Supplier Order Allocation by hhu69322


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									                 Order allocation issues among
           multiple suppliers in B2B E-market places
                    Jishnu Hazra (
                 B. Mahadevan (
                 Sudhi Seshadri (

           Indian Institute of Management Bangalore
         Bannerghatta Road, Bangalore 560 076, INDIA

We analyze a problem where a market maker on behalf of buyers allocates a
given amount of order among N suppliers each with finite capacity.
Specifically, we analyze a situation in which suppliers have the option of
going to open market and selling their capacity at a market price. However, the
supplier will incur search costs. Moreover, the demand for the supplier
capacity in the open market is stochastic. Based on these, we derive the
capacity-price curve (supply curve for capacity) for each supplier. The
capacity-price curve of the suppliers provides a basis for the market maker to
allocate the order among the suppliers so as to minimize his cost. We also
identify the key parameters that will influence the overall performance of the
electronic market place.

                  Fifth National Conference of
             The Society of Operations Management.

                              December 2001
                              Mumbai, INDIA.

    Order allocation issues among multiple suppliers in B2B E-market places

1. Introduction

Electronic market places for Business-to-Business (B2B) applications take several

forms. Amongst these, “Aggregators” is one popular application. The market maker in

the case of an aggregator brings together fragmented supply and demand and helps

allocate supply to meet demand.

We study key strategic issues that vendors and the aggregator variety of market maker

face in the process of price discovery in the electronic B2B exchange. Specifically,

we analyze a situation in which suppliers have both options of the electronic exchange

and the open market, and address several questions. What would suppliers prefer?

What business environment aspects would influence their pricing and B2B exchange

participation? Is there an optimal number of vendors on such an exchange? The recent

experience of several B2B exchanges where critical numbers of vendors and buyers

have failed to come on board and have opted to use the open market instead makes

these important questions for the market maker to address.

Let us say vendors have the option to sell capacity in the open market; the transaction

costs are primarily due to search costs, uncertain demand, and predetermined price

levels. On the electronic B2B exchange transaction costs are different. Here the

market maker (MM) imposes a price discovery mechanism, such as a reverse auction.

The number of competitors is likely to be different since the e-marketplace has a

wider reach and may be able to attract non-local vendors as well. Transaction costs

involve the risks of non-selection due to competitive bids, and asymmetric

information on costs.

In this setting, the vendor’s level of capacity utilization will affect pricing behavior.

Under utilization will add to overhead costs being divided over smaller number of

units. This leads to a higher per unit cost as capacity is under utilized. Therefore,

direct and indirect costs attributed to manufacture will depend upon capacity

utilization. Search costs could be quite high, especially if the local business

environment is experiencing a slack period and local capacity utilization is low.

Search costs include capacity and inventory carrying costs that are necessary for

expected business, costs of marketing and sales, and general administrative costs

involved in customer acquisition. These can often add up to 10-15 percent of


Moreover, the demand for the supplier capacity in the open market is stochastic. The

random nature of demand imposes a burden of financial risk on vendors. If the

uncertainties are high, the transaction cost for risk averse bidders can be high due to

dedicated capacity carrying costs. Vendors cannot adjust capacity arbitrarily. There is

a maximum capacity. Buyers know this and will seek to cultivate multiple suppliers to

avoid problems of stock-out. Therefore, vendors will seek to sell as much capacity as

possible on the B2B exchange and the rest (if any is left over) on the open market,

rather than to put all their eggs in one basket.

Based on these considerations, we derive the capacity-price curve (supply curve for

capacity) for each supplier. The capacity-price curve of the suppliers would be the

basis for the market maker to allocate the order among the suppliers so as to

maximize his earnings.

Under special circumstances, there may not be any information asymmetry between

vendors and the MM on the price – capacity curve. The assumption here is that

capacity, search and manufacturing costs as functions of capacity, and demand faced

by vendor is common knowledge. When these are not common knowledge, the MM is

faced with information asymmetry problems and needs a reverse auction mechanism

to achieve price-capacity curve discovery. Often the auction will be unit contract to m

vendors, with a uniform award price. Here, m is the number of accepted vendors and

is based on considerations that will minimize the total procurement cost. n is

announced in advance as a RFQ, along with the reverse auction rules, such as sealed

bid, duration of the bid taking period, other terms of fulfillment.

We find that the number of vendors, m will influence price-capacity curves as well as

bid prices. The higher m is, the lower the price capacity curve for any given vendor,

since the unit contract is smaller. The sum of the unit contracts, assuming no scale

effects are important, leads to a lower overall price-capacity curve. However, greater

numbers of accepted bidders reduce the risk of selection and induce higher bids on

that account. Moreover, vendors with increasingly higher price-capacity curves will

need to be brought on board the program, and this will certainly increase price. The

net trade-off will yield an interior optimal number of bids, m*. The sensitivity of this

optimal solution to key parameters will influence the overall performance of the

electronic market place.

The rest of the paper is organized as follows. The next section briefly reviews the

relevant literature; Section 3 presents the formal model and analyses; and Section 4

concludes with a discussion of our key results.

2. Literature review

Elmaghraby (2000) reviews the trends in the sourcing literature and identifies a

central issue of multiple sourcing: the number of vendors to be selected. In particular,

the review identifies one of four key questions as being the endogenous determination

of the number of suppliers through an auction process. Elmaghraby also examines the

growth of electronic procurement markets over the Internet, and points out that

reduction in transaction costs over the internet would potentially affect the number of

relationships a buyer could engage in with vendors.

In an early paper, Seshadri et al. (1991) note that multiple sourcing increases the

probability of more bidders; which leads to the higher dedicated capacity critical to

insure the buyer against surges in demand. They identify the tradeoff between short-

term price increases in creating slack capacity in the supplier base and the long-term

advantages in supply assurance and price reductions. Our model builds upon this

insight by explicitly determining how slack capacity in the supplier base affects


There is a paucity of papers that incorporate the informational aspects of the sourcing

problem (Elmaghraby 2000) Most approaches treat the supply relationship as a single-

decision maker optimization issue, rather than an interactive vendor – buyer situation.

A supply chain, however, rarely is managed by a central planner (ibid.). Sourcing

strategies depend on market environments and buyers must adjust them for best

results; one size does not fit all situations Elmaghraby’s review identifies several

under-researched areas, among them costs, vendor controls on costs, asymmetry and

un-observability of costs, and supplier balance and variability across the supplier

base. Our model includes search costs along with manufacturing cost, and addresses

incentive compatibility issues that arise from unobservability of costs. We also

account for varying supplier capacity and some degree of stochastic differences across

the supplier base.

Peng-Sheng You (2000) deals with a sequential buying process, where purchase of a

certain number of units of an item at the lowest total purchasing cost must happen

within a given number of time periods. He develops a dynamic model of pricing

policies and search rules for the decision maker. In another sequential model, Gallien

and Wein (2000) examine information asymmetry and incentive compatibility in

industrial procurement under capacity constraints. Unlike our model, these

approaches allow the buyer to (i) allocate variable quantity to vendors using

information from (ii) multiple rounds of bidding.

Variable quantity allocations have a downside in repeated procurement situations.

Klotz and Chatterjee (1996) and Khai Sheang Lee (2000) study learning effects in

sequential multiple sourcing arrangements. Their models confirm that unequal splits

would create non-symmetric cost structures in future periods, due to asymmetric

learning, which may have detrimental effects on future competition. Therefore, we

develop our model assuming equal splits.

Fath and Sarvary (2001) argue with a model where B2B exchanges reduce search

costs that the size of the exchange is not zero or infinite, but finite and stable. They

find that sellers’ prices may not necessarily decrease with lower search costs; but

buyers’ surplus usually increases. They call for further research to explicitly model

the actual market mechanisms -- such as auctions, demand aggregation, etc. In a

recent paper, Vulcano et al. (2001) analyze multi-period auctions as a tool in revenue

management, and propose a list price, capacity-controlled mechanism for setting the

opportunity cost of capacity in a single award auction by a single seller. In contrast,

our model proposes a multi-award bidding competition among multiple vendors based

on endogenously derived price-capacity curves.

In sum, the cited literature demonstrates the need for an explicit incentive compatible

model of capacity constrained vendor participation in B2B exchanges, where the

optimal number of vendors may be endogenously determined.

3. Model Description

We state the assumptions in the model.

1. There are n suppliers who are competing for the job. The buyer will select the m

   suppliers at the price of (m+1) lowest supplier’s bid. All m suppliers will get an

   equal allocation of the job.

2. The total amount of capacity the buyer is seeking is denoted by Λ. We denote by

    λi the amount of capacity that will be allocated to the ith supplier.

3. Every supplier has an alternative option in the open market, however, the demand

   for her capacity is uncertain and sampled from a uniform distribution U(0,b)

   denoted by random variable X.

4. The supplier is capacitated and the ith supplier’s capacity is given by µ i . The

   variable cost of production is given by vi . We assume that µ i ≤ b.

5. To find buyers in the open market the supplier incurs a search cost that is

   dependent on the amount of capacity it wants to sell. The search cost is given by

   S(x), where x is amount of capacity it wants to sell in the open market. We assume

   that S(x) is increasing in x. In our model we assume S ( x ) = kx 2 , k ≥ 0 .

6. The open market price per unit of capacity is P.

7. The suppliers are identical in all respect except that they have different capacities.

Each supplier can sell her capacity in the open market at price P. However, because of

uncertainty in the open market she may not be able to sell her entire capacity. Due to

this risk of not being able to sell her entire capacity in the market she would be

willing to sell part of her capacity to MM at a lower price as long as her expected total

profit remains the same as the case when she offers her entire capacity to the open

market. Her expected profit if she offers her entire capacity in the open market is

given by

PE{min( X i , µ i )}− S i ( µ i ) − C i ( E{min( X i , µ i )}) .                            (1)

Here X is the random variable denoting the amount of capacity she is able sell in the

open market. The first term in the above expression is the revenue earned, the second

term is the search cost incurred and the final term is the production cost.

Now suppose the supplier offers λi units of capacity (subscript i refers to the ith

supplier) to MM, in which case she will offer µ i − λi units of capacity in the open

market. Her expected profit will therefore be

Pi (λi )λi + PE{min( X i , µ i − λi )}− S i ( µ i − λi ) − C i (λi + E{min( X i , µ i − λi )}) .   (2)

In expression (2), Pi (λi ) is the price the supplier quotes to MM. This price is a

function of units of capacity the MM procures from this supplier. The supplier will

commit λi units of capacity provided (2) is at least as large as (1). Therefore the

reserve price-capacity curve is given by:

            P{E{min( X i , µ i )}− E{min( X i , µ i − λi )} − {S i ( µ i ) − S i ( µ i − λi )}
Pi (λi ) =                                                                                     / λi (3)
           − {C i ( E{min( X i , µ i )}) − C i (λi + E{min( X i , µ i − λi )})}               

Lemma 1 If the search cost S(x) is increasing and convex and production cost is given

by C(x) = vx, v>0, then the unit price quoted by the supplier increases with capacity

committed to MM.

The proof is available in the Appendix.

The above lemma is valid for any probability distribution of demand for supplier’s

capacity in the open market. At first glance the result is counter-intuitive. One should

recall that the supplier does not have economies of scale in production. Therefore the

benefits of purchasing large units of capacity by MM do not result in production cost

savings per se but it reduces risk to the supplier. If MM purchases more capacity from

the supplier then the probability of having unsold capacity in the open market is low.

The supplier offers a lower price to MM because of the risk reduction. But the

supplier is also forgoing potential profit by selling to MM at a lower price and this

leads to reserve price increasing with capacity committed to MM.

Lemma 2 Assume that the demand for capacity in the open market for the supplier is

uniformly distributed (0,b), b>0. When S ( x) = kx 2 , C(x) = vx, k ≥ 0, v > 0 and µ i ≤ b

then the price quoted by the suppliers increases linearly with capacity. The price is

given by:

Pi ( λi ) = P − ( 2 µ i − λi )(       + k)                                                   (4)

The proof is available in the Appendix.

The Bidding Model

We will assume that the demand distribution for supplier’s capacity is given by

assumption 3 above. MM gets quote from participating suppliers in the form of price-

capacity curve. The suppliers, however, does not reveal the parameters such as unit

production cost, search cost, etc. MM will a priori announce the number of suppliers

that will be selected after he learns the number of participating suppliers in the fray.

The market maker will give equal allocation to the best m suppliers at the same

uniform price of (m+1)th lowest bid. This is the price of the first rejected supplier.

The reservation price of the supplier is a function of capacity, demand distribution,

production cost and the search cost parameter. We assume that all have access to the

same production technology in which case the unit production cost is identical for all

suppliers. The demand distribution for supplier’s capacity (which is uniform) is

independent and identically distributed and the search cost parameter is also identical

across all suppliers. The suppliers differ from each other only in terms of capacity.

Let Λ be the amount of capacity required by MM. If he selects m suppliers and

allocates his requirement equally amongst the selected suppliers then these suppliers

is allocated     units of capacity. As suppliers are identical in terms of production cost

and demand distribution the reserve price in terms of capacity allocated can be written


    Λ       Λ P − v           P − v      
Pi ( ) = P +        + k  − µi       + 2k                                  (4′)
    m       m  2b               b        

We have assumed vi = v; bi = b; λi =        in equation (4). From expression (4′) it is

clear that the large supplier will have a lower reserve price. As the mean demand (that

is, 0.5b) for supplier’s capacity increases, the reserve price also increases. This

happens because the risk of not being able to sell the entire capacity in the open

market reduces. When MM’s requirement increases (that is, Λ ) then again the

reserve increases.

We now analyze the problem from the market maker’s perspective. MM has full

complete information on unit production cost and parameters of the demand

distribution. However, MM does not have complete information on supplier’s

capacity but knows that it is sampled from a uniform distribution U ( µ l , µ h ) . As the

MM has to announce the number of suppliers that will be selected (that is, m), it will

do in a fashion that will minimize its total procurement costs.

MM’s problem can therefore be formulated as Minm >0 λE ( P[ m +1] ) , where we use the

notation for order statistics: P[1] ≤ P[2 ] ≤ P[3] ≤  ≤ P[ m ] ≤ P[m +1]  ≤ P[ n ] . P[ m +1] is the

lowest bid price amongst the rejected suppliers. From equation (4′), it can be shown

                          Λ P − v                    P − v
                             2b + k  − E (µ[n −m ] ) b + 2k  . Here again we use a
                                                              
that E ( P[m+1] ) = P +                               
                          m                                  

similar notation for order statistics for the random variable µ ,

µ [1] ≤ µ [2 ] ≤ l ≤ µ [m ] ≤ l µ [ n −1] ≤ µ[ n ] . Substituting E (µ [n − m ] ) = µ l +        (µ h − µ l )
                                                                                            n +1

in the above equation and simplifying we get an expression for expected bid price.

                   P−v      Λ           n−m               
E (P[ m+1] ) = P + 
                        + k  − 2  µ l +      ( µ h − µ l )  ,                               (6)
                    2b      m          n +1              

The market maker’s optimization problem stated above can now be solved by simple

calculus and the optimum value of m can be derived. We therefore have Lemma 3.

Lemma 3 The optimum number of suppliers that the MM will select is given by

        Λ (n + 1)
m∗ =                  .
       2( µ h − µ l )

Proof : See Appendix.

The optimum number of suppliers is dependent on three parameters which are total

capacity requirement of MM, number of participating suppliers and MM’s knowledge

of the range of supplier’s capacity. All three parameters are known to MM and are

independent of parameters that are supplier specific such as unit production cost and

search cost parameter. In the above expression the only estimation error by MM that

can occur are µ h and µ l . The value of the optimum number of suppliers is less

sensitive to these two parameters.

4. Conclusions

Electronic markets provide a new channel for the buyers and suppliers to match

demand with supply in the short run. While suppliers and buyers experience several

benefits of utilizing the electronic market place, the operating characteristics and

pricing strategies differ from traditional open market. Our model suggests that

suppliers would quote a higher price for committing higher levels of capacity in the

electronic market. Under certain conditions, MM could a priori announce the number

of suppliers to be selected for award of contract through a reverse auction mechanism

and obtain low costs for the items procured through the contract.

The proposed model could be extended to provide additional insights into the

problem. Replacing uniform distribution for demand and capacity variations among

the participating suppliers with alternative distributions will be a useful extension of

the model. Moreover, at this stage, we have assumed that only the maximum capacity

available with each supplier (µi) varies. However, in reality the manufacturing cost

parameter (v) also could vary. For instance, Asian suppliers will have a different cost

structure compared to that of European and US suppliers and could have a much

lower price – capacity threshold. Incorporating this aspect will allow us to assess the

impact of International supplier participation typical to an electronic market.


1. Gabor Fath and Miklos Sarvary (2001) “A model of B2B exchanges” Working

   Paper, Review of Marketing Science.

2. Khai Sheang Lee (2000) “Production cost, transaction cost, and outsourcing

   strategy,” working paper, Marketing Department, National University of


3. Klotz, D.R . and K. Chatterjee (1996), “Dual sourcing in repeated procurement

   competitions,” Management Science, 41, 8. 1317-27.

4. Seshadri, S., K. Chatterjee and G.L.Lilien (1991),”Multiple source procurement

   competitions,” Marketing Science, 10, 3. 246-63.

5. Elmaghraby, W. (2000), “Supply contract competition and sourcing policies,”

   Manufacturing & Service Operations Management, 2, 4. 350-71.

6. Peng-Sheng You, (2000) “Sequential buying policies,” European Journal of OR,

   120, 3. 535-544

7. J. Gallien and L.M. Wein ( 2000) “Design and analysis of a smart market for

   industrial procurement,” Working Paper 122, Center for eBusiness at MIT.

8. Vulcano, G., G. van Ryzin, and C. Magleras (2001), “Optimal dynamic auctions

   for revenue management,” Working Paper, Graduate School of Business,

   Columbia University.


Proof of Lemma 1: If X is a non-negative random with density function f(x) and

distribution function F(x), and ξ is a constant then it can be shown that

E{min(ξ , X )} = ξ (1 − F (ξ ) ) + ∫ xf ( x)dx                                      (A1)

                                                         ∂P (λi )
To prove Lemma 1 we need to show that                             ≥ 0 , where P(λi ) is given by

equation (3). We suppress the supplier’s index i for brevity.

           ( P − v){E{min( X , µ ) − min( X , µ − λ )} S ( µ ) − S ( µ − λ )
P (λ ) =                                              −                      . We consider the
                               λ                                 λ

first expression on the rhs. Ignoring the (P-v) which is a positive term (otherwise the

supplier will go out of business), it suffices to show that the function

           E{min( X , µ ) − min( X , µ − λ )
G (λ ) =                                     is an increasing function of λ . Substituting

(A1) in G (λ ) we get

                             µ                                          µ − λi
           µ{1 − F ( µ )} + ∫ xf ( x)dx − ( µ − λ ){1 − F ( µ − λ ) −     ∫ xf ( x)dx
G (λ ) =                     0                                            0

            µ{1 − F ( µ )} ∫
                               xf ( x)dx
∂G (λ )
        =−                 −0
 ∂λ              λ2              λ2
          λ{1 − F ( µ − λ )} − λ ( µ − λ ) f ( µ − λ ) + ( µ − λ ){1 − F ( µ − λ )
               µ −λ

                ∫ xf ( x)dx + λ ( µ − λ ) f ( µ − λ )
           +    0


After algebraic simplification we get the expression below:

                      µ                         µ −λ
                 − ∫ xf ( x)dx + µF ( µ ) +      ∫ xf ( x)dx − µF ( µ − λ )
∂G (λ )
        =             0                          0

 ∂λ                                            λ2
             µ                    µ −λ

             ∫ (µ - x)f(x)dx −        ∫ (µ − x) f ( x)dx
         =   0                         0
                                  λ   2

                                              S (µ ) − S (µ − λ )
The second expression on the rhs is H (λ ) = −                    . To show that this is
an increasing function we first need a theorem from Bazaraa and Shetty (1979).

Theorem (Bazaraa and Shetty, Chapter 3, page 91): let S be a non-empty open
convex set in En and let f : S → E1 be differentiable on S. Then f is convex if and
only if for any x ∈ S , we have
                  −           −            −
f ( x ) ≥ f ( x ) + ∇f ( x ) t ( x − x )            for each x ∈ S .                     (A3)

 ∂H      λS ' ( µ − λ ) − S ( µ ) + S ( µ − λ )
    =−                                          . As S(.) is increasing and convex then from
 ∂λ                       λ2
inequality (A3) it is straightforward to show that                > 0.
As both G (λ ) and H (λ ) are increasing, therefore P(λ ) is also increasing.

Proof of Lemma 2: Let us assume that f(x) is a Uniform (0,b) where b > 0 . Then
after simplification we get
                  2bξ − ξ 2
 E{min(ξ , X )} =           , 0 ≤ξ ≤b
               = ,      ξ >b                             (A4).

Substituting (A4) in (3) we get after simplification equation (4).

Proof of Lemma 4: MM minimizes her cost by minimizing the expected bid price.

The expected bid price is given by equation (9). The optimality condition is given by

dE ( P[ m +1] )                                      dE ( P[ m+1] )        Λ P−v     B
                  = 0. Thus we have                                   =−    2 
                                                                                  +k+     (µ h − µ l ) = 0 ,
     dm                                                  dm                m  2b    n +1


  P − v      
B=      + 2k .
    b        

                                        P − v     
                                       Λ      + k  (n + 1)
                                         2b                      Λ(n + 1)
Solving for optimum m we get m ∗ =                            =                  .
                                     P − v                      2( µ h − µ l )
                                      b + 2k  ( µ h − µ l )
                                               

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