Supply Diversification with Isoelastic Demand

Tao Li†, Suresh P. Sethi∗, Jun Zhang‡

†Leavey School of Business, Santa Clara University, Santa Clara, CA 95053, tli1@scu.edu

∗Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, TX 75080,sethi@utdallas.edu

‡School of Management, Fudan University, Shanghai 200433, China , jzhang4@gmail.com

March 5, 2014

Abstract

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-costsupplier. The firm also orders from the high-cost supplier if and only if the effective purchasecost from the low-cost supplier is greater than the actual purchase cost from the high-cost sup-plier. We also find that when the firm orders from both suppliers, the total order quantitydecreases as the correlation between the suppliers’ capacities increases.

Keywords: supply diversification, supply uncertainty, isoelastic demand

i

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-first-reliability-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), Federgruen

and Yang (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 ex-

ample, 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 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

1

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 demand-curve functions can lead to very different results in a multi-echelon

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 a firm’s operational strategies significantly. Since Li et al. (2013)

derive a firm’s sourcing strategy under a linear demand, it is important to show if the 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 obtaine 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 (i = 1, 2) has a random capacity Ri. On an order of quantity Qi by the firm from

supplier i, the supplier’s deliver quantity is min{Qi, Ri}. We assume that Ri has the distribution

function Gi(r) and the corresponding density function gi(r) ≥ 0 for r > 0. Denote Gi(r) ≡ 1−Gi(r)

and g(r1, r2) 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(Qi) = min{Qi, Ri} at the end of the first stage. Denote Q ≡ (Q1, Q2) as the vector of the

order quantities. Let S(Q) = S1(Q1) + S2(Q2) 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 S(Q), the firm decides the unit retail price p for the product. We

assume the demand to be price-dependent and isoelastic, that is, D(p) = ap−b with a > 0 and b > 1.

Unsold products are salvaged at a unit price γ < 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

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is resolved. Possible examples are those of a big food processor/retailer purchasing produce 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 (Q), which is equal to its expected second-stage

profit E[Π2 (Q)] less its expected purchase cost in the first stage. Consequently, the firm’s problem

can be formulated as follows:

maxQ≥0

{Π (Q) = E

[Π2 (Q)−

2∑i=1

ciSi(Qi)

]}, (1)

where

Π2 (Q) = maxp≥0

π(p) = p ·min{D(p), S(Q)}+ γ · (S(Q)−D(p))+ − δ · (D(p)− S(Q))+. (2)

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.

3. Analysis

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(p) is strictly concave and the optimal retail price for

a given S is

p∗ =

bγ

b− 1, if S ≥ a

(bγ

b− 1

)−b,(

S

a

)−1/b

, otherwise.

(3)

That is, when the total delivery is less than a (bγ/(b− 1))−b, the firm sets the price to sell all.

Otherwise, the firm sets the price at bγ/(b− 1) and salvages the leftover products. In our analysis,

the quantity a (bγ/(b− 1))−b plays a significant role; let A ≡ a (bγ/(b− 1))−b. Let fQ(s) be the

conditional density of the random variable S given Q. By (1) and (3), the firm’s first-stage problem

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can be reformulated as follows:

maxQ≥0

Π(Q) =

∫ A

0s(sa

)− 1bfQ(s)ds+

∫ ∞A

γ (A− s+ bs)

b− 1fQ(s)ds−

2∑i=1

ciE[Si(Qi)]. (4)

Note that the firm never sells more than A units of the product in total. Consequently, the

optimal order quantity must satisfy Q1 ≤ A and Q2 ≤ A. As can be seen from Equation (4), Π(Q)

has different expressions for Q1 + Q2 ≤ A and Q1 + Q2 > A. 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 (Q1, Q2) as the solutions to the first-order condition for Q1 +Q2 ≤ A:

Gi(Qi)

[(Qi +Q3−i)

− 1b − bci

b− 1a−

1b

]+

∫ Q3−i

0

∫ ∞Qi

[(Qi + r3−i)

− 1b − (Qi +Q3−i)

− 1b

]g(r1, r2)dridr3−i = 0 for i = 1, 2. (5)

Similarly, denote Q̂i (i = 1, 2) as the solution to the first-order condition for Q1 +Q2 > A:

b− 1

ba

1b

∫ A−Qi

0

∫ ∞Qi

[(Qi + r3−i)

− 1b −A−

1b

]g(r1, r2)dridr3−i−(ci−γ)Gi(Qi) = 0 for i = 1, 2. (6)

Define hi(·) = gi(·)/(1 − Gi(·)) 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∫ Q3−i

0

∫ ∞Qi

(Qi + r3−i)− 1

b−1g(r1, r2)dridr3−i + b

∫ Q3−i

0

[(Qi + r3−i)

− 1b − (Qi +Q3−i)

− 1b

]g(xi)dr3−i

− bhi(Qi)

∫ Q3−i

0

∫ ∞Qi

[(Qi + r3−i)

− 1b − (Qi +Q3−i)

− 1b

]g(r1, r2)dridr3−i ≥ 0, (7)

a

∫ A−Q̂i

0

∫ ∞Q̂i

(Q̂i + r3−i)− 1

b−1g(r1, r2)dridr3−i + b

∫ A−Q̂i

0

[(Q̂i + r3−i)

− 1b −A−

1b

]g(xi)dr3−i

− bhi(Q̂i)

∫ A−Q̂i

0

∫ ∞Q̂i

[(Q̂i + r3−i)

− 1b −A−

1b

]g(r1, r2)dridr3−i ≥ 0, (8)

where x1 = (Q1, r2) and x2 = (r1, Q2). Then, Π(Q) is unimodal.

The conditions in Lemma 1 ensure the unimodality of Π(Q), and are assumed throughout

the paper. Verifying these conditions, however, requires solving (5) and (6). Therefore, we provide

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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 G(r1, r2) = P{R1 > r1, R2 > r2} be the joint survival function of (R1, R2). The hazard

function of (R1, R2) is then defined as H = − logG. H’s gradient h = ∇H is called the hazard

gradient. Note that hi(r1,r2) can be interpreted as the conditional hazard rate of Ri evaluated at ri,

given Rj > rj for all j 6= i. That is, hi(r1,r2) = gi(ri | Rj > rj , j 6= i)/Gi(ri | Rj > tj , j 6= i), where

gi(· | Rj > rj , j 6= i) and Gi(· | Rj > rj , j 6= i) are, respectively, the conditional density and survival

function of Ri, given Rj > rj for all j 6= i. Refer to Johnson and Kotz (1975) and Marshall (1975)

for details. R1 and R2 are associated if Cov[f1(R1), f2(R2)] ≥ 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. RTI(Ri | Rj), if

P [Ri > ri | Rj > rj ] is increasing in rj for all ri; (b) Ri and Rj are right-corner-set increasing, i.e.,

RCSI(Ri, Rj), if P [Ri > ri, Rj > rj | Ri > r′i, Rj > r′j ] is increasing in r′i and r′j for each fixed

ri, rj ; (c) Ri and Rj are TP2(Ri, Rj), if the joint probability density g(ri, rj) is totally positive of

order 2, that is, g(ri, rj)g(r′i, r′j) ≥ g(ri, r

′j)g(r′i, rj) for all ri < r′i, rj < r′j . 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 i = 1, 2, any Qi ∈ [0, A] satisfies∫ Q3−i

0

∫ ∞Qi

(Qi + r3−i)− 1

b−1g(r1, r2)dridr3−i + b

∫ Q3−i

0

[(Qi + r3−i)

− 1b − (Qi +Q3−i)

− 1b

]g(xi)dr3−i

− bhi(Qi)

∫ Q3−i

0

∫ ∞Qi

[(Qi + r3−i)

− 1b − (Qi +Q3−i)

− 1b

]g(r1, r2)dridr3−i ≥ 0

where x1 = (Q1, r2) and x2 = (r1, Q2).

(ii) For i = 1, 2, hi(Qi) + h3−i(Q1, Q2) ≥ hi(Q1, Q2).

(iii) For i = 1, 2, (R1, R2) satisfies RTI(Ri | R3−i).

(iv) (R1, R2) satisfies RCSI(R1, R2).

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(v) (R1, R2) satisfies TP2(R1, R2).

(vi) (R1, R2) is a bivariate normal vector with non-negative correlation.

The conditions become stronger as we go down the list.

By Lemma 2, Π(Q) 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

Ci ≡ ci +

∫ a(

bcib−1

)−b

0

(b− 1

b

(ra

)− 1b − ci

)dGi(r). (9)

Theorem 1. The firm’s optimal order quantities are

Q∗1 = a

(bc1

b− 1

)−b, Q∗2 = 0, if c2 ≥ C1,

Q∗1 = 0, Q∗2 = a

(bc2

b− 1

)−b, if c1 ≥ C2,

Q∗1 = Q1, Q∗2 = Q2, if c1 < C2 and c2 < C1 and Q̂1 + Q̂2 ≤ A,

Q∗1 = Q̂1, Q∗2 = Q̂2, otherwise.

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 not 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(X)] ≥ E[f(Y )] holds for all supermodular functions f(·); 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 the firm diversifies. Then, Q∗1 + Q∗2 decreases as the capacity correlation

increases.

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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

This paper explores a firm’s sourcing strategy when facing unreliable suppliers and isoelastic de-

mand. 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, a 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.

Appendix: Proofs

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 (Q∗1, Q∗2) satisfy

Q∗1 = a (bc1/(b− 1))−b , Q∗2 = 0, if c2 ≥ C1,

Q∗1 = 0, Q∗2 = a (bc2/(b− 1))−b , if c1 ≥ C2,

Q∗1 = Q1, Q∗2 = Q2, if c1 < C2 , c2 < C1 and Q1 +Q2 ≤ A,

Q∗1 +Q∗2 > A , otherwise.

Proof of Lemma 3. Consider the order quantity such that Q1 + Q2 ≤ A. For i = 1, 2, let

H1i (Q1, Q2) ≡ b

b−1a− 1

b∂Π(Q1,Q2)

∂Qi. Next, we analyze the curves H1

i (Q1, Q2) = 0 in the (Q1, Q2)-

plane. Applying the implicit function theorem to H11 (Q1, Q2) = 0 and the unimodality condi-

7

tion, we know that on the curve H11 (Q1, Q2) = 0, −1 ≤ dQ1/dQ2 < 0; furthermore, the points

(a (bc1/(b− 1))−b , 0) and (0, Q̃2) lie on the curve, where

[∫∞Q̃2Q̃− 1

b2 g2(r2)dr2 +

∫ Q̃2

0 r− 1

b2 g2(r2)dr2

]−b=

a (bc1/(b− 1))−b < Q̃2.

By symmetry, on curve H12 (Q1, Q2) = 0, −1 ≤ dQ2/dQ1 < 0; the points (a (bc2/(b− 1))−b , 0) and

(Q̃1, 0) lie on the curve, where

[∫∞Q̃1Q̃− 1

b1 g1(r1)dr1 +

∫ Q̃1

0 r− 1

b1 g1(r1)dr1

]−b= a (bc2/(b− 1))−b < Q̃1.

If c1 > c2, then Q̃1 > a (bc2/(b− 1))−b > a (bc1/(b− 1))−b. Thus, only if Q̃2 > a (bc2/(b− 1))−b,

which is equivalent to c1 < C2, that there exists a unique interior solution for H11 (Q1, Q2) = 0 and

H12 (Q1, Q2) = 0. So, when c2 < c1 < C2, the optimal order quantities are (Q1, Q2), the unique

interior solution of (5). When c1 ≥ C2, the optimal order quantities are on the boundary, which

implies Q∗2 = 0. When c1 ≤ c2 < C1, the proof is similar. Notice that Q1 +Q2 may be greater than

A. We will show later that if Q1 +Q2 > A, then the optimal order quantities satisfy Q∗1 +Q∗2 > 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{Q∗1 = Q̂1, Q

∗2 = Q̂2, if Q̂1 + Q̂2 ≥ A,

Q∗1 +Q∗2 < A, otherwise.

Proof of Lemma 4. Assume that Q1 + Q2 > A. For i = 1, 2, let H2i (Qi) ≡ ∂Π(Q1, Q2)/∂Qi.

Recall that Q̂i = 0. If Q̂1 + Q̂2 ≥ A, then Q̂1 and Q̂2 are the optimal order quantities such

that Q1 + Q2 > A. Otherwise, Q∗1 + Q∗2 < 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.

Lemma 5. (i) Q1 + Q2 = A ⇔ Q̂1(c1) + Q̂2(c2) = A; (ii) Q1 + Q2 > A ⇔ Q̂1 + Q̂2 > A; (iii)

Q1 +Q2 < A⇔ Q̂1 + Q̂2 < A.

Proof of Lemma 5. We only prove the “⇒ ” part of (ii). The “⇒ ” part of (iii) can be proved

by interchanging “< ” and “> ”. The “⇒ ” of (i) then follows directly.

AssumeQ1(c1)+Q2(c2) > A, that is, Q2 > A−Q1. SinceH11 (Q1, Q2) = 0 and ∂H1

1 (Q1, Q2)/∂Q2 <

0, we can conclude that H11 (Q1, A−Q1) > 0. Since H2

1 (Q̂1) = 0 and H21 (Q1) is unimodal, Q1 < Q̂1.

8

Similarly, Q2 < Q̂2. Thus, Q̂1(c1) + Q̂2(c2) > Q1(c1) +Q2(c2) > 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.

Proof of Lemma 6. It is completed by proving the following three claims. Define Ci(γ) as the

effective purchase cost of supplier i assuming the actual purchase cost from it is γ.

Claim 1: Q∗1 + Q∗2 ≤ A if c1 ≥ C2(γ) or c2 ≥ C1(γ). Proof of Claim 1. Since Π(Q1, Q2)

is unimodal for Q1 + Q2 > A, to have an interior optimal solution, we must have H21 (0) > 0 ⇔

c1 < C2(γ). Otherwise, H21 (Q1) < 0 for any Q1 ∈ (0, A]. That is, the optimal solution satisfying

Q1 +Q2 > A is a boundary solution. The Claim in the case c2 ≥ C1(γ) can be proved similarly.

Claim 2: Q∗1 + Q∗2 ≤ A if c1 < C2(γ), c2 < C1(γ), and Q̂1 + Q̂2 ≤ A. Proof of Claim 2.

If c1 < C2(γ), then, H21 (0) > 0. Since H2

1 (A) = (γ − c1)G1(A) < 0, there exists a unique point

Q̂1 ∈ (0, A) such that H21 (Q̂1) = 0. By Symmetry, if c2 < C1(γ), then there exists a unique point

Q̂2 ∈ (0, A) such that H22 (Q̂2) = 0. However, if Q̂1 + Q̂2 ≤ A, then for any (Q1, Q2) such that

Q1 +Q2 > A with Q1 > Q̂1, we have H21 (Q1) < 0; and for any (Q1, Q2) such that Q1 +Q2 > A with

Q2 > Q̂2, we have H22 (Q2) < 0. These two conditions covers all (Q1, Q2)’s such that Q1 +Q2 > A.

Thus the optimal solution satisfying Q∗1 +Q∗2 ≥ A must satisfy Q∗1 +Q∗2 = A.

Claim 3: Q∗1 + Q∗2 ≥ A if c1 < C2(γ), c2 < C1(γ), and Q̂1 + Q̂2 > A. Proof of Claim

3. By Lemma 5, when Q̂1 + Q̂2 > A, Q1 + Q2 > A. Since the profit function is unimodal for

Q1 + Q2 ≤ A, for any Q1 < Q1 such that Q1 + Q2 ≤ A, H11 (Q1, Q2) > 0; and for any Q2 < Q2

such that Q1 + Q2 ≤ A, H12 (Q1, Q2) > 0. These two conditions cover all (Q1, Q2)’s such that

Q1 +Q2 > A. Thus, the optimal solution satisfying Q∗1 +Q∗2 ≤ A must satisfy Q∗1 +Q∗2 = A.

Proof of Theorem 1. Follows from Lemmas 3-6.

Proof of Lemma 2. (i)⇒ unimodality conditions: By the proof of Theorem 1, any (Q1, Q2)

solving H11 (Q1, Q2) = H1

2 (Q1, Q2) = 0 and any Q1 solving H21 (Q1) = 0 satisfy the property that

Q1 ∈ [0, A]. When Q1 ∈ [0, A] and we fix Q2 = A−Q1, then Q2 ∈ [0, A]. It can be easily seen that

(i) implies the unimodality conditions.

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(ii)⇒(i): Define

Θ(Q2 | Q1) ≡∫ Q2

0

∫ ∞Q1

(Q1 + r2)−1b−1g(r1, r2)dr1dr2 + b

∫ Q2

0

[(Q1 + r2)−

1b − (Q1 +Q2)−

1b

]g(Q1, r2)dr2

− bh1(Q1)

∫ Q2

0

∫ ∞Q1

[(Q1 + r2)−

1b − (Q1 +Q2)−

1b

]g(r1, r2)dr1dr2.

Then Θ(0 | Q1) = 0, and Θ(Q2 | Q1) ≥ 0 if ∂Θ(Q2 | Q1)/∂Q2 ≥ 0, that is,∫∞Q1g(r1, Q2)dr1∫∞

Q2

∫∞Q1g(r1, r2)dr1dr2

+ h1(Q1) ≥∫∞Q2g(Q1, r2)dr2∫∞

Q2

∫∞Q1g(r1, r2)dr1dr2

. (10)

Similarly, we can show that the proof goes through in the other direction. (10) is equivalent to

condition (ii).

(iii)⇒(ii): A sufficient condition for h1(Q1) + h2(Q1, Q2) ≥ h1(Q1, Q2) is h1(Q1) ≥ h1(Q1, Q2).

From Theorem 2.1 of Karia and Deshpande (1999), for any Q1 and Q2, h1(Q1) ≥ h1(Q1, Q2) ⇔

RTI(R2 | R1). So, RTI(R1 | R2) is a sufficient condition for h1(Q1) + h2(Q1, Q2) ≥ h1(Q1, Q2).

Similarly for RTI(R2 | R1).

The implications (vi)⇒(v)⇒(iv)⇒(iii) follow from the facts that TP2(R1, R2)⇒ RCSI(R1, R2)⇒

RTI(Ri | R3−i) 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∗1 + Q∗2 ≤ A. Then, Q∗1 + Q∗2 decreases as the capacity correlation

increases. (ii) Assume that Q∗1 +Q∗2 > A. Then, Q∗i , i = 1, 2, decreases as the capacity correlation

increases.

Proof of Lemma 7. To prove (i), we analyze how H11 (Q1, Q2) changes w.r.t. the capacity

correlation. Define

φ1(r1, r2) ≡{

(Q1 + r2)−1b − (Q1 +Q2)−

1b , if 0 ≤ r2 ≤ Q2, r1 ≥ Q1,

0 , otherwise.

It can be verified that φ1(r1, r2) is a submodular function. So, E[φ1(R1, R2)] decreases as the

correlation between R1 and R2 increases (Shaked and Shanthikumar, 2007, 9.A.4, p.395). Note

that H11 (Q1, Q2) = G1(Q1)

[(Q1 +Q2)−

1b − bc1

b−1a− 1

b

]+ E[φ1(r1, r2)]. So, for any fixed Q1 and Q2,

H11 (Q1, Q2) decreases as the capacity correlation increases. Since ∂H1

1 (Q1, Q2)/∂Q2 < 0, then for

any fixedQ1, Q′2 obtained from H11 (Q1, Q

′2) = 0 decreases as the capacity correlation increases. This

10

means that the curve H11 (Q1, Q2) = 0 will shift downward. However, the two ending points (0, Q̃2)

and (a (bc1/(b− 1))−b , 0) stay the same. From the proof of Lemma 3, we know that on the curve

H11 (Q1, Q2) = 0, we have −1 ≤ dQ1/dQ2 < 0. By symmetry, we can show that when the capacity

correlation increases, the curve H12 (Q1, Q2) = 0 shifts downward as well, while the two ending points

(0, a (bc2/(b− 1))−b) and (Q̃1, 0) stay the same. From the proof of Lemma 3, we know that on the

curve H12 (Q1, Q2) = 0, −1 ≤ dQ2/dQ1 < 0. Consequently, when the capacity correlation increases,

the crossing point of H11 (Q1, Q2) = 0 and H1

2 (Q1, Q2) = 0, i.e., (Q1, Q2), can only move inside the

region bounded by H11 (Q1, Q2) = 0, H1

2 (Q1, Q2) = 0, and Q1 + Q2 = a (bmin{c1, c2}/(b− 1))−b.

In this region, the value Q1 +Q2 of any point (Q1, Q2) is less than the original (Q1 +Q2).

To prove (ii), we analyze how H21 (Q1) changes w.r.t the capacity correlation. Define

φ2(r1, r2) ≡{

(Q1 + r2)−1b −A−

1b , if 0 ≤ r2 ≤ A−Q1, r1 ≥ Q1,

0 , otherwise.

It can be verified that φ2(r1, r2) is submodular. Thus, when the capacity correlation increases,

E[φ2(r1, r2)] decreases. Note that H21 (Q1) = −(c1 − γ)G1(Q1) + b−1

b a1bE[φ2(r1, r2)]. So for any

fixed Q1, H21 (Q1) decreases as the capacity correlation increases. By the unimodality conditions,

Π(Q1, Q2) is unimodal for Q1 +Q2 > A, and achieves its maximum at (Q̂1, Q̂2). Thus, H21 (Q1) > 0

if Q1 > Q̂1; H21 (Q1) < 0 if Q1 < Q̂1. So, as the capacity correlation increases, Q̂1 decreases.

Similarly, as the capacity correlation increases, Q̂2 decreases.

Lemma 8. As the capacity correlation increases, the curve {(c1, c2) : Q̂1 + Q̂2 = A} shifts towards

the point (γ, γ) , while its two ending points (γ,C1(γ)) and (C2(γ), γ) remain the same.

Proof of Lemma 8. 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 the capacity

correlation increases, Q̂1 decreases. Therefore, Q̂2 has to increase on the curve Q̂1 + Q̂2 = A. If c2

is fixed, then Q̂2 decreases as the capacity correlation increases. When the capacity correlation is

fixed, Q̂2 decreases as c2 increases. Thus, when the capacity correlation increases, c2 must decrease

to have an increased Q̂2. So, when c1 is fixed, c2 will decrease as the capacity correlation increases

on the curve Q̂1 + Q̂2 = A. Similarly, when c2 is fixed, c1 will decrease as the capacity correlation

increases on the same curve.

11

References

Anupindi, R., R. Akella. 1993. Diversification under supply uncertainty. Management Science

39(8) 944–963.

Babich, V., A. N. Burnetas, P. H. Ritchken. 2007. Competition and diversification effects in supply

chains with supplier default risk. Manufacturing & Service Operations Management 9(2) 123–

146.

Barlow, R. E., F. Proschan. 1975. Statistical Theory of Reliability and Life-testing . Holt Rinehart

& Wilson, New York.

Dada, M., N. Petruzzi, L. B. Schwarz. 2007. A newsvendor’s procurement problem when suppliers

are unreliable. Manufacturing & Service Operations Management 9(1) 9–32.

Esary, J. D., F. Proschan, D. W. Walkup. 1967. Association of random variables, with applications.

The Annals of Mathematical Statistics 38(5) 1466–1474.

Federgruen, A., N. Yang. 2009. Optimal supply diversification under general supply risks. Opera-

tions Research 57(6) 1451–1468.

Federgruen, A., N. Yang. 2011. Procurement strategies with unreliable suppliers. Operations

Research 59(4) 1033–1039.

Feng, Q., R. Shi. 2012. Sourcing from multiple suppliers for price-dependent demands. Production

and Operations Management 21(3) 547–563.

Gerchak, Y., M. Parlar. 1990. Yield variability, cost tradeoffs and diversification in the EOQ model.

Naval Research Logistics 37(5) 341–354.

Hill, T. 2000. Manufacturing Strategy . 3rd ed. McGraw-Hill/Irwin, Boston, MA.

Hoch, S.J., B.D. Kim, A.L. Montgomery, P.E. Rossi. 1995. Determinants of store-level price elas-

ticity. Journal of Marketing Research 32(1) 17–29.

Johnson, N. L., S. Kotz. 1975. A vector multivariate hazard rate. Journal of Multivariate Analysis

5(1) 53–66.

12

Karia, S. R., J. V. Deshpande. 1999. Bounds for the hazard gradients in the competing risks set

up. Journal of Statistical Planning and Inference 75(2) 363–377.

Lau, A.H.L., H-S. Lau. 2003. Effects of a demand-curve’s shape on the optimal solutions of a multi-

echelon inventory/pricing model. European Journal of Operational Research 147(3) 530–548.

Li, J., S. Wang, T.C.E. Cheng. 2010. Competition and cooperation in a single-retailer two-supplier

supply chain with supply disruption. International Journal of Production Economics 124(1)

137–150.

Li, T., S. P. Sethi, J. Zhang. 2013. Supply diversification with responsive pricing. Production and

Operations Management 22(2) 447–458.

Marshall, A. W. 1975. Some comments on the hazard gradient. Stochastic Processes and their

Applications 3(3) 293–300.

Mulhern, F.J., R.P. Leone. 1991. Implicit price bundling of retail products: a multiproduct approach

to maximizing store profitability. Journal of Marketing 55(4) 63–76.

Nilsen, J. 2013. Delayed production and raw materials inventory under uncertainty. International

Journal of Production Economics 146(1) 337–345.

Parlar, M., D. Wang. 1993. Diversification under yield randomness in two simple inventory models.

European Journal of Operational Research 66(1) 52–64.

Petruzzi, N., M. Dada. 1999. Pricing and the newsvendor problem: A review with extensions.

Operations Research 47(2) 183–194.

Ramasesh, R., J. K. Ord, J. C. Hayya, A. Pan. 1991. Sole versus dual sourcing in stochastic

leadtime (s, Q) inventory models. Management Science 37(4) 428–443.

Shaked, M., J. G. Shanthikumar. 2007. Stochastic orders. Springer, New York.

Shi, R., J. Zhang, J. Ru. 2013. Impacts of power structure on supply chains with uncertain demand.

Production and Operations Management 22(5) 1232–1249.

Tang, C. S., R. Yin. 2007. Responsive pricing under supply uncertainty. European Journal of

Operational Research 182(1) 239–255.

13

Tellis, G.J. 1988. The price elasticity of selective demand: A meta-analysis of econometric models

of sales. Journal of Marketing Research 25(4) 331–341.

Tramontana, F. 2010. Heterogeneous duopoly with isoelastic demand function. Economic Modelling

27(1) 350–357.

Wang, J-C., A.H.L. Lau, H-S. Lau. 2012. Practical and effective contracts for the dominant re-

tailer of a newsvendor product with price-sensitive demand. International Journal of Production

Economics 138(1) 46–54.

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