Research ArticleA Framework for Online Reverse Auction Based onMarket MakerLearning with a Risk-Averse Buyer

Hojat Tayaran and Mehdi Ghazanfari

School of Industrial Engineering Iran University of Science and Technology Tehran Iran

Correspondence should be addressed to Mehdi Ghazanfari mehdiiustacir

Received 5 June 2020 Revised 20 August 2020 Accepted 22 September 2020 Published 6 October 2020

Academic Editor Rosa M Benito

Copyright copy 2020 Hojat Tayaran and Mehdi Ghazanfari is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

e online reverse auction is considered as a new e-commerce approach to purchasing and procuring goods and materials in thesupply chain With the rapid and ever-expanding development of information technology as well as the increasing usage of theInternet around the world the use of an online reverse auction method to provide the required items by organizations hasincreased Accordingly in this paper a new framework for the online reverse auction process is provided that takes both sides ofthe procurement process namely buyer and sellere proposed process is a multiattribute semisealed multiround online reverseauction e main feature of the proposed process is that an online market maker facilitates the sellerrsquos bidding process by theestimation of the buyerrsquos scoring function For this purpose a multilayer perceptron neural network was used to estimate thescoring function In this case in addition to hiding the buyerrsquos scoring function sellers can improve their bids using the estimatedscoring function and a nonlinear multiobjective optimization model e NSGA II algorithm has been used to solve the sellermodel To evaluate the proposed model the auction process is simulated by considering three scoring functions (additivemultiplicative and risk-aversion) and two types of open and semisealed auctionse simulation results show that the efficiency ofthe proposed model is not significantly different from the open auction and in addition unlike the open auction the buyerinformation was not disclosed

1 Introduction

An essential aspect of supply chain systems which is exten-sively discussed in the literature is the procurement of therequired materials and logistics In this regard the reverseauction method in the supply chain management is used toselect the best suppliers for the procurement [1] More pre-cisely a reverse auction has a similar structure to a forwardauction with the difference that in a reverse auction a buyerrequests several potential supplierssellers to make their bidsto sell one or more products en the buyer investigates thebids and selects one or more suppliers [2] e growth ofe-commerce since the late 1990s significantly enhanced theuse of online environments to organize reverse auctionsMany companies employ electronic reverse auction platformsto supply their productse use of this approach has resultedin significant savings in comparison to other common

product supply approaches particularly in business-to-business (B2B) procedures [3] us different companies andresearchers have developed online platforms with differentfeatures to facilitate the execution of reverse auctions

Despite the advantages of online reverse auctions theexperience shows that the transaction sides become lesswilling to use online reverse auctions after a while [4 5]Some of the reasons for this unwillingness are

(1) Online reverse auction focuses on buyerrsquos prefer-ences and ignores sellerrsquos preferences In otherwords sellers often do not have the opportunity toimprove their bids in an auction while buyers havemultiple alternatives from different sellers to opti-mize their purchases [6]

(2) Sellers need a multiround process to have the op-portunity to improve their bids In this case they can

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5604246 13 pageshttpsdoiorg10115520205604246

optimize their bid in the new round based on buyerrsquosfeedback in the previous round However due to thebuyerrsquos privacy information is generally not pro-vided to him other than the sellerrsquos rank or rating sothere is little possibility of optimizing the bid

Accordingly this study attempts to propose a mechanismthat incorporates the challenges mentioned above and findssolutions to mitigate or solve them In the proposed model inaddition to selecting the best bid by the buyer sellers also havethe opportunity to improve their bid and utility in a multi-period process based on the limited disclosed informationInformation is disclosed in such a way that the privacy of boththe buyer and the seller is maintained

In summary a major difference between the currentstudy and other multiattribute reverse auction studies suchas [7ndash12] is the information disclosure mechanism In theproposed model an online auction service provider (OASP)as an interface between the buyer and seller estimates thebuyerrsquos scoring function It then provides this information tothe seller to optimize their offer using the proposed newoptimization model OASP employs a multilayer perceptronneural network to predict a buyer scoring function andsellers solve a nonlinear mathematical programming modelusing the NSGAII multiobjective genetic algorithm

e remainder of this paper is organized as follows InSection 2 the review of literature related to multiattributereverse auctionmodeling is carried out In Section 3 the mainproblem is described and then in Section 4 in a simulatedenvironment several numerical examples are presented tocompare the different revelation policies and scoring func-tions Finally in Section 5 the overall conclusion of the paperand suggestions for future research are presented

2 Literature Review

An auction problem can have several characteristics In ageneral classification Teich et al [13] have divided thecharacteristics of an auction into 16 categories by reviewingthe literature (Table 1) Since the proposedmodel in this paperis structured based onmultiattribute reverse auction researchrelated to this structure has been considered in this section

e multiattribute reverse auction can be categorizedbased on either the buyerrsquos problem the sellerrsquos problemnumber of rounds of the auction number of attributes orthe type of disclosure of the buyerrsquos or sellerrsquos informationSome studies have focused on both the buyer and the sellerside [14ndash20] while in some research [21ndash25] only the buyerrsquosside has been taken into consideration On the one hand theaim of buyerrsquos problem is generally seeking to maximize theexpected scoring function [15 19 25 26] maximizing thebuyerrsquos utility [14 17 23] maximizing auction performance[18] minimizing purchase costs [20] or maximizing theprobability of winner determination in the next round [22]On the other hand the sellerrsquos problem in a competitivecondition is considered by equilibrium solutions such as theBayesian game [15] In monopoly conditions the maximi-zation of utility [14 19] or profit [17] is considered as theproblem of seller It should be noted that reverse auctions

can be single-round [7 18ndash20 25 26] two-round [23] ormultiround [14 15 17 22] In most cases more rounds inthe auction process can help to find the optimal price for thebuyerrsquos side and also increase the level of competition be-tween sellers It can be conducted on the basis of the in-formation provided to sellers in each round erefore thelevel of information disclosure is an influential factor inassessing the performance of an auction [27]

When it comes to the information disclosure in thereverse auction it can be divided into three categories (1)the disclosure of sellerrsquos information for other sellers (2) thedisclosure of sellerrsquos information for the buyer and (3) thedisclosure of buyerrsquos information for sellers e disclosureof sellerrsquos information for other sellers mainly includes theinformation about the bids and the rank of each seller Atmost real auctions the reverse auction is executed in a sealedform It means that the sellersrsquo information is not disclosedand the result of the ranking is privately announced to sellers[7 19 20 22 23 25] However it has been seen in someresearch [14 17 18] that this process has been consideredopenly In [15] to model the Bayesian games the costdistribution function of the seller is revealed for other sellers

Also in real conditions some information such as costfunction and how to determine the bid is not generallyavailable to the buyer e sealed type of disclosure forsellerrsquos information to the buyer is taken into considerationby [7 15 17 20 23 25] However in some studies theinformation such as the sellerrsquos cost function [18] theparametric form of the sellerrsquos cost function [14] the level ofattributes for the previous winner of the bid the thresholdlevel for nonprice attributes and the relative importance ofprevious attributes [22] and the distribution function of bids[19] has been revealed to the buyer Buyerrsquos information thatcan be revealed in a reverse auction process for sellers isrelated to how the buyer prioritizes or scores sellerserefore in some research it is assumed that the sellershave complete information on how to prioritize or score thebuyer [15 18 20] while in others no information is pro-vided from the buyer to sellers [14 22 25]

Some studies have tried to develop a mechanism forproviding information to sellers that are able to improve theirbids For example in [23] buyers propose their attributes intwo groups as competitive information (private information)as well as generic ones to the seller In [23] the sellerrsquosproblem has not been investigated and since the algorithmfor determining the level of information disclosure is de-pendent on the sellerrsquos bids the interactive relationship be-tween the sellers and the buyer has been neglected In anotherstudy of [17] in each round sellers using information aboutother sellers bids and their scores predict the buyerrsquos utilityfunction (weights of the utility function) and by solving amathematical programming model with the objective ofmaximizing profit submit their new bids In this study thebuyerrsquos scoring function is considered as the Lα metric dis-tance function and the process of forecasting does not allowthe use of other scoring functions

In the present paper a multiattribute multiround onlinereverse auctionmodel is provided In the proposedmodel inaddition to simultaneously modeling the buyerrsquos and sellerrsquos

2 Mathematical Problems in Engineering

problems without disclosing the sellerrsquos information toother sellers a level of buyerrsquos information is revealed tosellers In this framework an online market maker whoorganizes an auction on an online platform estimates thebuyerrsquos scoring function in each round and gives it to sellerse estimation of the scoring function is made using theartificial neural network and it is possible to predict thisfunction independently of the type of buyerrsquos scoringfunction e estimation of the scoring function is based onthe information acquired from the previous buyerrsquos auctioninformation on the online platform (if any) as well as theinformation obtained from the previous round of auction InTable 2 the summary of research related to multiattributereverse auctions as well as the suggested model of thisresearch (last line) is presented

3 Problem definition

e structure of a reverse auction is remarkably differen-tiated from the forward auction In the forward auction thebuyer offers hisher bid to sellers who aim to sell theirproducts such as artworks and antiques to select the bestbid In contrast in a reverse auction sellers offer their bids tothe buyer that they should choose the best bidder to buy theirproduct e proposed framework of this research is basedon a single product multiround multiattribute reverseauction Participation in the auction process is free and theauction is managed by a marketer which is here called theOnline Auction Service Provider (OASP) In the proposedapproach OASP plays an essential role In each round of theauction process OASP provides an estimate of the buyerrsquospriority function to the seller in order to offer more effectivebids in the next round Figure 1 shows the information flowof the auction process

Moreover Figure 2 describes the process of each flowbetween the buyer the OASP and the sellers

A market maker as an intermediary has a history ofbuyersrsquo records in auction operations Hence if the buyerdoes not make a change in his scoring function the marketmaker can use a better estimation of the scoring function

using the buyerrsquos past auction data If a seller uses the auctionplatform for the first time determining the auction winnerswill be postponed for more rounds (for example the secondround) to obtain sufficient data to fit the scoring function Inthis case it is sometimes possible to resample the dataobtained from the initial round In the following we describethe mathematical formulation of the auction process

Suppose there are n sellers K attributes and one buyere seller i will give OASP a bid vector S

j

i (aj

i1 aj

ik) ineach round until announcing the end of the auction anddetermining the winner where a

j

ik is attribute k of seller i inround j In the first round each seller will provide an initialbid on the OASP platform based on hisher capabilities isbid is made available to the buyer by OASP In this for-mulation in order to calculate the sellerrsquos score by the buyerthree scenarios are considered e problems that the buyerOASP and seller face are discussed in Figure 2 as follows

31 Buyerrsquos Problem Buyer in each round after receivingbids S

ji (a

ji1 a

j

ik) calculates the score of each sellerHere three types of scoring functions are calculated by thebuyer the additive scoring function the multiplicativescoring function and the risk-aversion scoring function

311 Additive and Multiplicative Scoring Function In theadditive scoring function the scoring of sellers is based onthe weighted sum of the values for the attributes (equation(1)) and the multiplicative scoring function is based on theproduct of the attributes (equation (2))

Scoreji 1113944

K

k1wkaprimejik (1)

Scoreji 1113945

K

k11 + wka

primejik1113874 1113875 minus 1 (2)

where wk is the weight of the kth attribute

Table 1 Classification of auction types

No Feature Description1 Number of items of a good One or more goods2 Nature of goods Homogenousheterogeneous3 Attributes One or more attributes4 Type of auction Forwardreverse5 Nature of auction One-roundmultiround6 English vs Dutch auction Ascendingdescending price7 Participants Invitationopen8 Market maker Existdoes not exist9 e price paid by the winner First pricesecond priceetc10 Is price discrimination applied Yesno11 Do constraints exist Explicitlyimplicitly12 Is there a follow-up negotiation Yesno13 Is a value function elicited for the buyer Yesno14 Nature of bids Opensemisealedsealed15 Are bids divisible Yesno16 Are bundle bids allowed Yesno

Mathematical Problems in Engineering 3

Table 2 Researches related to the multiattribute reverse auction and proposed model

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

[15]Determining the

highest score by thescoring function

Equilibriumsolution for

bayesian gamesMultiround Two Open

Costdistributionfunction

Sealed

[14]

Predicting the costfunction of sellers by

using reverseoptimization andmaximizing utility

Utility function Multiround Two ormore Sealed Open

Estimating theparametric form of the

cost function

[18] Maximizing theefficiency

Maximizing theefficiency

Single-round

Two ormore Open Open Cost function

[23]

Data envelopmentanalysis model withgoal of maximizingutility function

mdash Two-round Two ormore

e function ofprioritizing infirst round andprivate attributesin second round

Sealed Sealed

[22]

e dataenvelopment

analysis model aimsto maximize theprobability of

determining thewinner in the next

round

mdash Multiround Two ormore Sealed Sealed

Providing the level ofattributes for previous

winning bidsminimum threshold

levels fornoncompetitive

attributes and therelative importance of

attributes by theonline platform

[19] Maximizing theexpected score

Maximizingexpected utility

Single-round Two

Scoringdistributionfunction

Sealed Bids distributionfunction

[17] Maximizing theutility function

Maximizing theprofit Multiround Two or

more

Estimation ofthe Lα metric

scoring functionbased on

previous scores

Sealed Sealed

[7]

Fuzzy minimizationof cost of purchase(price) and expected

delivery timedifference withproposed deliverytime and objectivefunction of the seller

Profitmaximizationconsidering thesatisfaction of theconstraints on thesellerrsquos production

program

Single-round

Two ormore

Masterproductionschedule

Sealed Sealed

[20] Minimizing thepurchase costs

Maximizing theprofit

Single-round

Two ormore Open Sealed Sealed

[16] Maximizing theutility function

Maximizing theutility function Multiround Two or

more Sealed Sealed Sealed

[26]

Maximizing thescore by BOCR-

uRTODIMframework

mdash Single-round

Two ormore Sealed Sealed Sealed

[21]Maximizing thepayoff based on

evolutionary gamemdash Multiround Two or

more Sealed Sealed Sealed

4 Mathematical Problems in Engineering

312 Risk-Aversion Scoring Function In this scenario thebehavior of a risk-averse buyer is defined based on theprospect theory [28] In this case the buyer specifies the ideal

value Sjlowasti (a

jlowasti1 a

jlowastik ) for each of the attributes First

each of the bids is converted into a risk-aversion bid by

Aj

ik a

j

ik minus ajlowastik1113872 1113873

α a

j

ik minus ajlowastik ge 0

minusθ ajlowastik minus a

j

ik1113872 1113873β a

j

ik minus ajlowastik lt 0

⎧⎪⎨

⎪⎩(3)

In equation (3) 0lt αlt 1 and 0lt βlt 1 is the profit andloss parameters respectively θ indicates the buyerrsquos riskbehavior θgt 1 represents risk-aversion 0lt θlt 1 is riskseeking and θ 1 is risk-neutral behavior of the buyerSimilar to equation (3) the initial score is obtained from

iScorej

i 1113944

K

k1wkA

j

ik (4)

Since it is probable to obtain a negative score the finalscore is calculated by

Scoreji

iScoreji minus min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967lt 0

iScoreji + min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967gt 0

⎧⎪⎨

⎪⎩(5)

313 OASP Problem After calculating the score of eachseller the obtained information is given to OASP OASPestimates the buyerrsquos scoring function using a multilayerperceptron neural network e structure of estimating thebuyerrsquos scoring function using the neural network is shownin Figure 3 In this structure the sellerrsquos bid information upto round j is the input of the model and the target is thescore of bids Training of the neural network OASP providesthe function netj(S

j

i ) to improve sellerrsquos bids

314 Sellerrsquos Problem e seller uses the estimation of thescoring function performed by OASP to determine its newbid e problem of determining a new bid for a seller in thejth round is designed as equations (6)ndash(9)

max πji a

j

i1 minus C Sji1113872 1113873 (6)

mindi netj Sji1113872 1113873 minus netj S

lowast ji1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 (7)

st netj Sji1113872 1113873genetj S

jminuski1113872 1113873 for k 1 j minus 1 (8)

aj

ik le aj

ik le aj

ik for k 1 K (9)

Assuming that aj

i1 is the proposed price in the jth roundthe objective function in equation (6) ie πj

i of seller ishould be maximized where C(S

ji ) is the cost function of the

bid Sji Additionally the objective function in equation (7)

the distance between the estimation of the current bid scoreand the ideal score from the buyerrsquos point of view S

jlowasti are

minimized Ideal bid Sjlowasti is the upper bound of attribute a

j

ikfor the positive attributes and the lower bound is for thenegative attributes Sellers use these values if the buyer offerstheir ideal values for each attribute otherwise each selleruses its estimation for the ideal buyerrsquos value Constraint (8)states that the estimated score for the current bid must behigher than the score in the previous round Constraint (9)indicates the bounds of the buyerrsquos ability to provide anattribute where a

j

ik is the lower bound and aj

ik is the upperbound of attribute a

j

ik

Table 2 Continued

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

ecurrentpaper

Determining thehighest score by thescoring function

Profitmaximization andminimizing thedistance to theideal buyerrsquos bid

Multiround Two ormore

Estimation ofscoring function

by onlineauction serviceprovider usingartificial neural

network

Sealed Sealed

Buyer

Seller 1 Seller 2 Seller nhellip

Flow 2

Flow 1

Flow 3

Flow 4

OASP

Figure 1 Information flow of the proposed auction model

Mathematical Problems in Engineering 5

4 Model Assessments

In order to evaluate the mathematical model the proposedauction process is simulated by taking into account severalsellers and a buyer For this purpose the MATLAB softwareversion 2016 has been used e main purpose of thesimulation is to evaluate the performance of the model indifferent situations Evaluation is carried out in two ways

(1) e disclosure of buyerrsquos scoring function for sellers(an open auction)

(2) e estimation of the scoring function by OASP (asemisealed auction)

For each of these two auctions three types of additivemultiplicative and risk-aversion scoring functions areconsidered In the open auction the problem of the seller isdefined as follows

max πji a

ji1 minus C S

ji1113872 1113873

min di Score Sj

i1113872 1113873 minus Score Slowast j

i1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

st Score Sji1113872 1113873ge Score S

jminuski1113872 1113873

aji1 minus C S

ji1113872 1113873gt 0

aj

ik le aj

ik le aj

ik

(10)

e following criteria are also considered as indicatorsfor evaluating the model

(1) e average score of the winner(2) e average of the winnerrsquos profit(3) Number of rounds in each auction process

e pseudocode of the simulation process is presented inTable 3

In Table 4 the related parameters of simulation processincluding the maximum number of iterations (I) thenumber of sellers (n) the improvement threshold (Δ) and

Flow 1

The score and rank of each seller becomeavailable for himself

The estimation of the scoring function foreach seller is available for all sellers

Flow 2

Flow 3

Flow 4

Round N = 1 N ge 1

The request for a bid is submitted by the seller inthe provided OASP platform It includes types ofrequested product price and expected featuresfrom the buyer viewpoint and auction stoppingcondition

After receiving the sellersrsquo bids through OASPthe buyer determines the score and rank of eachseller In the second round the buyer providesthe mentioned information to OASP In thenext rounds if the highest score does notimprove by a given amount the information isprovided to OASP Nevertheless announcingthe highest score as the winner the reverseauction process stops

OASP discloses the submitted bids in an open waySellers are requested to provide their bids inaccordance with buyers requirements in OASPplatform during a given period

Sellers submit their bids through OASP platformbased on the circumstances and provided requests

OASP discloses all information obtained fromsellers for the buyer including identifyinginformation and offered bids

Using the information obtained from OASPsellers determine the optimal bid and submit itthrough OASP

OASP discloses the information of buyerrsquosnew bid

Using the information about the sellerrsquos bid andscore evaluated by buyers OASP makes thefollowing information available for sellersbull

bull

Figure 2 Problem under investigation

ANN function estimator

S11 Sn1 S1

jndash1 Snjndash1 S1j Snj

Score11 Scoren1 Score1

jndash1 Scorenjndash1 Score1j Scorenj

Figure 3 Structure of the estimation of the buyerrsquos scoringfunction using the neural network

6 Mathematical Problems in Engineering

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

problems without disclosing the sellerrsquos information toother sellers a level of buyerrsquos information is revealed tosellers In this framework an online market maker whoorganizes an auction on an online platform estimates thebuyerrsquos scoring function in each round and gives it to sellerse estimation of the scoring function is made using theartificial neural network and it is possible to predict thisfunction independently of the type of buyerrsquos scoringfunction e estimation of the scoring function is based onthe information acquired from the previous buyerrsquos auctioninformation on the online platform (if any) as well as theinformation obtained from the previous round of auction InTable 2 the summary of research related to multiattributereverse auctions as well as the suggested model of thisresearch (last line) is presented

3 Problem definition

e structure of a reverse auction is remarkably differen-tiated from the forward auction In the forward auction thebuyer offers hisher bid to sellers who aim to sell theirproducts such as artworks and antiques to select the bestbid In contrast in a reverse auction sellers offer their bids tothe buyer that they should choose the best bidder to buy theirproduct e proposed framework of this research is basedon a single product multiround multiattribute reverseauction Participation in the auction process is free and theauction is managed by a marketer which is here called theOnline Auction Service Provider (OASP) In the proposedapproach OASP plays an essential role In each round of theauction process OASP provides an estimate of the buyerrsquospriority function to the seller in order to offer more effectivebids in the next round Figure 1 shows the information flowof the auction process

Moreover Figure 2 describes the process of each flowbetween the buyer the OASP and the sellers

A market maker as an intermediary has a history ofbuyersrsquo records in auction operations Hence if the buyerdoes not make a change in his scoring function the marketmaker can use a better estimation of the scoring function

using the buyerrsquos past auction data If a seller uses the auctionplatform for the first time determining the auction winnerswill be postponed for more rounds (for example the secondround) to obtain sufficient data to fit the scoring function Inthis case it is sometimes possible to resample the dataobtained from the initial round In the following we describethe mathematical formulation of the auction process

Suppose there are n sellers K attributes and one buyere seller i will give OASP a bid vector S

j

i (aj

i1 aj

ik) ineach round until announcing the end of the auction anddetermining the winner where a

j

ik is attribute k of seller i inround j In the first round each seller will provide an initialbid on the OASP platform based on hisher capabilities isbid is made available to the buyer by OASP In this for-mulation in order to calculate the sellerrsquos score by the buyerthree scenarios are considered e problems that the buyerOASP and seller face are discussed in Figure 2 as follows

31 Buyerrsquos Problem Buyer in each round after receivingbids S

ji (a

ji1 a

j

ik) calculates the score of each sellerHere three types of scoring functions are calculated by thebuyer the additive scoring function the multiplicativescoring function and the risk-aversion scoring function

311 Additive and Multiplicative Scoring Function In theadditive scoring function the scoring of sellers is based onthe weighted sum of the values for the attributes (equation(1)) and the multiplicative scoring function is based on theproduct of the attributes (equation (2))

Scoreji 1113944

K

k1wkaprimejik (1)

Scoreji 1113945

K

k11 + wka

primejik1113874 1113875 minus 1 (2)

where wk is the weight of the kth attribute

Table 1 Classification of auction types

No Feature Description1 Number of items of a good One or more goods2 Nature of goods Homogenousheterogeneous3 Attributes One or more attributes4 Type of auction Forwardreverse5 Nature of auction One-roundmultiround6 English vs Dutch auction Ascendingdescending price7 Participants Invitationopen8 Market maker Existdoes not exist9 e price paid by the winner First pricesecond priceetc10 Is price discrimination applied Yesno11 Do constraints exist Explicitlyimplicitly12 Is there a follow-up negotiation Yesno13 Is a value function elicited for the buyer Yesno14 Nature of bids Opensemisealedsealed15 Are bids divisible Yesno16 Are bundle bids allowed Yesno

Mathematical Problems in Engineering 3

Table 2 Researches related to the multiattribute reverse auction and proposed model

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

[15]Determining the

highest score by thescoring function

Equilibriumsolution for

bayesian gamesMultiround Two Open

Costdistributionfunction

Sealed

[14]

Predicting the costfunction of sellers by

using reverseoptimization andmaximizing utility

Utility function Multiround Two ormore Sealed Open

Estimating theparametric form of the

cost function

[18] Maximizing theefficiency

Maximizing theefficiency

Single-round

Two ormore Open Open Cost function

[23]

Data envelopmentanalysis model withgoal of maximizingutility function

mdash Two-round Two ormore

e function ofprioritizing infirst round andprivate attributesin second round

Sealed Sealed

[22]

e dataenvelopment

analysis model aimsto maximize theprobability of

determining thewinner in the next

round

mdash Multiround Two ormore Sealed Sealed

Providing the level ofattributes for previous

winning bidsminimum threshold

levels fornoncompetitive

attributes and therelative importance of

attributes by theonline platform

[19] Maximizing theexpected score

Maximizingexpected utility

Single-round Two

Scoringdistributionfunction

Sealed Bids distributionfunction

[17] Maximizing theutility function

Maximizing theprofit Multiround Two or

more

Estimation ofthe Lα metric

scoring functionbased on

previous scores

Sealed Sealed

[7]

Fuzzy minimizationof cost of purchase(price) and expected

delivery timedifference withproposed deliverytime and objectivefunction of the seller

Profitmaximizationconsidering thesatisfaction of theconstraints on thesellerrsquos production

program

Single-round

Two ormore

Masterproductionschedule

Sealed Sealed

[20] Minimizing thepurchase costs

Maximizing theprofit

Single-round

Two ormore Open Sealed Sealed

[16] Maximizing theutility function

Maximizing theutility function Multiround Two or

more Sealed Sealed Sealed

[26]

Maximizing thescore by BOCR-

uRTODIMframework

mdash Single-round

Two ormore Sealed Sealed Sealed

[21]Maximizing thepayoff based on

evolutionary gamemdash Multiround Two or

more Sealed Sealed Sealed

4 Mathematical Problems in Engineering

312 Risk-Aversion Scoring Function In this scenario thebehavior of a risk-averse buyer is defined based on theprospect theory [28] In this case the buyer specifies the ideal

value Sjlowasti (a

jlowasti1 a

jlowastik ) for each of the attributes First

each of the bids is converted into a risk-aversion bid by

Aj

ik a

j

ik minus ajlowastik1113872 1113873

α a

j

ik minus ajlowastik ge 0

minusθ ajlowastik minus a

j

ik1113872 1113873β a

j

ik minus ajlowastik lt 0

⎧⎪⎨

⎪⎩(3)

In equation (3) 0lt αlt 1 and 0lt βlt 1 is the profit andloss parameters respectively θ indicates the buyerrsquos riskbehavior θgt 1 represents risk-aversion 0lt θlt 1 is riskseeking and θ 1 is risk-neutral behavior of the buyerSimilar to equation (3) the initial score is obtained from

iScorej

i 1113944

K

k1wkA

j

ik (4)

Since it is probable to obtain a negative score the finalscore is calculated by

Scoreji

iScoreji minus min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967lt 0

iScoreji + min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967gt 0

⎧⎪⎨

⎪⎩(5)

313 OASP Problem After calculating the score of eachseller the obtained information is given to OASP OASPestimates the buyerrsquos scoring function using a multilayerperceptron neural network e structure of estimating thebuyerrsquos scoring function using the neural network is shownin Figure 3 In this structure the sellerrsquos bid information upto round j is the input of the model and the target is thescore of bids Training of the neural network OASP providesthe function netj(S

j

i ) to improve sellerrsquos bids

314 Sellerrsquos Problem e seller uses the estimation of thescoring function performed by OASP to determine its newbid e problem of determining a new bid for a seller in thejth round is designed as equations (6)ndash(9)

max πji a

j

i1 minus C Sji1113872 1113873 (6)

mindi netj Sji1113872 1113873 minus netj S

lowast ji1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 (7)

st netj Sji1113872 1113873genetj S

jminuski1113872 1113873 for k 1 j minus 1 (8)

aj

ik le aj

ik le aj

ik for k 1 K (9)

Assuming that aj

i1 is the proposed price in the jth roundthe objective function in equation (6) ie πj

i of seller ishould be maximized where C(S

ji ) is the cost function of the

bid Sji Additionally the objective function in equation (7)

the distance between the estimation of the current bid scoreand the ideal score from the buyerrsquos point of view S

jlowasti are

minimized Ideal bid Sjlowasti is the upper bound of attribute a

j

ikfor the positive attributes and the lower bound is for thenegative attributes Sellers use these values if the buyer offerstheir ideal values for each attribute otherwise each selleruses its estimation for the ideal buyerrsquos value Constraint (8)states that the estimated score for the current bid must behigher than the score in the previous round Constraint (9)indicates the bounds of the buyerrsquos ability to provide anattribute where a

j

ik is the lower bound and aj

ik is the upperbound of attribute a

j

ik

Table 2 Continued

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

ecurrentpaper

Determining thehighest score by thescoring function

Profitmaximization andminimizing thedistance to theideal buyerrsquos bid

Multiround Two ormore

Estimation ofscoring function

by onlineauction serviceprovider usingartificial neural

network

Sealed Sealed

Buyer

Seller 1 Seller 2 Seller nhellip

Flow 2

Flow 1

Flow 3

Flow 4

OASP

Figure 1 Information flow of the proposed auction model

Mathematical Problems in Engineering 5

4 Model Assessments

In order to evaluate the mathematical model the proposedauction process is simulated by taking into account severalsellers and a buyer For this purpose the MATLAB softwareversion 2016 has been used e main purpose of thesimulation is to evaluate the performance of the model indifferent situations Evaluation is carried out in two ways

(1) e disclosure of buyerrsquos scoring function for sellers(an open auction)

(2) e estimation of the scoring function by OASP (asemisealed auction)

For each of these two auctions three types of additivemultiplicative and risk-aversion scoring functions areconsidered In the open auction the problem of the seller isdefined as follows

max πji a

ji1 minus C S

ji1113872 1113873

min di Score Sj

i1113872 1113873 minus Score Slowast j

i1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

st Score Sji1113872 1113873ge Score S

jminuski1113872 1113873

aji1 minus C S

ji1113872 1113873gt 0

aj

ik le aj

ik le aj

ik

(10)

e following criteria are also considered as indicatorsfor evaluating the model

(1) e average score of the winner(2) e average of the winnerrsquos profit(3) Number of rounds in each auction process

e pseudocode of the simulation process is presented inTable 3

In Table 4 the related parameters of simulation processincluding the maximum number of iterations (I) thenumber of sellers (n) the improvement threshold (Δ) and

Flow 1

The score and rank of each seller becomeavailable for himself

The estimation of the scoring function foreach seller is available for all sellers

Flow 2

Flow 3

Flow 4

Round N = 1 N ge 1

The request for a bid is submitted by the seller inthe provided OASP platform It includes types ofrequested product price and expected featuresfrom the buyer viewpoint and auction stoppingcondition

After receiving the sellersrsquo bids through OASPthe buyer determines the score and rank of eachseller In the second round the buyer providesthe mentioned information to OASP In thenext rounds if the highest score does notimprove by a given amount the information isprovided to OASP Nevertheless announcingthe highest score as the winner the reverseauction process stops

OASP discloses the submitted bids in an open waySellers are requested to provide their bids inaccordance with buyers requirements in OASPplatform during a given period

Sellers submit their bids through OASP platformbased on the circumstances and provided requests

OASP discloses all information obtained fromsellers for the buyer including identifyinginformation and offered bids

Using the information obtained from OASPsellers determine the optimal bid and submit itthrough OASP

OASP discloses the information of buyerrsquosnew bid

Using the information about the sellerrsquos bid andscore evaluated by buyers OASP makes thefollowing information available for sellersbull

bull

Figure 2 Problem under investigation

ANN function estimator

S11 Sn1 S1

jndash1 Snjndash1 S1j Snj

Score11 Scoren1 Score1

jndash1 Scorenjndash1 Score1j Scorenj

Figure 3 Structure of the estimation of the buyerrsquos scoringfunction using the neural network

6 Mathematical Problems in Engineering

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

Table 2 Researches related to the multiattribute reverse auction and proposed model

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

[15]Determining the

highest score by thescoring function

Equilibriumsolution for

bayesian gamesMultiround Two Open

Costdistributionfunction

Sealed

[14]

Predicting the costfunction of sellers by

using reverseoptimization andmaximizing utility

Utility function Multiround Two ormore Sealed Open

Estimating theparametric form of the

cost function

[18] Maximizing theefficiency

Maximizing theefficiency

Single-round

Two ormore Open Open Cost function

[23]

Data envelopmentanalysis model withgoal of maximizingutility function

mdash Two-round Two ormore

e function ofprioritizing infirst round andprivate attributesin second round

Sealed Sealed

[22]

e dataenvelopment

analysis model aimsto maximize theprobability of

determining thewinner in the next

round

mdash Multiround Two ormore Sealed Sealed

Providing the level ofattributes for previous

winning bidsminimum threshold

levels fornoncompetitive

attributes and therelative importance of

attributes by theonline platform

[19] Maximizing theexpected score

Maximizingexpected utility

Single-round Two

Scoringdistributionfunction

Sealed Bids distributionfunction

[17] Maximizing theutility function

Maximizing theprofit Multiround Two or

more

Estimation ofthe Lα metric

scoring functionbased on

previous scores

Sealed Sealed

[7]

Fuzzy minimizationof cost of purchase(price) and expected

delivery timedifference withproposed deliverytime and objectivefunction of the seller

Profitmaximizationconsidering thesatisfaction of theconstraints on thesellerrsquos production

program

Single-round

Two ormore

Masterproductionschedule

Sealed Sealed

[20] Minimizing thepurchase costs

Maximizing theprofit

Single-round

Two ormore Open Sealed Sealed

[16] Maximizing theutility function

Maximizing theutility function Multiround Two or

more Sealed Sealed Sealed

[26]

Maximizing thescore by BOCR-

uRTODIMframework

mdash Single-round

Two ormore Sealed Sealed Sealed

[21]Maximizing thepayoff based on

evolutionary gamemdash Multiround Two or

more Sealed Sealed Sealed

4 Mathematical Problems in Engineering

312 Risk-Aversion Scoring Function In this scenario thebehavior of a risk-averse buyer is defined based on theprospect theory [28] In this case the buyer specifies the ideal

value Sjlowasti (a

jlowasti1 a

jlowastik ) for each of the attributes First

each of the bids is converted into a risk-aversion bid by

Aj

ik a

j

ik minus ajlowastik1113872 1113873

α a

j

ik minus ajlowastik ge 0

minusθ ajlowastik minus a

j

ik1113872 1113873β a

j

ik minus ajlowastik lt 0

⎧⎪⎨

⎪⎩(3)

In equation (3) 0lt αlt 1 and 0lt βlt 1 is the profit andloss parameters respectively θ indicates the buyerrsquos riskbehavior θgt 1 represents risk-aversion 0lt θlt 1 is riskseeking and θ 1 is risk-neutral behavior of the buyerSimilar to equation (3) the initial score is obtained from

iScorej

i 1113944

K

k1wkA

j

ik (4)

Since it is probable to obtain a negative score the finalscore is calculated by

Scoreji

iScoreji minus min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967lt 0

iScoreji + min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967gt 0

⎧⎪⎨

⎪⎩(5)

313 OASP Problem After calculating the score of eachseller the obtained information is given to OASP OASPestimates the buyerrsquos scoring function using a multilayerperceptron neural network e structure of estimating thebuyerrsquos scoring function using the neural network is shownin Figure 3 In this structure the sellerrsquos bid information upto round j is the input of the model and the target is thescore of bids Training of the neural network OASP providesthe function netj(S

j

i ) to improve sellerrsquos bids

314 Sellerrsquos Problem e seller uses the estimation of thescoring function performed by OASP to determine its newbid e problem of determining a new bid for a seller in thejth round is designed as equations (6)ndash(9)

max πji a

j

i1 minus C Sji1113872 1113873 (6)

mindi netj Sji1113872 1113873 minus netj S

lowast ji1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 (7)

st netj Sji1113872 1113873genetj S

jminuski1113872 1113873 for k 1 j minus 1 (8)

aj

ik le aj

ik le aj

ik for k 1 K (9)

Assuming that aj

i1 is the proposed price in the jth roundthe objective function in equation (6) ie πj

i of seller ishould be maximized where C(S

ji ) is the cost function of the

bid Sji Additionally the objective function in equation (7)

the distance between the estimation of the current bid scoreand the ideal score from the buyerrsquos point of view S

jlowasti are

minimized Ideal bid Sjlowasti is the upper bound of attribute a

j

ikfor the positive attributes and the lower bound is for thenegative attributes Sellers use these values if the buyer offerstheir ideal values for each attribute otherwise each selleruses its estimation for the ideal buyerrsquos value Constraint (8)states that the estimated score for the current bid must behigher than the score in the previous round Constraint (9)indicates the bounds of the buyerrsquos ability to provide anattribute where a

j

ik is the lower bound and aj

ik is the upperbound of attribute a

j

ik

Table 2 Continued

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

ecurrentpaper

Determining thehighest score by thescoring function

Profitmaximization andminimizing thedistance to theideal buyerrsquos bid

Multiround Two ormore

Estimation ofscoring function

by onlineauction serviceprovider usingartificial neural

network

Sealed Sealed

Buyer

Seller 1 Seller 2 Seller nhellip

Flow 2

Flow 1

Flow 3

Flow 4

OASP

Figure 1 Information flow of the proposed auction model

Mathematical Problems in Engineering 5

4 Model Assessments

In order to evaluate the mathematical model the proposedauction process is simulated by taking into account severalsellers and a buyer For this purpose the MATLAB softwareversion 2016 has been used e main purpose of thesimulation is to evaluate the performance of the model indifferent situations Evaluation is carried out in two ways

(1) e disclosure of buyerrsquos scoring function for sellers(an open auction)

(2) e estimation of the scoring function by OASP (asemisealed auction)

For each of these two auctions three types of additivemultiplicative and risk-aversion scoring functions areconsidered In the open auction the problem of the seller isdefined as follows

max πji a

ji1 minus C S

ji1113872 1113873

min di Score Sj

i1113872 1113873 minus Score Slowast j

i1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

st Score Sji1113872 1113873ge Score S

jminuski1113872 1113873

aji1 minus C S

ji1113872 1113873gt 0

aj

ik le aj

ik le aj

ik

(10)

e following criteria are also considered as indicatorsfor evaluating the model

(1) e average score of the winner(2) e average of the winnerrsquos profit(3) Number of rounds in each auction process

e pseudocode of the simulation process is presented inTable 3

In Table 4 the related parameters of simulation processincluding the maximum number of iterations (I) thenumber of sellers (n) the improvement threshold (Δ) and

Flow 1

The score and rank of each seller becomeavailable for himself

The estimation of the scoring function foreach seller is available for all sellers

Flow 2

Flow 3

Flow 4

Round N = 1 N ge 1

The request for a bid is submitted by the seller inthe provided OASP platform It includes types ofrequested product price and expected featuresfrom the buyer viewpoint and auction stoppingcondition

After receiving the sellersrsquo bids through OASPthe buyer determines the score and rank of eachseller In the second round the buyer providesthe mentioned information to OASP In thenext rounds if the highest score does notimprove by a given amount the information isprovided to OASP Nevertheless announcingthe highest score as the winner the reverseauction process stops

OASP discloses the submitted bids in an open waySellers are requested to provide their bids inaccordance with buyers requirements in OASPplatform during a given period

Sellers submit their bids through OASP platformbased on the circumstances and provided requests

OASP discloses all information obtained fromsellers for the buyer including identifyinginformation and offered bids

Using the information obtained from OASPsellers determine the optimal bid and submit itthrough OASP

OASP discloses the information of buyerrsquosnew bid

Using the information about the sellerrsquos bid andscore evaluated by buyers OASP makes thefollowing information available for sellersbull

bull

Figure 2 Problem under investigation

ANN function estimator

S11 Sn1 S1

jndash1 Snjndash1 S1j Snj

Score11 Scoren1 Score1

jndash1 Scorenjndash1 Score1j Scorenj

Figure 3 Structure of the estimation of the buyerrsquos scoringfunction using the neural network

6 Mathematical Problems in Engineering

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

312 Risk-Aversion Scoring Function In this scenario thebehavior of a risk-averse buyer is defined based on theprospect theory [28] In this case the buyer specifies the ideal

value Sjlowasti (a

jlowasti1 a

jlowastik ) for each of the attributes First

each of the bids is converted into a risk-aversion bid by

Aj

ik a

j

ik minus ajlowastik1113872 1113873

α a

j

ik minus ajlowastik ge 0

minusθ ajlowastik minus a

j

ik1113872 1113873β a

j

ik minus ajlowastik lt 0

⎧⎪⎨

⎪⎩(3)

In equation (3) 0lt αlt 1 and 0lt βlt 1 is the profit andloss parameters respectively θ indicates the buyerrsquos riskbehavior θgt 1 represents risk-aversion 0lt θlt 1 is riskseeking and θ 1 is risk-neutral behavior of the buyerSimilar to equation (3) the initial score is obtained from

iScorej

i 1113944

K

k1wkA

j

ik (4)

Since it is probable to obtain a negative score the finalscore is calculated by

Scoreji

iScoreji minus min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967lt 0

iScoreji + min iScorej

i i 1 n1113966 1113967 + 1 min iScoreji i 1 n1113966 1113967gt 0

⎧⎪⎨

⎪⎩(5)

313 OASP Problem After calculating the score of eachseller the obtained information is given to OASP OASPestimates the buyerrsquos scoring function using a multilayerperceptron neural network e structure of estimating thebuyerrsquos scoring function using the neural network is shownin Figure 3 In this structure the sellerrsquos bid information upto round j is the input of the model and the target is thescore of bids Training of the neural network OASP providesthe function netj(S

j

i ) to improve sellerrsquos bids

314 Sellerrsquos Problem e seller uses the estimation of thescoring function performed by OASP to determine its newbid e problem of determining a new bid for a seller in thejth round is designed as equations (6)ndash(9)

max πji a

j

i1 minus C Sji1113872 1113873 (6)

mindi netj Sji1113872 1113873 minus netj S

lowast ji1113872 1113873

11138681113868111386811138681113868

11138681113868111386811138681113868 (7)

st netj Sji1113872 1113873genetj S

jminuski1113872 1113873 for k 1 j minus 1 (8)

aj

ik le aj

ik le aj

ik for k 1 K (9)

Assuming that aj

i1 is the proposed price in the jth roundthe objective function in equation (6) ie πj

i of seller ishould be maximized where C(S

ji ) is the cost function of the

bid Sji Additionally the objective function in equation (7)

the distance between the estimation of the current bid scoreand the ideal score from the buyerrsquos point of view S

jlowasti are

minimized Ideal bid Sjlowasti is the upper bound of attribute a

j

ikfor the positive attributes and the lower bound is for thenegative attributes Sellers use these values if the buyer offerstheir ideal values for each attribute otherwise each selleruses its estimation for the ideal buyerrsquos value Constraint (8)states that the estimated score for the current bid must behigher than the score in the previous round Constraint (9)indicates the bounds of the buyerrsquos ability to provide anattribute where a

j

ik is the lower bound and aj

ik is the upperbound of attribute a

j

ik

Table 2 Continued

Reference Buyer problem Seller problem e numberof rounds

enumber

ofattributes

Disclosure ofbuyerrsquos

information

Disclosure of sellerrsquos information

For othersellers Buyer

ecurrentpaper

Determining thehighest score by thescoring function

Profitmaximization andminimizing thedistance to theideal buyerrsquos bid

Multiround Two ormore

Estimation ofscoring function

by onlineauction serviceprovider usingartificial neural

network

Sealed Sealed

Buyer

Seller 1 Seller 2 Seller nhellip

Flow 2

Flow 1

Flow 3

Flow 4

OASP

Figure 1 Information flow of the proposed auction model

Mathematical Problems in Engineering 5

4 Model Assessments

In order to evaluate the mathematical model the proposedauction process is simulated by taking into account severalsellers and a buyer For this purpose the MATLAB softwareversion 2016 has been used e main purpose of thesimulation is to evaluate the performance of the model indifferent situations Evaluation is carried out in two ways

(1) e disclosure of buyerrsquos scoring function for sellers(an open auction)

(2) e estimation of the scoring function by OASP (asemisealed auction)

For each of these two auctions three types of additivemultiplicative and risk-aversion scoring functions areconsidered In the open auction the problem of the seller isdefined as follows

max πji a

ji1 minus C S

ji1113872 1113873

min di Score Sj

i1113872 1113873 minus Score Slowast j

i1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

st Score Sji1113872 1113873ge Score S

jminuski1113872 1113873

aji1 minus C S

ji1113872 1113873gt 0

aj

ik le aj

ik le aj

ik

(10)

e following criteria are also considered as indicatorsfor evaluating the model

(1) e average score of the winner(2) e average of the winnerrsquos profit(3) Number of rounds in each auction process

e pseudocode of the simulation process is presented inTable 3

In Table 4 the related parameters of simulation processincluding the maximum number of iterations (I) thenumber of sellers (n) the improvement threshold (Δ) and

Flow 1

The score and rank of each seller becomeavailable for himself

The estimation of the scoring function foreach seller is available for all sellers

Flow 2

Flow 3

Flow 4

Round N = 1 N ge 1

The request for a bid is submitted by the seller inthe provided OASP platform It includes types ofrequested product price and expected featuresfrom the buyer viewpoint and auction stoppingcondition

After receiving the sellersrsquo bids through OASPthe buyer determines the score and rank of eachseller In the second round the buyer providesthe mentioned information to OASP In thenext rounds if the highest score does notimprove by a given amount the information isprovided to OASP Nevertheless announcingthe highest score as the winner the reverseauction process stops

OASP discloses the submitted bids in an open waySellers are requested to provide their bids inaccordance with buyers requirements in OASPplatform during a given period

Sellers submit their bids through OASP platformbased on the circumstances and provided requests

OASP discloses all information obtained fromsellers for the buyer including identifyinginformation and offered bids

Using the information obtained from OASPsellers determine the optimal bid and submit itthrough OASP

OASP discloses the information of buyerrsquosnew bid

Using the information about the sellerrsquos bid andscore evaluated by buyers OASP makes thefollowing information available for sellersbull

bull

Figure 2 Problem under investigation

ANN function estimator

S11 Sn1 S1

jndash1 Snjndash1 S1j Snj

Score11 Scoren1 Score1

jndash1 Scorenjndash1 Score1j Scorenj

Figure 3 Structure of the estimation of the buyerrsquos scoringfunction using the neural network

6 Mathematical Problems in Engineering

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

4 Model Assessments

In order to evaluate the mathematical model the proposedauction process is simulated by taking into account severalsellers and a buyer For this purpose the MATLAB softwareversion 2016 has been used e main purpose of thesimulation is to evaluate the performance of the model indifferent situations Evaluation is carried out in two ways

(1) e disclosure of buyerrsquos scoring function for sellers(an open auction)

(2) e estimation of the scoring function by OASP (asemisealed auction)

For each of these two auctions three types of additivemultiplicative and risk-aversion scoring functions areconsidered In the open auction the problem of the seller isdefined as follows

max πji a

ji1 minus C S

ji1113872 1113873

min di Score Sj

i1113872 1113873 minus Score Slowast j

i1113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

st Score Sji1113872 1113873ge Score S

jminuski1113872 1113873

aji1 minus C S

ji1113872 1113873gt 0

aj

ik le aj

ik le aj

ik

(10)

e following criteria are also considered as indicatorsfor evaluating the model

(1) e average score of the winner(2) e average of the winnerrsquos profit(3) Number of rounds in each auction process

e pseudocode of the simulation process is presented inTable 3

In Table 4 the related parameters of simulation processincluding the maximum number of iterations (I) thenumber of sellers (n) the improvement threshold (Δ) and

Flow 1

The score and rank of each seller becomeavailable for himself

The estimation of the scoring function foreach seller is available for all sellers

Flow 2

Flow 3

Flow 4

Round N = 1 N ge 1

The request for a bid is submitted by the seller inthe provided OASP platform It includes types ofrequested product price and expected featuresfrom the buyer viewpoint and auction stoppingcondition

After receiving the sellersrsquo bids through OASPthe buyer determines the score and rank of eachseller In the second round the buyer providesthe mentioned information to OASP In thenext rounds if the highest score does notimprove by a given amount the information isprovided to OASP Nevertheless announcingthe highest score as the winner the reverseauction process stops

OASP discloses the submitted bids in an open waySellers are requested to provide their bids inaccordance with buyers requirements in OASPplatform during a given period

Sellers submit their bids through OASP platformbased on the circumstances and provided requests

OASP discloses all information obtained fromsellers for the buyer including identifyinginformation and offered bids

Using the information obtained from OASPsellers determine the optimal bid and submit itthrough OASP

OASP discloses the information of buyerrsquosnew bid

Using the information about the sellerrsquos bid andscore evaluated by buyers OASP makes thefollowing information available for sellersbull

bull

Figure 2 Problem under investigation

ANN function estimator

S11 Sn1 S1

jndash1 Snjndash1 S1j Snj

Score11 Scoren1 Score1

jndash1 Scorenjndash1 Score1j Scorenj

Figure 3 Structure of the estimation of the buyerrsquos scoringfunction using the neural network

6 Mathematical Problems in Engineering

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

the parameters of the buyerrsquos risk-aversion scoring (θ α β)

are presentede cost function of equation (6) is consideredas the weight combination for the attributes that isC(S

ji ) 1113936

Kk1 uka

primejik e unit cost uk for calculating the

sellerrsquos cost function weights wk the buyerrsquos scoringfunctions for each of the attributes and the ideal bid S

jlowasti are

provided in the Appendix (please see Tables 5 and 6) In eachiteration the values of the parameters in Table 4 as well as thespecified parameters (uk wk and S

lowast j

i ) are constant and onlythe initial selection vector for each bid would be changed

Also these constant values are considered for thecomparison of the two different types of disclosure for thebuyerrsquos function (open and semisealed) It is noteworthy thatthe three types of buyerrsquos scoring function (additive mul-tiplicative and risk-aversion) are assumed the same e useof nonlinear activation function in the multilayer Perceptronneural network generally has higher performance [29]hence the activation function tanh(x) is used to map theinput and the target variable erefore the function netj()

is nonlinear and consequently problems (6)ndash(9) are anonlinear two-objective model due to the objective function(7) and constraint (8) Here the NSGAII multiobjectivegenetic algorithm is used to solve the sellerrsquos problem Sincethe solutions of the model are obtained as the Pareto optimalfrontier the profit (first objective function) is prioritized toselect the optimal solution from the Pareto frontier ehidden layer size for the neural network (the number ofhidden layer neurons) is 10 and the neural network learning

function is LevenbergndashMarquardt For the NSGAII algo-rithm the population size is 200 the crossover ratio is 08and the Pareto ratio is 035

41 Simulation Results In this section the results of thesimulation process are presented Tables 7ndash9 show onesimulation iteration for an open auction using the additivemultiplicative and risk-aversion functions respectively

In the first round the vector of the bid is the same for allthe situations In each table the seller who has earned thehighest score is marked in bold For example in Table 7which belongs to the open auction results with an additivefunction in the first round the sixth seller obtained thehighest score en in the second round sellers have solvedtheir problems and eventually the fifth seller has earned thehighest score As can be seen the scores for all sellers havebeen upgraded Since Δ 0066 is larger than the thresholdΔT 005 the auction goes to the next round In the thirdround the improvement of score Δ 00046 is lower thanthe threshold and so the auction process is stopped ewinner of the auction is the fifth seller

As demonstrated in Table 7 the profit of the fifth seller inthe second round has risen alongside the first rounderefore for this seller there was a possibility to

Table 3 Pseudocode of the simulation process

Step Implementation1 Initialization11 Number of iterations I12 Number of sellers n13 Improvement thresholdΔT

14 S 12 Do the steps 3ndash12 until Slt I3 Round(j) 14 Generate n random bids S

ji (a

ji1 a

j

ik)

5 Compute the Scoreji for each bid

6 Determine the highest Scoreji and save it

7 Train a neural network for input (Sj1 S

jn S

jminus11 S

jminus1n S11 S1n) and target

(Scorej1 Scorej

n Scorejminus11 Scorejminus1

n Score11 Score1n)8 j + 1 j

9 Solve seller problem and obtain n new bids Sj+1i (a

j+1i1 a

j+1ik )

10 Compute the Scorej+1i for each new bid

11 Determine the highest Scorej+1i and save it

12 If Δ ((Scorej+1i minus Scorej

i )Scoreji )geΔT go to 8 otherwise go to 13

13 End

Table 4 Simulation parameters

Description Parameter ValueMaximum number of iterations I 50Number of sellers n 6Improvement threshold Δt 005

Risk-aversion cost functionθ 2α 05β 05

Table 5 Attributersquos unit cost and ideal bid level for each sellers

Parameter Seller 1 Seller 2 Seller 3 Seller 4 Seller 5 Seller 6u2 047 042 027 025 014 029u3 054 036 068 048 059 018u4 032 027 023 034 020 053Slowast [0 10]

Table 6 Buyer-scoring function weights

w1 w2 w3 w4

025 020 025 030

Mathematical Problems in Engineering 7

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

simultaneously improve the offer and profit in the secondround In Table 8 which belongs to the open auction with amultiplicative function the sixth seller is identified as thewinner of the auction in the first round Sellers 5 and 6 havenot been able to improve the bids for the foreseeable futurebut other sellers have upgraded their bids Table 9 shows theopen auction results with a risk-aversion function Similar tothe two previous scoring types in the first round the bestsellerrsquos bid is the sixth In the second round the fifth selleroffered the best bid Given that Δ 0486 the sellers havepresented their new bids In the third round none of thesellers have been able to improve their bids and Δ 0erefore the winner seller is the fifth seller

Before running a semisealed auction simulation as-suming that the seller has a history of using the auctionplatform for a specified product with given attributes amultilayer perceptron neural network is trained with 100

random bids for each scoring function (100 bids with theirscores) In each round new information (new bids and newscores) is added to the previous dataset and the network isthen retrained Tables 10ndash12 show the results for the sem-isealed auction In Table 10 the results of the semisealedauction are presented with the additive scoring function

Herein similar to the open auction the sixth seller winsthe first round In the second round the highest scorebelongs to the sixth seller but in the third round the fifthseller obtains the highest score and as the improvement inthe next round is less than the threshold the fifth seller isannounced as the winner As can be seen by comparing twoauctions with additive scoring functions the final score ofthe winning seller in the semisealed auction is higher andthe profit of winning seller is lower than that of the openauction Table 11 summarizes the results of a semisealedauction with a multiplicative scoring function Similar to the

Table 7 Results of the first iteration of simulation for open auction and additive scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2850 2449 9934 0946 2084 7162 2875 6024 9976 0081 1259 8464 2876 64972 8 2 1 6 2950 5117 9973 1367 4646 5028 2950 6296 9952 0166 4671 6045 3027 64883 8 2 6 3 3300 2614 9985 7155 0145 7389 3688 6186 9991 5814 0100 8331 3689 63684 5 2 2 2 2750 2829 9985 2790 0610 6790 2751 6639 9999 6250 0319 4735 2750 66295 8 5 2 5 3500 5067 10000 7212 1597 6480 3786 6684 9944 4759 1519 8193 3803 66636 7 2 6 3 3550 3708 9998 7436 6811 1195 3550 5939 9986 7095 8222 0238 3550 6281Δ mdash Δ 0066 Δ 00046

Table 8 Results of the first iteration of simulation for open auction and multiplicative scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 700 2449 5889 0005 0014 9816 7035 26822 8 2 1 6 635 5117 6164 0062 0028 9173 6492 35683 8 2 6 3 898 2614 4011 0011 0003 9980 9004 16534 5 2 2 2 656 2829 6364 0007 0001 9865 6570 29575 8 5 2 5 1025 5067 8000 5000 2000 5000 1025 50676 7 2 6 3 1064 3708 7000 2000 6000 3000 1064 3708

Δ mdash Δ 0

Table 9 Results of the first iteration of simulation for open auction and risk-aversion scoring function

First roundSellerrsquosscore

Sellerrsquosprofit

Second roundSellerrsquosscore

Sellerrsquosprofit

ird roundSellerrsquosscore

SellerrsquosprofitSeller

Attributes Attributes Attributes1 2 3 4 1 2 3 4 1 2 3 4

1 6 5 1 2 2507 2449 9984 0095 0044 8027 3007 7301 9984 0095 0044 8027 3007 73012 8 2 1 6 2341 5117 9994 0944 0559 7941 3841 7176 9994 0944 0559 7941 3841 71763 8 2 6 3 2464 2614 9976 8351 0007 3727 3964 6809 9976 8351 0007 3727 3964 68094 5 2 2 2 2568 2829 9993 8709 0040 4101 2568 6357 9963 9434 0216 3546 2568 62495 8 5 2 5 2536 5067 10000 8040 0001 4741 4036 7874 10000 8040 0001 4741 4036 78746 7 2 6 3 2714 3708 9999 1482 9988 0791 3714 7311 9999 1482 9988 0791 3714 5815Δ mdash Δ 0486 Δ 0

8 Mathematical Problems in Engineering

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

Tabl

e10R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandadditiv

escoringfunctio

n

Firstr

ound

Sellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

22852

2449

10000

0327

0846

8604

5358

6584

9980

1493

8245

9989

7851

1546

9980

1493

8245

9989

7851

1546

28

21

62953

5117

9815

2533

3051

5445

5356

6112

10000

0698

9475

9514

7863

3588

10000

0698

9475

9514

7863

3588

38

26

33301

2614

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

9968

4652

0399

7544

5785

6642

45

22

22754

2829

9993

0592

0012

8755

5246

6816

10000

9530

1889

9556

7745

3372

10000

9530

1889

9556

7745

3372

58

52

53502

5067

9991

7370

0078

6731

6010

7498

9995

8760

5307

9769

8508

3581

9995

8760

5307

9769

8508

3581

67

26

33551

2449

9996

0684

9191

3713

6048

6584

9996

0684

9191

3713

6048

6584

9966

0041

0028

0026

2515

1546

Δndash

Δ0703

Δ0406

Δ0

Mathematical Problems in Engineering 9

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

Tabl

e11R

esults

ofthefirst

iteratio

nof

simulationforsemise

aled

auctionandmultip

licativescoringfunctio

n

Firstroun

dSellerrsquos

score

Sellerrsquos

profi

t

Second

roun

dSellerrsquos

score

Sellerrsquos

profi

t

irdroun

dSellerrsquos

score

Sellerrsquos

profi

t

Fourth

roun

dSellerrsquos

score

Sellerrsquos

profi

tSeller

Attributes

Attributes

Attributes

Attributes

12

34

12

34

12

34

12

34

16

51

27000

2449

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

9870

4822

1524

6222

25965

4744

28

21

66350

5117

9999

1492

4544

5590

24981

6149

9998

5827

8222

9017

84784

2019

9998

5827

8222

9017

84784

2019

38

26

38975

2614

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

9977

6310

0893

8825

34267

5555

45

22

26560

2829

9989

5113

2616

4208

25468

5969

9988

9245

5749

8506

85249

1913

9988

9245

5749

8506

85249

1913

58

52

510250

5067

9999

7579

2322

6120

38462

6279

9242

9999

6893

9996

107144

1668

9242

9999

6893

9996

107144

1668

67

26

310

638

370

8994

7568

3775

6272

638

799

540

5993

2998

09913

825

412

516

8077

79932

9980

9913

8254

125168

0777

ΔmdashΔ

2647Δ

2226Δ

0

10 Mathematical Problems in Engineering

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

open auction the sixth seller is the winner of the auction Inthe third round none of the sellers have the ability to im-prove the bid and thus the winner is determined in thesecond round of the auction Similarly the final score of thewinning seller in semisealed is greater and the profit of thewinner is less than those in the open auction In Table 12 thesemisealed auction results are presented with a risk-aversionscoring function After the first round sellers have not beenable to improve their bids and so the sixth seller with thehighest score in the first round has been set as the winner ofthe auction Unlike additive and multiplicative scoring inthe semisealed auction with risk-aversion scoring thewinnerrsquos score is greater and the winnerrsquos profit is less thanthose in the open auction

After the completion of the simulation process theaverage winning seller score average profit of winning sellerand the average number of rounds of the auction in bothopen and semisealed auctions as well as three types ofadditive scoring multiplicative scoring and risk-aversionscoring are compared To this end nonparametric Man-nndashWhitney U test or Wilcoxon rank tests are used tocompare the open and semisealed auctions and also theKruskalndashWallis test is devised to compare three types ofscoring functionsese two tests are equivalent to t-test andone-way ANOVA in parametric conditions and are based onthe median of two independent populations e null hy-pothesis of both tests is the equality of themedians of the twopopulations An alternative hypothesis in the Krus-kalndashWallis test is the inequality between at least one pair ofpopulations For large samples the MannndashWhitney U teststatistic follows the normal distribution and the Krus-kalndashWallis test statistic follows the chi2 distribution with

N minus 1 degrees of freedom where N is the number ofsamples

Table 13 shows the results of the MannndashWhitney U testfor comparing the open and semisealed auctions in each ofthe scoring functions As can be seen at 95 confidencelevel there is a significant difference between the two typesof the open and semisealed auctions only in the risk-aversionscoring function and in the winning sellerrsquos profit and thenumber of rounds of the auction In the risk-aversionscoring the average profit in a semisealed auction is less thanthat in the open auction and according to the test result thisdifference is almost significant On the contrary the averagenumber of auction rounds in a semisealed auction is con-siderably higher than that in the open auction taking intoaccount the risk-aversion scoring function

In Table 14 the results of the KruskalndashWallis test arepresented to compare three scoring functions in each of theopen and semisealed auctions As can be seen there is ameaningful difference in the confidence level of 95 be-tween the profit of winning seller in the three types of thescoring function either in an open auction or a semisealedauction however this distinction is not strong in a semi-sealed auction (p-value is close to 005) In the open auctiona significant difference is found between the average profitsof the winning seller in the three scoring functions ehighest median profit belongs to the multiplicative scoring(977) then the additive scoring (6503) and finally therisk-aversion scoring (4802) According to results obtainedfrom the KruskalndashWallis test in Table 14 the differencebetween the average profits of the winning seller for allscoring functions is significant e highest average profit inthe open auction belongs to the multiplicative scoring

Table 12 Results of the first iteration of simulation for semisealed auction and risk-aversion scoring function

First roundSellerrsquos score Sellerrsquos profit

Second roundSellerrsquos score Sellerrsquos profit

SellerAttributes Attributes

1 2 3 4 1 2 3 41 6 5 1 2 2507 2449 600 500 100 200 2507 24492 8 2 1 6 2341 5117 800 200 100 600 2341 51173 8 2 6 3 2464 2614 800 200 600 300 2464 26144 5 2 2 2 2568 2829 500 200 200 200 2568 28295 8 5 2 5 2536 5067 800 500 200 500 2536 50676 7 2 6 3 2714 3708 700 200 600 300 2714 3708Δ mdash Δ 0

Table 13 e results of the MannndashWhitney U test for comparing the open auction and the semisealed auction in each of the scoringfunctions

Scoring function Criterion MannndashWhitney U test statistics p-value Results

AdditiveWinnerrsquos profit 16477 00994 ere is no significant differenceWinnerrsquos score 02665 07899 ere is no significant difference

e number of auction rounds 04175 06763 ere is no significant difference

MultiplicativeWinnerrsquos profit minus07281 04666 ere is no significant differenceWinnerrsquos score minus09506 03418 ere is no significant difference

e number of auction rounds minus06868 04922 ere is no significant difference

Risk-aversionWinnerrsquos profit minus34391 0 ere is a significant differenceWinnerrsquos score minus10338 03012 ere is no significant difference

e number of auction rounds minus31171 00018 ere is a significant difference

Mathematical Problems in Engineering 11

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

(7750) then the risk-aversion scoring (43181) and finallythe additive scoring (42330)

e results of the model show that in both open andsemisealed auctions the utility of the buyer and the selleralong with the number of auction rounds under the additiveand multiplicative scoring functions are not significantlydifferent Hence in the semisealed auction in addition to nodisclosure of the main buyerrsquos scoring function for the sellerand the OASP an appropriate approximation of thisfunction by OASP is provided to improve the sellerrsquos bidsduring the auction On the contrary the use of the multi-plicative scoring function has gained higher profits for theseller is is true for both open and semisealed auctions aswell

5 Conclusion

Regarding the importance and wide application of onlineauctions in this paper a multiround multiattribute onlinereverse auction in an e-commerce framework was proposedemain feature of the proposed model in comparison withother models is the design of an information disclosuremechanism that allows the improvement of sellersrsquo bids ineach round of auction while also considering the infor-mation privacy of buyers and sellers In this model an onlineauction service provider estimates the buyerrsquos scoringfunction and makes it available to sellers Based on thisestimated function sellers optimize their bid by solving anonlinear optimization model e process of bidding willcontinue until a stopping condition that is a predefinedimprovement in the score of the best bid is achieved Inorder to evaluate the proposed model the auction processwas simulated In this simulation three scoring functionsadditive multiplicative and risk-aversion were consideredfor the buyer Also the simulation was performed in twotypes open (seller knows the buyerrsquos scoring function) andsemisealed (the proposed model) e simulation resultsshow that in both open and semisealed auctions the utilityof the buyer and the seller as well as the number of auctionrounds with considering the additive and multiplicativescoring functions was not significantly different

Drab et al [27] with a statistical study of 11000 onlinereverse auctions have shown that disclosure of informationleads to increased auction efficiency but buyers and sellersare less willing to participate in an open auction In theproposed model of this paper in addition to the fact thatthere is no concern about explicit disclosure of informationthe efficiency of the auction has not changed significantlycompared to the open type ere is very little research onthe limited disclosure of buyer information Saroop et al

[23] have designed a mechanism to explicitly disclose thescoring weights of some of the criteria in the second round ofthe auction In the proposed model of this paper there is noexplicit disclosure of scoring weights Also Karakaya andKoksalan [17] assumed the Lα metric scoring function forthe buyer and provided an estimate of this function forsellers but based on our proposed model it is possible toestimate different scoring functions using neural networks

When it comes to the direction for future research theproposed framework for online auctions can be developed inmany ways e proposed framework of this article can beused in the auction processes with a single attribute or withsome items of a specific productservice Moreover differentcurve fitting methods such as regression and support vectorregression can be employed to fit the scoring function andevaluate their performance e use of new multiple-attri-bute decision-makingmethods such as the proposedmethodin [30] can be considered to model the buyerrsquos side (buyerselection) in combination with the model presented in thisarticle

Data Availability

All data used to support the findings of this study are in-cluded in the article

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] C-H Chen-Ritzo T P Harrison A M Kwasnica andD Jomas ldquoBetter faster cheaper an experimental analysisof a multiattribute reverse auction mechanism with restrictedinformation feedbackrdquo Management Science vol 51 no 12pp 1753ndash1762 2005

[2] M Huang X Qian S-C Fang and X Wang ldquoWinner de-termination for risk aversion buyers in multi-attribute reverseauctionrdquo Omega vol 59 pp 184ndash200 2016

[3] L Pham J Teich H Wallenius and J Wallenius ldquoMulti-attribute online reverse auctions recent research trendsrdquoEuropean Journal of Operational Research vol 242 no 1pp 1ndash9 2015

[4] A Gupta S T Parente and P Sanyal ldquoCompetitive biddingfor health insurance contracts lessons from the online HMOauctionsrdquo International Journal of Health Care Finance andEconomics vol 12 no 4 pp 303ndash322 2012

[5] R Tassabehji W A Taylor R Beach and A Wood ldquoReversee-auctions and supplier-buyer relationships an exploratorystudyrdquo International Journal of Operations amp ProductionManagement vol 26 no 2 pp 166ndash184 2006

Table 14e comparison of results obtained from the KruskalndashWallis test for multiplicative additive and risk-aversion scoring in the openand semisealed auction

Type of auction Criterion KruskalndashWallis test statistics p-value Results

Open Winnerrsquos profit 3174 0 ere is a significant differencee number of auction rounds 1433 00008 ere is no significant difference

Semisealed Winnerrsquos profit 832 00156 ere is a significant differencee number of auction rounds 25 02866 ere is no significant difference

12 Mathematical Problems in Engineering

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13

[6] L Peng and R Calvi ldquoWhy donrsquot buyers like electronic reverseauctions Some insights from a French studyrdquo InternationalJournal of Procurement Management vol 5 no 3 pp 352ndash367 2012

[7] C-B Cheng ldquoReverse auction with buyer-supplier negotia-tion using bi-level distributed programmingrdquo EuropeanJournal of Operational Research vol 211 no 3 pp 601ndash6112011

[8] G Karakaya and M Koksalan ldquoAn interactive approach forBi-attribute multi-item auctionsrdquo Annals of Operations Re-search vol 245 no 1-2 pp 97ndash119 2016

[9] B Mansouri and E Hassini ldquoA Lagrangian approach to thewinner determination problem in iterative combinatorialreverse auctionsrdquo European Journal of Operational Researchvol 244 no 2 pp 565ndash575 2015

[10] M Mouhoub and F Ghavamifar ldquoManaging constraints andpreferences for winner determination in multi-attribute re-verse auctionsrdquo in Proceedings of the 2016 15th IEEE Inter-national Conference on Machine Learning and Applications(ICMLA) 2016

[11] A K Ray M Jenamani and P K J Mohapatra ldquoAn efficientreverse auction mechanism for limited supplier baserdquo Elec-tronic Commerce Research and Applications vol 10 no 2pp 170ndash182 2011

[12] J E Teich H Wallenius J Wallenius and A Zaitsev ldquoAmulti-attribute e-auction mechanism for procurement the-oretical foundationsrdquo European Journal of Operational Re-search vol 175 no 1 pp 90ndash100 2006

[13] J E Teich H Wallenius J Wallenius and O R KoppiusldquoEmerging multiple issue e-auctionsrdquo European Journal ofOperational Research vol 159 no 1 pp 1ndash16 2004

[14] D R Beil and L M Wein ldquoAn inverse-optimization-basedauction mechanism to support a multiattribute RFQ processrdquoManagement Science vol 49 no 11 pp 1529ndash1545 2003

[15] Y-K Che ldquoDesign competition through multidimensionalauctionsrdquo 8e RAND Journal of Economics vol 24 no 4pp 668ndash680 1993

[16] Y Hu Y Wang Y Li and X Tong ldquoAn incentive mechanismin mobile crowdsourcing based on multi-attribute reverseauctionsrdquo Sensors vol 18 no 10 p 3453 2018

[17] G Karakaya and M Koksalan ldquoAn interactive approach formulti-attribute auctionsrdquo Decision Support Systems vol 51no 2 pp 299ndash306 2011

[18] D C Parkes and J Kalagnanam ldquoModels for iterative mul-tiattribute procurement auctionsrdquo Management Sciencevol 51 no 3 pp 435ndash451 2005

[19] G Perrone P Roma and G Lo Nigro ldquoDesigning multi-attribute auctions for engineering services procurement innew product development in the automotive contextrdquo In-ternational Journal of Production Economics vol 124 no 1pp 20ndash31 2010

[20] S X Xu and G Q Huang ldquoEfficient multi-attribute multi-unit auctions for B2B E-commerce logisticsrdquo Production andOperations Management vol 26 no 2 pp 292ndash304 2017

[21] X Liu Z ZhangW Qi and DWang ldquoAn evolutionary gamestudy of the behavioral management of bid evaluations inreserve auctionsrdquo IEEE Access vol 8 pp 95390ndash95402 2020

[22] R Narasimhan S Talluri and S Mahapatra ldquoEffective re-sponse to RFQs and supplier development a supplierrsquosperspectiverdquo International Journal of Production Economicsvol 115 no 2 pp 461ndash470 2008

[23] A Saroop S K Sehgal and K Ravikumar ldquoA multi-attributeauction format for procurement with limited disclosure ofbuyerrsquos preference structurerdquo in Annals of Information

Systems Decision Support for Global Enterprises pp 257ndash267Springer Berlin Germany 2007

[24] S Wang S Qu M Goh M I M Wahab and H ZhouldquoIntegrated multi-stage decision-making for winner deter-mination problem in online multi-attribute reverse auctionsunder uncertaintyrdquo International Journal of Fuzzy Systemsvol 21 no 8 pp 2354ndash2372 2019

[25] N Yang X Liao and W W Huang ldquoDecision support forpreference elicitation in multi-attribute electronic procure-ment auctions through an agent-based intermediaryrdquo Deci-sion Support Systems vol 57 pp 127ndash138 2014

[26] X Qian S-C Fang M Huang and X Wang ldquoWinner de-termination of loss-averse buyers with incomplete informa-tion in multiattribute reverse auctions for clean energy deviceprocurementrdquo Energy vol 177 pp 276ndash292 2019

[27] R Drab T Stofa and R Delina ldquoAnalysis of the efficiency ofelectronic reverse auction settings big data evidencerdquo Elec-tronic Commerce Research vol 20 pp 1ndash24 2020

[28] X Peng and Y Yang ldquoAlgorithms for interval-valued fuzzysoft sets in stochastic multi-criteria decision making based onregret theory and prospect theory with combined weightrdquoApplied Soft Computing vol 54 pp 415ndash430 2017

[29] L Zhang and N Ponnuthurai ldquoA survey of randomized al-gorithms for training neural networksrdquo Information Sciencesvol 364 pp 146ndash155 2016

[30] L Wang and N Li ldquoPythagorean fuzzy interaction powerBonferroni mean aggregation operators in multiple attributedecision makingrdquo International Journal of Intelligent Systemsvol 35 no 1 pp 150ndash183 2020

Mathematical Problems in Engineering 13