Robust tests for the bullwhip effect in supply chains with stochastic dynamics

European Journal of Operational Research 185 (2008) 340–353 www.elsevier.com/locate/ejor

O.R. Applications

Robust tests for the bullwhip effect in supply chains with stochastic dynamics Yanfeng Ouyang a

a,*

, Carlos Daganzo

b

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, United States b Institute of Transportation Studies and Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, United States Received 14 February 2006; accepted 31 October 2006 Available online 21 December 2006

Abstract This paper analyzes the bullwhip effect in single-echelon supply chains driven by arbitrary customer demands and operated nondeterministically. The supply chain, with stochastic system parameters, is modeled as a Markovian jump linear system. The paper presents robust analytical conditions to diagnose the bullwhip effect and bound its magnitude. The tests are independent of the customer demand. Examples are given. Ordering policies that pass these tests, and thus avoid the bullwhip effect in random environments for arbitrary customer demands, are shown to exist. The paper also presents possible extensions to multi-echelon chains.  2006 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Systems dynamics

1. Introduction In a supply chain, the fluctuations in a supplier’s order sequence are usually greater if the supplier is farther away from the customer. This phenomenon, often referred to as ‘‘the bullwhip effect,’’ has been found not only in industry operations [16,17,30,31], but also in macroeconomic data [2,3,22,26,35,36] and simulations studies (such as the Beer Game) [21,27,39]. The presence of the bullwhip effect results in huge extra operation costs [8,28,29]. Extensive research has been conducted to identify operational causes for the bullwhip effect. Earlier studies focus on single-echelon chains and specific families of customer demand processes, where the variances of orders placed and received by the supplier are analyzed and compared [2,6,7,20,29]. For multi-echelon chains, results exist for specific customer demand (e.g., ARMA) and certain ordering policies (e.g., order-up-to policy) [1,18–20,41]. Managerial insights are drawn to reveal how the assumed demand processes and ordering policies influence the bullwhip effect. *

Corresponding author. Tel.: +1 217 333 9858; fax: +1 217 333 1924. E-mail address: [email protected] (Y. Ouyang).

0377-2217/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.10.046

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Recently, efforts have been made to obtain robust diagnostic tests for the bullwhip effect. Because customer demand is often difficult to predict or control, it is desirable to derive analytical results that hold independently of the customer demand. Refs. [10–12] first present robust analytical results for multi-echelon chains with general demand inputs. Refs. [13,14] derive demand-dependent variance formulae and numerically illustrate the bullwhip effect. Refs. [33,34] present exact formulae to characterize the bullwhip effect at any stage of the chain with or without knowledge of the demand process. All these works assume that the factors determining the system dynamics (e.g., shipping reliability, business environment) are deterministic and known. In reality, however, supply chain operations are often subject to random variations. For example, the transportation network may suffer from unexpected congestion and hence yield random lead times; price discounts may be offered from time to time partly due to alternating growth and recession seasons. When the economy is growing (or seasonal discounts are offered), suppliers are more likely to use aggressive operating (production or ordering) policies that maintain higher inventory levels. During recessions or when discounts are over, suppliers are likely to be more conservative. Earlier research on the bullwhip effect, e.g., that in [28,29], finds that stochastic supplier behavior and price fluctuations exaggerate the bullwhip effect for specific ordering policies and customer demand processes. This paper presents some robust results for general policies, allowing for stochastic supplier behavior and operating uncertainties; i.e., for stochastic system dynamics. Randomness introduces complications that eliminate the advantages of working in the frequency domain with transform methods as in [10–14,33,34]. Therefore, we work here in the time domain, modeling the chain as a Markovian jump linear system (MJLS). We focus on a single echelon, but multi-echelon models are also presented. Section 2 introduces the basic notation, problem formulation, and a proposed bullwhip effect metric for robust analysis. Section 3 presents various analytical results for general policies, including diagnostics for the bullwhip effect and robust bounds for its magnitude. Section 4 gives several examples. Section 5 discusses the generalization of this work to multi-echelon chains. 2. Formulation We first present the basic notation for deterministic chains, and then introduce modifications to capture randomness. 2.1. Basic notation for deterministic linear time-invariant (LTI) systems The notation and policies in this subsection are the same as in [34]. Consider a single-echelon chain where a retailer satisfies demand u0(t) from a customer and places orders u1(t) to a vendor at discrete times t =. . ., 2, 1, 0, 1, 2, . . . Shipments from the vendor are received after a constant lead time l = 0, 1, 2, . . . The conservation equations for the retailer’s inventory position at time t (cumulative orders placed minus orders received, x(t)) and for the in-stock inventory at time t (cumulative items received minus orders received, y(t)) are: xðt þ 1Þ ¼ xðtÞ þ u1 ðtÞ  u0 ðtÞ;

ð1Þ

and yðt þ 1Þ ¼ yðtÞ þ u1 ðt  lÞ  u0 ðtÞ:

ð2Þ 0

0

0

The retailer places orders based on its inventory records {(x(t ), y(t )): "t 6 t} and the histories of orders received and placed {(u0(t 0 ), u1(t 0 )): "t 0 6 t  1}. Since (1) implies: u1(t 0 ) = x(t 0 + 1)  x(t 0 ) + u0(t 0 ), if follows that the history of orders placed, {u1(t 0 ), "t 0 6 t  1}, is redundant information. Thus, all the information available to the retailer at time t can be encapsulated in the following information set: IðtÞ :¼ fxðtÞ; xðt  1Þ; . . . ; xð1Þ; yðtÞ; yðt  1Þ; . . . ; yð1Þ; u0 ðt  1Þ; u0 ðt  2Þ; . . . ; u0 ð1Þg: A linear time-invariant (LTI) policy is a recipe for determining u1(t) from the elements of IðtÞ. The most general LTI policy can be written as follows:

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u1 ðtÞ ¼ c þ AðP ÞxðtÞ þ BðP ÞyðtÞ þ CðP Þu0 ðt  1Þ;

ð3Þ

where the parameter c is a real number, and A(Æ), B(Æ), C(Æ) are polynomials with real coefficients. The symbol P is a backward lag operator; i.e., Pkx(t) = x(t  k), " k = 0, 1, 2, . . . The polynomials A(P) and B(P) indicate the influence of the inventory history, and C(P) the history of orders received. We assume that policy (3) is proper in the sense of [10]; i.e., that if the retailer receives and places orders of constant size u1, the system tends to a nominal equilibrium state in which the retailer holds steady inventory position x1 and in-stock inventory y1, and such that: u1 ¼ c þ AðP Þx1 þ BðP Þy 1 þ CðP Þu1 :

ð4Þ

We assume for convenience that the system is in one of these equilibria for all t 6 0. To facilitate analysis, the system dynamics are expressed in terms of deviations (errors) from an equilibrium. (This is standard practice, especially for nonlinear policies.) To this end, let xðtÞ :¼ xðtÞ  x1 ; y ðtÞ :¼ yðtÞ  y 1 , and  ui ðtÞ :¼ ui ðtÞ  u1 , i = 0, 1. Note that xðtÞ ¼ y ðtÞ ¼ u1 ðtÞ ¼ u0 ðtÞ ¼ 0 for all t = 1, . . ., 0, since the system is assumed to start from equilibrium. The usual manipulations (e.g., subtracting (4) from (3), etc.) yield: xðt þ 1Þ ¼ xðtÞ þ  u1 ðtÞ   u0 ðtÞ; y ðt þ 1Þ ¼ y ðtÞ þ u 1 ðt  li Þ   u0 ðtÞ; 1 ðtÞ ¼ AðP ÞxðtÞ þ BðP Þy ðtÞ þ CðP Þ u u0 ðt  1Þ:

ð5Þ ð6Þ ð7Þ

Note that properness in this case implies that u1 ðtÞ ! 0 if u0 ðtÞ ! 0 as t ! 1. These equations were analyzed in [34]. We now recast them in a way that will allow us to treat the lead time l and the coefficients of A, B, C as random variables. 2.2. Matrix representation of deterministic LTI systems We express (5)–(7) in a form that directly relates order sequences fu1 ðtÞg and fu0 ðtÞg. To do this, write (7) for periods t + 1 and t and take the difference:  u1 ðt þ 1Þ   u1 ðtÞ ¼ AðP Þ½xðt þ 1Þ  xðtÞ þ BðP Þ½y ðt þ 1Þ  y ðtÞ þ CðP Þ½u0 ðtÞ  u0 ðt  1Þ: Then, using (5) and (6) to eliminate ½xðt þ 1Þ  xðtÞ and ½y ðt þ 1Þ  y ðtÞ, we obtain:  u1 ðtÞ þ WðP Þ u0 ðtÞ; u1 ðt þ 1Þ ¼ UðP Þ

ð8Þ

where U and W are polynomials related to A, B, C by U(P) :¼ [1 + A(P) + PlB(P)] and W(P) :¼ [(1  P)C(P)  B(P)  A(P)]. We now eliminate the dependence of (8) on history prior to t by augmenting the state. Assume both U(P) and W(P) have finite degrees; i.e., K :¼ max{degU(Æ), degW(Æ)} 6 1. Define a (K + 1) · 1 column vector ui ðtÞ;  ui ðt  1Þ; . . . ;  ui ðt  KÞT ; ui ðtÞ :¼ ½

i ¼ 0; 1;

ð9Þ

to represent the augmented ‘‘order state’’ for the customer and retailer at time t. The system dynamics (8) can now be written in terms of a single time step: u1 ðt þ 1Þ ¼ R  u1 ðtÞ þ S  u0 ðtÞ; where R and S are (K + 1) · (K + 1) matrices: 3 2 a0 a1    aK1 aK 2 b0 7 6 1 0  0 0 7 6 60 7 6 6 0 1  0 0 7; S :¼ 6 . R :¼ 6 7 6 6 . .. 7 .. .. . . 6 .. 4 . 4 . . 5 . . . 0 0 0  1 0

ð10Þ

b1



bK

0 .. .

 .. .

0 .. .

0



0

3 7 7 7; 7 5

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and ui(t) = 0, "t 6 0, i = 0, 1. The entries in {R, S}, i.e., ak and bk, k = 0, 1, . . ., K, are given by the coefficients of Pk in U(P) and W(P), respectively. They are functions of system parameters such as the lead time l and policy parameters in A(Æ), B(Æ), and C(Æ). The advantage of formulation (10) is that stability results exist for systems where the matrix pair {R, S} changes randomly over time [9,25]. We shall build on these results. This will allow us to diagnose systems with, for example, random lead times. The following subsection describes the random variations that are allowed. 2.3. Markovian jump linear systems (MJLS) A MJLS is a stochastic system that jumps randomly among several linear modes of operation; see for example [15,32,37,40]. In the supply chain context, the parameters of the system, {ak, bk}, define the modes. We assume that (i) there is a finite set of modes, M ¼ f1; 2; . . . ; Mg; and (ii) the modes evolve as a Markov chain {h(t): "t P 0}. The stochastic system dynamics are then represented by: u1 ðt þ 1Þ ¼ RhðtÞ  u1 ðtÞ þ S hðtÞ  u0 ðtÞ;

ð11Þ

where h(t) is the mode in effect at time t. When hðtÞ ¼ m 2 M, the system is in mode m, and we shall use Rm :¼ Rh(t) and Sm :¼ Sh(t) to denote the matrices of that specific mode. Finally, the transition probability matrix of the chain is defined as P ¼ ½pmn MM ; i.e., pmn ¼ Prfhðt þ 1Þ ¼ n j hðtÞ ¼ mg; 8m; n 2 M. The stochastic dynamics described by (11) can be used to model many supply chain operations that are subject to uncertainty (e.g., random delay and ordering policy alternations). Detailed applications of this model will be shown with numerical examples in Section 4. We now discuss the properties of this system that are relevant to the bullwhip effect. We define properness as follows: Definition 1 (Properness). For the system described by (11) with u0(t)  0, "t, the equilibrium state at u1(t) = 0 is proper if for every possible value of the initial state {u1(0),h(0)}, lim EfhðtÞg fku1 ðtÞk2 ju1 ð0Þ; hð0Þg ¼ 0;

t!1

or equivalently ( ) 1 X 2 EfhðtÞg ku1 ðtÞk ju1 ð0Þ; hð0Þ < 1; t¼0

where k Æ k denotes Euclidean norm and the expectations are taken across possible realizations of the Markov chain. For deterministic chains, Definition 1 reduces to the conventional definition of properness. Definition 1 is called ‘‘stochastic stability’’ in the MJLS literature. Therefore, we will use the words ‘‘properness’’ and ‘‘stability’’ interchangeably from now on. It is known that the introduction of parameter uncertainty may lead a system toward instability. Each mode being stable is neither necessary nor sufficient for an MJLS to be stable [24]. Ref. [9] introduced several necessary and sufficient conditions for stochastic stability in the form of Lyapunov functions. We now define a robust metric for the bullwhip effect. 2.4. A robust metric for the bullwhip effect in MJLS Note from (11) that any realization of customer demand fu0 ðtÞg1 t¼0 yields a random retailer order sequence 1 1 f u1 ðtÞgt¼0 that is dependent on the Markov chain fhðtÞgt¼0 . Thus, an easily understood metric for the bullwhip effect – intimately related to the conventional variance amplification measures – is the ratio of (i) the expected 1 root mean square errors (RMSE) of the retailer order sequence, fu1 ðtÞgt¼0 , and (ii) the RMSE of the customer 1 demand, f u0 ðtÞgt¼0 . This metric, however, depends on the customer demand. Therefore, for robust analysis, we adopt the worst-case expected RMSE amplification factor, W, across all possible customer demand sequences:

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Definition 2 (Worst-case metric). The proposed bullwhip effect metric is "  P1 #1=2 E u21 ðtÞ t¼0   P1 2  W :¼ sup ; u0 ðtÞ 8f u0 ðtÞg6¼0 t¼0 

ð12Þ

where f u0 ðtÞg is square summable. The expectation in the numerator of (12) (and all expectations without subscripts in this paper) is taken across realizations of the Markov chain {h(t)}. The worst-case metric will allow us to certify (with the condition W 6 1 from (12)) that the RMSE is not amplified under any customer demand whatsoever; i.e., that the bullwhip effect does not arise. We are now ready to introduce our main results. 3. Bullwhip effect bounds and tests Section 3.1 presents upper bounds for W in the form of linear matrix inequalities, and Section 3.2 describes their numerical evaluations. 3.1. Analytical results Theorem 1 presents sufficient conditions that bound the RMSE amplification W. The conditions are based on positive semi-definite matrices and non-strict matrix inequalities. It uses ideas in the MJLS literature [38]. Theorem 1. The supply chain is proper and W 6 c if there exist non-zero positive semi-definite matrices, Gn P 0; 8n 2 M and H P 0, that satisfy either of the following non-strict linear matrix inequality sets: (a) 



Gn

0

0

c2 H

Gn

0

0

c2 H



M X

 pnm

m¼1

(b) 



 

Rn

Sn

E

0

Rm

Sm

E

0

T "

T 

Gn

0

0

H

Gm

0

0

H

#



Rn

Sn

E

0

Rm

Sm

E

0

 P 0;

 P 0;

Gn :¼

8n 2 M;

M X

pnm Gm ;

ð13Þ

8n 2 M;

ð14Þ

m¼1

where E is the (K + 1) · (K + 1) identity matrix. Proof. See Appendix A. h For c = 1, Theorem 1 provides a test for the existence of the bullwhip effect. In the deterministic case (M = 1), Theorem 1 reduces to the following: Corollary 1. Deterministic LTI supply chains (with matrices R and S) do not experience the bullwhip effect if there exist non-zero positive semi-definite matrices G P 0 and H P 0 that satisfy 

G

0

0

H



 

R

S

E

0

T 

G

0

0

H



R

S

E

0

 P 0:

ð15Þ

This test complements those presented in [11,33]. Stronger sufficient and necessary conditions can also be developed to provide bounds for the bullwhip effect, as shown in the theorem below. The major difference is that matrices Gn(" n) and H have to be positive definite, the linear matrix inequalities (16) and (17) are strict, and the system have to be weakly controllable in the sense of [23]; i.e., for any initial state/mode fu01 ; h0 g and final state/mode fuf1 ; hf g there exists a finite time sf and customer order sequence {u0(t)} such that Prfu1 ðsf Þ ¼ uf1 ; hðsf Þ ¼ hf g > 0.

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Theorem 2. A supply chain is proper and W < c if and only if: (i) system (11) is weakly controllable; and (ii) there exist (K + 1) · (K + 1) positive definite matrices, Gn > 0; n 2 M and H > 0, that satisfy either of the following linear matrix inequality sets: (a) 

Gn

0

0

c2 H

Gn 0

0 2 cH

 

M X

 pnm

m¼1

(b) 





Rn  E

Sn 0

Rm

Sm

E

0

T "

T 

Gn

0

0

H

Gm

0

0

H

#

Rn E



Sn 0

Rm

Sm

E

0

 > 0;

 > 0;

Gn :¼

8n 2 M;

M X

pnm Gm ;

ð16Þ

8n 2 M:

ð17Þ

m¼1

Proof. See Appendix B. h 3.2. Implementation Theorems 1 and 2 both define a feasibility problem with matrix variables subject to linear matrix inequalities (LMI). The search for feasible matrices Gn(" n), and H can be conducted by convex optimization. Taking Theorem 2 as an example, we could pick a c (e.g., c = 1) and then solve the following optimization problem: min

1

s:t:

ð16Þ or ð17Þ TrðH Þ ¼ 1ðscalingÞ; Gn > 0; H > 0:

ð18Þ

8n 2 M;

If this problem is feasible, W < c. For any given c, problem (18) has Mþ1 ðK þ 1ÞðK þ 2Þ scalar variables and is 2 solvable in polynomial time [4]. This is also true of the related problem with non-strict inequalities (13), (14) or (15). Algorithms (e.g., ellipsoid and interior-point methods) and solvers have been developed [5]. To determine the tightest bound, we can conduct a binary search for the infimum of all feasible c. 4. Examples 4.1. Varying policies The business environment often exhibits alternating seasons of growth and recession. This cyclic phenomenon is often modeled as an exogenous Markov chain. Suppose, in any period of economic recession, the probability of starting growth in the next period is p1 = 0.1; we assume that the retailer uses a conservative order-based policy [10] (mode m = 1) with A(P) = 0.5, B(P) = 0, C(P) = 0.2, which yields:  u1 ðt þ 1Þ ¼ 0:5 u1 ðtÞ þ 0:3 u0 ðtÞ þ 0:2 u0 ðt  1Þ: We know that this policy does not incur the bullwhip effect in LTI chains [34]; i.e., W = 1 for this policy. During a growth period, the probability of ending growth in the next period is p2 = 0.2; the retailer uses an orderup-to policy (mode m = 2) where the ‘‘up-to level’’ is forecasted by a moving-average of demand received over 2 recent periods; A(P) = 1,B(P) = 0,C(P) = 1 + P and:  u1 ðt þ 1Þ ¼ 2 u0 ðtÞ   u0 ðt  2Þ: This policy is known to incur the bullwhip effect in LTI chains [34] with W = 3. We now apply our theorems to the full MJLS.

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We define u1(t) = [u1(t), u1(t  1), u1(t  2)]T and u0(t) = [u0(t), u0(t  1), u0(t  2)]T. In the recession mode (m = 1), the matrices representing the system dynamics are 2 3 2 3 0:5 0 0 0:3 0:2 0 6 7 6 7 R1 ¼ 4 1 0 0 5 ; S 1 ¼ 4 0 0 0 5: 0

1

0

0

0

0

In the growth mode (m = 2), the matrices are: 2 3 2 3 0 0 0 2 0 1 6 7 6 7 R2 ¼ 4 1 0 0 5; S 2 :¼ 4 0 0 0 5: 0

1 0

0

0

0

The corresponding transition probability matrix is     1  p1 0:9 0:1 p1 P¼ ¼ : 0:2 0:8 p2 1  p2 1 The long-run probabilities for the two modes are p1 1 ¼ 2=3; p 2 ¼ 1=3. It is trivial to verify that this system is stochastically stable and weakly controllable. Given the data, we cannot find matrix variables to satisfy the conditions in Theorem 2 with c = 1; the bullwhip effect will exist. A search over the solutions of (18) with different c quickly reveals that the worst-case bound for RMSE amplification is c = 2.7054. Interestingly, this exceeds the long-term average of the bounds of the deterministic 2 1 1=2 modes, ð1  p1 ¼ 1:91. 1 þ 3  p2 Þ

4.2. Delivery delay We consider a simple case where lead time l = 2 under normal conditions. With probability p = 0.1 every transportation shipment is independently subject to delay by exactly one time period. This varying lead time (l 2 {2, 3}) influences physical flow of items, and hence affects the performance of policies based on in-stock inventory; e.g., the general kanban policy [34] with A(P) = 1/8, B(P) = 1/8, and C(P) = (1 + P)/2:  u1 ðtÞ ¼ xðtÞ=8  y ðtÞ=8 þ  u0 ðt  1Þ=2 þ  u0 ðt  2Þ=2;

8t;

There are four possible modes of operation that would affect the in-stock inventory, depending on whether the orders scheduled to arrive in the previous and the current periods arrive on time or not. In each of these modes the in-stock inventory is governed by a different dynamic equation which leads to different matrix pairs. For example, if the previous and the current arrivals are both on-time (mode 1), the in-stock inventory changes according to (2) with l = 2: y 1 ðt þ 1Þ ¼ y 1 ðtÞ þ u1 ðt  2Þ  u0 ðtÞ: It can be calculated that for dynamics are 2 7=8 0 1=8 6 1 0 0 6 R1 :¼ 6 4 0 1 0 0

0

1

LTI chains in this mode, W = 1.6763. The matrices representing the system 3 0 07 7 7; 05 0

2

3=4 6 0 6 S 1 :¼ 6 4 0 0

0 0 0 0

3 1=2 0 0 07 7 7: 0 05 0

0

If the previous arrival is on-time and the current arrival is delayed (mode 2), no shipment arrives from upstream of the chain: y 1 ðt þ 1Þ ¼ y 1 ðtÞ  u0 ðtÞ: If the previous arrival is delayed and the current arrival is on-time (mode 3), two shipments arrive at the same time:

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347

y 1 ðt þ 1Þ ¼ y 1 ðtÞ þ u1 ðt  2Þ þ u1 ðt  3Þ  u0 ðtÞ: Finally, if the previous and the current arrivals are both delayed (mode 4), the in-stock inventory changes according to (2) with l = 3: y 1 ðt þ 1Þ ¼ y 1 ðtÞ þ u1 ðt  3Þ  u0 ðtÞ: In modes 2–4, W = 2.0000, 2.4395, 1.8716, respectively. The matrices for these modes are 2 3 2 3 2 3 7=8 0 0 0 7=8 0 1=8 1=8 7=8 0 0 1=8 6 1 6 1 6 1 0 0 07 0 0 0 7 0 0 0 7 6 7 6 7 6 7 R2 :¼ 6 7; R3 :¼ 6 7; R4 :¼ 6 7; 4 0 4 0 4 0 1 0 05 1 0 0 5 1 0 0 5 0 0 1 0 S 2 :¼ S 3 :¼ S 4 :¼ S 1 :

0

0

1

0

0

0

1

0

The stochasticity in lead times can be captured by a Markov chain with M = 4 and transition probability matrix: 3 2 3 2 0:9 0:1 0 0 1p p 0 0 6 6 0 0 0:9 0:1 7 0 1p p7 7 6 7 6 0 P¼6 7: 7¼6 41  p p 0 5 0 0 5 4 0:9 0:1 0 0

0 1p

0

p

0

0:9

0:1

The long-term probability for the system to be in these fours modes are (1  p)2 = 0.81, p(1  p) = 0.09, p(1  p) = 0.09, p2 = 0.01, respectively. It is trivial to verify that this system is also stochastically stable and weakly controllable. Given the data, we cannot find matrix variables to satisfy the conditions in Theorem 2 with c = 1; the bullwhip effect will exist. A search quickly reveals that the bound for RMSE amplification is c = 1.6970, which in this case is smaller than the long-term average of the bounds of the deterministic modes, (1.67632 Æ (1p)2 + 2.00002p(1p) + 2.43952p(1  p) + 1.87162p2)1/2 = 1.7909. In fact, when p varies between 0 to 1, the MJLS bound c never exceeds the long-term average; see Fig. 1. As is obvious from the above two numerical examples, the bullwhip effect magnitude of the stochastic chain is not trivially related to the magnitude of either of the two modes, nor the long-term average. This example verifies that (i) the performance of MJLS cannot be simply predicted from the performance of its individual modes; and (ii) operational uncertainty may degrade system performance, as is shown in the varying policy case.

2.1 2.05

RMSE bounds

2 1.95 1.9 1.85 1.8 1.75

γ long—term average

1.7 1.65 0

p 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 1. RMSE bounds for supply chain with varying lead time.

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4.3. Order-based policies The above results demonstrate that operational randomness degrades performance and also suggest that it may be difficult to find policies robust enough to avoid the bullwhip effect in random operating environments. Therefore it is reasonable to ask whether policies with this nice property exist, and whether they can be certified with the proposed tests. The following example answers these questions in the affirmative. Consider a supply chain operated with two order-based policies (M = 2). The first policy (mode m = 1) has A(P) = 0.5, B(P) = 0, C(P) = 0.2; i.e.,  u1 ðt þ 1Þ ¼ 0:5 u1 ðtÞ þ 0:3 u0 ðtÞ þ 0:2 u0 ðt  1Þ: The corresponding matrices are: 2 3 2 0:5 0 0 0:3 6 7 6 R1 ¼ 4 1 0 0 5; S 1 :¼ 4 0 0 1 0 0

0:2 0 0

0

3

7 0 5: 0

The second policy (mode m = 2) has A(P) = 0.7, B(P) = 0, C(P) = 0.4  0.2P; i.e.,  u1 ðtÞ þ 0:3 u0 ðtÞ þ 0:2 u0 ðt  1Þ þ 0:2u0 ðt  2Þ: u1 ðt þ 1Þ ¼ 0:3 The matrices are: 2 0:3 0 6 R2 ¼ 4 1 0 0

1

3 0 7 0 5;

2

0:3 6 S 2 :¼ 4 0

0

0

0:2 0

3 0:2 7 0 5:

0

0

Both policies can be shown to avoid the bullwhip effect when considered individually.1 If the transition matrix  0:5 0:5 is P ¼ , then Theorem 1 shows that W 6 1. This result establishes that non-trivial policies that 0:5 0:5 avoid the bullwhip effect in random operating environments exist. Thus, the search for practical policies of this type is not a chimera. The examples also show that the fitness of any policy can be easily tested. 5. Discussion: Extension to multi-echelon chains The results in this paper can be extended to multi-echelon chains and networks with similar metric and Markovian dynamics. For example, in a decentralized supply chain (where suppliers do not share any information) with I + 1 suppliers and a customer, the system dynamics at every stage should be similar to (10): ui ðt þ 1Þ ¼ Ri  u1 ðtÞ þ S i  u0 ðtÞ;

8i ¼ 1; 2; . . . ; I:

ð19Þ

We define based on (9) the state vector of the entire chain:  T T T T T UI ðtÞ :¼ uI ðtÞ ; uI1 ðtÞ ; . . . ; ui ðtÞ ; . . . ; u1 ðtÞ ; and the demand vector T

T

U0 ðtÞ :¼ ½u0 ðtÞ ; 0; . . . ; 0 : Assume now that the state space of Markov chain {h(t)} has multiple dimensions that capture the stochasticities at all supplier stages; i.e., as represented in matrix pairs {(RI, SI), (RI1, SI1), . . ., (R1, S1)}. Then it is trivial to show that the system dynamics of the entire chain, with uncertainty, can be represented by UI ðt þ 1Þ ¼ RhðtÞ  UI ðtÞ þ ShðtÞ  U0 ðtÞ; 1

ð20Þ

It is shown in Appendix C of [33] that in deterministic LTI chains the bullwhip effect is avoided if, as occurs with our two policies, the dynamics (8) satisfy: U(P) = a, W(P) = c0 + c1 Æ P +    + cK Æ PK, where a + W(1) = 1, 0 6 a < 1, and ck P 0, "k = 0, . . ., K.

Y. Ouyang, C. Daganzo / European Journal of Operational Research 185 (2008) 340–353

where

2

RhðtÞ

6 6 6 6 :¼ 6 6 6 4

RI;hðtÞ

3

2

7 7 7 7 7; 7 7 S 2;hðtÞ 5 R1;hðtÞ

6 6 6 :¼ 6 6 6 4

S I;hðtÞ RI1;hðtÞ

S I1;hðtÞ .. .

..

.

R2;hðtÞ

ShðtÞ

0 0 .. . 0

0  0  .. . 0 

3 0 07 7 .. 7 .7 7: 7 05

S 1;hðtÞ

0 

0

349

We may define the bullwhip effect metric for the most upstream order sequence fuI ðtÞg as "  P1 #1=2 E u2I ðtÞ t¼0   P1 2  W I :¼ sup : u0 ðtÞ 8f u0 ðtÞg6¼0 t¼0  It is clear by comparing (10) and (20) that all the results for single-echelon chains in previous sections should continue to2hold for multi-echelon chains, if only R, S, and H are replaced by R; S and a diagonal block ma3 H 6 7 0 7, respectively. trix H :¼ 6 .. 4 5 . 0 6. Conclusions and future work This paper presented a control framework to analyze the bullwhip effect in single-echelon supply chains under exogenous Markovian uncertainty. By formulating the supply chains as MJLS, we derived robust analytical conditions that diagnose the bullwhip effect and bound its magnitude. The results can help predict performance in uncertain operating environments. We also showed how the results can be extended to decentralized multi-echelon chains. The Markovian uncertainties in this work are assumed to be exogenous to the suppliers and independent of the suppliers’ (order, inventory) state. They allow us to model variable lead time and supplier behavior correlated with exogenous conditions, such as the economic climate. The tests in this paper are robust in the sense that they hold for any customer demand (the demand does not have to be known), but assume knowledge of the transition probabilities underlying the exogenous conditions of the operational environment. Future research should identify tests and policies that would relax the requirement for this knowledge. Another direct extension would make the Markov chain endogenous and state-dependent. This would enable us to address a broader spectrum of supplier behaviors. Another realistic extension would relax the order-conservation underlying the system dynamics. This would allow for supply chains with random shipment losses or imperfect production yields. Appendix A. Proof for Theorem 1 Before proving Theorem 1, we show the following useful lemmas: Lemma 1. Suppose system (11) is stochastically stable. For any non-zero square-summable fu0 ðtÞg and diagonal matrix D with Tr(D) > 0: ! ! ! 1 1 1 1 X X X X 2 2 T T 2 2 E ½u1 ðtÞ 6 c ½u0 ðtÞ () E u1 ðtÞ Du1 ðtÞ 6 c u0 ðtÞ Du0 ðtÞ : t¼0

t¼0

t¼0

Proof. Note that u1(t) = u0(t) = 0 for all t 6 0. From (9), we have s s K 1 X K X X X    ui ðtÞT Dui ðtÞ ¼ TrðDÞ  d k  u2i ðs  k 0 Þ ; u2i ðtÞ  t¼0

t¼0

k 0 ¼0 k¼k 0 þ1

t¼0

i ¼ 0; 1:

350

Y. Ouyang, C. Daganzo / European Journal of Operational Research 185 (2008) 340–353

For an arbitrary realization of {u0(t)}, s X 

s K 1 X K X   2  X   u1 ðtÞ  c2 u20 ðtÞ  u1 ðtÞT Du1 ðtÞ  c2 u0 ðtÞT Du0 ðtÞ ¼ TrðDÞ  d k  u21 ðs  k 0 Þ  c2 u20 ðs  k 0 Þ :

t¼0

k 0 ¼0 k¼k 0 þ1

t¼0

ð21Þ 1 f u0 ðtÞgt¼0

lims!1 u20 ðs

0

0

is square summable, so  k Þ ¼ 0 for all 0 6 k 6 K  1. When the system The sequence is stochastically stable, Ef u21 ðsÞg ! 0 as  u20 ðsÞ ! 0 for s ! 1. Given that K < 1 and 0 < Tr(D) < 1, we have ( ) K1 X K X  2  0 0 2 2 E dk   u1 ðs  k Þ  c  u0 ðs  k Þ ! 0 when s ! 1: k 0 ¼0 k¼k 0 þ1

Letting s ! 1, the expectation of (21) gives E

1 X

T

u1 ðtÞ D u1 ðtÞ  c

t¼0

2

1 X

(

T

u0 ðtÞ D u0 ðtÞ ¼ TrðDÞ 

t¼0

E

1 X t¼0

! u21 ðtÞ

c

2

1 X

) u20 ðtÞ

:

t¼0

The lemma follows. h Note that the identity matrix is a special case of D. Lemma 1 leads to the following: Lemma 2. Suppose system (11) is stochastically stable and fu0 ðtÞg is square summable. Then W 6 c if and only if there exists a symmetric matrix H with Tr(H) > 0 such that for all {u0(t)} 5 0,

E

1 X

! T

u1 ðtÞ H u1 ðtÞ

6 c2

t¼0

1 X

! T

u0 ðtÞ H u0 ðtÞ :

ð22Þ

t¼0

Proof. Symmetric matrix H can be orthogonally diagonalized; i.e., H = UTDU, where D is diagonal, Tr(D) = Tr(H) > 0, and U is orthogonal. Note that wi(t) :¼ Uui(t), i = 0, 1, satisfies w1 ðt þ 1Þ ¼ R0hðtÞ  w1 ðtÞ þ S 0hðtÞ  w0 ðtÞ; R0hðtÞ

T

S 0hðtÞ

8hðtÞ;

ð23Þ R0hðtÞ ; S 0hðtÞ

T

¼ URhðtÞ U , ¼ US hðtÞ U . Note the similarity between systems (11) and (23), and that where and Rh(t), Sh(t) are different only by a rotation of the coordinates in the linear space (i.e., orthogonal transformation), we thus have " P1 " P1 #1=2 #1=2 T T E E t¼0 w1 ðtÞ D w1 ðtÞ t¼0 u1 ðtÞ D u1 ðtÞ sup ¼ sup : P1 P1 T T 8fw0 ðtÞg6¼0 8fu0 ðtÞg6¼0 t¼0 w0 ðtÞ D w0 ðtÞ t¼0 u0 ðtÞ D u0 ðtÞ Note that wi(t)TD wi(t) = ui(t)TH ui(t), i = 0, 1. Lemma 2 then follows from Lemma 1.

h

1 Lemma 2 also implies that fu1 ðtÞT H u1 ðtÞg1 t¼0 is summable if and only if fu1 ðtÞgt¼0 is square summable. Now we are ready to prove Theorem 1. We use ideas from the proofs for Theorem 5.3 in [37] and for Theorem 2 in [38]. T Define quadratic functions Vðu1 ðtÞ; nÞ :¼ u1 ðtÞ Gn u1 ðtÞ; 8n 2 M. By zero initial conditions, u1(1) = 0 and Vðu1 ð1Þ; hð0ÞÞ ¼ 0.

Ps (a) Note that EfhðtÞgst¼0 t¼1 ½Vðu1 ðt þ 1Þ; hðtÞÞ  Vðu1 ðtÞ; hðt  1ÞÞ ¼ EfhðtÞgst¼0 Vðu1 ðs þ 1Þ; hðsÞÞ P 0. Therefore, s s X X T T u1 ðtÞ H u1 ðtÞ  c2 u0 ðtÞ H u0 ðtÞ EfhðtÞgs1 t¼0

t¼0

6 EfhðtÞgst¼0

t¼0 s X  t¼1

T

T

u1 ðtÞ H u1 ðtÞc2 u0 ðtÞ H u0 ðtÞ þ Vðu1 ðt þ 1Þ; hðtÞÞ  Vðu1 ðtÞ; hðt  1ÞÞ



Y. Ouyang, C. Daganzo / European Journal of Operational Research 185 (2008) 340–353

¼

s X

351

  Ehð0Þ;;hðtÞ u1 ðtÞT H u1 ðtÞ  c2 u0 ðtÞT H u0 ðtÞ þ u1 ðt þ 1ÞT GhðtÞ u1 ðt þ 1Þ  u1 ðtÞT Ghðt1Þ u1 ðtÞ

t¼1

¼

s X

 Ehð0Þ;;hðtÞ





T

T

u1 ðtÞ u0 ðtÞ F hðt1Þ;hðtÞ

t¼1

u1 ðtÞ u0 ðtÞ

 ð24Þ

;

where  F hðt1Þ;hðtÞ :¼

Ghðt1Þ

0

0

c2 H



 

RhðtÞ

S hðtÞ

E

0

T 

GhðtÞ

0

0

H



RhðtÞ

S hðtÞ

E

0

 :

For "t, 0 6 t 6 s, equality (i) below holds by conditional expectation. Equality (ii) below holds because given {h(0), . . ., h(t  1)} and initial conditions, u1(t) and u0(t) are fixed vectors and they do not depend on h(t). Equality (iii) is straightforward from the memoryless property of Markov chain {h(t)}. ( " #) u1 ðtÞ  T T Ehð0Þ;...;hðtÞ u1 ðtÞ u0 ðtÞ F hðt1Þ;hðtÞ u0 ðtÞ ( " " ##) u1 ðtÞ  T T ¼ Ehð0Þ;...;hðt1Þ EhðtÞ u1 ðtÞ u0 ðtÞ F hðt1Þ;hðtÞ ðiÞ u0 ðtÞ ( " #) u1 ðtÞ  T T ¼ Ehð0Þ;...;hðt1Þ u1 ðtÞ u0 ðtÞ ½EhðtÞ F hðt1Þ;hðtÞ  ðiiÞ u0 ðtÞ !" ( #) M X u1 ðtÞ  T T phðt1Þm F hðt1Þ;m ¼ Ehð0Þ;...;hðt1Þ u1 ðtÞ u0 ðtÞ ðiiiÞ u0 ðtÞ m¼1 ðivÞ

P 0:

ð25Þ

P  M When hðt  1Þ ¼ n 2 M, we find m¼1 p hðt1Þm F hðt1Þ;m equals the left-hand side of (16), and thus is positive semi-definite. Then inequality (iv) in (25) holds and (24) leads to: ! ! s s X X T T 2 u1 ðtÞ H u1 ðtÞ 6 c u0 ðtÞ H u0 ðtÞ : EfhðtÞgs1 t¼0

t¼0

t¼0

When s ! 1, Lemma 1 holds. By Definition 2, the bullwhip effect is avoided when c = 1. This completes the proof for (a). (b) The proof is very similar to that for (a). Note that EfhðtÞgsþ1 t¼0

s X

½Vðu1 ðt þ 1Þ; hðt þ 1ÞÞ  Vðu1 ðtÞ; hðtÞÞ ¼ EfhðtÞgsþ1 Vðu1 ðs þ 1Þ; hðs þ 1ÞÞ P 0: t¼0

t¼1

We have, similar to (24), the following: EfhðtÞgs1

s X

t¼0

u1 ðtÞT H u1 ðtÞ  c2

t¼0

t¼0

¼

t¼1

u0 ðtÞT H u0 ðtÞ

t¼0

6 EfhðtÞgsþ1 s X

s X

s X 

u1 ðtÞT H u1 ðtÞ  c2 u0 ðtÞT H u0 ðtÞ þ Vðu1 ðt þ 1Þ; hðt þ 1ÞÞ  Vðu1 ðtÞ; hðtÞÞ

t¼1

Ehð0Þ;...;hðtÞ

(

" #) h i u1 ðtÞ  T T 0 u1 ðtÞ u0 ðtÞ Ehðtþ1Þ FhðtÞ;hðtþ1Þ ; u0 ðtÞ



352

Y. Ouyang, C. Daganzo / European Journal of Operational Research 185 (2008) 340–353

where F 0hðtÞ;hðtþ1Þ :¼



GhðtÞ

0

0

c2 H



 

RhðtÞ

S hðtÞ

E

0

T 

Ghðtþ1Þ

0

0

H



RhðtÞ

S hðtÞ

E

0

 ;

and Ehðtþ1Þ F 0hðtÞ;hðtþ1Þ equals the left-hand side of (17), which is positive semi-definite. The rest is similar to the proof for (a). Appendix B. Proof for Theorem 2 For H > 0 there exist matrices H1/2 and H1/2 such that H1/2H1/2 = H and H1/2H1/2 = H1/2H1/2 = E. Let u01 ðtÞ :¼ H 1=2 u1 ðtÞ and u00 ðtÞ :¼ H 1=2 u0 ðtÞ. System (11) can be written as   0   0  u1 ðt þ 1Þ RhðtÞ S 0hðtÞ u01 ðtÞ ¼ ; ð26Þ u00 ðtÞ u01 ðtÞ E 0 where R0hðtÞ ¼ H 1=2 RhðtÞ H 1=2 ; S 0hðtÞ ¼ H 1=2 S hðtÞ H 1=2 ; 8hðtÞ, and ! ! 1 1 X X T 0 T 0 EfhðtÞg1 u1 ðtÞ u1 ðtÞ ¼EfhðtÞg1 u1 ðtÞ H u1 ðtÞ ; t¼0 t¼0 t¼0 1 X t¼0

t¼0

u00 ðtÞT u00 ðtÞ ¼

1 X

u0 ðtÞT H u0 ðtÞ:

t¼0

Eq. (26) is in the form of system (1) in [38] and system (5.3) in [37]. Obviously, weakly controllability (i) continues to hold for (26). These reference show that the system is stochastically stable with ! 1 1 X X T T 0 0 2 EfhðtÞg1 u ðtÞ u ðtÞ < c u00 ðtÞ u00 ðtÞ ð27Þ 1 1 t¼0 t¼0

t¼0

if and only if there exist matrices G0n > 0; n 2 M that satisfy   0  X  0 T  0  M Gn 0 Rm S 0m Gm 0 R0m S 0m pnm  > 0; 0 c2 E 0 E E 0 E 0 m¼1

8n 2 M;

ð28Þ

or

#   0 T " 0  0 Rn S 0n Gn 0 R0n S 0n  > 0; 8n 2 M; ð29Þ c2 E E 0 E 0 0 E  1=2  PM 0 H where G0n ¼ m¼1 pnm G0m . Multiply (28) and (29) on the left and right by , they become (16) and 0 H 1=2 P M (17) with Gn :¼ H 1=2 G0n H 1=2 P 0; 8n and Gn :¼ m¼1 pnm Gm ¼ H 1=2 G0n H 1=2 . This, in light of Lemma 1 and (27), completes the proof. h 

G0n 0

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