Forming Buyer Coalitions with Bundles of Items

KES AMSTA 2009, June 5, 20091

Forming Buyer Coalitions with Bundles of Items

Laor Boongasame, Department of Computer Engineering, Bangkok University, Bangkok, Thailand

Ho-fung Leung, Department of Computer Science and Engineering, The Chinese University of Hong Kong, Hong Kong, P.R. China

Veera Boonjing, Department of Mathematics and Computer Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok,

ThailandDickson K. W. Chiu, Dickson Computer Systems, 7 Victory Avenue,

Kowloon, Hong Kong, P.R. China


Outline of Presentation

I. Motivation for the research

II. Forming Buyer Coalitions with Bundles of ItemsI. Example and Problem Formulation

II. Algorithm and Example Revisited

III. SimulationI. Setup of Experiments

II. Results and Analysis

IV. Discussion and Conclusion


I. Motivation for the Research


I. Motivation for the Research

A buyer coalition is a group of buyers who join together to negotiate with sellers for a bulk purchasing of items at a larger discount (Tsvetovat, M., Sycara, K. P., Chen, Y., 2001).

There are several existing buyer coalition schemes (He, L., Ioerger T., 2005; Anand, K.S., Aron, R., 2003; Tsvetovat, M., Sycara, K. P., Chen, Y., Ying, 2001, Hyodo, M., Matsuo, T., Ito, T., 2003). However, these schemes do not consider forming a buyer coalition with bundles of items.


I. Motivation for the Research (Cont.)

This practice can be observed very often in the real world such as restaurants (e.g., McDonald's Happy Meal), durable consumer goods (e.g., personal computer options), and non-durable consumer goods (e.g., dishwasher detergent and rinse aid packages).


II. Forming Buyer Coalitions with Bundles of Items


II. Forming Buyer Coalitions with Bundles of Items: Example and Problem Formulation

Item Package/Item Unit Price ($)

1 (pk1) Printer, CPU, Monitor 2700

2 (pk2) Printer, CPU, RAM 2700

3 … Printer 1000

4 CPU 1000

5 Monitor 1000

6 RAM 1000

Pk


II. Forming Buyer Coalitions with Bundles of Items: Example and Problem Formulation (Cont.)

Table 2 shows the subsidiaries' required computer equipment and their reservation prices, i.e., the maximum price that the buyer bk is willing to pay for a unit of each item. For instance, subsidiary Bangkok-B wants to purchase a unit of CPU at $900 or lower and a unit of monitor at $900 or lower.



subsidiaries Rsi Of Printer

Rsi Of CPU

Rsi Of Monitor

Rsi Of RAM

Bkk-A 1000

Bkk-B 900 900

Bkk-C 800 1000 800

Bkk-D 900 1000 900



The problem that we solve is forming buyer coalitions for purchasing item packages ,such that can purchase required items and is maximal, where is the joint utility that members of Ci can reach by cooperating via coalitional activity for purchasing a specic bundles of items.

PkPk l BCi Bbi i iV

iV


II. Forming Buyer Coalitions with Bundles of Items: Algorithm and Example Revisited

Step 1: Calculate all the permutations that include up to k buyers. This is the set of all potential coalitions PC.


II. Forming Buyer Coalitions with Bundles of Items: Algorithm and Example Revisited (Cont.)

The number of members in any buyer coalitions

The set of potential coalitions PC

1 {Bkk-A}, {Bkk-B}, {Bkk-C}, {Bkk-D}

2 {Bkk-A, Bkk-B}, {Bkk-A, Bkk-C}, {Bkk-A, Bkk-D}, {Bkk-B, Bkk-C}, {Bkk-B, Bkk-D}, {Bkk-C, Bkk-D}

3 {Bkk-A, Bkk-B, Bkk-C},{Bkk-A, Bkk-B, Bkk-D}, {Bkk-A, Bkk-C, Bkk-D}, {Bkk-B, Bkk-C, Bkk-D}

4 {Bkk-A, Bkk-B, Bkk-C, Bkk-D}



Step 2: In each coalition, the coalitional potential items vector is calculated by summing up the unused items of the numbers of the coalition. Formally,

Cb i

PCC

iGdGd

PCCGd




printer CPU Monitor Memory

{Bkk-A} √

{Bkk-B} √ √

{Bkk-C} √ √ √

{Bkk-D} √ √ √

{Bkk-A, Bkk-B} √ √ √

{Bkk-B, Bkk-C} √ √√ √ √

…

PCCGd



Step 3: For each bundle of items ,perform: Step 3.1: Check what items are wanted for the satisfaction of

Pkpk l

lGd

lpk



packages Gdl

printer CPU Monitor Memory

pk1√ √ √

pk2√ √ √

pk3√

pk4√

pk5√

pk6√



Step 3.2: Compare to the sum of the unused items of the members of the coalition , thus finding the packages that can be satisfied by coalition C. The coalition C will be formed to purchase the package if there is at least one member of coalition C wants to purchase any items in the package or when and .

lGd

PCCGd

lpk

lpk

lpk

lih Gdgd Cbi lh Gdgd




packages

printer CPU Monitor Memory pk1 pk2 pk3 pk4 pk5 pk6

{Bkk-A} √ √ √ √{Bkk-B} √ √ √ √ √ √{Bkk-C} √ √ √ √ √ √ √ √{Bkk-D} √ √ √ √ √ √ √ √{Bkk-A, Bkk-B}

√ √ √ √ √ √ √ √

{Bkk-B, Bkk-C}

√ √√ √ √ √ √ √ √ √ √

…

PCCGd



Step 3.3: , a discount that the coalition C get from forming to purchase the packages , will be calculated. Formally, when condition1) for and and condition2)

ClV

lpk

lihCl rsV Pr

jh

ih rsrs Cb j lh pkgd

Cbi




Vcl

v1 v2 v3 v4 v5 v6

{Bkk-A} 1000-2700 = -1700 -1700 0

{Bkk-B} -900 -1800 -100 -100

{Bkk-C} -900 -100 -200 0 -200

{Bkk-D} -800 -900 -100 0 -100

{Bkk-A, Bkk-B} 1000+900+900-2700=100

-800

{Bkk-B, Bkk-C} 800+1000+900-2700=0

…



Step 4: which give the maximum will be chosen.

clV



The set of potential coalitions PC vcl

Bkk-A 0

…

Bkk-A, Bkk-B 100

…

Bkk-A, Bkk-B, Bkk-C 200

Bkk-A, Bkk-B, Bkk-D 200

Bkk-A, Bkk-C, Bkk-D 300



Step 5: Update the items-vectors of all of the members of C” according to their contribution to the package-execution.



Buyer-Items

rs of Printer

rs of CPU rs of Monitor

rs of RAM

Bkk-A 1000

Bkk-B 900 900

Bkk-C 800 1000 800

Bkk-D 900 1000 900



Total discount . We repeat this process until all buyers in the group of buyers can purchase items that they require, like Table in next slide.

)(300 1pkVtot



Buyer-Items

rs of Printer

rs of CPU rs of Monitor

rs of RAM

Bkk-A 1000

Bkk-B 900 900

Bkk-C 800 1000 800

Bkk-D 900 1000 900



))(200())(200())(100()(0)(300 63611 pkpkpkpkpkVtot


III. Simulation


III. Simulation: Setup of Experiments

Table 3: summarizes the simulation parameters.


III. Simulation: Setup of Experiments (Cont.)

Parameter Range Range

Seller The number of sellers 1

Packages The number of packages

6

The number of single item

4

The number of bundles of items

2

The number of items necessary for each bundle of item

3

The average number of both item necessary for bundles of items and single item of the seller (AIS)

2

Buyers The number of buyers 4, 6, 8

Required items for each buyer

The average number of required items for each buyer (IB)

1, 2, 4


III. Simulation: Results and Analysis

The results of the simulation are divided into three categories:

1) the number of buyers is smaller than the number of packages

2) the number of buyers is equal to the number of packages, and

3) the number of buyers is greater than the number of packages.


III. Simulation: Results and Analysis: Fig. 1

-120

-100

-80

-60

-40

-20

0

0.5 1 1.5

IB/AIS

Mea

n of

the

tota

l dis

coun

t of

any

coal

ition

s in

the

Gro

upB

uyP

acka

ge s

chem

e

1

2

3


III. Simulation: Results and Analysis : Fig 1 (Cont.)

From Fig. 1, it is observed that the mean of the total discount of any coalition in the GroupBuyPackage scheme with the third category are higher than that in the GroupBuyPackage scheme with the first category and the second category.


III. Simulation: Results and Analysis: Fig. 2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.5 1 1.5

IB/AIS

The

ratio

of b

undl

e of

item

s on

e by

sin

gle

of it

em

1

2

3


III. Simulation: Results and Analysis: Fig. 2 (Cont.)

From Fig. 2, it is observed that the ratio of the number of formed bundles of items to one by single of item in the GroupBuyPackage scheme with the third category is higher than that in the GroupBuyPackage scheme with the first category and the second category.


III. Simulation: Results and Analysis (Cont.)

From both graphs, we conclude that the total discounts of any coalition in this schemes is high as the number of buyers were more than the number of packages (third category).

This is because the ratio of the number of formed bundles of items to one by single of item in the GroupBuyPackage scheme with the third category is higher than that in the GroupBuyPackage scheme with the first category and the second category.


III. Simulation: Results and Analysis (Cont.)

We denote the mean of the total discount of any coalition as TD and the difference between TD of the GroupBuyPackage scheme and TD of the optimal scheme as DIFF. The performance ratio of TD of the GroupBuyPackage scheme to DIFF is illustrated in Fig. 3.

The horizontal axis represents IB/AIS, while the vertical axis represents the ratio of TD of the GroupBuyPackage scheme to that by DIFF.

The value 1 means that the two schemes have the same performance; a value below 1 indicates that the optimal scheme is better; and a value above 1 shows the opposite.



0

0.2

0.4

0.6

0.8

1

1.2

0.5 1 1.5

IB/AIS

The

ratio

of T

D o

f the

G

roup

Buy

Pac

kage

sch

eme

one

by D

IFF 1

2

3



From Fig. 3, it is observed that the ratio of TD of the GroupBuyPackage scheme to one by DIFF is close to one on all values of IB/AIS.

In other words, the mean of the total discount of any coalition of the GroupBuyPackage scheme is close to that of the optimal scheme.





This algorithm is suitable for cases where buyers cooperate in order to maximize a total discount, especially when individual buyers cannot buy a whole bundle of items by themselves.

Nevertheless, they may get more discounts (or utilities) when the discounts from buying the items individually is lower than the discounts from purchasing bundles of items.


IV. Discussion and Conclusion (Cont.)

To guarantee the performance of this algorithm, we compare its results with those of the optimal algorithm.

In the simulation, main effective forming factors were IB, and AIS.

From Fig. 1-3, the total discount of any coalition in this algorithm is close to that in the optimal algorithm in almost all values of IB and AIS.


IV. Discussion and Conclusion (Cont.)

In our future research, pure bundling, whereby only the item bundle is offered and the individual items in the bundle cannot be purchased on their own, is considered in forming the buyer coalition.


Thank you for your participation

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Forming Buyer Coalitions with Bundles of Items