Research Article A Novel Nonadditive Collaborative...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 957184 10 pageshttpdxdoiorg1011552013957184

Research ArticleA Novel Nonadditive Collaborative-Filtering Approach UsingMulticriteria Ratings

Yi-Chung Hu

Department of Business Administration Chung Yuan Christian University Chung-Li 32023 Taiwan

Correspondence should be addressed to Yi-Chung Hu ychucycuedutw

Received 13 June 2013 Accepted 23 September 2013

Academic Editor Jung-Fa Tsai

Copyright copy 2013 Yi-Chung HuThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Although single-criterion recommender systems have been successfully used in several applications multicriteria rating systemswhich allow users to specify ratings for various content attributes of individual items are gaining importance in recommendationcontext An overall rating of an unrated item is often obtained by the weighted average method (WAM) when criterion weightsare available However the assumption of additivity for the WAM is not always reasonable For this reason this paper presents anew collaborative-filtering approach using multicriteria ratings in which a nonadditive technique inMulticriteria decisionmaking(MCDM) namely the Choquet integral is used to aggregate multicriteria ratings for unrated items Subsequently the system canrecommend items with higher overall ratings for each user The degrees of importance of the respective criteria are determined bya genetic algorithm In contrast to the additive weighted average aggregation the Choquet integral does not ignore the interactionamong criteria The applicability of the proposed approach to the recommendation of the initiators on a group-buying website isexamined Experimental results demonstrate that the generalization ability of the proposed approach performs well compared withother similarity-based collaborative-filtering approaches using multicriteria ratings

1 Introduction

It is well known that personalized recommender systemscan avoid information overload by providing items whichare more relevant to consumers [1ndash3] These recommendeditems with higher predicted overall ratings enable greatercross-selling to be achieved In particular single-criterionrecommender systems have been successful in a numberof personalization applications The key property of suchsystems is that users are required to offer only a single-criterion or overall rating for each consumed item In otherwords users cannot express their preference for each crite-rion of these items However practical problems are oftencharacterized by several criteria [4] Recommender systemsshould benefit from leveraging multicriteria informationbecause it can potentially improve recommendation accuracy[5] Several online sites such as Zagatrsquos Guide for restaurantsBuycom and Yahoo Movies provide a recommendationservice that uses multicriteria ratings for each object Incontrast to Yahoo Movies the rating on each criterion for a

restaurant in Zagatrsquos Guide is the same for all users whichmeans that multicriteria ratings in Zagatrsquos Guide are notpersonalized The uniqueness of Yahoo Movies indicatesthat personalizationmulticriteria recommender systemsmaybecome an important component in personalization applica-tions

Multicriteria rating problems could be an important issuefor the next generation of recommender systems [5 6]Multicriteria recommender systems have been addressed inseveral approaches For instance Schafer [7] presented ametarecommendation system with DynamicLens which letsusers express their preference and relative importance foreach criterion As a result the system will filter the preferreditems on the basis of their requirements Lee et al [8]proposed intelligent agent-based systems for personalizedrecommendations Because they assume that the value orrank of each criterion is the same for all users such systemsdo not take personalization into account Ricci andWerthner[9] employed case-based querying to recommend travelplanning with multiple criteria (eg location and activity)

2 Mathematical Problems in Engineering

These recommendation systems do not allow users to specifytheir subjective perception of the various criteria of individ-ual items

From the viewpoint of multicriteria decision making(MCDM) the overall rating has a certain relationship withthe multicriteria ratings for an item For this in contrastto the multicriteria recommendation systems previouslydescribed Adomavicius and Kwon [5] presented a frame-work for an aggregation-function-based approach to leveragemulticriteria rating information in which the rating for eachcriterion can be estimated by the traditional similarity-basedapproach using single-criterion ratings Then the output ofan aggregation function is regarded as a predicted value ofthe overall rating of an unrated item The weighted averagemethod (WAM) is often used as an aggregation function andcan aggregate ratings on different criteria when the criterionweights are available [5 10] Note that the criterion weights inWAM are interpreted as the relative importance of the crite-ria The WAM is a simple decomposed method and assumesthat criteria do not interact [11] However because the criteriaare not always independent the assumption of additivity isnot always reasonable [12] and may affect recommendationperformance Thus this motivates us to use a nonadditivetechnique the Choquet integral [13ndash16] as an aggregationfunction Actually the Choquet integral is a generalizationof the weighted average [17 18] To address this this paperfurther proposes a novel nonadditive multicriteria recom-mendation method on the basis of the Choquet integralFurthermore because the goal of the proposed approach is torecommend correctly a set of a few relevant items to each usera variation of the popular 119865-measure is presented to evaluaterecommendation performance To achieve high accuracy agenetic algorithm (GA) is implemented to determine theparameter specifications

The remainder of the paper is organized as followsSection 2 briefly introduces several collaborative-filteringapproaches using multicriteria ratings including thesimilarity-based and the aggregation-function-basedapproaches Section 3 describes the proposed aggregation-function-based collaborative-filtering approach using theChoquet integral and presents an accuracy metric forrecommendation performance evaluation A GA-basedmethod for constructing a nonadditive recommendationmodel is demonstrated in Section 4 Section 5 applies theproposed nonadditive approach to initiator recommendationon a group-buying website in Taiwan Section 6 contains thediscussion and conclusions

2 Collaborative-Filtering Approaches UsingMulticriteria Ratings

Collaborative-filtering approaches rely on the ratings of auser as well as those of other users in the system [19] Thekey idea is to recommend items that users with similarpreferences have liked in the past Traditional collaborative-filtering approaches using single-criterion rating can be cat-egorized into two classes neighborhood-based and model-based approaches Adomavicius and Kwon [5] presented

the similarity-based and aggregation-function-based meth-ods within neighborhood-based approaches The latter is thefocus of this paper

21 Similarity-Based Approach Assume that a system askseach user to offer feedback on 119899 criteria with respect to aconsumed item or a person with whom he or she has aconnection Let 119877

0denote the set of possible overall ratings

and let119877119894denote the set of possible ratings for each individual

criterion (1 le 119894 le 119899) For the (user item) pairs the ratingfunction 119877 in a multicriteria recommender system is definedas follows

119877 (user item) 997888rarr 1198770times 1198771times sdot sdot sdot times 119877

119899 (1)

For instance for simplicity only the reputation (criterion 1)and the response (criterion 2) are used to evaluate an initiatoron a group-buying website (ie 119899 = 2) for the initiatorrecommendation User Randy might assign ratings of 5 7and 6 to the reputation the response and the overall ratingrespectively for initiator Ryan Of course it is necessarythat Randy has already joined the group confirmed by RyanTherefore 119877(Randy Ryan) can be denoted by (119903RandyRyan

0

119903RandyRyan1

119903RandyRyan2

) = (6 5 7) If user Frances has notyet joined the group confirmed by Ryan the recommendersystem would directly estimate the overall rating that Franceswould give to Ryan (ie 119903FrancesRyan

0) by estimating 119877

In this example without losing generality the estimate ofthe overall rating that Frances would give to Ryan is basedon the similarity between Frances and user 119906 denoted bysim(Frances 119906) who rated Ryan meanwhile the similarityis calculated according to the initiators that Frances and user119906 have both rated previouslyThemore similar Frances and 119906

are the more 119903119906Ryan0

would contribute to 119903FrancesRyan0

The cosine-based similarity measure is most commonly

used to derive sim119894(Frances 119906) on criterion 119894 sim

119894(Frances 119906)

is defined as

sim119894 (Frances 119906) = ( sum

119895isin119868(Frances 119906)119903Frances 119895119894

119903119906 119895

119894)

times (radic sum

119895isin119868(Frances 119906)(119903

Frances 119895119894

)2

times radic sum

119895isin119868(Frances 119906)(119903119906 119895

119894)2

)

minus1

= 1 119899

(2)

where 119868(Frances 119906) represents the sets of initiators rated byboth Frances and user 119906 sim(Frances 119906) is a used-basedaverage obtained by aggregating the individual similarities inseveral ways as follows

(1) average similarity

sim (Frances 119906) = 1

119899 + 1

119899

sum

119894=0

sim119894(Frances 119906) (3)

Mathematical Problems in Engineering 3

(2) worst-case similarity

sim (Frances 119906) = min119894=0119899

sim119894(Frances 119906) (4)

In addition to the cosine-based similarity measure adistance-based similarity can be formulated as follows

sim (Frances 119906)

= 1 times (1 +1

|119868 (Frances 119906)|

times sum

119895isin119868(Frances 119906)119889 (119877 (Frances 119895) 119877 (119906 119895)))

minus1

(5)

where 119889(119877(Frances 119895) 119877(119906 119895)) can be derived by variousdistance metrics for example

(i) Manhattan distance

119889 (119877 (Frances 119895) 119877 (119906 119895)) =

119899+1

sum

119894=0

10038161003816100381610038161003816119903Frances 119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (6)

(ii) Euclidean distance

119889 (119877 (Frances 119895) 119877 (119906 119895)) = radic

119899+1

sum

119894=0

(119903Frances119895119894

minus 119903119906 119895

119894)2

(7)

(iii) Chebyshev distance

119889 (119877 (Frances 119895) 119877 (119906 119895)) = max119894=0 119899

10038161003816100381610038161003816119903Frances119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (8)

The predicted overall rating 119903FrancesRyan0

can then be definedby a weighted average of 119903119906Ryan

0

119903FrancesRyan0

= 119890(Frances)

+

sum119906sim (Frances 119906) (119903119906Ryan

0minus 119890(119906))

sum119906sim (Frances 119906)

(9)

where 119890(119906) represents the average overall rating of user 119906Thisformulation has been used by the well-known GroupLens[20] to provide personalized predictions forUsenet news [21]

As for the single-criterion rating systems the ratingfunction 119877 for the (user item) pairs is defined as follows

119877 (user item) 997888rarr 1198770 (10)

where sim(Frances 119906) is simply specified as sim0(Frances 119906)

to obtain 119903FrancesRyan0

22 Aggregation-Function-Based Approach In contrast to thesimilarity-based approach the aggregation-function-basedapproach assumes that a certain relationship exists betweenthe overall rating and the multicriteria ratings of itemsUndoubtedly the aggregation function plays an importantrole for an aggregation-function-based approach The ratingfunction 119877 is defined as follows

119877 (user item) 997888rarr 119877119894 119894 = 1 119899 (11)

Following the example in the previous subsection instead ofcomputing the individual similarity weights it is necessaryto estimate that ratings of the reputation (ie 119903FrancesRyan

1)

and the response (ie 119903FrancesRyan2

) that Frances would give toRyan can be estimated by a user-based deviation-from-meanmethod as follows

119903FrancesRyan119894

= 119890119894(Frances)

+

sum119906sim119894(Frances 119906) (119903119906Ryan

119894minus 119890119894(119906))

sum119906sim119894(Frances 119906)

119894 = 1 2

(12)

where 119890119894(119906) represents the average overall rating of user 119906

for criterion 119894 119903FrancesRyan119894

can be obtained by considering thecosine-based similarity measure Then 119903

FrancesRyan0

is furtherpredicted by aggregating 119903

FrancesRyan1

and 119903FrancesRyan2

There-fore the focus of the similarity-based and the aggregation-function-based approaches is quite different The WAM isoften used to aggregate the partial ratings (ie 119903FrancesRyan

1and

119903FrancesRyan2

) when 1199081and 119908

2have been assigned

119903FrancesRyan0

= 1199081119903FrancesRyan1

+ 1199082119903FrancesRyan2

(13)

where 1199081+ 1199082

= 1 This means that a classical setfunction 120583 can be defined on 119903

FrancesRyan1

119903FrancesRyan2

with 120583(119903FrancesRyan1

) = 1199081and 120583(119903

FrancesRyan2

) = 1199082

such that 120583(119903FrancesRyan1

119903FrancesRyan2

) = 120583(119903FrancesRyan1

) +

120583(119903FrancesRyan2

) The additivity of 120583 indicates that there existsno interactions among 119903

FrancesRyan1


Unfortu-nately as mentioned above this assumption is not warrantedin many applications [4] Because the fuzzy integral does notassume the independence of elements [17 18] it is reasonableto obtain 119903

FrancesRyan0

by using a nonadditive Choquet integralto aggregate 119903FrancesRyan

1and 119903

FrancesRyan2

3 Non-Additive Aggregation-Function-BasedCollaborative-Filtering Approach

In this section the fuzzy measure used for describing theinteraction among attributes in a set is first described inSection 31 Section 32 presents the proposed approach usingthe Choquet integral and an accuracy metric for recommen-dation performance evaluation


31 Description of the Interaction Using a Fuzzy Measure Let119875(119883) denote the power set of119883 = 119909

1 1199092 119909

119899 where119883 is

called the feature spaceThen (119883 119875(119883)) is ameasurable spaceA nonadditive and nonnegative set function 120583 119875(119883) rarr

[0 1] is a fuzzymeasure that satisfies the following conditions[22ndash24]

(1) 120583(Oslash) = 0(2) for all 119877 119878 isin 119875(119883) if 119877 sub 119878 then 120583(119877) le 120583(119878)

(monotonicity)(3) for every sequence of subsets of 119883 if either 119877

1sube

1198772sube sdot sdot sdot or 119877

1supe 1198772supe sdot sdot sdot then lim

119894rarrinfin120583(119877119894) =

120583(lim119894rarrinfin

119877119894) (continuity)

When 120583(119883) = 1 120583 is said to be regular The fuzzy measureis developed to consider the interaction among attributestowards the objective attribute [17] by replacing the usualadditive property with the monotonic property

Let 120583119896denote 120583(119909

119896) which is called a fuzzy density and

119864119896

= 119909119896 119909119896+1

119909119899 (1 le 119896 le 119899) where 0 le 120583

119896le

1 Interaction among the attributes of 119864119896can be described

using 120583(119864119896) which expresses the relative importance or

discriminatory power of 119864119896 This means that 120583 can be

regarded as an importance measure and then 120583119896can be

interpreted as the degree of importance of 119909119896 120583(119864119896) may be

less than or greater than 120583119896+120583119896+1

+sdot sdot sdot+120583119899 thereby expressing

an interaction among the elements 119909119896 119909119896+1

119909119899[12] For

instance if 1205831= 03 120583

2= 05 and 120583(119909

1 1199092) = 09 then

120583(1199091 1199092) gt 120583

1+ 1205832indicates that the joint contribution of

1199091and 119909

2to the decision or the objective attribute is greater

than the sum of their individual contributionsThis indicatesthat they would enhance each other

Among the various options for 120583 a 120582-fuzzy measure is aconvenientmeans of computing the fuzzy integral [12 23 25]For all 119877 119878 isin 119875(119883) with 119877 cap 119878 = Oslash 120583 is a 120582-fuzzy measure ifit satisfies the following property

(1)120583 (Oslash) = 0 120583 (119883) = 1 (14)

(2)120583 (119877 cup 119878) = 120583 (119877) + 120583 (119878) + 120582120583 (119877) 120583 (119878) 120582 isin (minus1infin)

(15)

The advantage of using the 120582-fuzzy measure is that afterdetermining the fuzzy densities 120583

1 1205832 120583

119899 120582 can be

uniquely determined from the condition 120583(119883) = 1 120583(119864119896) can

be further determined by 120582 and 120583119895as follows

120583 (119864119896) =1

120582[ prod

119894=119896sdotsdotsdot119899

(1 + 120582120583119894) minus 1] (16)

As mentioned above 120583(119877 cup 119878) expresses the importance of119877cup 119878 The value of 120582 determines the nature of the interactionbetween 119877 and 119878 If 120582 gt 0 there is a multiplicative effectbetween 119877 and 119878 (ie 119877 and 119878 are superadditive) if 120582 lt 0

there is a substitutive effect between 119877 and 119878 (ie 119877 and 119878

are subadditive) If 120582 = 0 then 119877 and 119878 are not interactive120583(119877cup119878) = 120583(119877)+120583(119878) Actually the sign of 120582 can be identifiedby sum119899119894=1

120583119894 In other words if sum119899

119894=1120583119894gt 1 then minus1 lt 120582 lt 0 if

sum119899

119894=1120583119894lt 1 then 120582 gt 0 and if sum119899

119894=1120583119894= 1 then 120582 = 0

120583(En) 120583(Enminus1) middot middot middot

120583(E2) 120583(E1)

f(i) (Xn)

f(i) (Xnminus1)

f(i) (X2)

f(i) (X1)

Figure 1 Graphical representation of Choquet fuzzy integral

32 The Proposed Non-Additive Approach

321 Aggregating Multicriteria Ratings Using the ChoquetIntegral To consider interactions among criteria a nonaddi-tive collaborative-filtering approach is proposed to estimatefor instance 119903119906 V

0 using the Choquet integral where V is

an initiator but has not been rated by 119906 The Choquetintegral which is a generalization of the linear Lebesgueintegral (eg the weighted average method) [17] can berepresented in terms of fuzzy measures [12 18] Let x

119894=

(1199091198941 1199091198942 119909

119894119899) = (119903

119906 V1

119903119906 V2

119903119906 V119899

) For the proposednonadditive aggregation-function-based approach the syn-thetic evaluation of x

119894can be further obtained by the Choquet

integral Let 119891(119894) with respect to x119894be a non-negative real-

valuedmeasurable function defined on119883 such that119891(119894)(119909119896) =

119909119894119896(119896 = 1 2 119899) falls into a certain range The element

in 119883 with min119891(119894)(119909119896) | 119896 = 1 2 119899 is renumbered as

one where 119891(119894)(119909119896) denotes the performance or observation

value of 119909119896with respect to x

119894 In other words all elements

119909119896are rearranged in order of descending 119891

(119894)(119909119896) so that

119891(119894)(1199091) le 119891(119894)(1199092) le sdot sdot sdot le 119891

(119894)(119909119899) As illustrated in Figure 1

the Choquet integral (119888) int 119891(119894)d120583 over 119883 of 119891 with respect to

120583 is defined as follows

(119888) int119891(119894)d120583 =

119899

sum

119896=1

119891(119894)(119909119896) (120583 (119864

119896) minus 120583 (119864

119896+1)) (17)

where 120583(119864119899+1

) is specified as zero and 1198770(119906 V) = (119888) int119891

(119894)d120583If sum119895120583119895is equal to one the Choquet integral coincides with

the WAM

322 Evaluating Recommendation Performance The perfor-mance of a recommendation approach can be evaluated bydecision-support accuracymetricswhich determine howwellthe recommendation approach can predict items the userwould rate highly Commonly used metrics are precisionrecall and the 119865-measure Precision is the number of trulyhigh overall ratings expressed as a fraction of the totalnumber of ratings that the system predicted they would behigh recall is the number of correctly predicted high ratingsexpressed as a fraction of all the ratings known to be highwhile the 119865-measure is the harmonic mean of precisionand recall Often there is an inverse relationship betweenprecision and recall because it is possible to increase one at


the cost of reducing the other Usually precision and recallscores are not discussed in isolationTherefore it pays to takeinto account the 119865-measure [1] as follows

119865 =2precision sdot recallprecision + recall

(18)

where recall and precision are evenly weighted However theactual weights on precision and recall should be dependenton the goal of a recommendation approach van Rijsbergen[26] further proposed the 119865

120573measure as follows

119865120573= (1 + 120573

2)

precision sdot recall1205732precision + recall

(19)

where 120573 ge 0 119865120573weights recall higher than precision when

120573 gt 1 and weights precision higher than recall when 120573 lt 1Clearly the original 119865-measure is simply the 119865

1measure

From the viewpoint of practicality many users of recom-mendation applications are typically interested in only thefew highest-ranked item recommendations [5] Thereforethe popular metric related to precision namely precision-in-top-119873 (119873 = 1 2 3 ) should be taken into account for theproposed method This metric is defined as the fraction oftruly high overall ratings among those the system predictedwould be the119873 relevant items for each user Furthermore itis reasonable to place more emphasis on precision than onrecall to highlight the significance of precision for users Forthis a variation of the 119865

120573measure that places emphasis on

precision-in-top-119873 is presented to evaluate the performanceof the proposed nonadditive recommendation approach

V119865120573= (1 + 120573

2)

precision-in-top-119873 sdot recall1205732precision-in-top-119873 + recall

(20)

where 0 le 120573 le 1

4 A GA-Based Method for Constructing aNon-Additive Recommendation Model

41 Constructing a Non-Additive Recommendation ModelUsing Multicriteria Ratings The proposed model does notinvolve any complicated mechanisms for selecting the freeparameters Because decision-makers cannot easily pre-specify the criterion weights a real-valued GA-basedmethodis used here involving the basic operations of selectioncrossover and mutation [27 28] to determine the optimalvalues of the criterion weights (ie 120583

1 1205832 120583

119899) Let 119899size

and 119899max denote the population size and the total numberof generations respectively The following steps are usedto construct a recommendation model using the proposednonadditive collaborative-filtering approach

Algorithm 1 Construct a nonadditive recommendationmodel using the collaborative-filtering approach

Input Population size (119873pop) stopping condition (119873con ietotal number of generations) number of elite chromosomes(119873del) crossover probability (Pr

119888) mutation probability

(Pr119898) the value of 120573 for V119865

120573measure the value of 119873 for the

precision-in-top-119873measure a set of training patternsOutput A nonadditive recommendation model using thecollaborative-filtering approach with a higher V119865

120573measure

MethodStep 1 Population Initialization Generate an initial popula-tion of 119899size chromosomes each consisting of 119899 real-valuedparameters Randomly assign a real value chosen from theinterval [0 1] to each parameter in a chromosomeStep 2 Chromosome EvaluationCompute the fitness value foreach chromosome Because the objective of the algorithm isto construct a nonadditive recommendation model using thecollaborative-filtering approach with a higher V119865

120573measure

the V119865120573measure is used as the fitness function for evaluating

a chromosomeStep 3 Generation of New Chromosomes Generate a newgeneration of 119899size chromosomes by selection crossover andmutationStep 4 Elitist Strategy Randomly remove 119899del (0 le 119899del le

119899size) of the newly generated chromosomes Insert 119899del copiesof the chromosome with maximum fitness in the previousgenerationStep 5 Termination Test Terminate the algorithm if 119899maxgenerations have been generated otherwise return to Step 2

When the stopping condition is satisfied the algorithmis terminated and whichever chromosome has the maximumfitness among all generations serves as the desired solutionIt is noted that the above algorithm can also be appliedto construct an additive recommendation model using theWAM

42 Genetic Operations Let 119875119896denote the population gener-

ated in generation 119896 (1 le 119896 le 119899max) Chromosome 119894 (1 le

119896 le 119899size) generated in 119875119896is represented by 120583

119896

1198941120583119896

1198942sdot sdot sdot 120583119896

119894119899

After evaluating the fitness value for each chromosomein 119875119896 selection crossover and mutation are applied until

119899size new chromosomes have been generated for 119875119896+1

Thesegenetic operations are described in more detail below Incontrast to a nonadditive recommendation model for anadditive recommendation model using the WAM 120583119896

119894119895can be

specifically set as follows before evaluating the fitness value

120583119896

119894119895=

120583119896

119894119895

120577 (21)

where 120577 = 120583119896

1198941+ 120583119896

1198942+ sdot sdot sdot + 120583

119896

119894119899

421 Selection Using the binary tournament selection twochromosomes from the current population are randomlyselected and the one with the higher fitness is placed in amating pool This process is repeated until there are 119899sizechromosomes in the mating pool Next 119899size pairs of chro-mosomes from the pool are randomly selected for mating


The crossover and mutation operations are applied to theparents to generate children

422 Crossover For each mated pair of chromosomes 119894 and119895 1205831198961198941120583119896

1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896

119895119899 each pair of genes has a

probability Pr119888of undergoing the crossover operation The

operations are performed as 120583119896

119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a

randomnumber in the interval [0 1] Two new chromosomesare thereby generated which will replace their parents ingeneration 119875

119896+1

423 Mutation There is a probability Pr119898that the mutation

operationwill be performed on each real-valued parameter innew chromosomes generated by the crossover operation Toavoid excessive perturbation of the gene pool a lowmutationrate is used If a mutation occurs for a gene it will be changedby adding a number randomly selected from a specifiedinterval After crossover and mutation 119899del (0 le 119899del le 119899size)chromosomes in 119875

119896+1are randomly removed from the set of

new chromosomes (those formed by genetic operations) tomake room for additional copies of the chromosome withmaximum fitness value in 119875

119896

5 Application to Initiator Recommendationon a Group-Buying Website

Group-buyingwebsites play the role of a transaction platformbetween businesses and consumersThe websites call a groupof consumers with the same needs for the purchase ofitems and then negotiate with vendors to obtain the bestprice or to get a special discount Groupon which has thehigh market share is a representative group-buying websiteWith group-buying activities increasing and their associatedwebsites expanding rapidly a market research institute inVirginia BIAKelsey predicted that the group-buyingmarketin the United States will reach US$ 393 billion in 2015[29] Undoubtedly group-buying has become an importanttransaction model for online shopping

In Taiwan the Institute for Information Industry (MIC)of Taiwan reported that the group-buying market reachedapproximately US$ 239 billion in 2010 and is expectedto reach US$ 3 billion by 2011 [30] In the applicationthe proposed recommendation approach has been appliedto initiator recommendation on one popular group-buyingwebsite in Taiwan Its sales volume was over US$ 017 billionin 2010 On this website users often search for appropriateinitiators who can bargain with vendors over the price forcertain items However due to the large number of searchresults users have suffered from the problem of informationoverload especially for hot items Furthermore applicationdomains in previous research do not involve the initiatorrecommendations for the group-buyingThis motivates us toapply the recommendation techniques to this website

The computer simulations were programmed in Delphi70 and executed on a 2GHz dual-CPU Pentium computerSection 51 describes the data collected Section 52 describes

the parameter specification of the GA for the computersimulations In Section 53 the performance of the proposednonadditive recommendation approach is compared withseveral recommendation approaches using multicriteria rat-ings

51 Data Description A total of 211 undergraduates withbusiness administration as their major subject and who werefamiliar with group-buying participated in the experimentEach subject was asked to rate twenty experienced initiatorsselected from the above-mentioned website Each subjectassigned an overall rating and five criterion ratings namelyability reputation responsiveness trust and interaction toeach initiator These criteria are described belowAbility This represents knowledge and techniques that canbe used to solve problems for customers [31] Initiators areexpected not only to have the experience in confirminggroups but also to solve any problems that arise in group-buyingReputation This indicates whether the initiators have theability and intention to fulfill their promises [32]Responsiveness In a virtual community members alwaysexpect to receive responses to their posted informationfrom other members [33] In the group-buying context thiscan promote the establishment of trust among initiatorsand group members Responsiveness indicates whether theinitiators tried to respond to any problems reported by thegroup membersTrust Based on the viewpoint of trust in [34] it is consideredthat the initiators are expected to have a positive expectationof the intentionInteraction Regular communication between sellers andbuyers is helpful in building up the trust of buyers in sellers[35] Therefore initiators are expected to discuss progressand problems actively with group members during a group-buying session

All ratings range from zero representing ldquovery unsatis-factoryrdquo to ten representing ldquovery satisfactoryrdquo The overallrating indicates how much a user appreciates the initiatorBecause the decision-support measures are used to estimateaccuracy it is necessary to define every overall rating on abinary scale [21] as ldquohigh-rankedrdquo or ldquonot high-rankedrdquo Itshould be reasonable to translate these overall ratings intoa binary scale by treating the ratings greater than or equalto seven as high-ranked and those less than seven as nothigh-ranked In other words the initiators whose ratings aregreater than or equal to seven are relevant to the users

The leave-ten-out technique is used to examine thegeneralization ability of different recommendation In each ofthe iterations ten evaluations given by a subject are randomlyselected to serve as test data and the remaining evaluationswere chosen as the training data This means that test dataare produced for each subject in each of the iterations Thenthe overall rating was predicted for each evaluation in thetest data based on the information in the training data Theprecision-in-top-119873 for the test data can also be obtained


Table 1 V119865120573of different methods

Classification method 120573

1 12 13 14 15 16 17 18Single-criterion rating 0545 0676 0735 0762 0776 0784 0789 0793Average similarity 0546 0681 0742 0770 0785 0794 0799 0803Worst-case similarity 0536 0671 0733 0762 0777 0785 0791 0795Manhattan distance 0563 0707 0773 0803 0820 0829 0835 0839Euclidean distance 0553 0691 0753 0782 0797 0806 0812 0815Chebyshev distance 0553 0688 0749 0777 0792 0801 0806 0810WAM 0569 0682 0735 0756 0769 0793 0789 0792Non-additive approach 0657 0719 0751 0779 0797 0822 0837 0851

This procedure is iterated until the evaluations of each of thesubjects have been used as the test data Because the resultsof a random sampling procedure may be dependent on theselection of evaluations the above procedure is repeated fivetimes

52 GA Parameter Specifications A number of factors caninfluence GA performance including the size of the pop-ulation and the probabilities of applying the crossover andmutation operators Unfortunately there is no standardprocedure for choosing optimal GA specifications Based onthe principles suggested by Osyczka [36] and Ishibuchi et al[37] the parameters are specified as follows

(i) 119899size = 50 GA populations commonly range from50 to 500 individuals Hence 50 individuals is anacceptable size

(ii) 119899max = 500 the stopping condition is specifiedaccording to the available computing time

(iii) 119899del = 2 only a small number of elite chromosomesare used

(iv) Pr119888= 10 Pr

119898= 001 since Pr

119888controls the range of

exploration in the solution space most sources rec-ommend choosing a large value To avoid generatingan excessive perturbation Pr

119898should be set to a small

value(v) 119873 = 5 it is assumed that users required the system

to recommend the five most highly ranked initiatorsIn other words the precision-in-top-5 is incorporatedinto the V119865

120573measure

(vi) 120573 is specified as 1 12 13 18 A recommendersystem is constructed for each 120573 value

Although the above specifications are somewhat subjectivethe experimental results show that they are acceptableTherefore customized parameter tuning is not considered forthe proposed approach

53 Performance Evaluation The proposed nonadditiveapproach is compared with several collaborative-filteringapproaches introduced in the previous section the traditionalsingle-criterion rating approach similarity-based approachesusing multicriteria ratings with average similarity worst-case

similarity distance-based similarity as described in [5] andan aggregation-function-based approach using theWAM Bya random sampling procedure with 50 training data and50 test data the average results of these approaches areobtained from five trials For each evaluation in the test datathe evaluated initiator is treated as the target item Eachapproach considered is used to predict the overall ratingthat the target item would have by using the training dataThen ldquohigh-rankedrdquo or ldquonot high-rankedrdquo for each targetitem could be determined by its predicted overall rating

By maximizing V119865120573with the precision-in-top-5 measure

for a certain 120573 it is interesting to examine the precision-in-top-1 and precision-in-top-3 for the proposed approachThe idea is to identify 120573 for which the proposed method canperform well compared with the similarity-based methodsconsidered for V119865

120573and various precision-in-top-119873measures

(119873 = 1 3 5) Table 1 shows that the V119865120573value of all the

methods improved from 120573 = 1 to 120573 = 18 This isreasonable because V119865

120573approaches precision-in-top-119873when

120573 is sufficiently small The notable results in Tables 1 2 and 3can be summarized as follows

(1) An approach with a higher precision-in-top-5 mea-sure is not guaranteed to have a higher V119865

120573 For

instance for 120573 = 1 and 12 although the V119865120573

value of the proposed nonadditive approach exceedsthe similarity-based approaches considered it can beseen in Table 2 that the precision-in-top-5 measuresobtained by the proposed approach are inferior tothose obtained by the similarity-based approachesand the WAM

(2) When 120573 le 15 the proposed nonadditive approachperforms well compared with the similarity-basedmethods using the average similarity and the worst-case similarity for V119865

120573and different precision-in-top-

119873measures (119873 = 1 3 5)

(3) When 120573 le 17 the proposed nonadditive approachperforms well compared with similarity-based meth-ods using distance-based similarities for V119865

120573 the

precision-in-top-3 measure and the precision-in-top-5 measure

(4) The proposed nonadditive approach is inferior tothe approaches using distance-based similarities for


Table 2 Various precisions () of different methods

Method Precision-in-top-1 Precision-in-top-3 Precision-in-top-5Single-criterion rating 8202 8290 8045Average similarity 8302 8335 8148Worst-case similarity 8352 8357 8081Manhattan distance 9371 8855 8522Euclidean distance 9151 8644 8277Chebyshev distance 8928 8536 8217

the precision-in-top-1 measure (ie Manhattan andEuclidean distances) regardless of the value of 120573

(5) When 120573 le 15 the proposed nonadditive approachperforms well compared with the single-criterionrating approach for V119865

120573and various precision-in-top-

119873measures (119873 = 1 3 5)(6) The proposed nonadditive approach outperforms the

WAM for V119865120573and various precision-in-top-119873 mea-

sures (119873 = 1 3 5) when 120573 le 14(7) Among the similarity-based approaches the single-

criterion rating approach seems to perform worst forV119865120573and various precision-in-top-119873 measures (119873 =

1 3 5) regardless of the value of 120573(8) The WAM outperforms the single-criterion rating

approach for the precision-in-top-5 measure when120573 le 13The precision-in-top-1 and precision-in-top-3 measures of the WAM are inferior to those of thesingle-criterion rating approach

Therefore by incorporating the precision-in-top-5 measureinto the fitness function (ie V119865

120573) with appropriate 120573 values

the proposed nonadditive approach is found to outper-form the traditional single-criterion rating approach andthe similarity-based approaches using multicriteria ratingsfor V119865

120573 the precision-in-top-3 and the precision-in-top-

5 measures Moreover the proposed nonadditive approachoutperforms the additive WAM when using appropriate 120573

values for V119865120573and different precision-in-top-119873 measures

The experimental results indicate that leveraging the mul-ticriteria ratings of initiators could be helpful in improvingthe prediction performance of the traditional single-criterionrating approach In addition it is found that120582 is less thanminus09for the best solution In other words there exists a substitutiveinteraction effect among the attributes Therefore it seemsquite reasonable to use the fuzzy integral with a 120582-fuzzymeasure as an aggregation function instead of the WAM

6 Discussion and Conclusions

It is known that the assumption of independence amongcriteria in the WAM is not always reasonable The mainissue that this paper addresses is the additivity of the WAMfor the multicriteria aggregation-function-based approachIn view of the nonadditivity of the fuzzy integral this papercontributes to present a nonadditive aggregation-function-based approach using multicriteria ratings by combining thesimilarity-based approach using single-criterion ratings with

theChoquet fuzzy integralThe former is used to predictmul-ticriteria ratings and the latter to generate an overall rating foran item Becausemany users of recommendation applicationsare typically interested in only the few highest-ranked itemrecommendations a variant of the 119865

120573measure has been

designed by incorporating the precision-in-top-119873 metricinto the 119865

120573measure to assess prediction performance Both

the proposed nonadditive approach and the similarity-basedapproacheswith averageworst-case similarity use the cosine-based similarity measure to obtain the similarity on eachcriterion between two users Whereas the former (ie theproposed nonadditive approach) derives the rating for eachcriterion from individual similarities and then estimates theoverall rating for an item the latter (ie the similarity-basedapproaches with averageworst-case similarity) aggregatesthe individual similarities to compute the overall similaritybetween two users

The proposed nonadditive approach is applied to initiatorrecommendation on a group-buying website in Taiwan inorder to examine its prediction performance The criterionweights for the fuzzy integral are determined automaticallyby a GA Compared with the traditional single-criterionrating approach several collaborative-filtering approachesusing multicriteria ratings and the aggregation-function-based approach using the WAM it can be seen that theproposed nonadditive collaborative-filtering approach yieldsencouraging V119865

120573 precision-in-top-3 and precision-in-top-5

measures with appropriate 120573 values Therefore the proposedapproach is able to improve recommendation performanceby leveraging multicriteria information This indicates thatthe proposed nonadditive approach has applicability to otherapplication domains Note that the choice of 120573 is eventuallydependent on proprietors of websites Besides identificationof the best collaborative-filtering approach is not possiblebecause there is no such thing as the ldquobestrdquo approach [25]

It is notable that when a traditional single-criterionrating approach is treated as a baseline collaborative-filtering approach the experimental results indicate thatthe similarity-based approaches using multicriteria ratingsof initiators perform better than the traditional single-criterion rating approach Moreover the proposed nonad-ditive approach and the WAM outperform the traditionalsingle-criterion rating approach with appropriate 120573 valuesHowever it is not possible to conclude that recommendationapproaches using multicriteria ratings outperform the tradi-tional single-criterion rating approach in all domains wheremulticriteria information exists


Table 3 Various precisions () of WAM and non-additive approach

Method Precision 120573

1 12 13 14 15 16 17 18

WAMPrecision-in-top-1 7701 7862 7981 8072 8103 8104 8166 8198Precision-in-top-3 8009 8115 8161 8096 8029 8164 8137 8136Precision-in-top-5 7837 8043 8134 8125 8119 8220 8246 8207

Non-additive approachPrecision-in-top-1 7616 7884 8021 8391 8616 9245 9015 8844Precision-in-top-3 7603 7819 8033 8289 8532 8868 8867 8858Precision-in-top-5 7488 7768 7942 8308 8550 8867 8863 8914

As mentioned above the precision-in-top-1 andprecision-in-top-3 measures for the proposed nonadditiveapproach are generated by a system that is trained byincorporating the precision-in-top-5 measure into the fitnessfunction whereas the precision-in-top-1 measures obtainedby the proposed nonadditive approach are inferior to thoseobtained by approaches using distance-based similarities Itwould be interesting to examine whether the precision-in-top-1 measure obtained by the proposed approach can beimproved when the system is trained by incorporating theprecision-in-top-1 measure into the fitness function

Acknowledgments

The author would like to thank the anonymous referees fortheir valuable commentsThis research is partially supportedby the National Science Council of Taiwan under grant NSC102-2410-H-033-039-MY2

References

[1] D Jannach M Zanker A Felfernig and G Friedrich Recom-mender Systems An Introduction Cambridge University PressNew York NY USA 2010

[2] M Gao Z Wu and F Jiang ldquoUserrank for item-based col-laborative filtering recommendationrdquo Information ProcessingLetters vol 111 no 9 pp 440ndash446 2011

[3] S-L Huang ldquoDesigning utility-based recommender systemsfor e-commerce evaluation of preference-elicitation methodsrdquoElectronic Commerce Research and Applications vol 10 no 4pp 398ndash407 2011

[4] 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

[5] G Adomavicius and Y Kwon ldquoNew recommendation tech-niques formulticriteria rating systemsrdquo IEEE Intelligent Systemsvol 22 no 3 pp 48ndash55 2007

[6] G Adomavicius and A Tuzhilin ldquoToward the next generationof recommender systems a survey of the state-of-the-art andpossible extensionsrdquo IEEE Transactions on Knowledge and DataEngineering vol 17 no 6 pp 734ndash749 2005

[7] J B Schafer ldquoDynamicLens a dynamic user-interface fora meta-recommendation systemrdquo in Proceedings of BeyondPersonalization 2005 A Workshop on the Next Stage of Recom-mender Systems Research pp 72ndash76

[8] W-P Lee C-H Liu and C-C Lu ldquoIntelligent agent-basedsystems for personalized recommendations in Internet com-mercerdquo Expert Systems with Applications vol 22 no 4 pp 275ndash284 2002

[9] F Ricci and HWerthner ldquoCase-based querying for travel plan-ning recommendationrdquo Information Technology and Tourismvol 4 no 3-4 pp 215ndash226 2002

[10] N Manouselis and C Costopoulou ldquoExperimental analysis ofdesign choices in multi-attribute utility collaborative filteringrdquoin Personalization Techniques and Recommender G Uchyigitand M Y Ma Eds pp 111ndash134 World Scientific River EdgeNJ USA 2008

[11] T Onisawa M Sugeno Y Nishiwaki H Kawai and Y HarimaldquoFuzzy measure analysis of public attitude towards the use ofnuclear energyrdquo Fuzzy Sets and Systems vol 20 no 3 pp 259ndash289 1986

[12] W Wang ldquoGenetic algorithms for determining fuzzy measuresfrom datardquo Journal of Intelligent and Fuzzy Systems vol 6 no 2pp 171ndash183 1998

[13] T Murofushi and M Sugeno ldquoAn interpretation of fuzzymeasures and the Choquet integral as an integral with respectto a fuzzy measurerdquo Fuzzy Sets and Systems vol 29 no 2 pp201ndash227 1989

[14] T Murofushi and M Sugeno ldquoA theory of fuzzy measuresrepresentations the Choquet integral and null setsrdquo Journal ofMathematical Analysis and Applications vol 159 no 2 pp 532ndash549 1991

[15] T Murofushi and M Sugeno ldquoSome quantities represented bythe Choquet integralrdquo Fuzzy Sets and Systems vol 56 no 2 pp229ndash235 1993

[16] M Sugeno Y Narukawa and T Murofushi ldquoChoquet integraland fuzzy measures on locally compact spacerdquo Fuzzy Sets andSystems vol 99 no 2 pp 205ndash211 1998

[17] Z Wang K-S Leung and G J Klir ldquoApplying fuzzy measuresand nonlinear integrals in dataminingrdquo Fuzzy Sets and Systemsvol 156 no 3 pp 371ndash380 2005

[18] Z Wang K-S Leung and J Wang ldquoA genetic algorithm fordetermining nonadditive set functions in information fusionrdquoFuzzy Sets and Systems vol 102 no 3 pp 463ndash469 1999

[19] C Desrosiers and G Karypis ldquoA comprehensive survey ofneighborhood-based recommendation methodsrdquo in Recom-mender Systems Handbook F Ricci L Rokach B Shapira andP B Kantor Eds pp 107ndash144 Springer New York NY USA2011

[20] J A Konstan B N Miller D Maltz J L Herlocker L RGordon and J Riedl ldquoApplying collaborative filtering to usenetnewsrdquo Communications of the ACM vol 40 no 3 pp 77ndash871997


[21] J Herlocker J Konstan A Borchers and J Riedl ldquoAn algo-rithmic framework for performing collaborative filteringrdquo inProceedings of the 22nd Annual International ACM SIGIR Con-ference on Research and Development in Information Retrievalpp 230ndash237 1999

[22] M Sugeno Theory of fuzzy integrals and its applications [PhDthesis] Tokyo Institute of Technology Tokyo Japan 1974

[23] M Sugeno ldquoFuzzy measures and fuzzy integralsmdasha surveyrdquoin Fuzzy Automata and Decision Processes pp 89ndash102 North-Holland New York NY USA 1977

[24] Z Y Wang and G J Klir Fuzzy Measure Theory Plenum PressNew York NY USA 1992

[25] L I Kuncheva Fuzzy Classifier Design Physica HeidelbergGermany 2000

[26] C J van Rijsbergen Information Retrieval Butterworth Lon-don UK 1979

[27] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learning Addison-Wesley Reading Mass USA 1989

[28] K F Man K S Tang and S Kwong Genetic AlgorithmsConcepts and Designs Springer London UK 1999

[29] B I A Kelsey 2011 httpwwwbiakelseycom[30] ldquoForeseeing Innovative New Digiservices in Institute for

Information Industryrdquo 2011 httpwwwfindorgtwfindhomeaspxpage=manyampid=290

[31] J Singh and D Sirdeshmukh ldquoAgency and trust mechanismsin consumer satisfaction and loyalty judgmentsrdquo Journal of theAcademy of Marketing Science vol 28 no 1 pp 150ndash167 2000

[32] S Ganesan ldquoDeterminants of long-term orientation in buyer-seller relationshipsrdquo Journal of Marketing vol 58 pp 1ndash19 1994

[33] C M Ridings D Gefen and B Arinze ldquoSome antecedentsand effects of trust in virtual communitiesrdquo Journal of StrategicInformation Systems vol 11 no 3-4 pp 271ndash295 2002

[34] D M Rousseau S Sitkin R S Burt and C Camerer ldquoNot sodifferent after all a cross-discipline view of trustrdquo Academy ofManagement Review vol 23 pp 393ndash404 1998

[35] P M Doney and J P Cannon ldquoAn examination of the nature oftrust in buyer-seller relationshipsrdquo Journal of Marketing vol 61no 2 pp 35ndash51 1997

[36] A Osyczka Evolutionary Algorithms for Single andMulticriteriaDesign Optimization Physica New York NY USA 2002

[37] H Ishibuchi T Nakashima and M Nii Classification andModeling with Linguistic Information Granules AdvancedApproaches to Linguistic Data Mining Springer 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


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


These recommendation systems do not allow users to specifytheir subjective perception of the various criteria of individ-ual items

From the viewpoint of multicriteria decision making(MCDM) the overall rating has a certain relationship withthe multicriteria ratings for an item For this in contrastto the multicriteria recommendation systems previouslydescribed Adomavicius and Kwon [5] presented a frame-work for an aggregation-function-based approach to leveragemulticriteria rating information in which the rating for eachcriterion can be estimated by the traditional similarity-basedapproach using single-criterion ratings Then the output ofan aggregation function is regarded as a predicted value ofthe overall rating of an unrated item The weighted averagemethod (WAM) is often used as an aggregation function andcan aggregate ratings on different criteria when the criterionweights are available [5 10] Note that the criterion weights inWAM are interpreted as the relative importance of the crite-ria The WAM is a simple decomposed method and assumesthat criteria do not interact [11] However because the criteriaare not always independent the assumption of additivity isnot always reasonable [12] and may affect recommendationperformance Thus this motivates us to use a nonadditivetechnique the Choquet integral [13ndash16] as an aggregationfunction Actually the Choquet integral is a generalizationof the weighted average [17 18] To address this this paperfurther proposes a novel nonadditive multicriteria recom-mendation method on the basis of the Choquet integralFurthermore because the goal of the proposed approach is torecommend correctly a set of a few relevant items to each usera variation of the popular 119865-measure is presented to evaluaterecommendation performance To achieve high accuracy agenetic algorithm (GA) is implemented to determine theparameter specifications

The remainder of the paper is organized as followsSection 2 briefly introduces several collaborative-filteringapproaches using multicriteria ratings including thesimilarity-based and the aggregation-function-basedapproaches Section 3 describes the proposed aggregation-function-based collaborative-filtering approach using theChoquet integral and presents an accuracy metric forrecommendation performance evaluation A GA-basedmethod for constructing a nonadditive recommendationmodel is demonstrated in Section 4 Section 5 applies theproposed nonadditive approach to initiator recommendationon a group-buying website in Taiwan Section 6 contains thediscussion and conclusions

2 Collaborative-Filtering Approaches UsingMulticriteria Ratings

Collaborative-filtering approaches rely on the ratings of auser as well as those of other users in the system [19] Thekey idea is to recommend items that users with similarpreferences have liked in the past Traditional collaborative-filtering approaches using single-criterion rating can be cat-egorized into two classes neighborhood-based and model-based approaches Adomavicius and Kwon [5] presented

the similarity-based and aggregation-function-based meth-ods within neighborhood-based approaches The latter is thefocus of this paper

21 Similarity-Based Approach Assume that a system askseach user to offer feedback on 119899 criteria with respect to aconsumed item or a person with whom he or she has aconnection Let 119877

0denote the set of possible overall ratings

and let119877119894denote the set of possible ratings for each individual

criterion (1 le 119894 le 119899) For the (user item) pairs the ratingfunction 119877 in a multicriteria recommender system is definedas follows

119877 (user item) 997888rarr 1198770times 1198771times sdot sdot sdot times 119877

119899 (1)

For instance for simplicity only the reputation (criterion 1)and the response (criterion 2) are used to evaluate an initiatoron a group-buying website (ie 119899 = 2) for the initiatorrecommendation User Randy might assign ratings of 5 7and 6 to the reputation the response and the overall ratingrespectively for initiator Ryan Of course it is necessarythat Randy has already joined the group confirmed by RyanTherefore 119877(Randy Ryan) can be denoted by (119903RandyRyan

0

119903RandyRyan1

119903RandyRyan2

) = (6 5 7) If user Frances has notyet joined the group confirmed by Ryan the recommendersystem would directly estimate the overall rating that Franceswould give to Ryan (ie 119903FrancesRyan

0) by estimating 119877

In this example without losing generality the estimate ofthe overall rating that Frances would give to Ryan is basedon the similarity between Frances and user 119906 denoted bysim(Frances 119906) who rated Ryan meanwhile the similarityis calculated according to the initiators that Frances and user119906 have both rated previouslyThemore similar Frances and 119906

are the more 119903119906Ryan0

would contribute to 119903FrancesRyan0

The cosine-based similarity measure is most commonly

used to derive sim119894(Frances 119906) on criterion 119894 sim

119894(Frances 119906)

is defined as

sim119894 (Frances 119906) = ( sum

119895isin119868(Frances 119906)119903Frances 119895119894

119903119906 119895

119894)

times (radic sum

119895isin119868(Frances 119906)(119903

Frances 119895119894

)2

times radic sum

119895isin119868(Frances 119906)(119903119906 119895

119894)2

)

minus1

= 1 119899

(2)

where 119868(Frances 119906) represents the sets of initiators rated byboth Frances and user 119906 sim(Frances 119906) is a used-basedaverage obtained by aggregating the individual similarities inseveral ways as follows

(1) average similarity

sim (Frances 119906) = 1

119899 + 1

119899

sum

119894=0

sim119894(Frances 119906) (3)



sim (Frances 119906) = min119894=0119899

sim119894(Frances 119906) (4)



= 1 times (1 +1

|119868 (Frances 119906)|

times sum


minus1

(5)



119889 (119877 (Frances 119895) 119877 (119906 119895)) =

119899+1

sum

119894=0

10038161003816100381610038161003816119903Frances 119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (6)


119889 (119877 (Frances 119895) 119877 (119906 119895)) = radic

119899+1

sum

119894=0

(119903Frances119895119894

minus 119903119906 119895

119894)2

(7)


119889 (119877 (Frances 119895) 119877 (119906 119895)) = max119894=0 119899

10038161003816100381610038161003816119903Frances119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (8)



0

119903FrancesRyan0

= 119890(Frances)

+


0minus 119890(119906))


(9)



119877 (user item) 997888rarr 1198770 (10)




119877 (user item) 997888rarr 119877119894 119894 = 1 119899 (11)


1)




= 119890119894(Frances)

+


119894minus 119890119894(119906))

sum119906sim119894(Frances 119906)

119894 = 1 2

(12)




FrancesRyan0


FrancesRyan1



1and

119903FrancesRyan2

) when 1199081and 119908

2have been assigned

119903FrancesRyan0



(13)

where 1199081+ 1199082


FrancesRyan1

119903FrancesRyan2


) = 1199081and 120583(119903

FrancesRyan2

) = 1199082


119903FrancesRyan2

) = 120583(119903FrancesRyan1

) +



FrancesRyan1



FrancesRyan0


1and 119903

FrancesRyan2





1 1199092 119909

119899 where119883 is





1sube



119894rarrinfin120583(119877119894) =


119877119894) (continuity)


Let 120583119896denote 120583(119909


119864119896

= 119909119896 119909119896+1

119909119899 (1 le 119896 le 119899) where 0 le 120583

119896le









119909119899[12] For

instance if 1205831= 03 120583

2= 05 and 120583(119909

1 1199092) = 09 then

120583(1199091 1199092) gt 120583


1199091and 119909




(1)120583 (Oslash) = 0 120583 (119883) = 1 (14)


(15)


1 1205832 120583

119899 120582 can be



120583 (119864119896) =1

120582[ prod

119894=119896sdotsdotsdot119899

(1 + 120582120583119894) minus 1] (16)






sum119899


119894=1120583119894= 1 then 120582 = 0


120583(E2) 120583(E1)

f(i) (Xn)

f(i) (Xnminus1)

f(i) (X2)

f(i) (X1)






119894=

(1199091198941 1199091198942 119909

119894119899) = (119903

119906 V1

119903119906 V2

119903119906 V119899











(119894)(119909119896) so that





(119888) int119891(119894)d120583 =

119899

sum

119896=1

119891(119894)(119909119896) (120583 (119864

119896) minus 120583 (119864

119896+1)) (17)

where 120583(119864119899+1



the WAM





(18)



119865120573= (1 + 120573

2)


(19)



1measure




V119865120573= (1 + 120573

2)


(20)




1 1205832 120583

119899) Let 119899size








120573measure


120573measure








119896

1198941120583119896

1198942sdot sdot sdot 120583119896

119894119899




119894119895can be


120583119896

119894119895=

120583119896

119894119895

120577 (21)

where 120577 = 120583119896

1198941+ 120583119896

1198942+ sdot sdot sdot + 120583

119896

119894119899





1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896




119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a


119896+1





119896


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











sim (Frances 119906) = min119894=0119899

sim119894(Frances 119906) (4)



= 1 times (1 +1

|119868 (Frances 119906)|

times sum


minus1

(5)



119889 (119877 (Frances 119895) 119877 (119906 119895)) =

119899+1

sum

119894=0

10038161003816100381610038161003816119903Frances 119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (6)


119889 (119877 (Frances 119895) 119877 (119906 119895)) = radic

119899+1

sum

119894=0

(119903Frances119895119894

minus 119903119906 119895

119894)2

(7)


119889 (119877 (Frances 119895) 119877 (119906 119895)) = max119894=0 119899

10038161003816100381610038161003816119903Frances119895119894

minus 119903119906 119895

119894

10038161003816100381610038161003816 (8)



0

119903FrancesRyan0

= 119890(Frances)

+


0minus 119890(119906))


(9)



119877 (user item) 997888rarr 1198770 (10)




119877 (user item) 997888rarr 119877119894 119894 = 1 119899 (11)


1)




= 119890119894(Frances)

+


119894minus 119890119894(119906))

sum119906sim119894(Frances 119906)

119894 = 1 2

(12)




FrancesRyan0


FrancesRyan1



1and

119903FrancesRyan2

) when 1199081and 119908

2have been assigned

119903FrancesRyan0



(13)

where 1199081+ 1199082


FrancesRyan1

119903FrancesRyan2


) = 1199081and 120583(119903

FrancesRyan2

) = 1199082


119903FrancesRyan2

) = 120583(119903FrancesRyan1

) +



FrancesRyan1



FrancesRyan0


1and 119903

FrancesRyan2





1 1199092 119909

119899 where119883 is





1sube



119894rarrinfin120583(119877119894) =


119877119894) (continuity)


Let 120583119896denote 120583(119909


119864119896

= 119909119896 119909119896+1

119909119899 (1 le 119896 le 119899) where 0 le 120583

119896le









119909119899[12] For

instance if 1205831= 03 120583

2= 05 and 120583(119909

1 1199092) = 09 then

120583(1199091 1199092) gt 120583


1199091and 119909




(1)120583 (Oslash) = 0 120583 (119883) = 1 (14)


(15)


1 1205832 120583

119899 120582 can be



120583 (119864119896) =1

120582[ prod

119894=119896sdotsdotsdot119899

(1 + 120582120583119894) minus 1] (16)






sum119899


119894=1120583119894= 1 then 120582 = 0


120583(E2) 120583(E1)

f(i) (Xn)

f(i) (Xnminus1)

f(i) (X2)

f(i) (X1)






119894=

(1199091198941 1199091198942 119909

119894119899) = (119903

119906 V1

119903119906 V2

119903119906 V119899











(119894)(119909119896) so that





(119888) int119891(119894)d120583 =

119899

sum

119896=1

119891(119894)(119909119896) (120583 (119864

119896) minus 120583 (119864

119896+1)) (17)

where 120583(119864119899+1



the WAM





(18)



119865120573= (1 + 120573

2)


(19)



1measure




V119865120573= (1 + 120573

2)


(20)




1 1205832 120583

119899) Let 119899size








120573measure


120573measure








119896

1198941120583119896

1198942sdot sdot sdot 120583119896

119894119899




119894119895can be


120583119896

119894119895=

120583119896

119894119895

120577 (21)

where 120577 = 120583119896

1198941+ 120583119896

1198942+ sdot sdot sdot + 120583

119896

119894119899





1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896




119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a


119896+1





119896


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 1199092 119909

119899 where119883 is





1sube



119894rarrinfin120583(119877119894) =


119877119894) (continuity)


Let 120583119896denote 120583(119909


119864119896

= 119909119896 119909119896+1

119909119899 (1 le 119896 le 119899) where 0 le 120583

119896le









119909119899[12] For

instance if 1205831= 03 120583

2= 05 and 120583(119909

1 1199092) = 09 then

120583(1199091 1199092) gt 120583


1199091and 119909




(1)120583 (Oslash) = 0 120583 (119883) = 1 (14)


(15)


1 1205832 120583

119899 120582 can be



120583 (119864119896) =1

120582[ prod

119894=119896sdotsdotsdot119899

(1 + 120582120583119894) minus 1] (16)






sum119899


119894=1120583119894= 1 then 120582 = 0


120583(E2) 120583(E1)

f(i) (Xn)

f(i) (Xnminus1)

f(i) (X2)

f(i) (X1)






119894=

(1199091198941 1199091198942 119909

119894119899) = (119903

119906 V1

119903119906 V2

119903119906 V119899











(119894)(119909119896) so that





(119888) int119891(119894)d120583 =

119899

sum

119896=1

119891(119894)(119909119896) (120583 (119864

119896) minus 120583 (119864

119896+1)) (17)

where 120583(119864119899+1



the WAM





(18)



119865120573= (1 + 120573

2)


(19)



1measure




V119865120573= (1 + 120573

2)


(20)




1 1205832 120583

119899) Let 119899size








120573measure


120573measure








119896

1198941120583119896

1198942sdot sdot sdot 120583119896

119894119899




119894119895can be


120583119896

119894119895=

120583119896

119894119895

120577 (21)

where 120577 = 120583119896

1198941+ 120583119896

1198942+ sdot sdot sdot + 120583

119896

119894119899





1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896




119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a


119896+1





119896


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(18)



119865120573= (1 + 120573

2)


(19)



1measure




V119865120573= (1 + 120573

2)


(20)




1 1205832 120583

119899) Let 119899size








120573measure


120573measure








119896

1198941120583119896

1198942sdot sdot sdot 120583119896

119894119899




119894119895can be


120583119896

119894119895=

120583119896

119894119895

120577 (21)

where 120577 = 120583119896

1198941+ 120583119896

1198942+ sdot sdot sdot + 120583

119896

119894119899





1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896




119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a


119896+1





119896


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198942sdot sdot sdot 120583119896

119894119899and 120583

119896

1198951120583119896

1198952sdot sdot sdot 120583119896




119894119908

1015840

= 119886119908120583119896

119894119908+ (1 minus 119886

119908)120583119896

119895119908

120583119896

119895119908

1015840

= (1 minus 119886119908)120583119896

119894119908+ 119886119908120583119896

119895119908(1 le 119908 le 119899) where 119886

119908is a


119896+1





119896


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















(iv) Pr119888= 10 Pr

119898= 001 since Pr






120573measure













120573 For





119873measures (119873 = 1 3 5)


120573 the




































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































1 12 13 14 15 16 17 18




Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Novel Nonadditive Collaborative...

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