Multi-unit Combinatorial Reverse Auctions with

Transformability Relationships among Goods

Andrea GiovannucciJuan A. Rodríguez-Aguilar

Jesús Cerquides

TFG-MARA. Budapest 16-11-2005

Institut d’Investigació en Intel.ligència Artificial

Consejo Superior de Investigaciones Científcias

2

Motivations & Goals

Modeling Transformation Relationships

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

3

Motivation

Combinatorial Auctions have recently deserved much attention in the literature.

The literature has considered the possibility to express relationships among assets on the bidder side (as complementarity and substitutability).

The impact of eventual relationships among different assets on the bid-taker side has not been addressed so far: a bid-taker may desire to express transformability relationships among the goods at auction.

4

Example. Parts purchasingFRONT SUSPENSION, FRONT WHEEL BEARING ACQUISITION

PART NUMBER

DESCRIPTION UNITS

1 FRONT HUB 2

7 LOWER CONTROL ARM BUSHINGS

3

8 STRUT 4

9 COIL SPRING 2

14 STABILIZER BAR 1

GOAL: BUY PARTS TO PRODUCE 200 SUSPENSIONS

TRANSFORMATION COST: 90$/UNIT

5

Motivations: WDP and Transformability RelationshipsSUSPENSION

FRONT HUB

LOWER CONTROL ARM BUSHINGS

STRUT

COIL SPRING

STABILIZER BAR

Transformation Cost90 $

200 Suspensions

2

3

4

2

RFQ

OFFERS

ALLOCATION

1

100 5000 $

100

400

600 $

PROVIDER 1 PROVIDER 2

100 * 90$ =

9000$

6

Motivations

Thus the buyer/auctioneer faces a decision problem:• Shall he buy the required components to assemble them in house

into suspensions?• Or buy already-assembled motherboards?• Or maybe opt for a mixed-purchase solution?

This concern is reasonable since the cost of components plus the assembly costs may be eventually higher than the cost of already assembled suspensions.

7

Goals

The Buyer requires a combinatorial auction mechanism that provides:

• A language to express required goods along relationships that hold among them.

• A winner determination solver that not only assesses what goods to buy and to whom, but also the transformations to apply to such goods in order to obtain the initially required ones.

8

MUCRAtR

We extend the notion of RFQ (Request-For-Quotation) to allow for the introduction of transformation relationships

(t-relationships) We extend the formalization of the well known Multi Unit

Combinatorial Reverse Auction Winner Determination Problem to introduce transformability.

We provide a mapping of our formal model to integer programming that assesses the winning set of bids along with the transformations to apply.

9

Motivation & Goals

Modeling Transformation

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

10

Modeling the t-relationships

We need a model that expresses different configurations of goods, and the possibility of switching among them at a certain cost.

PETRI NETS is the model that best fits the requirements

11

Example of TNS Transformability Network Structure

(TNS)• Places represent the goods at auction.• Transitions represent t-relationships.• Arcs indicate how goods are related

through transformations.• Arc weights stand for the number of

goods either produced or consumed by a transformation.

• Each t-relationship is labeled with a transformation cost.

12

Modeling a Transformation

The activation of transformations is modeled as firing of transitions

2 1

190$

400$

400$+90$=490$

2

1

0

-2

-1

1

0

0

1

* 1+ =

M0 + T x = M’Sufficient Condition:

ACYCLIC PETRI NET

Item 1 Item 2

Item 3

Item 1

Item 2

Item 3

13

Motivation & Goals

Modeling Transformation

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

14

The Multi-Dimensional Knapsack Problem

It is a well known result in optimization theory that the winner determination problem in a multi-item multi unit combinatorial auction can be modeled as a MDKP:

BIDSjiijjITEMSi

BIDSjjj

uby

py

,

min

15

Extending the Multi-Dimensional Knapsack Problem

We extend this model considering that we can transform some of the items bought

BIDSji

tionsTransformakkkiijjITEMSi

tionsTransformakkk

BIDSjjj

uqtby

cqpy

,,

min

M0 + T x

16

Acyclic Petri Nets

ACYCLIC

17

Motivation & Goals

Modeling Transformation

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

18

Empirical Evaluation

In our preliminary experiments we compared the impact of introducing transformation relationships analyzing two main aspects:

• The added computational complexity.• The potential variation in the auctioneer cost.

With this aim we compared the new mechanism to a state-of-the-art combinatorial auction winner determination solver in terms of:

• CPU time• Auctioneer cost

19

Experimental Setting

We employed a modified version of a state-of-the-art multi-unit combinatorial bids generator (Leyton-Brown).

In these early experiments the only variable was the number of bids, whereas we fixed:

• Price distribution - Normal with variance 0.1• Number of items - 20• Number of t-relationships - 8• Maximum cardinality of an offer – 15

The number of bids ranged from 50 to 270000

20

Hardware Setting

Pentium IV, 3.1 GhZ. 1 Gb RAM. OS Windows XP Professional. MATLAB release 14.1 (To create the test set). ILOG OPL Studio and CPLEX 9.0. (Commercial

Optimization Library, www.ilog.com)

21

Experimental Results: Computational Hardness

22

Experimental Results: Scalability

23

Experimental Results: Auctioneer Cost

24

Experimental Results: Costs’ Ratios

Cost without Transformations

Cost without Transformations

25

Motivation & Goals

Modeling Transformation

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

26

e-CATS Demo

27

Motivation & Goals

Modeling Transformation

The Winner Determination Problem

Empirical Evaluation

Demo

Conclusions and Future Work

Agenda

28

Conclusions: pros

No significant burden in the computational complexity is added introducing transformations.

We experimented revenue savings ranging from 3% to 30% (Although we have to further study the variables that affect the phenomenon).

Competence among bidders is increased Providers of components vs. Providers of suspensions.

Efficiency is increased

29

Conclusions: cons

Bidding is more difficult.

The auctioneer has to reveal private information about his internal production process.

30

Conclusions

We presented a new type of combinatorial auction in which it is possible to express transformability relationships on the auctioneer side.

To the best of our knowledge it is the first system that introduces this type of information into a combinatorial auction.

We studied the associated winner determination problem providing an integer programming solution to it.

We empirically evaluated it comparing with a state-of-the-art solver:

• The scalability.• The difference in the auctioneer revenue.

31

Future Work

Design and analysis of the auction mechanism. Decision support to bidders to elaborate winning bids. Theoretical analysis of the auctioneer’s cost of our

mechanism with respect to multi-unit combinatorial auctions.

Extending the model in order to support combinatorial offers over range of units.

32

Thank you ... Any question?

33

Backup/Extra Slides

34

Electronic Negotiation

Negotiation of bundles of items

35

Conclusions on Experiments

Auctioneer revenues increased by 10 % to 30 % in medium-small scenarios (< 200 bids).

Solving times of around 0.3 sec. in middle-large scenarios (2500 bids).

Largest instance solved: 270000 bids.

36

Experimental Setting

An important consideration is that when transformation relationships hold among goods, the price distribution must take them into account.

2 1

190$

400$/unit 300$/unit

400$*2 + 300$ +90$ = 1190$