IIE Transactions (2002) 34, 717–728
Supply chain capacity and outsourcing decisions: the dynamic interplay of demand and supply uncertainty PANOS KOUVELIS and JOSEPH M. MILNER* John M. Olin School of Business, Washington University, St. Louis, MO 63130, USA E-mail:
[email protected] or
[email protected] Received January 2001 and accepted January 2002
We study the interplay of demand and supply uncertainty in capacity and outsourcing decisions in multi-stage supply chains. We consider a firm’s investment in two stages of a supply chain (Stage 1 models the ‘‘core’’ activities of the firm, while Stage 2 are the ‘‘non-core’’ activities). The firm invests in these two stages in order to maximize the multi-period, discounted profit. We consider how non-stationary stochastic demand affects the outsourcing decisions. We also consider how investment levels are affected by non-stationary stochastic supply when the market responds to the firm’s investments. We characterize the optimal capacity investment decisions for the single- and multi-period versions of our model and focus on how changes in supply and demand uncertainty affect the extent of outsourcing. We find that as the responsiveness of the market to investments made by the firm increases, the reliance on outsourcing generally increases. While greater supply and greater demand have the expected effect on investments, decreases in variability are not as straightforward. Greater supply uncertainty increases the need for vertical integration while greater demand uncertainty increases the reliance on outsourcing. In the multi-period model, we find that the nature of adjustments in capacity based on changes in demand or supply follows from the comparative statics of the single-period model, although whether outsourcing increases or decreases depends on the costs of adjusting capacity.
1. Introduction Capacity determination and expansion for both new products in existing markets and existing products in new markets is influenced by expectations regarding the demand for the product as well as the available supply of needed goods and services for its production. Uncertainty in supply in existing markets may be attributed to supplier availability and quality for the newly developed product. In new markets, especially emerging markets such as China, Malaysia, Brazil and Russia, such uncertainties are further compounded by additional concerns such as scarcity of materials because of import restrictions, inadequate transportation and distribution infrastructure, and centralized, and in many instances, irrational allocation of available supply capacity by local manufacturers (Dornier et al. 1998). At the same time, the very nature of non-existing or limited previous market penetration of a firm’s product in the new market, or the novelty of the newly developed product not only points to the presence of the growth opportunity but also to the inherent difficulty in accurately forecasting demand.
*Corresponding author 0740-817X
Ó 2002 ‘‘IIE’’
Firms respond to uncertain supply and demand by both increasing vertical integration and by outsourcing. We study such decisions in this paper. Companies undertake a variety of creative approaches in their supply chain design and development to overcome supply problems. A common reaction is a decreased reliance on outsourcing as compared with similar operations where supply is more assured. For example, recent uncertainty of supply in the Flat-Panel Display (FPD) market has led to creative investment, partnering and purchase of capacity by Original Equipment Manufacturers (OEMs) in a critical, though generally not considered core, technology. Giffi et al. (1990) identified FPDs as being outside the core activity of the major laptop producer Toshiba. However, as the market has changed from 1000 to 1200 and recently to 1400 , 1500 and larger displays, uncertainty in supply has increased. According to Omid Milani, senior marketing manager for flat-panel products at NEC Electronics, uncertainty in price is ‘‘dominated by rapid demand/supply fluctuations; target markets are diverse and their requirements are constantly changing’’. Major producers such as Philips, Apple and Toshiba and IBM have responded to this uncertainty by increasing their vertical integration and by making investments in suppliers to ensure a captive supply (Lieberman 1999).
718 Similarly, expansion into new markets often is accompanied by investment in activities not usually undertaken in domestic US operations. For example, Tambrands Inc., a Fortune 500 manufacturer of personal care products, faced substantial problems in its tampon and fiber processing operations in the Ukraine (Emmons, 1991). The most crucial input material to their production process is high quality, bleached cotton fiber. After realizing the unreliability of supply from the Ukrainian supplier, Tambrands acquired a Russian bleaching facility in St. Petersburg to cover the majority of their supply needs. It was the first time in Tambrands’ operating history that bleaching operations became part of the company. In a story of a similar nature, ABB, the international conglomerate in the areas of power generation and power distribution acquired Zamech, the Polish monopolist in the design, construction and maintenance of large turbines (Chambers et al. 1998). Quite uncharacteristic for ABB operations, however, was their insistence on acquiring not only the power generation operations but also the foundry, the only one in the ABB system. In this paper we study the interplay between demand and supply uncertainty in dynamic capacity expansion decisions of multi-stage supply chains. The intention of our study is to provide a theoretical and, to the extent possible, conceptually simple foundation on which to base the planning and evaluation of supply chain expansion strategies. Our model uses a linear, two-stage supply chain (see Fig. 1). For the purposes of our model, we refer to ‘‘Stage 1’’ as the ‘‘core activities’’ of the firm. As usually implied by the term, these are activities in which the firm exhibits strong competency, has effectively competed on, and demonstrates efficiency in executing. ‘‘Stage 2’’ are the ‘‘non-core activities’’ of the firm, i.e., activities in which the firm has not traditionally exhibited competency and/or invested to a substantial extent in the past. In our simple setting, the firm considers the capacity investment decision in both types of activities, core and non-core, in an environment characterized by demand and supply uncertainty. In the absence of adequate investments in non-core activities, the firm has to rely on outside suppliers, who according to the basic premise of our study have uncertain capacity. Of interest is the dy-
Fig. 1. Cases.
Kouvelis and Milner namic adjustment to capacity of the various activities in a simple two-period setting. We find that as the responsiveness of the market to investments made by the firm increases, the reliance on outsourcing generally increases. While greater supply and greater demand have the expected affect on investments, we find that demand uncertainty and supply uncertainty lead to alternate conclusions regarding the degree of outsourcing. Greater supply uncertainty increases the need for vertical integration while greater demand uncertainty increases the reliance on outsourcing. We also consider changes in sourcing in a two-period model. The nature of adjustments based on changes in demand or supply is in accordance with the comparative statics of the single-period model, though whether outsourcing increases or decreases depends on the marginal cost of adjusting capacity. The capacity planning literature is rich in single- and multi-facility and single- and multiple-region expansion models with deterministic demand and supply conditions. Representative references to this literature are Fong and Srinivasan (1981), Klincewitz et al. (1988), and Li and Tirupati (1994). The stochastic capacity expansion literature addresses the issue of demand uncertainty in expanding the activities of a single-stage in the supply chain, but allowing for the possibility that multiple facilities might be serving such demand. The emphasis in this literature is on discovering conditions under which it is possible to solve a deterministic equivalent in place of the original stochastic problem. For references in the stochastic capacity expansion literature see Manne (1961), Giglio (1970), Luss (1982), Davis et al. (1987), Eppen et al. (1989), Erlenkotter et al. (1989), Paraskevopoulos et al. (1991). Our work contributes to this literature on two main dimensions: (i) explicitly dealing with uncertain supply conditions and their dynamic interplay with demand uncertainty; and (ii) accounting for the capacity expansion of two-stage supply chains in a stochastic environment. Our work is along the lines of recent research on multi-resource capacity investments under demand uncertainty by Harrison and Van Mieghem (1999), where the need for imbalanced capacity among resources is proved to be optimal for an infinite horizon model. We distinguish our work from theirs in two important ways: First, they do not consider supply uncertainty, a factor we show important in determining supply chain structure. Second, their results, while posed in a dynamic environment, are reduced to a static environment to obtain their results. We specifically investigate how changes in the environment may affect investment and reliance on outsourcing. Other recent work includes Angelus and Porteus (2000) who study the capacity expansion and reduction policy for perishable and durable goods for a short-life cycle product for which demand expands initially and then contracts. In this model the non-stationary aspect of supply is not considered. Also, Van Mieghem
Dynamic interplay of demand and supply uncertainty
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(1999) studies capacity investment by two parties in a competitive game-theoretic model, showing that volatility and negative correlation of demands in two markets drive outsourcing. The inventory management/production planning literature has dealt with the interplay of supply and demand uncertainty in the context of ordering and lot sizing decisions. The review paper by Yano and Lee (1995) discusses the references in multi-period inventory systems with random yields. Ciarallo et al. (1994) study a multiperiod production planning problem of a single-product single-stage manufacturer with random demand and random capacity. They show that under negligible fixed costs of replenishments, linear carrying and backorder costs, and stationary distributions of demand and capacity, the optimal policy for a finite horizon problem is of an order-up-to level form. Jain and Silver (1995) consider a single-period variant of Ciarallo et al. (1994) where dedicated capacity can be ensured by paying a premium charge to the supplier. Our modeling of supply uncertainty is along the lines of the work of Ciarallo et al. (1994) and Jain and Silver (1995). However, beyond a conceptual resemblance of a restricted case of the singleperiod version of our model, both the structure and the results of our studied capacity expansion problem are distinctively different. Other relevant research also includes some of the work done in the global supply chain management literature. The work of Huchzermeier and Cohen (1996) demonstrates how flexible facility networks with excess capacity can provide real options to hedge exchange rate fluctuations in the long-term. Our work follows in the spirit of that work, as we consider the use of capacity investment in non-core activities in creating operational flexibility which protects the firm against bad outcomes resulting from uncontrollable aspects in the supply. Other papers include Kogut and Kulatilaka (1994) and Li and Kouvelis (1998). The paper’s structure is as follows: In Section 2 we present our stylized model and discuss its solution under the assumption of stationary demand and supply. In Section 3 we characterize the optimal investment decisions for the single-period version of our model and discuss how changes in demand and supply affect outsourcing. In Section 4 we characterize the optimal capacity investment policies for the two-period version of our model. We conclude with a summary of our research and its managerial insights in Section 5.
is of the order of several months to a year. Core and noncore capacity differ in that the firm cannot obtain outside supply of core production capacity while there may be some supply of non-core capacity which can be obtained in each period if needed. In order to maintain a proper perspective, we assume that non-core activities are those which are crucial to the operation of the firm but traditionally have been performed by outside suppliers. The outsourced supply will complement any non-core output the firm produces from its own facilities. As noted, this supply is also non-stationary and stochastic. In particular we assume there is a known distribution governing the supply in each period. In contrast to the more operational work of inventory and production planning, we do not assume that the supply is based on the yield of a contracted amount or reflects the quality in a batch. Rather, we assume that supply is determined by investment and allocation decisions made by a number of suppliers over the medium-term (several months to a year). We assume independence between periods for both supply and demand. For expositional ease, we assume no maintenance cost for capacity nor any penalty beyond lost revenue for unmet demand, although these could be easily incorporated. We also assume no penalty for unused capacity or supply and do not allow the firm to sell unutilized noncore capacity. Prior to each period, the firm determines its core and non-core capacities, and (dis) invests to achieve these levels. Supply and demand conditions are then realized and operating decision are made. We assume outside supply is purchased for use only in the current period, i.e., it is not an investment which carries through to the next period, nor may it be held as inventory to buffer future uncertainty. Again, the medium-term decision time scale precludes the use of inventory, especially in the context of just-in-time production of goods such as FPDs. We assume the core and non-core capacity are non-depreciating. We assume stationary, linear investment (disinvestment) costs (revenues) and supply costs. Further we assume there is no initial investment or fixed charge associated with adjusting either core or non-core capacity. By assuming supply costs are linear, we implicitly assume a perfectly competitive market. Previous researchers have made similar assumptions (see Jain and Silver, 1995) in considering stochasticity in supply. We make this assumption in order to consider the role of the stochastic supply in an operational environment with a simple, but insightful and robust model. We use the following notation:
2. Model and assumptions We consider the multi-period, discounted profit maximization problem of a firm that invests in core and non-core capacity and faces non-stationary, stochastic supply and demand. The investment decision is a medium-term decision that, depending on the product, implies each period
xtc
= the total installed core capacity (in product units) in period t; xtn = the total installed non-core capacity (in product units) in period t; Pðxtc ; xtn Þ = the expected operating profit for period t (excluding the investment cost);
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= the revenue per unit sold; = the per unit cost of core, non-core capacity investment; rc , rn = the per unit revenue generated by core, noncore capacity disinvestment; cs = the per unit cost of purchased supply; yt = units sold in period t; yst = units of outside supply purchased in period t; b = responsiveness factor of the non-core supply; d = periodic discount factor; ðxÞþ = maxð0; xÞ. Let xt ¼ ðxtc ; xtn Þ be the capacity investment vector in period t. We define the net capacity cost in period t to be
p cc , cn
This may be due to the fact that the firm is a significant size player in the industry to demand and gets supplier attention to its capacity expansion efforts. Clearly, such responsiveness on the part of a supplier should be reflected in the investment level of the firm as well as the degree of outsourcing. Let D be the r.v. for the demand with CDF FD ðÞ and pdf fD ðÞ. Let S be the r.v. for the outside supply with CDF FS ðÞ and pdf fS ðÞ. We assume FD ð0Þ ¼ 0 and FS ð0Þ ¼ 0. Let FD ðÞ and FS ðÞ be the complementary CDF for D and S, respectively. To investigate such a case consider the following model of the firm’s operating decision: Pðxc ; xn Þ ¼ ES;D ½zp ðxc ; xn Þ
t
Cðx ; x
t1
Þ¼
þ t t1 þ xt1 c Þ þ cn ðxn xn Þ t þ rc ðxt1 xtc Þþ rn ðxt1 c n xn Þ :
cc ðxtc
While we assume a linear form, the resulting optimal policy has the same structure under a convex cost assumption. Let Pðxc ; xn Þ be the expected operating profit when demand and supply are realized simultaneously. The multi-period problem for periods 1 to T is max
x1 ;...;xT
T X
dt1 ðPt ðxt Þ Cðxt ; xt1 ÞÞ;
t¼1
0
where x ¼ 0. We solve this problem through dynamic programming. Let J t ðxt1 Þ be the maximum profit from period t onward given initial investment level xt1 . We assume J T þ1 ðxT Þ ¼ rc xTc þ rn xTn . Then J t ðxt1 Þ ¼ maxxt Pt ðxt Þ Cðxt ; xt1 Þ þ dJ tþ1 ðxt Þ. For the case of stationary demand and supply, Proposition 1 of Eberly and Van Mieghem (1997) applies: Proposition 1. If the demands and supply periods are IID, no adjustment to capacity is made after the first period. (All proofs appear in an unabridged working paper, Kouvelis and Milner (2000).) We define the operating environment more explicitly in the discussion of the single-period problem which follows.
3. A single-period model with supply responsive to investment levels
where zP ðxc ; xn Þ ¼ max py cs ys ; y;ys
y xc
subject to
ð1Þ
y x n þ ys
ð2Þ
y D
ð3Þ
ys S þ bðxc xn Þ
ð4Þ
y; ys 0 where 0 < b < 1 represents the responsiveness of the supply. As b increases, the total supply in the market increases. Thus by setting an investment level, the firm signals additional demand to the market which may be partially met. Note also that while we do not include costs for use of core and non-core capacity in a period, we do so without loss of generality. In particular if we define yc and yn as the amount of core and non-core capacity utilized and define jc (jn ) as the cost of utilizing a unit of core (noncore) capacity, the objective function above would be maxy;yc ;yn ;ys py jc yc jn yn cs ys . However, we know yc ¼ y in any optimal solution and yn ¼ y ys > 0 assuming cs > jn (the cost of purchasing outside supply is greater than the cost of utilizing purchased capacity). Letting p0 ¼ p jc jn and c0s ¼ cs jn , we have the previous formulation. That is, p0 reflects the revenue less internal production costs and c0s reflects the incremental cost of purchasing outside non-core capacity. Based on the defined expected operating profit and assuming no initial capacity, the firm’s investment problem is: maximize Pðxc ; xn Þ cc xc cn xn ;
Consider a firm entering a new market where some (uncertain) level of supply is available and demand is also unknown. The firm needs to establish its core and noncore capacity. However, the supply may respond to the entrance of the firm by increasing capacity. In this model, we assume that the supply increases to reflect the difference between the firm’s core and non-core investments.
xc ; xn
subject to
xn xc ; xc 0; xn 0:
Letting k be the Lagrangian multiplier associated with the constraint, xn xc , the Kuhn–Tucker conditions are
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Dynamic interplay of demand and supply uncertainty
sequent to determining the core capacity, the firm may determine the level of non-core capacity by considering the uncertainty in the supply. We highlight the nonoptimality of doing so by considering such a heuristic approach. Suppose the core capacity, xH c , is determined by maximizing the expected revenue less the cost of core capacity as H max pED ½minðxH c ; DÞ cc xc ;
@P cc þ k 0; @xc @P cn k 0; @xn kðxc xn Þ ¼ 0; @P xc cc þ k ¼ 0; @xc @P cn k ¼ 0; xn @xn
xH c 0
while the non-core capacity, xH n is determined by minimizing the expected lost revenue from insufficient supply less the expected cost of supply and non-core capacity as
and k 0: We show the following:
H min ES ½p maxð0; ðxH c xn Þð1 bÞ SÞ
H 0 xH n xc
H H H þcs minðS þ bðxH c xn Þ; xc xn Þ þ cn xn ;
Proposition 2. Pðxc ; xn Þ is concave and @Pðxc ; xn Þ=@xc ¼ ðp cs ÞFD ðxc ÞFS ðð1 bÞðxc xn ÞÞ þb
Z
xc
which imply the solution 1 xH c ¼ FD ððp cc Þ=pÞ;
FD ðnÞ
1 1 H xH n ¼ xc ð1 bÞ FS ððcn cs Þ=ðð1 bÞðp cs ÞÞÞ:
xn þbðxc xn Þ
fS ðn ðxn þ bðxc xn ÞÞÞdn: Z xc FD ðnÞ @Pðxc ; xn Þ=@xn ¼ ðp cs Þð1 bÞ
ð8Þ
ð5Þ
xn þbðxc xn Þ
fS ðn ðxn þ bðxc xn ÞÞÞdn þ cs FD ðxn Þ: ð6Þ
Equations (5) and (6) provide the marginal benefit for increasing the core and non-core capacities. By our assumption of risk neutrality and the concavity of the profit function, the K–T conditions for the single-period problem imply that if x c and x n satisfy @Pðx c ; x n Þ=@xc ¼ cc , @Pðx c ; x n Þ=@xn ¼ cn and 0 < x n < x c , they are the optimal capacities. Observe x c > 0 for non-trivial demand cases, while it is possible that x n ¼ 0. Comment 1. We observe that if xc ¼ xn ¼ x, @P ¼ ðp cs ÞFD ðxÞ; @xc and
ð7Þ
@P ¼ cs FD ðxÞ; @xn
which imply FD ðxÞ ¼ ðp cc cn Þ=p and k ¼ cs ðcc þ cn Þ= p cn . Therefore, k > 0 implies cs > pcn =ðcc þ cn Þ. This highlights the option value of outsourcing: a firm will outsource some of its supply unless the cost of the supply is significantly larger than the cost of establishing noncore capacity (by a factor of p=ðcc þ cn Þ). Comment 2. Firms may approach the problem of determining both core and non-core capacity by decomposing the problem into two steps. A firm may regard the core capacity to be a function of the demand and choose to invest in an appropriate amount of core capacity. Sub-
Observe that the heuristic implies that the non-core capacity will approach the core capacity as the cost of the outsourced supply approaches the cost of the non-core capacity. Thus the heuristic ignores the contingent value of outsourced supply in contrast to the previous comment. We can further observe that the heuristic will overestimate both the core and non-core needs vis-a-vis the optimal. Because the heuristic decomposes the two random effects, when choosing the core capacity, the heuristic ignores the chance of insufficient supply. Similarly, in determining the non-core capacity the heuristic assumes that demand will exceed the core capacity. That is, Proposition 3. Assuming @xc =@b > 0 if ðx c ; x n Þ is the op H timal solution, x c xH c and xn xn . (See Proposition 4 for a discussion of why the assumption is necessary.)
3.1. Comparative statics While the absolute levels of core and non-core investment can be determined through (5) and (6) we present some comparative statics which are of interest in providing insight in this stylized model. Of main concern to us are the absolute and relative changes in the level of core and non-core investment as the demand and/or the supply distributions change. Throughout we let q ¼ xn =xc be the ratio of the optimal level of non-core to core capacity, which is a measure of the degree of vertical integration of the supply chain. Then, 1 q is the degree of outsourcing. The following comparative statics are made assuming outsourcing is worthwhile (0 < xn < xc or, rather, cs < pcn =ðcc þ cn Þ).
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First, intuitively, as b increases, one would expect that the degree of outsourcing would increase. However, in general, this is not the case. We can show the following: Proposition 4. As the responsiveness b increases, the noncore capacity decreases. Further if Z xc b fS ðz ðxn þ bðxc xn ÞÞÞfD ðzÞdz xn þbðxc xn Þ
ð1 bÞFD ðxc ÞfS ðð1 bÞðxc xn ÞÞ;
ð9Þ
core capacity increases and q decreases as b increases. Assumption (9) is required as we can construct examples which violate the restriction for which xc is decreasing (as the supplier becomes more responsive, the firm reduces capacity). For example, suppose D and S have a Betað0:5; 1Þ distribution (f ðxÞ ¼ 0:5x0:5 for 0 x 1). Let p ¼ 10, cc ¼ 1, cn ¼ 3 and cs ¼ 3. For b ¼ 0:75, x c ¼ 0:752, x n ¼ 0:002 and @xc =@b < 0. Note also that for D ¼ xc xn , @D=@b, implying the absolute level of outsourcing is decreasing as b increases. In this case, the left-hand-side of the restriction is 0.20 while the righthand-side is 0.04 in violation of the constraint. It is instructive to understand why the example results in a reduction in the amount of outsourcing even as the market is promising to provide a greater percentage of the difference between the core and non-core investments. Because there is a high likelihood of S being close to zero, as the firm reduces its non-core capacity, it must rely mostly on the responsiveness of the market for supply. But because there is also a high likelihood of demand being small, one cannot justify increasing the core capacity. In fact, we see here a reduction in the core results in a higher profit as more of the core investment is guaranteed to be covered by the increase in b. While this is clearly a constructed example, it does indicate how the responsiveness of the outside supply can be viewed as an increase in option value. Under normal cases (those that satisfy Equation (9)), an increase in responsiveness is met by an increase in core capacity. However, we see in this case, the option value of the responsive supply is limited by the high likelihood of low demand. Next we consider how changes in the demand and supply distributions affect the investments. To motivate the analysis, Fig. 2 presents the value of q as function of the ratio of the average supply (lS ) to average demand (lD ) for several supply and demand distributions assuming b ¼ 0. Demand and supply are assumed to have a Gamma Distribution with a SCV (Squared Coefficient of Variation) of either 0.2 or one (C½5; x or C½1; x ). The model parameters are p ¼ 20, cc ¼ 4, cn ¼ 8 and cs ¼ 4. We use Gamma distributions in our examples because they have sufficient flexibility to provide the needed stochastic dominance for our examples. For most of our examples we use an SCV of 0.2 which is a moderate de-
Fig. 2. q versus lS =lD .
gree of variability in keeping with the uncertainty associated with new markets. Finally, and quite candidly, they are easy to manipulate in Equations (5) and (6). We observe the following: (i) as the demand mean increases, q increases; (ii) as the supply mean increases, q decreases; (iii) as the variance of D decreases, q increases; (iv) as the variance of S decreases, q decreases; and (v) when both the supply and variance decrease, q increases. We formalize these observations through the following propositions. Proposition 5. If the demand increases either through a shift in the mean or through a rescaling of the demand distribution by a factor of c (Dc ¼ D þ c or Dc ¼ cD, respectively), the core and non-core capacity increase. If demand increases through a shift, q is increasing. Further, if demand increases through a rescaling and ðð1 bÞ= bÞðp cs Þ=cs FS ðð1 bÞðxc xn ÞÞ > qfD ðxc =cÞ=fD ðxn =cÞ, q is increasing. The proposition implies the growth rate of the non-core must be at least as large as that of the core. Without additional supply, the fraction supplied internally must increase. For the case of demand rescaling, the sufficient condition holds when the market responds weakly to investment (b small) and when lost marginal revenue exceeds the marginal supply cost: cases where non-core investment would be expected. Similarly, we would expect that as the supply increases stochastically, the non-core capacity would decrease while the core capacity would increase. Proposition 6. If the supply increases through either a shift in the mean or a rescaling of the supply distribution (Sc ¼ S þ c or Sc ¼ cS, respectively), then the non-core capacity decreases, and assuming (9) holds, the core capacity increases, so that q decreases. We next consider changes in the variance of the demand and supply. Ross (1983) shows that, for non-negative random variables X and Y with E½X ¼ E½Y , X is more
Dynamic interplay of demand and supply uncertainty
723
variable than Y (X V Y ) iff E½hðX Þ E½hðY Þ for all convex functions h. Consider the mean preserving transformations
supply and demand in contrast to the increased need as demand uncertainty is resolved. Suppose that both the demand and supply variances are reduced simultaneously using a common a. If it is the case that q is currently less than what would be needed in the deterministic case, then q will increase. This would be the case when the core capacity is greater than expected demand while the non-core capacity is less than the expected shortfall of supply (xc > lD and xn < lD lS ).
Da ¼ aD þ ð1 aÞlD ; Sa ¼ aS þ ð1 aÞlS ; for a 0. Then E½Da ¼ lD , and for a1 > a2 , Da1 V Da2 and similar for Sa (Gerchak and Mossman, 1992). As a decreases to zero, Sa and Da approach deterministic values. For a ¼ 0, intuitively, x c ¼ lD and if lS < lD then x n ¼ lD lS and x n ¼ 0 otherwise. Proposition 7. As the demand variance decreases through the mean preserving transformation: (i) if the core capacity is less than the mean demand, both the core and non-core capacities increase; (ii) if the non-core capacity exceeds the mean demand, both decrease; and (iii) q increases. The final point, that the degree of outsourcing decreases as the demand variance decreases, is somewhat counterintuitive. With a decreasing demand variance, there is a decreased level of uncertainty in the difference between the supply and demand. Since non-core capacity is used to cover this difference, one might expect that the relative level of non-core to core would similarly decrease. However, as the demand variance decreases, a relatively larger portion of uncertainty is attributed to the supply, so that the relative level of non-core capacity to core actually increases. Similarly for the supply distribution we have: Proposition 8. If the mean supply is greater than the difference between core and non-core capacity, as the supply variance decreases through the mean preserving transformation: (i) the non-core capacity decreases: and (ii) if (9) holds, q decreases. The condition holds when the firm will not let non-core supply determine production on average, as is typically the case. By reducing the uncertainty of supply, less noncore capacity is needed to provide for differences in
Proposition 9. For b ¼ 0, if q < 1 lS =lD , q increases as the supply and demand variances decrease simultaneously with a common a. Intuitively, when there is no variability, xn =xc ¼ ðlD lS Þ=lD . Therefore, as both the supply and demand uncertainty decrease, if q is less than ðlD lS Þ=lD , it will increase. Similarly, if q is larger than this value, it should decrease although it may not do so monotonically. From this discussion of variance we observe that increased demand variability is addressed through increased outsourcing while increased supply variability reduces the value of such. A simultaneous increase in both demand and supply variability is addressed by either increasing or decreasing outsourcing, depend upon whether the optimal solution implies an extent of outsourcing greater than or less than that which would hold in the case of no uncertainty. That is, if there is greater outsourcing than would be the case with no variability (a case that might be caused by demand uncertainty), then increasing both demand and supply uncertainty would further increase the extent of outsourcing. In Table 1 we compare the solutions for various values of b. We consider two non-core salvage values, rn ¼ 6 and rn ¼ 8, and set demand and supply to either C½5; 2 or C½2; 5 . In the responsive supply model, we observe that supply variability decreases the degree of outsourcing while demand variability increases it. We also studied the heuristic suggested in Comment 2 above using these parameter cases. We find that the average q for the heuristic solution is over twice as large as the optimal q and has a
Table 1. Parameters; p = 20, cc = 4, cn = 10, rc = 3, cs = 4, d ¼ 0:9 (SCV (Demand), SCV (Supply)) (0.2, 0.2)
(0.5, 0.2)
(0.2, 0.5)
(0.5, 0.5)
Parameter
rn
q
Profit
q
Profit
q
Profit
q
Profit
b¼0 b ¼ 0:4 b ¼ 0:8 b¼0 b ¼ 0:4 b ¼ 0:8
6 6 6 8 8 8
0.45 0.17 0.0 0.58 0.46 0.43
124.63 130.37 134.18 138.76 140.29 140.63
0.41 0.13 0.0 0.55 0.35 0.27
108.59 118.76 125.03 123.24 127.25 129.04
0.53 0.26 0.0 0.65 0.50 0.44
120.94 127.77 133.82 136.89 139.28 140.48
0.47 0.16 0.0 0.61 0.392 0.27
104.25 115.60 124.53 120.70 125.38 128.76
724 profit on average 16% less than the optimal. Thus disaggregating the problem in a natural way results in poor decisions and a significant reduction in the profit.
4. Sequential investment policies We now consider the problem of investing in capacity for two periods where capacity is partially reversible in the second period. Capacity may be purchased prior to the first and second periods and may be salvaged after the first and second periods. We assume that rc < cc , rn < cn and the initial capacity is zero. To ease notation, let c ¼ ðcc ; cn Þ be the vector of capacity investment costs and r ¼ ðrc ; rn Þ be the vector of revenues received from disinvestment. We characterize the optimal policy as follows: Let R ¼ fx ¼ ðxc ; xn Þ : ð1 dÞr rP2 ðxÞ c drg as displayed in Fig. 3 and Ri , i ¼ 1; . . . ; 8 correspond to the numbered regions. Recall x1 ðx2 Þ is the optimal solution in period 1 (2). It can be shown that there is a unique optimal solution for the two-period problem. Let x ¼ ðxc ; xn Þ solve rP1 ðxÞ þ drP2 ðxÞ ¼ c d2 r. If x 2 R, then x1 ¼ x2 ¼ x. Otherwise x1 2 = R and x2 lies on an edge or corner of R. For brevity we present the solutions when x1 lies in R1 and when x1 2 R2 . Other regions are analogous. If x1 2 R1 , then x1 and x2 solve rP1 ðx1 Þ ¼ c dr and rP2 ðx2 Þ ¼ c dr. Note that these are the single-period solutions for each period assuming no initial capacity. Similarly, if x1 2 R2 , then x1c ¼ x2c ¼ xc and xc , x1n and x2n solve @P1 ðxc ; x1n Þ=@xc þ d@P2 ðxc ; x2n Þ=@xc ¼ cc d2 rc , @P1 ðxc ; x1n Þ=@xn ¼ cn drn , @P2 ðxc ; x2n Þ=@xn ¼ cn drn . The core capacity is found considering both periods simultaneously while the non-core capacity for each period (given the core capacity) is found as in the single-period problem.
Fig. 3. Optimal second period policy.
Kouvelis and Milner In brief, within R, the cost of adjusting the first period’s investment levels is not justified by the marginal benefit while outside the region such adjustment is justified. Note that along either of the two coordinate dimensions the possible investment actions exhibit a monotone order: invest, stay-put, and disinvest, which is the so-called ISD policy. Previous research has focused on showing that such a policy is optimal under general distributional assumptions (Eberly and Van Mieghem, 1997) and on characterizing the region of inaction (Dixit 1997). In summary, Region R exists because investment costs are not fully reversible (r < c) and there is uncertainty in supply and demand. Note that since the profit function, Pðxc ; xn Þ, is supermodular, the borders of region R are upward sloping. One implication of this is: Proposition 10. The second period core and non-core capacities are always at least as large as their values given by the solution to the myopic single-period problem for the second period, i.e., the solution to rP2 ðx2c ; x2n Þ ¼ c dr. It is our desire to characterize how changes in the demand and supply distributions affect changes in the capacities from the first to the second period. By Proposition 1, if the demand and supply distribution do not change from the first to the second period, x1 ¼ x2 2 R. We therefore consider cases where either the demand or supply distributions change. By the comparative statics results of the previous section, we know in which regions the first period solution may lie and therefore the possible directions of change. We have the following proposition: Proposition 11. If there is a change in the core (non-core) capacity from the first to the second period, it is of the same sign as given by the comparative statics of the single-period model for a given change in the demand or supply distribution. The proposition rules out the cases which are intuitively impossible, e.g., if demand increases, neither core nor non-core may decreases, although it is possible that neither would increase. Note that Proposition 11 does not imply that the comparative statics of the previous section are immediately transferable. This is definitely not the case for q. The proposition asserts that the nature of the solution is not clear without solving for the optimal solution given above. To better understand how changes in the demand or supply distributions from the first to the second period affect the investments, we consider several examples and then give a general heuristic to apply for other cases. First, consider the case where demand is stochastically increasing. When the cost of increasing core capacity is low, one would expect that the optimal level of core
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capacity for each period would be purchased so that the core capacity would increase from the first to second period. However, when the marginal cost of core capacity is high, it is optimal to install a level of core capacity greater than that which would be installed for the second period if considered alone. By doing so the purchase cost is amortized over two periods. Similar statements can be made with regard to the non-core cost and the two-period solution. Unfortunately, more definite statements can only be made considering particular instances. To demonstrate the example for given distributions and cost scenarios, we consider the case where outside supply acts independent of investment (b ¼ 0) and assume demand increases stochastically from C½5; 1:5 to C½5; 2 . Let supply be distributed C½5; 2 in both periods and set p ¼ 20, cc ¼ 4, cn ¼ 10, cs ¼ 4, rc ¼ 3, rn ¼ 9:5 and d ¼ 0:9, in the base case. In Fig. 4(a) we observe that the stochastic increase in the demand results in both xc
and xn increasing from the first to second period as indicated by the arrow. Here the single-period (myopic) solution holds for each period and x1 lies in R1 . Note that the dot in Region R corresponds to the solution had the distribution been C½5; 2 in each period. If, however, cc ¼ 6 and all other parameters are as in the base case, we find x1c ¼ x2c and only xn changes, i.e., x1 2 R2 in Fig. 4(b). Similarly, if we let cn ¼ 12 (all other parameters are as in the base case), we observe that the stochastic increase in demand results in only the core capacity increasing while the non-core capacity is set for two periods at the same level (see Fig. 4(c)). Next, consider the case of a stochastic decrease in demand from period 1 to period 2. This would result in either the decrease in both core and non-core capacity when rc and rn are large; in only core capacity or non-core capacity, when rn or rc , respectively, are small; or neither when disinvestment marginal revenues are small. Again
Fig. 4. Cases: (a) demand mean increasing, cn ¼ 9:5; (b) demand mean increasing, cc ¼ 6; (c) demand mean increasing; cn ¼ 12; (d) demand variance decreasing, rn ¼ 9; and (e) demand variance decreasing, rc ¼ 2.
726 the degree of stochastic decrease coupled with the cost vector determine the solution. If the demand variance decreases significantly from period 1 to period 2, we would expect the core and/or non-core capacity to decrease (assuming the initial core capacity exceeded the mean demand). Figure 4(d and e) present the case where D1 C½1; 10 and D2 C½5; 2 . In case (d) with rn ¼ 9, only xc decreases while in (e) with rc ¼ 2, only xn decreases. Reducing the marginal value of disinvestment from the base case has the expected result. Note that in these examples, the change from an exponential distribution (C½1; 10 ) to the stochastically less variable C½5; 2 distribution provided enough of a difference to ensure that both solutions were not in Region R. Distributions with shorter tails would result in smaller changes in investments. In the case of a stochastic increase in the supply one would expect either non-core capacity to decrease or core capacity to increase or both. Here the degree of the change combined with the value of disinvestment in the non-core would determine which is the case. In particular, a significantly low value of rn could result in the increase in the core alone. In Fig. 5(a–c) we consider the case of a significant increase in supply (from C½5; 0:2 to C½5; 2 ) for the cases of: (a) rn ¼ 9:5; (b) rn ¼ 9:0; and (c) rn ¼ 8:5 and observe the expected results. As rn decreases, the amount of non-core investment in the first period decreases as does the total for both periods. For the two-period problem, we can make the following observations. From Proposition 11, we know that the potential direction of adjustment for the core and non-
Kouvelis and Milner Table 2. Alternatives for investment change based on supply and demand changes as shown in Fig. 3 Case
Feasible alternatives
Increasing mean demand
R, 2, 8, 1
Increasing mean supply
R, 6, 7, 8
Decreasing demand variance core capacity < mean demand Decreasing demand variance core capacity > mean demand Decreasing supply variance
R, 2, 8, 1
Comment Higher cn implies 8 Higher cc implies 2 Higher cc implies 6 Lower rn implies 8 Higher cn implies 8 Higher cc implies 2
R, 4, 5, 6
Lower rn implies 4 Lower rc implies 6
R, 6, 7, 8
Lower rn implies 8 Higher cc implies 6
core capacity must follow the direction implied by the comparative statics of the single-period problem. We observe from the numerical results of this section that the degree of the adjustment depends on the costs and revenues associated with adjustment. To present the results systematically, we note that the solution can be characterized by the region in which x1 lies. In Table 2 we summarize for several cases the feasible alternate regions for x1 and note which particular region it will tend to lie as a function of one of the costs or revenues. For exam-
Fig. 5. Cases: (a) supply mean increasing, rn ¼ 9:5; (b) supply mean increasing, rn ¼ 9:0; (c) supply mean increasing, rn ¼ 8:5.
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ple, as we observed, if the mean demand increases, we would expect x1 2 R1 , R2 , R8 or R. We then indicate that higher cn would imply x1 would tend to lie in R8 . We have observed that upward adjustment of a capacity (core or non-core) occurs when the cost of doing so is low while higher costs result in no adjustment from the first to the second period, allowing the investment cost to be amortized. Similarly, reduction in capacity follows when the revenue generated by disinvestment is sufficiently high. Having made these observations, the reader may hypothesize the nature of alternate capacity adjustments by first noting the potential regions based on the comparative statics and then considering the costs.
on raw materials will decrease supply availability and increase supply uncertainty, thus again favoring vertical integration. The framework also allows us to interpret how product and market characteristics may affect the extent of outsourcing. For example, firms which have highly diverse product lines may face great uncertainty from their many suppliers. While vertical integration may be indicated by the framework, the capital needs would be overwhelming. In such cases, any effort to standardize or narrow the product line will lead to less supply uncertainty and therefore more outsourcing. Similarly, for high technology items, several effects determine the degree of integration. Often there is a scarcity of qualified suppliers and an associated greater degree of uncertainty, leading to a greater degree of integration. However, the rapid transformation of the product/process technology leads to a smaller demand for any given product generation. This would imply a greater degree of outsourcing. Typically the latter dominates and more outside suppliers are developed. However, the more complex the product/process technology, the higher the uncertainty, so a greater degree of integration is required. This was the case in the market for flat-panel displays in 1999.
5. Conclusions In the highly volatile global markets of today, firms are frequently faced with the issue of ‘‘right-sizing’’ the capacity of their various core and non-core activities in an effort to match supply with demand in the most profitable way. Our paper provides a simple conceptual framework to better understand the interplay of demand and supply uncertainty in deciding these capacities and the degree of outsourcing relied upon. According to our conceptual framework, firms divide their supply chain activities into ‘‘core’’ and ‘‘non-core’’ elements, with this division driven by the firm’s business strategy and its traditional strengths and capabilities. However, both core and noncore activities are essential and either can limit the firm’s ability to meet the demand. We show that both supply and demand play a role in determining capacity investment. Within our framework, for cases where supply responds to firm investments, we show demand variability increases outsourcing while supply variability decreases it. Within environments where investment in the supplier is possible, supply variability may be addressed by increasing such investment and so increases the reliance on outsourcing. As firms initiate operations in developing countries, they encounter substantial deficiencies in various infrastructural resources (transportation networks, telecommunications capabilities, worker skills, supplier quality, etc.), activities often perceived as essential, non-core inputs to firms’ production processes. According to our framework, the larger the market and the higher its rate of growth, the more likely the firm will source internally. For example, mature products nearing the end of their product life-cycle do not have enough demand strength to justify such vertical integration. Therefore, the development of local suppliers becomes a necessity when entering new markets with older products. Similarly, in emerging markets where governments institute import controls on finished goods, such regulations will contribute to an increased and more predictable demand which will lead to increased vertical integration by firms. Import controls
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Biographies Panos Kouvelis is the Emerson Distinguished Professor of Operations and Manufacturing Management at the Olin School of Business, Washington University in St. Louis. He previously taught at the Fuqua School of Business, Duke University, and the Management Department, Graduate School of Business, University of Texas at Austin. He has held visiting appointments at the Graduate School of Business, University of Chicago, and the Koblenz School of Management, Germany. He specializes in global supply chain management, e-fulfillment, operations strategy and lean manufacturing practices. He has published over 60 research articles in these topics, and has co-authored a textbook on Global Operations and Logistics- Text and Cases, (Wiley, 1998), and a research monograph on Robust Discrete Optimization (Kluwer, 1996). He is a department editor of IIE Transactions and an associate editor of Management Science. He is also on the editorial review boards of MSOM and POMS. He is the Co-Director of the Boeing Center on Technology, Information and Manufacturing at Washingston University. Joseph M. Milner is an Assistant Professor of Operations and Manufacturing Management in the Olin School of Business at Washington University in St. Louis. His research interests are in models of supply chain structure and contracting and the inclusion of economic and market-based considerations in modeling supply chain relationships. His research has been published in Management Science and Naval Research Logistics. He completed his Bachelors degree in Industrial Engineering at Cornell University in 1989 and received at Ph.D. in Operations Research at MIT in 1995.
