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COMBINATORIAL NEGOTI-AUCTION WITH
ADAPTIVE BIDDING FOR INVENTORYCOST
MINIMIZATION
.
By
Avinash Tripathy (06IM3007)
Under the supervision of
Prof. Mamata Jenamani
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INTRODUCTION
Procurement is one of the major activities in the Manufacturing
Resource Planning (MRP II), which is closely coupled with
inventory management.
Any improvement in this area will have a direct impact on the
performance of the entire supply chain. Reverse auction mechanism has proved itself a successful
procurement method when there are several potential suppliers
available.
The large number of available suppliers and the price options
helps in securing the best procurement deal with the most cost-efficient suppliers.
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INTRODUCTION
Combinatorial auctions are pricing mechanisms in which a
bundle of different goods or services (items) is sold/bought
in one auction.
Combinatorial auctions are best suited for selling/buying
items that are complements or substitutes. When the items are substitutes, the bidder only wants one
of the items (not more).
When items are complements, the value of the whole
bundle is larger than the sum of the values of its
components separately. In reverse auctions, complementarities translate into
economies of scope in the sellers production process and
hence result in lower prices for combinatorial bid bundles.
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LITERATURE SURVEY
Combinatorial
auctions &Supply Chain
procurement
Choi and Han (2007); Na et al.(2005), Sven de
Vries,Rakesh Vohray(2000); Chen,R., Roundy,R., Zhang,R.,& Janakiraman,G. (2005).
Multi-Attribute
E-auctions,
Negoti-auctions
Teich et al. (2006); Biel & Wein (2003); A. Peke, M.
Rothkopf(2003); Gallien, J.Wein,L.M.(2005); A.M.
Kwasnica, et al. (2005) ;
Synergy of
production,
Economies of
scope.
Murray & White(1983); Chernomaz & Dan Levin (2007);
Bidding Policies/ Winner
Determination
Holte (2001);Regan et al. (2003); Mulleret al.(2006).
Auction
Inventory
Models
Ertogral & Wu (2000); Farahvash et al. (2008);
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LITERATURE SURVEY
Ertogral and Wu (2000) introduce an optimization method
that is implemented for a multi-level, multi-item
capacitated lot sizing problem. They construct an auction-
like mechanism to coordinate production planning for
multi-facility supply chain. Beil and Wein (2003) study the case when a single
manufacturer uses an open ascending, multi round, multi
attribute reverse auction for supplier selection.
Reverse auctions for cost minimization
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LITERATURE SURVEY
Teich et al. (1996) suggest that the auction should provide
support not only to the bid taker but the bidders. They propose a
negotiation based mechanism to provide decision support to the
bidders.
o In a combinatorial reverse auction setting Peke &Rothkopf(2003) propose a mechanism to suggest the right
combinations to the loosing bidders to increase their probability
of winning.
o Kwasnica, et al. (2005) solve the above problem by modeling the
situation as a knapsack problem.
Decision support for the bidders
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LITERATURE SURVEY
In the multi unit combinatorial auctions Leskel et al.
(2007) design a decision support systems for loosing
bidders. They suggest a new price to the bidder for
activating his bid. They do not give any suggestion on
adjusting the quantity based in accordance with otherinactive bidders.
Farahvash et al. (2008) propose winner determination
model that takes the inventory costs into consideration.
There model is for a multi unit single item auction.
We extend the work of Farahvash et al. (2008) to a multiitem inventory scenario by integrating the winner
determination problem with the buyers inventory cost
minimization problem. Through this model, following
Leskel et al. (2007), we suggest right price and quantity
combination to the loosing bidders in a multi unit
combinatorial reverse auction setting.
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COMPLEXITY OF WINNER DETERMINATION
PROBLEM IN COMBINATORIAL AUCTION
PROBLEM
The Combinatorial reverse auction can be formulated as set
covering problem ( Hohner et al. 2003, Narahari and Dayama
2005) which is NP-complete (Xia et al. 2004, Narahari and
Dayama 2005) meaning that a polynomial-time algorithm to findthe optimal allocation is unlikely ever to be found.
Therefore, for large problems it can be solved by heuristic
methods only.
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PROBLEM DEFINITION
In a typical case of Reverse auction we have the
following scenario.
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Buyer
requirements
Item Quantity
A 100
B 70
C 80
Supplier
Bids
Bidder A B C Price
Bidder
1
100 20 p1
Bidder
2
60 40 p2
Bidder
3
80 20 p3
In this case none of the bids satisfy the item
demand requirements individually. Whereas
bid combinations may satisfy the total buyer
requirement.
Reserved
Price
P
The need for Combinatorial Auctions
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Bidder A B C Price
Bidder 1 100 20 p2
Bidder 2 60 40
Bidder 3 80 20 p3
180 100 40 p2+p3
The alternatives of the bid combinations.
Bidder A B C Price
Bidder 1 100 20 p1
Bidder 2 60 40 p2
100 80 40 p1+p2
Excess inventory
10 units
cost.
Trade-off
between
profit gained
below the
reserved
price and
extra holding
cost.
Bidder A B C Price
Bidder 2 60 40 p2
Bidder 3 80 20 p3
80 80 40 p2+p3Shortage of 20
units
Excess inventory
10 units
Trade-off
between profit
gained below
the reserved
price and
extra holding
Excess inventory
30 units
Excess inventory
80 units
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OBJECTIVE
Efficient Bid Allocation for the procurer:
To provide the best combination of bids to the
procurer fulfilling his demand for multiple items.
Minimizing the inventory cost of the procurer. Providing better bids to the procurer with the
proceeding of the auction rounds by new
combinations of updated bids.
Support for the Bidders: Providing decision support to the bidders to
increase their chance of winning.
Exploring better options for bidders to win the
bid with maximum possible profit margin.
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OBJECTIVE
To develop an alternate heuristic i.e. Genetic algorithm to solve the problem
in case of large number of bid bundle combinations and bidders.
Exploring better options for bidders to win the bid with maximum possible
profit margin.
To propose a new adaptive bidding strategy in multi-round combinatorial
auctions.
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PROPOSED
METHODOLOGYFLOW CHART
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PROPOSED METHODOLOGY
Accepting bids from the bidders.
Winner determination.
Generation of a list of active bids.
Decision Support to the inactive bids.
Estimation of the cost function of the bidders
and updating the cost as rounds of bids
progresses.
Inclusion of the inventory cost of the procurerinto the decision support.
Price Support to the inactive bidders.
Quantity support to the inactive bidders.
Analysis of the various parameters involved.
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PROPOSED METHODOLOGY
Terminology
Active Bids: Bids which have been preferred over other bids as being
optimal. These bids satisfy the demand and price requirements as laid
by the procurer.
Inactive Bids: Bids which dont satisfy the demand requirements. But
with suitable combinations as suggested by the (quantity, price)
support with other bids may satisfy the procurers requirements.
Excess Inventory Cost: The cost incurred by the procurer due to
acceptance of a bid bundle which can be procured at a lesser price due
to complimentaries in suppliers production costs but overshoots
procurer s demand requirements.
Cost Estimate: The estimation of the cost function, total cost of the
bidder from his acceptance of rejection of a bid bundle suggestion.
Auction Rounds: The auction proceeds through rounds. In any round
one bidder can have only one active bid.
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WINNER DETERMINATION PROBLEM
The winner determination problem is formulated as an IntegerProgramming problem as follows:
procure the bids at minimum prices.
only one bid from a supplier is active
represents if bids on an item in
the bundle is satisfied.
implies the bid is activated.
The criteria of meeting the reservation price can also be an addedconstraint.
1 1
1 1
1 1
1 1, 2,..
. .
{0,1}
1
n ni
ij ij
i j
n ni
ij
i j
n ni
ij ik k
i j
ij
ij
Min x p
x i N
S t x q d
x
x
! !
! !
! !
e !
"
!
!
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WINNER DETERMINATION TAKING INTO
ACCOUNT THE INVENTORYCOSTS.
Where hk is the inventory holding cost associated, with the item k.
1 1 1
1 1
1 1
[ ] (4)
. . 1,...., (5)
1 1, 2,.. (6)
{0,1}
n ni K
ij ij ijk k
i j k
n ni
ij ijk k
i j
n ni
ij
i j
ij
in x p q h
S t x q d k K
x i
x
! ! !
! !
! !
u !
e !
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As per our requirements when the auction proceeds from round 1 toround 2 we desire to give a chance to the losing bidders to bring them
back into the auction.
For the above objective we require that the new set of bidders should be
mutually exclusive construct a new constraint as follows-
Initially at the start of the auction all the bidders have their bid status
indicator as zi=0. After the completion of the round 1 we do the following
assignment zi=xi-1.
After round two is over at least one of the bids which got reformed in the
round two should be included in the subsequent round three else the WDPwill give the same solution as round one.
So, we do the following operation
And put the additional bidder participation constraint as
PARTICIPATION OF INACTIVE BIDDERS
1
0 1, 2,.. (7)N
i i i
i
x z i N z is bid status indicator !
e !
1 1 1, 2..i i z x i N ! !
1
1 1, 2,.. (8)N
i i
i
x z i N!
e !
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CALCULATING SUGGESTED PRICE FOR A
NEW BID COMBINATION
In an iterative auction we seek to decrease total cost to buyer from the
current round (denoted C*) to the next.
In the beginning of the auction when there are no bids, the buyer's
reservation price can be used instead of total cost.
We ask the buyer in the beginning of an auction to specify a desired bid
decrement (>0) in total cost C* from one iteration (bid) to the next. And
put up a constraint that the new total cost of the selected buyers in around should be less than the updated C* for that round.
Calculate new reservation price as Reservation Price=Reservation Price-
.
R represents the new updated reservation price, xi the current bid status
of the bidder, pi and pi the new suggested price and the old price of the
bidder respectively.
Cut off in cost Updated Reservation Price Total Combined Bid price
=R'- (9)i ii I
p x p
where I represents selected bids
!
( '(10)
i
i i
i i
i I
p
p p px p
! v (
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SUGGESTED QUANTITY FOR A NEW BID
COMBINATION
In addition to the price, it may happen that a bid combination may not be
able to outbid the previous winning bid due to an excess inventory
resulting in additional costs for the buyer.
These inventory costs if minimized may reduce the total cost of
procurement.
To mark out the items in which there is an excess inventory.
We construct a matrix for each of the individual items Yki and allocate
value to it as
If there is an inventory excess or shortage in item k then for the
corresponding bidders who have requested price and quantity support weassign
The overall excess inventory in each item is then found as
1, 2..ki iY x k K ! !
2kiY !
(11)
' . . 2
k ik
i A
ki
EI q
A set of all bidder i s for s t Y
!
!
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The new quantity support for the bidder is calculated as follows
We use the dual prices of the linear relaxation of the WDP as proxies for
the cost coefficients in a linear approximation of the cost function, which
would take the form
Where k's are the dual prices of the corresponding demand constraints.
After the total cost of the new suggested bid quantities is determined we
calculate the approximated total cost involved in the combined bids using
the dual prices and determine the overall profit as the difference between
the updated reservation price for that round and the total approximatedcosts.
Tis the approximate profit to be shared.
ci and pi are the approximated costs and new suggested price respectively.
' (12)ik
ik ik k i A
ik
i A
q
q q EI q
! v
,
1
( ) ' (13)K
i i new k ik
k
c Q qQ!
!
' (14)ii I
R cT
! ' (15)ii ii
i I
cp c
c
T
! v
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The information on the costs, becomes more accurate as
auction progresses into multiple rounds i.e. the more bids
there are from the bidders in the bid stream, the moreaccurate the estimate.
The Cost Estimation table is gradually updated with each
round. With the fixed costs Fis and the variable cost cis.
The profits can be shared among the inactive bidders in a
bid combination. This shall encourage them to accept a bid
to overtake a winning active bidder.
F1 F2 F3 F12 F23 F31 F123
c1 c2 c3
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EXAMPLE ON AUCTION PROCEEDINGS
Procurer
Requirement
600 600 600
bidder item1 item2 item3 price
1 0 260 205 28,435
2 150 0 260 25,761
3 205 370 370 52,273
4 0 260 0 18,657
5 0 0 205 15,622
6 425 260 205 49,675
7 315 0 0 22,560
8 0 315 370 40,655
9 370 370 0 43,743
10 205 205 150 33,937
11 0 0 260 18,222
12 315 0 315 38,060
13 0 370 0 25,646
14 0 0 425 28,017
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FORMULATION OF WINNER DETERMINATION
PROBLEM
n: total no. of items
Let G denotes the set of n no. of items
S: Bundles of items and SG
The total no. of possible bundles is 2n 1.
N: total no. of suppliers
bi(S) : a bid for bundle S from ith supplier.
xi(S) : 1 when bid of ith supplier for bundle S is selected else
0.
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GENETICALGORITHM ENCODING
A brief discussion is now presented on how the
chromosomes are encoded, and how the other operators of
GA namely selection, crossover and mutation are done in
the context of the present problem.
1= the bidder bids for the specified item.
0= the bidder doesnt bid for the specified item.
Supplie
r #
A B AB BC AC
1 1 0 0 1 0
2 0 1 1 0 1
3 0 1 0 0 1
4 0 0 1 1 0
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The Chromosome Encoding
All the item state configurations from the suppliers are lined up to
form the chromosome.
0 1 1 0 0 1 0 1 0 0 1 0
A B A
B
A B A
B
A B A
B
A B A
B
Bidder 1 Bidder 2 Bidder 3 Bidder 4
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REPAIRING THE SOLUTIONS
Random generation may produce infeasible solutions.
1) If sum of the product instances is equal to or more than 1, then preserve the
product order configuration.
2) If sum of the product instances is equal to 0, then a random number (r)
between 1 and N (No. of suppliers) is generated and for that supplier the
product state of that particular product is changed to one.
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REPAIRING THE SOLUTIONS1 0 0 0 0 1 0 0 1
Here product A and C occur in one or more instances among the suppliers, but
product B didnt occur at all.
So a random no. is generated between 1-N. (N is the number of suppliers)
Let the random number generated is 3 (r=3)This means that product B has to be ordered from Supplier 3.
That means the product order configuration of supplier 3 has to be changed from
001 to 011.
1 0 0 0 0 0 0 1 1
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CROSS OVER1 0 0 0 0 0 0 1 0
0 1 0 1 0 0 0 0 0
0 1 0 0 0 0 0 1 0
1 0 0 1 0 0 0 0 0
Select two parents from the
population. Here the crossover
represents crossing over the product
order configuration of a randomly
selected supplier (in this case
supplier no. 1)
The crossover part is represented in
blue and yellow colors
Parents
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MUTATION0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0
Mutation is done by interchanging product states between any two suppliers of
the same chromosome. The solutions are then repaired and they are now
added in the population.
0 1 0 1 0 0 0 0 0
Mutation Bits
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FITNESS FUNCTION
Minimize:
Procurement Cost + Demand Overshoot Penalty
Demand Overshoot Penalty =
di is the demand of the buyer for item i.
ai is the sum of items from the winning bidders for item i.
n is the total number of items to be procured.
The value of M (i.e. 10000) is kept very large to prevent all the solutionsfor which the combined procurement from bidders is more than thedemand.
A safety margin of 50% within the demand is observed to prevent sub
optimal solutions from being discarded.
Only those solutions are accepted into the mating pool for which thedemand for all the products is met.
1
( 1.5 ) *n
i i
i
a d!
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The profit margin used by the bidder adopting this kind of strategy can be
adjusted constantly according to bidding histories, and finally approaches to
the optimal profit margin in the current market environment.
Reference record of a bid bfor a bidder for a bidder i is(pmb, loseb, winb ).
pmb is the profit margin for bid b, loseb is the number of rounds the bidder
keeps on bidding before bid b is won or dropped and winb is a integer of 0 or 1
denoting whether this bid is won or dropped.
The minimum value of loseb is 0, when the bidder wins the requested resource
bundle at the first round after he submits it.
A Bidding History of a bidder, denoted as bhis the sequence of recent k
reference records.
ADAPTIVE BIDDING
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A consistent bidding history of a bidder, denoted as cbh, is a bidding history
in which all reference records share the same profit margin.
The expected utility function of bidder i on a consistent bidding history cbh,
denoted as
wherepmcbh is the common profit margin used in this consistent
bidding history, and waitb and winb are the same as in the definition of
reference record.
( )( )
b
b
rr cbh b
ex cbh
rr cbh b b
inu cbh pm
in ait
! v
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Every time when a new consistent bidding history is formed, the profit
margin used by the bidder is increased or decreased according to thebidders 1st and 2nd most recent consistent bidding histories.
The new profit margin is used by the bidder when he bids in subsequent
rounds until the next consistent bidding history is formed.
The increase and decrease of the profit margin as a positive andnegative adjustment respectively, and use a -1 or 1 variable to indicate
the previous adjustment of the profit margin: if = 1, then the previous
adjustment is positive, otherwise negative.
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pm = , step = , = 1 and u = 0.
while auction does not finish do
Use profit margin ofpm to bid for the currentround
if a new consistent bidding history cbh is formed
Compute uex(cbh).
ifu < u then
pm =pm step
else ifu u thenpm =pm + step
end if
ifpm >pm then
= 1
else ifpm
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COST MODEL FOR THE BIDDER WITH
SYNERGY
The cost for the bidder consists of fixed (Fi)andvariable cost(ci) for an item i.
The fixed cost of combination of items is greater thanthat for a single item.
F12>F1; F12>F1 The combined fixed cost is less that the sum of
individual fixed cost.
F1+F2>F12 ;
F12+F3>F123 ;
All bidders are assumed to be glocal meaning thatthey have production capacity for all three items butthey are willing to settle for any item and unitcombination as long as it is profitable.
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REFERENCES Na A.,Wehad E., Pinar K. (2005) Bidding Strategies and their impact on
revenues. Revenues and pricing management.
Choi J.& Han I. (2007). Combinatorial auction based collaborative procurement.
omputer Information Systems.
Farahvash P. & Altiok T. (2008) Application of multi-dimensional procurement
auction in single-period inventory models.A
nn Operations Research 164: 229251
Leskel R., Teich J., Wallenius H., Wallenius J. (2007) Decision support for
multi-unit combinatorial bundle auctions. Decision Support Systems (Vol.43)
420434.
Vries S.& Vohray R. (2000); Combinatorial Auctions a Survey.
Hohner G. , Rich J. , Ng E., Reid G., Davenport A.J., Kalaganam V., Lee H.,An C., Combinatorial and quantity discount procurement auctions benefit
Mars, Incorporated and its suppliers. Interfaces 33 (2003) 2335.
Ertogral, K., & Wu, S. D. (2000). Auction-theoretic coordination of production
planning in the supply chain. IIETransaction, (32), 931940.
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