Understanding supply chain dynamics: A chaos perspective

European Journal of Operational Research 184 (2008) 1163–1178 www.elsevier.com/locate/ejor

O.R. Applications

Understanding supply chain dynamics: A chaos perspective H. Brian Hwarng *, Na Xie Department of Decision Sciences, National University of Singapore, BIZ1, 1 Business Link, Singapore 117592, Singapore Received 1 June 2006; accepted 5 December 2006 Available online 22 December 2006

Abstract Variability in orders or inventories in supply chain systems is generally thought to be caused by exogenous random factors such as uncertainties in customer demand or lead time. Studies have shown, however, that orders or inventories may exhibit significant variability even if customer demand and lead time are deterministic. In this paper, we investigate how this class of variability, chaos, may occur in a multi-level supply chain and offer insights into how to manage relevant supply chain factors to eliminate or reduce system chaos. The supply chain is characterized by the classical beer distribution model with some modifications. We observe the supply chain dynamics under the influence of various factors: demand pattern, ordering policy, demand-information sharing, and lead time. Through proper decision-region formation, the effect of various factors on system chaos is investigated using a factorial design. The degree of system chaos is quantified using the Lyapunov exponent across all levels of the supply chain. This study shows that, to reduce the degree of chaos in the supply chain system, the adjustment parameters for both inventory and supply line discrepancies should be more comparable in magnitude. Counter-intuitively, in certain decision regions, sharing demand information can do more harm than good. Similar to the bullwhip effect observed previously in demand, we discover the phenomenon of ‘‘chaos-amplification’’ in inventory across supply chain levels.  2007 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Chaos theory; Beer distribution game; System dynamics

1. Introduction A supply chain is a complex system which involves multiple entities encompassing activities of moving goods and adding value from the raw material stage to the final delivery stage. Along the chain, there exist various types of uncertainties, e.g., demand uncertainty, production uncertainty, * Corresponding author. Tel.: +65 6516 6449; fax: +65 6779 2621. E-mail address: [email protected] (H.B. Hwarng).

and delivery uncertainty. Making decisions as to how much and when to replenish, often involves a feedback process triggering interaction between system entities, which may result in system nonlinearity. A time delay is observed when there is a lag between when a decision is made and when its effect is felt, which often further complicates the interaction between entities. Feedback, interaction, and time delay are inherent to many processes in a supply chain, making it a dynamic system. For instance, the two widely used inventory replenishment policies, i.e., the continuous-review

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policy and the periodic-review policy, involve a negative feedback mechanism. For the continuousreview policy, the inventory is reviewed continuously and when the inventory drops to the reorder point, Q units are ordered to restore the inventory level. The replenishment process creates a negative feedback loop. A similar feedback process is observed in the periodic-review policy. Feedback demands actions which must involve interactions between entities in a system or between the effects of these actions or decisions made. This is seen in a typical automobile supply chain such as Toyota’s wherein a dealer’s (with a local view) pre-set inventory level and the automaker’s (with a global view) allocation policy will affect the velocity of parts going to the dealers. Besides feedback and interaction, time delays between cause and effect often further complicate the interaction between system entities. For example, ideally, the safety stock levels at a centralized hub such as Dell’s should be kept as low as possible. However, time delays in the form of long and uncertain lead time coupled with build-toorder practice and decision making behaviors adversely contribute to the variability of the safety stock. Indeed, feedback, interaction, and time delay induce variability, instability, and complex behaviors that make supply chain management even more challenging. The above-mentioned dynamics and complex behaviors in a supply chain can be appropriately modeled by adopting the system dynamics approach, particularly from a chaos perspective. Chaos is disorderly looking long-term evolution occurring in a deterministic nonlinear system (Williams, 1997). Chaos theory is concerned with chaotic behavior/chaos in nonlinear dynamical systems from a number of aspects, e.g., the principles and mathematical operations underlying chaos, chaos characteristics, and the methods to identify chaos. The origin of chaos theory dates back to Lorenz’s (1963) study in weather forecasting systems. The work of Feigenbaum (1978) and Mandelbrot (1982) has contributed to greater interest in studying and applying chaos theory. A system of chaos is often characterized by a number of distinct features (Williams, 1997; Wilding, 1998), such as: (1) non-randomness and nonlinearity; (2) apparent disorder: the motion of the variables looks disorganized and irregular; (3) strange attractor: order, structure, or pattern can be found in phase space; (4) bounds: the ranges of the variables have finite bounds, hence, the attractor

is bounded; and (5) sensitivity to initial conditions: a small change in initial conditions can have a large effect on the evolution of the system. To investigate the existence and formation of chaotic behaviors, it is necessary to constrain parameters to deterministic. Mosekilde and Larsen (1988), Thomsen et al. (1992), Sosnovtseva and Mosekilde (1997), and Larsen et al. (1999) adopted such an approach to studying various nonlinear chaotic behaviors in a distribution system. Their foci, nevertheless, remain on showing the existence and categorization of chaotic behaviors in the system. The cause-and-effect and dynamic nature of various system factors contributing to the chaotic behavior remains an interesting topic for further research. In this paper, we intend to investigate how various supply chain factors contribute to the complex dynamics and chaotic behaviors. We are interested in a general class of multi-level supply chains that can be represented by the well-known beer distribution model (Jarmain, 1963). Various supply chain factors are considered, such as demand pattern, ordering policy, demand-information sharing, and lead time, with different options or levels. A simulation model is developed to observe system dynamics, particularly the inventory across all levels of the supply chain. Using the Lyapunov exponent (LE), we quantify the degree of system chaos in terms of inventory across all supply chain levels. The objectives of this paper are: (1) to understand, from a chaos perspective, how the different levels of a multi-level supply chain behave under the influence of factors such as demand pattern, lead time, demand-information sharing, and ordering policy; (2) to investigate how these factors act or interact to affect the system dynamics which in many cases lead to chaos; and (3) to offer some insights into more effective management of supply chains. 2. Related past studies It is well studied and recognized that demand pattern, ordering policy, lead time, and information sharing have direct impact on the performance of supply chains. Lead time reduction is found to be very beneficial and can reduce inventory and demand variability, and improve customer service and responsiveness (see for example, Chen et al., 2000; Karmarkar, 1987; Lee et al., 1997a,b; Oke and Szwejczewski, 2005; Ryu and Lee, 2003; Steckel

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et al., 2004; Treville et al., 2004). The Forrester effect or the bullwhip effect is an interesting phenomenon observed in supply chains manifested in demand amplification (see for example, Forrester, 1961; Sterman, 1989; Lee et al., 1997a,b) or inventory amplification (see for example, Forrester, 1961; Mosekilde and Larsen, 1988; Sterman, 1989; Larsen et al., 1999). The Forrester effect is usually caused or aggravated by long lead time, a lack of information visibility across levels of a supply chain, or a mixture of feedback, time delays and interactions as discussed earlier. Studies have shown that the benefits of information sharing may contribute to inventory reduction, more reliable delivery, higher order fulfillment rate, cost reduction, and reduction of demand amplification (see for example, Cachon and Fisher, 2000; Chen et al., 2000; D’Amours et al., 1999; Lee et al., 1997a,b, 2000; Simchi-Levi and Zhao, 2003; Steckel et al., 2004; Zhao et al., 2002). In a multi-level supply chain, different supply chain levels have different visibility or exposure to customer demand, and the amplification of demand or inventory also grows as it moves upstream the supply chain. These factors are bound to complicate the chaotic behaviors in a complex supply chain system. There has been considerable interest in applying chaos theory to finance, economics and management studies. Studies in the domains of inventory management and supply chain management, however, are still limited. By applying a widely known re-order and stock control algorithm, Wilding (1998) observed chaos in supply chain systems and discussed the implications of chaos theory for supply chain management, e.g., sensitivity to initial conditions, islands of stability, patterns, and invalidation of the reductionist view. Larsen et al. (1999) found that unstable behaviors, such as chaos/hyperchaos, generate higher cost than stable/periodic behavior. A majority of the literature focuses on showing the existence and categorization of chaotic/hyperchaotic behaviors exhibited in a supply

chain system represented by the classical beer distribution model (see for example, Mosekilde and Larsen, 1988; Thomsen et al., 1992; Sosnovtseva and Mosekilde, 1997; Larsen et al., 1999). These studies usually adopt the anchoring and adjustment heuristic for the ordering decision that is originally proposed and tested by Sterman (1989). In short, the above studies highlight the negative effects of chaotic behaviors on effective management of supply chain systems. Since none of the above studies investigate how the dynamics of a supply chain system result in chaos, the current study is designed to fill this research gap. 3. The model 3.1. The beer distribution model The supply chain of interest is characterized by the beer distribution model (Jarmain, 1963) which includes four levels: brewery or factory, distributor, wholesaler, and retailer, as shown in Fig. 1. The beer distribution model represents a multi-level supply chain and is more realistic than well explored two-level supply chains. The purpose of such a cascaded distribution system is to reach a widespread market. In this system, orders propagate from customers to the factory. A retailer estimates customers’ demands and places an order with a regional wholesaler; the wholesaler then decides how many units to order from a distributor; next the distributor places an order with the factory and the factory will make a production decision based on the distributor’s order and other related information. Conversely, products flow from the factory to the retailer. The factory ships the goods to the distributor if it has sufficient inventory; the distributor receives the goods and transports them to the wholesaler; the wholesaler gets the goods and allocates them to the retailer; and the final customers buy the goods from the retailer. Inventory is held at each supply chain level. For ease of analysis

Distributor

Wholesaler

Products

Fig. 1. The beer distribution model.

Retailer

Customers

Orders

Brewery/ Factory

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and discussion, the model considers only one type of product. The production and shipment of goods involve time delays, and the same applies to the transmission of orders. The production time is three time periods, and the production capacity is assumed to be unlimited. Both the transmission delay of orders and shipment delay of the product between two successive levels are one time period. Customer demand is exogenous and like a step function. The demand is four units per time period in the first four time periods and is increased to eight per period from the fifth period till the end of the experiment or simulation. The model is governed by the following rules: (1) orders must be filled if there is sufficient inventory; (2) unfilled orders are kept in backlog and shall be filled when the inventory is sufficient; and (3) placed orders cannot be cancelled and shipments made cannot be returned. The only decision variable for all supply chain levels is the number of units to be ordered in each period. This decision is made based on information, such as the desired and actual inventory levels, backlog, expected orders, and incoming shipments, locally available to the decision makers. Each supply chain level manages the inventory to minimize holding costs while attempting to avoid out of stock situations. The inventory holding cost is half of the stockout cost, and there is no sharing of the demand information in the system. Each sector does not know the state of other sectors, and decision makers do not know how the time delays and system structure affect the dynamics. Building upon this classical beer distribution model, we will incorporate additional supply chain factors to reflect more comprehensive settings. 3.2. The ordering heuristic The ordering heuristic used in this study is an anchoring and adjustment heuristic for stock management which applies a feedback mechanism (Sterman, 1989). To facilitate the model description, the following notations are introduced: b Lt Lt St S* ASt SLt

the expected demand at time t, the actual demand at time t, the actual stock level at time t, the desired stock level, the adjustment for the stock level at time t, the actual supply line (orders placed but not yet received) at time t,

SL* ASLt h aS

aSL

Ot

the desired supply line, the adjustment for the supply line at time t, a constant which determines how fast expectations are updated, 0 6 h 6 1, the rate at which the discrepancy between actual and desired stock levels is eliminated, 0 6 aS 6 1, the rate at which the discrepancy between actual and desired supply line is eliminated, 0 6 aSL 6 1, the order quantity at time t.

Assuming that the decision makers have adaptive expectations, the expected demand at time t can be defined as follows: b L t ¼ hLt1 þ ð1  hÞ b L t1 ;

0 6 h 6 1:

ð1Þ

To regulate the stock and supply line, a negative feedback mechanism is used. The adjustment is linear in the discrepancy between the desired stock S* and the actual stock St and in the discrepancy between the desired supply line SL* and the actual supply line SLt. That is ASt ¼ as ðS   S t Þ; ASLt ¼ aSL ðSL  SLt Þ:

ð2Þ ð3Þ

Usually aSL 6 aS as it is logical to pay more attention to the inventory than the supply line. Therefore, the generic decision rule for each supply chain level for the order quantity at time t is defined below (Sterman, 1989):   Ot ¼ max 0; b L t þ ASt þ ASLt : ð4Þ For ease of modeling and testing, Sterman (1989) combined the desired stock S* and the desired supply line SL* into a new variable S 0 , i.e., S 0 = S* + bSL* where b = aSL/aS. Others modeled S 0 as a constant (Thomsen et al., 1992). In this study, since aS and aSL are varied for us to study the chaotic behavior, b is also varied. In fact, it is more revealing to determine the desired stock S* and the desired supply line SL* separately rather than combining them as one variable. Similar modification is found in Larsen et al. (1999). Since both S* and SL* are proportional to the expected demand as well as the lead time, we define the desired stock S* and the desired supply line SL* as the lead time multiplied by the expected demand. We adopt the above ordering heuristic for the following reasons: (1) the rule explains the subjects’ behavior well (Sterman, 1989); (2) the negative

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feedback mechanism applied in this heuristic is widely adopted in actual ordering or replenishment decisions, e.g., the continuous review policy and the periodic review policy; and (3) with the changes of the two parameters, aS and aSL, the heuristic can represent many possible ordering decisions. A wide variety of ordering decisions can thus reflect many realistic situations and assist us in drawing more valid conclusions with regard to the effect of the supply chain factors on the system behavior in the supply chain. 3.3. Characterization of chaos Generally there are two ways, namely, graphical methods and quantifiers, to identify chaos or show whether a system is stable, periodic, quasi-periodic, or chaotic. Graphs and plots are visually efficient in showing trends and patterns. Various types of plots such as time series, phase plots, phase space plots, return maps and power spectrum (Hilborn, 1994; Sprott and Rowlands, 1995) are used to study system behavior; however, graphical methods are limited in their accuracy. A more accurate alternative is to calculate some quantifiers. Widely used quantifiers include the Lyapunov exponent (LE), entropy, fractal dimension, capacity dimension, and correlation dimension (Hilborn, 1994; Sprott and Rowlands, 1995). We adopt the LE as the principal quantifier as it is a proven measure for determining and classifying nonlinear system behavior (Wolf et al., 1985). The LE measures the rate at which nearby trajectories in phase space diverge. Interpreted differently, it measures the sensitivity to initial conditions. Though LEs can be defined from different perspectives, the LE adopted here is defined in the manner most relevant to spectral calculations (Wolf et al., 1985). For a continuous dynamical system in an n-dimensional phase space, an infinitesimal n-sphere of initial conditions will become an n-ellipsoid in the long term. The ith LE is defined as: ki ¼ lim

t!1

1 p ðtÞ log2 i ; t pi ð0Þ

ð5Þ

where pi(t) is the length of the ellipsoidal principal axis at time t and the ki are ordered from largest to smallest (Wolf et al., 1985). If at least one LE or the largest LE is positive, the system is chaotic; otherwise the system is stable, periodic or quasiperiodic. The calculation of the LE is a timeconsuming and difficult task, especially when the

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examined systems are complex. Wolf et al. (1985) first proposed algorithms to estimate non-negative LEs from an experimental time series. Since then the algorithms have been widely used. Here we use Chaos Data Analyzer (Sprott and Rowlands, 1995) which implements the algorithms to calculate the largest LE. 4. General description of supply chain factors Five major supply chain factors, namely, demand pattern, ordering policy, order-information sharing, lead time, and supply chain level, have direct impact on the supply chain dynamics and are considered in the model. 4.1. Demand pattern The demand pattern experienced at the retailerend drives the order information flow upward. Two types of demand patterns are considered, i.e., a step function and a broad-pulse function. A step demand function, for example, may occur as a result of price reductions. When the price of a product is significantly reduced for an extended period of time, the demand of this product may jump to a higher level also for an extended period of time. On the other hand, if the stimuli for demand increase are rather temporary, thus the demand increase lasts only for a short period of time; the demand function will appear like a broad pulse. For instance, for some promoted items, the demand is increased to a higher level and sustains that level only for a short period. After the promotion ends, the demand goes back to the normal level. For products which have some high-demand periods such as Christmas, the demand pattern is also like a broad pulse. For a step function, the demand stays at an original level up to a certain instant and thereafter is increased to a shifted level. Contrastingly in a broad pulse, the demand stays at an original level for some time, then is increased to a shifted level for a short period, and finally returns to the original level (Fig. 2). In this study, without losing generality, for both demand patterns the original demand level is four units, whereas the shifted level is eight units. The demand is four units per time period in the first four periods for both demand patterns. In addition, the demand is increased to eight units per period from the fifth period till the end of the simulation for the step-function pattern, while it stays at eight per period for 50 periods and returns to four units

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Demand

Demand

Time

Time

Fig. 2. Two demand patterns. (a) Step function. (b) Broad pulse.

per period thereafter for the broad-pulse pattern. As shown in Fig. 2, one difference between the two demand patterns is the duration of the demand staying at the shifted level. 4.2. Ordering policy The ordering policy determines how the orders will be passed upward along the supply chain levels. The baseline policy adopted here is one single ordering heuristic for all supply chain levels, denoted as Ordering Policy 1 (OP1), i.e., the same ordering heuristic (Sterman, 1989) as presented in Section 3.2 is used for all supply chain levels. Additionally, we consider the second ordering policy, denoted as Ordering Policy 2 (OP2), which allows different ordering heuristics for different supply chain levels. With reference to the stationary beer game model (Chen and Samroengraja, 2000), the minimum total cost for the supply chain can be achieved if each level applies an installation, base-stock policy with a constant target level. In our model, with the parameters being deterministic, such a policy would be similar to ordering the exact quantity placed by the customer and is termed the pass-order heuristic. In OP2, the pass-order heuristic is adopted by the retailer and the ordering heuristic presented in Section 3.2 is adopted by the other three supply chain levels, namely, wholesaler, distributor, and factory. 4.3. Demand-information sharing In the classical beer distribution model, no information sharing is considered. With the advance of information technology, sharing information becomes easier and less costly. In global supply chain management, the importance and value of information sharing cannot be overstated. To reflect

this reality and investigate the efficiency of sharing demand information, we consider two situations in the model, i.e., without information sharing and with information sharing. ‘With information sharing’ allows the retailer to share the customer demand information, or quantity, with the other supply chain levels, while ‘without information sharing’ prohibits the demand information to be transmitted to other levels. 4.4. Lead time We consider two lead time options, namely, short lead time, which is used in the classical beer distribution model, and long lead time. The lead time between two successive levels includes the transmission delay of orders and the shipment delay of the product. The transmission delay of orders between two successive levels is one time period. The two lead time options, however, differ in the length of shipment delay, i.e., one time period for the short lead time option and two periods for the long lead time option. Hence, the lead time between two successive levels is two-thirds of the production time for the short lead time option, whereas it equals the production time for the long lead time option. 4.5. Supply chain level Different supply chain levels have different visibility and exposure to customer demand. To investigate the system behavior at different supply chain levels, these levels are included as a factor in the experimental design. There are four levels, i.e., brewery or factory, distributor, wholesaler, and retailer, in the beer distribution model. To facilitate easy reference to various supply chain factors and scenarios, Table 1 summarizes

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Table 1 Experimental design and scenario coding Demand pattern:

Broad pulse

Ordering policy:

OP1

Information sharing:

With

Without

With

Without

With

Without

With

Without

R W D F

1 2 3 4

9 10 11 12

17 18 19 20

25 26 27 28

33 34 35 36

41 42 43 44

49 50 51 52

57 58 59 60

R W D F

5 6 7 8

13 14 15 16

21 22 23 24

29 30 31 32

37 38 39 40

45 46 47 48

53 54 55 56

61 62 63 64

Long lead time Supply chain level

Short lead time Supply chain level

Step function OP2

OP1

OP2

Notation: R, retailer; W, wholesaler; D, distributor; F, factory; OP1, ordering policy 1; OP2, ordering policy 2.

the factors and scenario coding. There are two levels for the first four factors and four supply chain levels, thus there are 64 (2 · 2 · 2 · 2 · 4 = 64) scenarios in total. Numbers 1–64 in Table 1 represent the 64 scenarios. 5. Simulation design A simulation model was developed to investigate the system behavior in a supply chain characterized by the beer distribution model. The supply chain system is simulated under the 64 scenarios described in Section 4. We adopt the same initial conditions as those used by Sterman (1989). The initial values of variables and the values or ranges of parameters are presented in Table 2. The model was built using well-known system dynamics simulation software, Vensim DSS (Ventana Simulation Incorporation, 2003). In the simulation, h is set to 0.25 for all levels. The two decision parameters, aS and aSL, are varied to study different system behaviors. For simplicity, the same parameter set is used at different supply

chain levels. For each parameter, 50 different values with an increment of 0.02 from 0.00 to 1.00 are used to simulate various ordering decisions (see Table 2). Since aSL 6 aS, there are 1275 parameter sets or ordering decisions in total for each scenario. The run length for each simulation scenario is 2000 time periods. We concentrate on studying the system behavior exhibited in the effective inventory at different levels in the supply chain system. Effective inventory is defined as the inventory level after fulfilling the backlog. In each scenario, we collect the effective inventory data for each parameter set and then calculate the largest LE from the effective inventory data. A total of 1275 LEs are calculated for each scenario. To illustrate how minor changes in the decision parameters can affect the dynamics of the system, we present the following scenarios. Without losing generality, we use the effective inventories at the wholesaler and distributor to draw time series plots and phase plots to show two types of behavior under the same condition, i.e., the step function

Table 2 Initial values of variables and values or ranges of parameters Variable

Initial value (unit)

Parameter

Value or range

Inventory Order Shipment Transmission delay Shipment delay

12 4 4 4 4 for the short lead time option 8 for the long lead time option

Updating parameter for expectations (h) Adjustment parameter for inventory discrepancies (aS) Adjustment parameter for supply line discrepancies (aSL)

0.25 0.00–1.00, an increment of 0.02 0.00–1.00, an increment of 0.02

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900 650

Inventory

Inventory

500 300 100 -100

400 150 -100

-300 0

200 400

-350

600 800 1000 1200 1400 1600 1800 2000

0

200 400

600 800 1000 1200 1400 1600 1800 2000

Time WEI

Time

DEI

50

WEI

DEI

100 DEI

150

90 80 70

40

WE I

WE I

60 30

50 40 30

20

20 10

10

0 0

20

40

60

80 100 DEI

120

140

160

180

-50

0

50

200

250

Fig. 3. Two types of system behaviors with a minor change in decision parameters. (a) Time series plot, aS = 0.74, aSL = 0.28. (b) Phase plot, aS = 0.74, aSL = 0.28. (c) Time series plot, aS = 0.74, aSL = 0.24. (d) Phase plot, aS = 0.74, aSL = 0.24.

demand pattern, OP1, no information sharing, and the long lead time option. In the plots (see Fig. 3), WEI and DEI represent the effective inventory at the wholesaler and the distributor, respectively. As the system began with a ‘cold state’, both WEI and DEI remained at the initial level of 12 units for six periods then oscillated violently until around the 100th period; it then gradually stabilized. To exhibit the system behaviors more clearly, we exclude the inventories of the transient period (the first 200 time units) from the phase plots. Fig. 3a and b shows the system behavior when aS = 0.74 and aSL = 0.28. In the time series plot, both WEI and DEI fluctuate in the first 100 hours and become stable thereafter. In the phase plot, the pattern is quite clear with some systematic dotted curves and then converges to a trended line. More calculations show that the largest LEs are negative and thus the system is stable. If we decrease aSL slightly, i.e., by 0.04, the system becomes chaotic as shown in Fig. 3c and d. The inventories exhibit significant variability or random-like fluctuations although customer demand remains deterministic. In addition, the attractor becomes more complex though some patterns are noticeable. We are interested in further investigating how these dynam-

ics evolve as various factors and parameters are changed. 6. Decision region formation To study the effect of the supply chain factors on the system behavior, strictly speaking, we need to analyze the largest LEs calculated from the effective inventories for 1275 parameter sets in 64 scenarios. However, with so many parameter sets, the analysis becomes intractable if we analyze the LE for each parameter set in 64 scenarios separately. On the other hand, we cannot simply use the average LEs of all parameter sets in each scenario since the system behavior can be quite different in different parameter-set regions. To study the effect of the supply chain factors efficiently and accurately, we may cluster the parameter sets into different regions following two criteria: (1) in each region, the system behaviors or the LEs are homogeneous or similar; and (2) the number of regions should be reasonably manageable. The average LEs in each region in 64 scenarios can then be analyzed. The typical distribution of the LEs over the (aS, aSL) plane in each scenario can be seen in a contour plot as shown in Fig. 4 (for Scenario 1). In

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Fig. 4. The distribution of the LEs over the (aS, aSL) plane (for Scenario 1).

1 0. 9 0. 8 0. 7 11

AlphaSL

total, 64 contour plots are produced. The general pattern is that bands are oriented along the diagonal with the slope of the upper bands larger than that of the lower bands. In addition, the LE values tend to decrease in the upper bands. Based on the contour plot, one reasonable region formation is to break the (aS, aSL) plane into different bands or regions along the diagonal. With reference to the two criteria, we evaluate how to form regions using different clustering methods, including K-means. Eventually, the coefficient of variation (CV), which is defined as standard deviation divided by mean, is used as a measure of homogeneity of the LEs in each region. Given a certain formation, the CV for each region is derived. The smaller the CV, the more homogeneous the LEs are in that region. We repeat the calculation for all 64 scenarios. For a suitable formation, in each region the CV values in most of the scenarios should be below the pre-set upper limit. There is a tradeoff between homogeneity of the LEs in each region and the number of regions. With the consideration of the two criteria and the available data, we find that 1.00 is a good upper limit for the CV. Considering all scenarios, a suitable formation is shown in Fig. 5. In this formation, the (aS, aSL) plane is divided into 11 regions. In each region, the CV values in more than 70% of the 64 scenarios are less than 1.00; for all regions at least 80% of the CV values are less than 1.00. As shown in Fig. 5, the

0. 6 8

0. 5 0. 4

7 6

0. 3

10

5 4 3

0. 2 0. 1

2

9

1 0

0

0. 1

0. 2

0. 3

0. 4

0. 5 AlphaS

0. 6

0. 7

0. 8

0. 9

1

Fig. 5. Region formation.

regions parallel the diagonal approximately and the slope of the upper regions increases slightly. The distribution of regions is consistent with the typical distribution of the LEs over the (aS, aSL) plane (Fig. 4). The number of parameter sets in each region is listed in Table 3. 7. Results and discussion A mixed full-factorial experimental design is described as follows. Of the five factors, demand pattern, ordering policy, information sharing, and

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Table 3 Number of parameter sets in each region in Fig. 5 Regions

1

2

3

4

5

6

7

8

9

10

11

Number of data

35

102

53

52

66

63

203

246

75

130

250

lead time each has two levels, whereas supply chain has four levels. The response variable is the average LE which is calculated for each region in 64 scenarios (tabulated in Table 4). We primarily test the main effects and two-factor interactions, since there is only one average LE in each region in 64 scenarios (equivalent to a single replicate of the experimental design). Based on the ‘sparsity of effects principle’, three-factor and other higher-order interactions are negligible and thus can be combined as an estimate of error (Montgomery, 2001). The analysis of variance shows that the main effects and twofactor interactions explain more than 80% of the variance. In addition, normal probability plots of residuals and plots of residuals against factor levels or fitted values, show that the normality assumption and independence assumption made in the analysis of variance are satisfied. The mixed full-factorial experimental design is valid. With this design, the main effects and two-factor interactions in 11 regions are calculated using Minitab software (Minitab Incorporation, 2005) and summarized in Table 5. We use a significance level of a = 0.05. A p-value < a means that the corresponding main effect or interaction effect is significant. As shown in Table 4, the system behavior is stable in most of the scenarios in Regions 7–11; thus, we are not concerned with the magnitude of the LE in these decision regions. Instead, the following analysis and discussion will mainly focus on the decision regions which typically exhibit chaotic behaviors, i.e., Regions 1–6. 7.1. Effect of decision region The average LEs listed in Table 4 are graphically presented in Fig. 6. In most of the scenarios, the average LE becomes smaller as it goes up from lower regions to upper regions. In addition, most of the average LEs are positive in Regions 1–6 (Fig. 6a and c), whereas most of them are negative in Regions 7–11 (Fig. 6b and d). The results show that the system is more chaotic in lower regions, e.g., Regions 1–6. In Regions 1–6, aS varies from 0.40 to 1.00, whereas aSL ranges from 0.00 to 0.52 (see Fig. 5).

The difference between aS and aSL ranges from 0.40 to 1.00 in these regions. Consider a case such as aS = 0.80 and aSL = 0.02 which falls in Region 2; the disparity in magnitude between the two adjustment parameters could cause the system to be unstable or chaotic. This is generally due to unusually large or small values of ASt, Ot and ASLt+1 occurring in a repeating fashion. To avoid unstable and chaotic behavior, decision makers should select the parameter sets in upper regions where the adjustment parameters for inventory discrepancies and supply line discrepancies, i.e., aS and aSL, are more comparable in their magnitude. 7.2. Effect of demand pattern In general, the system becomes more chaotic when the customer demand follows a step function. The phenomenon is more obvious in the two lowest regions, i.e., Regions 1 and 2, where the effect of demand pattern is significant with p-value = 0.001 and 0.006, respectively (Table 5) and there are only 2–3 scenarios in which the system is more stable for a step function demand. In these two regions, aS ranges from 0.65 to 1.00 and aSL varies from 0.00 to 0.24. Such a disparity in parameter values could cause the similar problem highlighted in Section 7.1. The problem could be intensified due to the sustaining shifted demand level (Fig. 2). In Regions 6–11, the effect of demand pattern is significant and the system is more stable for a broadpulse demand pattern in most of the scenarios. In Regions 3–5, the difference between the average LEs for these two demand patterns is not significant (Table 5) although the system is generally more chaotic when the customer demand is a step function. The above analysis shows that, generally, the system is more chaotic when the customer demand follows a step function. In other words, the degree of system chaos increases when the impact of a demand shift is sustained for a longer period. Under the circumstances, decision makers should manage other factors more carefully, e.g., diligently compress the lead time, so as to avoid the chaotic behavior as exhibited in the inventories.

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Table 4 Average Lyapunov exponents in Scenarios 1–64 Scenario

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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

Region 1

2

3

4

5

6

7

8

9

10

11

0.0864 0.1095 0.1936 0.2840 0.0883 0.0804 0.1685 0.2108 0.0606 0.0923 0.1441 0.2480 0.0412 0.0503 0.0362 0.2338 0.0088 0.1104 0.1491 0.3597 0.0178 0.1318 0.1000 0.2475 0.0150 0.0781 0.0936 0.2482 0.0027 0.0714 0.0693 0.1789 0.1166 0.1395 0.2548 0.3434 0.1303 0.1102 0.2041 0.2899 0.0820 0.1024 0.2107 0.0452 0.0781 0.0657 0.1226 0.0154 0.0888 0.1295 0.1788 0.4071 0.1341 0.1692 0.1358 0.3175

0.1026 0.1056 0.2129 0.3327 0.0417 0.0494 0.0885 0.1439 0.0626 0.0648 0.1566 0.0650 0.0467 0.0447 0.1263 0.0541 0.0035 0.0964 0.0912 0.2651 0.0068 0.0515 0.0638 0.1097 0.0151 0.0750 0.0663 0.2274 0.0086 0.0604 0.0603 0.1729 0.1212 0.1451 0.2677 0.3674 0.0591 0.0666 0.1046 0.1481 0.0686 0.0818 0.2010 0.1832 0.0661 0.0627 0.1553 0.1229 0.0386 0.1089 0.1149 0.3059 0.1078 0.0743 0.0985 0.1548

0.0721 0.0821 0.1329 0.2877 0.0035 0.0144 0.0398 0.0792 0.0328 0.0260 0.1789 0.1204 0.0120 0.0225 0.0601 0.0766 0.0049 0.0381 0.0360 0.0831 0.0054 0.0075 0.0186 0.0408 0.0150 0.0376 0.0295 0.2310 0.0068 0.0158 0.0348 0.0661 0.0847 0.1122 0.1735 0.3154 0.0082 0.0239 0.0439 0.0655 0.0314 0.0530 0.2228 0.2198 0.0268 0.0482 0.0728 0.0921 0.0350 0.0291 0.0323 0.0826 0.0998 0.0115 0.0280 0.0532

0.0390 0.0453 0.0670 0.2323 0.0184 0.0102 0.0302 0.0636 0.0297 0.0274 0.1816 0.1915 0.0116 0.0004 0.0378 0.0674 0.0088 0.0241 0.0250 0.0443 0.0114 0.0169 0.0059 0.0236 0.0143 0.0260 0.0251 0.1882 0.0108 0.0120 0.0064 0.0393 0.0453 0.0688 0.1075 0.2392 0.0041 0.0164 0.0348 0.0469 0.0331 0.0492 0.1938 0.2357 0.0070 0.0309 0.0482 0.0693 0.0060 0.0231 0.0235 0.0442 0.0530 0.0195 0.0068 0.0286

0.0205 0.0269 0.0402 0.1184 0.0540 0.0399 0.0154 0.0380 0.0323 0.0449 0.1138 0.1750 0.0490 0.0345 0.0141 0.0302 0.0078 0.0173 0.0209 0.0366 0.0142 0.0512 0.0343 0.0019 0.0148 0.0322 0.0365 0.1167 0.0134 0.0375 0.0246 0.0096 0.0218 0.0377 0.0535 0.1310 0.0184 0.0070 0.0222 0.0312 0.0348 0.0620 0.1195 0.1946 0.0100 0.0115 0.0316 0.0510 0.0060 0.0163 0.0237 0.0438 0.0303 0.0552 0.0098 0.0080

0.0058 0.0105 0.0294 0.0542 0.0941 0.0667 0.0100 0.0264 0.0080 0.0147 0.0354 0.0840 0.0928 0.0707 0.0080 0.0183 0.0076 0.0040 0.0107 0.0219 0.0145 0.0915 0.0674 0.0275 0.0149 0.0045 0.0118 0.0286 0.0147 0.0800 0.0653 0.0243 0.0021 0.0165 0.0354 0.0631 0.0266 0.0018 0.0140 0.0198 0.0017 0.0277 0.0435 0.0825 0.0246 0.0038 0.0223 0.0336 0.0059 0.0019 0.0070 0.0267 0.0190 0.0974 0.0208 0.0012

0.0369 0.0260 0.0166 0.0312 0.1656 0.0931 0.0070 0.0149 0.0374 0.0264 0.0171 0.0310 0.1647 0.1066 0.0049 0.0168 0.0074 0.0396 0.0261 0.0082 0.0138 0.1593 0.1345 0.0554 0.0150 0.0382 0.0274 0.0106 0.0125 0.1600 0.1290 0.0518 0.0500 0.0073 0.0183 0.0261 0.0227 0.0026 0.0090 0.0105 0.0520 0.0085 0.0223 0.0285 0.0295 0.0046 0.0189 0.0274 0.0059 0.0475 0.0185 0.0071 0.0107 0.1721 0.0290 0.0081

0.1060 0.0831 0.0104 0.0166 0.2993 0.1083 0.0044 0.0072 0.1072 0.0870 0.0072 0.0144 0.2936 0.1363 0.0032 0.0064 0.0066 0.1092 0.0860 0.0023 0.0118 0.2979 0.2570 0.0715 0.0144 0.1086 0.0858 0.0570 0.0095 0.2990 0.2448 0.0803 0.1094 0.0032 0.0093 0.0130 0.0204 0.0012 0.0059 0.0050 0.1115 0.0034 0.0119 0.0152 0.0290 0.0006 0.0097 0.0161 0.0060 0.1215 0.0258 0.0008 0.0045 0.3030 0.0211 0.0088

0.2163 0.2077 0.2088 0.0884 0.1850 0.1780 0.0075 0.0088 0.0142 0.0453 0.0092 0.0173 0.2269 0.3897 0.4039 0.2160 0.2312 0.0252 0.0066 0.0075 0.0071 0.0496 0.0007 0.0092 0.2272 0.2114 0.2094 0.1768 0.1675 0.1592 0.0218 0.0055 0.0120 0.0128 0.0036 0.0157 0.2051 0.3904 0.3993 0.1449 0.3008 0.0608 0.0206 0.0025 0.0135 0.0509 0.0089 0.0089 0.0064 0.0140 0.0029 0.2155 0.2103 0.2100 0.1895 0.1848 0.1752 0.1785 0.0574 0.0043 0.0025 0.0069 0.0097 0.2779 0.3898 0.4009 0.2712 0.3724 0.4081 0.1122 0.0458 0.0156 0.0057 0.0134 0.0148 0.2175 0.2108 0.2081 0.2046 0.1671 0.1598 0.1572 0.1280 0.0735 0.0051 0.0016 0.0118 0.2637 0.3454 0.3846 0.2137 0.3601 0.3404 0.1678 0.1016 0.0355 0.2252 0.2143 0.1406 0.0067 0.0020 0.0021 0.0093 0.0052 0.0057 0.0012 0.0047 0.0058 0.0096 0.0143 0.0267 0.0066 0.0140 0.0040 0.0076 0.0021 0.0024 0.0047 0.0054 0.0027 0.2324 0.2183 0.1461 0.0017 0.0018 0.0047 0.0040 0.0075 0.0089 0.0090 0.0060 0.0078 0.0393 0.0232 0.0246 0.0026 0.0058 0.0036 0.0028 0.0042 0.0038 0.0140 0.0096 0.0088 0.0033 0.0060 0.0059 0.2362 0.2282 0.2263 0.0702 0.0341 0.0188 0.0211 0.0106 0.0017 0.0015 0.0030 0.0033 0.2587 0.4459 0.5216 0.0186 0.0148 0.0088 0.0203 0.0043 0.0033 (continued on next page)

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Table 4 (continued) Scenario

57 58 59 60 61 62 63 64

Region 1

2

3

4

5

6

7

8

9

10

11

0.0843 0.0920 0.1239 0.3345 0.1371 0.0978 0.0930 0.2572

0.0279 0.0785 0.0851 0.3154 0.0058 0.0763 0.0874 0.2309

0.0240 0.0303 0.0422 0.3325 0.0917 0.0271 0.0543 0.0853

0.0063 0.0213 0.0372 0.2754 0.1333 0.0074 0.0298 0.0500

0.0060 0.0351 0.0577 0.1514 0.1131 0.0487 0.0022 0.0113

0.0059 0.0005 0.0133 0.0359 0.0911 0.0943 0.0240 0.0171

0.0058 0.0474 0.0205 0.0100 0.1013 0.1723 0.0391 0.0470

0.0057 0.1206 0.0347 0.0496 0.1213 0.3060 0.0368 0.0658

0.0045 0.2453 0.1067 0.1615 0.2466 0.1630 0.0029 0.1456

0.0050 0.2279 0.0479 0.1050 0.2301 0.4411 0.0259 0.1506

0.0060 0.2267 0.0230 0.0586 0.1064 0.5285 0.0121 0.0549

Table 5 Main effects and two-factor interaction effects of supply chain factors in 11 regions Factors

Demand pattern Ordering policy Information sharing Lead time Supply chain level Demand pattern * ordering policy Demand pattern * information sharing Demand pattern * lead time Demand pattern * supply chain level Ordering policy * information sharing Ordering policy * lead time Ordering policy * supply chain level Information sharing * lead time Information sharing * supply chain level Lead time * supply chain level

p-Value Region Region Region Region Region Region Region Region Region Region 1 2 3 4 5 6 7 8 9 10

Region 11

0.001 0.081 0.000 0.245 0.000 0.213 0.346

0.006 0.000 0.014 0.000 0.000 0.243 0.310

0.441 0.000 0.367 0.000 0.000 0.202 0.420

0.493 0.000 0.044 0.000 0.000 0.290 0.801

0.103 0.000 0.000 0.000 0.000 0.034 0.942

0.045 0.000 0.699 0.000 0.000 0.030 0.684

0.013 0.000 0.568 0.000 0.000 0.079 0.653

0.003 0.001 0.489 0.001 0.000 0.271 0.720

0.001 0.001 0.364 0.798 0.002 0.360 0.357

0.004 0.006 0.389 0.045 0.000 0.145 0.610

0.014 0.002 0.772 0.062 0.000 0.368 0.767

0.298 0.076 0.004

0.621 0.028 0.001

0.346 0.229 0.025

0.412 0.424 0.448

0.910 0.244 0.420

0.036 0.642 0.381

0.013 0.679 0.468

0.037 0.520 0.560

0.194 0.302 0.521

0.163 0.371 0.541

0.488 0.163 0.770

0.236 0.000 0.897 0.000

0.059 0.000 0.000 0.056

0.038 0.217 0.530 0.538

0.072 0.302 0.173 0.078

0.251 0.074 0.000 0.009

0.209 0.000 0.257 0.746

0.021 0.000 0.713 0.943

0.030 0.000 0.834 0.963

0.407 0.000 0.835 0.749

0.105 0.000 0.666 0.853

0.040 0.000 0.973 0.953

0.033

0.001

0.004

0.002

0.005

0.109

0.066

0.192

0.869

0.360

0.581

7.3. Effect of ordering policy

7.4. Effect of information sharing

The effect of ordering policy is significant (Table 5) in Regions 2–11. OP2 tends to reduce the system chaos in most of the scenarios in these decision regions. Although OP2 can be beneficial, the benefit of OP2 is decreased for Region 1 where the disparity between the adjustment parameters for inventory discrepancies and supply line discrepancies is more pronounced. The analysis shows that the whole supply chain system benefits when the retailer uses the pass-order heuristic. In practice, decision makers at different levels should collaborate and try to understand the effect of different ordering policies on system behavior, then select the ordering policies appropriate for reducing system chaos.

The effect of sharing customer demand information is generally insignificant except for Regions 1, 2, 4 and 5 (see Table 5). However, some cases are likely significant due to the interaction effects of other factors, e.g., lead time and supply chain level. Upon closer examination (see Table 4), it is interesting to note that sharing demand information seems to make the system more chaotic in Regions 1 and 2, but less chaotic in Regions 4 and 5. With all the other conditions fixed, this observation suggests that the decision region, i.e., the replenishment policy or decision-making behavior, plays a significant role in the effect on the system chaos. It is beneficial to share demand information only when a proper decision region, e.g., Region 4 or 5, is adopted. The

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Lyapunov exponent

0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 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 26 27 28 29 30 31 32 Scenario Region 1

Region 2

Region 3

Region 4

Region 5

Region 6

Lyapunov exponent

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 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 26 27 28 29 30 31 32 Scenario Region 7

Region 8

Region 9

Region 10

Region 11

Lyapunov exponent

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Scenario Region 1

Region 2

Region 3

Region 4

Region 5

Region 6

Lyapunov exponent

0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Scenario Region 7

Region 8

Region 9

Region 10

Region 11

Fig. 6. Average Lyapunov exponents for all scenarios and decision regions. (a) Scenarios 1–32, Regions 1–6. (b) Scenarios 1–32, Regions 7–11. (c) Scenarios 33–64, Regions 1–6. (d) Scenarios 33–64, Regions 7–11.

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above finding, in terms of system chaos, is somewhat dissimilar to the general notion that information sharing is beneficial, for example, in reducing inventory, supply chain cost, and demand amplification.

The effect of lead time is consistently significant in all cases except for Region 1 (see Table 5). The benefit of lead time reduction is apparently offset by the magnitude disparity between aS and aSL in Region 1. Other than this extreme case, lead time reduction generally compresses time delay, thus diminishes the negative impact of inventory discrepancies and supply line discrepancies and reduces the degree of chaos of the supply chain system.

regions. Given the experimental setting, supply chain level and lead time play a more significant role in affecting the extent of system chaos. A closer examination shows that the higher the supply chain level and the longer the lead time, the more chaotic the system is likely to be. Due to interactions, the benefit of adopting ordering policy OP2 becomes more obvious only when the lead time is long and at upper supply chain levels. This is also true in the case of sharing demand information. However, the interaction effects involving information sharing appears to be significant primarily in Region 5. For OP2 to be effective, the retailer has to share customer demand information with other supply chain levels. Overall, system chaos can be reduced by reducing lead time.

7.6. Effect of supply chain level

8. Key findings and managerial implications

The effect of the supply chain level is significant for all regions and the degree of system chaos (exhibited in the inventory) generally increases as it goes upstream in the supply chain (see Table 5). The amplification of system chaos in inventory observed in this study echoes the phenomenon of inventory amplification that was observed long ago.

First, the system becomes more chaotic when the disparity between the adjustment parameter for inventory discrepancies (the difference between the desired and actual inventories) and that for supply line discrepancies (the difference between the desired and actual supply lines) grows bigger. To reduce the degree of chaos in the supply chain system, the adjustment parameters for both inventory and supply line discrepancies should be more comparable in magnitude. Second, the system is generally more chaotic when the shift in customer demand is sustaining,

7.5. Effect of lead time

7.7. Dynamics among factors Fig. 7 highlights all significant interaction effects between or among factors for indicated decision Regions 5,6

Regions 5,6

Demand Pattern

Region 2

Regions 1,2,3,4,5

Regions 1,2,6

Ordering Policy

Regions 1,5

Regions 2,5

Supply Chain Level

Lead Time

Regions 5,6

Region 5

Region 3

Information Sharing

Regions 1,2,3

Region 6

Fig. 7. Significant interaction effects between/among factors for indicated regions. The number of significant interaction effects associated with demand pattern, ordering policy, supply chain level, lead time, and information sharing (regardless of regions) are 3, 6, 7, 7, and 4, respectively (refer to Regions 1–6 in Table 5).

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i.e., a step function, rather than temporal, i.e., a broad pulse. Under such circumstances, decision makers should manage the other factors, e.g., lead time, more carefully to avoid chaotic behavior. Of course, a better solution is to collaborate with other supply chain levels to jointly respond to the shifted level of customer demand. Third, system chaos can generally be reduced when the retailer adopts the pass-order heuristic even though the upper levels still adopt the feedback heuristic. This shows that, when the customer demand is deterministic, additional feedback incorporated at the customer-end level does not help stabilize the system, but makes it more chaotic. It is a good practice, with customer order visibility, that the retailer orders only what is needed (ordered) by the customer. Fourth, the effect of sharing customer demand information is generally insignificant except for Regions 1, 2, 4 and 5. It is beneficial to share information only when a certain decision region, e.g., Region 4 or 5, is adopted. Counter-intuitively, in regions of high-disparity, e.g., Regions 1 and 2, sharing demand information can do more harm than good. Fifth, apart from the extreme-disparity case (Region 1), lead time reduction generally compresses time delay, thus diminishes the negative impact of inventory discrepancies and supply line discrepancies and reduces the degree of chaos of the supply chain system. When lead time reduction does not contribute to supposedly positive results, management must painstakingly investigate other decision parameters to adjust the system in order to reap the deserving benefits. Finally, the ‘‘chaos-amplification’’ phenomenon across supply chain levels is clearly observed. That is, the inventory becomes more chaotic at the upper levels of the supply chain system. This phenomenon echoes the phenomena of demand and inventory amplification that were observed long ago. 9. Conclusion This research studies the chaotic behaviors exhibited at all levels of a supply chain under the influence of demand pattern, lead time, ordering policy, and information sharing. The multi-level supply chain is characterized by the widely known beer distribution model. Employing a set of rather sophisticated and robust order heuristics, the model allows us to investigate supply chain dynamics in a

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wide domain of decision parameter space. From a nonlinear dynamic or chaotic perspective, Lyapunov exponents are calculated based on inventories to quantify system chaos. As shown in this study, chaotic behaviors in supply chain systems can be generated by deterministic exogenous factors, e.g., customer demand, coupled with deterministic endogenous factors, e.g., ordering policy, lead time and information sharing. This is contrary to the conventional perception that system variability is mainly caused by unpredictable external conditions. Such a presumption would certainly obscure the path to deriving the right solutions for reducing inventory. It is therefore important to recognize deterministic chaos versus stochastic randomness. To manage the supply chain effectively, decision makers must understand not only the effect of various supply chains factors on the system behavior, but also the interactions between exogenous and endogenous factors in order to reduce inventory variability and system chaos. Acknowledgments The authors thank the two referees for their valuable comments and suggestions. This research was supported in part by National University of Singapore research Grant No. R-314-000-060-112, Graduate Research Scholarship, and President’s Graduate Fellowship. References Cachon, G., Fisher, M., 2000. Supply chain inventory management and the value of shared information. Management Science 46, 1032–1048. Chen, F., Samroengraja, R., 2000. The stationary beer game. Production and Operations Management 9, 19–30. Chen, F., Drezner, Z., Ryan, J.K., Simichi-Levi, D., 2000. Quantifying the bullwhip effect in a single supply chain: The impact of forecasting, lead times, and information. Management Science 46, 436–443. D’Amours, S., Montreuil, B., Lefrancois, P., Soumis, F., 1999. Networked manufacturing: The impact of information sharing. International Journal of Production Economics 58, 63– 79. Feigenbaum, M.J., 1978. Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics 19, 25–52. Forrester, J.W., 1961. Industrial Dynamics. MIT Press, Cambridge. Hilborn, R.C., 1994. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers. Oxford University Press, New York.

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