Mythri A (mythri@csa.iisc.ernet.in)
Sandhya G (sandhya@csa.iisc.ernet.in)
Auction
Auction can be defined as a market institution with explicit
rules determining resource allocation and payments depending on the bids offered
from the market participants. In an auction we will have a set of traders who
would like to sell the product and set of traders who would like to buy the
product. One set of traders will place the bids and other set can choose from
the bids and payment is made depending on the rules.
Auctions can be classified into various types depending on
the rules followed and number of buyers and sellers involved and number of
items involved. There can be single unit auctions, multiple units of same
product called multi unit auctions or auctions for multiple products called
combinatorial auctions.
Auctions reduce the complexity of negotiations because there
are fixed set of rules to be followed; they are ideal for the implementation in
computer. Online auctions have an advantage that many traders can participate.
Internet may make auctions more secure as bidders may come from all over the
world and the algorithm can be designed not to reveal buyers and suppliers
private information. Rapid advances in computational capability make it
possible to conduct sophisticated allocation rules quickly and at low cost.
Auctions can be classified into various types depending on number of buyers and
sellers and the rules used in the auction. There can be also one seller and a
set of buyers or vice versa. The techniques used to deal them are the same.
To participate in the auction, each supplier (buyer) is
required to submit a bid to an auctioneer (also called a market intermediary or
an agent). Based on these bids, the quantities to purchase (sell) from, and the
payment to each supplier (buyer), are determined by the auctioneer through
pre-specified rules known as a mechanism. A mechanism is incentive compatible
(or induces truth-telling) if each supplier's dominant strategy is to submit
her true production cost as her bid. Incentive compatibility is important
because it is usually a requirement for an auction to be efficient. An
efficient mechanism is one that guarantees an allocation of production
quantities that minimizes the total system cost.
There is a recent emphasis on using auctions in the supply
chain as an effective and efficient means of achieving lower acquisition costs,
lower barriers for new suppliers to enter a market, and consequently better
market efficiency. Exploiting recent advances in information technology, such
auctions can be carried out through the Internet, referred to as online
auctions. Online auctions allow geographically diverse buyers and sellers to
exchange goods, services, and information, and to dynamically determine prices
that reflect the demand and supply at a certain point of time so that efficient
matches of supply and demand can be realized.
Procurement
Auction
In Procurement Auctions there will be a single buyer who
wants to procure some product and a set of sellers who are interested in
selling the product. The buyer might require a single product or multiple
products. The auction, which we use, will support the requirement of buyer for
multiple units.
There are three dominant types of mechanisms for multi-unit
auctions, known as Pay as-You-Bid, uniform-price, and VCG auctions (e.g.
second-price auctions in the single-unit setting, also known as Vickrey auctions) (Klemperer
1999). The Pay-as-you-Bid auction is self-explanatory. In a uniform auction a
uniform price is paid for each unit purchased. The price can be either the
first rejected or the last accepted bid. In economics literature it is well
known that neither Pay-as-You-Bid nor uniform-price auctions is incentive
compatible or efficient, whereas the family of mechanisms known as VCG
auctions, attributed to Vickrey (1961), Clarke
(1971), and Groves (1973), are both incentive compatible and efficient. In a
VCG auction, the buyers' payment to a supplier is based not only on the bids
submitted, but also on the contribution that the supplier makes to the system
by participating in the auction.
Though
VCG auctions are incentive compatible, VCG auctions are rarely used because of
possible cheating by the auctioneer. In an online auction, however, the
auctioneer can be a web site (virtual computer agent), which receives bids, and
decides awards and payments based on some pre specified algorithm so that this
problem can be minimized or eliminated. Furthermore, the Internet may make auctions
more secure as bidders may come from all over the world and the algorithm can
be designed not to reveal buyers and suppliers private information. Rapid
advances in computational capability make it possible to conduct sophisticated
allocation rules quickly and at low cost. With the recent advances in
information technology, VCG auctions have the potential to become an important
mechanism for online auctions.
Procurement
Auction Model
Most auctions are price-driven. However, there are other
costs associated with integrating a supply chain, e.g., costs associated with
transportation, capacity management, inventories, etc. The model described
takes into consideration the transportation costs involved in the shipment of
the products.
In the model described in the paper[1]
a buyer can have set of distribution sites. Each buyer has his own private
values called consumption quantities. Consumption quantities at each site form
the consumption vector. This model also considers the transportation costs from
the source to destination. It tries to minimize the overall cost i.e. the
production cost and the consumption cost together. Each supplier has a
production cost, which can be described as a convex function of the quantities
that she produces at her production locations. Such a cost can already include
a reasonable profit margin of the industry. It also assumes that each supplier
is a rational, self-interest player who is trying to maximize her own profit
(i.e. the payment received minus her production cost). With general convex
production costs, one cannot simply combine the production and transportation
costs and treat them as a single cost function. In addition, the buyer pays the
transportation costs associated with every shipment from a supplier location to
a buyer location. We assume that the buyer knows the per-unit transportation
cost along each arc of the network.
The payment to be made by the buyer is determined by the Vickrey strategy. The allocations are made with all the
sellers. Once an allocation is made, each winner is removed from the scene and
the allocation problem is solved again with the remaining sellers. The
difference between the prices in the two cases is the contribution made by the
winner to the market and is given as an incentive to him. The use of Vickrey strategy ensures incentive compatibility and allocative efficiency.
N = total number of supplier production facilities
K = number of suppliers
M = total number of buyer locations
Nk = set of production
facilities owned by supplier k
kn = index for the supplier that owns production facility n
qm = consumption at demand center m, qm
> 0
xn = production quantity at production facility n
ynm = quantity shipped from production facility n to demand
center m
zkm = S Nk ynm,
total quantity shipped to demand center m by supplier k
Ck(xk) = production cost function for supplier k (R|Nk|® R)
Fk(xk) =
bidding function from supplier k (R|Nk|
® R)
Tnm = cost for shipping one unit from production facility n to
demand center m
Boldfaced letters represent vectors or matrices.
In Auction T the buyer submits a fixed consumption vector q
to the auctioneer. Supplier k submits to the auctioneer a bid function Fk(xk) for supplying xk
units, for which she incurs a production cost Ck(xk),
xk R|Nk|.
The suppliers may or may not see the consumption vector. The auctioneer will
decide the quantities awarded to each of the suppliers, the amount transported
from each supplier location to each of the buyer locations, and the payments
made by the buyer to the sellers. This is not an iterative process and only a
single round is involved. This auction can handle multiple units of the
products. The bidding used is closed bidding. The sellers do not know what
others have bid. They might or might not know the consumption vector of the
buyer. Since the payment strategy used is Vickrey
which is incentive compatible we can assume all the sellers would bid their
true values.
Under Auction T, the auctioneer will minimize the sum of the
accepted bids and the transportation costs, for a given consumption vector q,
as Min. Sk( Fk(xk) + Sn Sm
tnm * ynm
Subject to
S n ynm = qm, m= 1, . . .,M
Sm. ynm = xn, n = 1, . . .,N
ynm > 0, m= 1, . . .,M, n = 1, . . . , N.
Let P(q)
be the optimal value of the objective function for a given q. Define Q = {q : q > 0, P(q) < infinity }
and we restrict qÎ
Q to ensure sufficient supply capacity. Since Fk(·) is closed, for any q ÎQ, an optimal solution exists.
The objective function states that we want to minimize the
total payment to be made by the buyer. This includes both the transportation
costs and the production costs. Fk(xk) denotes the bid function
provided by the seller. It represents the amount to be paid if the supplier
supplies xk amount of goods. ynm denotes the amount transferred from n
to m and tnm represents the transportation
cost of the same. So when we add both production and transportation costs we
will get the required objective function.
The first constraint says that amount supplied from all the
production sites to each consumption site should be equal to the amount
required at the consumption site.
The second constraint states quantities transported from
each site to all other sites should sum up to the quantity produced at that
site.
The payment is based on Vickrey
strategy, so the winners would pay the second highest price. To calculate this
price we should solve n linear programming problems if the number of
winners is n. For each winner we will solve the above problem with
additional constraint that Xk=0, to get
the price buyer should have paid if the seller was not present in the auction.
Let (xT , yT ) be
an optimal solution, and P(q)
be the optimal value of the objective function with the additional constraint xk = 0 (i.e., supplier k does not participate in the
auction).
The buyer will pay supplier
Y T k (q) = P-k(q) - Pk(q) + Fk(xTk )
where P-k(q) - Pk(q) is the bonus
payment made to supplier k, representing the value she adds to the system by
participating in the auction. The buyer pays supplier k her bid Fk(xTk ) plus her contribution to the system.
This payment scheme belongs to the general truth-inducing
VCG family described in Nisan and Ronen (2001).
Consequently, rational suppliers will bid their costs, Fk(xk)
= Ck(xk), irrespective of other suppliers bids.
Therefore, Auction T is incentive compatible for all suppliers.
Other
variations of the Procurement Model
Auction
R
In
this auction, the buyer submits a utility function (a proxy for her true
consumption utility function) to the auctioneer, rather than a fixed
consumption vector. The auctioneer treats this as if it were the profit
(excluding acquisition costs) that the buyer makes by consuming q, a
consumption vector, and considers it when making production and transportation
decisions. The quantities that the buyer will be awarded and
her payments are the outputs of the auction. Under the VCG payment structure, Auction R is incentive
compatible for the suppliers, but not for the buyer. The auction mechanism meets
the consumption vector q at minimum production and transportation cost and
hence, is efficient for providing q. It also typically results in a consumption
vector q that fails to optimize the total supply chain. Under Auction R, we
show that the buyer’s payments are always less than or equal to those in
Auction T for providing the same level of consumption.
Auction
S
In
this the buyer submits a fixed consumption vector to the auctioneer who will
decide the production quantities at all supplier locations and the buyer’s
payments to the suppliers by minimizing the total production cost in the
system. Transportation decisions are subsequently made to match the demand and
supply at the lowest total transportation cost. It is obvious that Auction T
achieves a lower total supply chain cost than Auction S. However, the buyer may
prefer Auction S under certain circumstances. It is seen that typical
regular-overtime production cost structure can lead to higher payments in efficient
auctions to distort buyer behavior.
References
Efficient
Auction Mechanism For Supply Chain Management Rachel R. Chen, Ganesh
Janakiraman Robin Roundy, and Rachel Q. Zhang
Vickrey, W., 1961, Counterspeculation,
Auctions, and Competitive Sealed Tenders , Journal of Finance, 16, 8-37.