pp 8980ndash89852009

[29] H Deng C H Yeh and R J Willis ldquoInter-company compari-son using modified TOPSIS with objective weightsrdquo Computersand Operations Research vol 27 no 10 pp 963ndash973 2000

[30] T C Wang H D Lee and C C Wu ldquoA fuzzy topsis approachwith subjective weights and objective weightsrdquo in Proceedings ofthe 6th WSEAS International Conference on Applied ComputerScience 2007

[31] E U Choo and W C Wedley ldquoOptimal criterion weightsin repetitive multicriteria decision makingrdquo Journal of theOperational Research Society vol 36 no 11 pp 983ndash992 1985

[32] C E Shannon and W Weaver The Mathematical Theory ofCommunication The University of Illinois Press Urbana IllUSA 1947

[33] M Zeleny Multiple Criteria Decision Making Springer NewYork NY USA 1996

[34] A A Muley and V H Bajaj ldquoA comparative FMADM methodused to solve real life problemrdquo International Journal ofMachineIntelligence vol 2 no 1 pp 35ndash39 2010

[35] B Kosko ldquoFuzzy entropy and conditioningrdquo Information Sci-ences vol 40 no 2 pp 165ndash174 1986

[36] J Burg ldquoMaximum entropy spectral analysisrdquo in Proceedingsof the 37th Meeting of the Society of Exploration GeophysicistsOklahoma City Okla USA

[37] A Golan G Judge andDMillerMaximumEntropy Economet-rics Robust Estimation with Limited Data John Wiley amp SonsNew York NY USA 1996

[38] M Dhar ldquoOn some properties of entropy of fuzzy numbersrdquoInternational Journal of Intelligent Systems and Applications vol5 no 3 pp 66ndash73 2013

[39] A J Chaghooshi M R Fathi and M Kashef ldquoIntegrationof fuzzy Shannonrsquos entropy with fuzzy TOPSIS for industrialrobotic system selectionrdquo Journal of Industrial Engineering andManagement vol 5 no 1 pp 102ndash114 2012

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