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Supply diversification with isoelastic demand
Int. J. Production Economics 157 (2014) 2–6
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Supply diversification with isoelastic demand Tao Li a, Suresh P. Sethi b,n, Jun Zhang c a b c
Leavey School of Business, Santa Clara University, Santa Clara, CA 95053, United States Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080, United States School of Management, Fudan University, Shanghai 200433, China
art ic l e i nf o
a b s t r a c t
Available online 22 July 2014
We study a firm's sourcing strategy when facing two unreliable suppliers and a price-dependent isoelastic demand. At optimality, the firm always orders at least from the low-cost supplier. The firm also orders from the high-cost supplier if and only if the effective purchase cost from the low-cost supplier is greater than the actual purchase cost from the high-cost supplier. We also find that when the firm orders from both suppliers, the total order quantity decreases as the correlation between the suppliers' capacities increases. & 2014 Elsevier B.V. All rights reserved.
Keywords: Supply diversification Supply uncertainty Isoelastic demand
1. Introduction We study a firm's optimal sourcing strategy with two suppliers for a product. The suppliers may be unreliable due to their random capacities. The demand for the product is deterministic and price dependent with constant elasticity. We show that the cost-firstreliability-second (CFRS) decision rule continues to be optimal when deciding which supplier to source from, as in Hill (2000), Anupindi and Akella (1993), Dada et al. (2007), Federgruen and Yang (2009, 2011), and Li et al. (2013). Moreover, whether the firm should diversify (order from both suppliers) depends on how low the cost of high-cost supplier is in comparison to the cost of the low-cost supplier, but not on the correlation structure between the suppliers' capacities. The capacity correlation only affects the order quantities when the optimal sourcing strategy is to diversify. As the suppliers' capacities become more correlated in the sense of the supermodular order, the firm's optimal total order quantity decreases. These results corroborate those obtained with deterministic linear demand (Li et al., 2013) as well as price-independent stochastic demand (Dada et al., 2007). Therefore, our paper provides evidence toward the robustness of the results with respect to demand specifications. A firm's optimal sourcing strategy with unreliable suppliers has been widely studied; see, for example, Gerchak and Parlar (1990), Ramasesh et al. (1991), and Parlar and Wang (1993). Recently, the firm's pricing decision has been taken into consideration in exploring the optimal sourcing strategy with supply uncertainty. Tang and Yin (2007) study the benefit of responsive pricing with n
Corresponding author. E-mail addresses: [email protected] (T. Li), [email protected] (S.P. Sethi), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.ijpe.2014.07.014 0925-5273/& 2014 Elsevier B.V. All rights reserved.
supply uncertainty. Interestingly, when the firm can price its product based on the supplier's capacity realization, the CFRS sourcing rule may not yield the optimal supplier set. For example, Feng and Shi (2012) demonstrate that the CFRS rule is no longer optimal when the firm can adjust prices dynamically. Li et al. (2013) show that the CFRS sourcing rule is not optimal when there are more than two suppliers and their capacities are correlated. Whereas Li et al. (2013) assume a demand linear in price primarily for tractability, we consider a more realistic isoelastic demand having a great deal of empirical support. Indeed, the extant literature is replete with empirical estimation of demand functions for a wide variety of products including food items (such as soft drinks or juices) and nonfood items (such as detergents or paper towels), where we find that the functions estimated are often isoelastic, and seldom linear, in price. See, e.g., Tellis (1988), Mulhern and Leone (1991), and Hoch et al. (1995). According to Mulhern and Leone (1991), linear demand models have the undesirable property of having lower elasticities for deeply discounted prices, and modeling price/quantity relationships using them is erroneous. Not surprisingly, we see a wide use of isoelastic demands in the production economics literature as can be seen in the papers of Petruzzi and Dada (1999), Tramontana (2010), Wang et al. (2012), and Nilsen (2013). Our use of an isoelastic demand also finds its justification in observations made by Lau and Lau (2003) that different demandcurve functions can lead to very different results in a multiechelon system and that, in some situations, a very small change in the demand-curve appearance can lead to large changes in the optimal solutions for the system. Shi et al. (2013) argue that the form of the demand function may affect firm's operational strategies significantly. Since Li et al. (2013) derive firm's sourcing strategy under a linear demand, it is important to show if the
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
insights obtained there hold for the more realistic but less tractable isoelastic demand. By showing that they do, this paper testifies for the robustness of the results obtained in Li et al. (2013).
2. Model Consider a firm that may order a product from two suppliers to sell to customers in a single selling season. Supplier i ði ¼ 1; 2Þ has a random capacity Ri. On an order of quantity Qi by the firm from supplier i, the supplier's deliver quantity is minfQ i ; Ri g. We assume that Ri has the distribution function Gi(r) and the corresponding density function g i ðrÞ Z 0 for r 4 0. Denote G i ðrÞ 1 Gi ðrÞ and gðr 1 ; r 2 ) as the joint probability density function of (R1 ; R2 ). The selling season consists of two stages. In the first stage, the firm orders Qi from supplier i and receives Si ðQ i Þ ¼ minfQ i ; Ri g at the end of the first stage. Denote Q ðQ 1 ; Q 2 Þ as the vector of the order quantities. Let SðQ Þ ¼ S1 ðQ 1 Þ þS2 ðQ 2 Þ denote the total delivered quantity. The firm pays a supplier only for the quantity delivered at the unit purchase cost ci. In the second stage, based on the total delivered quantity SðQ Þ, the firm decides the unit retail price p for the product. We assume the demand to be pricedependent and isoelastic, that is, DðpÞ ¼ ap b with a 4 0 and b 4 1. Unsold products are salvaged at a unit price γ o ci and the cost of lost goodwill is δ for each unit of unsatisfied demand. We assume that the firm has pricing power and is therefore able to adjust the retail price depending on the amount delivered from the suppliers. On the other hand, the wholesale prices are specified in purchase contracts signed with the suppliers before the supply uncertainty is resolved. Possible examples are those of a big food processor/ retailer purchasing product from small farmers whose yields depend on weather and a multinational corporation purchasing from fringe overseas suppliers subject to disruptions in shipping their products. In a year when the delivered amount is low, the food processor can adjust the retail price while the farmers cannot as they have contractually agreed to supply at the wholesale prices determined before the yields are realized. In the second example, it is relatively easier for the multinational to adjust its retail price than it is for the overseas suppliers to adjust their wholesale prices. The firm's objective is to choose the order quantities Q in the first stage and the retail price p in the second stage to maximize its expected profit Π ðQ Þ, which is equal to its expected second-stage profit E½Π 2 ðQ Þ less its expected purchase cost in the first stage. Consequently, the firm's problem can be formulated as follows: ( " #) 2
max Π ðQ Þ ¼ E Π 2 ðQ Þ ∑ ci Si ðQ i Þ Q Z0
i¼1
;
ð1Þ
a given S is 8 > bγ > > > < b1 pn ¼ 1=b > S > > > : a
¼0
; ð3Þ
otherwise:
for i ¼ 1; 2:
ð5Þ
b (i¼1,2) as the solution to the first-order Similarly, denote Q i condition for Q 1 þ Q 2 4 A: Z Z i b 1 1=b A Q i 1 h a ðQ i þ r 3 i Þ 1=b A 1=b gðr 1 ; r 2 Þ dr i dr 3 i b Qi 0 ðci γ ÞGi ðQ i Þ ¼ 0
for i ¼ 1; 2:
ð6Þ
Define hi ðÞ ¼ g i ðÞ=ð1 Gi ðÞÞ as the hazard rate function of Ri. To ensure that the profit function is unimodal, we make some assumptions specified in the following lemma. All proofs are relegated to the appendix. Lemma 1 (Unimodality conditions). Assume that Z
Q 3i
Z Z
1 Qi
ðQ i þ r 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i
Q 3i
þb 0
pZ0
h
i ðQ i þr 3 i Þ 1=b ðQ i þ Q 3 i Þ 1=b gðxi Þ dr 3 i
Z
bhi ðQ i Þ
ð2Þ
Q 3i
Z
0
1 Qi
h ðQ i þ r 3 i Þ 1=b
i ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z0;
In this formulation, the firm's second-stage profit is equal to the sum of its sales and salvage revenues from any leftover products, or equal to its sales revenue less the shortage cost.
Z
bi AQ
a
Z
1
bi Q
0
We first solve the firm's second-stage problem to obtain the optimal retail price for a given total delivered quantity S. From (2), we see that π ðpÞ is strictly concave and the optimal retail price for
b
Qi
0
þ
3. Analysis
bγ b 1
Note that the firm never sells more than A units of the product in total. Consequently, the optimal order quantity must satisfy Q 1 r A and Q 2 r A. As can be seen from Eq. (4), Π ðQ Þ has different expressions for Q 1 þ Q 2 r A and Q 1 þ Q 2 4A. So the optimal order quantities are obtained by first finding the best order quantities under either of these two conditions and then selecting the better ones. Denote ðQ 1 ; Q 2 Þ as the solutions to the first-order condition for Q 1 þ Q 2 r A: bci 1=b a G i ðQ i Þ ðQ i þ Q 3 i Þ 1=b b1 Z Q3 i Z 1 h i ðQ i þ r 3 i Þ 1=b ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i þ
where
δ ðDðpÞ SðQ ÞÞ þ :
if S Za
That is, when the total delivery is less than aðbγ =ðb 1ÞÞ b , the firm sets the price to sell all. Otherwise, the firm sets the price at bγ =ðb 1Þ and salvages the leftover products. In our analysis, the quantity aðbγ =ðb 1ÞÞ b plays a significant role; let A aðbγ =ðb 1ÞÞ b . Let f Q ðsÞ be the conditional density of the random variable S given Q . By (1) and (3), the firm's first-stage problem can be reformulated as follows: Z A 1=b s s f Q ðsÞ ds max Π ðQ Þ ¼ a Q Z0 0 Z 1 2 γ ðA s þ bsÞ f Q ðsÞ ds ∑ ci E½Si ðQ i Þ: þ ð4Þ b1 A i¼1
0
Π 2 ðQ Þ ¼ maxπ ðpÞ ¼ p minfDðpÞ; SðQ Þg þ γ ðSðQ Þ DðpÞÞ
3
Z
b þ r 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i ðQ i
bi AQ
þb 0
b Þ bhi ðQ i
h i b þ r 3 i Þ 1=b A 1=b gðxi Þ dr 3 i ðQ i
Z
bi AQ 0
Z
1
bi Q
h
b þ r 3 i Þ 1=b ðQ i
ð7Þ
4
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
i A 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z 0;
ð8Þ
where x1 ¼ ðQ 1 ; r 2 Þ and x2 ¼ ðr 1 ; Q 2 Þ. Then, Π ðQ Þ is unimodal. The conditions in Lemma 1 ensure the unimodality of Π ðQ Þ, and are assumed throughout the paper. Verifying these conditions, however, requires solving (5) and (6). Therefore, we provide several easily verifiable conditions under which the unimodality conditions (7) and (8) hold a priori. First, we present some concepts commonly used in multivariate analysis. For our purpose, we only present these concepts in terms of (R1,R2), while noting that they apply to general random vectors. Let Gðr 1 ; r 2 Þ ¼ PfR1 4 r 1 ; R2 4 r 2 g be the joint survival function of (R1 ; R2 ). The hazard function of (R1 ; R2 Þ is then defined as H ¼ log G. H's gradient h ¼ ∇H is called the hazard gradient. Note that h i ðr 1; r 2 Þ can be interpreted as the conditional hazard rate of Ri evaluated at ri, given Rj 4 r j for all j a i. That is, h i ðr 1; r 2 Þ ¼ g i ðr i ∣Rj 4 r j ; j a iÞ=G i ðr i ∣Rj 4 t j ; j a iÞ; where g i ð∣Rj 4 r j ; j aiÞ and G i ð∣Rj 4 r j ; j a iÞ are, respectively, the conditional density and survival function of Ri, given Rj 4 r j for all j ai. Refer to Johnson and Kotz (1975) and Marshall (1975) for details. R1 and R2 are associated if Cov f 1 ðR1 Þ; f 2 ðR2 Þ Z 0 for all nondecreasing functions f1 and f2 (Esary et al., 1967). The notion of association among random variables is just one among many notions of multivariate dependence. Next, we present some alternative notions of positive dependence which imply association; for more in-depth discussions on these concepts, see Barlow and Proschan (1975, Sec. 5.4). (a) Ri is right tail increasing in Rj, i.e. RTIðRi ∣Rj Þ, if P Ri 4 r i ∣Rj 4 r j is increasing in rj for all ri; (b) Ri and Rj are right-corner-set increasing, i.e., RCSIðRi ; Rj Þ, if h i P Ri 4 r i ; Rj 4 r j ∣Ri 4r 0i ; Rj 4 r 0j is increasing in r 0i and r 0j for each fixed ri, rj; (c) Ri and Rj are TP 2 ðRi ; Rj Þ, if the joint probability density gðr i ; r j Þ is totally positive of order 2, that is, gðr i ; r j Þ gðr 0i ; r 0j Þ Z gðr i ; r 0j Þgðr 0i ; r j Þ for all r i o r 0i , r j o r 0j . With these preliminary concepts, we introduce Lemma 2 that provides the sufficient conditions under which the conditions in Lemma 1 hold. Lemma 2. Conditions (7) and (8) are satisfied if any of the following conditions is satisfied: (i) For i¼1,2, any Q i A ½0; A satisfies Z Q3 i Z 1 ðQ i þr 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i 0
Z þb
Qi Q3 i
0
h
ðQ i þ r 3 i Þ
Z
Q3 i
Z
1
1=b
ðQ i þ Q 3 i Þ
1=b
i
0
gðx Þ dr 3 i
h
Qi
where x1 ¼ ðQ 1 ; r 2 Þ and x2 ¼ ðr 1 ; Q 2 Þ. (ii) For i¼ 1,2, hi ðQ i Þ þ h 3 i ðQ 1 ; Q 2 Þ Z h i ðQ 1 ; Q 2 Þ. (iii) For i¼1,2, ðR1 ; R2 Þ satisfies RTIðRi ∣R3 i Þ. (iv) ðR1 ; R2 Þ satisfies RCSIðR1 ; R2 Þ. (v) ðR1 ; R2 Þ satisfies TP 2 ðR1 ; R2 Þ. (vi) ðR1 ; R2 Þ is a bivariate normal vector with non-negative correlation. The conditions become stronger as we go down the list. By Lemma 2, Π ðQ Þ is unimodal when R1 and R2 are associated or positively correlated. Next, we present the main result of the paper. To simplify the notation, define the (unit) effective purchase cost from supplier i as Z
aðbci =ðb 1ÞÞ b
C i ci þ 0
b 1 r 1=b ci dGi ðrÞ: b a
Comparing Theorem 1 with Theorem 2 in Li et al. (2013), we see that the CFRS decision rule in picking suppliers continues to hold with two unreliable suppliers and constant-elasticity demand. Specifically, the firm always orders from the low-cost supplier. Whether the supplier orders from the high-cost supplier depends on the effective purchase cost from the low-cost supplier and the actual purchase cost of the high-cost supplier. As in the linear demand case, whether the firm diversifies or does not depend on the correlation between the two suppliers' capacities because the effective purchase cost from a supplier depends on the marginal distribution of its capacity. Next, we study the impact of capacity correlation on the firm's ordering decision, using the concept of the supermodular order. A random vector X is said to be greater than another random vector Y in the supermodular order if E f ðXÞ Z E f ðYÞ holds for all supermodular functions f ðÞ; see, for example, Shaked and Shanthikumar (2007, Sec. 9. A.4) for further discussions on the supermodular order. Throughout this paper, an increase in correlation is in the sense of supermodular order. Our next result characterizes the impact of suppliers' capacity correlation on the firm's optimal order quantities. Theorem 2. Assume that the firm diversifies. Then, Q n1 þ Q n2 decreases as the capacity correlation increases. Theorem 2 implies that suppliers should try to differentiate more from each other to reduce their capacity dependence in order to obtain larger orders from the firm.
4. Concluding remarks
i
ðQ i þ r 3 i Þ 1=b i ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z 0 bhi ðQ i Þ
Theorem 1. The firm's optimal order quantities are 8 b > > > Q n1 ¼ a bc1 ; Q n2 ¼ 0 if c2 Z C 1 ; > > > b1 > > > b > < n bc2 if c1 Z C 2 ; Q 1 ¼ 0; Q n2 ¼ a b1 > > > > n n >Q ¼Q ;Q ¼Q ; b þQ b rA; > if c1 o C 2 and c2 o C 1 and Q > 1 2 1 2 1 2 > > > : Qn ¼ Q b ; Qn ¼ Q b ; otherwise: 1 2 1 2
ð9Þ
This paper explores firm's sourcing strategy when facing unreliable suppliers and isoelastic demand. We show that the key insights derived from the model with linear demand continue to hold with an isoelastic demand. Given that the results obtained in this paper hold for linear and isoelastic demands, firm's optimal sourcing strategy appears to be robust with demand function specifications. In order to focus on the impacts of supply uncertainty and demand function forms, we have assumed that the demand is deterministic. As a future research topic, it would be of interest to see if the main insights developed in this paper would hold for stochastic demands. In this paper, the wholesale prices are assumed to be exogenous. Such an assumption may not be realistic when the suppliers are not price-takers, in the sense that the suppliers might be able to set their wholesale prices strategically to compete with each other. For example, Babich et al. (2007) examine the competition and diversification effects in supply chains with supplier default risk. Li et al. (2010) investigate the sourcing strategy of a retailer and the pricing strategies of two suppliers in a supply chain under an environment of supply disruption. In this spirit, it would be interesting, as a topic for future research, to extend out model to allow the suppliers compete with each other via their wholesale prices.
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
Appendix A. Proofs
Q 1 ðc1 Þ þ Q 2 ðc2 Þ 4 A,
Assume
Proof of Lemma 1. The proof can be found in the proof of Theorem 1. □ Lemmas 3–6, stated and proved below, are required for the proof of Theorem 1. Lemma 3. The unique optimal order quantities ðQ n1 ; Q n2 Þ satisfy
8 > Q n1 ¼ aðbc1 =ðb 1ÞÞ b ; Q n2 ¼ 0 > > > > < Q n ¼ 0; Q n ¼ aðbc =ðb 1ÞÞ b 2 1 2 n n > > Q 1 ¼ Q 1; Q 2 ¼ Q 2 > > > : Q n þ Q n 4A
if c2 Z C 1 ; if c1 Z C 2 ;
5
that
Q 2 4 A Q 1.
is,
Since
H 11 ðQ 1 ; Q 2 Þ ¼ 0 and ∂H 11 ðQ 1 ; Q 2 Þ=∂Q 2 o 0; we can conclude that b Þ ¼ 0 and H 2 ðQ Þ is unimodal, H 1 ðQ ; A Q Þ 4 0. Since H 2 ðQ 1
1
1
1
1
1
1
b . Similarly, Q o Q b . Thus, Q b ðc1 Þ þ Q b ðc2 Þ 4 Q ðc1 Þ þ Q 1 oQ 1 2 2 1 2 1 Q 2 ðc2 Þ 4 A. The proof of “)” part of (ii) is completed. With the “)” parts of all three claims proved, the “(” can be proved by contradiction. □ Lemma 6. There exists a unique pair of optimal order quantities for the firm.
ality condition, we know that on the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0,
Proof. It is completed by proving the following three claims. Define C i ðγ Þ as the effective purchase cost of supplier i assuming the actual purchase cost from it is γ. Claim 1 : Q n1 þ Q n2 r A if c1 Z C 2 ðγ Þ or c2 Z C 1 ðγ Þ. Proof: Since Π ðQ 1 ; Q 2 Þ is unimodal for Q 1 þ Q 2 4 A, to have an interior optimal solution, we must have H 21 ð0Þ 40 3 c1 o C 2 ðγ Þ. Otherwise, H 21 ðQ 1 Þ o 0 for any Q 1 A ð0; A. That is, the optimal solution satisfying Q 1 þ Q 2 4A is a boundary solution. The claim in the case c2 Z C 1 ðγ Þ can be proved similarly. b þQ b r A. Claim 2: Q n þQ n r A if c1 o C 2 ðγ Þ, c2 oC 1 ðγ Þ, and Q
1 r dQ 1 =dQ 2 o 0; furthermore, the points ðaðbc1 =ðb 1ÞÞ b ; 0Þ R 1 1=b and ð0; Q~ 2 Þ lie on the curve, where ½ Q~ 2 Q~ 2 g 2 ðr 2 Þ dr 2 þ R Q~ 2 1=b b b ~ g 2 ðr 2 Þ dr 2 ¼ aðbc1 =ðb 1ÞÞ o Q 2 : 0 r2
b A ð0; AÞ symmetry, if c2 o C 1 ðγ Þ, then there exists a unique point Q 2 2 b b b such that H ðQ Þ ¼ 0. However, if Q þ Q r A, then for any
1
2
if c1 o C 2 ; c2 o C 1 and Q 1 þ Q 2 r A; otherwise:
Proof. Consider the order quantity such that Q 1 þ Q 2 r A. For i¼1,2, let H 1i ðQ 1 ; Q 2 Þ b=ðb 1Þa 1=b ∂Π ðQ 1 ; Q 2 Þ=∂Q i : Next, we analyze the curves H 1i ðQ 1 ; Q 2 Þ ¼ 0 in the ðQ 1 ; Q 2 Þ-plane. Applying the implicit function theorem to H 11 ðQ 1 ; Q 2 Þ ¼ 0 and the unimod-
By symmetry, on curve H 12 ðQ 1 ; Q 2 Þ ¼ 0, 1 rdQ 2 =dQ 1 o 0; the points ðaðbc2 =ðb 1ÞÞ b ; 0Þ and ðQ~ 1 ; 0Þ lie on the curve, where R 1 1=b R Q~ 1=b ½ Q~ 1 Q~ 1 g 1 ðr 1 Þ dr 1 þ 0 1 r 1 g 1 ðr 1 Þ dr 1 b ¼ aðbc2 =ðb 1ÞÞ b o Q~ 1 : ~ If c1 4 c2 , then Q 4 aðbc2 =ðb 1ÞÞ b 4aðbc1 =ðb 1ÞÞ b . Thus, 1
only if Q~ 2 4 aðbc2 =ðb 1ÞÞ b , which is equivalent to c1 o C 2 , that
there exists a unique interior solution for H 11 ðQ 1 ; Q 2 Þ ¼ 0 and
1
1
2
2
Proof: If c1 o C 2 ðγ Þ, then, H 21 ð0Þ 4 0. Since H 21 ðAÞ ¼ ðγ c1 ÞG 1 ðAÞ o 0, b A ð0; AÞ such that H 2 ðQ b Þ ¼ 0. By there exists a unique point Q 1
2
2
(Q 1 ; Q 2 Þ
such
1
Q1 þQ2 4A
that
1
1
with
2
b , Q1 4Q 1
we
have
H 21 ðQ 1 Þ o 0; and for any ðQ 1 ; Q 2 Þ such that Q 1 þ Q 2 4 A with b , we have H 2 ðQ Þ o0. These two conditions cover all Q 4Q 2
2
2
2
(Q 1 ; Q 2 Þ's such that Q 1 þ Q 2 4 A. Thus the optimal solution satisfying Q n1 þ Q n2 Z A must satisfy Q n1 þ Q n2 ¼ A. b þQ b 4 A. Claim 3: Q n þ Q n Z A if c1 oC 2 ðγ Þ, c2 o C 1 ðγ Þ, and Q 1
1
2
2
H 12 ðQ 1 ; Q 2 Þ ¼ 0. So, when c2 o c1 o C 2 , the optimal order quantities
b þQ b 4 A, Q þ Q 4 A. Since the Proof: By Lemma 5, when Q 1 2 1 2
are ðQ 1 ; Q 2 Þ, the unique interior solution of (5). When c1 Z C 2 , the optimal order quantities are on the boundary, which implies Q n2 ¼ 0. When c1 rc2 oC 1 , the proof is similar. Notice that
that Q 1 þ Q 2 r A, H 11 ðQ 1 ; Q 2 Þ 4 0; and for any Q 2 o Q 2 such that
Q 1 þ Q 2 may be greater than A. We will show later that if Q 1 þ Q 2 4 A, then the optimal order quantities satisfy Q n1 þ Q n2 4 A. By the unimodality conditions, if the FOC has an interior solution, then the Hessian at the FOC solution is negative definite. Thus, the profit function is directionally unimodal, which implies that it is jointly unimodal and is maximized at the local optimal point. □ Lemma 4. The optimal order quantities satisfy ( n b ; Qn ¼ Q b b þQ b Z A; if Q Q1 ¼ Q 1 2 1 2 2 Q n1 þ Q n2 o A
otherwise:
For i¼ 1,2, let Proof. Assume that Q 1 þ Q 2 4 A. b ¼ 0. If Q b þQ b Z A, then H 2i ðQ i Þ ∂Π ðQ 1 ; Q 2 Þ=∂Q i . Recall that Q i 1 2 b are the optimal order quantities such that Q þQ 4 A. b and Q Q 1 2 1 2 Otherwise, Q n1 þ Q n2 o A must be true, which will be proved later. By the unimodality conditions, it can be verified that if the FOC has an interior solution, then the Hessian at the FOC solution is negative definite. Thus, the profit function is jointly unimodal and it is maximized at the local optimal point. □ b ðc1 Þ þ Q b ðc2 Þ ¼ A; (ii) Q þ Q 4 Lemma 5. (i) Q 1 þ Q 2 ¼ A 3 Q 1 2 1 2 b þQ b 4 A; (iii) Q þQ oA 3 Q b þQ b o A. A3Q 1 2 1 2 1 2 Proof. We only prove the “)” part of (ii). The “)” part of (iii) can be proved by interchanging “ o ” and “ 4 ”. The “)” of (i) then follows directly.
profit function is unimodal for Q 1 þ Q 2 r A, for any Q 1 o Q 1 such Q 1 þ Q 2 r A, H 12 ðQ 1 ; Q 2 Þ 4 0. These two conditions cover all (Q 1 ; Q 2 Þ'ssuch that Q 1 þ Q 2 4 A. Thus, the optimal solution satisfying Q n1 þ Q n2 r A must satisfy Q n1 þ Q n2 ¼ A. □ Proof of Theorem 1. Follows from Lemmas 3 to 6.
□
Proof of Lemma 2. (i) ) unimodality conditions: by the proof of Theorem 1, any ðQ 1 ; Q 2 Þ solving H 11 ðQ 1 ; Q 2 Þ ¼ H 12 ðQ 1 ; Q 2 Þ ¼ 0 and any Q1 solving H 21 ðQ 1 Þ ¼ 0 satisfy the property that Q 1 A ½0; A. When Q 1 A ½0; A and we fix Q 2 ¼ A Q 1 , then Q 2 A ½0; A. It can be easily seen that (i) implies the unimodality conditions. (ii) ) (i): Define Z Q2 Z 1 ΘðQ 2 ∣Q 1 Þ ðQ 1 þr 2 Þ 1=b 1 gðr 1 ; r 2 Þ dr 1 dr 2 Q1
0
Z
Q2
þb 0
½ðQ 1 þ r 2 Þ 1=b ðQ 1 þ Q 2 Þ 1=b gðQ 1 ; r 2 Þ dr 2 Z
bh1 ðQ 1 Þ
Q2
Z
0
1 Q1
½ðQ 1 þ r 2 Þ 1=b ðQ 1 þQ 2 Þ 1=b gðr 1 ; r 2 Þ dr 1 dr 2 :
Then Θð0∣Q 1 Þ ¼ 0, and ΘðQ 2 ∣Q 1 Þ Z 0 if ∂ΘðQ 2 ∣Q 1 Þ=∂Q 2 Z 0, that is, R1 R1 Q 1 gðr 1 ; Q 2 Þ dr 1 Q gðQ 1 ; r 2 Þ dr 2 R1 R1 þ h1 ðQ 1 Þ Z R 1 R 12 : ð10Þ gðr ; r Þ dr dr 1 2 1 2 Q2 Q1 Q 2 Q 1 gðr 1 ; r 2 Þ dr 1 dr 2 Similarly, we can show that the proof goes through in the other direction. Eq. (10) is equivalent to condition (ii). (iii)
)
(ii):
A
sufficient
condition
for
h1 ðQ 1 Þ þ h 2
ðQ 1 ; Q 2 Þ Z h 1 ðQ 1 ; Q 2 Þ is h1 ðQ 1 Þ Zh 1 ðQ 1 ; Q 2 Þ: From Theorem 2.1 of Karia and Deshpande (1999), for any Q1 and Q2,
6
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
h1 ðQ 1 Þ Z h 1 ðQ 1 ; Q 2 Þ 3 RTIðR2 ∣R1 Þ. So, RTIðR1 ∣R2 Þ is a sufficient condition for h1 ðQ 1 Þ þ h 2 ðQ 1 ; Q 2 Þ Z h 1 ðQ 1 ; Q 2 Þ. Similarly for RTIðR2 ∣R1 Þ. The implications (vi) ) (v) ) (iv) ) (iii) follow from the facts that TP 2 ðR1 ; R2 Þ ) RCSIðR1 ; R2 Þ ) RTIðRi ∣R3 i Þ and a bivariate normal distribution with nonnegative correlation is TP2 (Barlow and Proschan, 1975). □ Proof of Theorem 2. Follows from Lemmas 7 and 8, which are stated and proved below. □ Lemma 7. (i) Assume that Q n1 þ Q n2 r A. Then, Q n1 þ Q n2 decreases as the capacity correlation increases. (ii) Assume that Q n1 þ Q n2 4 A: Then, Q ni , i¼1,2, decreases as the capacity correlation increases. Proof. To prove (i), we analyze how H 11 ðQ 1 ; Q 2 Þ changes w.r.t. the capacity correlation. Define ( ðQ 1 þ r 2 Þ 1=b ðQ 1 þ Q 2 Þ 1=b if 0 r r 2 rQ 2 ; r 1 ZQ 1 ; ϕ1 ðr1 ; r2 Þ 0 otherwise: It can be verified that ϕ1 ðr 1 ; r 2 Þ is a submodular function. So, E ϕ1 ðR1 ; R2 Þ decreases as the correlation between R1 and R2 increases (Shaked and Shanthikumar, 2007, 9.A.4, p. 395). Note h
i
that H 11 ðQ 1 ; Q 2 Þ ¼ G 1 ðQ 1 Þ ðQ 1 þ Q 2 Þ 1=b bc1 =ðb 1Þa 1=b þ E ϕ1 ðr 1 ; r2 Þ : So, for any fixed Q1 and Q2, H 11 ðQ 1 ; Q 2 Þ decreases as the capacity correlation increases. Since ∂H 11 ðQ 1 ; Q 2 Þ=∂Q 2 o 0, then for any fixed Q1, Q 02 obtained from H 11 ðQ 1 ; Q 02 Þ ¼ 0 decreases as the capacity correlation increases. This means that the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0 will shift downward. However, the two ending points ð0; Q~ Þ and 2
ðaðbc1 =ðb 1ÞÞ b ; 0Þ stay the same. From the proof of Lemma 3, we know that on the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0, we have 1 r dQ 1 =dQ 2 o 0. By symmetry, we can show that when the capacity correlation increases, the curve H 12 ðQ 1 ; Q 2 Þ ¼ 0 shifts downward as well, while the two ending points ð0; aðbc2 =ðb 1ÞÞ b Þ and ðQ~ ; 0Þ stay the same. From the proof of 1
Lemma 3, we know that on the curve H 12 ðQ 1 ; Q 2 Þ ¼ 0, 1 r dQ 2 =dQ 1 o 0. Consequently, when the capacity correlation increases, the crossing point of H 11 ðQ 1 ; Q 2 Þ ¼ 0 and H 12 ðQ 1 ; Q 2 Þ ¼ 0, i.e., ðQ 1 ; Q 2 Þ, can only move inside the region bounded by H 11 ðQ 1 ; Q 2 Þ ¼ 0, fc1 ; c2 g=ðb 1ÞÞ
b
H 12 ðQ 1 ; Q 2 Þ ¼ 0,
and
Q1 þQ2
¼ aðb min
. In this region, the value Q 1 þQ 2 of any point
ðQ 1 ; Q 2 Þ is less than the original ðQ 1 þ Q 2 Þ. To prove (ii), we analyze how H 21 ðQ 1 Þ changes w.r.t. the capacity correlation. Define ( ðQ 1 þ r 2 Þ 1=b A 1=b if 0 r r 2 r A Q 1 ; r 1 Z Q 1 ; ϕ2 ðr1 ; r2 Þ 0 otherwise: It can be verified that ϕ2 ðr 1 ; r 2 Þ is submodular. Thus, when the capacity correlation increases, E ϕ2 ðr 1 ; r 2 Þ decreases. Note that H 21 ðQ 1 Þ ¼ ðc1 γ ÞG 1 ðQ 1 Þ þ ððb 1Þ=bÞa1=b E ϕ2 ðr 1 ; r 2 Þ : So for any fixed Q1, H 21 ðQ 1 Þ decreases as the capacity correlation increases. By the unimodality conditions, Π ðQ 1 ; Q 2 Þ is unimodal for b ;Q b Þ. Thus, Q þ Q 4A, and achieves its maximum at ðQ 1
2
1
2
b ; H 2 ðQ Þ o 0 if Q o Q b .So, as the capacity H 21 ðQ 1 Þ 4 0 if Q 1 4 Q 1 1 1 1 1 b correlation increases, Q 1 decreases. Similarly, as the capacity b decreases. □ correlation increases, Q 2
Lemma 8. As the capacity correlation increases, the curve b þQ b ¼ Ag shifts towards the point ðγ ; γ Þ , while its two fðc1 ; c2 Þ : Q 1 2 ending points ðγ ; C 1 ðγ ÞÞ and ðC 2 ðγ Þ; γ Þ remain the same.
Proof. Note that the two ending points of the boundary curve are not affected by the capacity correlation. To see how the curve changes w.r.t. the capacity correlation, we fix c1 and check how c2 changes as the capacity correlation changes. From Lemma 7, when b decreases. Therefore, Q b has the capacity correlation increases, Q 1 2 b þQ b ¼ A. If c2 is fixed, then Q b to increase on the curve Q 1 2 2 decreases as the capacity correlation increases. When the capacity b decreases as c2 increases. Thus, when the correlation is fixed, Q 2 capacity correlation increases, c2 must decrease to have an b . So, when c1 is fixed, c2 will decrease as the capacity increased Q 2 b þQ b ¼ A. Similarly, when c2 correlation increases on the curve Q 1 2 is fixed, c1 will decrease as the capacity correlation increases on the same curve. □
References Anupindi, R., Akella, R., 1993. Diversification under supply uncertainty. Manag. Sci. 39 (8), 944–963. Babich, V., Burnetas, A.N., Ritchken, P.H., 2007. Competition and diversification effects in supply chains with supplier default risk. Manuf. Serv. Oper. Manag. 9 (2), 123–146. Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life-testing. Holt Rinehart & Wilson, New York. Dada, M., Petruzzi, N., Schwarz, L.B., 2007. A newsvendor's procurement problem when suppliers are unreliable. Manuf. Serv. Oper. Manag. 9 (1), 9–32. Esary, J.D., Proschan, F., Walkup, D.W., 1967. Association of random variables, with applications. Ann. Math. Stat. 38 (5), 1466–1474. Federgruen, A., Yang, N., 2009. Optimal supply diversification under general supply risks. Oper. Res. 57 (6), 1451–1468. Federgruen, A., Yang, N., 2011. Procurement strategies with unreliable suppliers. Oper. Res. 59 (4), 1033–1039. Feng, Q., Shi, R., 2012. Sourcing from multiple suppliers for price-dependent demands. Prod. Oper. Manag. 21 (3), 547–563. Gerchak, Y., Parlar, M., 1990. Yield variability, cost tradeoffs and diversification in the EOQ model. Naval Res. Logist. 37 (5), 341–354. Hill, T., 2000. Manufacturing Strategy, 3rd ed. McGraw-Hill/Irwin, Boston, MA. Hoch, S.J., Kim, B.D., Montgomery, A.L., Rossi, P.E., 1995. Determinants of store-level price elasticity. J. Mark. Res. 32 (1), 17–29. Johnson, N.L., Kotz, S., 1975. A vector multivariate hazard rate. J. Multivar. Anal. 5 (1), 53–66. Karia, S.R., Deshpande, J.V., 1999. Bounds for the hazard gradients in the competing risks set up. J. Stat. Plan. Inference 75 (2), 363–377. Lau, A.H.L., Lau, H.-S., 2003. Effects of a demand-curve's shape on the optimal solutions of a multi-echelon inventory/pricing model. Eur. J. Oper. Res. 147 (3), 530–548. Li, J., Wang, S., Cheng, T.C.E., 2010. Competition and cooperation in a single-retailer two-supplier supply chain with supply disruption. Int. J. Prod. Econ. 124 (1), 137–150. Li, T., Sethi, S.P., Zhang, J., 2013. Supply diversification with responsive pricing. Prod. Oper. Manag. 22 (2), 447–458. Marshall, A.W., 1975. Some comments on the hazard gradient. Stoch. Process. Appl. 3 (3), 293–300. Mulhern, F.J., Leone, R.P., 1991. Implicit price bundling of retail products: a multiproduct approach to maximizing store profitability. J. Mark. 55 (4), 63–76. Nilsen, J., 2013. Delayed production and raw materials inventory under uncertainty. Int. J. Prod. Econ. 146 (1), 337–345. Parlar, M., Wang, D., 1993. Diversification under yield randomness in two simple inventory models. Eur. J. Oper. Res. 66 (1), 52–64. Petruzzi, N., Dada, M., 1999. Pricing and the newsvendor problem: a review with extensions. Oper. Res. 47 (2), 183–194. Ramasesh, R., Ord, J.K., Hayya, J.C., Pan, A., 1991. Sole versus dual sourcing in stochastic leadtime (s, Q) inventory models. Manag. Sci. 37 (4), 428–443. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer, New York. Shi, R., Zhang, J., Ru, J., 2013. Impacts of power structure on supply chains with uncertain demand. Prod. Oper. Manag. 22 (5), 1232–1249. Tang, C.S., Yin, R., 2007. Responsive pricing under supply uncertainty. Eur. J. Oper. Res. 182 (1), 239–255. Tellis, G.J., 1988. The price elasticity of selective demand: a meta-analysis of econometric models of sales. J. Mark. Res. 25 (4), 331–341. Tramontana, F., 2010. Heterogeneous duopoly with isoelastic demand function. Econ. Model. 27 (1), 350–357. Wang, J.-C., Lau, A.H.L., Lau, H.-S., 2012. Practical and effective contracts for the dominant retailer of a newsvendor product with price-sensitive demand. Int. J. Prod. Econ. 138 (1), 46–54.
Contents lists available at ScienceDirect
Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
Supply diversification with isoelastic demand Tao Li a, Suresh P. Sethi b,n, Jun Zhang c a b c
Leavey School of Business, Santa Clara University, Santa Clara, CA 95053, United States Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080, United States School of Management, Fudan University, Shanghai 200433, China
art ic l e i nf o
a b s t r a c t
Available online 22 July 2014
We study a firm's sourcing strategy when facing two unreliable suppliers and a price-dependent isoelastic demand. At optimality, the firm always orders at least from the low-cost supplier. The firm also orders from the high-cost supplier if and only if the effective purchase cost from the low-cost supplier is greater than the actual purchase cost from the high-cost supplier. We also find that when the firm orders from both suppliers, the total order quantity decreases as the correlation between the suppliers' capacities increases. & 2014 Elsevier B.V. All rights reserved.
Keywords: Supply diversification Supply uncertainty Isoelastic demand
1. Introduction We study a firm's optimal sourcing strategy with two suppliers for a product. The suppliers may be unreliable due to their random capacities. The demand for the product is deterministic and price dependent with constant elasticity. We show that the cost-firstreliability-second (CFRS) decision rule continues to be optimal when deciding which supplier to source from, as in Hill (2000), Anupindi and Akella (1993), Dada et al. (2007), Federgruen and Yang (2009, 2011), and Li et al. (2013). Moreover, whether the firm should diversify (order from both suppliers) depends on how low the cost of high-cost supplier is in comparison to the cost of the low-cost supplier, but not on the correlation structure between the suppliers' capacities. The capacity correlation only affects the order quantities when the optimal sourcing strategy is to diversify. As the suppliers' capacities become more correlated in the sense of the supermodular order, the firm's optimal total order quantity decreases. These results corroborate those obtained with deterministic linear demand (Li et al., 2013) as well as price-independent stochastic demand (Dada et al., 2007). Therefore, our paper provides evidence toward the robustness of the results with respect to demand specifications. A firm's optimal sourcing strategy with unreliable suppliers has been widely studied; see, for example, Gerchak and Parlar (1990), Ramasesh et al. (1991), and Parlar and Wang (1993). Recently, the firm's pricing decision has been taken into consideration in exploring the optimal sourcing strategy with supply uncertainty. Tang and Yin (2007) study the benefit of responsive pricing with n
Corresponding author. E-mail addresses: [email protected] (T. Li), [email protected] (S.P. Sethi), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.ijpe.2014.07.014 0925-5273/& 2014 Elsevier B.V. All rights reserved.
supply uncertainty. Interestingly, when the firm can price its product based on the supplier's capacity realization, the CFRS sourcing rule may not yield the optimal supplier set. For example, Feng and Shi (2012) demonstrate that the CFRS rule is no longer optimal when the firm can adjust prices dynamically. Li et al. (2013) show that the CFRS sourcing rule is not optimal when there are more than two suppliers and their capacities are correlated. Whereas Li et al. (2013) assume a demand linear in price primarily for tractability, we consider a more realistic isoelastic demand having a great deal of empirical support. Indeed, the extant literature is replete with empirical estimation of demand functions for a wide variety of products including food items (such as soft drinks or juices) and nonfood items (such as detergents or paper towels), where we find that the functions estimated are often isoelastic, and seldom linear, in price. See, e.g., Tellis (1988), Mulhern and Leone (1991), and Hoch et al. (1995). According to Mulhern and Leone (1991), linear demand models have the undesirable property of having lower elasticities for deeply discounted prices, and modeling price/quantity relationships using them is erroneous. Not surprisingly, we see a wide use of isoelastic demands in the production economics literature as can be seen in the papers of Petruzzi and Dada (1999), Tramontana (2010), Wang et al. (2012), and Nilsen (2013). Our use of an isoelastic demand also finds its justification in observations made by Lau and Lau (2003) that different demandcurve functions can lead to very different results in a multiechelon system and that, in some situations, a very small change in the demand-curve appearance can lead to large changes in the optimal solutions for the system. Shi et al. (2013) argue that the form of the demand function may affect firm's operational strategies significantly. Since Li et al. (2013) derive firm's sourcing strategy under a linear demand, it is important to show if the
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
insights obtained there hold for the more realistic but less tractable isoelastic demand. By showing that they do, this paper testifies for the robustness of the results obtained in Li et al. (2013).
2. Model Consider a firm that may order a product from two suppliers to sell to customers in a single selling season. Supplier i ði ¼ 1; 2Þ has a random capacity Ri. On an order of quantity Qi by the firm from supplier i, the supplier's deliver quantity is minfQ i ; Ri g. We assume that Ri has the distribution function Gi(r) and the corresponding density function g i ðrÞ Z 0 for r 4 0. Denote G i ðrÞ 1 Gi ðrÞ and gðr 1 ; r 2 ) as the joint probability density function of (R1 ; R2 ). The selling season consists of two stages. In the first stage, the firm orders Qi from supplier i and receives Si ðQ i Þ ¼ minfQ i ; Ri g at the end of the first stage. Denote Q ðQ 1 ; Q 2 Þ as the vector of the order quantities. Let SðQ Þ ¼ S1 ðQ 1 Þ þS2 ðQ 2 Þ denote the total delivered quantity. The firm pays a supplier only for the quantity delivered at the unit purchase cost ci. In the second stage, based on the total delivered quantity SðQ Þ, the firm decides the unit retail price p for the product. We assume the demand to be pricedependent and isoelastic, that is, DðpÞ ¼ ap b with a 4 0 and b 4 1. Unsold products are salvaged at a unit price γ o ci and the cost of lost goodwill is δ for each unit of unsatisfied demand. We assume that the firm has pricing power and is therefore able to adjust the retail price depending on the amount delivered from the suppliers. On the other hand, the wholesale prices are specified in purchase contracts signed with the suppliers before the supply uncertainty is resolved. Possible examples are those of a big food processor/ retailer purchasing product from small farmers whose yields depend on weather and a multinational corporation purchasing from fringe overseas suppliers subject to disruptions in shipping their products. In a year when the delivered amount is low, the food processor can adjust the retail price while the farmers cannot as they have contractually agreed to supply at the wholesale prices determined before the yields are realized. In the second example, it is relatively easier for the multinational to adjust its retail price than it is for the overseas suppliers to adjust their wholesale prices. The firm's objective is to choose the order quantities Q in the first stage and the retail price p in the second stage to maximize its expected profit Π ðQ Þ, which is equal to its expected second-stage profit E½Π 2 ðQ Þ less its expected purchase cost in the first stage. Consequently, the firm's problem can be formulated as follows: ( " #) 2
max Π ðQ Þ ¼ E Π 2 ðQ Þ ∑ ci Si ðQ i Þ Q Z0
i¼1
;
ð1Þ
a given S is 8 > bγ > > > < b1 pn ¼ 1=b > S > > > : a
¼0
; ð3Þ
otherwise:
for i ¼ 1; 2:
ð5Þ
b (i¼1,2) as the solution to the first-order Similarly, denote Q i condition for Q 1 þ Q 2 4 A: Z Z i b 1 1=b A Q i 1 h a ðQ i þ r 3 i Þ 1=b A 1=b gðr 1 ; r 2 Þ dr i dr 3 i b Qi 0 ðci γ ÞGi ðQ i Þ ¼ 0
for i ¼ 1; 2:
ð6Þ
Define hi ðÞ ¼ g i ðÞ=ð1 Gi ðÞÞ as the hazard rate function of Ri. To ensure that the profit function is unimodal, we make some assumptions specified in the following lemma. All proofs are relegated to the appendix. Lemma 1 (Unimodality conditions). Assume that Z
Q 3i
Z Z
1 Qi
ðQ i þ r 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i
Q 3i
þb 0
pZ0
h
i ðQ i þr 3 i Þ 1=b ðQ i þ Q 3 i Þ 1=b gðxi Þ dr 3 i
Z
bhi ðQ i Þ
ð2Þ
Q 3i
Z
0
1 Qi
h ðQ i þ r 3 i Þ 1=b
i ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z0;
In this formulation, the firm's second-stage profit is equal to the sum of its sales and salvage revenues from any leftover products, or equal to its sales revenue less the shortage cost.
Z
bi AQ
a
Z
1
bi Q
0
We first solve the firm's second-stage problem to obtain the optimal retail price for a given total delivered quantity S. From (2), we see that π ðpÞ is strictly concave and the optimal retail price for
b
Qi
0
þ
3. Analysis
bγ b 1
Note that the firm never sells more than A units of the product in total. Consequently, the optimal order quantity must satisfy Q 1 r A and Q 2 r A. As can be seen from Eq. (4), Π ðQ Þ has different expressions for Q 1 þ Q 2 r A and Q 1 þ Q 2 4A. So the optimal order quantities are obtained by first finding the best order quantities under either of these two conditions and then selecting the better ones. Denote ðQ 1 ; Q 2 Þ as the solutions to the first-order condition for Q 1 þ Q 2 r A: bci 1=b a G i ðQ i Þ ðQ i þ Q 3 i Þ 1=b b1 Z Q3 i Z 1 h i ðQ i þ r 3 i Þ 1=b ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i þ
where
δ ðDðpÞ SðQ ÞÞ þ :
if S Za
That is, when the total delivery is less than aðbγ =ðb 1ÞÞ b , the firm sets the price to sell all. Otherwise, the firm sets the price at bγ =ðb 1Þ and salvages the leftover products. In our analysis, the quantity aðbγ =ðb 1ÞÞ b plays a significant role; let A aðbγ =ðb 1ÞÞ b . Let f Q ðsÞ be the conditional density of the random variable S given Q . By (1) and (3), the firm's first-stage problem can be reformulated as follows: Z A 1=b s s f Q ðsÞ ds max Π ðQ Þ ¼ a Q Z0 0 Z 1 2 γ ðA s þ bsÞ f Q ðsÞ ds ∑ ci E½Si ðQ i Þ: þ ð4Þ b1 A i¼1
0
Π 2 ðQ Þ ¼ maxπ ðpÞ ¼ p minfDðpÞ; SðQ Þg þ γ ðSðQ Þ DðpÞÞ
3
Z
b þ r 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i ðQ i
bi AQ
þb 0
b Þ bhi ðQ i
h i b þ r 3 i Þ 1=b A 1=b gðxi Þ dr 3 i ðQ i
Z
bi AQ 0
Z
1
bi Q
h
b þ r 3 i Þ 1=b ðQ i
ð7Þ
4
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
i A 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z 0;
ð8Þ
where x1 ¼ ðQ 1 ; r 2 Þ and x2 ¼ ðr 1 ; Q 2 Þ. Then, Π ðQ Þ is unimodal. The conditions in Lemma 1 ensure the unimodality of Π ðQ Þ, and are assumed throughout the paper. Verifying these conditions, however, requires solving (5) and (6). Therefore, we provide several easily verifiable conditions under which the unimodality conditions (7) and (8) hold a priori. First, we present some concepts commonly used in multivariate analysis. For our purpose, we only present these concepts in terms of (R1,R2), while noting that they apply to general random vectors. Let Gðr 1 ; r 2 Þ ¼ PfR1 4 r 1 ; R2 4 r 2 g be the joint survival function of (R1 ; R2 ). The hazard function of (R1 ; R2 Þ is then defined as H ¼ log G. H's gradient h ¼ ∇H is called the hazard gradient. Note that h i ðr 1; r 2 Þ can be interpreted as the conditional hazard rate of Ri evaluated at ri, given Rj 4 r j for all j a i. That is, h i ðr 1; r 2 Þ ¼ g i ðr i ∣Rj 4 r j ; j a iÞ=G i ðr i ∣Rj 4 t j ; j a iÞ; where g i ð∣Rj 4 r j ; j aiÞ and G i ð∣Rj 4 r j ; j a iÞ are, respectively, the conditional density and survival function of Ri, given Rj 4 r j for all j ai. Refer to Johnson and Kotz (1975) and Marshall (1975) for details. R1 and R2 are associated if Cov f 1 ðR1 Þ; f 2 ðR2 Þ Z 0 for all nondecreasing functions f1 and f2 (Esary et al., 1967). The notion of association among random variables is just one among many notions of multivariate dependence. Next, we present some alternative notions of positive dependence which imply association; for more in-depth discussions on these concepts, see Barlow and Proschan (1975, Sec. 5.4). (a) Ri is right tail increasing in Rj, i.e. RTIðRi ∣Rj Þ, if P Ri 4 r i ∣Rj 4 r j is increasing in rj for all ri; (b) Ri and Rj are right-corner-set increasing, i.e., RCSIðRi ; Rj Þ, if h i P Ri 4 r i ; Rj 4 r j ∣Ri 4r 0i ; Rj 4 r 0j is increasing in r 0i and r 0j for each fixed ri, rj; (c) Ri and Rj are TP 2 ðRi ; Rj Þ, if the joint probability density gðr i ; r j Þ is totally positive of order 2, that is, gðr i ; r j Þ gðr 0i ; r 0j Þ Z gðr i ; r 0j Þgðr 0i ; r j Þ for all r i o r 0i , r j o r 0j . With these preliminary concepts, we introduce Lemma 2 that provides the sufficient conditions under which the conditions in Lemma 1 hold. Lemma 2. Conditions (7) and (8) are satisfied if any of the following conditions is satisfied: (i) For i¼1,2, any Q i A ½0; A satisfies Z Q3 i Z 1 ðQ i þr 3 i Þ 1=b 1 gðr 1 ; r 2 Þ dr i dr 3 i 0
Z þb
Qi Q3 i
0
h
ðQ i þ r 3 i Þ
Z
Q3 i
Z
1
1=b
ðQ i þ Q 3 i Þ
1=b
i
0
gðx Þ dr 3 i
h
Qi
where x1 ¼ ðQ 1 ; r 2 Þ and x2 ¼ ðr 1 ; Q 2 Þ. (ii) For i¼ 1,2, hi ðQ i Þ þ h 3 i ðQ 1 ; Q 2 Þ Z h i ðQ 1 ; Q 2 Þ. (iii) For i¼1,2, ðR1 ; R2 Þ satisfies RTIðRi ∣R3 i Þ. (iv) ðR1 ; R2 Þ satisfies RCSIðR1 ; R2 Þ. (v) ðR1 ; R2 Þ satisfies TP 2 ðR1 ; R2 Þ. (vi) ðR1 ; R2 Þ is a bivariate normal vector with non-negative correlation. The conditions become stronger as we go down the list. By Lemma 2, Π ðQ Þ is unimodal when R1 and R2 are associated or positively correlated. Next, we present the main result of the paper. To simplify the notation, define the (unit) effective purchase cost from supplier i as Z
aðbci =ðb 1ÞÞ b
C i ci þ 0
b 1 r 1=b ci dGi ðrÞ: b a
Comparing Theorem 1 with Theorem 2 in Li et al. (2013), we see that the CFRS decision rule in picking suppliers continues to hold with two unreliable suppliers and constant-elasticity demand. Specifically, the firm always orders from the low-cost supplier. Whether the supplier orders from the high-cost supplier depends on the effective purchase cost from the low-cost supplier and the actual purchase cost of the high-cost supplier. As in the linear demand case, whether the firm diversifies or does not depend on the correlation between the two suppliers' capacities because the effective purchase cost from a supplier depends on the marginal distribution of its capacity. Next, we study the impact of capacity correlation on the firm's ordering decision, using the concept of the supermodular order. A random vector X is said to be greater than another random vector Y in the supermodular order if E f ðXÞ Z E f ðYÞ holds for all supermodular functions f ðÞ; see, for example, Shaked and Shanthikumar (2007, Sec. 9. A.4) for further discussions on the supermodular order. Throughout this paper, an increase in correlation is in the sense of supermodular order. Our next result characterizes the impact of suppliers' capacity correlation on the firm's optimal order quantities. Theorem 2. Assume that the firm diversifies. Then, Q n1 þ Q n2 decreases as the capacity correlation increases. Theorem 2 implies that suppliers should try to differentiate more from each other to reduce their capacity dependence in order to obtain larger orders from the firm.
4. Concluding remarks
i
ðQ i þ r 3 i Þ 1=b i ðQ i þ Q 3 i Þ 1=b gðr 1 ; r 2 Þ dr i dr 3 i Z 0 bhi ðQ i Þ
Theorem 1. The firm's optimal order quantities are 8 b > > > Q n1 ¼ a bc1 ; Q n2 ¼ 0 if c2 Z C 1 ; > > > b1 > > > b > < n bc2 if c1 Z C 2 ; Q 1 ¼ 0; Q n2 ¼ a b1 > > > > n n >Q ¼Q ;Q ¼Q ; b þQ b rA; > if c1 o C 2 and c2 o C 1 and Q > 1 2 1 2 1 2 > > > : Qn ¼ Q b ; Qn ¼ Q b ; otherwise: 1 2 1 2
ð9Þ
This paper explores firm's sourcing strategy when facing unreliable suppliers and isoelastic demand. We show that the key insights derived from the model with linear demand continue to hold with an isoelastic demand. Given that the results obtained in this paper hold for linear and isoelastic demands, firm's optimal sourcing strategy appears to be robust with demand function specifications. In order to focus on the impacts of supply uncertainty and demand function forms, we have assumed that the demand is deterministic. As a future research topic, it would be of interest to see if the main insights developed in this paper would hold for stochastic demands. In this paper, the wholesale prices are assumed to be exogenous. Such an assumption may not be realistic when the suppliers are not price-takers, in the sense that the suppliers might be able to set their wholesale prices strategically to compete with each other. For example, Babich et al. (2007) examine the competition and diversification effects in supply chains with supplier default risk. Li et al. (2010) investigate the sourcing strategy of a retailer and the pricing strategies of two suppliers in a supply chain under an environment of supply disruption. In this spirit, it would be interesting, as a topic for future research, to extend out model to allow the suppliers compete with each other via their wholesale prices.
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
Appendix A. Proofs
Q 1 ðc1 Þ þ Q 2 ðc2 Þ 4 A,
Assume
Proof of Lemma 1. The proof can be found in the proof of Theorem 1. □ Lemmas 3–6, stated and proved below, are required for the proof of Theorem 1. Lemma 3. The unique optimal order quantities ðQ n1 ; Q n2 Þ satisfy
8 > Q n1 ¼ aðbc1 =ðb 1ÞÞ b ; Q n2 ¼ 0 > > > > < Q n ¼ 0; Q n ¼ aðbc =ðb 1ÞÞ b 2 1 2 n n > > Q 1 ¼ Q 1; Q 2 ¼ Q 2 > > > : Q n þ Q n 4A
if c2 Z C 1 ; if c1 Z C 2 ;
5
that
Q 2 4 A Q 1.
is,
Since
H 11 ðQ 1 ; Q 2 Þ ¼ 0 and ∂H 11 ðQ 1 ; Q 2 Þ=∂Q 2 o 0; we can conclude that b Þ ¼ 0 and H 2 ðQ Þ is unimodal, H 1 ðQ ; A Q Þ 4 0. Since H 2 ðQ 1
1
1
1
1
1
1
b . Similarly, Q o Q b . Thus, Q b ðc1 Þ þ Q b ðc2 Þ 4 Q ðc1 Þ þ Q 1 oQ 1 2 2 1 2 1 Q 2 ðc2 Þ 4 A. The proof of “)” part of (ii) is completed. With the “)” parts of all three claims proved, the “(” can be proved by contradiction. □ Lemma 6. There exists a unique pair of optimal order quantities for the firm.
ality condition, we know that on the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0,
Proof. It is completed by proving the following three claims. Define C i ðγ Þ as the effective purchase cost of supplier i assuming the actual purchase cost from it is γ. Claim 1 : Q n1 þ Q n2 r A if c1 Z C 2 ðγ Þ or c2 Z C 1 ðγ Þ. Proof: Since Π ðQ 1 ; Q 2 Þ is unimodal for Q 1 þ Q 2 4 A, to have an interior optimal solution, we must have H 21 ð0Þ 40 3 c1 o C 2 ðγ Þ. Otherwise, H 21 ðQ 1 Þ o 0 for any Q 1 A ð0; A. That is, the optimal solution satisfying Q 1 þ Q 2 4A is a boundary solution. The claim in the case c2 Z C 1 ðγ Þ can be proved similarly. b þQ b r A. Claim 2: Q n þQ n r A if c1 o C 2 ðγ Þ, c2 oC 1 ðγ Þ, and Q
1 r dQ 1 =dQ 2 o 0; furthermore, the points ðaðbc1 =ðb 1ÞÞ b ; 0Þ R 1 1=b and ð0; Q~ 2 Þ lie on the curve, where ½ Q~ 2 Q~ 2 g 2 ðr 2 Þ dr 2 þ R Q~ 2 1=b b b ~ g 2 ðr 2 Þ dr 2 ¼ aðbc1 =ðb 1ÞÞ o Q 2 : 0 r2
b A ð0; AÞ symmetry, if c2 o C 1 ðγ Þ, then there exists a unique point Q 2 2 b b b such that H ðQ Þ ¼ 0. However, if Q þ Q r A, then for any
1
2
if c1 o C 2 ; c2 o C 1 and Q 1 þ Q 2 r A; otherwise:
Proof. Consider the order quantity such that Q 1 þ Q 2 r A. For i¼1,2, let H 1i ðQ 1 ; Q 2 Þ b=ðb 1Þa 1=b ∂Π ðQ 1 ; Q 2 Þ=∂Q i : Next, we analyze the curves H 1i ðQ 1 ; Q 2 Þ ¼ 0 in the ðQ 1 ; Q 2 Þ-plane. Applying the implicit function theorem to H 11 ðQ 1 ; Q 2 Þ ¼ 0 and the unimod-
By symmetry, on curve H 12 ðQ 1 ; Q 2 Þ ¼ 0, 1 rdQ 2 =dQ 1 o 0; the points ðaðbc2 =ðb 1ÞÞ b ; 0Þ and ðQ~ 1 ; 0Þ lie on the curve, where R 1 1=b R Q~ 1=b ½ Q~ 1 Q~ 1 g 1 ðr 1 Þ dr 1 þ 0 1 r 1 g 1 ðr 1 Þ dr 1 b ¼ aðbc2 =ðb 1ÞÞ b o Q~ 1 : ~ If c1 4 c2 , then Q 4 aðbc2 =ðb 1ÞÞ b 4aðbc1 =ðb 1ÞÞ b . Thus, 1
only if Q~ 2 4 aðbc2 =ðb 1ÞÞ b , which is equivalent to c1 o C 2 , that
there exists a unique interior solution for H 11 ðQ 1 ; Q 2 Þ ¼ 0 and
1
1
2
2
Proof: If c1 o C 2 ðγ Þ, then, H 21 ð0Þ 4 0. Since H 21 ðAÞ ¼ ðγ c1 ÞG 1 ðAÞ o 0, b A ð0; AÞ such that H 2 ðQ b Þ ¼ 0. By there exists a unique point Q 1
2
2
(Q 1 ; Q 2 Þ
such
1
Q1 þQ2 4A
that
1
1
with
2
b , Q1 4Q 1
we
have
H 21 ðQ 1 Þ o 0; and for any ðQ 1 ; Q 2 Þ such that Q 1 þ Q 2 4 A with b , we have H 2 ðQ Þ o0. These two conditions cover all Q 4Q 2
2
2
2
(Q 1 ; Q 2 Þ's such that Q 1 þ Q 2 4 A. Thus the optimal solution satisfying Q n1 þ Q n2 Z A must satisfy Q n1 þ Q n2 ¼ A. b þQ b 4 A. Claim 3: Q n þ Q n Z A if c1 oC 2 ðγ Þ, c2 o C 1 ðγ Þ, and Q 1
1
2
2
H 12 ðQ 1 ; Q 2 Þ ¼ 0. So, when c2 o c1 o C 2 , the optimal order quantities
b þQ b 4 A, Q þ Q 4 A. Since the Proof: By Lemma 5, when Q 1 2 1 2
are ðQ 1 ; Q 2 Þ, the unique interior solution of (5). When c1 Z C 2 , the optimal order quantities are on the boundary, which implies Q n2 ¼ 0. When c1 rc2 oC 1 , the proof is similar. Notice that
that Q 1 þ Q 2 r A, H 11 ðQ 1 ; Q 2 Þ 4 0; and for any Q 2 o Q 2 such that
Q 1 þ Q 2 may be greater than A. We will show later that if Q 1 þ Q 2 4 A, then the optimal order quantities satisfy Q n1 þ Q n2 4 A. By the unimodality conditions, if the FOC has an interior solution, then the Hessian at the FOC solution is negative definite. Thus, the profit function is directionally unimodal, which implies that it is jointly unimodal and is maximized at the local optimal point. □ Lemma 4. The optimal order quantities satisfy ( n b ; Qn ¼ Q b b þQ b Z A; if Q Q1 ¼ Q 1 2 1 2 2 Q n1 þ Q n2 o A
otherwise:
For i¼ 1,2, let Proof. Assume that Q 1 þ Q 2 4 A. b ¼ 0. If Q b þQ b Z A, then H 2i ðQ i Þ ∂Π ðQ 1 ; Q 2 Þ=∂Q i . Recall that Q i 1 2 b are the optimal order quantities such that Q þQ 4 A. b and Q Q 1 2 1 2 Otherwise, Q n1 þ Q n2 o A must be true, which will be proved later. By the unimodality conditions, it can be verified that if the FOC has an interior solution, then the Hessian at the FOC solution is negative definite. Thus, the profit function is jointly unimodal and it is maximized at the local optimal point. □ b ðc1 Þ þ Q b ðc2 Þ ¼ A; (ii) Q þ Q 4 Lemma 5. (i) Q 1 þ Q 2 ¼ A 3 Q 1 2 1 2 b þQ b 4 A; (iii) Q þQ oA 3 Q b þQ b o A. A3Q 1 2 1 2 1 2 Proof. We only prove the “)” part of (ii). The “)” part of (iii) can be proved by interchanging “ o ” and “ 4 ”. The “)” of (i) then follows directly.
profit function is unimodal for Q 1 þ Q 2 r A, for any Q 1 o Q 1 such Q 1 þ Q 2 r A, H 12 ðQ 1 ; Q 2 Þ 4 0. These two conditions cover all (Q 1 ; Q 2 Þ'ssuch that Q 1 þ Q 2 4 A. Thus, the optimal solution satisfying Q n1 þ Q n2 r A must satisfy Q n1 þ Q n2 ¼ A. □ Proof of Theorem 1. Follows from Lemmas 3 to 6.
□
Proof of Lemma 2. (i) ) unimodality conditions: by the proof of Theorem 1, any ðQ 1 ; Q 2 Þ solving H 11 ðQ 1 ; Q 2 Þ ¼ H 12 ðQ 1 ; Q 2 Þ ¼ 0 and any Q1 solving H 21 ðQ 1 Þ ¼ 0 satisfy the property that Q 1 A ½0; A. When Q 1 A ½0; A and we fix Q 2 ¼ A Q 1 , then Q 2 A ½0; A. It can be easily seen that (i) implies the unimodality conditions. (ii) ) (i): Define Z Q2 Z 1 ΘðQ 2 ∣Q 1 Þ ðQ 1 þr 2 Þ 1=b 1 gðr 1 ; r 2 Þ dr 1 dr 2 Q1
0
Z
Q2
þb 0
½ðQ 1 þ r 2 Þ 1=b ðQ 1 þ Q 2 Þ 1=b gðQ 1 ; r 2 Þ dr 2 Z
bh1 ðQ 1 Þ
Q2
Z
0
1 Q1
½ðQ 1 þ r 2 Þ 1=b ðQ 1 þQ 2 Þ 1=b gðr 1 ; r 2 Þ dr 1 dr 2 :
Then Θð0∣Q 1 Þ ¼ 0, and ΘðQ 2 ∣Q 1 Þ Z 0 if ∂ΘðQ 2 ∣Q 1 Þ=∂Q 2 Z 0, that is, R1 R1 Q 1 gðr 1 ; Q 2 Þ dr 1 Q gðQ 1 ; r 2 Þ dr 2 R1 R1 þ h1 ðQ 1 Þ Z R 1 R 12 : ð10Þ gðr ; r Þ dr dr 1 2 1 2 Q2 Q1 Q 2 Q 1 gðr 1 ; r 2 Þ dr 1 dr 2 Similarly, we can show that the proof goes through in the other direction. Eq. (10) is equivalent to condition (ii). (iii)
)
(ii):
A
sufficient
condition
for
h1 ðQ 1 Þ þ h 2
ðQ 1 ; Q 2 Þ Z h 1 ðQ 1 ; Q 2 Þ is h1 ðQ 1 Þ Zh 1 ðQ 1 ; Q 2 Þ: From Theorem 2.1 of Karia and Deshpande (1999), for any Q1 and Q2,
6
T. Li et al. / Int. J. Production Economics 157 (2014) 2–6
h1 ðQ 1 Þ Z h 1 ðQ 1 ; Q 2 Þ 3 RTIðR2 ∣R1 Þ. So, RTIðR1 ∣R2 Þ is a sufficient condition for h1 ðQ 1 Þ þ h 2 ðQ 1 ; Q 2 Þ Z h 1 ðQ 1 ; Q 2 Þ. Similarly for RTIðR2 ∣R1 Þ. The implications (vi) ) (v) ) (iv) ) (iii) follow from the facts that TP 2 ðR1 ; R2 Þ ) RCSIðR1 ; R2 Þ ) RTIðRi ∣R3 i Þ and a bivariate normal distribution with nonnegative correlation is TP2 (Barlow and Proschan, 1975). □ Proof of Theorem 2. Follows from Lemmas 7 and 8, which are stated and proved below. □ Lemma 7. (i) Assume that Q n1 þ Q n2 r A. Then, Q n1 þ Q n2 decreases as the capacity correlation increases. (ii) Assume that Q n1 þ Q n2 4 A: Then, Q ni , i¼1,2, decreases as the capacity correlation increases. Proof. To prove (i), we analyze how H 11 ðQ 1 ; Q 2 Þ changes w.r.t. the capacity correlation. Define ( ðQ 1 þ r 2 Þ 1=b ðQ 1 þ Q 2 Þ 1=b if 0 r r 2 rQ 2 ; r 1 ZQ 1 ; ϕ1 ðr1 ; r2 Þ 0 otherwise: It can be verified that ϕ1 ðr 1 ; r 2 Þ is a submodular function. So, E ϕ1 ðR1 ; R2 Þ decreases as the correlation between R1 and R2 increases (Shaked and Shanthikumar, 2007, 9.A.4, p. 395). Note h
i
that H 11 ðQ 1 ; Q 2 Þ ¼ G 1 ðQ 1 Þ ðQ 1 þ Q 2 Þ 1=b bc1 =ðb 1Þa 1=b þ E ϕ1 ðr 1 ; r2 Þ : So, for any fixed Q1 and Q2, H 11 ðQ 1 ; Q 2 Þ decreases as the capacity correlation increases. Since ∂H 11 ðQ 1 ; Q 2 Þ=∂Q 2 o 0, then for any fixed Q1, Q 02 obtained from H 11 ðQ 1 ; Q 02 Þ ¼ 0 decreases as the capacity correlation increases. This means that the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0 will shift downward. However, the two ending points ð0; Q~ Þ and 2
ðaðbc1 =ðb 1ÞÞ b ; 0Þ stay the same. From the proof of Lemma 3, we know that on the curve H 11 ðQ 1 ; Q 2 Þ ¼ 0, we have 1 r dQ 1 =dQ 2 o 0. By symmetry, we can show that when the capacity correlation increases, the curve H 12 ðQ 1 ; Q 2 Þ ¼ 0 shifts downward as well, while the two ending points ð0; aðbc2 =ðb 1ÞÞ b Þ and ðQ~ ; 0Þ stay the same. From the proof of 1
Lemma 3, we know that on the curve H 12 ðQ 1 ; Q 2 Þ ¼ 0, 1 r dQ 2 =dQ 1 o 0. Consequently, when the capacity correlation increases, the crossing point of H 11 ðQ 1 ; Q 2 Þ ¼ 0 and H 12 ðQ 1 ; Q 2 Þ ¼ 0, i.e., ðQ 1 ; Q 2 Þ, can only move inside the region bounded by H 11 ðQ 1 ; Q 2 Þ ¼ 0, fc1 ; c2 g=ðb 1ÞÞ
b
H 12 ðQ 1 ; Q 2 Þ ¼ 0,
and
Q1 þQ2
¼ aðb min
. In this region, the value Q 1 þQ 2 of any point
ðQ 1 ; Q 2 Þ is less than the original ðQ 1 þ Q 2 Þ. To prove (ii), we analyze how H 21 ðQ 1 Þ changes w.r.t. the capacity correlation. Define ( ðQ 1 þ r 2 Þ 1=b A 1=b if 0 r r 2 r A Q 1 ; r 1 Z Q 1 ; ϕ2 ðr1 ; r2 Þ 0 otherwise: It can be verified that ϕ2 ðr 1 ; r 2 Þ is submodular. Thus, when the capacity correlation increases, E ϕ2 ðr 1 ; r 2 Þ decreases. Note that H 21 ðQ 1 Þ ¼ ðc1 γ ÞG 1 ðQ 1 Þ þ ððb 1Þ=bÞa1=b E ϕ2 ðr 1 ; r 2 Þ : So for any fixed Q1, H 21 ðQ 1 Þ decreases as the capacity correlation increases. By the unimodality conditions, Π ðQ 1 ; Q 2 Þ is unimodal for b ;Q b Þ. Thus, Q þ Q 4A, and achieves its maximum at ðQ 1
2
1
2
b ; H 2 ðQ Þ o 0 if Q o Q b .So, as the capacity H 21 ðQ 1 Þ 4 0 if Q 1 4 Q 1 1 1 1 1 b correlation increases, Q 1 decreases. Similarly, as the capacity b decreases. □ correlation increases, Q 2
Lemma 8. As the capacity correlation increases, the curve b þQ b ¼ Ag shifts towards the point ðγ ; γ Þ , while its two fðc1 ; c2 Þ : Q 1 2 ending points ðγ ; C 1 ðγ ÞÞ and ðC 2 ðγ Þ; γ Þ remain the same.
Proof. Note that the two ending points of the boundary curve are not affected by the capacity correlation. To see how the curve changes w.r.t. the capacity correlation, we fix c1 and check how c2 changes as the capacity correlation changes. From Lemma 7, when b decreases. Therefore, Q b has the capacity correlation increases, Q 1 2 b þQ b ¼ A. If c2 is fixed, then Q b to increase on the curve Q 1 2 2 decreases as the capacity correlation increases. When the capacity b decreases as c2 increases. Thus, when the correlation is fixed, Q 2 capacity correlation increases, c2 must decrease to have an b . So, when c1 is fixed, c2 will decrease as the capacity increased Q 2 b þQ b ¼ A. Similarly, when c2 correlation increases on the curve Q 1 2 is fixed, c1 will decrease as the capacity correlation increases on the same curve. □
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