Research Article A Rough VIKOR-Based QFD for...

Research ArticleA Rough VIKOR-Based QFD for Prioritizing DesignAttributes of Product-Related Service

Xiuzhen Li1 and Wenyan Song2

1School of Mechanical Engineering Shanghai Jiao Tong University Shanghai 200240 China2School of Economics and Management Beihang University Beijing 100191 China

Correspondence should be addressed to Wenyan Song 198212swy163com

Received 5 April 2016 Revised 16 September 2016 Accepted 28 September 2016

Academic Editor Ibrahim Zeid

Copyright copy 2016 X Li and W Song 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

Many manufacturers today are striving to offer high value-added product-related services (PRS) due to increasing competitionand environmental pressure PRS can reduce the negative impact on the environment because it extends the life of productsand minimizes the cost Product and service planning has been considered as the critical factor to the success of PRS Qualityfunction deployment (QFD) has been recognized as an efficient planning tool which can convert customer needs (CNs) into designattributes of PRS involving product attributes (PAs) and service attributes (SAs) However the subjective and vague informationin the design of PRS with QFD may lead to inaccurate priority of PAs and SAs To solve this problem a novel rough VIKOR-(VIseKriterijumska Optimizaciji I Kompromisno Resenje-) based QFD is proposedThe proposed approach integrates the strengthof rough number (RN) in manipulating vague concepts with less a priori information and the merit of VIKOR in structuringframework of compromise decision-making Finally an application in compressor-based service design is presented to illustratethe potential of the proposed method

1 Introduction

Product-related services (PRS) are services that are closelyassociated with goods in products Expanding the servicecontent of products has been for years a major trend inbusiness strategy [1ndash3] The physical goods are more andmore associated with complex services that enhance theproduct value for customers and provide interesting businessmodels for producers [4] Besides PRS canminimize the costfor long-lasting well-functioning products and aim to sustaina balance between environmental economic and socialdimensions [5] Developing PRS is therefore a major concernfor firms in awide range of industries To ensure the success ofPRS it is necessary to satisfy various customer needs (CNs)Quality function deployment (QFD) is often used to translateCNs into technical attributes (TAs) of product [6 7] How-ever the prioritizing TAs in the conventional QFD containvagueness and impreciseness (eg linguistic judgments onthe CN importance and the CN-TA relationships) whichwill affect the accuracy of the TA importance Furthermore

comparedwith product design the design of PRS always linksto vague customer perception and subjective experienceThismakes the conventional QFD not suitable for prioritizingdesign attributes of PRS which include product attributes(PAs) and service attributes (SAs) Some researchers haveapplied the fuzzy set theory to improve conventionalQFD forexample computation using fuzzy variables [8 9] fuzzy inte-gral [10] and fuzzy goal programming [11] However fuzzyQFDhas been criticized for the pre-setmembership functionthe fixed fuzzy interval and the inefficiency in manipulatingvague and subjective information Thus the conventionalQFD and fuzzy QFD cannot effectively deal with the vagueprioritization process of PAs and SAs in PRS design

To solve the above problems this work proposes a newapproach for prioritizing PAs and SAs of PRS by integratingVIKOR (VIseKriterijumska Optimizaciji I KompromisnoResenje) and rough number (RN) In the rough QFD therelationships among CNs PAs and SAs are mapped andVIKORandRNare used to convert the roughCN importanceinto the rough PA importance and then into the rough SA

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9642018 11 pageshttpdxdoiorg10115520169642018

2 Mathematical Problems in Engineering

importance Thus designers can make reasonable decisionswith vague subjective and limited information

The rest of this paper is organized as follows Section 2briefly reviews the related work Section 3 introduces theframework and the detailed steps of the proposed roughVIKOR-based QFD approach Section 4 uses a case of aircompressor service design to illustrate the proposed methodComparisons and discussions are also made in this sectionSection 5 concludes the research

2 Related Work

21 QFD in Service Design Some researchers have appliedthe conventional QFD to the field of service design An etal [12] developed an integrated product service roadmapwith QFD Lin and Pekkarinen [13] proposed a frameworkof QFD-based logistics service design to integrate the houseof quality (HOQ) technique and modular logic to help indesigning logistics services with high quality In J A Fitzsim-mons and M J Fitzsimmons [14] QFD is used to match theprocess between service factors and CNs Li et al [15] appliedQFD to map CNs to service function and map the relation-ships between product function and structure Geum et al[16] modified and applied HOQ to offer a service-focusedmodularization method The relationships between moduledrivers and decomposed service components were analyzedin the Strategic Modularability Matrix (SMM) whereas theinterrelationships among service components are identifiedin Interrelated Components Modularability Matrix (ICMM)Selen and Schepers [17] implemented a service QFD analysisat a federal police station in Belgium that could match thedemands and needs of the general public and authorities tothe activities deployed by the police service Chen and Chou[18] integrated Grey Relational Analysis (GRA) into QFDto improve service techniques for an academic library Inconventional QFD determining the weights of WHATs andHOWs and the relationships between theWHATs andHOWsis an art But the decision-making information in the conven-tional QFD is usually expressed in form of expertsrsquo linguisticjudgements which are subjective or vague [19] This may leadto the final analysis results being inaccurate

Therefore to deal with vague and subjective informationin the conventional QFD for service design the fuzzy settheory is utilized Geng et al [20] improved fuzzy QFD forproduct service system planning First they used the fuzzypairwise comparison to obtain the initial weights of engineer-ing characteristics considering CNs After that they appliedthe data envelopment analysis (DEA) approach to obtainthe final weights of engineering characteristics Song et al[21] proposed a QFD based on rough-grey relational analysisapproach to prioritize the design attributes In order to iden-tify the critical determinants relating to customer satisfactionSu and Lin [22] analyzed the correlation between the impre-cise requirements from customers and the determinants ofservice quality with fuzzy QFD Bottani [23] proposed anapproach to manage customer service based on fuzzy QFDFuzzy logic is also adopted to handle the ill-defined natureof the qualitative linguistic judgments required in the fuzzyQFD Ding [24] applied a fuzzy QFD model to identify

solutions of service delivery system (SDS) for a port fromthe viewpoints of customers Although fuzzy QFD uses fuzzynumber to deal with vague information in service designits inherent deficiencies will influence the final results in theQFD analysis For example fuzzy interval indicating the esti-mation range may be enlarged after fuzzy arithmetic opera-tions [25] which have impact on the result of QFD Besidesthe membership functions in the fuzzy QFD are usually sub-jectively and intuitively selected by experts with experience[26]

22 Rough Set Theory Pawlak [27] proposed the rough settheory (RST) and considered it as an effective tool to copewith subjective and vague information even if the data distri-bution is unknown Pawlak [28] considered that each vagueconcept can be represented with a pair of precise conceptsusing the lower and upper approximations in RST Assumethat there is a set of 119899 classes of human judgments 119877 =1198691 1198692 119869119899 ordered in the manner of 1198691 lt 1198692 lt sdot sdot sdot lt 119869119899and 119884 is an arbitrary object of 119880 then the lower approxima-tion of 119869119894 the upper approximation of 119869119894 and boundary regionare defined [29]

Lower approximation

Apr (119869119894) = ⋃119884 isin 119880 | 119877 (119884) le 119869119894 (1)

Upper approximation

Apr (119869119894) = ⋃119884 isin 119880 | 119877 (119884) ge 119869119894 (2)

Boundary region

Bnd (119869119894) = ⋃119884 isin 119880 | 119877 (119884) = 119869119894= 119884 isin 119880 | 119877 (119884) gt 119869119894cup 119884 isin 119880 | 119877 (119884) lt 119869119894

(3)

For a concept 119869119894 the greatest definable set contained inthe concept is called the lower approximation of 119869119894 The leastdefinable set that contained concept 119869119894 is called the upperapproximation of 119869119894 Elements belonging only to the upperapproximation compose the boundary region

Zhai et al [25] proposed that the class 119869119894 can be repre-sented by a RN which is defined by its lower limit Lim(119869119894) andupper limit Lim(119869119894) The calculation principles are as follows[30]

Lim (119869119894) = [119873119871prod119896=1

119877 (119884) | 119884 isin Apr (119869119894)]1119873119871

Lim (119869119894) = [119873119880prod119896=1

119877 (119884) | 119884 isin Apr (119869119894)]1119873119880

(4)

where 119873119871 and 119873119880 are the number of objects included inthe lower approximation and upper approximation of 119862119894respectively The human judgments can be represented byRNs on the basis of lower limit (Lim(119869119894)) and upper limit

Mathematical Problems in Engineering 3

(Lim(119869119894)) The RN and interval of boundary region areexpressed by the following equations

Rough number

RN (119869119894) = [Lim (119869119894) Lim (119862119894)] (5)

Interval of boundary region

IBR (119869119894) = Lim (119869119894) minus Lim (119869119894) (6)

The arithmetic operations of interval analysis can also beused in RNs as follows [25 30]

23 VIKOR Method VIKOR (VIseKriterijumska Optimiza-ciji I Kompromisno Resenje) is an effective tool in multicrite-ria decision-making (MCDM) It is proposed by Opricovic[31] who introduced the multicriteria ranking index todetermine the compromise ranking-list The alternatives areranked by comparing the measure of closeness to the idealsolution [32] VIKOR is developed from the 119871119901-metric incompromise programming

119871119901119894 = [[119898sum119895=1

(120596119895 times 119891lowast119895 minus 119891119894119895119891lowast119895 minus 119891minus119895 )119901]]1119901

1 le 119901 le infin 119894 = 1 2 3 119899

(7)

where 119871119901119894 is an aggregating function 120596119895 is the weight of the119895th criterion119891119894119895 is the evaluation value of the 119895th criterion forthe 119894th alternative 119891lowast119895 and 119891minus119895 are the best and worst values ofthe 119895th criterion respectively and 119898 and 119899 are the numbersof criteria and alternatives respectively

In the VIKOR method 1198711119894 is defined as 119878119894 and 119871infin119894 isdefined as 119877119894 which are used to formulate ranking measureWhile the optimal compromise solution is determined amaximum group utility (min119899119894=1119878119894) of the majority and aminimum individual regret (min119899119894=1119877119894) of the opponent arealso considered

The VIKOR method is suitable for the situation wherethe decision-maker is not able or does not know to expresshisher preference at the early stage of solution selection [33]The ranking order of solutions is determined by the aggre-gating function119876 in which the formats of inputs and outputsare identical For example the inputs are interval numbersand the outputs are also interval numbers However in theTOPSIS no matter whether the inputs are precise numbersor interval numbers the outputs are precise numbers If theoutputs of a process are the new inputs of the next process(eg QFD) and the original inputs are interval numbers theresult from TOPSIS is inaccurate because the original inputsare interval numbers and the new inputs are precise numbersVIKOR can deal with this problem Therefore the VIKORmethod is widely used in industry

AlthoughVIKOR is a simple and straightforwardMCDMtechnique it cannot well reflect the vague and subjectiveinformation contained in the process of QFD analysis forPRS that is it lacks the capability of capturing and reflectingthe subjective perceptions of designers in the analyzing pro-cess Thus a new method should be developed to effectivelymanipulate the vague and subjective information in the QFD

3 The Proposed Rough VIKOR-BasedQFD Method

In order to solve the problem of vagueness and subjectivity inthe early design of PRS a QFD framework based on roughVIKOR is proposed as shown in Figure 1 The proposedmethod is composed of two phasesThe first phase is to iden-tify CNs PAs and SAs CNs are classified PAs are selected tosatisfy CNs and SAs are identified to ensure the normal oper-ation of PAsThe second phase is to calculate the importanceof PAs and SAs Customers evaluate the CN importance andexperts judge the CN-PA relationships and PA-SA relation-ships These crisp ratings are then converted into RNs Indi-vidual RNs are aggregated to generate group RNs The grouprough importance and relationships are normalized Basedon the CN importance and CN-PA relationships the PAimportance is calculated and prioritized by the proposedroughVIKOR Similarly the SA importance is also calculatedand ranked based on the PA importance and PA-SA relation-ships

31 Analyze CNs PAs and SAs In order to improve customersatisfaction of new products CNs are collected and classifiedMeanwhile experts identify PAs and SAs

311 Classify CNs CNs are the crucial inputs for the successof new product development A CN is a description in thecustomerrsquos own words of the benefit to be fulfilled by theproduct or service [34] Usually CNs are too general or toodetailed to be directly used for new product developmentSome tools and methods are chosen to classify CNs suchas an affinity diagram [35] Kano model [36] and Maslowmodel [37] In this paper CNs of PRS are identified withthe method of I-CAC (Industrial customer activity analysiscycle) proposed in Song et al [38]The identificationmethodof I-CAC can systematically consider the full stages of CNsBecause of the limited space here the specific methods arenot repeated Interested readers are encouraged to read thework of Song et al [38] Here CNs are denoted as follows

CN = CN1CN2CN3 CN119894 CN119898 (8)

where CN119894 represents the 119894th CN for new product 119898 is thenumber of CNs forallCN119894CN119895 existCN119894 cap CN119895 = Oslash (119894 = 119895)312 Identify PAs and SAs To satisfy CNs PAs are identifiedThen SAs are also identified to improve the design They areexpressed as follows

PA = PA1PA2PA3 PA119894 PA119899 SA = SA1 SA2 SA3 SA119894 SA119896 (9)

where PA119894 represents the 119894th PA of new product 119899 is thenumber of PAs SA119894 represents the 119894th SA of new product and119896 is the number of SAs forallTA119894TA119895 existTA119894 cap TA119895 = Oslash (119894 = 119895)and forallSA119894 SA119895 existSA119894 cap SA119895 = Oslash (119894 = 119895)32 Calculate the Importance of PAs and SAs In the roughQFD the importance of PAs and SAs is calculated byintegrating VIKOR and RNThe process is as follows


Phase 1 analyze CNs PAs and SAs

Classify CNs

Identify PAs

Identify SAs

Step 1 evaluate the CN importance CN-PA andPA-SA relationships with crisp judgement

Phase 2 calculate the design attribute importance

Step 2 convert the crisp judgement into roughnumber (RN)

Step 3 calculate the average RN

Step 4 normalize RN

Step 5 calculate the PA importance with roughVIKOR

Step 6 calculate the SA importance with roughVIKOR

① Construct a rough decision matrix

② Determine the best and worst valuesof criteria

③ Calculate the maximum group utility(S) and minimum of individual regret (R)

④ Calculate the aggregating function (Q)

⑤ Rank S R and Q

⑥ Rank PAs and SAs

The process of rough VIKOR

Figure 1 The proposed QFD framework based on rough VIKOR

Step 1 (evaluate the CN importance the CN-PA relationshipsand PA-SA relationships with crisp judgment) Customersevaluate the CN importance with the 9-point subscale (1-3-5-7-9) Scores of 1 3 5 7 and 9 are define as very lowlow moderate high and very high importance respectivelySimilarly experts evaluate the CN-PA relationship and thePA-SA relationship with the 9-point subscale Scores of 13 5 7 and 9 represent very weak weak moderate strongand very strong relationshipTheCN importance the CN-PArelationship and the PA-SA relationship are obtained

120596119894 = 1205961119894 1205962119894 1205963119894 120596ℎ119894 120596119867119894 119903119894119895 = 1199031119894119895 1199032119894119895 1199033119894119895 119903119897119894119895 119903119871119894119895 (10)

where 120596ℎ119894 represents the ℎth customerrsquos evaluation on theimportance of the 119894th CN 119867 is the number of customers119903119897119894119895 represents the 119897th expertrsquos evaluation on the relationshipbetween the 119894th CN and the 119895th PA or the 119894th PA and the 119895thSA and 119871 is the number of experts

Step 2 (convert the crisp judgement into RN) The crispimportance and crisp relationships are converted into RNswith formula (4)

RN (120596119896119894 ) = [120596ℎ119871119894 120596ℎ119880119894 ] RN (119903119904119894119895) = [119903119897119871119894119895 119903119897119880119894119895 ] (11)

where120596ℎ119871119894 and 119903119897119871119894119895 are the lower limits of RNs and120596ℎ119880119894 and 119903119897119880119894119895are the upper limits of RNs

Step 3 (aggregate individual RN to generate group RN) Thegroup RN is aggregated as follows

120596119871119894 = 119867radic 119867prodℎ=1

120596ℎ119871119894 120596119880119894 = 119867radic 119867prod

ℎ=1

120596ℎ119880119894 119903119871119894119895 = 119878radic 119878prod

119904=1

119903119904119871119894119895 119903119880119894119895 = 119878radic 119878prod

119904=1

119903119904119880119894119895

(12)

Step 4 (normalize the rough importance and rough relation-ships) The rough importance and rough relationships arenormalized as follows

1205961015840119871119894 = 120596119871119894max119898119894=1 max [120596119871119894 120596119880119894 ]

1205961015840119880119894 = 120596119880119894max119898119894=1 max [120596119871119894 120596119880119894 ]

(13)

119891119871119894119895 = 119903119871119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

119891119880119894119895 = 119903119880119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

(14)


Step 5 (calculate the PA importance with roughVIKOR) Theimportance of PAs is calculated with roughVIKOR as follows[39]

A Construct a Rough Decision Matrix 119863 The CN-PA rela-tionships are defined as the alternatives and the importanceof CNs is the evaluation criterionThe rough decision matrix119863 is

119863 =[[[[[[[[

[11989111987111 11989111988011] [11989111987112 11989111988012] sdot sdot sdot [1198911198711119898 1198911198801119898][11989111987121 11989111988021] [11989111987122 11989111988022] sdot sdot sdot [1198911198712119898 1198911198802119898] [1198911198711198991 1198911198801198991] [1198911198711198992 1198911198801198992] sdot sdot sdot [119891119871119899119898 119891119880119899119898]

]]]]]]]] (15)

B Determine the Best 119891+119895 and Worst 119891minus119895 For the benefitcriterion the larger the value of 119891119895 is the better the result isFor the cost criterion the smaller the value of 119891119895 is the betterthe result is

119891+119895 = max119899119894=1119895isin119861

119891119880119894119895 or min119899119894=1119895isin119862

119891119871119894119895 119891minus119895 = min119899119894=1

119895isin119861

119891119871119894119895 or max119899119894=1119895isin119862

119891119880119894119895 (16)

where 119861 is associated with the benefit criterion and 119862 isassociated with the cost criterion

C Calculate 119878119894 and 119877119894 119878 and 119877 are calculated as follows

119878119871119894 = sum119895isin119861

1205961015840119871119895 119891+119895 minus 119891119880119894119895119891+119895 minus 119891minus119895 + sum119895isin1198621205961015840119871119895119891119871119894119895 minus 119891+119895119891minus119895 minus 119891+119895 (17)

119878119880119894 = sum119895isin119861


119877119871119894 = max119898119895=1119895isin119861

1205961015840119871119895 119891+119895 minus 119891119880119894119895119891+119895 minus 119891minus119895 or max119898119895=1119895isin119862

1205961015840119871119895 119891119871119894119895 minus 119891+119895119891minus119895 minus 119891+119895 (19)

119877119880119894 = max119898119895=1119895isin119861



D Calculate the Aggregating Function 119876119894 Consider thefollowing

119876119871119894 = 120592119878119871119894 minus 119878+119878minus minus 119878+ + (1 minus 120592) 119877119871119894 minus 119877+119877minus minus 119877+


(21)

where 119878+ = min119899119894=1119878119871119894 119878minus = max119899119894=1119878119880119894 119877+ = min119899119894=1119877119871119894 119877minus =max119899119894=1119877119880119894 120592 is the weight of the strategy of the majority ofcriteria (120592 isin [0 1]) and usually 120592 = 05E Rank 119878 119877 and 119876 119878 119877 and 119876 are ranked in decreasingorder So three ranking lists are obtained

For any two interval numbers [Lim(120572) Lim(120572)] and[Lim(120573) Lim(120573)] the ranking rule is described as follows

(a) If the interval of a RN is not strictly contained byanother

(i) if Lim(120573) ge Lim(120572) and Lim(120573) gt Lim(120572)or Lim(120573) gt Lim(120572) and Lim(120573) ge Lim(120572)then RN(120573) ≻ RN(120572) where ldquo≻rdquo means ldquomoreimportant thanrdquo

(ii) if Lim(120573) = Lim(120572) and Lim(120573) = Lim(120572) thenRN(120573) = RN(120572)

(b) If the interval of a RN is strictly contained by another

(i) if Lim(120573) gt Lim(120572) and Lim(120573) lt Lim(120572)if 119872(120573) le 119872(120572) then RN(120573) ≺ RN(120572)where 119872(120572) and 119872(120573) are the median ofRN(120572) and RN(120573) respectivelyif119872(120573) gt 119872(120572) then RN(120573) ≻ RN(120572)

(ii) If Lim(120573) lt Lim(120572) and Lim(120573) gt Lim(120572)if119872(120573) le 119872(120572) then RN(120573) ≺ RN(120572)if119872(120573) gt 119872(120572) then RN(120573) ≻ RN(120572)

F Rank PAs Assume that PA119894 is ranked the best by cal-culating 119876 (minimum) if the following two conditions aresatisfied

(C1) Acceptable advantage

radic [119876119871 (PA119895) minus 119876119871 (PA119894)]2 + [119876119880 (PA119895) minus 119876119880 (PA119894)]22ge 1119899 minus 1

(22)

where PA119895 is the second PA ranked by calculating 119876(C2) Acceptable stability in decision-making PA119894 must

also be ranked the best by calculating 119878 orand119877Thiscompromise solution is stable in the decision-makingprocess When 120592 gt 05 it could be the strategy ofmaximum group utility or ldquoby consensusrdquo (120592 asymp 05)or ldquowith vetordquo (120592 lt 05)

If (C1) or (C2) is not satisfied a set of PAs is proposed asfollows

(1) PA119894 and PA119895 if only (C2) is not satisfied(2) PA119894PA119895 PA119904 if (C1) is not satisfied PA119904 is calcu-

lated by

radic [119876119871 (PA119904) minus 119876119871 (PA119894)]2 + [119876119880 (PA119904) minus 119876119880 (PA119894)]22lt 1119904 minus 1

(23)


Therefore the ranking order of PAs is determined by theaggregating function 119876 However the smaller the value of 119876is the larger the importance of PA becomes For example themedian of 119876119901 is the maximum and the median of 119876119894 is theminimum The importance of PA119894 is [119876119871119901 119876119880119901 ] and the PA119901importance is [119876119871119894 119876119880119894 ] The importance of PAs is the keyinput of calculating the SA importance in the next processof rough QFD

Step 6 (calculate the SA importance with rough VIKOR)Similarly the SA-PA relationshipmatrix is the rough decisionmatrix and the PA importance is the evaluation criterionAccording to Step 5 the importance of SAs is calculated

4 Case Study

In this section the design of the compressor-related servicesis taken as an example to illustrate the application of theproposed method The compressor is the heart of refrigera-tion system It can compress and transport refrigerant vaporand make the refrigerant workThe design of the compressoraffects the performance of a refrigerator directly The infor-mation of the compressor is provided by companyA who hasdeveloped the compressor for more than 40 years It mainlyprovides the compressor and related services to its customers

41 Analyze CNs PAs and SAs Before developing the com-pressor a team consisting of 20 investigators in company Atake more than two months to collect CNs These investi-gators are divided into five groups Three groups interviewkey customers one group communicates with their vendorsand the other exchanges the information of the compressorwith the relevant enterprises After collecting CNs the teamrefines them and six key CNs are determinedThey are safety(CN1) lower energy consumption (CN2) lower noise (CN3)lower failure rate (CN4) being easy to maintain (CN5) andenvironmental protection (CN6)

To satisfy the six key CNs design team identifies PAs ofthe compressor In the concurrent and collaborative designall groups can work together at the same time For exampleone group involving 25 persons designs the parts or com-ponents one group including 10 people develops the powersystem and another group consisting of 8 people designsthe hydraulic system According to the existing knowledgeexperience and CNs these designers exchange the informa-tion and then identify seven key PAs that is refrigeratingcapacity (PA1) cylinder volume (PA2) rated power (PA3)performance coefficient (PA4) structure (PA5) noise (PA6)and air discharge (PA7) Similarly service team consisting of22 people identify SAs to improve the design of the compres-sor Seven key SAs are determined depending on the existingknowledge CNs PAs and so forthThe final determined SAsare diagnosing failure timely (SA1) less repair time (SA2)lower repair cost (SA3) supplying spare parts timely (SA4)supplying spare parts with lower cost (SA5) professionalcleaning (SA6) and timely lubrication (SA7)

Table 1 The crisp ratings for the CN importance

CN C1 C2 C3 C4 C5CN1 9 7 9 9 9CN2 5 9 7 7 7CN3 5 7 9 5 7CN4 7 9 7 9 9CN5 3 3 5 3 5CN6 5 5 7 5 7

42 Calculate the PA Importance and SA Importance ThePA importance and SA importance are calculated in thefollowing steps

Step 1 (evaluate the CN importance the CN-PA relationshipsand PA-SA relationships with crisp judgment) Five keycustomers are invited to evaluate the CN importance of thecompressor with the 9-point subscale as shown in Table 1Similarly five key experts from the design team evaluatethe CN-PA relationships as shown in Table 2 Note that 0indicates that CN and PA and PA and SA are uncorrelatedThen five experts from the service team evaluate the PA-SArelationships as shown in Table 3

Step 2 (convert the crisp ratings into RNs) The crisp ratingsare converted into RNs with formula (4) For example theCN2 importance is 5 9 7 7 7 Lim(5) = 1radic5 = 5 Lim(5) =5radic5 times 7 times 7 times 7 times 9 = 688 Lim(9) = 5radic9 times 7 times 7 times 7 times 5 =688 Lim(9) = 1radic9 = 9 Lim(7) = 4radic7 times 7 times 7 times 5 = 644and Lim(7) = 4radic7 times 7 times 7 times 9 = 745 The rough importanceof CN2 is [500 688] [688 900] [644 745] [644 745][644 745]Step 3 (aggregate individual RN to generate group RN)According to (12) the group rough importance and grouprough relationships are aggregated For the rough importanceof CN2 Lim(1205962) = 5radic500 times 688 times 644 times 644 times 644 = 620and Lim(1205962) = 5radic688 times 900 times 745 times 745 times 745 = 761 Thegroup rough importance of CN2 is [620 761]

Step 4 (normalize the group rough importance and grouprough relationships) The group rough importance andgroup rough relationships are normalized with formula (13)-(14) respectively

Step 5 (calculate the PA importance with rough VIKOR)According to Step 4 the CN importance is [092 100] [070085] [063 083] [086 097] [037 047] [059 070] andthe rough decision matrix of PAs is determined (shownin Table 4) The best 119891+ and worst 119891minus are identified withformula (16) as shown in Table 5 119878 and 119877 are calculated withformulas (17)-(18) and (19)-(20) respectively (see Table 6)119876 is calculated with formula (21) as shown in Table 6According to E and F in Section 32 the PA importance isdetermined (see Table 6)

Step 6 (calculate the SA importance with rough VIKOR)Similarly the rough decision matrix of SAs is determined


Table 2 The crisp ratings for the relationships between CNs and PAs

PA1 PA2 PA3 PA4 PA5 PA6 PA7CN1 7 3 7 5 5 1 1 1 1 1 5 7 7 5 7 9 7 9 9 9 5 5 5 3 5 3 1 1 3 1 1 1 3 1 1CN2 9 9 7 9 7 0 0 0 0 0 5 1 3 3 3 7 9 7 7 9 3 1 1 1 1 0 0 0 0 0 3 1 1 3 1CN3 3 1 3 3 5 0 0 0 0 0 3 3 1 5 1 7 5 7 9 5 5 3 5 7 5 9 9 9 9 9 1 1 1 3 1CN4 5 5 5 5 3 0 0 0 0 0 3 1 5 3 5 5 3 7 7 7 5 3 3 3 3 1 3 1 3 1 1 3 1 1 1CN5 0 0 0 0 0 5 3 5 5 3 0 0 0 0 0 0 0 0 0 0 7 5 5 5 3 0 0 0 0 0 0 0 0 0 0CN6 7 5 5 3 5 0 0 0 0 0 0 0 0 0 0 3 5 3 7 5 3 3 3 1 1 5 1 3 5 3 3 1 3 3 1

Table 3 The crisp ratings for the relationships between PAs and SAs

SA1 SA2 SA3 SA4 SA5 SA6 SA7PA1 5 3 3 7 5 7 5 5 7 7 5 7 5 7 5 0 0 0 0 0 0 0 0 0 0 1 3 1 3 1 3 1 3 3 1PA2 1 3 1 5 3 5 5 5 7 5 9 7 9 9 7 3 3 1 3 3 9 5 7 7 5 0 0 0 0 0 0 0 0 0 0PA3 5 7 5 5 7 7 7 7 7 5 7 5 7 9 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 3 5 7 3PA4 3 3 3 5 3 5 3 5 3 5 5 7 5 5 7 3 1 3 5 3 7 5 5 3 5 7 7 5 5 7 5 5 7 9 7PA5 5 3 5 5 7 7 5 7 7 7 9 9 9 7 9 5 3 3 3 3 9 7 5 9 7 5 3 5 5 3 5 7 5 7 7PA6 5 3 1 3 5 3 3 5 5 5 7 5 5 5 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 7 5 5PA7 1 5 1 3 3 5 3 5 5 3 5 5 7 3 5 0 0 0 0 0 0 0 0 0 0 5 7 5 7 5 5 7 5 7 3

Table 4 The rough decision matrix of PAs

CN1 CN2 CN3 CN4 CN5 CN6PA1 [048 069] [089 100] [022 039] [063 075] [000 000] [074 100]PA2 [011 011] [000 000] [000 000] [000 000] [065 083] [000 000]PA3 [063 074] [023 041] [016 035] [032 062] [000 000] [000 000]PA4 [092 100] [084 095] [063 074] [069 100] [000 000] [064 095]PA5 [046 055] [012 017] [046 062] [047 055] [074 100] [027 045]PA6 [013 023] [000 000] [100 100] [018 031] [000 000] [037 073]PA7 [012 017] [014 023] [012 017] [016 023] [000 000] [027 045]

Table 5 The best 119891+ and worst 119891minus of each criterion (PAs)

CN1 CN2 CN3 CN4 CN5 CN6119891+119895 100 100 100 100 100 100119891minus119895 011 000 000 000 000 000

(shown in Table 7) the best 119891+ and worst 119891minus are shown inTable 8 and 119878 119877 and119876 and the SA importance are shown inTable 9

43 Comparisons and Discussion To reveal the advantages ofthe proposed method the conventional QFD (using precisenumbers) and fuzzy QFD (using symmetrical triangularfuzzy numbers) are applied (see Tables 10 11 and 12) Thecriteria of comparisons between rough QFD conventionalQFD and fuzzy QFD are uncertainty manipulation mecha-nism prior information requirement and flexibility

(1) Comparisons between the Rough QFD and ConventionalQFD The conventional QFD is a systematic and operationalmethod which realizes CNs to drive the product design and

production processThe precise numbers are usually adoptedto translate ldquovoice of customerrdquo into ldquovoice of technicianrdquoThe relationship between CNs and PAs is evaluated preciselywhich can improve customer satisfactionTherefore the con-ventional QFD using the precise numbers is used widely inthe product development However decision-makers expresstheir perceptions with the vague and subjective informationThe precise numbers deal with these information inaccu-rately For example the importance of PA2 is 000 in Table 11and the importance of SA4 is also 000 in Table 12 Thisindicates that PA2 and SA4 should not be considered in thenext stage of development However this is inconsistent withthe expectation of designers in the case company Actuallythe importance of PA2 and SA4 is very low rather than 000The rough and fuzzy QFD can provide the result in Tables 11and 12 In this respect the rough QFD considers vague andsubjective information in the product development and it ismore practical than the conventional QFD

(2) Comparisons between the Rough QFD and Fuzzy QFDAlthough the rough and fuzzy QFD handle the vague andsubjective information effectively their mechanisms of deal-ing with vague and subjective information are different The


Table 6 The 119878 119877 and 119876 and the weights of PAs

119878 119877 119876 Weight Rank119878119894 Rank 119877119894 Rank 119876119894 RankPA1 [129 232] 2 [038 065] 2 [015 044] 2 [073 096] 2PA2 [376 452] 7 [092 100] 7 [084 100] 7 [000 020] 7PA3 [238 360] 4 [059 070] 3 [040 065] 4 [040 065] 4PA4 [060 156] 1 [037 047] 1 [000 020] 1 [084 100] 1PA5 [200 295] 3 [058 075] 4 [035 060] 3 [060 086] 3PA6 [262 353] 5 [080 098] 5 [060 086] 5 [035 060] 5PA7 [328 425] 6 [086 099] 6 [073 096] 6 [015 044] 6

Table 7 The rough decision matrix of SA

PA1 PA2 PA3 PA4 PA5 PA6 PA7SA1 [054 080] [017 031] [073 086] [040 047] [047 063] [037 073] [023 044]SA2 [085 100] [059 065] [086 096] [047 060] [070 078] [065 082] [058 074]SA3 [080 094] [089 100] [086 100] [068 080] [092 100] [074 100] [070 090]SA4 [000 000] [023 033] [000 000] [026 046] [034 040] [000 000] [000 000]SA5 [000 000] [067 089] [000 000] [054 073] [071 092] [000 000] [000 000]SA6 [018 030] [000 000] [000 000] [073 086] [041 052] [000 000] [085 100]SA7 [022 038] [000 000] [050 080] [075 100] [063 074] [064 095] [069 099]

Table 8 The best 119891+ and worst 119891minus of each criterion (SAs)

PA1 PA2 PA3 PA4 PA5 PA6 PA7119891+119895 100 100 100 100 100 100 100119891minus119895 000 000 000 026 034 000 000

roughQFD fully consider the vague and subjective evaluationof decision-makers The fuzzy QFD is affected by the pre-set membership function The interval of fuzzy number isfixed which is determined by the types of membershipfunctions Moreover the predetermination of the member-ship function increases additional subjective informationwhich can enlarge the vagueness of fuzzy number It canbe clearly seen from Figure 2 that the interval of roughnumber is more flexible and smaller than that of fuzzynumber For instance the crisp ratings of the CN2 impor-tance are 5 9 7 7 7 The corresponding fuzzy numbersare [4 6] [8 10] [6 8] [6 8] [6 8] with the fixed intervalof 2 respectively The aggregated group fuzzy interval is[59 79] This is not true in the real world because thefuzzy method does not consider decision-makersrsquo differentknowledge and experience and thus the fuzzyQFD considersthat all the judgements have the same uncertainty (the fixedinterval of 2) On the contrary the rough numbers are [500688] [688 900] [644 745] [644 745] [644 745] withthe flexible interval respectivelyThe aggregated group roughinterval is [620 761] which is more flexible and smaller than[59 79]

Although the three methods produce the same rankingsthey have differentmechanisms of decision-making informa-tion manipulation Firstly different from the conventional

QFD both fuzzyQFDand roughQFDconsider the subjectiv-ity and vagueness in the decision-making process Secondlycompared with fuzzy QFD rough QFD does not needmuch a priori information for example pre-set membershipfunction in the fuzzymethodsMore importantly roughQFDuses flexible intervals to describe vague and subjective infor-mation while fuzzy QFD uses fixed intervals The weightsfrom the former have smaller intervals than that of the latterwhich indicates that the result of rough QFD is more preciseIn fact the precise weights of design attributes are importantin the design decision-making process Designers always setdifferent threshold values of weights to determine whetherthe design attributes can be considered in the next stage ofdevelopment For example PA2 will be not considered in thenext stage of conventional QFD because its weight is 000However PA2 will be still considered in the fuzzy QFD androughQFD because theweights in the twomethods are [000033] and [000 020] respectively

The differences of the three methods are summarized inTable 13

5 Conclusions

This paper presents an improvedQFDmethod for PRS designbased on the rough set theory and VIKOR The proposedapproach uses rough VIKOR to prioritize design attributesof PRS in the vague and subjective situation The validationof the proposed method in compressor-related service showsthat it is an effective decision support tool for design of PRSTo sum up the approach reveals the following features

The proposed QFD method provides a progressive map-ping process for PRS design That is mapping relationshipsbetween CNs and PAs and then mapping relationships


Table 9 The 119878 119877 and 119876 and the weights of SAs

119878 119877 119876 Weight Rank119878119894 Rank 119877119894 Rank 119876119894 RankSA1 [133 300] 4 [060 081] 5 [036 064] 4 [036 064] 4SA2 [077 182] 2 [045 072] 2 [020 049] 2 [051 085] 2SA3 [029 113] 1 [023 043] 1 [000 022] 1 [060 100] 1SA4 [280 480] 7 [073 100] 7 [060 100] 7 [000 022] 7SA5 [195 372] 6 [073 096] 6 [051 085] 6 [020 049] 6SA6 [186 344] 5 [051 079] 4 [036 071] 5 [022 058] 5SA7 [094 245] 3 [045 075] 3 [022 058] 3 [036 071] 3

CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance

(a) Comparison of the CNsrsquo importance

PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance

Rough importanceFuzzy importance

02 04 06 08 1 120

(b) Comparison of the PAsrsquo importance

SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance


(c) Comparison of the SAsrsquo importance

Figure 2 Comparison of the importance of CN PA and SA


Table 10 Ranking of CNs with precise fuzzy and rough numbers

CNPrecisenumbers Fuzzy numbers Rough numbers

120596119894 Rank 120596119894 Rank 120596119894 RankCN1 100 1 [079 100] 1 [092 100] 1CN2 080 3 [061 083] 3 [070 085] 3CN3 075 4 [056 078] 4 [063 083] 4CN4 095 2 [075 096] 2 [086 097] 2CN5 043 6 [028 049] 6 [037 047] 6CN6 067 5 [049 070] 5 [059 070] 5

Table 11 Ranking of PAs in the conventional fuzzy and roughQFD(120592 = 05)PA

ConventionalQFD Fuzzy QFD Rough QFD

120596119895 Rank 120596119895 Rank 120596119895 RankPA1 092 2 [064 095] 2 [073 096] 2PA2 000 7 [000 033] 7 [000 020] 7PA3 053 4 [035 068] 4 [040 065] 4PA4 100 1 [070 100] 1 [084 100] 1PA5 078 3 [053 090] 3 [060 086] 3PA6 045 5 [032 065] 5 [035 060] 5PA7 021 6 [015 052] 6 [015 044] 6

Table 12 Ranking of SAs in the conventional fuzzy and roughQFD(120592 = 05)SA


120596119895 Rank 120596119895 Rank 120596119895 RankSA1 064 4 [031 082] 4 [036 064] 4SA2 083 2 [047 087] 2 [051 085] 2SA3 100 1 [053 100] 1 [060 100] 1SA4 000 7 [000 040] 7 [000 022] 7SA5 029 6 [027 065] 6 [020 049] 6SA6 035 5 [025 064] 5 [022 058] 5SA7 061 3 [036 078] 3 [036 071] 3

between PAs and SAs which is not presented in previousliterature of PRS PRS designers can systematically makereasonable planning of product and service in the early designof PRS

RN with flexible boundary is used to manipulate thevagueness and subjectivity in the QFD analysis process toreduce lost information because it can comprehensivelyreflect decision-makerrsquos subjective judgment and preference

The rough VIKOR provides a structured framework ofcompromise decision-making in PRS design under vague andsubjective environment

The proposed approach for PRS planning can be imple-mented without large amount of data and much a prioriinformation (eg pre-set membership function)

Table 13 Main differences between the rough QFD conventionalQFD and fuzzy QFD

Method Manipulation ofuncertainty

Reliance on muchprior information Flexibility

ConventionalQFD No No Low

Fuzzy QFD Partial Yes LowRough QFD Yes No High

Although the rough VIKOR-based QFD has merits indealing with vagueness and subjectivity it does not considerdifferent weights of decision-makers in the QFD groupTherefore to better reflect the actual situation of decision-making in QFD implementation process it is necessaryto develop suitable aggregation operators for judgmentsaggregation The aggregation operatorsrsquo influence on therough VIKOR-based QFD would also be explored in futureresearches Besides more testing work is necessitated to gainexternal validity

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work described in this paper was supported by theNational Natural Science Foundation of China (Grant no71501006) It was also partially supported by the NationalNatural Science Foundation of China (Grants nos 7133200371632003 and 71420107025) and the Fundamental ResearchFunds for the Central Universities

References

[1] Magnusson and R Peter Customer-Oriented Product Develop-ment Experiments Involving Users in Service Innovation 2003

[2] M A CusumanoThe Business of Software Free PressSimon ampSchuster Cambridge Mass USA 2004

[3] W Song ZWu X Li and Z Xu ldquoModularizing product exten-sion services an approach based on modified service blueprintand fuzzy graphrdquoComputers and Industrial Engineering vol 85pp 186ndash195 2015

[4] W Song and F T S Chan ldquoMulti-objective configurationoptimization for product-extension servicerdquo Journal of Manu-facturing Systems vol 37 pp 113ndash125 2015

[5] T S Baines H W Lightfoot S Evans et al ldquoState-of-the-artin product-service systemsrdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 221 no 10 pp 1543ndash1552 2007

[6] Y Akao Quality Function Deployment Integrating CustomerRequirements into Product Design Productivity Press Cam-bridge Mass USA 1990

[7] C-T Wu T-S Pan M-H Shao and C-S Wu ldquoAn extensiveQFD and evaluation procedure for innovative designrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 935984 7pages 2013


[8] M Li ldquoThe method for product design selection with incom-plete linguistic weight information based on quality functiondeployment in a fuzzy environmentrdquoMathematical Problems inEngineering vol 2013 Article ID 943218 10 pages 2013

[9] S Yang J Liu K Wang and Y Miao ldquoAn uncertain QFDapproach for the strategic management of logistics servicesrdquoMathematical Problems in Engineering vol 2016 Article ID1486189 10 pages 2016

[10] C-Y Tsai C-C Lo and A C Chang ldquoUsing fuzzy QFDto enhance manufacturing strategic planningrdquo Journal of theChinese Institute of Industrial Engineers vol 20 no 1 pp 33ndash41 2003

[11] L-H Chen and M-C Weng ldquoAn evaluation approach to engi-neering design inQFDprocesses using fuzzy goal programmingmodelsrdquo European Journal of Operational Research vol 172 no1 pp 230ndash248 2006

[12] Y An S Lee and Y Park ldquoDevelopment of an integratedproduct-service roadmap with QFD a case study on mobilecommunicationsrdquo International Journal of Service IndustryManagement vol 19 no 5 pp 621ndash638 2008

[13] Y Lin and S Pekkarinen ldquoQFD-basedmodular logistics servicedesignrdquo Journal of Business and IndustrialMarketing vol 26 no5 pp 344ndash356 2011

[14] J A Fitzsimmons and M J Fitzsimmons Service Man-agement Operations Strategy and Information TechnologyIrwinMcGraw-Hill 2006

[15] H Li Y Ji X Gu G Qi and R Tang ldquoModule partition processmodel andmethod of integrated service productrdquoComputers inIndustry vol 63 no 4 pp 298ndash308 2012

[16] Y Geum R Kwak and Y Park ldquoModularizing services amodified HoQ approachrdquo Computers amp Industrial Engineeringvol 62 no 2 pp 579ndash590 2012

[17] W J Selen and J Schepers ldquoDesign of quality service systems inthe public sector use of quality function deployment in policeservicesrdquo Total Quality Management vol 12 no 5 pp 677ndash6872001

[18] Y-T Chen andT-Y Chou ldquoApplyingGRA andQFD to improvelibrary service qualityrdquo The Journal of Academic Librarianshipvol 37 no 3 pp 237ndash245 2011

[19] H-Y Wu and H-Y Lin ldquoA hybrid approach to developan analytical model for enhancing the service quality of e-learningrdquo Computers and Education vol 58 no 4 pp 1318ndash1338 2012

[20] X Geng X Chu D Xue and Z Zhang ldquoA systematic decision-making approach for the optimal product-service system plan-ningrdquo Expert Systems with Applications vol 38 no 9 pp 11849ndash11858 2011

[21] W Song X Ming and Y Han ldquoPrioritising technical attributesinQFDunder vague environment a rough-grey relational anal-ysis approachrdquo International Journal of Production Research vol52 no 18 pp 5528ndash5545 2014

[22] C-T Su and C-S Lin ldquoA case study on the application of fuzzyQFD in TRIZ for service quality improvementrdquo Quality andQuantity vol 42 no 5 pp 563ndash578 2008

[23] E Bottani ldquoA fuzzy QFD approach to achieve agilityrdquo Interna-tional Journal of Production Economics vol 119 no 2 pp 380ndash391 2009

[24] J-F Ding ldquoApplying fuzzy quality function deployment (QFD)to identify solutions of service delivery system for port ofKaohsiungrdquo Quality amp Quantity vol 43 no 4 pp 553ndash5702009

[25] L-Y Zhai L-P Khoo and Z-W Zhong ldquoA rough set enhancedfuzzy approach to quality function deploymentrdquo The Interna-tional Journal of Advanced Manufacturing Technology vol 37no 5-6 pp 613ndash624 2008

[26] L C Jain Knowledge Based Intelligent Techniques in Industryvol 1 CRC Press New York NY USA 1999

[27] Z Pawlak ldquoRough setsrdquo International Journal of Computer ampInformation Sciences vol 11 no 5 pp 341ndash356 1982

[28] Z Pawlak Rough Sets Theoretical aspects of Reasoning aboutData Kluwer Academic Dordrecht The Netherlands 1991

[29] L P Khoo S B Tor and L Y Zhai ldquoRough-set-based approachfor classification and rule inductionrdquo The International Journalof Advanced Manufacturing Technology vol 15 no 6 pp 438ndash444 1999

[30] C Lee H Lee H Seol and Y Park ldquoEvaluation of new serviceconcepts using rough set theory and group analytic hierarchyprocessrdquo Expert Systems with Applications vol 39 no 3 pp3404ndash3412 2012

[31] S Opricovic ldquoMulticriteria optimization of civil engineeringsystemsrdquo Faculty of Civil Engineering Belgrade vol 2 no 1 pp5ndash21 1998

[32] S Opricovic and G-H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[33] S Opricovic and G-H Tzeng ldquoExtended VIKOR method incomparison with outranking methodsrdquo European Journal ofOperational Research vol 178 no 2 pp 514ndash529 2007

[34] A Griffin and J R Hauser ldquoThe voice of the customerrdquoMarketing Science vol 12 no 1 pp 1ndash27 1993

[35] C K Kwong and H Bai ldquoDetermining the importance weightsfor the customer requirements in QFD using a fuzzy AHP withan extent analysis approachrdquo IIE Transactions vol 35 no 7 pp619ndash626 2003

[36] NKanoN Seraku F Takahashi and S Tsuji ldquoAttractive qualityand must-be quality Hinshitsurdquo The Journal of the JapaneseSociety for Quality Control vol 14 no 2 pp 39ndash48 1984

[37] A H Maslow ldquoA theory of human motivationrdquo PsychologicalReview vol 50 no 4 pp 370ndash396 1943

[38] W Song XMing Y Han and ZWu ldquoA rough set approach forevaluating vague customer requirement of industrial product-service systemrdquo International Journal of Production Researchvol 51 no 22 pp 6681ndash6701 2013

[39] S Liao M-J Wu C-Y Huang Y-S Kao and T-H LeeldquoEvaluating and enhancing three-dimensional printing serviceproviders for rapid prototyping using the DEMATEL basednetwork process and VIKORrdquoMathematical Problems in Engi-neering vol 2014 Article ID 349348 16 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


importance Thus designers can make reasonable decisionswith vague subjective and limited information

The rest of this paper is organized as follows Section 2briefly reviews the related work Section 3 introduces theframework and the detailed steps of the proposed roughVIKOR-based QFD approach Section 4 uses a case of aircompressor service design to illustrate the proposed methodComparisons and discussions are also made in this sectionSection 5 concludes the research

2 Related Work

21 QFD in Service Design Some researchers have appliedthe conventional QFD to the field of service design An etal [12] developed an integrated product service roadmapwith QFD Lin and Pekkarinen [13] proposed a frameworkof QFD-based logistics service design to integrate the houseof quality (HOQ) technique and modular logic to help indesigning logistics services with high quality In J A Fitzsim-mons and M J Fitzsimmons [14] QFD is used to match theprocess between service factors and CNs Li et al [15] appliedQFD to map CNs to service function and map the relation-ships between product function and structure Geum et al[16] modified and applied HOQ to offer a service-focusedmodularization method The relationships between moduledrivers and decomposed service components were analyzedin the Strategic Modularability Matrix (SMM) whereas theinterrelationships among service components are identifiedin Interrelated Components Modularability Matrix (ICMM)Selen and Schepers [17] implemented a service QFD analysisat a federal police station in Belgium that could match thedemands and needs of the general public and authorities tothe activities deployed by the police service Chen and Chou[18] integrated Grey Relational Analysis (GRA) into QFDto improve service techniques for an academic library Inconventional QFD determining the weights of WHATs andHOWs and the relationships between theWHATs andHOWsis an art But the decision-making information in the conven-tional QFD is usually expressed in form of expertsrsquo linguisticjudgements which are subjective or vague [19] This may leadto the final analysis results being inaccurate

Therefore to deal with vague and subjective informationin the conventional QFD for service design the fuzzy settheory is utilized Geng et al [20] improved fuzzy QFD forproduct service system planning First they used the fuzzypairwise comparison to obtain the initial weights of engineer-ing characteristics considering CNs After that they appliedthe data envelopment analysis (DEA) approach to obtainthe final weights of engineering characteristics Song et al[21] proposed a QFD based on rough-grey relational analysisapproach to prioritize the design attributes In order to iden-tify the critical determinants relating to customer satisfactionSu and Lin [22] analyzed the correlation between the impre-cise requirements from customers and the determinants ofservice quality with fuzzy QFD Bottani [23] proposed anapproach to manage customer service based on fuzzy QFDFuzzy logic is also adopted to handle the ill-defined natureof the qualitative linguistic judgments required in the fuzzyQFD Ding [24] applied a fuzzy QFD model to identify

solutions of service delivery system (SDS) for a port fromthe viewpoints of customers Although fuzzy QFD uses fuzzynumber to deal with vague information in service designits inherent deficiencies will influence the final results in theQFD analysis For example fuzzy interval indicating the esti-mation range may be enlarged after fuzzy arithmetic opera-tions [25] which have impact on the result of QFD Besidesthe membership functions in the fuzzy QFD are usually sub-jectively and intuitively selected by experts with experience[26]

22 Rough Set Theory Pawlak [27] proposed the rough settheory (RST) and considered it as an effective tool to copewith subjective and vague information even if the data distri-bution is unknown Pawlak [28] considered that each vagueconcept can be represented with a pair of precise conceptsusing the lower and upper approximations in RST Assumethat there is a set of 119899 classes of human judgments 119877 =1198691 1198692 119869119899 ordered in the manner of 1198691 lt 1198692 lt sdot sdot sdot lt 119869119899and 119884 is an arbitrary object of 119880 then the lower approxima-tion of 119869119894 the upper approximation of 119869119894 and boundary regionare defined [29]

Lower approximation

Apr (119869119894) = ⋃119884 isin 119880 | 119877 (119884) le 119869119894 (1)

Upper approximation

Apr (119869119894) = ⋃119884 isin 119880 | 119877 (119884) ge 119869119894 (2)

Boundary region

Bnd (119869119894) = ⋃119884 isin 119880 | 119877 (119884) = 119869119894= 119884 isin 119880 | 119877 (119884) gt 119869119894cup 119884 isin 119880 | 119877 (119884) lt 119869119894

(3)

For a concept 119869119894 the greatest definable set contained inthe concept is called the lower approximation of 119869119894 The leastdefinable set that contained concept 119869119894 is called the upperapproximation of 119869119894 Elements belonging only to the upperapproximation compose the boundary region

Zhai et al [25] proposed that the class 119869119894 can be repre-sented by a RN which is defined by its lower limit Lim(119869119894) andupper limit Lim(119869119894) The calculation principles are as follows[30]

Lim (119869119894) = [119873119871prod119896=1

119877 (119884) | 119884 isin Apr (119869119894)]1119873119871

Lim (119869119894) = [119873119880prod119896=1

119877 (119884) | 119884 isin Apr (119869119894)]1119873119880

(4)

where 119873119871 and 119873119880 are the number of objects included inthe lower approximation and upper approximation of 119862119894respectively The human judgments can be represented byRNs on the basis of lower limit (Lim(119869119894)) and upper limit



Rough number

RN (119869119894) = [Lim (119869119894) Lim (119862119894)] (5)


IBR (119869119894) = Lim (119869119894) minus Lim (119869119894) (6)



119871119901119894 = [[119898sum119895=1


1 le 119901 le infin 119894 = 1 2 3 119899

(7)









CN = CN1CN2CN3 CN119894 CN119898 (8)






Classify CNs

Identify PAs

Identify SAs





Step 4 normalize RN







⑤ Rank S R and Q





120596119894 = 1205961119894 1205962119894 1205963119894 120596ℎ119894 120596119867119894 119903119894119895 = 1199031119894119895 1199032119894119895 1199033119894119895 119903119897119894119895 119903119871119894119895 (10)



RN (120596119896119894 ) = [120596ℎ119871119894 120596ℎ119880119894 ] RN (119903119904119894119895) = [119903119897119871119894119895 119903119897119880119894119895 ] (11)



120596119871119894 = 119867radic 119867prodℎ=1

120596ℎ119871119894 120596119880119894 = 119867radic 119867prod

ℎ=1

120596ℎ119880119894 119903119871119894119895 = 119878radic 119878prod

119904=1

119903119904119871119894119895 119903119880119894119895 = 119878radic 119878prod

119904=1

119903119904119880119894119895

(12)


1205961015840119871119894 = 120596119871119894max119898119894=1 max [120596119871119894 120596119880119894 ]

1205961015840119880119894 = 120596119880119894max119898119894=1 max [120596119871119894 120596119880119894 ]

(13)

119891119871119894119895 = 119903119871119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

119891119880119894119895 = 119903119880119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

(14)




119863 =[[[[[[[[


]]]]]]]] (15)


119891+119895 = max119899119894=1119895isin119861

119891119880119894119895 or min119899119894=1119895isin119862

119891119871119894119895 119891minus119895 = min119899119894=1

119895isin119861

119891119871119894119895 or max119899119894=1119895isin119862

119891119880119894119895 (16)



119878119871119894 = sum119895isin119861


119878119880119894 = sum119895isin119861


119877119871119894 = max119898119895=1119895isin119861



119877119880119894 = max119898119895=1119895isin119861






(21)












(22)





lated by


(23)




4 Case Study





































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Rough number

RN (119869119894) = [Lim (119869119894) Lim (119862119894)] (5)


IBR (119869119894) = Lim (119869119894) minus Lim (119869119894) (6)



119871119901119894 = [[119898sum119895=1


1 le 119901 le infin 119894 = 1 2 3 119899

(7)









CN = CN1CN2CN3 CN119894 CN119898 (8)






Classify CNs

Identify PAs

Identify SAs





Step 4 normalize RN







⑤ Rank S R and Q





120596119894 = 1205961119894 1205962119894 1205963119894 120596ℎ119894 120596119867119894 119903119894119895 = 1199031119894119895 1199032119894119895 1199033119894119895 119903119897119894119895 119903119871119894119895 (10)



RN (120596119896119894 ) = [120596ℎ119871119894 120596ℎ119880119894 ] RN (119903119904119894119895) = [119903119897119871119894119895 119903119897119880119894119895 ] (11)



120596119871119894 = 119867radic 119867prodℎ=1

120596ℎ119871119894 120596119880119894 = 119867radic 119867prod

ℎ=1

120596ℎ119880119894 119903119871119894119895 = 119878radic 119878prod

119904=1

119903119904119871119894119895 119903119880119894119895 = 119878radic 119878prod

119904=1

119903119904119880119894119895

(12)


1205961015840119871119894 = 120596119871119894max119898119894=1 max [120596119871119894 120596119880119894 ]

1205961015840119880119894 = 120596119880119894max119898119894=1 max [120596119871119894 120596119880119894 ]

(13)

119891119871119894119895 = 119903119871119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

119891119880119894119895 = 119903119880119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

(14)




119863 =[[[[[[[[


]]]]]]]] (15)


119891+119895 = max119899119894=1119895isin119861

119891119880119894119895 or min119899119894=1119895isin119862

119891119871119894119895 119891minus119895 = min119899119894=1

119895isin119861

119891119871119894119895 or max119899119894=1119895isin119862

119891119880119894119895 (16)



119878119871119894 = sum119895isin119861


119878119880119894 = sum119895isin119861


119877119871119894 = max119898119895=1119895isin119861



119877119880119894 = max119898119895=1119895isin119861






(21)












(22)





lated by


(23)




4 Case Study





































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Classify CNs

Identify PAs

Identify SAs





Step 4 normalize RN







⑤ Rank S R and Q





120596119894 = 1205961119894 1205962119894 1205963119894 120596ℎ119894 120596119867119894 119903119894119895 = 1199031119894119895 1199032119894119895 1199033119894119895 119903119897119894119895 119903119871119894119895 (10)



RN (120596119896119894 ) = [120596ℎ119871119894 120596ℎ119880119894 ] RN (119903119904119894119895) = [119903119897119871119894119895 119903119897119880119894119895 ] (11)



120596119871119894 = 119867radic 119867prodℎ=1

120596ℎ119871119894 120596119880119894 = 119867radic 119867prod

ℎ=1

120596ℎ119880119894 119903119871119894119895 = 119878radic 119878prod

119904=1

119903119904119871119894119895 119903119880119894119895 = 119878radic 119878prod

119904=1

119903119904119880119894119895

(12)


1205961015840119871119894 = 120596119871119894max119898119894=1 max [120596119871119894 120596119880119894 ]

1205961015840119880119894 = 120596119880119894max119898119894=1 max [120596119871119894 120596119880119894 ]

(13)

119891119871119894119895 = 119903119871119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

119891119880119894119895 = 119903119880119894119895max119899119895=1 max [119903119871119894119895 119903119880119894119895 ]

(14)




119863 =[[[[[[[[


]]]]]]]] (15)


119891+119895 = max119899119894=1119895isin119861

119891119880119894119895 or min119899119894=1119895isin119862

119891119871119894119895 119891minus119895 = min119899119894=1

119895isin119861

119891119871119894119895 or max119899119894=1119895isin119862

119891119880119894119895 (16)



119878119871119894 = sum119895isin119861


119878119880119894 = sum119895isin119861


119877119871119894 = max119898119895=1119895isin119861



119877119880119894 = max119898119895=1119895isin119861






(21)












(22)





lated by


(23)




4 Case Study





































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119863 =[[[[[[[[


]]]]]]]] (15)


119891+119895 = max119899119894=1119895isin119861

119891119880119894119895 or min119899119894=1119895isin119862

119891119871119894119895 119891minus119895 = min119899119894=1

119895isin119861

119891119871119894119895 or max119899119894=1119895isin119862

119891119880119894119895 (16)



119878119871119894 = sum119895isin119861


119878119880119894 = sum119895isin119861


119877119871119894 = max119898119895=1119895isin119861



119877119880119894 = max119898119895=1119895isin119861






(21)












(22)





lated by


(23)




4 Case Study





































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












4 Case Study





































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































5 Conclusions






CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












CN1

CN2

CN3

CN4

CN5

CN6

Crisp importance

Rough importance

02 04 06 08 1 120

Fuzzy importance


PA1

PA2

PA3

PA4

PA5

PA6

PA7

Crisp importance


02 04 06 08 1 120


SA1

SA2

SA3

SA4

SA5

SA6

SA7

02 04 06 08 1 120

Crisp importance
























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Competing Interests


Acknowledgments


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Rough VIKOR-Based QFD for...

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