A Multi-Sourcing Model from a Cost Heterogeneous Supply Base Jishnu Hazra

Department of Operations Management, Indian Institute of Management, Bangalore, India

Abstract: In this paper we analyze a scenario where a buyer procures capacity from one or more suppliers in the presence of demand uncertainty. The buyer does not have information of suppliers’ production cost function but assumes it is sampled from a known probability distribution. The supplier, of course, has knowledge of its own cost function but like the buyer does not have information on other suppliers’ cost function. Under this scenario we explicitly derive suppliers’ capacity reservation price, which is a function of their capacity, amount of capacity reserved by the buyer and other parameters. The buyer’s decisions are how much capacity to reserve and from how many suppliers. Under certain conditions we develop solutions and show that the model is robust to the number of suppliers from whom capacity is procured through reservation. When the parameters of demand distribution changes the supply base is likely to remain more or less the same.

Keywords – multi-sourcing, capacity, contracts

I. INTRODUCTION

Rapid globalization of markets aided by advances in the Internet and electronic markets has in recent times thrown new opportunities for Supply Chain Management (SCM). Several studies predicted that electronic markets will play a significant role in redefining buyer – supplier relationships [1]. A variety of Business-to-Business (B2B) market places are available today and organizations could put a combination of these to use through an E-procurement plan [2]. E-procurement offers several new advantages to industrial buyers. Many of them stem from the ability to reach a wider set of suppliers than before. For instance, a registered user of Alibaba.com gains access to several thousand trade leads in 27 industry categories and 900 product categories posted by potential buyers and suppliers. The flip side to the increased globalization of the supply chain is the new set of challenges that firms face today. Consider the following trends in SCM. A recent study by The Economist Intelligence Unit based on a world-wide survey of 1,656 executives belonging to the

auto component sector suggested that there will be a big shift in the manufacture of auto components to countries like India and China [3]. The Swedish car manufacturer Volvo reported that the casting and forging components sourced from India are 50% cheaper than those sourced from Europe [4]. One sees a similar trend even in other sectors such as Aerospace and Defence. Anselmo [5] suggested that a major shift in spending by aerospace manufacturers is likely in the next three years towards countries such as India, China and Eastern Europe. We analyze a typical situation involving suppliers who differ form one another with respect to the cost of capacity to be procured and identify useful strategies for the buyers to minimize their procurement cost. Our work is at the intersection of supplier selection and capacity allocation using supply contracts. While supply contracting literature addresses various contracting mechanisms employed between a chosen set of suppliers and a buyer, we extend this work by studying the effect of cost heterogeneous mix of the supply base typical of today’s global markets. A few studies in the recent past have studied SCM problems pertaining to a cost heterogeneous supply base. Li and Debo [6] analyze optimum strategies for contracting with a high cost and a low cost supplier and concluded that in periods of demand uncertainty the high cost supplier is more favored. In another paper [7], they model a situation involving multiple suppliers whose cost information is private and explores the value of commitment through long term contracts with suppliers. They use a competitive bidding process to elicit suppliers’ cost. A supplier selection scenario in which the buyer does not know the competing suppliers’ cost and a queuing model to minimize its procurement and operating cost is analyzed in [8]. The suppliers are assumed to have a similar capacity but vary in their costs. They model a situation in which the buyer will sample the participating suppliers’ cost from a uniform distribution ],[ hl bb . In this paper we make a similar set of assumptions about suppliers’ capacities and costs.

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Reference [1] reviews the trends in the sourcing literature and identifies a central issue of multiple sourcing: the number of vendors to be selected. In a multiple–supplier, single-buyer setting, the presence of multiple suppliers will permit the buyer to setup a competitive mechanism such as a bidding process for capacity allocation among the selected suppliers. An excellent review of the supply chain contract literature appears in [9]. We refer the readers to the respective papers for details. Our study differs from the earlier studies on the issue of capacity reservation price. Earlier studies in capacity contracts often assumed the capacity reservation price to be exogenous. We however, explicitly model this situation and derive price capacity curves for suppliers. Although suppliers have an option to sell capacity in both the spot market and through pre-negotiated contract with buyers, they face different sets of costs [10]. Consequently, suppliers will deploy alternative strategies for pricing their capacity in these markets. We believe that business environment aspects such as search cost differentials and risks associated with selling capacity in the open market would influence suppliers’ pricing strategies and capacity allocation through contractual agreement with the buyers. We therefore derive an expression for pricing of supplier’s capacity. The rest of the paper is organized as follows: We begin by motivating the research problem in section 2 with a review of pertinent literature. In section 3 we characterise the cost heterogeneity of the supply base and describe the problem. We also present the context and the decision issues for the buyer and the supplier. In section 5 we analyze the cost heterogeneity of supply base and derive solution procedures and useful policy framework. We finally conclude the paper by discussing key managerial implications of the study in section 6.

II. MODEL DESCRIPTION Consider an item or a component that a buyer would like to procure for which there are npre-qualified suppliers available to supply as per specifications. The buyer’s demand is a stochastic variable D, sampled from a distribution F(x). The decision to reserve capacity is made at the beginning of the season when it has information only of demand distribution. Similar to [8], the suppliers who can provide the required item have a finite capacity denoted by μ . However, there could be significant variations in the unit cost of the

capacity offered by suppliers. The supplier’s manufacturing cost is private information and not known to the buyer or to other suppliers. However, the buyer knows that the suppliers’ unit cost are sampled from a distribution

).,( hl vvU Cost differentials occur primarily as a result of variations in input factor cost. Shifting of major manufacturing activities to China, software development activities to India and textile manufacturing to China, Latin and Central America and South Asia are attributed to cost structure differentials. The salient features of the supplier selection process are as follows: The buyer pre-qualifies the number of suppliers (denoted by n> 1) to participate in a competitive bidding process, through a RFQ, along with bidding rules and other terms of fulfillment. A preliminary qualification for the supplier to be enlisted for participating in the auction is that the supplier’s capacity is at least as large as the capacity sought to be reserved through the contract. The assumption is necessary to avoid infeasibility of the solution in the event the number of suppliers selected for award of contract is just one. Similar assumptions have been made in [11]. The buyer will also a priori announce the number of suppliers, m ),11( −≤≤ nm that will be selected before the start of the bidding process. The buyer utilizes a pay-to-delaycontract to reserve capacity from a set of suppliers. Let us denote the capacity procured through contract by Ω . All m suppliers, finally selected, will get an equal allocation of the capacity at the price of the first rejected supplier. Therefore, the capacity awarded to each supplier is m/Ω . Equal allocation has been adopted by earlier researchers. (See for example, the review paper by [1] and [12] for details). The buyer obtains quotes from the participating suppliers in the form of a price-capacity curve and uses it as the basis for allocating the order among the suppliers so as to minimize the total expected cost of the items procured. The spot market is not capacity constrained in the near term. After observing the demand the buyer will procure additional capacity from the market in case her demand exceeds the total capacity reserved with the suppliers. The enlisted suppliers, on the other hand, will also sell the remaining capacity (the unsuccessful suppliers will sell the entire capacity) in the spot market. In our model we assume that demand for suppliers’ capacity is

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uncertain and is independent and identically distributed. Therefore the suppliers may not be able sell his entire capacity in the spot market. We also assume the contract between the buyer and the suppliers will not have any impact on average unit price in the spot market which is

denoted by .oP

III. THE SUPPLIERS’ PROBLEMWe first derive the price a supplier is likely to quote in the competitive bidding process for obtaining a contract. Using this information, we further determine the price for awarding the contract. Consider the ith supplier who has the option of selling his capacity to a specific buyer through a contract and also through the spot market. While attempting to sell his capacity in the spot market, the supplier finds that the demand is stochastic. We denote the demand for supplier’s capacity in the spot market by a random variable X. The demand distribution for supplier’s capacity, in the spot market, is assumed to be independently and identically distributed. The random nature of demand imposes a financial burden on suppliers because of the risk of unsold capacity. We assume that the supplier will sell his capacity in the spot market at an average unit price of oP . The supplier’s production cost is linear in number of units produced and is given by xvi , where )0(, >ii vv is the unit production cost and x is number of units produced. Since the supplier is likely to face uncertain demand in the spot market he will consider selling some capacity to the buyer through a contract at a lower price, as long as his total expected profit is at least as large as the case when he sells his entire capacity in the spot market. Based on these considerations, we derive the price – capacity curve for a supplier who would like to sell his capacity to the buyer through contract. The expected profit for the ith supplier offers if he offers his entire capacity in the spot market is given by

{ } { }( ) (1)),min(),min( μμ iiio XEvXEP − The first term in the above expression is the expected revenue earned and the second term is the expected production cost. The E in the equation is the expectation operator. Suppose the supplier offers ω (where

μω ≤ ) units of capacity to the buyer through a

capacity reservation contract. He will have to sell ωμ − units of capacity in the spot market. His

expected profit, if he wins the contract, will then be:

{ }{ }( ) (2)),min(

),min()(ωμω

ωμωω−+

−−+

ii

ioi

XEvXEPP

In (2), )(ωiP is the price supplier iquotes to the buyer, which is a function of units of capacity, the buyer procures from this supplier through a capacity reservation contract. The second term is the expected revenue the supplier earns from the spot market and last term is the expected production cost. The supplier will commit ω units of capacity provided (2) is at least as large as (1). Therefore the price-capacity curve is given by:

( ) ( ){ } (3)),min(),min()()( iiiio

i vXEXEvPP +−−−=ω

ωμμω

We now assume that the demand for capacity in the spot market for the suppliers isuniformly distributed (0,b), b>0. For such a demand distribution it can be shown that the price quoted by the ith supplier when b≤μ as:

(4))2(2

)(b

vPPP iooi ωμω −

−−=

It is evident that the price quoted by the supplier is always less than the average spot market price. Furthermore, this price decreases as the supplier’s capacity increases. This is evident due to the fact that with an increase in ω , the risk of unsold capacity is reduced, as the supplier may not be able to sell (X-ω ) in the spot market always. Hence, the supplier would be satisfied with a lower capacity reservation price than oP .

IV. THE BUYER’S PROBLEMThe total procurement cost optimization problem for the buyer is given by:

0,1

0subject to

(5)),0max(),( ]1[

≥Ω−≤

≤−Ω

Ω−+ΩΩ=Ω +

mnmm

DEPm

PEmCMin om

μ

In equation set (5), Ω+ m

PE m ]1[ is the

expected unit capacity reservation price of the first rejected supplier. This will be the unit price that the buyer will pay to the m selected

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suppliers to reserve capacity. Using the price curve of the suppliers as derived earlier, we develop an expression for capacity reservation price of (m+1)th supplier before we proceed with our analysis of the buyer’s problem. The constraint ensures that the reservation quantity

)/( mΩ from each selected supplier does not exceed its capacity )(μ . Let us now consider the multiple suppliers setting. Once the buyer makes an appropriate choice of Ω , she will allocate this equally among the m selected suppliers. However, in order to determine the price at which the buyer reserves the capacity, she needs to know the price of the first rejected supplier. In (4), we assume that the buyer knows the value of all parameters except each supplier’s cost. We proceed to determine the expected price of the first rejected supplier.

Let

[ ] [ ] [ ] )()()()( ][121 mP

mP

mP

mP nm

Ω≤Ω≤≤Ω≤Ω+

denote the order statistics of the unit capacity

reservation price (bids) received and )(]1[ mPm

Ω+

is the lowest bid amongst the rejected suppliers. Similarly, let

[ ] [ ] [ ] ][]1[21 nnm vvvvv ≤≤≤≤≤ −

denote the order statistics of suppliers’ costs. From (4), it is clear that.

( )]1[]1[ (221)( ++ −Ω−−=Ω

moom vEPmb

Pm

PE μ

From the buyer’s perspective the s'ivare uniformly distributed between hv and lv ,one can get closed form expression for the expected value of the order statistics. Using the order statistics for v, we obtain,

)(11)( ]1[ lhlm vv

nmvvE −

+++=+

Substituting it in the above equation and simplifying we get an expression for expected unit price to be paid for reserving the capacity of the selected suppliers.

−++−−Ω−−=Ω

+ )(112

21)(]1[ lhloom vv

nmvP

mbP

mPE μ

Therefore, (6) gives the price at which the buyer will award the contract to the first m lowest cost

suppliers. Substituting (6) into the buyer’s objective function in (5), we can re-write the cost equation, for the buyer, as

),0max(

)(11

221

),(

Ω−+

−++

−−Ω

−−Ω=Ω

DEP

vvnmvP

mbPmC

o

lhloo μ

V. MODEL ANALYSIS We now analyze the case for the stochastic demand. We assume that D, the buyer’s demand, is a random variable representation by a distribution F(x). The buyer would like to minimize the cost of procurement given the stochastic nature of the demand and the existence of n qualified suppliers to choose from. The buyer’s problem boils down to an optimal choice of Ω and m. We consider two cases of this problem and derive expressions and insights into the problem. We assume that the buyer’s demand distribution is continuous and differentiable in its domain.

Proposition 1: If the buyer’s demand distribution is continuous and differentiable in its domain then the first constraint in the buyer’s optimization problem in (5) holds as a strict inequality and 0>Ω∗ . If the constraint becomes an equality, that is, μ∗∗ =Ω m , then it would imply that the buyer does not reserve any capacity in advance from the suppliers and instead sources entirely from the spot market ( 0=Ω∗ and m = 0).However, this is not possible because for

ε≤Ω<0 , for some

,ε om Pm

PE <Ω+ )(]1[ . Therefore the buyer

will always source a certain quantity through the capacity reservation quantity with a subset of suppliers. In general, it is never optimal to exhaust the supplier’s capacity. There is, however, a special case when the buyer may procure the entire capacity of the selected suppliers. This case has is analyzed in [13].

Proposition 2: The optimum number of suppliers, ∗m , and the quantity reserved, ∗Ω , is given by

−−−

+−Ω=∗

∗ 1,1)(

)1)((2

min nvvnvPmlh

lo

μand

(6)

(7)

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o

lhlo

bPmn

vvmvP

F

Ω−+−

+−−=Ω

∗

∗∗

∗

μ1

)1()(

For a general demand distribution it is not possible to get an explicit solution for ∗Ω but it is possible to obtain the solution for ∗m and

∗Ω numerically. However, explicit solution can be derived when D is Uniformly distributed.

Corollary 1: If the buyer’s demand D is deterministic and is given by Ω (that is

1)Pr( =Ω=D ) then the optimum number of

suppliers, ∗m , is given by

−−

+−Ω=∗ 1)(

)1()(2 lh

lo

vvnvPm

μ.

In Corollary 1, the buyer’s mean demand forecast is relatively accurate and is given by Ω . In this model the buyer does not have to procure from the spot market and instead gets her entire requirement from the reservation contract with the suppliers. The size of the

supply base increases with Ω which makes the contract relatively stable in the sense that small variation in demand forecast will not change the size of the supply base.

VI. CONCLUSIONS This paper addresses the capacity reservation issue in detail and analyses alternative strategies that a buying firm would like to adopt. Suppliers participating in such contracts are likely to quote higher price at higher levels of utilisation and will have several motivational elements to participate in the contract. The optimal number of suppliers to contract capacity remains robust to changes in model parameters. However, variability of demand distribution has significant effect on the optimal capacity to contract. Future work on extending the model includes analysing for other probability distributions for demand and suppliers’ capacities. Further, modelling transaction costs related to the pre-selection process and dealing with multiple suppliers in a post-selection setting will sharpen the insights.

Note: We omit the proof and algebraic details for all propositions in this paper for the sake of brevity. The proofs are available in [13].

ACKNOWLEDGMENT The author gratefully acknowledge the funding support received from the EADS-SMI Endowed Chair for Sourcing and Supply Management for this research.

REFERENCES

[1] W. Elmaghraby, Supply contract competition and sourcing policies. Manufacturing & Service Operations Management, 2 (4), 2000; 350 – 71.

[2] B. Mahadevan, “Business models for Internet based E-Commerce: An anatomy,” California Management Review, 42, 4, 55 – 69, 2000

[3] EIU, Driving the Market”, The Economist Intelligence Unit, Business India Intelligence, July 5, 2006, pp 3 – 5.

[4] EIU, Another spoke for the hub, The Economist Intelligence Unit, Business India Intelligence, August 30, 2006, pp 5 – 6.

[5] J.C. Anselmo, Distant (Off) Shores, Aviation Week & Space Technology, 163 (20), 2005

[6] C. Li and L. Debo, Second Sourcing vs. Sole Sourcing: the Impact of Demand Uncertainty, Working paper, Carnegie Mellon University, 2007

[7] C. Li and L. Debo, Strategic Dynamic Sourcing from Competing Suppliers: The Value of Commitment, Working paper, Carnegie Mellon University, 2006

[8] G.P. Cachon and F. Zhang. Obtaining fast service in a queuing system via performance-based allocation of demand. Management Science, 52(6), 881-896, 2006

[9] G.P. Cachon, “Supply chain coordination with contract,” in Handbooks in Operations Research and Management Science: Supply Chain Management. Steve Graves and Ton de Kok (eds). North Holland, 2003, ch. 6, pp 229-340

[10]W. Grey, T. Olavson and D. Shi. The role of E-Market places in relationship based supply chains. International Conference on Global Supply Chain Management August 5-7, 2002, Beijing, China

[11]F. Bernstein and F. De Vericourt, Allocation of Supply Contracts with Service Guarantees. Operations Research, 2007. (forthcoming).

[12]S. Seshadri, Sourcing Strategy: Principles, Policy and Designs, Springer, 2005

[13] J. Hazra, “A Multi-sourcing Procurement Model,” Working paper, IIM Bangalore, India.

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