rial Negoti-Auction With Inventory Cost Minimization

8/9/2019 rial Negoti-Auction With Inventory Cost Minimization

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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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rial Negoti-Auction With Inventory Cost Minimization

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