Bullwhip Effect in Supply ChainsDr Stephen Disney

Cardiff Business School, Cardiff University

Last updated on 27 April 2009

Abstract

We review a range of methodological approaches to solving the bullwhip problem. The bullwhip problem is a dynamic consequence of supply chain structures and replenishment policies. The roles of the structure of the demand process, the treatment of time (continuous v discrete), forecasting techniques and lead-times will be reviewed. In practice, and in the theory, a variety of techniques have been used to smooth the dynamics of supply chains. These include, the use of sophisticated forecasting, pooling of demand and inventories, proportional feedback controllers and full-state feedback systems. Multi-echelon supply chains also present a number of interesting innovations. From the traditional, arms-length trading relationships, information sharing, vendor managed inventory and echelon stock policies can be developed. More sophisticated collaboration and co-ordination mechanisms may also lead to altruistic behaviour and result in superior performance. The impact of these procedures will be examined. Finally thoughts on new directions in bullwhip research is presented.

Keywords: Bullwhip, supply chains, inventory, multi-echelon, order-up-to policy

Introduction: The bullwhip effect in supply chains

The bullwhip effect is a dynamical phenomenon in supply chains. It refers to the tendency of the variability of orders rates to increase as they pass through the echelons of a supply chain towards producers and raw material suppliers. A classic example of the effect is baby nappies or diapers. Babies are fairly regular in their use of nappies - they have a new nappy (almost) every time they feed. Sure, there is seasonal variation in the birth rates as more babies are conceived in spring (when male sperm count is significantly higher than in any other season; however this is not globally consistent and the

there is some debate over the role of both temperature and the day length [1]). Neither-the-less, this seasonal variation is small compared to the widely fluctuating and erratic production rates experienced by the diaper manufacturer after the orders have passed through the supermarkets and distribution centres.

There are various measures of the bullwhip effect proposed in the literature [2]. The most common measure is the ratio of the variance of the order rate to the variance of the demand rate, see equation (1). It is the measure that we will focus on herein, and works best for stationary, stochastic, discrete time demand processes. However there are other measures. Standard deviations could be used instead. Indeed, as we will reveal later, this is more natural when the economics of the bullwhip (and inventory) are considered. Another (practical) measure is ratios of the co-efficient of variation ( COV=variance/mean) of order and demand rates. This is a useful measure when there are multiple products going through multiple routes to market and some comparison is needed across different products, businesses or routes to market.

(1)

Bullwhip creates unstable production schedules. These unstable production schedules are the cause of a range of unnecessary costs in supply chains. Companies have to invest in extra capacity to meet the high variable demand. This capacity is then under-utilised when demand drops. Unit labour costs rise in periods of low demand, over-time, agency and sub-contract costs rise in periods of high demand. The highly variable demand increases the requirements for safety stock in the supply chain. Additionally, companies may decide to produce to stock in periods of low demand to increase productivity. If this is not managed properly this will lead to excessive obsolescence. Highly variable demand also increases lead-times. These inflated lead-times lead to increased stocks and bullwhip effects. Thus the bullwhip effect can be quite exasperating for companies; they invest in extra capacity, extra inventory, work over-time one week and stand idle the next, whilst at the retail store the shelves of popular products are empty, and the shelves with products that aren’t selling are full.

There are various causes to the bullwhip effect. Lee, Padmanabhan and Whang (1997) [3] made ground-breaking contributions and re-ignited interest in the subject [4]. Their main contribution was to analyse four different causes of the bullwhip effect; batching, shortage gaming, lead-times and demand signal processing. However, there are other sources of the bullwhip effect [5]. Together demand signal

processing (the way that replenishment decisions are made) and the impact of lead-times have previously been called the Forrester Effect [6]. We will mainly focus herein on the bullwhip problems generated by the so-called Forrester Effects.

Methodological approaches to solving the bullwhip problem

The biggest decision to make is whether to study the bullwhip problem in discrete or continuous time. In discrete time, system states (demand rates, inventory and WIP levels) and replenishment orders are made at the equally spaced moments of time. In between these moments of time, nothing is known about the system. In continuous time the systems states are monitored at all moments of time and the order rate is continuously adjusted.

Neither representation of time is incorrect; it is just that one representation of time may be more suitable for a given situation than the other. For example, in a grocery supply chain, supermarkets total up demand that has occurred during the day, a replenishment order is generated and a delivery is despatched from the distribution centre overnight. This scenario is very suitable for a discrete time analysis. A petrochemical plant, on the other hand, may be able to continuously adjust its production of different grades of product to reflect the current demand rates for each grade. This type of scenario is more amenable to a continuous analysis.

Continuous time methods

The Laplace transform was originally developed by Laplace and Euler in the 17 th century for studying the orbits of planets. However, electronic engineers have developed a whole range of tools, loosely termed control theory, for studying continuous time systems based on Laplace transformed transfer functions. These transfer function techniques work very well if the system is linear, time invariant (LTI - a common assumption) and the system has no initial conditions (IC). Simon (1952) [7] seems to have been the first to apply the Laplace transform to a production and inventory control problem. Transform approaches work well in Single Input and Single Output (SISO) scenarios as then only a single transfer function is required. Transforms also contain complete information about frequency response of the system. Interestingly, the transforms that describe cash flows are directly related to the Net Present Value of that cash flow [8].

If the two assumptions of LTI and zero IC do not hold, then the analytical approaches have to resort back to the (non-linear) differential equation forms. Unfortunately there is no “standard approach” for analysis of such systems. Indeed many systems have no known solution, and even when we can obtain a solution there is often an infinite number of them, one for each set of IC and non-linearity. Linear differential equations are also readily handled by state-space techniques. These are essentially matrix representations of systems of equations. State-space methods are especially good at handling Multiple Input, Multiple Output (MIMO) systems and can be easily extended to include non-zero IC’s.

Another important type of system is known as the differential-delay equation. These are systems that contain a pure time delay in them (as supposed to a lag, which can be readily handled by differential equations and Laplace transforms). Pure time delays occur in supply chain settings when there is a transport delay, whereas lags have been shown to be a good representation of factory output when there are multiple stages of production. The principle problem with differential-delay equations is that they generate an infinite number of complex solutions to the characteristic equation and thus have a transcendental nature [9]. However, the Lambert W function has been successfully applied to obtain solutions to delay differential equations [10]. The Lambert W function is the inverse function of f( w)= we w. The general strategy is to re-arrange to differential equation to make it look like Y=Xe X and then use the W function to provide the solution, X=W( Y).

Discrete time methods

The discrete time analogue of the Laplace transform is the z-transform. It was developed independently by scholars from the UK [11] and Russia [12] during the Second World War for controlling such things radar and gun targeting systems and other applications that involved the newly available digital computers. The first book that brought together all of the developments of the z-transform was by Jury (1964) [13], but the first person to apply the z-transform to a production and inventory control problem appears to have been Vassian (1955) [14]. The advantages of using the z-transform over the time domain difference equations are the same as for the continuous case; convolution in the time domain is multiplication in the frequency domain. However, the disadvantages are that it has to be LTI and possess zero IC. However, problems with the pure-time delay are completely avoided in discrete time as it forms the kernel of the z-transform. State space methods (with the same advantages) are also available in discrete time.

In discrete time a lot can be done with stochastic techniques using the expectation operator. However, the calculation of the co-variances can become very tedious when complex systems are studied. Interestingly this difficultly is completely avoided with transform approaches. Martingales have also

been used to study inventory problems, for example, see Graves (1999) [15]. Martingales are useful tools as they can yield insights into magnitude of infinite variances that occur in non-stationary time series.

A particularly useful difference equation approach was developed by Box and Jenkins (1970) [16]. Known as ARIMA modelling, Box and Jenkins developed a generalised time series model that consisted of an arbitrary number of three types of terms. That is, Auto-regressive, Integrated and Moving Average terms. Their generalised time series model has been found to represent a wide range of stochastic time series. The general ARIMA( p,d,q) model is given by equation 2. The Box and Jenkins approach copes with non-stationary processes by differencing the time series.

(2)

Other approaches

Any time series, continuous or discrete, can be analysed using variations of the Fourier transform. This is a frequency response method, where a time series is broken up into a series of harmonics. Harmonics are sine waves of different frequencies, amplitudes and phase lags. Understanding how replenishment rules respond to the complete spectrum of individual harmonic frequencies allow us to understand how they react any demand signal, thus the tool is particularly powerful [17].

The beer game, a table top management game, is also very good demonstrator of the bullwhip effect. It may also be used to generate knowledge and insights into how actual people manage supply chains, Sterman (1989) [18]. The beer game has been used extensively in business schools world-wide and many forms of it exist. Some variants of the beer game are simply electronic or internet based [19] versions of the original game, others have been adapted to represent particular industries [20] or to investigate particular supply chain strategies such as information sharing or VMI [21].

Systems dynamics is an intuitively based computer simulation technique that essentially relies on animating causal loop diagrams. It was originally advocated by Forrester (1961) [22] as a means of investigating large non-linear systems without resorting to complex mathematical models. Another form of simulation is discrete event simulation. It actually has the power to investigate, at least numerically,

very realistic models of supply chains. It is possible to explicitly model such things as capacity constraints, non-negative inventory and WIP levels, actual real-life demand patterns, process uncertainties (machine breakdowns), quality losses, process time variation, rework and even quality control procedures [23]. The real value from system dynamics (and simulation approaches in general) is from the act of building the model itself as the process formalises a lot of tacit knowledge. However, simulation based approaches suffer from the drawback of being cumbersome, time consuming and only providing limited insight.

Supply chain strategies for taming the bullwhip effect

The bullwhip effect is greatly influenced by the structure of the material and information flow in a supply chain and the how the replenishment decisions are made. In this section we will briefly review the impact of the supply chain structure and how it influences the bullwhip effect. We will do this with a “water tank model” as it very easily illustrates the main principles [24]. In the water tank model, water represents inventory, tanks represent companies, flow of water out of a tank represents sales, flow in, deliveries. The replenishment or ordering decision is conceptualised with a “ball cock” that regulates the flow.

Traditional supply chains

In a traditional supply chain each echelon makes its own replenishment decision, based on it own local information. This local information often includes sales (flow out of its tank), inventory (water in the tank) and orders placed, but not yet received or WIP (water in the pipe flowing towards the tank), see Figure 1. Most supply chains operate in this mode. The problem is bullwhip increases geometrically in traditional supply chains. Every time the order passes through an echelon of the supply chain, the variance of the order rates “multiples up”, and this causes inefficiencies.

Figure 1. Schematic of a traditional supply chain

Information sharing

Some supply chains have the ability to share point of sale data to the end consumer with other members of the supply chain. Consider, for example, the supermarket supply chains in the UK. As the barcodes are scanned at checkouts an electronic file is populated from which sales patterns for particular products can be determined. These sales patterns are then transferred (sold even) to suppliers. Often suppliers use this in capacity planning activities, but the real benefit comes from using it in their replenishment / ordering decisions (see Figure 2). Using the POS data solely in the capacity decisions has no effect on the dynamics of the supply chain; the bullwhip effect still exists, and it still increases geometrically. Thus suppliers are left dismayed, wondering why the retailer’s orders fluctuate widely despite fairly steady sales.

However, if suppliers are sophisticated enough to incorporate this information into the forecasts that they use inside their replenishment decisions, then the bullwhip effect can be greatly reduced. This is especially true for echelons further away from the end consumer as now the bullwhip effect will only increase linearly as it proceeds up the supply chain.

VMI- type supply arrangements

The gold standard in supply chains structures however comes from completely eliminating replenishment decisions. This can be done via Vendor Managed Inventory (VMI) arrangements. In VMI supply chains suppliers have complete visibility of the downstream supply chain; they can see end

consumer demand, their customer’s inventory levels and the contents of the pipeline (WIP). They can then base their replenishment rules on the state of the complete downstream supply chain. It is then possible to get a multi-echelon supply chain to act dynamically as a single echelon. This effectively removes the bullwhip problem from the supply chain.

Figure 2. Schematic of a supply chain with information sharing

To achieve this, the retailer and the supplier need to come to some arrangement about how the retailers inventory levels are managed. Often a minimum and maximum allowable inventory level is agreed upon and freedom is given to the supplier to replenish the retailers inventory whenever he desires, as long as the inventory levels stay within the pre-defined limits. This is akin to deciding how big the retailers tank is, see Figure 3.

Figure 3 . Schematic of a VMI supply chain

It is often assumed that the supplier is managing the retailer’s inventory. This is simply unacceptable for many retailers, as they consider their inventory management skill to be a core competency of their business. However, the supplier does not need to be in direct control of the retailer’s inventory. Consider the second water tank model of Figure 4. It does not matter who’s “hand” is controlling the “cup” (the cup is an analogy for a truckload of product) to move the inventory from one company to the other. It could be either the retailer or the supplier. What is important is not who owns inventory or who makes the decisions. What is important is how decisions are formed and what information is used within them.

Figure 4 . Alternative schematic of a VMI supply chain

Replenishment rules for solving the bullwhip problem

In order to illustrate a range of solutions to the bullwhip problem we will consider a simple example. It will be based on some assumptions, although we will highlight the impact of these assumptions when it is appropriate. Indeed some solutions to the bullwhip effect come from breaking down these assumptions. The replenishment rule operates in discrete time, uses exponential smoothing as a forecasting method, demand is a stochastic, independent and identically distributed random variable and the Order-Up-To (OUT) policy is used to formulate the replenishment decision.

The OUT policy makes replenishment decisions based on three factors; a forecast of future demand, an inventory level discrepancy term and a WIP level discrepancy term, see equation (3).

(3)

Notice in (3) we have defined the Target Net Stock as a time invariant constant. This need not always be the case, it maybe appropriate to assume that it is a function of time, t. The forecast of demand can be done in a variety of ways, but here it is convenient for demonstration purposes to use simple exponential smoothing, see equation (4).

(4)

To complete the definition of the OUT policy we will also need the inventory and WIP balance equations. These are

, (5)

. (6)

In equations (5) and (6) there is a new term, T p, which is the replenishment lead-time. This lead-time is the time between placing an order and receiving the products ordered into stock. This order can be placed either on a supplier or on a production system.

It is also obvious from (5) that there is a link between the bullwhip effect and “net stock variance amplification” or NSAmp. NSAmp is the analogy of bullwhip in the inventory levels. Clearly, when designing a replenishment rule, we want to avoid both bullwhip (capacity related) costs and NSAmp (inventory related) costs.

(7)

Better forecasting

An important skill for companies to master in order to tame the bullwhip effect is to create good forecasts. By “good” we mean forecasts that minimise mean squared error (MMSE) between the forecast of demand over the lead-time (and review period in discrete time systems) and the actual,

realised demand over the lead-time (and review period). Assuming an i.i.d. demand and simple exponential smoothing forecasting is used inside the OUT policy, then the bullwhip is given by,

. (8)

In equation (8) when T a=0 then Bullwhip=5+6T p+2T p 2 and as T a approaches infinity the Bullwhip ratio approaches unity. The value of T a that minimizes all future forecast errors for i.i.d. demand is T a=infinity. Thus, we can see that we make more accurate forecasts the bullwhip problem is reduced (but is not eliminated in this scenario) with more accurate forecasts.

It is also interesting to see what happens to the inventory bullwhip effect, NSAmp, in our stylised example. It is given by equation (9),

(9)

Here we can see that the inventory variance is always greater than unity. Hence inventory levels will always vary more than the demand signal and it is not possible to achieve zero inventory policy without making the customer wait for the product with this ordering decision. The last part of equation (9) tends to zero as Ta approaches infinity. Thus, more accurate forecasting also leads to lower inventory levels.

Reduce lead-times

Reducing lead-times will also reduce bullwhip. For example, see equation (8). Here T p, the replenishment lead-time, occurs in the numerator of the bullwhip expression and thus smaller T p’s will

mean less bullwhip is produced. Furthermore, for i.i.d. demand and MMSE forecasting, the lead-time drops out of the order variance ratio expression. Unfortunately this is not a general conclusion. For other types of demand processes (such as ARIMA demand) and other forecasting mechanisms, lead-time does play a role. What is interesting though is the fact that bullwhip will never increase when lead-time decreases.

Reducing lead-time will also have a positive impact on the inventory costs in a supply chain. Inspection of equation (9) reveals that the inventory variance is made up of three components. The first unit of inventory variance is due to the order of events delay. That is, because demand for this planning period is not known until the end of the planning period, inventory has got to vary at least as much as the demand levels (in a continuous system this order-of-events delay does not occur and thus this statement does not hold anymore). Inventory variance increases as the production lead-time increases. In fact, we can see that inventory levels will also vary by at least ((1+ Tp)*(Variance of demand)), no matter how well the forecasts are generated. The third component of the inventory variance (equation (9)) is to cover the forecasts errors introduced by the non-optimal forecasts.

Proportional feedback controllers and smoothing replenishment rules

A control engineer may look at equation (3) and not be at all surprised by the fact that the OUT policy produces bullwhip effects. Scrutiny of equation (3) shows that it has two feedback loops, one for the net stock and one for the WIP. Both of these feedback loops have a gain of unity. That is, all of the discrepancy between actual and target inventory (and WIP) is incorporated into the order rate.

It is almost second nature to a control engineer that it always possible to eliminate the bullwhip effect if the feedback loops or gains are properly designed. There are various options for designing these feedback loops. The simplest and most common approach is to use a “proportional feedback controller”. It may take the form of a fraction, 1/ T i for example (although a PID [25] controller could also have been used, amongst others) and including into the OUT policy may result in a new form of equation (3);

(10)

In equation (9) we have added two independent feedback controllers, T i and T w. T i is for regulating the net stock error feedback and T w is for regulating the WIP error feedback. Allowing the feedback loops to be independent allows the natural frequency and damping ratio of the OUT policy to be decoupled from each other [26]. However, when T w= T i the mathematics involved become considerably simpler [27], although the independent case is possible to study [28]. When T w= T i, and T a = infinity, the bullwhip expression, in our stylized example becomes

(11)

In (11) we notice that bullwhip is now independent of the lead-time. Under the same conditions the NSAmp expression becomes

(12)

where can now see that both T i>1 (smoothing) and T i<1 (bullwhip) increases inventory requirements. However, the influence of the lead-time on the inventory has remained the same; longer lead-times mean that the variance of the inventory levels increases.

An interesting trade-off situation now occurs. If we assume costs were equal to the order and inventory variances we may wish to minimise the sum of the bullwhip and NSAmp expressions. Adding together equations (11) and (12), differentiating them with respect to T i and solving for zero gradient shows that the optimal value of T i is the golden ratio, 1.618034. This is illustrated graphically in Figure 5.

The use of proportional controllers need not be limited to feedback loops either. By exploiting optimal control theory and state-space approaches it is possible to derive policies that take full account of all the states in the system. These types of systems are known as a full-state feedback systems [29] and can

account for the structure of complex demand process in a superior manner when compared to the traditional (and the proportional) order-up-to policy.

Figure 5. Variance ratios with the golden feedback controller

Multi-echelon supply chain policies

When we consider multi-echelon supply chains then there a whole new range of options available to us. For example, we have already spoke of VMI supply chains. Here downstream supply chain states are communicated to suppliers and this information is used in their replenishment decisions. This type of arrangement is closely related to what is known as the echelon-stock policy or the echelon order-up-to policy, Hoberg, Bradley and Thonemann (2007) [30].

Collaboration and coordination mechanisms for multi-echelon supply chains also have very good economic performance. For example, if a retailer is willing and able to smooth his replenishment orders he places on his supplier, the supplier will be able to manufacture product more efficiently. However, this is effectively an altruistic contribution on behalf of the retailer if the supplier does not share the gains with the retailer. Disney, Lambrecht, Towill and Van de Velde (2007) [31] have been investigating the role that matched proportional feedback controllers can have in this type of coordination scheme when both bullwhip and inventory costs exist. Hosoda and Disney (2006) [32] considered the case when only inventory costs exist.

Finally, there is a very innovative and novel method currently being developed by Gaalman and Disney (2007) [33] for co-ordinating a multi-echelon supply chain. This policy has been derived using optimal control theory and, in a sense, is like a VMI supply chain in reverse. The core of the idea is that because the retailer’s lead-time is shorter than the manufacturer’s lead-time, he can correct some of the manufacturer’s forecast errors. In this way, the retailer is accounting for the state of the “upstream” supply chain. The analysis of this policy is not yet complete but initial findings are very interesting.

Economics of the bullwhip problem

In this section we will quickly explore an economic approach to looking at the bullwhip problem. To capture the lost capacity and over-time costs that may reasonably be assumed to be dependent on the order rate we will assume that demand is normally distributed and a discrete time, linear system exists. Other costs may be assumed to be constant or independent of the order rate and we simply ignore them here. Such costs may be materials, energy and administration overheads, for example.

Let S be an amount of (slack) capacity above the mean demand rate (

) and N be the cost per unit per period of not producing to the available production capacity. Then there will be piece-wise linear, convex lost capacity costs in each period of

. (13)

Let P be the cost per unit per period of producing in over-time, so there are piece-wise linear, convex over-time costs in each period of

. (14)

We can invest in an optimal amount of capacity (above or below average demand) to achieve an economic over-time probability. The desired amount of capacity is driven by the critical fractile and turn out be

. (15)

When this optimal amount of capacity exists then the bullwhip related costs are

. (16)

Equation (16) shows us that the bullwhip costs are linearly related to the standard deviation of the order rates.

The square root law for bullwhip

Let’s turn our focus now to a distribution network. Assume that we have multiple retailers being served by a number of distribution centres. All transportation lead-times are the unity, regardless of the number of distribution centres that exist. Each retailer faces i.i.d. demand with the same mean and variance. Furthermore each retailer and distribution centre employs a traditional OUT policy with MMSE forecasting and unit feedback gains. For such a scenario Maister (1976) [34] introduced the “Square Root Law for Inventory” when consolidation occurs in the distribution network. Quoting directly from Maister,

“If the inventories of a single product (or stock keeping unit) are originally maintained at a number ( n) of field locations (refereed to as the decentralised system) but are then consolidated into one central inventory then the ratio

exists”, Maister (1976).

Amazingly, the square root law also exists for bullwhip costs. Consider the capacity related costs at the DC echelon. Equation (16) shows us that bullwhip or capacity costs are given by

. (17)

Y is a constant determined by the lost capacity and overtime costs. It is easy too prove the square root law for bullwhip exists by considering that in the decentralised supply chain the standard deviation of the orders at each of the n distribution centres is

, and in the centralised supply chain (with only one distribution centre) the standard deviation of the orders is

. Thus,

, (18)

which is the “Square Root Law for Bullwhip” [35]. This result surprised us, as intuitively, we expected it to have the opposite impact. This result also suggests that reasons to consolidate distribution networks are actually a lot stronger than previously thought (as this is often based solely on inventory costs). The likely impact of this is to force companies to consolidate even further than they have in the past, increasing the amount of traffic on the road. Thus, internalising the external costs transportation causes is now even more important.

Some ideas on future bullwhip research

We have reviewed the main methodological approaches to solving the bullwhip effect and highlighted the role that the supply chain structure and the replenishment rules have upon it. We have briefly considered the economics of the bullwhip effect and what it might mean in a divergent distribution network. We have also given a short summary of an industrial case study where we designed out bullwhip from a supply chain. We will now conclude with some thoughts on future bullwhip research.

Joint replenishment polices (JRP) which control more than one product in the same inventory replenishment decision is a very promising but complex issue. It is closely related to the inventory routing problem (IRP) and there are some initial results [36] on both the JRP and IRP that have some interesting properties and show improved economic performance. This is especially true when order set-up or transportation costs are considered.

Recently we have discovered that there is a link between smoothing and lead-times. We have been using queuing theory to model a manufacturer. The manufacturer works on a make-to-order, first come, first served basis. When a retailer smoothes his replenishment order that is placed on a manufacturer, the lead-time that the manufacturer needs to produce and deliver the products is reduced. Thus, there is a link between bullwhip and lead-time variability [37]. The smoother the retailer’s order is, the quicker the manufacturer can replenish his order, on the average. Thus there is actually a mechanism to break the bullwhip, inventory trade-off we spoke of earlier. Other research that has recently considered similar problems was conducted by Chatfield, Kim, Harrison and Hayya (2004) [38] who investigated the impact of stochastic lead-times, information quality and information sharing in the OUT policy via a simulation experiment. This approach is further clarified with analytical insights in Kim et al (2006) [39].

Multi-echelon policies offer a very promising route for future bullwhip analysis. However, they will require a significant industrial engineering effort to implement in practice. Collaboration and co-ordination mechanisms are also needed. However, the biggest problem with multi-echelon research is properly capturing the effect of lost sales and capacity constraints. This is very difficult to achieve as these systems are non-linear. However, Markov Chains do offer a means of analysis and will even cope with quantised systems where only integer amount of products can be ordered from a supplier or production system.

Finally, if you would like to explore more about the bullwhip effect, please go to www.bullwhip.co.uk. There you will find a collection of simulations, java explorers, table top games and reference lists associated with the bullwhip effect.

References