- Email: [email protected]
The stochastic ordering of mean-preserving transformations and its applications
Accepted Manuscript The stochastic ordering of mean-preserving transformations and its applications Wanshan Zhu, Zhengping Wu PII: DOI: Reference:
S0377-2217(14)00500-1 http://dx.doi.org/10.1016/j.ejor.2014.06.017 EOR 12361
To appear in:
European Journal of Operational Research
Received Date: Accepted Date:
18 June 2012 16 June 2014
Please cite this article as: Zhu, W., Wu, Z., The stochastic ordering of mean-preserving transformations and its applications, European Journal of Operational Research (2014), doi: http://dx.doi.org/10.1016/j.ejor.2014.06.017
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The Stochastic Ordering of Mean-Preserving Transformations and Its Applications
Wanshan Zhu Department of Industrial Engineering Tsinghua University Beijing 100084, China Email: [email protected]
Zhengping Wu∗ Whitman School of Management Syracuse University Syracuse, NY 13244-2450 [email protected] ∗
corresponding author
March 2014
The Stochastic Ordering of Mean-Preserving Transformations and Its Applications
Abstract
The stochastic variability measures the degree of uncertainty for random demand and/or price in various operations problems. Its ordering property under mean-preserving transformation allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. Based on Chebyshev’s algebraic inequality, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of an important result in the inventory literature, which has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management.
Key words: uncertainty modeling; stochastic variability; mean-preserving transformation; Chebyshev’s algebraic inequality; inventory management; procurement risk management
1
1.
Introduction In the study of a variety of operations problems in the presence of uncertainties (including
demand, purchase price, and/or lead-time uncertainties), it is often of great interest to assess the impact of the degree of randomness, also known as stochastic variability. However, this could be a challenging task without an easily tractable way of ordering randomness. A useful approach commonly used in the literature is to employ a mean-preserving transformation of the random variable, which not only factors out the effect of the mean, but also parameterizes the variability of the random variable by a single scalar. In this paper, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and then apply it to a broad class of specific transformations, including the mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of a fundamental result in [1] that has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management. The remainder of the paper is organized as follows: Section 2 reviews related literature. Section 3 provides a general framework for stochastic variability ordering under mean-preserving transformations, which is our main technical results. Section 4 discusses in great detail three different applications of our general framework. The paper concludes in Section 5.
2
2.
Literature review Stochastic variability plays a pivotal role in supply chain management because it directly
affects the inventory policies and the associated total operational costs. [2] studies the order variability in the upstream of the supply chain when the downstream uses different lot sizing rules: one to minimize the average cost per time and the other to minimize the average cost per unit. [3] studies a multi-echelon inventory system under both decentralized and centralized controls, and compares their impact on the demand variance amplification (the so-called bullwhip effect) from the end to the beginning of the supply chains. Both [2] and [3] use variance as the measure of stochastic variability. In general, however, variance is not a proper measure of risk (see [1]; an example is available on page 137 of [4]). A rigorous way to rank variability calls for theories of stochastic ordering. The theoretical study of stochastic ordering can find its root in mathematics, and has drawn attention of researchers in operations engineering and economics. The underlying mathematical structure of stochastic ordering is driven by the concept of Majorization [5], which formalizes the intuitive notion that the components of a vector are more spread out than those of another vector. Many basic mathematical inequalities, including a special case of Chebyshev’s inequality, are used to characterize the Majorization. A comprehensive collection of stochastic ordering properties and their applications in operations engineering can be found [6], which dedicates its chapter 3 to variability orders. The convex order of random variables discussed in that chapter is the concept we adopt in this paper. The convex order is closely related to the stochastic dominance, which is a subject studied in economics for decision making under uncertainty [4]. In particular, the second order stochastic dominance
3
is equivalent to the convex order when two random variables have equal means. Most of the above-mentioned literature study general stochastic ordering properties. [5] is more akin to our paper as it studies families of scalar parameterized distributions, but it limits its attention to a few specific distributions, e.g., exponential, whereas our paper develops a general framework to parameterize stochastic variability ordering for any distribution. The study of the mean-preserving transformation first appears in [7], which shows that a riskier (stochastically more variable) random variable can be constructed by adding a white noise of zero mean to the original random variable. However, the limitation of this mean-preserving transformation is the lack of a convenient means to assess the impact of stochastic variability. [1] overcomes this limitation by constructing a mean-preserving affine transformation that can be parameterized by a single scalar. It performs a mean-preserving transformation of a random variable in a linear fashion, and shows that the stochastic order of the transformed variables is monotone in the parameter. The same affine mean-preserving transformation is also used to study the value of inventory pooling in [8]. Our paper differs in that we provide a general framework, which not only covers the mean-preserving affine transformation, but also extends to other types of transformations, i.e., nonlinear transformations. These nonlinear mean-preserving transformations of random variables are commonly seen in operations applications. For example, under a scheduled ordering scheme in a supply chain [9], the demand distribution in the upstream is transformed from the truncation of the downstream demand by a pre-agreed fixed shipment quantity. In procurement with options [10], the purchase price distribution is transformed from the uncertain market price, with a cap on the strike price specified in the option contract. Our unified framework enables us to use a single scalar as the parameter to study the stochastic 4
variability ordering properties of all the aforementioned mean-preserving transformations. 3.
A general framework for stochastic variability ordering under mean-preserving transformations The general framework makes use of a generalized Chebyshev’s Algebraic Inequality [11],
which is replicated in Lemma 1 below for ease of reference. Note that throughout the text, the terms increasing and decreasing should be taken in their weak sense; e.g., increasing means nondecreasing. Lemma 1. (Mitrinovi´c et al. [11], p. 248) Let u, g : [a, b] → R and F : [a, b] → [0, 1] be a distribution function. Suppose that u is monotonically increasing. Define GF : (a, b] → R, GF (t) =
t a
g(s)dF (s)/
GF (t) ≤ GF (b)
t a
dF (s). If
(1)
for all t ∈ (a, b], then a
b
u(s)g(s)dF (s) ≥
b
u(s)dF (s) a
b
g(s)dF (s)
(2)
a
If (1) holds and u is monotonically decreasing, the reverse of (2) holds. In addition, as a special case, (2) holds if g(s) is increasing. We are now ready to present our main result below in Theorem 1, which identifies unifying sufficient conditions for using a scalar parameter to order the stochastic variability of random variables under any mean-preserving transformation that can be parameterized by a single scalar. Let X be a nonnegative random variable defined on the support [a, b] with cumulative distribution function F : [a, b] → [0, 1], a ≥ 0, and Y (X, α) be a mean-preserving transfor5
mation of X, i.e., for all α ≥ 0, E[Y (X, α)] = E[X] = μ,
(3)
where Y (X, α) is assumed to be piecewise differentiable. As we shall see below in Theorem 1, the stochastic variability of Y (X, α) is parameterized by the single scalar α. In the following text, for notational convenience, we use a subscript to specify the range of the integration whenever an expectation is not taken on the entire support of a random variable. For instance, we use EX≤t [∂Y (X, α)/∂α] to denote
t a
∂Y (X, α)/∂αdF (x).
Theorem 1. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: (a) Y (X, α) is increasing in X; (b) for any a ≤ t ≤ b, EX≤t [∂Y (X, α)/∂α] ≤ 0.
Proof. It is well known that Y (X, α1 ) ≥v Y (X, α2 ) if and only if E[h(Y (X, α1 ))] ≥ E[h(Y (X, α2 ))] for any convex function h (see Theorem D.1(c) of [12]), where the operator ≥v denotes second order stochastic dominance, i.e., Y (X, α1 ) ≥v Y (X, α2 ) means Y (X, α1 ) is stochastically more variable than Y (X, α2 ). Therefore, it suffices to show that for any convex function h, E[h(Y (X, α))] is increasing in α, i.e., ∂E[h(Y (X, α))]/∂α ≥ 0, under the two conditions stated in the theorem.
6
By the chain rule, ∂E[h(Y (X, α))] ∂h(Y (X, α)) ∂Y (X, α) = E ∂α ∂Y (X, α) ∂α ∂Y (X, α) ∂h(Y (X, α)) E ≥ E ∂Y (X, α) ∂α
(4)
= 0,
where the last step follows from the fact that E [∂Y (X, α)/∂α] = ∂E [Y (X, α)]/∂α = 0, because E [Y (X, α)] = μ is independent of α. The inequality (4) follows from Lemma 1, which is applicable because if we let ∂h(Y (X, α))/∂Y (X, α) and ∂Y (X, α)/∂α correspond to functions u and g respectively in Lemma 1, then the two sufficient conditions in Lemma 1 are implied by (a) and (b) of this theorem. In particular, the first condition of Lemma 1 requires ∂h(Y (X, α))/∂Y (X, α) to be monotonically increasing in X, which is true because h is convex and Y (X, α) is increasing in X (condition (a)); the second condition of Lemma 1 is
t a
[∂Y (x, α)/∂α]dF (x)/F (t) ≤ E[∂Y (X, α)/∂α], which is implied by (b) and the afore
mentioned fact that E [∂Y (X, α)/∂α] = 0.
While Theorem 1 only requires rather weak conditions (especially condition (b)), this very reason makes it somewhat difficult to be used directly. We next develop a more readily applicable corollary.
Corollary 1. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: (a) Y (X, α) is increasing in X; (b) ∂Y (X, α)/∂α is increasing in X.
7
Proof. Condition (a) is the same as that of Theorem 1. Next, we will show that condition (b) implies Theorem 1(b) by contradiction. Suppose Theorem 1(b) is not implied by part (b) of this corollary, i.e., there exists a t ∈ [a, b] such that EX≤t [∂Y (X, α)/∂α] > 0. It then must be the case that ∂Y (t, α)/∂α > 0, because ∂Y (t, α)/∂α is the largest value of the increasing function ∂Y (X, α)/∂α in the region [a, t]. As a result, we have
E[∂Y (X, α)/∂α] = EX≤t [∂Y (X, α)/∂α] + Et EX≤t [∂Y (X, α)/∂α] + Et 0,
where the first inequality follows from the increasing property of ∂Y (X, α)/∂α, and the second inequality holds because both terms in the previous step are positive. This result clearly contradicts the fact that E[∂Y (X, α)/∂α] = 0, which has been established in the
proof of Theorem 1.
It is worth noting that a similar logic can be used to obtain the following: if condition (b) of Corollary 1 changes to ∂Y (X, α)/∂α is decreasing in X, then Y (X, α) becomes stochastically less variable as α increases. In what follows, we will see how the above results can be applied to various operations problems to examine the impact of stochastic variability on the optimal decisions and the associated objective values.
8
4.
Applications
4.1 The mean-preserving affine transformation The mean-preserving affine transformation is defined as follows. Let X be a random variable with mean μ, and α be a nonnegative scalar. Consider the family of random variables XαA ≡ Y A (X, α) = αX + (1 − α)μ.
(5)
Clearly, E[XαA ] = μ for all α. The beauty of this transformation is that the extent of randomness is simply parameterized by the scalar α, as can be seen in Lemma 2 below. Lemma 2. If α1 > α2 , then XαA1 ≥v XαA2 . The mean-preserving affine transformation was first introduced by [1] to provide a neat and useful tool for analyzing cardinal effects of the demand randomness on optimal inventory levels and the associated expected costs. Their key findings hinge on Lemma 2. However, their proof of this result, replicated below, is questionable. For any convex function h, E{h[αX + (1 − α)μ]} ≤ E[αh(X) + (1 − α)h(μ)] = h(μ) + α{E[h(X)] − h(μ)}, which is increasing in α because the bracketed quantity is positive (by Jensen’s inequality). The above argument is based on two facts: the right-hand side is increasing in α; in addition, it is no less than the left-hand side. Nevertheless, these two facts taken together do not imply that the left-hand side is necessarily increasing in α. For instance, for any n > 0,
9
we have (n − α)2 ≤ α2 + n2 . While the right-hand side is increasing in α, the left-hand side is evidently decreasing in α for 0 ≤ α ≤ n. It is worth noting that [1] is a well-known work in the inventory literature and attracted a great deal of attention [13, 14, 15]. The same mean-preserving affine transformation together with the fundamental property claimed in Lemma 2 has been adopted by others in a wide range of contexts, including outsourcing logistics [16], pricing setting newsvendor problem [17], and lead-time randomness reduction [18]. Another notable application of this transformation is that it is used in [8] to establish the condition under which the benefit of risk pooling increases with the demand variability. Since Lemma 2 lays the foundation of all the aforementioned papers, it is important to supply a proper proof. Our result in Corollary 1 can be easily used to rectify the problem in [1] that has gone unnoticed for more than two decades. It is evident that the two conditions required by the corollary are satisfied by the mean-preserving affine transformation defined in Equation (5). Hence the claim in Lemma 2. 4.2 The mean-preserving truncation Corollary 1 can also be applied to another different type of mean-preserving transformation (dubbed mean-preserving truncation) defined below. XαT ≡ Y T (X, α) = (X − α)+ + μ − E (X − α)+ .
(6)
Figure 1 sketches the mean-preserving truncation. Clearly, XαT = μ − E [(X − α)+ ], for X ≤ α; and X − α + μ − E [(X − α)+ ] otherwise (note that the expectation E [(X − α)+ ] is a constant independent of the realization of X). It is straightforward to see ∂XαT /∂X = 0, for X ≤ α; and 1 otherwise. Thus condition (a) 10
XαT
μ − E[(X − α)+]
α
X
Figure 1: Visualizing the Mean-Preserving Truncation XαT
of Corollary 1 is satisfied. Furthermore, if α1 ≥ α2 , then ∂XαT1 /∂X ≤ ∂XαT2 /∂X, implying ∂XαT /∂X is decreasing in α. This is equivalent to ∂XαT /∂α being decreasing in X, which means that the reverse of condition (b) in Corollary 1 is satisfied. As a result, Corollary 1 leads to Lemma 3 below. Lemma 3. As α increases, XαT becomes stochastically less variable. The mean-preserving truncation can find interesting applications in inventory models dealing with inefficiency associated with supply chain decentralization. In decentralized supply chains, the upstream usually sees a magnified demand uncertainty due to the widely recognized bullwhip effect. To combat the bullwhip effect, [9] analyze scheduled ordering policies for a two-stage decentralized supply chain consisting of a single supplier and multiple retailers who face uncertain end-customer demands. They consider the Scheduled Balanced Ordering Policy (SBOP) and the Scheduled Synchronized Ordering Policy (SSOP), both 11
allowing retailers to order from the supplier freely only in one period in an m-period ordering cycle. In other periods, retailers receive fixed shipments δ. Retailers take turns to order freely under SBOP, while under SSOP all retailers order freely in the same period. To evaluate the performance of the policies, it is crucial to examine the impact of the fixed shipment quantity δ on the uncertainty faced by the supplier. Let Dij be the end-customer demand at retailer i in period j of an ordering cycle and dij be its realization. The supplier sees an + uncertain demand ( m−1 j=1 dij + Dim − (m − 1)δ) from retailer i in the free-ordering period under both policies. Letting X =
m−1 j=1
dij + Dim and α = (m − 1)δ, we then have ( m−1 j=1 dij + Dim − (m −
1)δ)+ = (X − α)+ , which possesses the same stochastic ordering properties as XαT , because the constant terms μ − E [(X − α)+ ] in Equation (6) do not affect the variability of XαT . It then follows from Lemma 3 that, as expected from intuition, the uncertain demand faced by the supplier becomes stochastically less variable as δ increases, enabling considerable cost savings. Since [9] is silent on the impact of the fixed shipment quantity δ, we now conduct a numerical study to explore this issue. Our numerical study employs the same system parameters as those in [9], and focuses on quantifying the impact of the the fixed shipment quantity δ on the optimal ordering decisions and total costs. Similar to [9], we consider one supplier with two identical retailers; the cycle length m = 2; the unit holding cost is 1 for all parties; the backlogging costs are 6 and 9 for the retailer and the supplier, respectively. Figures 2 and 3 plot the impact of the fixed shipment quantity δ on the supplier’s optimal average order-up-to level and total inventory costs with different degree of demand uncertainty. Note that the average order-up-to level is used because the supplier makes order 12
Supplierorderupto level 115
Demandvar=600
Demandvar=400
Demandvar=100
Demandvar=50
105 95 85 75 65 55 45 35 25 1
3
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Fixed shipmentquantiy Figure 2: The impact of the fixed shipment quantity δ on supplier’s average order-up-to level
Suppliercost
Demandvar=600
Demandvar=400
30
Demandvar=100
Demandvar=50
25 20 15 10 5 1
3
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Fixed shipmentquantiy Figure 3: The impact of the fixed shipment quantity δ on supplier’s total inventory cost
13
decisions after observing the retailers’ inventory level (see [9]). These two figures show that as the fixed shipment quantity δ increases, the supplier’s optimal average order-up-to level and total inventory costs both decrease, which is not hard to predict based on insights from [1] and our previous finding that the supplier’s demand becomes stochastically less variable as δ increases. Notably, Figure 3 shows that the supplier benefits more from increasing the fixed shipment quantity δ when the end demand has a larger variance. Under favorable conditions, the inventory cost reduction can be as high as one-third.
Retailerorderupto level 60
Demandvar=600
Demandvar=400
Demandvar=100
Demandvar=50
55 50 45 40 35 30 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Fixed shipmentquantiy Figure 4: The impact of the fixed shipment quantity δ on retailers’ order-up-to level
On the retailer side, as the fixed shipment quantity δ increases, Figure 4 indicates that the retailers’ optimal order-up-to levels decrease, which is intuitive. The retailers’ total inventory cost, however, responds in a different way. In particular, it is convex in the fixed shipment quantity δ, as illustrated in Figure 5. This is because when δ is small, the retailers 14
Retailer cost 105
Demandvar=600
Demandvar=400
Demandvar=100
Demandvar=50
95 85 75 65 55 45 35 25 1
3
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Fixed shipmentquantiy Figure 5: The impact of the fixed shipment quantity δ on retailers’ total inventory cost
benefit from the stability of the system enabled by the fixed shipment arrangement. But when δ grows too big, the over-commitment eventually hurts the retailers. 4.3 The mean-preserving capping Our results also apply to a third type of mean-preserving transformation, termed meanpreserving capping, as defined below. XαC ≡ Y C (X, α) = where E[min(X, α)] =
α a
xdF (x) +
b α
min(X, α) μ, E[min(X, α)]
(7)
αdF (x).
Taking partial derivative with respect to α yields ∂XαC = ∂α where EX≤α [X] =
α a
x[F (b)−F (α)] − (E[min(X,α)])2 μ, if x < α; EX≤α [X] μ, (E[min(X,α)])2
otherwise,
xdF (x). Since this derivative decreases in X for x < α, as shown 15
in Figure 6, the transformation defined in Equation (7) does not meet condition (b) in Corollary 1. Therefore, we need to develop from Theorem 1 another corollary to facilitate the study of the mean-preserving capping. ∂XαC ∂α EX≤α [X] (E[min(X,α)])2 μ
a
α
b
X
a(F (b)−F (α)) − (E[min(X,α)]) 2μ
α(F (b)−F (α)) − (E[min(X,α)]) 2μ
Figure 6: Visualizing the Weak Single-Crossing Property of ∂XαC /∂α
The derivative ∂XαC /∂α depicted in Figure 6 satisfies the so-called weak single-crossing (WSC) property, which can be utilized to show Theorem 1(b), as we shall see shortly. A function ψ : [a, b] → R is said to satisfy WSC on [a, b] if there exits x0 ∈ [a, b] such that ψ(x) ≤ 0 for all x < x0 and ψ(x) ≥ 0 for all x > x0 . When the WSC property is met, the following result can be derived from Theorem 1. Corollary 2. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: 16
(a) Y (X, α) is increasing in X; (b) ∂Y (X, α)/∂α satisfies WSC on [a, b].
Proof. It suffices to show that part (b) leads to Theorem 1(b). When WSC is satisfied, if t < x0 , then by definition ∂Y (X, α)/∂α ≤ 0 on [a, t]. Hence we know EX≤t [∂Y (X, α)/∂α] ≤ 0. If t ≥ x0 , then ∂Y (X, α)/∂α ≥ 0 on [t, b], and thus EX≤t [∂Y (X, α)/∂α] ≤ E[∂Y (X, α)/∂α] =
0.
It is straightforward to verify that condition (a) of the above corollary is also satisfied by the mean-preserving capping XαC , because ∂XαC /∂X = μ/E[min(X, α)] if x ≤ α, and 0 otherwise. Therefore, Corollary 2 leads to the following. Lemma 4. As α increases, XαC becomes stochastically more variable. The mean-preserving capping can find important applications in procurement risk management (PRM), especially when option contracts are used. [10] reports that HewlettPackard (HP) obtained $425 million in costs savings cumulatively over six years by using the PRM approach. This approach hinges upon structured contracts with suppliers to manage the supply chain risks on the procurement side. At the core of the approach are two types of option contracts used to mitigate the risks posed by the volatile spot market price of components (e.g., flash memory): One is fixed price option contract and the other is discount-off-of-market price option contract. The fixed price option contract puts a cap on the cost and has two parameters: a unit reservation cost paid in advance by HP to its supplier, and a unit execution cost α paid later if the option is exercised after market price X realizes, i.e., if x ≥ α. In this contract, HP’s 17
exercising cost is min(X, α) because when the market price realizes it pays α by exercising the option if x ≥ α, and x by buying from the spot market if x < α. Note that α ≤ b (the upper support of the market price) for the contract to be viable, otherwise it will degenerate to simply buying at spot market price. The discount-off-of-market price option contract differs from the fixed price option contract only in the second parameter. It has a unit exercising cost ratio β, in place of the unit execution cost α. In this contract, HP’s exercising cost is βX when the market price realizes. The key issue for HP to address in order to optimize its portfolio of contracts is to understand which of the above two option contracts poses smaller risks, if both contracts have the same reservation cost and the expected exercising costs. It turns out that the stochastic variability ordering property proved in Lemma 4 can be used to tell that the fixed price option contract always has less variable exercising costs than the discount-off-of-market price option contract, as shown in the following lemma. Lemma 5. If E[βX] = E[min(X, α)], then βX ≥v min(X, α). Proof. From the equal expectation condition, i.e., the condition E[βX] = E[min(X, α)], we have β = E[min(X, α)]/E [X] = E[min(X, α)]/μ. Hence, it suffices to show that X ≥v μ min(X, α)/E[min(X, α)], which follows directly from Lemma 4, because X = XbC according to Equation (7), and b ≥ α.
4.3.1 Comparison between total costs of the fixed price option and the discount-off-of-market price option contracts Early papers studying the option contract (e.g., [19]) typically consider the capacity purchase decision from suppliers under demand uncertainty, but not market price uncertainty. 18
[20] considers market price uncertainty, but not demand uncertainty. [21] integrates both uncertainties and shows that with a single supplier, a mixed procurement strategy consisting of a long-term contract and spot market procurement is optimal. [22] also incorporates both demand and market price uncertainties into the procurement problem using option contracts, but they consider multiple suppliers. [23] further extends [22] to account for the fixed procurement costs. However, all these papers study only the fixed price option contract. The discount-off-of-market price option contract that has been used by HP for PRM [10] has not received due research attention. To fill the gap in the literature, we next study the discount-off-of-market price option contract and compare it with the fixed price option contract. As a benchmark, the fixed price option contract studied in [22] is described in greater detail as follows: let c denote the unit cost of capacity reservation, D denote the random demand, and X denote the random spot market price. The decision for the buyer firm (e.g., HP) is how much capacity Q to reserve from the supplier. Under the fixed price option contract, the buyer reserves capacity Q at unit cost c upfront. After the spot market price and demand realize, the buyer can buy up to Q at the price min(α, X) from the supplier and the rest at the spot price X to meet all her demand. Let 1Z be an indicator function, i.e., 1Z = 1 if Z is true; and 0 otherwise. The total expected procurement cost of the fixed price option contract can then be expressed as follows: cQ + ED,X [1D≤Q min(α, X)D + 1D≥Q (min(α, X)Q + X(D − Q))],
(8)
where ED,X [·] denotes the expectation over both random variables D and X. In this expression, the first term cQ is the total reservation cost. The entire second term (i.e., ED,X [·]) 19
is the expected cost of exercising the option and spot market procurement, which is derived from two cases: if the realized demand D is below the reserved capacity Q, the buyer will purchase D units of capacity at the unit cost min(α, X); otherwise, the buyer will buy all the reserved capacity Q at min(α, X) each, and purchase an additional capacity D − Q at the unit spot price X to cover the shortfall. We now turn our attention to the discount-off-of-market price option contract with an exercising cost ratio β, which results in the following total expected procurement cost: cQ + ED,X [1D≤Q βXD + 1D≥Q (βXQ + X(D − Q))].
(9)
A close look at Equations (8) and (9) reveals that the total expected procurement costs of the two contracts closely resemble each other and have similar interpretations. The only difference between them is the unit exercising cost of the reserved capacity: min(α, X) under the fixed price option, and βX under the discount-off-of-market price option. As shown in Lemma 5, the unit exercising cost βX is stochastically more variable than min(α, X). As a result, one might draw a parallel between [1] and our model, and intuit that the discount-off-of-market price option contract yields a higher total expected procurement cost. However, this intuition is disproved by the following proposition.
Proposition 1. For independent demand and spot market price, if the fixed price option contract and the discount-off-of-market price option contract have the same expected unit exercising cost, i.e., EX [min(α, X)] = EX [βX], then the following holds: (a) The two contracts have the same optimal capacity reservation level Q∗ ; (b) The two contracts have the same total expected procurement cost. 20
Proof. For any reserved capacity Q, the difference (9) − (8) is ED,X [1D≤Q βXD] − ED,X [1D≤Q min(α, X)D] + ED,X [1D≥Q βXQ] − ED,X [1D≥Q min(α, X)Q] = ED [1D≤Q D]{EX [βX] − EX [min(α, X)]} + QED [1D≥Q ]{EX [βX] − EX [min(α, X)]} = 0,
where the first equality follows from the assumption of independence between D and X, and the second equality follows from EX [min(α, X)] = EX [βX]. In other words, if the conditions stated in the proposition are satisfied, then the two contracts behave identically for any reserved capacity Q. As a result, the optimal capacity reservation level Q∗ and the associated total expected procurement costs are the same.
Proposition 1 makes a nice contribution to the literature. It highlights an important difference between the impact of demand uncertainty and unit exercising cost uncertainty on total inventory (or procurement) costs. In particular, unlike demand uncertainty, unit exercising cost uncertainty in general is not a major driver of the total inventory cost, which is an insight not developed in the prior literature. The fundamental reason behind the difference can be explained as follows: demand uncertainty makes it difficult to match supply and demand, whereas unit exercising cost uncertainty does not. While the fixed price option contract and the discount-off-of-market price option contract are equivalent for independent demand and spot market price, it remains an open question whether the equivalence remains unchanged when demand and spot market price are correlated (correlation is common in many industries, especially those with a few dominant players). We now perform a numerical study to explore this question. In our study, we employ the same parameters as the those used in [22]: demand and market price are mod21
eled by a truncated bivariate normal distribution with demand D ∼ N (100, 50) and spot market price X ∼ N (20, 7), c = 5.3272, α = 6.7810 (note that the aggregate unit capacity cost c + α = 12.1082, which falls in the range of the price distribution). In addition, for the discount-off-of-market price option contract, we set β = 0.335 to guarantee equal mean of the unit exercising cost between the two contracts, i.e., EX [min(α, X)] = EX [βX].
Opimtal reservation quantity 140
Fixedpriceoption
Discountpriceoption
130 120 110 100 90 80 1
0.5
0
0.5
1
Correlationbetweendemandandmarketprice Figure 7: Comparison of the optimal capacity reservation level between the fixed price option contract and discount-off-of-market price option contract
Figures 7 and 8 compare the optimal capacity reservation levels and the optimal total expected procurement costs, respectively, of the two contracts. As we can see, the comparison hinges on the correlation between demand and the spot market price. It is worth noting that in both figures, the two curves cross each other when the correlation is 0. In other words, the two contracts have the same optimal capacity reservation decisions and 22
the corresponding optimal total expected procurement costs when demand and spot market price are independent, which corroborates our results in Proposition 1.
Opimtal procurement costs
Fixedpriceoption
Discountpriceoption
170 160 150 140 130 120 1
0.5
0
0.5
1
Correlationbetweendemandandmarketprice Figure 8: Comparison of the total expected procurement costs between the fixed price option contract and discount-off-of-market price option contract
However, the two contracts no longer behave the same way when demand and spot market price are correlated. Conceivably, the buyer is most vulnerable when the realized demand and spot market price are both high, which is possible when they are positively correlated. Under such a circumstance, the fixed price option better protects the buyer than the discount-off-of-market price option does, because the unit exercise cost of the former is capped at α, whereas for the latter, the unit exercise cost can get out of hand. This explains why in Figure 7, the optimal reserved capacity for the fixed price option contract is greater than that for the discount-off-of-market price option contract when the correlation 23
is positive, because the buyer can only cash in on the better protection provided by the fixed price option contract if sufficient capacity is reserved. Therefore, the buyer tends to reserve more capacity under the fixed price option contract to reap more benefits from it. The same logic also explains why the total expected procurement cost is lower for the fixed price option contract than for the discount-off-of-market price option contract when the correlation is positive, as can be seen in Figure 8. When demand and market price are negatively correlated, the relationship between the two contracts is totally reversed. In the presence of negative correlation between demand and market price, market price tends to be low when realized demand is high. In such a case, the fixed price option contract is not very useful because the low market price reduces the odds to exercise the option. Therefore, the buyer tends to reserve less capacity in order not to waste the reservation cost. In contrast, the discount-off-of-market price option contract is more effective because the buyer can always enjoy the discount within the reserved capacity. As a result, reserving more capacity makes sense. On the other hand, market price tends to be high when realized demand is low. In such a case, the reserved capacity would be low for both contracts – the demand is low to begin with. Overall, when demand and market price are negatively correlated, the fixed price contract is less beneficial to the buyer and results in a lower reserved capacity and a high total expected procurement cost, as shown in Figures 7 and 8. In sum, our analysis shows that although the fixed price option contract has a stochastically less variable unit exercising cost than the discount-off-of-market price option contract, the stochastic variability of the unit exercising cost is not really the determinant of the total procurement cost. Instead, the correlation between demand and spot market price plays a 24
more important role. 5.
Conclusion The stochastic variability measures the degree of uncertainty for random demand and/or
price in various operations problems. Its ordering property allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. However, stochastic ordering of random variables is typically challenging without a tractable approach to capture variability. In this paper, we provide a unifying framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. We first apply our framework to the mean-preserving affine transformation, which is then used to rectify an incorrect proof of a fundamental result in the inventory literature [1] that has gone unnoticed for more than two decades. The second application, to mean-preserving truncation, addresses inventory strategies in decentralized supply chains. Specifically, we use it to investigate the effectiveness of using fixed shipment quantity to mitigate inefficiency associated with supply chain decentralization. We conduct a numerical study to complete the work of [9], and illustrate that the potential cost reduction could be as high as one-third under certain parametric conditions. Lastly, we also apply our general framework to the mean-preserving capping, which sheds light on using option contracts for procurement risk management. To the best of our knowledge, we are the first to analyze the discount-off-of-market price option contract,
25
which is used by HP to manage its procurement risk. We use our approach to theoretically compare the stochastic variability of the unit exercising cost of two contracts: the fixed price option contract and the discount-off-of-market price option contract. We also conduct a numerical study to assess the relative performance of the two contracts. Surprisingly, we find that the stochastic variability of the unit exercising cost is not really the determinant of the total procurement cost. Instead, what drives the disparity between the two option contracts is the correlation between demand and spot market price. References References [1] Y. Gerchak, D. Mossman, On the effect of demand randomness on inventories and costs, Operations Research 40 (4) (1992) 804–807. [2] I. Nyoman Pujawan, The effect of lot sizing rules on order variability, European Journal of Operational Research 159 (3) (2004) 617–635. [3] C. A. Garcia Salcedo, A. Ibeas Hernandez, R. Vilanova, J. Herrera Cuartas, Inventory control of supply chains: Mitigating the bullwhip effect by centralized and decentralized internal model control approaches, European Journal of Operational Research 224 (2) (2013) 261–272. [4] H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, Springer, 2006. [5] A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, 2010.
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[6] M. Shaked, J. G. Shanthikumar, Stochastic orders, Springer, 2007. [7] M. Rothschild, J. E. Stiglitz, Increasing risk: I. a definition, Journal of Economic Theory 2 (3) (1970) 225–243. [8] Y. Gerchak, Q.-M. He, On the relation between the benefits of risk pooling and the variability of demand, IIE Transactions 35 (11) (2003) 1027–1031. [9] L. G. Chen, S. Gavirneni, Using scheduled ordering to improve the performance of distribution supply chains, Management Science 56 (9) (2010) 1615–1632. [10] V. Nagali, J. Hwang, D. Sanghera, M. Gaskins, M. Pridgen, T. Thurston, P. Mackenroth, D. Branvold, P. Scholler, G. Shoemaker, Procurement risk management (PRM) at Hewlett-Packard company, Interfaces 38 (1) (2008) 51–60. [11] D. Mitrinovi´c, J. Peˇcari´c, A. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht, 1993. [12] E. Porteus, Foundations of Stochastic Inventory Theory, Stanford University Press, CA, 2002. [13] V. Agrawal, S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management 2 (4) (2000) 410–423. [14] J.-S. Song, C. A. Yano, P. Lerssrisuriya, Contract assembly: Dealing with combined supply lead time and demand quantity uncertainty, Manufacturing & Service Operations Management 2 (3) (2000) 287–296. 27
[15] A. V. Iyer, A. Jain, The logistics impact of a mixture of order-streams in a manufacturerretailer system, Management Science 49 (7) (2003) 890–906. [16] O. Alp, N. K. Erkip, R. Gullu, Outsourcing logistics: Designing transportation contracts between a manufacturer and a transporter, Transportation Science 37 (1) (2003) 23–39. [17] Q. Li, D. Atkins, On the effect of demand randomness on a price/quantity setting firm, IIE Transactions 37 (12) (2005) 1143 – 1153. [18] Y. Gerchak, M. Parlar, Investing in reducing lead-time randomness in continuous-review inventory models, Engineering Costs and Production Economics 21 (2) (1991) 191–197. [19] D. A. Serel, M. Dada, H. Moskowitz, Sourcing decisions with capacity reservation contracts, European Journal of Operational Research 131 (3) (2001) 635–648. [20] D. J. Wu, P. R. Kleindorfer, J. E. Zhang, Optimal bidding and contracting strategies for capital-intensive goods, European Journal of Operational Research 137 (3) (2002) 657–676, 00225. [21] R. W. Seifert, U. W. Thonemann, W. H. Hausman, Optimal procurement strategies for online spot markets, European Journal of Operational Research 152 (3) (2004) 781–799, 00178. [22] Q. Fu, C.-Y. Lee, C.-P. Teo, Procurement management using option contracts: random spot price and the portfolio effect, IIE Transactions 42 (11) (2010) 793–811. [23] C.-Y. Lee, X. Li, Y. Xie, Procurement risk management using capacitated option contracts with fixed ordering costs, IIE Transactions 45 (8) (2013) 845–864. 28
2ESEARCH (IGHLIGHTS
x x x x
A unified theory that enables variability ordering for any meanpreserving transformation Application to affine transformation and correction of an error in the literature Application to truncation and inventory models in supply chain management Application to capping and procurement risk management.
S0377-2217(14)00500-1 http://dx.doi.org/10.1016/j.ejor.2014.06.017 EOR 12361
To appear in:
European Journal of Operational Research
Received Date: Accepted Date:
18 June 2012 16 June 2014
Please cite this article as: Zhu, W., Wu, Z., The stochastic ordering of mean-preserving transformations and its applications, European Journal of Operational Research (2014), doi: http://dx.doi.org/10.1016/j.ejor.2014.06.017
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The Stochastic Ordering of Mean-Preserving Transformations and Its Applications
Wanshan Zhu Department of Industrial Engineering Tsinghua University Beijing 100084, China Email: [email protected]
Zhengping Wu∗ Whitman School of Management Syracuse University Syracuse, NY 13244-2450 [email protected] ∗
corresponding author
March 2014
The Stochastic Ordering of Mean-Preserving Transformations and Its Applications
Abstract
The stochastic variability measures the degree of uncertainty for random demand and/or price in various operations problems. Its ordering property under mean-preserving transformation allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. Based on Chebyshev’s algebraic inequality, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of an important result in the inventory literature, which has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management.
Key words: uncertainty modeling; stochastic variability; mean-preserving transformation; Chebyshev’s algebraic inequality; inventory management; procurement risk management
1
1.
Introduction In the study of a variety of operations problems in the presence of uncertainties (including
demand, purchase price, and/or lead-time uncertainties), it is often of great interest to assess the impact of the degree of randomness, also known as stochastic variability. However, this could be a challenging task without an easily tractable way of ordering randomness. A useful approach commonly used in the literature is to employ a mean-preserving transformation of the random variable, which not only factors out the effect of the mean, but also parameterizes the variability of the random variable by a single scalar. In this paper, we provide a general framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and then apply it to a broad class of specific transformations, including the mean-preserving affine transformation, truncation, and capping. The application to mean-preserving affine transformation rectifies an incorrect proof of a fundamental result in [1] that has gone unnoticed for more than two decades. The application to mean-preserving truncation addresses inventory strategies in decentralized supply chains, and the application to mean-preserving capping sheds light on using option contracts for procurement risk management. The remainder of the paper is organized as follows: Section 2 reviews related literature. Section 3 provides a general framework for stochastic variability ordering under mean-preserving transformations, which is our main technical results. Section 4 discusses in great detail three different applications of our general framework. The paper concludes in Section 5.
2
2.
Literature review Stochastic variability plays a pivotal role in supply chain management because it directly
affects the inventory policies and the associated total operational costs. [2] studies the order variability in the upstream of the supply chain when the downstream uses different lot sizing rules: one to minimize the average cost per time and the other to minimize the average cost per unit. [3] studies a multi-echelon inventory system under both decentralized and centralized controls, and compares their impact on the demand variance amplification (the so-called bullwhip effect) from the end to the beginning of the supply chains. Both [2] and [3] use variance as the measure of stochastic variability. In general, however, variance is not a proper measure of risk (see [1]; an example is available on page 137 of [4]). A rigorous way to rank variability calls for theories of stochastic ordering. The theoretical study of stochastic ordering can find its root in mathematics, and has drawn attention of researchers in operations engineering and economics. The underlying mathematical structure of stochastic ordering is driven by the concept of Majorization [5], which formalizes the intuitive notion that the components of a vector are more spread out than those of another vector. Many basic mathematical inequalities, including a special case of Chebyshev’s inequality, are used to characterize the Majorization. A comprehensive collection of stochastic ordering properties and their applications in operations engineering can be found [6], which dedicates its chapter 3 to variability orders. The convex order of random variables discussed in that chapter is the concept we adopt in this paper. The convex order is closely related to the stochastic dominance, which is a subject studied in economics for decision making under uncertainty [4]. In particular, the second order stochastic dominance
3
is equivalent to the convex order when two random variables have equal means. Most of the above-mentioned literature study general stochastic ordering properties. [5] is more akin to our paper as it studies families of scalar parameterized distributions, but it limits its attention to a few specific distributions, e.g., exponential, whereas our paper develops a general framework to parameterize stochastic variability ordering for any distribution. The study of the mean-preserving transformation first appears in [7], which shows that a riskier (stochastically more variable) random variable can be constructed by adding a white noise of zero mean to the original random variable. However, the limitation of this mean-preserving transformation is the lack of a convenient means to assess the impact of stochastic variability. [1] overcomes this limitation by constructing a mean-preserving affine transformation that can be parameterized by a single scalar. It performs a mean-preserving transformation of a random variable in a linear fashion, and shows that the stochastic order of the transformed variables is monotone in the parameter. The same affine mean-preserving transformation is also used to study the value of inventory pooling in [8]. Our paper differs in that we provide a general framework, which not only covers the mean-preserving affine transformation, but also extends to other types of transformations, i.e., nonlinear transformations. These nonlinear mean-preserving transformations of random variables are commonly seen in operations applications. For example, under a scheduled ordering scheme in a supply chain [9], the demand distribution in the upstream is transformed from the truncation of the downstream demand by a pre-agreed fixed shipment quantity. In procurement with options [10], the purchase price distribution is transformed from the uncertain market price, with a cap on the strike price specified in the option contract. Our unified framework enables us to use a single scalar as the parameter to study the stochastic 4
variability ordering properties of all the aforementioned mean-preserving transformations. 3.
A general framework for stochastic variability ordering under mean-preserving transformations The general framework makes use of a generalized Chebyshev’s Algebraic Inequality [11],
which is replicated in Lemma 1 below for ease of reference. Note that throughout the text, the terms increasing and decreasing should be taken in their weak sense; e.g., increasing means nondecreasing. Lemma 1. (Mitrinovi´c et al. [11], p. 248) Let u, g : [a, b] → R and F : [a, b] → [0, 1] be a distribution function. Suppose that u is monotonically increasing. Define GF : (a, b] → R, GF (t) =
t a
g(s)dF (s)/
GF (t) ≤ GF (b)
t a
dF (s). If
(1)
for all t ∈ (a, b], then a
b
u(s)g(s)dF (s) ≥
b
u(s)dF (s) a
b
g(s)dF (s)
(2)
a
If (1) holds and u is monotonically decreasing, the reverse of (2) holds. In addition, as a special case, (2) holds if g(s) is increasing. We are now ready to present our main result below in Theorem 1, which identifies unifying sufficient conditions for using a scalar parameter to order the stochastic variability of random variables under any mean-preserving transformation that can be parameterized by a single scalar. Let X be a nonnegative random variable defined on the support [a, b] with cumulative distribution function F : [a, b] → [0, 1], a ≥ 0, and Y (X, α) be a mean-preserving transfor5
mation of X, i.e., for all α ≥ 0, E[Y (X, α)] = E[X] = μ,
(3)
where Y (X, α) is assumed to be piecewise differentiable. As we shall see below in Theorem 1, the stochastic variability of Y (X, α) is parameterized by the single scalar α. In the following text, for notational convenience, we use a subscript to specify the range of the integration whenever an expectation is not taken on the entire support of a random variable. For instance, we use EX≤t [∂Y (X, α)/∂α] to denote
t a
∂Y (X, α)/∂αdF (x).
Theorem 1. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: (a) Y (X, α) is increasing in X; (b) for any a ≤ t ≤ b, EX≤t [∂Y (X, α)/∂α] ≤ 0.
Proof. It is well known that Y (X, α1 ) ≥v Y (X, α2 ) if and only if E[h(Y (X, α1 ))] ≥ E[h(Y (X, α2 ))] for any convex function h (see Theorem D.1(c) of [12]), where the operator ≥v denotes second order stochastic dominance, i.e., Y (X, α1 ) ≥v Y (X, α2 ) means Y (X, α1 ) is stochastically more variable than Y (X, α2 ). Therefore, it suffices to show that for any convex function h, E[h(Y (X, α))] is increasing in α, i.e., ∂E[h(Y (X, α))]/∂α ≥ 0, under the two conditions stated in the theorem.
6
By the chain rule, ∂E[h(Y (X, α))] ∂h(Y (X, α)) ∂Y (X, α) = E ∂α ∂Y (X, α) ∂α ∂Y (X, α) ∂h(Y (X, α)) E ≥ E ∂Y (X, α) ∂α
(4)
= 0,
where the last step follows from the fact that E [∂Y (X, α)/∂α] = ∂E [Y (X, α)]/∂α = 0, because E [Y (X, α)] = μ is independent of α. The inequality (4) follows from Lemma 1, which is applicable because if we let ∂h(Y (X, α))/∂Y (X, α) and ∂Y (X, α)/∂α correspond to functions u and g respectively in Lemma 1, then the two sufficient conditions in Lemma 1 are implied by (a) and (b) of this theorem. In particular, the first condition of Lemma 1 requires ∂h(Y (X, α))/∂Y (X, α) to be monotonically increasing in X, which is true because h is convex and Y (X, α) is increasing in X (condition (a)); the second condition of Lemma 1 is
t a
[∂Y (x, α)/∂α]dF (x)/F (t) ≤ E[∂Y (X, α)/∂α], which is implied by (b) and the afore
mentioned fact that E [∂Y (X, α)/∂α] = 0.
While Theorem 1 only requires rather weak conditions (especially condition (b)), this very reason makes it somewhat difficult to be used directly. We next develop a more readily applicable corollary.
Corollary 1. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: (a) Y (X, α) is increasing in X; (b) ∂Y (X, α)/∂α is increasing in X.
7
Proof. Condition (a) is the same as that of Theorem 1. Next, we will show that condition (b) implies Theorem 1(b) by contradiction. Suppose Theorem 1(b) is not implied by part (b) of this corollary, i.e., there exists a t ∈ [a, b] such that EX≤t [∂Y (X, α)/∂α] > 0. It then must be the case that ∂Y (t, α)/∂α > 0, because ∂Y (t, α)/∂α is the largest value of the increasing function ∂Y (X, α)/∂α in the region [a, t]. As a result, we have
E[∂Y (X, α)/∂α] = EX≤t [∂Y (X, α)/∂α] + Et
where the first inequality follows from the increasing property of ∂Y (X, α)/∂α, and the second inequality holds because both terms in the previous step are positive. This result clearly contradicts the fact that E[∂Y (X, α)/∂α] = 0, which has been established in the
proof of Theorem 1.
It is worth noting that a similar logic can be used to obtain the following: if condition (b) of Corollary 1 changes to ∂Y (X, α)/∂α is decreasing in X, then Y (X, α) becomes stochastically less variable as α increases. In what follows, we will see how the above results can be applied to various operations problems to examine the impact of stochastic variability on the optimal decisions and the associated objective values.
8
4.
Applications
4.1 The mean-preserving affine transformation The mean-preserving affine transformation is defined as follows. Let X be a random variable with mean μ, and α be a nonnegative scalar. Consider the family of random variables XαA ≡ Y A (X, α) = αX + (1 − α)μ.
(5)
Clearly, E[XαA ] = μ for all α. The beauty of this transformation is that the extent of randomness is simply parameterized by the scalar α, as can be seen in Lemma 2 below. Lemma 2. If α1 > α2 , then XαA1 ≥v XαA2 . The mean-preserving affine transformation was first introduced by [1] to provide a neat and useful tool for analyzing cardinal effects of the demand randomness on optimal inventory levels and the associated expected costs. Their key findings hinge on Lemma 2. However, their proof of this result, replicated below, is questionable. For any convex function h, E{h[αX + (1 − α)μ]} ≤ E[αh(X) + (1 − α)h(μ)] = h(μ) + α{E[h(X)] − h(μ)}, which is increasing in α because the bracketed quantity is positive (by Jensen’s inequality). The above argument is based on two facts: the right-hand side is increasing in α; in addition, it is no less than the left-hand side. Nevertheless, these two facts taken together do not imply that the left-hand side is necessarily increasing in α. For instance, for any n > 0,
9
we have (n − α)2 ≤ α2 + n2 . While the right-hand side is increasing in α, the left-hand side is evidently decreasing in α for 0 ≤ α ≤ n. It is worth noting that [1] is a well-known work in the inventory literature and attracted a great deal of attention [13, 14, 15]. The same mean-preserving affine transformation together with the fundamental property claimed in Lemma 2 has been adopted by others in a wide range of contexts, including outsourcing logistics [16], pricing setting newsvendor problem [17], and lead-time randomness reduction [18]. Another notable application of this transformation is that it is used in [8] to establish the condition under which the benefit of risk pooling increases with the demand variability. Since Lemma 2 lays the foundation of all the aforementioned papers, it is important to supply a proper proof. Our result in Corollary 1 can be easily used to rectify the problem in [1] that has gone unnoticed for more than two decades. It is evident that the two conditions required by the corollary are satisfied by the mean-preserving affine transformation defined in Equation (5). Hence the claim in Lemma 2. 4.2 The mean-preserving truncation Corollary 1 can also be applied to another different type of mean-preserving transformation (dubbed mean-preserving truncation) defined below. XαT ≡ Y T (X, α) = (X − α)+ + μ − E (X − α)+ .
(6)
Figure 1 sketches the mean-preserving truncation. Clearly, XαT = μ − E [(X − α)+ ], for X ≤ α; and X − α + μ − E [(X − α)+ ] otherwise (note that the expectation E [(X − α)+ ] is a constant independent of the realization of X). It is straightforward to see ∂XαT /∂X = 0, for X ≤ α; and 1 otherwise. Thus condition (a) 10
XαT
μ − E[(X − α)+]
α
X
Figure 1: Visualizing the Mean-Preserving Truncation XαT
of Corollary 1 is satisfied. Furthermore, if α1 ≥ α2 , then ∂XαT1 /∂X ≤ ∂XαT2 /∂X, implying ∂XαT /∂X is decreasing in α. This is equivalent to ∂XαT /∂α being decreasing in X, which means that the reverse of condition (b) in Corollary 1 is satisfied. As a result, Corollary 1 leads to Lemma 3 below. Lemma 3. As α increases, XαT becomes stochastically less variable. The mean-preserving truncation can find interesting applications in inventory models dealing with inefficiency associated with supply chain decentralization. In decentralized supply chains, the upstream usually sees a magnified demand uncertainty due to the widely recognized bullwhip effect. To combat the bullwhip effect, [9] analyze scheduled ordering policies for a two-stage decentralized supply chain consisting of a single supplier and multiple retailers who face uncertain end-customer demands. They consider the Scheduled Balanced Ordering Policy (SBOP) and the Scheduled Synchronized Ordering Policy (SSOP), both 11
allowing retailers to order from the supplier freely only in one period in an m-period ordering cycle. In other periods, retailers receive fixed shipments δ. Retailers take turns to order freely under SBOP, while under SSOP all retailers order freely in the same period. To evaluate the performance of the policies, it is crucial to examine the impact of the fixed shipment quantity δ on the uncertainty faced by the supplier. Let Dij be the end-customer demand at retailer i in period j of an ordering cycle and dij be its realization. The supplier sees an + uncertain demand ( m−1 j=1 dij + Dim − (m − 1)δ) from retailer i in the free-ordering period under both policies. Letting X =
m−1 j=1
dij + Dim and α = (m − 1)δ, we then have ( m−1 j=1 dij + Dim − (m −
1)δ)+ = (X − α)+ , which possesses the same stochastic ordering properties as XαT , because the constant terms μ − E [(X − α)+ ] in Equation (6) do not affect the variability of XαT . It then follows from Lemma 3 that, as expected from intuition, the uncertain demand faced by the supplier becomes stochastically less variable as δ increases, enabling considerable cost savings. Since [9] is silent on the impact of the fixed shipment quantity δ, we now conduct a numerical study to explore this issue. Our numerical study employs the same system parameters as those in [9], and focuses on quantifying the impact of the the fixed shipment quantity δ on the optimal ordering decisions and total costs. Similar to [9], we consider one supplier with two identical retailers; the cycle length m = 2; the unit holding cost is 1 for all parties; the backlogging costs are 6 and 9 for the retailer and the supplier, respectively. Figures 2 and 3 plot the impact of the fixed shipment quantity δ on the supplier’s optimal average order-up-to level and total inventory costs with different degree of demand uncertainty. Note that the average order-up-to level is used because the supplier makes order 12
Supplierorderupto level 115
Demandvar=600
Demandvar=400
Demandvar=100
Demandvar=50
105 95 85 75 65 55 45 35 25 1
3
5
7
9
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13
15
17
19
21
23
25
Fixed shipmentquantiy Figure 2: The impact of the fixed shipment quantity δ on supplier’s average order-up-to level
Suppliercost
Demandvar=600
Demandvar=400
30
Demandvar=100
Demandvar=50
25 20 15 10 5 1
3
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9
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13
15
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21
23
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Fixed shipmentquantiy Figure 3: The impact of the fixed shipment quantity δ on supplier’s total inventory cost
13
decisions after observing the retailers’ inventory level (see [9]). These two figures show that as the fixed shipment quantity δ increases, the supplier’s optimal average order-up-to level and total inventory costs both decrease, which is not hard to predict based on insights from [1] and our previous finding that the supplier’s demand becomes stochastically less variable as δ increases. Notably, Figure 3 shows that the supplier benefits more from increasing the fixed shipment quantity δ when the end demand has a larger variance. Under favorable conditions, the inventory cost reduction can be as high as one-third.
Retailerorderupto level 60
Demandvar=600
Demandvar=400
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Demandvar=50
55 50 45 40 35 30 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Fixed shipmentquantiy Figure 4: The impact of the fixed shipment quantity δ on retailers’ order-up-to level
On the retailer side, as the fixed shipment quantity δ increases, Figure 4 indicates that the retailers’ optimal order-up-to levels decrease, which is intuitive. The retailers’ total inventory cost, however, responds in a different way. In particular, it is convex in the fixed shipment quantity δ, as illustrated in Figure 5. This is because when δ is small, the retailers 14
Retailer cost 105
Demandvar=600
Demandvar=400
Demandvar=100
Demandvar=50
95 85 75 65 55 45 35 25 1
3
5
7
9
11
13
15
17
19
21
23
25
Fixed shipmentquantiy Figure 5: The impact of the fixed shipment quantity δ on retailers’ total inventory cost
benefit from the stability of the system enabled by the fixed shipment arrangement. But when δ grows too big, the over-commitment eventually hurts the retailers. 4.3 The mean-preserving capping Our results also apply to a third type of mean-preserving transformation, termed meanpreserving capping, as defined below. XαC ≡ Y C (X, α) = where E[min(X, α)] =
α a
xdF (x) +
b α
min(X, α) μ, E[min(X, α)]
(7)
αdF (x).
Taking partial derivative with respect to α yields ∂XαC = ∂α where EX≤α [X] =
α a
x[F (b)−F (α)] − (E[min(X,α)])2 μ, if x < α; EX≤α [X] μ, (E[min(X,α)])2
otherwise,
xdF (x). Since this derivative decreases in X for x < α, as shown 15
in Figure 6, the transformation defined in Equation (7) does not meet condition (b) in Corollary 1. Therefore, we need to develop from Theorem 1 another corollary to facilitate the study of the mean-preserving capping. ∂XαC ∂α EX≤α [X] (E[min(X,α)])2 μ
a
α
b
X
a(F (b)−F (α)) − (E[min(X,α)]) 2μ
α(F (b)−F (α)) − (E[min(X,α)]) 2μ
Figure 6: Visualizing the Weak Single-Crossing Property of ∂XαC /∂α
The derivative ∂XαC /∂α depicted in Figure 6 satisfies the so-called weak single-crossing (WSC) property, which can be utilized to show Theorem 1(b), as we shall see shortly. A function ψ : [a, b] → R is said to satisfy WSC on [a, b] if there exits x0 ∈ [a, b] such that ψ(x) ≤ 0 for all x < x0 and ψ(x) ≥ 0 for all x > x0 . When the WSC property is met, the following result can be derived from Theorem 1. Corollary 2. As α increases, Y (X, α) becomes stochastically more variable if the following conditions hold: 16
(a) Y (X, α) is increasing in X; (b) ∂Y (X, α)/∂α satisfies WSC on [a, b].
Proof. It suffices to show that part (b) leads to Theorem 1(b). When WSC is satisfied, if t < x0 , then by definition ∂Y (X, α)/∂α ≤ 0 on [a, t]. Hence we know EX≤t [∂Y (X, α)/∂α] ≤ 0. If t ≥ x0 , then ∂Y (X, α)/∂α ≥ 0 on [t, b], and thus EX≤t [∂Y (X, α)/∂α] ≤ E[∂Y (X, α)/∂α] =
0.
It is straightforward to verify that condition (a) of the above corollary is also satisfied by the mean-preserving capping XαC , because ∂XαC /∂X = μ/E[min(X, α)] if x ≤ α, and 0 otherwise. Therefore, Corollary 2 leads to the following. Lemma 4. As α increases, XαC becomes stochastically more variable. The mean-preserving capping can find important applications in procurement risk management (PRM), especially when option contracts are used. [10] reports that HewlettPackard (HP) obtained $425 million in costs savings cumulatively over six years by using the PRM approach. This approach hinges upon structured contracts with suppliers to manage the supply chain risks on the procurement side. At the core of the approach are two types of option contracts used to mitigate the risks posed by the volatile spot market price of components (e.g., flash memory): One is fixed price option contract and the other is discount-off-of-market price option contract. The fixed price option contract puts a cap on the cost and has two parameters: a unit reservation cost paid in advance by HP to its supplier, and a unit execution cost α paid later if the option is exercised after market price X realizes, i.e., if x ≥ α. In this contract, HP’s 17
exercising cost is min(X, α) because when the market price realizes it pays α by exercising the option if x ≥ α, and x by buying from the spot market if x < α. Note that α ≤ b (the upper support of the market price) for the contract to be viable, otherwise it will degenerate to simply buying at spot market price. The discount-off-of-market price option contract differs from the fixed price option contract only in the second parameter. It has a unit exercising cost ratio β, in place of the unit execution cost α. In this contract, HP’s exercising cost is βX when the market price realizes. The key issue for HP to address in order to optimize its portfolio of contracts is to understand which of the above two option contracts poses smaller risks, if both contracts have the same reservation cost and the expected exercising costs. It turns out that the stochastic variability ordering property proved in Lemma 4 can be used to tell that the fixed price option contract always has less variable exercising costs than the discount-off-of-market price option contract, as shown in the following lemma. Lemma 5. If E[βX] = E[min(X, α)], then βX ≥v min(X, α). Proof. From the equal expectation condition, i.e., the condition E[βX] = E[min(X, α)], we have β = E[min(X, α)]/E [X] = E[min(X, α)]/μ. Hence, it suffices to show that X ≥v μ min(X, α)/E[min(X, α)], which follows directly from Lemma 4, because X = XbC according to Equation (7), and b ≥ α.
4.3.1 Comparison between total costs of the fixed price option and the discount-off-of-market price option contracts Early papers studying the option contract (e.g., [19]) typically consider the capacity purchase decision from suppliers under demand uncertainty, but not market price uncertainty. 18
[20] considers market price uncertainty, but not demand uncertainty. [21] integrates both uncertainties and shows that with a single supplier, a mixed procurement strategy consisting of a long-term contract and spot market procurement is optimal. [22] also incorporates both demand and market price uncertainties into the procurement problem using option contracts, but they consider multiple suppliers. [23] further extends [22] to account for the fixed procurement costs. However, all these papers study only the fixed price option contract. The discount-off-of-market price option contract that has been used by HP for PRM [10] has not received due research attention. To fill the gap in the literature, we next study the discount-off-of-market price option contract and compare it with the fixed price option contract. As a benchmark, the fixed price option contract studied in [22] is described in greater detail as follows: let c denote the unit cost of capacity reservation, D denote the random demand, and X denote the random spot market price. The decision for the buyer firm (e.g., HP) is how much capacity Q to reserve from the supplier. Under the fixed price option contract, the buyer reserves capacity Q at unit cost c upfront. After the spot market price and demand realize, the buyer can buy up to Q at the price min(α, X) from the supplier and the rest at the spot price X to meet all her demand. Let 1Z be an indicator function, i.e., 1Z = 1 if Z is true; and 0 otherwise. The total expected procurement cost of the fixed price option contract can then be expressed as follows: cQ + ED,X [1D≤Q min(α, X)D + 1D≥Q (min(α, X)Q + X(D − Q))],
(8)
where ED,X [·] denotes the expectation over both random variables D and X. In this expression, the first term cQ is the total reservation cost. The entire second term (i.e., ED,X [·]) 19
is the expected cost of exercising the option and spot market procurement, which is derived from two cases: if the realized demand D is below the reserved capacity Q, the buyer will purchase D units of capacity at the unit cost min(α, X); otherwise, the buyer will buy all the reserved capacity Q at min(α, X) each, and purchase an additional capacity D − Q at the unit spot price X to cover the shortfall. We now turn our attention to the discount-off-of-market price option contract with an exercising cost ratio β, which results in the following total expected procurement cost: cQ + ED,X [1D≤Q βXD + 1D≥Q (βXQ + X(D − Q))].
(9)
A close look at Equations (8) and (9) reveals that the total expected procurement costs of the two contracts closely resemble each other and have similar interpretations. The only difference between them is the unit exercising cost of the reserved capacity: min(α, X) under the fixed price option, and βX under the discount-off-of-market price option. As shown in Lemma 5, the unit exercising cost βX is stochastically more variable than min(α, X). As a result, one might draw a parallel between [1] and our model, and intuit that the discount-off-of-market price option contract yields a higher total expected procurement cost. However, this intuition is disproved by the following proposition.
Proposition 1. For independent demand and spot market price, if the fixed price option contract and the discount-off-of-market price option contract have the same expected unit exercising cost, i.e., EX [min(α, X)] = EX [βX], then the following holds: (a) The two contracts have the same optimal capacity reservation level Q∗ ; (b) The two contracts have the same total expected procurement cost. 20
Proof. For any reserved capacity Q, the difference (9) − (8) is ED,X [1D≤Q βXD] − ED,X [1D≤Q min(α, X)D] + ED,X [1D≥Q βXQ] − ED,X [1D≥Q min(α, X)Q] = ED [1D≤Q D]{EX [βX] − EX [min(α, X)]} + QED [1D≥Q ]{EX [βX] − EX [min(α, X)]} = 0,
where the first equality follows from the assumption of independence between D and X, and the second equality follows from EX [min(α, X)] = EX [βX]. In other words, if the conditions stated in the proposition are satisfied, then the two contracts behave identically for any reserved capacity Q. As a result, the optimal capacity reservation level Q∗ and the associated total expected procurement costs are the same.
Proposition 1 makes a nice contribution to the literature. It highlights an important difference between the impact of demand uncertainty and unit exercising cost uncertainty on total inventory (or procurement) costs. In particular, unlike demand uncertainty, unit exercising cost uncertainty in general is not a major driver of the total inventory cost, which is an insight not developed in the prior literature. The fundamental reason behind the difference can be explained as follows: demand uncertainty makes it difficult to match supply and demand, whereas unit exercising cost uncertainty does not. While the fixed price option contract and the discount-off-of-market price option contract are equivalent for independent demand and spot market price, it remains an open question whether the equivalence remains unchanged when demand and spot market price are correlated (correlation is common in many industries, especially those with a few dominant players). We now perform a numerical study to explore this question. In our study, we employ the same parameters as the those used in [22]: demand and market price are mod21
eled by a truncated bivariate normal distribution with demand D ∼ N (100, 50) and spot market price X ∼ N (20, 7), c = 5.3272, α = 6.7810 (note that the aggregate unit capacity cost c + α = 12.1082, which falls in the range of the price distribution). In addition, for the discount-off-of-market price option contract, we set β = 0.335 to guarantee equal mean of the unit exercising cost between the two contracts, i.e., EX [min(α, X)] = EX [βX].
Opimtal reservation quantity 140
Fixedpriceoption
Discountpriceoption
130 120 110 100 90 80 1
0.5
0
0.5
1
Correlationbetweendemandandmarketprice Figure 7: Comparison of the optimal capacity reservation level between the fixed price option contract and discount-off-of-market price option contract
Figures 7 and 8 compare the optimal capacity reservation levels and the optimal total expected procurement costs, respectively, of the two contracts. As we can see, the comparison hinges on the correlation between demand and the spot market price. It is worth noting that in both figures, the two curves cross each other when the correlation is 0. In other words, the two contracts have the same optimal capacity reservation decisions and 22
the corresponding optimal total expected procurement costs when demand and spot market price are independent, which corroborates our results in Proposition 1.
Opimtal procurement costs
Fixedpriceoption
Discountpriceoption
170 160 150 140 130 120 1
0.5
0
0.5
1
Correlationbetweendemandandmarketprice Figure 8: Comparison of the total expected procurement costs between the fixed price option contract and discount-off-of-market price option contract
However, the two contracts no longer behave the same way when demand and spot market price are correlated. Conceivably, the buyer is most vulnerable when the realized demand and spot market price are both high, which is possible when they are positively correlated. Under such a circumstance, the fixed price option better protects the buyer than the discount-off-of-market price option does, because the unit exercise cost of the former is capped at α, whereas for the latter, the unit exercise cost can get out of hand. This explains why in Figure 7, the optimal reserved capacity for the fixed price option contract is greater than that for the discount-off-of-market price option contract when the correlation 23
is positive, because the buyer can only cash in on the better protection provided by the fixed price option contract if sufficient capacity is reserved. Therefore, the buyer tends to reserve more capacity under the fixed price option contract to reap more benefits from it. The same logic also explains why the total expected procurement cost is lower for the fixed price option contract than for the discount-off-of-market price option contract when the correlation is positive, as can be seen in Figure 8. When demand and market price are negatively correlated, the relationship between the two contracts is totally reversed. In the presence of negative correlation between demand and market price, market price tends to be low when realized demand is high. In such a case, the fixed price option contract is not very useful because the low market price reduces the odds to exercise the option. Therefore, the buyer tends to reserve less capacity in order not to waste the reservation cost. In contrast, the discount-off-of-market price option contract is more effective because the buyer can always enjoy the discount within the reserved capacity. As a result, reserving more capacity makes sense. On the other hand, market price tends to be high when realized demand is low. In such a case, the reserved capacity would be low for both contracts – the demand is low to begin with. Overall, when demand and market price are negatively correlated, the fixed price contract is less beneficial to the buyer and results in a lower reserved capacity and a high total expected procurement cost, as shown in Figures 7 and 8. In sum, our analysis shows that although the fixed price option contract has a stochastically less variable unit exercising cost than the discount-off-of-market price option contract, the stochastic variability of the unit exercising cost is not really the determinant of the total procurement cost. Instead, the correlation between demand and spot market price plays a 24
more important role. 5.
Conclusion The stochastic variability measures the degree of uncertainty for random demand and/or
price in various operations problems. Its ordering property allows us to study the impact of demand/price uncertainty on the optimal decisions and the associated objective values. However, stochastic ordering of random variables is typically challenging without a tractable approach to capture variability. In this paper, we provide a unifying framework for stochastic variability ordering under any mean-preserving transformation that can be parameterized by a single scalar, and apply it to a broad class of specific transformations, including the widely used mean-preserving affine transformation, truncation, and capping. We first apply our framework to the mean-preserving affine transformation, which is then used to rectify an incorrect proof of a fundamental result in the inventory literature [1] that has gone unnoticed for more than two decades. The second application, to mean-preserving truncation, addresses inventory strategies in decentralized supply chains. Specifically, we use it to investigate the effectiveness of using fixed shipment quantity to mitigate inefficiency associated with supply chain decentralization. We conduct a numerical study to complete the work of [9], and illustrate that the potential cost reduction could be as high as one-third under certain parametric conditions. Lastly, we also apply our general framework to the mean-preserving capping, which sheds light on using option contracts for procurement risk management. To the best of our knowledge, we are the first to analyze the discount-off-of-market price option contract,
25
which is used by HP to manage its procurement risk. We use our approach to theoretically compare the stochastic variability of the unit exercising cost of two contracts: the fixed price option contract and the discount-off-of-market price option contract. We also conduct a numerical study to assess the relative performance of the two contracts. Surprisingly, we find that the stochastic variability of the unit exercising cost is not really the determinant of the total procurement cost. Instead, what drives the disparity between the two option contracts is the correlation between demand and spot market price. References References [1] Y. Gerchak, D. Mossman, On the effect of demand randomness on inventories and costs, Operations Research 40 (4) (1992) 804–807. [2] I. Nyoman Pujawan, The effect of lot sizing rules on order variability, European Journal of Operational Research 159 (3) (2004) 617–635. [3] C. A. Garcia Salcedo, A. Ibeas Hernandez, R. Vilanova, J. Herrera Cuartas, Inventory control of supply chains: Mitigating the bullwhip effect by centralized and decentralized internal model control approaches, European Journal of Operational Research 224 (2) (2013) 261–272. [4] H. Levy, Stochastic Dominance: Investment Decision Making under Uncertainty, Springer, 2006. [5] A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, Springer, 2010.
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[6] M. Shaked, J. G. Shanthikumar, Stochastic orders, Springer, 2007. [7] M. Rothschild, J. E. Stiglitz, Increasing risk: I. a definition, Journal of Economic Theory 2 (3) (1970) 225–243. [8] Y. Gerchak, Q.-M. He, On the relation between the benefits of risk pooling and the variability of demand, IIE Transactions 35 (11) (2003) 1027–1031. [9] L. G. Chen, S. Gavirneni, Using scheduled ordering to improve the performance of distribution supply chains, Management Science 56 (9) (2010) 1615–1632. [10] V. Nagali, J. Hwang, D. Sanghera, M. Gaskins, M. Pridgen, T. Thurston, P. Mackenroth, D. Branvold, P. Scholler, G. Shoemaker, Procurement risk management (PRM) at Hewlett-Packard company, Interfaces 38 (1) (2008) 51–60. [11] D. Mitrinovi´c, J. Peˇcari´c, A. Fink, Classical and New Inequalities in Analysis, Kluwer, Dordrecht, 1993. [12] E. Porteus, Foundations of Stochastic Inventory Theory, Stanford University Press, CA, 2002. [13] V. Agrawal, S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management 2 (4) (2000) 410–423. [14] J.-S. Song, C. A. Yano, P. Lerssrisuriya, Contract assembly: Dealing with combined supply lead time and demand quantity uncertainty, Manufacturing & Service Operations Management 2 (3) (2000) 287–296. 27
[15] A. V. Iyer, A. Jain, The logistics impact of a mixture of order-streams in a manufacturerretailer system, Management Science 49 (7) (2003) 890–906. [16] O. Alp, N. K. Erkip, R. Gullu, Outsourcing logistics: Designing transportation contracts between a manufacturer and a transporter, Transportation Science 37 (1) (2003) 23–39. [17] Q. Li, D. Atkins, On the effect of demand randomness on a price/quantity setting firm, IIE Transactions 37 (12) (2005) 1143 – 1153. [18] Y. Gerchak, M. Parlar, Investing in reducing lead-time randomness in continuous-review inventory models, Engineering Costs and Production Economics 21 (2) (1991) 191–197. [19] D. A. Serel, M. Dada, H. Moskowitz, Sourcing decisions with capacity reservation contracts, European Journal of Operational Research 131 (3) (2001) 635–648. [20] D. J. Wu, P. R. Kleindorfer, J. E. Zhang, Optimal bidding and contracting strategies for capital-intensive goods, European Journal of Operational Research 137 (3) (2002) 657–676, 00225. [21] R. W. Seifert, U. W. Thonemann, W. H. Hausman, Optimal procurement strategies for online spot markets, European Journal of Operational Research 152 (3) (2004) 781–799, 00178. [22] Q. Fu, C.-Y. Lee, C.-P. Teo, Procurement management using option contracts: random spot price and the portfolio effect, IIE Transactions 42 (11) (2010) 793–811. [23] C.-Y. Lee, X. Li, Y. Xie, Procurement risk management using capacitated option contracts with fixed ordering costs, IIE Transactions 45 (8) (2013) 845–864. 28
2ESEARCH (IGHLIGHTS
x x x x
A unified theory that enables variability ordering for any meanpreserving transformation Application to affine transformation and correction of an error in the literature Application to truncation and inventory models in supply chain management Application to capping and procurement risk management.