Multi-unit Combinatorial Reverse Auctions with Transformability Relationships among Goods

1. Multi-unit Combinatorial Reverse Auctions with Transformability Relationships among Goods Andrea Giovannucci Juan A. Rodríguez-Aguilar Jesús Cerquides Institut d’Investigació en Intel.ligència Artificial Consejo Superior de Investigaciones Científcias TFG-MARA. Budapest 16-11-2005

2. Agenda Motivations & Goals Modeling Transformation Relationships The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

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. 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 Example. Parts purchasing FRONT SUSPENSION, FRONT WHEEL BEARING ACQUISITION GOAL: BUY PARTS TO PRODUCE 200 SUSPENSIONS TRANSFORMATION COST: 90$/UNIT

5. Motivations: WDP and Transformability Relationships RFQ 200 Suspensions 2 3 OFFERS Transformation Cost 90 $ 4 2 1 ALLOCATION PROVIDER 2 PROVIDER 1 600 $ 100 5000 $ 100 * 90$ = 9000$ 400 100

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. Agenda Motivation & Goals Modeling Transformation The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

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. 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. Example of TNS

12. Modeling a Transformation 400$+90$=490$ • The activation of transformations is modeled as firing of transitions Item 3 1 Item 1 Item 2 Item 3 -2 -1 1 2 1 0 0 0 1 90$ *1 = + 2 1 400$ M0 + T x = M’ Item 1 Item 2 Sufficient Condition: ACYCLIC PETRI NET

13. Agenda Motivation & Goals Modeling Transformation The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

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:

15. Extending the Multi-Dimensional Knapsack Problem • We extend this model considering that we can transform some of the items bought M0 + T x

16. Acyclic Petri Nets ACYCLIC

17. Agenda Motivation & Goals Modeling Transformation The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

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. Agenda Motivation & Goals Modeling Transformation The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

26. e-CATS Demo

27. Agenda Motivation & Goals Modeling Transformation The Winner Determination Problem Empirical Evaluation Demo Conclusions and Future Work

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. 400$*2 + 300$ +90$ = 1190$ 1 90$ 2 1 400$/unit 300$/unit