Efﬁcient Combinatorial Auction Mechanisms in Electronic Commerce

Tumpa Banerjee

, Dinesh Kumar Pradhan

and Prasenjit Choudhury

Department of Computer Application, Siliguri Institute of Technology, Siliguri, India

Department of CSE/IT, Dr. B. C. Roy Engineering College, Durgapur, India

Department of Computer Application, National Institute of Technology, Durgapur, India

Keywords:

Resource Allocation, Combinatorial Auction, New FixedPrice, New Combinatorial Auction using LinearPro-

gramming (New CA LP), Combinatorial Auction using Greedy, New Combinatorial Auction using Greedy

Approach ( CAGREEDY-MODIFIED).

Abstract:

Electronic commerce or e-commerce is the trading of products or services via internet. The product with little

demand is generally sold in ﬁxed price. However, when the demand of a product is huge, auction mechanism

can be used to maximize the proﬁt. Selling price of some inevitable products like medicine does not depend

on the demand. Auction is the best method for selling products which provide maximum possible proﬁt to

the sellers and the buyers get the product in reasonable price. Today, a large part of e-commerce uses online

auction for selling their products or to provide any service to the worldwide buyers. Winner determination

and payment value calculation of combinatorial auction is a very complex task. The solution to this problem

demands optimal result to the auctioneer within manageable time and the satisfaction of both the buyers and

sellers in terms of proﬁt. Most simple combinatorial auction already used by many websites for e-procurement

is ﬁxed price auction. Fixed price auction is not truthful and gives more proﬁt to the seller. In this paper we

study different auction mechanisms for item procurement in e-commerce and proposed a new truthful auction

strategy that outperforms the existing approaches in the context of time and truthfulness.

1 INTRODUCTION

Auction is the most important mechanism for dy-

namic pricing in e-commerce (Muller, 2001). The

seller offers a variety of items or service for sale with

the aim of obtaining more proﬁt. Online Auction is

useful to maximize the seller revenue and provides an

opportunity to a buyer to buy the products or services

at fair market value based on selling price and not

asking price. A seller can meet maximum numbers of

customer from different geographic area because of

the location independence of e-commerce. Combina-

torial auctions have been gaining signiﬁcant interest

as an automated mechanism for selling multiple

products or services to a single buyer (Narahari and

Dayama, 2005). Combinatorial auction is extremely

useful in numerous e-business applications such as e-

selling, e-procurement, e-logistic etc. Combinatorial

auctions are those those in which participants place

bids on combination of products or services, called

packages, rather than individual product (Cramton

et al., 2005). In this case, participants or buyers

either win all the products or services of the package

they have requested to buy or lose. Combinatorial

auctions are useful to a buyer to buy all the related

products or services altogether at one bid at fair

market value and seller also able to sell bundle of

products or services at a time at maximum proﬁt.

Auctions have gained much greater prominence as

the means of determining prices at which goods are

bought and sold as a result of improved commu-

nication and information processing capabilities of

the personal computers and the Internet (Morgan,

2002). At a consumer level, the auction site eBay

and Amazon have transformed the market for buying

and selling collectibles as well as a host of other

products and their sales revenue continue to grow

rapidly by more than 90 percent in last ﬁve years

(http://www.marketwatch.com/investing/stock/ebay/

ﬁnancials).

1.1 Some Applications of Combinatorial

Auctions

Numerous applications have been reported in various

290

Banerjee, T., Pradhan, D. and Choudhury, P.

Efﬁcient Combinatorial Auction Mechanisms in Electronic Commerce.

In Proceedings of the 18th International Conference on Enterprise Information Systems (ICEIS 2016) - Volume 2, pages 290-297

ISBN: 978-989-758-187-8

research papers for combinatorial auction. We have

only discussed few of the applications areas of com-

binatorial auction.

Spectrum Allocation: Wireless and mobile tech-

nologies are growing very rapidly in recent times.

Service providers need to provide emerging wireless

service and new wireless architecture to fulﬁll the re-

quirement of huge number of wireless/mobile devices

and applications. In static resource management sys-

tem spectrum is allocated for long term basis. Some

radio spectrum may leave idly where some wireless

systems are unable to work due to spectrum crisis.

Static resource management in wireless system is in-

efﬁcient in modern era since demand and supply may

not always match. Auction is widely used to dy-

namic allocation of spectrum to improve the chan-

nel/spectrum utilization(Zheng et al., 2014). Govern-

ment or authorities use auction mechanism to sell the

spectrum license among telecommunication compa-

nies.

Electricity Distribution: In (Zhongjing and Yingy-

ing, 2014)(Sarlati et al., 2013) hierarchical electricity

distribution retailers distribute electricity using auc-

tion mechanism. The electricity market consists of

one generation provider (owner of whole electricity),

several retailers (different companies) and many con-

sumers. Combinatorial auction mechanisms are used

by electric generation provider to assign electricity to

the retailers.

E-Procurement: Auctions are widely used in E-

procurement (Narahari and Dayama, 2007)(Shikui

et al., 2014)(Gregory and Molson, 2013). Procure-

ment is purchase of goods and services. Buyers look

for sellers who would sell the items and services. Sell-

ers with similar goods and services submit their bids;

Sellers also obtain the information regarding goods

and interest of other sellers which help them to price

negotiation. Buyers continue the bid until one seller

is ready to offer lower price than the one offered in

the last bid or the end of time. More than one round

is required to ﬁnalize the winner.

Resource allocation: In cloud computing, cloud

provider rent their various resources to the user for

their use over internet. Users from various location of

the world may hire resource from cloud. Combina-

torial auction mechanism is used by cloud providers

to rent their resources to the users (Zaman and Grosu,

2013a)(Iosiﬁdis and Koutsopoulos, 2010) on pay per

use basis.

1.2 Literature Review and Research

Motivation

There are three basic techniques of combinatorial auc-

tion; i.e. Fixed Price Auction, Combinatorial Auc-

tion Linear Programming (CA-LP) and Combinato-

rial Auction with Greedy approach (CA-GREEDY).

Shikui (Shikui et al., 2014) has proposed the

Fixed Price Auction mechanism which is the sim-

plest combinatorial auction and is used by many cloud

providers. It has been proved that it is not incentive

compatible (Zheng et al., 2014).

Archer (Archer et al., 2005) has proposed one

mechanism CA-LP of combinatorial auction using

linear programming problem and (Shikui et al.,

2014)has given extended version of CA-LP. Though

CA-LP is incentive compatible but it has been solved

by linear programming problem and results non linear

time complexity.

CA-GREEDY is another combinatorial auction

mechanism proposed by Archer (Archer et al., 2005),

Zaman (Zaman and Grosu, 2013b). This mechanism

takes less time to determine the winner and calculat-

ing payment value and it is incentive compatible but

not like CA-LP.

CA-Provision is another approach using greedy

mechanism proposed by (Zaman and Grosu, 2013a).

A plethora of research has been done on combina-

torial auctions. None of the existing literature com-

pares the performance parameter of the existing pop-

ular multi-item auction used in e-commerce. The ob-

jective of this paper is to compare the three auction

techniques and proposed a new mechanism design ap-

proach to improve the performance.

Rest of the paper is organized as follows: section

2 present the comparison of popular auction mech-

anisms in e-commerce. The multiple products sell-

ing is represented as mechanism design problem in

section 3, the new proposed Fixed Price, CA-LP and

CA-GREEDY-MODIFIED mechanism is discussed

in section 4. The simulation of proposed mechanisms

is detailed in section 5; ﬁnally we conclude the paper

and discuss future research direction in section 6.

2 COMPARISON OF

COMBINATORIAL AUCTION

MECHANISMS

Basic parameters for measuring performance of any

auction mechanism are truthfulness, execution time

and the value of utility function. Utility of each user

is the difference between the bid values she has asked

Efﬁcient Combinatorial Auction Mechanisms in Electronic Commerce

291

for her requested resources at the bidding time and

the original price she has been charged for using re-

sources. Execution time means total time taken to ex-

ecute the algorithm to determine the winner and for

calculating the payment value. Truthfulness means

that each bidder bids their true value of the resources.

We have implemented all the existing combinatorial

auction mechanisms in MATLAB environment and

tested for various numbers of inputs. For all simula-

tions, inputs are randomly generated within the given

range and the following values of performance param-

eter found.

Table 1: Performance comparison of combinatorial auction

mechanisms.

Truth

fullness

Value of the

utility

function

Time

Complex-

ity

Fixed

Price

No Zero Low

CA-LP Yes High High

CA-

GREEDY

Yes Average Average

The above table shows that Fixed Price auction is

not at all truthful and the value of the utility func-

tion is zero so the seller’s revenue is very high but

its execution time is low. The seller revenue is very

high in this case but not maximum. We have pro-

posed New FixedPrice combinatorial auction where

the seller may earn maximum possible proﬁt. But

CA-LP ensures that truthfulness is the dominant strat-

egy of each user and the value of utility function is too

high compared to other combinatorial auction. But

execution time of CA-LP is very high. So CA-LP

is the best combinatorial auction mechanism for all

performance parameters for small input values. CA-

LP takes longer time to execute a large number of

inputs. But payment method of CA-LP mechanism

is designed in such a way that the user pay zero for

the product with a little demand. But the product

with a little demand is generally sold at the ﬁxed

price. In this paper, we have proposed New CA LP

where we extended the payment part of CA-LP so

that seller’s revenue will not be zero for any prod-

uct, the buyer has to pay minimum ﬁxed price of

the product. CA-GREEDY mechanism also ensures

truthfulness, execution time and value of the util-

ity function at average. CA-GREEDY is very use-

ful for large number of inputs but value of the util-

ity function is not quite bright. In a research pa-

per Zaman (Zaman and Grosu, 2013a) has given one

new method; i.e. CA-PROVISION based on CA-

GREEDY where the value of utility function is bet-

ter than CA-GREEDY but not like CA-LP. In this pa-

per we have modiﬁed the CA-GREEDY mechanism

(CA-GREEDY-MODIFIED) to improve the value of

the utility function so that CA-GREEDY-MODIFIED

will be the best choice for combinatorial auction for a

large number of inputs and provides best value for all

performance parameters.

3 MULTIPLE PRODUCT

DISTRIBUTION AS A

MECHANISM DESIGN

PROBLEM

For formulating the multiple product or service sell-

ing process we consider the following notations and

assumptions:

1)Let

I = (u

,...,u

)

be the set of n buyers,

K = (r

,..., r

)

be the set of m available products or services and r

the total number of available products or resource of

type i and

W = (w

,...w

)

be the set of weights where w

is the weight or mini-

mum value per unit of ith product.

2) A set of winners WIN contains list of winners.

3) Each user or buyer precisely knows the value

that he bids but does not know the bid value of the

others. The bidder i submits the bid

,..., a

)

Where a

i j

denote the number of j

resource request

by user i and v

is the valuation of total items re-

quested by her. User i can put a

i j

as zero, if she does

not want any number of resource j.

4) Bidders are not aware of the type and number

of requested products and total valuation submitted by

other bidder.

5) Let

P = (p

, p

,...p

)

be the set of payment value paid by the buyer to the

seller.

6) The agents always try to maximize a utility

function. Utility function deﬁnes as the difference be-

tween bid value submitted by user and the value actu-

ally paid.

The below mentioned matrix represents the bid

proﬁles of all the bidders.

bid =







... a







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292

After collecting all the individual bids, combina-

torial auction mechanism determines the winners and

it also ﬁxes the charge paid by every winning bidder.

4 NEW PROPOSED MECHANISM

In this section, three mechanisms have been presented

to solve the problem of multiple product sale to a sin-

gle buyer in a single bid. The ﬁrst New FixedPrice

auction, is the extended version of Fixed Price Auc-

tion by Archer (Archer et al., 2005). The next two

mechanisms are the proposed New-CA-LP and CA-

GREEDY-MODIFIED based on linear programming

problem and Greedy mechanism.

4.1 New Fixed Price Auction

(New FixedPrice Auction)

Fixed Price Auction mechanism is the simplest

combinatorial auction and is used by many cloud

providers (Zaman and Grosu, 2013b). The Fixed-

Price auction mechanism deﬁnes a ﬁxed price vector

,...q

) where q

is the price a buyer has to pay

to buy one instance of i

product. Archer (Archer

et al., 2005) explains this mechanism as ﬁrst come

ﬁrst serve basis until the products are exhausted. In

this case during the opening of the market there will

be a huge trafﬁc to submit the request ﬁrst. The ﬁxed

price auction mechanism has been extended by re-

ordering the users in descending order of the respec-

tive bid value. The bidder with the highest bid value

will get the ﬁrst chance to win the products. Bidders

win the products depending on their bid values, no

need to compete for submitting the bid ﬁrst and the

auctioneer proﬁts more. It also makes sure that in or-

der to get the requested bundle of products, the valu-

ation vi of buyer u

is at least F

, where F

is the sum

of the ﬁxed prices of each products in her bundle. In

this case the bid value reﬂects necessity and urgency

of the products. The algorithm for implementing the

ﬁxed price auction is given below(Algorithm-1).

In the our algorithm (Algorithm-1), reordering of

the bids is included and time complexity increases

from O(n) to O(nlogn). But using this method an

auctioneer can avoid one time rush and the resource

providers have a chance to earn more revenue because

resource providers distribute resources from high to

low bid value. Fixed price method is not incentive

compatible (Zheng et al., 2014) resource allocation.

The Winners have to pay the value what they have de-

clared during bid submission. To encourage users to

participate in auction we need to charge less than the

value declared by them.

Algorithm 1. New FixedPrice Auction.

Step 1 : Collect the bid f romthe users.

Step 2 : Arrangethe users in order (u

,..., u

)

so that v

>= v

>= ...v

Step 3 : InitializeW IN = φ

Step 4 : f or i = 1to n

i f v

∑

j=1

i j

∗ q

then

i f A

<= R

thenW IN = W IN U NION

{

}

and R = R − A

whereA

= (a

,..., a

)

and R = (r

,..., r

)

[End I f ]

[End f or]

Step 5 : f or i = 1to n

I f i ∈ W IN then p

= v

Else p

= 0

[EndI f ]

[EndFor]

It is assumed that two types of products are offered

by the buyer. Relative weight of the products is W=[2

,1] and ﬁxed prices are 10 and 15 respectively. Total

ten users participate in the bidding process. Numbers

of products of each type are 10 and 15 respectively.

Table 2: Examples of Submitted bids.

No of Prod-

ucts (r1)

No of Prod-

ucts (r2)

Bid Value

2 1 40

1 2 30

2 5 60

3 3 35

2 4 50

3 1 33

2 0 20

0 5 60

1 3 55

4 2 85

Each users(i) bid is 3-tuple (r

) where r

is the number of requested instances of item 1, r

the number of requested instances of item 2 and v

the bid value that user i wants to pay for the requested

items.

Result: Existing Fixed Price auction and proposed

New Fixed Price auction have been implemented on

the above data set and the result is given below (Table-

3):

4.2 New CA LP

Combinatorial Auction-Linear Programming (CA-

Efﬁcient Combinatorial Auction Mechanisms in Electronic Commerce

293

Table 3: Result of New FixedPrice Auction.

Winner

set

Payment

set

Seller

rev-

enue

Existing

Fixed Price

auction

1, 2, 3, 5,

40, 30, 60,

50, 33

213

New Fixed

Price Auction

10, 8, 3, 9,

85, 60, 60,

55, 40

300

LP) represents as standard linear programming

problem. Archer (Archer et al., 2005) explained the

combinatorial auction just like the virtual machine

allocation problem in (Zaman and Grosu, 2013b),

the bidder can request maximum one (a

i j

ε {0,1})

resource of each type Where as in (Shikui et al.,

2014) user can request any number of resources (a

i j

ε {0,1,2,3,....}). Archer (Archer et al., 2005) reduced

the number of resources to prohibit oversell of the

products. In our case the total number of products is

not reduced because we consider oversell is impos-

sible. Winners are selected if the requested products

are available. The payment calculation part of the

algorithm (Zaman and Grosu, 2013a) is modiﬁed

and as a result of that the price of the less demand

product will be the ﬁxed value. According to (Zaman

and Grosu, 2013b) payment of each winner is the

minimum value to win the bid. In the situation while

demand is less than supplies, one buyer becomes

the winner irrespective of her bid value. The buyer

may become the winner even when the bid value is

zero; then that user pays zero for using resources.

Every seller asks at least ﬁxed price for low demand

products. For example, one seller has two types of

products (r1, r2). Total numbers of available products

are (2, 2). Four buyers submit their bids

bid =







2 3 60

1 3 70

3 2 90

2 0 80







In this situation only the requirements of 4

user

matche with the availability. Existing CA-LP calcu-

lates the payment value as the critical value to win and

the payment value of user u

is 0.

The existing CA-LP mechanism provides the

buyer with maximum value of utility function but the

sellers may become looser. So in our algorithm an

auctioneer calculates payment value as the minimum

price to win the product but it should not be less than

the ﬁxed price of the product. The algorithm is given

below

Algorithm 2. New CA LP.

Step 1 : Fixed prices o f the products are (y

,...,y

)

Step 2 : Collect the bid f rom the users.

(a)Solvethe f ollowing linear programming problem

Maximize

∑

i=1

∗ x

Sub ject to

∑

i=1

i j

∗ x

<= r

f or j = 1, 2, ...,m where x

>= 0

and the value o f x

is the winning

probability o f user u

Step 3 : WinnerDetermination

(a)Initialize W IN = φ

(b)For eachuser u

taken in descending order o f x

I f A

<= R

then W IN = W IN UNION

{

}

and R = R −A

where A

= (a

,...,a

)

and R

= (r

,...,r

)

[EndI f ]

[End f or]

Step 4 : Payment Calculation

(a)For eachuser u

∈ W IN

Per f orm binary search f or v

in [0,v

]

where v

is the critical value to win

I f v

∑

j=1

i j

∗ y

then

set p

= v

Else

set p

∑

j=1

i j

∗ y

[EndI f ]

[End f or]

(b)Set p

= 0 f or i does not belong to W IN

Result:Existing CA-LP auction and our proposed

New CA LP auction mechanisms have been executed

on the above data set where minimum service charges

for the resources are [3,3] and the result is given

below(Table-4):

Table 4: Result of New CA LP.

Winner

set

Payment

set

Value of the

utility

function

CA-LP

9, 10, 8, 1,

7, 2

35, 52, 0,

26, 15, 27

130

New CA

9, 10,

8,1,7,2

35, 52, 15,

26, 15, 27

120

4.3 CA-GREEDY-MODIFIED

The winner determination and payment mechanism

of CA-GREEDY and CA-PROVISION in (Zaman

and Grosu, 2013b)are extended in CA-GREEDY-

MODIFIED. In strategy-proof auction each user

would maximize her utility . CA-GREEDY and CA-

PROVISION provide the seller with more beneﬁt as

their users utility is less in comparison to CA-LP. For

ICEIS 2016 - 18th International Conference on Enterprise Information Systems

294

maximizing the value of the utility function, we have

changed the payment mechanism of greedy method.

We use Vickery-Clark-Groves (VCG) mechanism for

payment value calculation. The algorithm is given be-

low.

Algorithm 3. CA-GREEDY-MODIFIED.

Step 1 : Collect the bid f romthe users.

Step 2 : Winner determination

(a)W IN = φ

[Find out approax valuation based on

the number items requested and weight

o f the items]

(b) f or i = 1 to n

∑

j=1

i j

∗W

[End f or]

(c)Reorder the users suchthat

>= v

>= ... >= v

(d) f or i = 1to n

i f A

<= R

thenW IN = W IN U NION

{

}

and R = R − A

where A

= (a

,..., a

)

and R = (r

,..., r

)

[EndI f ]

[End f or]

Step 3 : Payment Calculation

(a)For each user i ∈ W IN

Per f orm binary search in 0to v

, such

that we f ind a critical value v

which isthe least value to win.

I f v

∑

j=1

i j

∗W

then set p

= v

Else

set p

∑

j=1

i j

∗W

[EndI f ]

[End f or]

(b)Set p

= 0 f or i does not belong toW IN

The CA-GREEDY-MODIFIED mechanism deter-

mines the winners in accordance with ranking the

buyers in descending order of their unit price (v

)

(Lehmann et al., 2002) and then they are greedily se-

lected as winners from the list. Before selecting the

winner the mechanism veriﬁes that the new combina-

tion of products does not exceed the number of avail-

able items of each type of products. The payment p

a winner u

pays is calculated as the minimum price

to win the bid but it should not be less than the ﬁxed

price of the products i.e. the winners pays the critical

value.

Result: We have implemented existing CA-

GREEDY (Narahari and Dayama, 2007), CA-

PROVISION(Zaman and Grosu, 2013a) auctions and

our proposed CA-GREEDY-MODIFIED auction on

the above data set where ﬁxed price for the products

is [3,3] and the result is given below:

Table 5: Result of CA-GREEDY-MODIFIED.

Winner

set

Payment

set

Value of the

utility

function

CA-

GREEDY

10, 9, 8, 1,

67, 54, 52,

39, 42

CA-

PROVISION

10, 9, 1, 8,

7, 2

83, 53, 36,

50, 20, 0

CA-

GREEDY-

MODIFIED

10, 9,

1,8,7

51, 12, 9,

15, 6,9

188

5 RESULTS AND DISCUSSION

We have implemented all the combinatorial auction

mechanisms in MATLAB environment. MATLAB

environment is convenient for matrix operation and

it exhibits the execution time of the entire program or

portion of the program. Input data sets are randomly

generated within a speciﬁed range for a given num-

ber of users. Once the data sets are generated, all the

strategies are tested on the same data sets. Execution

time, proﬁt and the values of the utility functions of

all the simulations are collected, to generate compar-

ative graph for the speciﬁed algorithms. We compare

the performance of existing Fixed Price auction with

the New Fixed Price auction, existing CA-GREEDY

with CA-GREEDY-MODIFIED and also the perfor-

mance of CA-GREEDY-MODIFIED with the perfor-

mance of other combinatorial auction based mecha-

nisms in respect of execution time and the value of

the utility function.

Fixed price auction that is the most simple auc-

tion mechanism has already been used by many e-

commerce web sites. In this case the users pay the ac-

tual value that she has submitted at the bidding time.

So the value of the utility function is zero. We com-

pare our New FixedPrice auction mechanism with the

existing ﬁxed price auction in respect of execution

time (Figure-1) and seller revenue(Figure-2). In com-

parison with the existing Fixed Price Auction, New

Fixed price Auction mechanism provides the service

provider with more proﬁt. The execution time of New

Fixed price Auction is higher than the existing one but

it is better as it is a polynomial time problem.

The same types of winner determination and pay-

ment method have been used in New CA LP mech-

Efﬁcient Combinatorial Auction Mechanisms in Electronic Commerce

295

Figure 1: Execution time of New FixedPrice Auction vs

Existing Fixed Price Auction.

Figure 2: Seller Revenue of New FixedPrice Auction and

existing Fixed Price Auction.

anism as in the existing CA-LP. Only difference

between New CA LP and existing CA-LP is that

New CA LP charge ﬁxed price for low demand prod-

ucts whereas CA-LP may charge very low price even

if it may be zero. So we are not conferring any com-

parative results of New CA LP and CALP because of

equal execution time and utility functions value.

Figure-(3) and Figure-(4) represent the value of

the utility function and require time of CA-GREEDY,

CA-PROVISION and CA-GREEDY-MODIFIED for

different number of users and resources. Figure-(3)

shows that our CA-GREEDY-MODIFIED takes max-

imum time to execute in comparing to other two auc-

Figure 3: Execution time of CA-GREEDY-MODIFIED,

CA-GREEDY and CA-PROVISION.

Figure 4: Utility function’s value of CA-GREEDY-

MODIFIED, CA-GREEDY and CA-PROVISION.

tion mechanisms but its utility functions value, an-

other performance parameter is also very high in com-

parison with other two. For small input values three

mechanisms perform the same approximately. But for

large inputs the value of the utility function of CA-

GREEDY-MODIFIED is 50 percent more than CA-

GREEDY and CA-PROVISION mechanisms but its

required 60 to 80 percent more execution time.

Figure-(5) and Figure-(6) shows the evaluation

results of all the combinatorial auction mechanisms

when there are different number of bids and dif-

ferent number of products. We can see that CA-

GREEDY-MODIFIED always outperforms the other

three auction mechanism. Figures-(5) and (6) demon-

strates the value of the utility function and execution

time of CA-LP, CA-GREEDY, CA-PROVISION and

CA-GREEDY-MODIFIED mechanisms. It shows the

value of the utility function of CA-LP is much better

than CA-GREEDY and CA-PROVISION but more or

less the same as CA-GREEDY-MODIFIED. But the

execution time is too high comparable to other auction

mechanisms. CA-LP auction mechanism takes more

than one hour for 500 users bidding for 50 differ-

ent items. CA-GREEDY-MODIFIED auction takes

Figure 5: Utility function’s value of CA-GREEDY-

MODIFIED, CA-GREEDY and CA-PROVISION and CA-

LP.

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296

Figure 6: Execution time of CA-GREEDY-MODIFIED,

CA-GREEDY and CA-PROVISION and CA-LP.

less time and utility functions value is also high. In

CA-GREEDY-MODIFIED auction mechanism users

pay minimum value to win the resources. So truth

telling is the dominant strategy of the CA-GREEDY-

MODIFIED auction.

6 CONCLUSION

The indispensability of combinatorial auction for sell-

ing a group of products has been look over for any

e-commerce application. Extensive simulation exper-

iments have been carried out to concluded that CA-

GREEDY-MODIFIED is clearly a better choice for

selling products or service over Internet. The value

of the utility function of New CA LP is better than

the other auction mechanisms. In CA-GREEDY ap-

proach execution time is less than the other truthful

auction mechanisms but the utility function value is

not so good as CA-LP. Moreover, CA-LP is solved

by linear programming problem which is a NP com-

plete problem. So execution time of CA-LP auc-

tion mechanism is inﬁnite for large number of in-

puts. The utility value of the proposed CA-GREEDY-

MODIFIED is better than CA-GREEDY and is closer

to CA-LP and takes less time than New CA LP. So

CA-GREEDY-MODIFIED is the best auction mech-

anism to sell multiple products via electronic devices.

Further work includes the deployment of the proposed

mechanisms on an experimental e-commerce testbed

and tries to reduce execution time of CA-GREEDY-

MODIFIED.

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