Risk pooling strategy in a multi-echelon supply chain with price-sensitive demand

Math Meth Oper Res (2008) 67:391–421DOI 10.1007/s00186-007-0203-8

ORIGINAL ARTICLE

Risk pooling strategy in a multi-echelon supply chainwith price-sensitive demand

Yue Dai · Shu-Cherng Fang · Xiaoli Ling ·Henry L. W. Nuttle

Published online: 11 January 2008© Springer-Verlag 2008

Abstract This paper studies a nonstationary inventory and pricing problem. Weconsider a two-echelon supply chain with one supplier and two retailers, in whichthe supplier carries all inventory to supply the retailers. Both the reserved and pooledinventory systems are analyzed. Results with normally distributed demands are com-pared. Assuming the random demand at each retailer is price-sensitive, we furtherconsider the cases when the retailers have and do not have service level requirements.We start with analyzing inventory and pricing strategies for the supplier in a one-periodscenario. Then we extend our analysis to both the backlogging and lost-sale scenariosin an infinite planning horizon.

The first author’s research is sponsored by Grant No. 70502009 and No. 70432001 of the Chinese NationalNatural Science Foundation and the second author’s research is sponsored by Grant #W911NF-04-D-0003of the US Army Research Office and Grant #DMI-0553310 of the US National Science Foundation.

Y. Dai (B)School of Management, Fudan University, Shanghai, 200433 Chinae-mail: [email protected]

S.-C. Fang · X. Ling · H. L. W. NuttleIndustrial Engineering and Operations Research, North Carolina State University,Raleigh, NC 27695-7906, USAe-mail: [email protected]

S.-C. FangMathematical Sciences and Industrial Engineering, Tsinghua University, Beijing, China

X. Linge-mail: [email protected]

H. L. W. Nuttlee-mail: [email protected]

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Keywords Supply chain · Pooled inventory system · Reserved inventory system ·Pricing

1 Introduction

In the late 1990s, to enhance customer service, Cadillac set up a new sales regime.Instead of letting dealers place order on what they thought customers might desire,Cadillac shipped only demonstration vehicles to the dealers, while a main inventorywas kept at a distribution center in central Florida. This is refereed to as a pooledinventory system, or risk pooling. In operations management, risk pooling is oftenachieved by consolidating a product with random demands into one location. Its bene-fits have been well known. Yet, other inventory carrying strategies are widely appliedin reality as well. For example, in the electronics industry, an electronics manufac-turing service provider (EMS) carries an inventory of CPU chips for two or moreoriginal equipment manufacturers (OEM). The current practice dictated by the OEMsis to carry physically separate inventory, which is referred to as reserved inventory, foreach company. An immediate question is that “should the OEMs prefer the pooled orreserved inventory policy?”

To answer the question, in this paper, we consider a supply chain with one supplierand two retailers, in which the supplier carries all inventory to supply the retailers. Therandom demand at each retailer is assumed to be price-sensitive following a multipli-cative format. Two inventory policies are studied: for the reserved inventory policy, thesupplier carries separate inventory for each retailer; for the pooled inventory policy,the supplier carries a central inventory shared by both retailers. We investigate boththe single period (selling period) and infinite horizon models. For the single-periodmodel, we consider the cases when the retailers have and do not have service levelrequirements. For the infinite horizon model, we study both the backlogging and lose-sale cases. The profits of the supplier and the two retailers for each scenario withnormally distributed demands are compared.

There are other papers investigating inventory pooling in a multi-echelon supplychain. Anupindi and Bassok (1999) considered a two-echelon supply chain with asingle supplier and two retailers. In their model, when a demand cannot be met byone retailer because of a stockout, the customer may go to the other retailer, which isreferred to as “market search.” The authors compared two scenarios: one in which theretailers hold stocks separately and the other in which retailers cooperate to centralizestocks at a single location. They found that whether one system is better than the otherdepends on the probability of customers’ switching to another retailer upon encounter-ing a stockout. They also discussed the possibility of optimizing the wholesale price orintroducing holding cost subsidies in order to coordinate the supply chain. Our modelis quite different from that of Anupindi and Bassok (1999). In their model, the inven-tory decision is made by the retailers, who bear all the inventory risk without servicelevel requirements. In contrast, we assume the supplier carries all inventory and studythe cases when the retailers have and do not have service level requirements. Anotherdifference is that they assumed the demand at each retailer is exogenous, while therandom demand is price-sensitive in our model.

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Risk pooling strategy in a multi-echelon supply chain 393

Netessine and Rudi (2006) considered two supply chain strategies, namely“traditional operation” and “drop shipping.” Similar to our work, in the former situ-ation, the retailer holds the inventory purchased from the supplier, while the supplierholds the inventory in the latter. They also considered a two-echelon system, in whichthe second echelon actually consists of a collection of identical retailers. The retailersare mainly intermediaries between the end customer and the supplier and they func-tion as a single joint retailer. Netessine and Rudi compared the traditional operationand drop shipping strategies with normally distributed demands and found that thesupply chain’s profit may be higher for either strategy. Cachon (2004) considered thecases of “push contract,” in which the retailer bears all inventory risk, and the “pullcontract,” in which the supplier carries all inventory for one period. The study focusedon identifying Pareto-optimal price-only contracts and supply chain efficiency undersuch contracts. Different from our model, there is only one retailer in Cachon’s modeland the random demand is exogenous.

Of the existing literature, our work is closely related to Bartholdi and Kemahlioglu-Ziya (2005). They also considered a supply chain with one supplier and two retailers,in which the supplier bears all inventory risk. They found that the total system profitwill increase after pooling the inventory. In addition, using the Shapely value to allo-cate the additional profit, they analyzed various schemes by which the supplier maypool inventory. By allocating the Shapely value, they could coordinate the whole sup-ply chain. Our model is different from theirs. Assuming the random demand is price-sensitive and follows a multiplicative format, we have analyzed both the single-periodand infinite horizon models. In addition, we have compared the profits of the supplierand retailers for different scenarios with normally distributed demands.

The analysis in this paper is related to our earlier work (Ling et al. 2005). A major dif-ference lies in the demand format. In Ling et al. (2005), all analysis were based on a spe-cific demand function in an additive format. Here we consider a multiplicative demandfunction to derive different results. Moreover, Ling et al. (2005) only considered asingle-period model, while this paper considers the infinite horizon model as well.Both the backlogging and lose-sale scenarios are studied with normally distributeddemands.

The rest of the paper is organized as follows: in Sect. 2, we present details of theproblem. In Sect. 3, assuming the retailers have no service level requirements, we startour study with a single-period model. We analyze the supplier’s decision in both thereserved inventory and pooled inventory systems. Section 4 considers the case whenthe retailers have the service level requirements. In Sect. 5, we extend the model to aninfinite horizon and study the optimal inventory level and performance of the supplychain for both the backlogging and lost-sale scenarios. We conclude the paper with adiscussion in Sect. 6. The Appendix gives all the proofs.

2 Problem description

Throughout this paper, we study the following two inventory systems for the supplier:

1. Reserved inventory system: the supplier holds a separate inventory for each ofretailers 1 and 2;

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2. Pooled inventory system: the supplier holds one central inventory to be shared bythe two retailers.

For the reserved inventory system, at the beginning of the period, the supplier hasa separate stock x1, x2 for retailers 1 and 2, respectively, at a manufacturing cost ofc per unit. After retailer i (i = 1, 2) observes the realization of its local demand,she places an order to the supplier. Retailer i receives the inventory immediately andpays a wholesale price of wi (wi > c > 0) for each unit received. Let mi be themarkup/margin on the wholesale price retailer i (i = 1, 2) charges. In other words,the retail price at retailer i is pi = wi + mi . The supplier carries all inventory toreplenish the retailers. If the stock xi of the supplier cannot satisfy the orders fromretailer i , the unsatisfied portion results in lost sales. If the stock xi exceeds retaileri’s order, stock left at the end of the season are disposed at a unit cost of h (|h| ≤ c).Note that h may be negative, in which case it represents the salvage value.

For the pooled inventory system, the supplier only has one central distribution cen-ter, and it is shared by the two retailers. At the beginning of the period, the supplierstocks x p for retailers 1 and 2 at a unit manufacturing cost of c. After the retailersobserve the realization of their demands, they place orders to the supplier and pay awholesale price wp for each unit received. If the stock x p of the supplier cannot satisfythe combined order, the unmet portion of the order becomes lost sales. We assume themarkup is m p for both of the retailers. In the pooled inventory case, when inventorycannot satisfy the total demand, the supplier needs to allocate the product to each of theretailers. There are a number of papers discussing inventory allocation for differencescenarios (Cachon and Lariviere 1999; Dai et al. 2005, 2006). Our model focuses onthe impact of the different policies on the profit of the supplier and the total profit ofthe retailers. Hence, we regard the two retailers as one joint retailer and ignore thedetails on how the capacity is allocated between the retailers.

The random demand at retailer i is denoted as Di with CDF Fi (·) and PDF fi (·).We assume D1 and D2 are independent and let Dp be the joint demand for the retailerswith PDF f p(·) and CDF Fp(·). Note that the joint demand Dp = D1 + D2, and Fp(·)is the convolution of F1(·) and F2(·).

We assume the random demand at each retailer is price-sensitive and follows amultiplicative format (Bernstein and Federgruen 2004):

Di (wi ) = y(wi )εi , i = 1, 2,

where y(wi ) is a deterministic, decreasing function of the wholesale price wi , takingthe following form of

y(wi ) = aw−bi where a > 0, b > 1,

and εi is a random variable defined on a range [A, B] (B > A ≥ 0) with CDF Gi (·)and PDF gi (·), whose mean value is µi and standard deviation is σi .

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In the pooled inventory case, we use Dp(wp) to denote the joint demand for theretailers at the wholesale price wp. In this case,

Dp(wp) = D1(wp) + D2(wp)

= y(wp)(ε1 + ε2)

= y(wp)εp,

where

εp = ε1 + ε2.

Notice that the random variable εp is defined on the range [2A, 2B] with a meanvalue of µp = µ1 + µ2. We use G p(·) and gp(·) to denote the CDF and PDF of εp,respectively, and G p(·) becomes the convolution of G1(·) and G2(·).

3 Retailers have no service requirements

In this section, we consider the case when the retailers do not have service levelrequirements. In Sect. 3.1, we analyze the inventory and pricing strategies of the sup-plier in the reserved inventory system, and the pooled inventory system is analyzedin Sect. 3.2. We compare the results of these two systems with normally distributeddemands in Sect. 3.3. The case when the retailers have the service level requirementsis analyzed in Sect. 4.

3.1 Reserved inventory system

In our model, the supplier decides the wholesale price and the inventory level for eachretailer in order to maximize the profit of his own, and these decisions must be madebefore the demand is observed. Note that the wholesale price must be greater thanmanufacturing cost. Otherwise, the supplier cannot make profit.

Following Petruzzi and Data (1999), we define the stocking factor for retailer i byzi = xi/y(wi ), where xi is the stock kept by the supplier for retailer i and y(wi ) isthe deterministic component of retailer i’s demand. Let �ri (zi , wi ) be the supplier’sexpected profit associated with retailer i when he carries the stocking factor zi forretailer i and charges her a wholesale price of wi . Assuming ε1 and ε2 are indepen-dent, and thus D1 and D2 are also independent. Since the supplier carries separateinventory for each retailer whose demand is independent, the supplier’s expected profit,�r (z1, z2, w1, w2), is separable.

We define the expected excess stock, �1(x), and the expected shortage, �1(x),when the inventory level is x as follows:

�1(x) =x∫

A

(x − u)g1(u)du,

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and

�1(x) =B∫

x

(u − x)g1(u)du,

where g1(·) is the PDF of ε1 as defined before.By definition, we know that �1(x) is a nonnegative function of x . Taking the first

derivative of �1(x) with respect to x , we have �′1(x) = G1(x)−1 ≤ 0. Hence �1(x)

is decreasing in x . In addition, it is easy to show that �1(x) and �1(x) satisfy thefollowing equation

�1(x) = �1(x) − x + µ1.

Since the supplier’s expected profit, �r (z1, z2, w1, w2), is separable, the supplier’sproblem can be separated into two problems

maxz1,w1≥c

�r1(z1, w1),

maxz2,w2≥c

�r2(z2, w2),

where

�ri (zi , wi ) =zi∫

A

{wi (y(wi )u) − hy(wi )(zi − u)} gi (u)du

+B∫

zi

wi y(wi )zi gi (u)du + cy(wi )zi

= (wi − c)(y(wi )µi ) − y(wi )(c + h)�i (zi )

−y(wi )(wi − c)�i (zi ), i = 1, 2. (1)

Due to the identical structure of �r1 and �r2, in the remainder of this section we focuson max

z1,w1≥c�r1(z1, w1) only.

Lemma 1 The supplier’s expected profit associated with retailer i , �r1(z1, w1), hasthe following properties:

1. For a given w1, �r1(z1, w1) is concave in z1.2. For a given z1, �r1(z1, w1) is a quasi-concave function in w1.3. For a given z1, the optimal price is determined by

w∗1(z1) = w0 + b

b − 1

(c + h)�1(z1)

µ1 − �1(z1),

where w0 = bcb−1 .

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4. For a given w1, the optimal stocking factor is determined by

z∗1(w1) = G−1

1

(w1 − c

w1 + h

).

Substituting w1 = w∗1(z1) into Eq. (1) for i = 1, the optimization problem becomes

a maximization problem over a single variable z1, i.e.,

maxz1

�r1(z1, w∗1(z1)), (2)

where

�r1(z1, w∗1(z1)) = y(w∗

1(z1))[µ1(w

∗1(z1) − c) − (c + h)�1(z1)

− (w∗1(z1) − c)�1(z1)

].

Assume z∗1 is the optimal solution of problem (2). Then by Lemma 1 the optimal

inventory and pricing decision in the reserved inventory system is to stock x∗1 =

y(w∗1)z∗

1 for retailer 1 and charge her the wholesale price of w∗1 .

Before we illustrate how to determine z∗1, we introduce the concept of failure rate.

For an arbitrary random variable with CDF F(·) and PDF f (·), we use h(u), u ∈ R,to denote the generalized failure rate

h(u) = u f (u)

1 − F(u),

and r(u) the classical failure rate

r(u) = f (u)

1 − F(u).

The classical failure rate gives roughly the percentage reduction in the stock-outprobability when increasing the stock by one unit, while the generalized failure rateshows roughly the percentage reduction in the stock-out probability when increasingthe stock by 1%. An increasing failure rate (IFR) has appealing implications. As asupplier holds more stock for a retailer, the retailer’s order quantity becomes less elas-tic, i.e., the probability of stock-out becomes smaller. Many distributions are of IFR,such as the normal and uniform distributions. If a distribution is of IFR, it is clearlyalso of IGFR (increasing general failure rate), but the reverse may not hold.

Back to our optimization problem, we now show the uniqueness conditions of theoptimal solution and illustrate how to calculate z∗

1.

Theorem 1 Let r(·) be the failure rate of random variable ε1. If r(z1) satisfies thefollowing inequality

2r(z1)2 + r ′(z1) > 0,

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then there is a unique optimal solution z∗1 within region [A,B], which satisfies

−c − h

1 − G1(z1)+ w∗

1(z1) + h = 0.

Recall that the retailers do not hold any inventory, and the margin of retailer i ismi . Let πri (xi ) be retailer i’s expected profit if the supplier stocks an inventory levelof xi for her, and we have

πri (xi ) = E[mi min(xi , Di )], for i = 1, 2.

3.2 Pooled inventory system

In this subsection, we consider the pooled inventory system in which the supplierholds a central inventory level of x p and charges each retailer a common wholesaleprice wp to maximize the profit of his own.

For this case, we define the inventory stocking factor as z p = x p/y(wp), and theexpected profit of the supplier can be written as

�p(z p, wp) =z p∫

A

{wp

(y(wp)u

) − hy(wp)(z p − u)}

gp(u)du

+B∫

z p

wp y(wp)z pgp(u)du + cy(wp)z p

= (wp−c)(y(wp)µp)−y(wp)(c + h)�p(z p)−y(wp)(wp−c)�p(z p).

(3)

The supplier makes the inventory and pricing decision to maximize his expected profit,and his decision problem is

maxz p,wp≥c

�p(z p, wp).

Lemma 2 The supplier’s expected profit in the pooled inventory system, �p(z p, wp),has the following properties:

1. For a given wp, �p(z p, wp) is concave in z p.2. For a given z p, �p(z p, wp) is a quasi-concave function in wp.3. For a given z p, the optimal price is determined by

w∗p(z p) = w0 + b

b − 1

(c + h)�p(z p)

µp − �p(z p),

where w0 = bcb−1 .

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4. For a given wp, the optimal stocking factor is determined by

z∗p(wp) = G−1

p

(wp − c

wp + h

).

Substituting wp = w∗p(z p) in Eq. (3), the optimization problem becomes a

maximization problem with a single variable z p

maxz p

�p(z p, w∗p(z p)), (4)

where

�p(z p, w∗p(z p)) = y(w∗

p(z p))[µp(w

∗p(z p) − c) − (c + h)�p(z p)

− (w∗p(z p) − c)�p(z p)

].

Assume z∗p is the optimal solution of problem (4). Then the optimal stocking and

pricing policy in the pooled inventory system is to stock x∗p = y(w∗

p)z∗p units and

charge the retailers a wholesale price of w∗p, determined by Lemma 2. The following

theorem gives the uniqueness condition for the optimal solution z∗p.

Theorem 2 Let r(z p) be the failure rate of the random variable εp. If r(z p) satisfiesthe following inequality

2r(z p)2 + r ′(z p) > 0,

there is a unique optimal solution z∗p within the region [2A, 2B] which satisfies

−c − h

1 − G p(z p)+ w∗

p(z p) + h = 0.

Recall that the retailers do not hold any inventory, and their margin is m p. Lettingπp(x p) be the profit of the retailers when the supplier keeps an inventory level of x p,we have

πp(x p) = E[m p min(x p, Dp)].

3.3 Comparative results

Sections 3.1 and 3.2 analyze the inventory and pricing decisions under the reservedand pooled inventory systems, respectively. In this subsection, we compare the resultsin these two systems with normally distributed demands.

Lemma 3 For both the reserved inventory system (i = 1, 2) and the pooled inventorysystem (i = p), the optimal stocking factor z∗

i (wi ) is nondecreasing in wi .

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Same as before, we defined the range of the random component of the demand εi

as [A, B] with A, B ≥ 0, which implies nonnegativity of the demand. And we usenormal distribution to approximate this demand distribution, by assuming εi is nor-mally distributed. In addition, we let � and φ be the CDF and the PDF of the standardnormal distribution, respectively, and �−1 the inverse function of �.

Lemma 4 Assume ε1 and ε2 are i.i.d. and normally distributed. Let �1(·) = �2(·) =�(·) and G1(·) = G2(·) = G(·), then we have

�p(2u) ≤ 2�(u) for any u ∈ R.

For the case in which ε1 and ε2 are i.i.d. and normally distributed, we can providea detailed comparison of the reserved and pooled inventory systems.

Theorem 3 Assume the supplier charges the same wholesale prices in the reservedand pooled inventory systems, i.e., w1 = w2 = wp = w. When ε1 and ε2 are i.i.d.and normally distributed, if w−c

w+h ≥ 0.5, then the sum of the optimal stocking factorsfor the retailers 1 and 2 in the reserved inventory system is at least as large as theoptimal stocking factor in the pooled inventory system, i.e., z∗

p(w) ≤ z∗1(w) + z∗

2(w).Otherwise, z∗

p(w) > z∗1(w) + z∗

2(w).

Theorem 4 If ε1 and ε2 are i.i.d. and normally distributed, the optimal profit of thesupplier in the pooled inventory system is at least as large as that in the reservedinventory system, i.e., �∗

p ≥ �∗r .

Theorem 5 Assume that w1 = w2 = wp = w and m1 = m2 = m p = m. If ε1 andε2 are independently, identically and normally distributed random variables, then theretailers’ total expected profit in the pooled inventory case is at least as large as thatin the reserved inventory case, i.e., πr1(x1) + πr2(x2) ≤ πp(x p).

In the absence of the service level requirements, when the demands are normally dis-tributed, the comparative results indicate that the supplier will always prefer the pooledinventory system. Meanwhile, if the supplier charges the same wholesale prices underthe reserved and pooled inventory systems, the retailers will also prefer the pooledinventory policy. In other words, pooling the inventory together becomes a win–winstrategy for the supplier and the two retailers.

4 Retailers have service level requirements

In this section, we consider the case when the retailers have service level require-ments. In Sect. 4.1, we analyze the inventory and pricing strategies of the supplier inthe reserved inventory system, and the pooled inventory system is studied in Sect. 4.2.We compare the results under these two systems with normally distributed demandsin Sect. 4.3.

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4.1 Reserved inventory system

In this subsection, we analyze the case when the retailers have the bargain powerto request the supplier to carry reserved inventory so that the retailers’ service levelrequirements are satisfied. We use the same notations as before, and let ρi be theservice level requirements for retailers i . The supplier maximizes his profit associatedwith retailer i by solving the following problem

maxxi ,wi ≥c

�ri (xi , wi )

s.t. P(Di (wi ) < xi ) ≥ ρi for i = 1, 2.

Introducing the stocking factor zi = xi/y(wi ), the problem becomes

maxzi ,wi ≥c

�ri (zi , wi )

s.t. Gi (zi ) ≥ ρi for i = 1, 2.

Recall that without service level requirements, we can obtain the optimal stockingfactor z∗

i (i = 1, 2) following the procedure in Sect. 3.1. For the problem with servicelevels constraints, we know that Gi (zi ) is nondecreasing in zi and �ri (zi , w

∗i (zi )) is

a quasi-concave function over the range [c,+∞]. Hence the optimal stocking factorfor the supplier with the service level requirement is max(z∗

i , G−1i (ρi )).

Similar as in Sect. 3.1, the retailers do not hold any inventory, and the margin ofretailer i is mi . Letting πri (xi ) be the expected profit of retailer i when the supplierkeeps an inventory level of xi for her, we have

πri (xi ) = E[mi min(xi , Di )] for i = 1, 2.

4.2 Pooled inventory system

For the pooled inventory system, the supplier sets up one common inventory x p andcharges each retailer a common unit wholesale price wp. Furthermore, the retailershave a joint service level requirement ρp. In this case, the supplier maximizes hisprofit by solving the following problem

maxx p,wp≥c

�p(x p, wp)

s.t. P(Dp(wp) < x p) ≥ ρp

Introducing the stocking factor z p = x p/y(wp), the problem becomes

maxx p,wp≥c

�p(z p, wp)

s.t. P(Dp(wp) < x p) ≥ ρp.

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Recall that without service level requirements, we can get the optimal stockingfactor z∗

p by the procedure in Sect. 3.2. With service levels constraints, since G p(z p)

is nondecreasing in z p and �p(z p, w∗p(z p)) is a quasi-concave function on the range

[c,+∞], the optimal stocking factor is max(z∗p, G−1

p (ρp)).Similar as in Sect. 3.2, the retailers do not hold any inventory, and their margin is

m p. Letting πp(x p) be the expected total retail profit given the supplier’s inventorylevel is x p, we have

πp(x p) = E[m p min(x p, Dp)].

4.3 Comparative results

In this subsection, we analyze the impact of pooling inventory when the demandsare normally distributed and the supplier needs to satisfy the retailer’s service levelrequirements. Assume that ε1 and ε2 are independently, identically and normally dis-tributed with the mean value µ and standard deviation σ . In addition, assume that theservice level requirement of each retailer under the reserved inventory system and thejoint service level in the pooled inventory system are the same, i.e., ρ1 = ρ2 = ρp = ρ.

Lemma 5 For both the reserved inventory case (i = 1, 2) and the pooled inventorycase (i = p), the optimal stocking factor z∗

i (wi ) is nondecreasing in wi .

Theorem 6 Assume that w1 = w2 = wp = w and ρ1 = ρ2 = ρp = ρ. If ε1and ε2 are independently, identically and normally distributed random variables andmax( w−c

w+h , ρ) ≥ 0.5, then the sum of the optimal total stocking factors for the retail-ers 1 and 2 in the reserved inventory case is at least as large as the optimal stock-ing factor in the pooled inventory case, i.e., z∗

p(w) ≤ z∗1(w) + z∗

2(w). Otherwise,z∗

p(w) > z∗1(w) + z∗

2(w).

Theorem 7 Assume that w1 = w2 = wp = w and ρ1 = ρ2 = ρp = ρ. If ε1 andε2 are i.i.d. and normally distributed, then the optimal profit of the supplier in thepooled inventory system is at least as large as that in the reserved inventory system,i.e., �∗

p ≥ �∗r .

Theorem 8 Assume that w1 = w2 = wp = w, ρ1 = ρ2 = ρp = ρ and m1 = m2 =m p = m. If ε1 and ε2 are independently, identically and normally distributed randomvariables, then the retailers’ total expected profit in the pooled inventory case is atleast as large as that in the reserved inventory case, i.e., πr1(x1)+πr2(x2) ≤ πp(x p).

The results above show that when the retailers have the same service level require-ments under the reserved and pooled inventory systems, with normally distributeddemands, the supplier prefers the pooled inventory policy. Meanwhile, if the suppliercharges the same wholesale prices under the reserved and pooled inventory systems,the retailers will also prefer the pooled inventory policy, which makes pooling inven-tory a win–win strategy for the supply chain. These comparative results are the sameas those in Sect. 3, the case without the service level requirements.

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5 Infinite horizon problem

In this section we extend the two-echelon one-period model to an infinite planninghorizon. We assume the retailers do not hold any inventory and the supplier makesthe production decision at the beginning of each period. After the retailers observe thedemand, they place an order to the supplier. If retailers’ order cannot be completelysatisfied by the supplier, backlogging or lost-sales could take place, and we study bothscenarios in this section. The supplier bears the inventory holding cost and the penaltyincurred at the end of each period, and his objective is to maximize the discountedprofit over the infinite horizon. We assume the demands in each period are i.i.d. andthe leadtime between the supplier and the retailers is zero.

According to Clark and Scarf (1960), when the leadtime is zero, no matter theexcess demand is completely backlogged or lost, the optimal inventory policy is anorder-up-to policy. We focus on analyzing the profits of the supplier and the retailersunder the reserved and pooled inventory policy. To our knowledge, this problem hasnot been analyzed in the inventory theory literature so far. For a thorough review ofinfinite horizon profit maximization problems, we refer to Bertsekas (1995). In thissection, we formulate a dynamic programming model and study the optimal policyunder the reserved and pooled inventory systems. In addition, we compare the resultsin these two systems.

5.1 Infinite horizon with backlogging

First, we consider the reserved inventory system where the supplier holds a separateinventory for each of the two retailers. The sequence of events in one period is asfollows:

1. At the beginning of period t , the supplier replenishes his inventory reserved forretailer i from an inventory level of xi,t to yi,t . As before, assume the supplierpays for the replenishment at a unit cost of c.

2. During period t , random demand Di,t occurs at retailer i . We assume Di,t is i.i.d.with CDF Fi (·) and PDF fi (·), whose mean value is µi and standard deviation isσi . After observing the realization of demand, the retailer places an order to thesupplier. And if the inventory at the supplier cannot satisfy the order, the unmetdemands are fully backlogged. As before, the margin of retailer i is mi . The sup-plier charges retailer i a wholesale price of wi and his inventory level for retaileri drops to yi,t − Di,t . Hence, (yi,t − Di,t )

− denotes the amount of unmet demandat the end of period t .

3. At the end of period t , the supplier is charged with a unit inventory holding costof h and a unit backlogging penalty of b. In other words, the supplier bears therisk of demand variation.

We assume the cost and price parameters are nonnegative, i.e.,

wi , mi , c, h, b ≥ 0 for i = 1, 2,

and the backlog cost is always greater than the manufacturing cost, i.e., b > c.

123

404 Y. Dai et al.

The pooled inventory system operates in a similar manner. At the beginning ofperiod t , the supplier replenishes his pooled inventory level from x p,t to yp,t . Dp,t

denotes the joint demand occurred at the retailers, and it can be expressed as Dp,t =D1,t +D2,t . Both retailers have a markup/margin of m p. The supplier charges a whole-sale price of wp for each unit the retailers order. Let α be the discount factor per period,and we assume 0 ≤ α < 1.

Suppose under policy π , the supplier replenishes his inventory level from xi,t toyπ

i,t for retailer i (i = 1, 2) in period t . Our objective is to find the optimal policy, i.e.,the optimal order-up-to level in each period, which maximizes the total profit of thesupplier over the infinite horizon. Let gi,t (·) represent the supplier’s profit from retaileri in period t . Here gi,t (·) is a function of demand Di,t and the inventory levels beforeand after the replenishment, namely, xi,t and yπ

i,t . Denoting the discounted profit thatthe supplier makes from retailer i over the infinite horizon as V π

i (xi,1), we have

V πi (xi,1) = lim

N→∞ E

[N∑

t=1

αt−1gi,t (yπi,t , xi,t , Di,t )

].

If the profit function gi,t (·) is independent of time, i.e.,

gi,1(·) = gi,2(·) = · · · = gi,N (·) = · · · ,

we can drop the subscript of t , and,

V πi (xi,1) = lim

N→∞ E

[N∑

t=1

αt−1gi (yπi,t , xi,t , Di,t )

].

The supplier is interested in finding the optimal policy π from the set of all feasiblepolicies �, and hence the optimal total profit from retailer i over the infinite horizonis

Vi (xi,1) = maxπ∈�

V πi (xi,1, Di,t ).

Dropping the time subscripts t for notational convenience, consider the reservedinventory system in which the supplier carries separate inventory for the retailers andmakes all inventory decisions to optimize his expected profit over the infinite horizon.For the supplier, the leftover inventory at the end of a period incurs a holding cost,and the backlogging incurs a penalty cost. The expected holding and penalty costsincurred in a period can be written as

L(yi ) = hE(yi − Di )+ + bE(yi − Di )

−.

Let Ui (·) be the supplier’s expected profit from retailer i in period t , and we have

Ui (yi , xi ) = Egi (yi , xi , Di,t )

= E[wi Di − L(yi ) − c(yi − xi )] for i = 1, 2.

123


The first term is the supplier’s revenue, the second term is the holding and penaltycosts, and the last term is the manufacturing cost. We can write Ui (yi , xi ) as

Ui (yi , xi ) = cxi + Gi (yi ),

where

Gi (yi ) = E[wi Di − L(yi ) − cyi ]. (5)

Notice that Gi (yi ) is a concave function of yi .From Clark and Scarf (1960), we know that the optimal policy for this model is an

order-up-to policy. Let Ri (Ri ≥ 0) be the order-up-to level for retailer i . At the begin-ning of each period, if retailer i’s initial inventory level is less than Ri , the supplierwill replenish the inventory level to Ri . And if the initial inventory level xi is greaterthan Ri , the supplier will not replenish retailer i’s inventory level until it drops to orbelow Ri .

Without loss of generality, we assume that the initial inventory level for retailer iat the beginning of the infinite horizon is less than Ri , i.e., xi < Ri . Since Ri is theorder-up-to level which maximizes the supplier’s expected profit from retailer i , wehave

Vi (xi ) = maxyi ≥xi

{cxi + Gi (yi ) + αEVi (yi − Di )} i = 1, 2 (6)

= cxi + Gi (Ri ) + αEVi (Ri − Di ).

Since the planning horizon is infinite, we can assume the order-up-to level Ri is appliedfrom the second period. Hence, in order to find the optimal order-up-to level yi in thefirst period, we can solve the following optimization problem


{cxi + Gi (yi ) + E

[lim

N→∞

N∑t=2

αt−1g(yi,t , xi,t )

]}

= maxyi ≥xi

E[cxi + Gi (yi ) + αc(yi − Di ) + αGi (Ri ) + α2Vi (Ri − Di )].

We know that E[cxi + Gi (yi ) + αc(yi − Di ) + αGi (Ri ) + α2Vi (Ri − Di )] is aconcave function of yi since G(yi ) is concave and αc(yi − Di ) is linear. To maximizethis concave function, letting Zi (yi ) = E[cxi + Gi (yi ) + αc(yi − Di ) + αGi (Ri ) +α2Vi (Ri − Di )], and taking the first derivative with respect to yi , we have

d Zi (yi )

yi= dGi (yi )

yi+ αc

= −(h + b)Fi (yi ) − c + b + αc.

123

406 Y. Dai et al.

Since Ri is also the optimal order-up-to level for the first period, which implies

d Zi (yi )

yi

∣∣yi =Ri = 0,

and so we have the optimal order-up-to level

Ri = F−1i

(b − (1 − α)c

h + b

).

Assumption b > c implies that 0 ≤ b−(1−α)ch+b < 1, which guarantees the feasibility

of Ri . In addition, notice that the optimal order-up-to inventory level is increasing in band decreasing in h. This goes with the intuition: when the penalty is high, the supplierwill maintain a high inventory level in order to reduce the quantity of lost sales; whenthe holding cost is high, the supplier will maintain a low inventory level in order toreduce the holding cost.

Given the initial inventory levels x1 and x2, the supplier’s optimal profit �r (x1, x2)

under the reserved inventory system can be expressed as

�r (x1, x2) = V1(x1) + V2(x2).

Since the planning horizon is infinite, by moving cxi back to the previous period(Veinott 1965), we have cxi = cα(yi − Di ). With (5) and (6), we can rewrite thesupplier’s expected profit from retailer i as follows

Vi (xi ) = E[wi Di + αc(Ri − Di ) − cRi − L(Ri ) + αVi (Ri − Di )]= E[(wi − αc)Di − (1 − α)cRi − L(Ri ) + αVi (Ri − Di )]. (7)

From (7), noticing that the expected profit does not depend on the initial inventorylevel, we can simplify the notation �r (x1, x2), V1(x1), V2(x2) as �∗

r , V ∗1 , V ∗

2 , and wehave

V ∗i = 1

1 − αE[(wi − αc)Di − (1 − α)cRi − L(Ri )] for i = 1, 2.

Hence, the supplier’s total expected profit �∗r can be written as

�∗r = 1

1 − αE[(w1−αc)D1+(w2 − αc)D2−(1 − α)c(R1+R2) − L(R1)−L(R2)].

Similarly, for the pooled inventory system, the supplier needs to solve the followingoptimization problem to maximize his profit:

Vp(x p) = maxyp≥x p

{cx p + G p(yp) + αEVp(yp − Dp)},

123


where

G p(yp) = E[wp Dp − L(yp) − cyp].Through the analysis above, we know the optimal order-up-to level for the pooledinventory system is

Rp = F−1p

(b − (1 − α)c

h + b

).

The supplier’s optimal profit under the pooled inventory system can be expressed as

�p(x p) = Vp(x p).

Simplifying the notation �p(x p) as �∗p, we have

�∗p = 1

1 − αE[(wp − αc)Dp − (1 − α)cRp − L(Rp)].

As for the retailers’ profit function, recall that the retailer places an order to thesupplier after observing the realization of the demand and unmet demands are fullybacklogged. Therefore, the retailers neither carry any inventory nor take any risk.Let π∗

r and π∗p be the retailers’ total profit under the reserved and pooled inventory

systems, respectively. Then π∗r and π∗

p are the same, namely

π∗r = π∗

p =∞∑

i=0

αi mi E[D1 + D2];

Now we extend our analysis by comparing the results with normally distributeddemands under the reserved and pooled inventory systems. We assume that D1 andD2 are independently and normally distributed with mean value µ1 and µ2, respec-tively, and standard deviation σ1 and σ2, respectively. As before, let �(·) be the CDFof the standard normal distribution and �−1(·) its inverse function.

Theorem 9 If D1 and D2 are independently and normally distributed, and the criti-cal ratio b−(1−α)c

h+b ≥ 0.5, then the supplier’s optimal inventory level under the pooledinventory system is at most as large as the total inventory level under the reservedinventory system, i.e., Rp ≤ R1 + R2. Otherwise, Rp > R1 + R2.

Theorem 10 Assume the supplier charges the same wholesale prices under thereserved and pooled inventory systems, i.e., w1 = w2 = wp = w. If D1 and D2

are independently and normally distributed and the critical ratio b−(1−α)ch+b ≥ 0.5,

then the supplier will be better off under the pooled inventory system, i.e., �∗p ≥ �∗

r .

Note that the condition b−(1−α)ch+b ≥ 0.5 is equivalent to b ≥ h + 2(1 − α)c. The

analysis above shows that, when the penalty cost of backlogging b is high, the suppliercan stock less by pooling the inventory together and still be better off. This goes withour intuition.

123

408 Y. Dai et al.

5.2 Infinite horizon with lost-sales

In the previous subsection, we have studied the supplier’s inventory decisions overan infinite horizon when the unsatisfied demand is backlogged. In this subsection weanalyze the case when the unsatisfied demand at each period is lost, and we use thesame notation as in the previous subsection.

Under the reserved inventory system, given the initial inventory level xi , we defineVi (xi ) as the optimal profit of retailer i , and we have


E[cxi + wi min(yi , Di ) − cyi − h(yi − Di )+

−b(yi − Di )− + αVi (yi − Di )

+].

The supplier’s profit under the reserved inventory system can be written as

�r (x1, x2) = V1(x1) + V2(x2).

By Clark and Scarf (1960) we known that the order-up-to policy is still the optimalpolicy for the lost-sales case. Similarly as before, assuming that the optimal order-up-tolevel is Ri and the initial inventory level xi < Ri , we have


E[cxi + wi min(yi , Di ) − cyi − h(yi − Di )+

−b(yi − Di )− + αVi (yi − Di )

+]= E[cxi + wi min(Ri , Di ) − cRi − h(Ri − Di )

+

−b(Ri − Di )− + αVi (Ri − Di )

+].

Following the same reasoning as before, we can write the problem as


E[cxi + wi min(yi , Di ) − cyi − h(yi − Di )+ − b(yi − Di )

−

+ α(c(yi −Di )+ + wi min(Ri , Di )−cRi −h(Ri −Di )

+ − b(Ri −Di )−)

+ αVi (R − Di )+)].

Letting Ui (yi ) = E[wi min(yi , Di ) − cyi + (αc − h)(yi − Di )+ − b(yi − Di )

−] andtaking the first derivative of Ui (yi ) with respect to yi , we have

dUi (yi )

yi= wi − c + b + (αc − h − wi − b)Fi (yi ).

Since Ri is also the optimal order-up-to level in the first period, it satisfies the followingequation

dUi (yi )

yi

∣∣yi =Ri = 0,

123


which implies that

Ri = F−1i

(wi + b − c

h + b + wi − αc

).

Following the same reasoning as in Sect. 5.1, by moving cxi back to the previousperiod, we can rewrite the supplier’s profit from retailer i as

Vi (xi ) = E[(αc − h)(Ri − Di )+ + wi min(Ri , Di ) (8)

− cRi − b(Ri − Di )− + αVi (Ri − Di )

+].

Noticing that Vi (xi ) does not depend on the initial inventory level xi , we can simplifythe notation �r (x1, x2), V1(x1), and V2(x2) as �∗

r , V ∗1 , and V ∗

2 . From Eq. (8), weknow that in the reserved inventory system, the supplier’s profit can be rewritten as

�∗r = V ∗

1 + V ∗2

where

V ∗i = 1

1 − αE[(αc − h)(Ri − Di )

+ + wi min(Ri , Di )

− cRi − b(Ri − Di )−], for i = 1, 2.

As for the pooled inventory system, the supplier needs to find the optimalorder-up-to level Rp in order to maximize his profit Vp(x p), where

Vp(x p) = E[cx p + wp min(Rp, Dp) − cRp − h(Rp − Dp)+

− b(Rp − Dp)− + αVp(Rp − Dp)

+].

We can show that the optimal order-up-to level in the pooled inventory system is

Rp = F−1p

(wp + b − c

h + b + wp − αc

),

and the supplier’s profit can be written as

�∗p = 1

1 − αE[(αc − h)(Rp − Dp)

+ + wp min(Rp, Dp) − cRp − b(Rp − Dp)−].

Theorem 11 When D1 and D2 are independently and normally distributed, if thecritical ratio w+b−c

h+b+w−αc ≥ 0.5, then the supplier’s order-up-to level under the pooledinventory system is at most as large as that under the reserved inventory system, i.e.,Rp ≤ R1 + R2. Otherwise, Rp > R1 + R2.

When demands are normally distributed and the wholesale prices charged underthe reserved and pooled inventory systems are the same, we find the same results forthe supplier’s profit as under the backlog scenario.

123

410 Y. Dai et al.

Theorem 12 Assume the supplier charges the same wholesale prices in the reservedand pooled inventory systems, i.e., w1 = w2 = wp = w. If D1 and D2 are inde-pendently and normally distributed and the critical ratio w+b−c

h+b+w−αc ≥ 0.5, then thesupplier will be better off in the pooled inventory system, i.e., �∗

p ≥ �∗r .

Now we consider the profits of the two retailers. Recall that each retailer places anorder to the supplier after observing the realization of the demand and the unmet de-mand is lost. Therefore, retailers’ total profits under the reserved and pooled inventorysystems are respectively

π∗r =

∞∑i=0

αi E[m1 min(R1, D1) + m2 min(R2, D2)],

π∗p =

∞∑i=0

αi E[m p min(Rp, Dp)].

Note that in contrast to the backlogging scenario, in the lost-sales scenario, retailers’total profits under the reserved and pooled inventory systems are different.

Theorem 13 If D1 and D2 are independently and normally distributed and all themarkup prices are the same, i.e., m1 = m2 = m p = m, then the total profit ofthe retailers under the pooled inventory system is at least as large as that under thereserved inventory system, i.e., π∗

p ≥ π∗r .

Note that the condition w+b−ch+b+w−αc ≥ 0.5 is equivalent to w + b ≥ h + (2 − α)c.

The analysis for the lost-sales scenario shows that, when the penalty cost of lostsales b is high, the supplier can stock less and be better off by pooling the inventorytogether. This goes with our intuition. In addition, different from the backloggingscenario, in the lost-sales scenario, retailers’ total profits is always higher under thepooled inventory system than under the reserved inventory system. In other words,when w+b−c

h+b+w−αc ≥ 0.5, pooling inventory together is a win–win strategy for both thesupplier and the two retailers.

6 Concluding remarks

We have studied a two-echelon supply chain with one supplier and two retailers. Thesupplier carries all inventory and replenishes the retailers as needed. Both the reservedand pooled inventory systems are considered. Assuming the random demand is price-sensitive and follows a multiplicative format, we have developed the inventory andpricing strategies for the supplier in the one-period model, and analyzed both back-logging and lost-sale scenarios in the infinite horizon model.

In order to obtain insights into the reserved and pooled inventory policies, we havecompared the profits of the supplier and the retailers in these two systems with nor-mally distributed demands. We have shown that the supplier and the retailers maybenefit from risk pooling. However, generally speaking, whether the supply chain canbenefit from pooling inventory and whether it is a win–win strategy are conditional

123


and depend on the value of cost parameters. We have shown that when the penaltycost is high, both the supplier and the retailers tend to be better off by pooling theinventory, which goes with the intuition.

There are some potentially interesting extensions. For example, in our paper, weassumed that there is no competition between the retailers. What if the retailers com-pete for a common customer base? Chen et al. (2004) and Bernstein and Federgruen(2003) considered a supply chain model with one supplier and multiple competingretailers in which each retailer makes decisions such as inventory level and retailprice. Bernstein and Federgruen (2004) considered a supply chain system in whichthe supplier decides how the retailers are replenished and bears all the production costincurred in the supply chain. However each retailer chooses her own retail price, andthe demand at each retailer is determined by a general demand function of the pricescharged by all retailers.

In this paper, we have demonstrated the benefit of the pooled inventory policy. Thisresult leads to the question of how to share the benefits between the retailers and sup-plier. We can use a contract to coordinate the retailers and the supplier. Cachon (2003)gave an excellent review on contract coordination. Wang et al. (2004) also analyzedthe consignment contract with revenue sharing in a supply chain with one supplierand one retailer. Another related research area is cooperative games. Anupindi et al.(2001) analyzed a snapshot allocation game (SAG). In SAG, the value of the gameis calculated based on each realization of the random demand. Another approach isthe allocation game in expectation (AGE) studied by Anupindi et al. (2001). Granotand Sosic (2002) mentioned the Shapley value as a profit-allocation mechanism thatmay induce optimal supply chain inventory decisions. Then Bartholdi and Kemah-lioglu-Ziya (2005) used the Shapely value as a value-sharing mechanism to affectthe operational decisions of supply chain partners. It would be interesting to applythese methods to allocate the benefits. However, the most difficult part may lie in theallocation of the inventory under the pooled inventory system.

Appendix

Proof of Lemma 1 Taking the first derivatives of �r1(z1, w1) with respect to z1 andw1, we have

∂�r1(z1, w1)

∂z1= y(w1)(−c − h) + y(w1)(w1 + h)(1 − G1(z1)),

∂�r1(z1, w1)

∂w1= aw

−(b+1)1 (b − 1)(µ1 − �1(z1))

×(

−w1 + bc

b − 1+ b

b − 1

(c + h)�1(z1)

µ1 − �(z1)

).

Taking the second partial derivatives of �r1(z1, w1) with respect to z1, we have

∂2�r1(z1, w1)

∂2z1= −y(w1)(w1 + h)g1(z1) ≤ 0,

123

412 Y. Dai et al.

which implies the first statement in Lemma 1. Since µ1 − �1(z1) ≥ A > 0,∂�r1(z1,w1)

∂w1= 0 if and only if −w1 + bc

b−1 + bb−1

(c+h)�1(z1)µ1−�1(z1)

= 0, which impliesthat there is only one local minimum or maximum, w∗

1(z1), which is

w∗1(z1) = w0 + b

b − 1

(c + h)�1(z1)

µ1 − �1(z1),

where w0 = bcb−1 . Taking the second derivative of �r1(z1, w1) with respect to z1 at

this local optimal point, we have

∂2�r1(z1, w1)

∂2w1

∣∣∣w1=w∗1 (z1) = −aw

−(b+1)1 (b − 1) < 0.

Hence, given a fixed z1, �r1(z1, w1) is a quasi-concave function in w1 . The third state-ment follows from equation ∂�r1(z1,w1)

∂w1= 0, and the fourth from ∂�r1(z1,w1)

∂z1= 0.

�

Proof of Theorem 1 By the chain rule, we have

d�r1(z1, w∗1(z1))

dz1= ∂�r1(z1, w

∗1(z1))

∂z1+ ∂�r1(z1, w

∗1(z1))

∂w∗1(z1)

dw∗1(z1)

dz1

= −y(w∗1(z1))(1 − G1(z1))

( −c − h

1 − G1(z1)+ w∗

1(z1) + h

).

Define

V (z1) = −c − h

1 − G1(z1)+ w∗

1(z1) + h,

and we know that the optimal solution z∗1 satisfies V (z∗

1) = 0. Taking the first and thesecond derivatives of V (z1) with respect to z1, we have

dV (z1)

dz1= b

b − 1

(c + h)(G1(z1)z1 − �1(z1))

(µ1 − �1(z1))2 − (c + h)r(z1)

1 − G1(z1),

d2V (z1)

dz21

= −2b

b − 1(1 − G1(z1))

(c + h)(G1(z1)z1 − �1(z1))

(µ1 − �1(z1))3

+ b

b − 1g1(z1)

(c + h)z1

(µ1 − �1(z1))2 −(c + h)

(r ′(z1)

1 − G1(z1)+ r2(z1)

1 − G1(z1)

)

= −(c + h)

(b − 2

b − 1

r(z1)

µ1 − �1(z1)+ 2r(z1)

2 + r(z1)′

1 − G1(z1)

)

−(

2(1 − G1(z1))

µ1 − �1(z1)+ r(z1)

)dV

dz1.

123


The optimal points of V (z1) satisfy

dV (z1)

dz1= 0,

and at those points, we have

d2V (z1)

dz21

= −(c + h)

(b − 2

b − 1

r(z1)

µ1 − �1(z1)+ 2r(z1)

2 + r(z1)′

1 − G1(z1)

).

If the distribution function satisfies 2r(z1)2 + r ′(z1) > 0, then d2V (z1)

dz21

< 0 at those

optimal points of V (z1). Hence V (z1) is a quasi-concave function, and there are at mosttwo roots for V (z1) = 0. Furthermore, V (B) = −∞ < 0 and V (A) = w(z1)+c > 0guarantee the uniqueness of z∗

1. �Proof of Lemma 2 The proof is similar to that of Lemma 1 and thus is omitted. �Proof of Theorem 2 The proof is similar to that of Theorem 1 and thus is omitted. �Proof of Lemma 3 From the results in Sects. 3.1 and 3.2, we know that z∗

i (wi ) satisfiesthe following equations

1 − Gi (z∗i ) = c + h

h + wi.

Taking the first derivative of z∗i with respect to wi , we have

dz∗i

dwi= c + h

(h + wi )2gi (zi ).

The nonnegativity of c+h(h+wi )

2 and gi (zi ) imply thatdz∗

idwi

≥ 0. Hence z∗i (wi ) is nonde-

creasing in wi . �Proof of Lemma 3 Assume that ε1 and ε2 have the same mean µ and standard devia-tion σ and define

�(u) = �p(2u) − 2�(u).

A sufficient condition for �p(2u) ≤ 2�(u) is �(u) ≤ 0. Taking the first derivativeof �(u) with respect to u, we obtain

�′(u) = −2(1 − G p(2u)) + 2(1 − G(u))

= √2σ

(√2�

(√2(u − µ)

σ

)− �

(u − µ

σ

)).

123

414 Y. Dai et al.

Since �(·) is a nondecreasing function, we have

�′(u) ={≥ 0 if u ≥ µ

< 0 if u < µ.

We show �(u) ≤ 0 by considering the following two cases:Case 1 : u ≥ µ

�′(u) ≥ 0 indicates that �(u) is nondecreasing in u. Hence a sufficient condition of�(u) ≤ 0 is that �(+∞) ≤ 0. However �(+∞) = 0 and �p(+∞) = 0, hence�(u) ≤ 0 for u ≥ µ.Case 2 : u < µ

�′(u) < 0 indicates that �(u) is decreasing in u. A sufficient condition for �(u) ≤ 0is �(−∞) = 0. From Hadley and Whitin (1963), we have

�(u) = σφ

(u − µ

σ

)− (u − µ)

(1 − �

(u − µ

σ

)).

Then

�(u) = √2σφ

(√2(u − µ)

σ

)− 2σφ

(u − µ

σ

)

+ 2(u − µ)

(�

(√2(u − µ)

σ

)− �

(u − µ

σ

)).

When u approaches −∞, the first two terms go to 0 and �(√

2(u−µ)σ

) − �(u−µ

σ)

converges to 0 with exponential speed. Hence the third term also converges to 0.Therefore, �p(2u) ≤ 2�(u) for u < µ. �Proof of Theorem 3 We know that

G1(z∗1) = G2(z

∗2) = G p(z

∗p) = 1 − c + h

h + w.

Therefore,

z∗i (w) = µ + σ�−1

(w − c

w + h

)for i = 1, 2,

z∗p(w) = 2µ + √

2σ�−1(

w − c

w + h

),

z∗1(w) + z∗

2(w) − z∗p(w) = (2 − √

2)σ�−1(

w − c

w + h

).

When w−cw+h ≥ 0.5, then �−1( w−c

w+h ) ≥ 0, and hence z∗1(w) + z∗

2(w) ≥ z∗p(w). When

w−cw+h < 0.5, then �−1( w−c

w+h ) < 0, and hence z∗1(w) + z∗

2(w) ≥ z∗p(w). �

123


Proof of Theorem 4 When ε1 and ε2 are i.i.d., �r1 is the same as �r2 and hence theyhave the same optimal value. If we can show that �p(2z1, w1) ≥ 2�r1(z1, w1) forany (z1, w1), then �∗

p ≥ �p(2z∗1, w

∗1) ≥ 2�r1(z∗

1, w∗1) = �∗

r . Note that

2�r1(z1, w1) − �p(2z1, w1) = y(w1)(−2(c + h)�1(z1) − 2(w1 − c)�1(z1)

+ (c + h)�p(2z1) + (w1 − c)�p(2z1))

= y(w1)(h + w1)[�p(2z1) − 2�1(z1)

].

From Lemma 4, we have

�p(2z1) − 2�1(z1)) ≤ 0.

In addition, y(w1)(h + w1) > 0. This implies that 2�r1(z1, w1) − �p(2z1, w1) ≤ 0for any (z1, w1). �Proof of Theorem 5 Given the inventory level x1, x2 and x p, the expected profit ofretailers in the reserved and pooled systems are

πri (xi ) = E[m min(xi , Di )] for i = 1, 2,

πp(x p) = E[m min(x p, Dp)].

Given the wholesale price w1 = w2 = wp = w, the optimal inventory levels arexi = y(w)z∗

i (w) (i = 1, 2) and x p = y(w)z∗p(w). Taking some algebraic manipula-

tion, we have

πr1(x1) + πr2(x2) = my(w)[2µ − 2�1(z∗1(w))],

πp(x p) = my(w)[2µ − �p(z∗p(w))].

Recall that, given the wholesale price w1 = w2 = wp = w, the optimal stockingfactors are

z∗1(w) = z∗

2(w) = µ + σ�−1(

w − c

w + h

),

z∗p(w) = 2µ + √

2σ�−1(

w − c

w + h

).

By Lemma 4, we have

�p

(2µ + √

2σ�−1(

w − c

w + h

))≤ 2�i

(µ +

√2

2σ�−1

(w − c

w + h

))

≤ 2�i

(µ + σ�−1

(w − c

w + h

)).

The second inequality follows from the increasing property of function �i . Therefore,we have πr1(x1) + πr2(x2) ≤ πp(x p). �

123

416 Y. Dai et al.

Proof of Theorem 5 We know that z∗i (wi ) satisfies the following equations

Gi (z∗i ) = max

(wi − c

wi + h, ρ

),

where wi −cwi +h is increasing in wi and Gi is a nondecreasing function. Hence z∗

i (wi ) isnondecreasing in wi . �

Proof of Theorem 6 We know that

G1(z∗1) = G2(z

∗2) = G p(z

∗p) = max

(w − c

w + h, ρ

).

Similar to the proof of Theorem 3, this theorem can be easily proved. �

Proof of Theorem 7 When ε1 and ε2 are independently, identically and normally dis-tributed,�r1 is the same as�r2. If we can show that�p(z∗

p(w),w) ≥ 2�r1(z∗1(w),w)

for any w, then �∗p ≥ �p(z∗

p(w),w∗) ≥ 2�r1(z∗1(w

∗), w∗) = �∗r .

Note that

z∗i (w) = µ + σ�−1

(max

(w − c

w + h, ρ

)), i = 1, 2,

z∗p(w) = 2µ + √

2σ�−1(

max

(w − c

w + h, ρ

)).

Similar to the proof of Theorem 4, this theorem can be proved. �

Proof of Theorem 8 Similar to the proof of Theorem 5, we have

2�1

(µ+σ�−1

(max

(w − c

w + h, ρ

)))≥�p

(2µ+√

2σ�−1(

max

(w − c

w + h, ρ

))).

Therefore, πr1(x1) + πr2(x2) ≤ πp(x p). �

Proof of Theorem 9 Under the reserved inventory system, the optimal inventory levelis given by

Ri = µi + σi�−1

(b − (1 − α)c

h + b

),

and the total inventory level is

R1 + R2 = µ1 + µ2 + (σ1 + σ2)�−1

(b − (1 − α)c

h + b

).

123


Under the pooled inventory system, Dp ∼ N (µ1+µ2,

√σ 2

1 + σ 22 ). Hence the optimal

total inventory level under the pooled inventory system is

Rp = µ1 + µ2 +√

σ 21 + σ 2

2 �−1(

b − (1 − α)c

h + b

).

We know that �−1(·) is a monotone nondecreasing function and �−1(0.5) = 0, whichimply that �−1(

b−(1−α)ch+b ) ≥ 0 given (

b−(1−α)ch+b ) ≥ 0.5. We know that σ1 + σ2 ≥√

σ 21 + σ 2

2 , hence Rp ≤ R1 + R2 if (b−(1−α)c

h+b ) ≥ 0.5. Otherwise, Rp > R1 + R2.�

Proof of Theorem 10 The supplier’s optimal profit under the reserved inventory sys-tem is

�∗r = 1

1 − αE[(w − αc)(D1 + D2) − (1 − α)c(R1 + R2) − L(R1) − L(R2)],

= 1

1 − αE[(w − αc − b)(D1 + D2) − ((1 − α)c + h)(R1 + R2)

+ (h + b)(min(R1, D1) + min(R2, D2))]

We have that

E[min(Ri , Di )] =Ri∫

−∞Di d F(Di ) +

∞∫

Ri

Ri d F(Di )

= µi −∞∫

Ri

(Di − Ri )d F(Di ).

In Zipkin (2000),∫ ∞

Ri(Di − Ri )d F(Di ) is referred to as loss function, and the standard

normal loss function

�1(x) =∞∫

x

(ε − x)d�(ε),

where ε has the standard normal distribution. By Zipkin (2000), we have

∞∫

Ri

(Di − Ri )d F(Di ) = 1

σi�1

(Ri − µi

σi

).

123

418 Y. Dai et al.

Therefore, we have

�∗r = 1

1 − α((w − αc − b)(µ1 + µ2) − ((1 − α)c + h)(R1 + R2))

+ 1

1 − α(h + b)

(µ1 + µ2 − 1

σ1�1

(R1 − µ1

σ1

)− 1

σ2�1

(R2 − µ2

σ2

))

Similarly, the supplier’s optimal profit under the pooled inventory system is

�∗p = 1

1 − αE[(w − αc)Dp − (1 − α)cRp − L(Rp)],

= 1

1 − α(w − αc − b)(µ1 + µ2) − ((1 − α)c + h)Rp

+ (h + b)

⎛⎝µ1 + µ2 − 1√

σ 21 + σ 2

2

�1

⎛⎝ Rp − µ1 − µ2√

σ 21 + σ 2

2

⎞⎠

⎞⎠

Calculating the difference between �∗r and �∗

p, we have

�∗r − �∗

p = 1

1 − α((1 − α)c + h)(Rp − R1 − R2)

+ 1

1 − α(h+b)

⎛⎝ 1√

σ 21 +σ 2

2

�1

⎛⎝ Rp−µ1−µ2√

σ 21 + σ 2

2

⎞⎠− 1

σ1�1

(R1 − µ1

σ1

)

− 1

σ2�1

(R2 − µ2

σ2

) ⎞⎠ .

By the definition of �1(x), we know that

�1

⎛⎝ Rp − µ1 − µ2√

σ 21 + σ 2

2

⎞⎠ = �1

(R1 − µ1

σ1

)= �1

(R2 − µ2

σ2

)≥ 0,

and it is easy to show that

1√σ 2

1 + σ 22

− 1

σ1− 1

σ2≤ 0.

Meanwhile, from Theorem 9, when the critical ratio b−(1−α)ch+b ≥ 0.5, we have

Rp ≤ R1 + R2.

123


Therefore when the critical ratio b−(1−α)ch+b ≥ 0.5, we have �∗

r ≤ �∗p. In other words,

the supplier is better off in the pooled inventory system if the critical ratio b−(1−α)ch+b ≥

0.5. �

Proof of Theorem 11 Under the reserved inventory system, the inventory level forretailer i is

Ri = µi + σi�−1

(w + b − c

h + b + w − αc

),

and the total inventory level is

R1 + R2 = µ1 + µ2 + (σ1 + σ2)�−1

(w + b − c

h + b + w − αc

).

Under the pooled inventory system, the total inventory level is

Rp = µ1 + µ2 +√

σ 21 + σ 2

2 �−1(

w + b − c

h + b + w − αc

).

Since �−1(·) is a monotone, nondecreasing function and �−1(0.5) = 0, we knowthat when w+b−c

h+b+w−αc ≥ 0.5, we have Rp ≤ R1 + R2. Otherwise, Rp > R1 + R2. �

Proof of Theorem 12 The supplier’s optimal profits in the reserved and pooled inven-tory systems are

�∗r =

2∑i=1

1

1 − αE[(αc − h)(Ri − Di )

+ + w min(Ri , Di ) − cRi − b(Ri − Di )−]

=2∑

i=1

1

1 − αE[(αc − h − c)Ri − (αc − h − w − b) min(Ri , Di ) − bDi ]

=2∑

i=1

1

1 − α

((αc − h − c) Ri − (αc − h − w − b)

(µi − 1

σi�1

(Ri − µi

σi

))− bµi

)

and

�∗p = 1

1 − αE

[(αc − h)(Rp − Dp)

+ + w min(Rp, Dp) − cRp − b(Rp − Dp)−]

= 1

1 − α

((αc − h − c)Rp − b(µ1 + µ2)

−(αc − h − w − b)

⎛⎝µ1 + µ2 − 1√

σ 21 + σ 2

2

�1

⎛⎝ Rp − µ1 − µ2√

σ 21 + σ 2

2

⎞⎠

⎞⎠

⎞⎠ .

123

420 Y. Dai et al.

Calculating the difference between �r and �p, we have

�∗r − �∗

p = 1

1 − α

⎛⎝ (αc − h − c) (Rp − R1 − R2)

+ (αc − h − w − b)

⎛⎝ 1√

σ 21 + σ 2

2

�1

⎛⎝ Rp − µ1 − µ2√

σ 21 + σ 2

2

⎞⎠

− 1

σ1�1

(R1 − µ1

σ1

)− 1

σ2�1

(R2 − µ2

σ2

))⎞⎠

Following the same reasoning as in Theorem 11, we know that when the critical ratiow+b−c

h+b+w−αc ≥ 0.5, we have

Rp ≤ R1 + R2.

Therefore when the critical ratio w+b−ch+b+w−αc ≥ 0.5 we have �∗

r ≤ �∗p. In other words,

the supplier is better off under the pooled inventory system. �

Proof of Theorem 13 We first show that in each period the expected total sales underthe pooled inventory system is higher than under the reserved inventory system.

E[min(R1, D1) + min(R2, D2) − min(Rp, Dp)]

= 1√σ 2

1 + σ 22

�1

⎛⎝ Rp − µ1 − µ2√

σ 21 + σ 2

2

⎞⎠ − 1

σ1�1

(R1 − µ1

σ1

)− 1

σ2�1

(R2 − µ2

σ2

)

≤ 0

Hence

π∗r =

∞∑i=0

αi m E[min(R1, D1) + min(R2, D2)];

≤∞∑

i=0

αi m E[min(Rp, Dp)]

= π∗p,

which completes the proof. �

123


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Risk pooling strategy in a multi-echelon supply chain with price-sensitive demand

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