Chaos Theory: Implications for Supply Chain Management

Since the late 1950’s it has been recognised that the systems used internally within supply chains can lead to oscillations in demand and inventory as orders pass through the system. The uncertainty generated can result in late deliveries, order cancellations and an increased reliance on inventory to buffer these effects. Despite the best efforts of organisations to stabilise the dynamics generated, industry still experiences a high degree of uncertainty. The failure to significantly reduce uncertainty through traditional approaches may in part be explained by chaos theory. This paper defines deterministic chaos and demonstrates that supply chains can display some of the key characteristics of chaotic systems, namely: Chaos exhibits sensitivity to initial conditions, it has “Islands of Stability”, generates patterns, invalidates the reductionist view and undermines computer accuracy. The implications for the management and design of supply chains are briefly discussed. INTRODUCTION The term chaos is currently much used within the management literature. The term chaos has been used to describe the seemingly random disorder of customer demands for products (e.g. Womack and Jones [1 p.81]) and by Tom Peters [2] in his book “Thriving on chaos” to describe disorganised yet responsive business structures that rapidly adapt and gain competitive advantage. Chaos is also used as a metaphor to describe how a small change can be amplified to have a large effect on the system. This has resulted from the popularisation of the “butterfly effect” where a butterfly flapping its wings creates tiny changes in the atmosphere that result in the creation of a tornado a few weeks later [3]. Authors [1 p.87, 4]) describing amplification within the supply chain have used the term chaos in this context. Using chaos as a metaphor for amplification within systems is an over simplification and can lead to misunderstanding. It has been shown that chaos and amplification are linked yet distinctly different types of behaviour [5]. Chaos theory has been applied and found in a variety of systems including cardiac systems [6], stock market data [7] and the management of digital telephone exchanges [8]. Chaos is even being exploited in advanced washing machines to improve cleaning [9] and in the manufacturing quality control of springs [10]. “Chaos Theory: Implications for supply chain management.” 4 CHAOS The Collins English dictionary describes chaos as meaning “complete disorder and confusion”. However, within this paper the term chaos describes deterministic chaos. The definition used in this work is adapted from that proposed by Kaplan and Glass [11 p.27] and Abarbanel [12 p.15]: Chaos is defined as aperiodic, bounded dynamics in a deterministic system with sensitivity dependence on initial conditions, and has structure in phase space. The key terms can be defined as follows:  Aperiodic; the same state in never repeated twice.  Bounded; on successive iterations the state stays in a finite range and does not approach plus or minus infinity.  Deterministic; there is a definite rule with no random terms governing the dynamics.  Sensitivity to initial conditions; two points that are initially close will drift apart as time proceeds.  Structure in Phase Space; Nonlinear systems are described by multidimensional vectors. The space in which these vectors lie is called phase space (or state space). The dimension of phase space is an integer [12]. Chaotic systems display discernible patterns when viewed. Stacey [13 p.228] emphasises this by defining chaos as; “order (a pattern) within disorder (random behaviour)”. Professor Ian Stewart proposes the following simplified definition [14 p.17]: “Stochastic behaviour occurring in a deterministic system”. Stochastic means random or lawless, deterministic systems are governed by exact unbreakable laws or rules. So chaos is “Random (or Lawless) behaviour governed entirely by laws!” Chaos is deterministic, generated by fixed rules that in themselves involve no element of chance (hence the term deterministic chaos). In theory, therefore, the system is predictable, but in practice the nonlinear effects of many causes make the system less predictable. The system is also extremely sensitive to the initial conditions, so an infinitesimal change to a system variable’s initial condition may result in a completely different response. This presents us with a good news / bad news scenario. The good news is that apparently random behaviour may be more predictable than was first thought, so information collected in the past, and subsequently filed as being too complicated, may now be explained in terms of simple rules. The bad “Chaos Theory: Implications for supply chain management.” 5 news is that due to the nature of the system there are fundamental limits to the horizon and accuracy of prediction. Past patterns of system behaviour are never repeated exactly but may reoccur within certain limits. For example the weather, a true chaotic system, has limits to the forecast horizon. Even if every variable was known exactly the theoretical maximum forecast is 2 to 3 weeks [14]. For non-linear dynamic systems the assumption of one-to-one, cause and effect relationships implicit in most human logic do not hold. For chaotic systems, a tiny change in conditions may result in an enormous change in system output, whereas a substantial change in conditions may be absorbed without significant effect to the system’s output. NON-LINEAR SYSTEMS AND CHAOS Mathematicians have discovered that non-linear feedback systems, are particularly prone to chaos. Information is fed back thus impacting on the outcome in the next period of time. Jay Forrester proposes the following definition for a feedback system [15]: “ An information-feedback system exists when the environment leads to a decision that results in action which affects the environment and thereby influences future decisions” The process is continuous and new results lead to new decisions that keep the system in continuous motion. All logistics and supply chain management systems are made up of a series of feedback control loops. This is the way the majority of business systems operate. Within logistics and supply chain management a large proportion of the feedback loops are non-linear. For example, the availability of inventory affects the shipment rate from a warehouse. When inventory is near the desired level, shipment rate can equal order rate but as inventory reduces the shipment rate can become halted or checked by the amount of available inventory. This in turn leads to the issue of “service level”. The relationship between service level and cost to an organisation is depicted as a steeply rising curve. This results from the high costs of carrying additional safety stock to cover those times of unexpectedly high demand. It is therefore possible that the systems of control developed for managing the supply chain exhibit chaotic behaviour. “Chaos Theory: Implications for supply chain management.” 6 CHARACTERISTICS OF CHAOTIC SYSTEMS There are five key characteristics of chaotic systems that have implications for supply chain management. These are: Chaos exhibits sensitivity to initial conditions The characteristic of sensitivity is a central concept of chaos theory. However it should be emphasised that sensitivity does not automatically imply chaos [16 p.512]. This misunderstanding is prevalent in much of the management literature and is linked to the popularisation of the “butterfly effect”. Sensitivity to initial conditions within chaotic systems is more distinct. Given a small deviation in initial conditions, this small difference or error becomes amplified until it is the same order of magnitude as the correct value. The amplitude of the error is magnified exponentially until there is no means of differentiating the actual signal from the signal generated by the error. This results in two systems with starting conditions varied by a fraction of one percent producing outcomes over time that are totally different. The error propagation of the system results in the system being inherently unpredictable and therefore long term forecasting of such systems is generally impossible. This error propagation can be quantified by the use of Lyapunov Exponents. Lyapunov exponents have proved to be one of the most useful diagnostic tools for detecting chaos. The exponent is a measurement of sensitivity to initial conditions. The maximum Lyapunov exponent can be described as the maximal average factor by which an error is amplified within a system. A system can be defined as chaotic if at least one positive Lyapunov exponent is present. If the maximum exponent is negative the system is stable or periodic [17]. The magnitude of the exponent gives a reflection of the time scale over which the dynamics of the system are predictable, so the exponent can be used to approximate the average prediction horizon of a system [17, 18]. After this prediction horizon has been reached the future dynamics of the system become unforecastable. This occurs because any small error is amplified so that the magnitude of the dynamics generated by the error exceeds the original dynamics being measured. This results in cause and effect relationship between current data and previous data becoming increasingly blurred and eventually lost. The concept of prediction horizons can enable managers to differentiate between short and long term strategic decisions and also define a limit to which any forecasting is effective [19]. The sensitivity to initial conditions also results in dramatic changes occurring unexpectedly. Stable behaviour can be followed by rapid change or a system behaving in a seemingly random manner may change to a stable form of behaviour without any warning. There are many sensitive systems that do not behave chaotically. A simple example of this is the equation: “Chaos Theory: Implications for supply chain management.” 7 X CX t t   1 where C is a parameter much greater than 1. Any small error is magnified by a factor of C during each iteration. This system is sensitive to initial conditions but in no way can be defined as chaotic. The error will remain proportionally identical as the system is iterated Chaos has “Islands of Stability”. Chaotic Systems are often related to aperiodic behaviour. Aperiodic behaviour is characterised by irregular oscillations that neither exponentially grow nor decay nor move to steady state [11 p.11]. These oscillations never repeat the same state twice. It is a behaviour that is neither periodic nor stochastic [12]. Chaotic systems typically can exhibit other domains of behaviour that may include stable convergent behaviour, oscillating periodic behaviour, and unstable behaviour [20]. A system may be operating in a stable manner but when a parameter is changed periodic or chaotic behaviour may be witnessed. Also, it is also not uncommon for chaotic systems to spontaneously switch between different modes of behaviour as the system evolves with time. Systems have been observed that will produce aperiodic behaviour for a long period of time and then spontaneously “lock” on to a stable periodic solution [21]. It is therefore possible for “Islands of stability” to be present in between areas of chaotic behaviour. This characteristic has been harnessed for some chaotic systems, by changing a parameter so that the chaotic system can be controlled to produce more regular behaviour. Chaos generates patterns. Despite the generation of apparently random data, chaotic systems produce patterns in the data. These patterns never repeat exactly but have characteristic properties. An example of this is a snowflake. The snowflake is generated by deterministic relationships within the environment but tiny changes become amplified. This results in every snowflake being different, but when observed it clearly belongs to the category of snow flakes. The patterns generated by systems are referred to as “attractors”. An attractor can be defined as [12 p.199]: “The set of points in phase space visited by the solution to an evolution equation long after (initial) transients have died out” Attractors are geometric forms that characterise the long-term behaviour of a system in phase space [22]. Systems have a stable state, the state to which all initial conditions tend to gravitate; this state serves as an attractor. In classical systems, if the system gravitates towards a single point it is said to have a point attractor, if it gravitates towards a stable cyclic response it has a periodic attractor, if the attractor results from a combination of 2 or more periods the system can be defined as a quasi-periodic attractor. The term “strange attractor” is used to describe the shape of the chaotic patterns generated. These strange attractors are a classic feature of chaotic systems [23]. “Chaos Theory: Implications for supply chain management.” 8 To understand the nature of chaotic attractors one needs to understand a simple stretching and folding operation. This results in a shuffling process being undertaken on the chaotic attractor, akin to a dealer shuffling a deck of cards. The randomness of the chaotic orbits is the result of this shuffling process [22]. The stretching and folding process can create patterns that reveal more detail as they are increasingly magnified, these are referred to as fractals. We cannot predict exactly the behaviour of chaotic systems, but by understanding the patterns generated some degree of prediction is possible. Chaos generates endless individual variety, which is recognisably similar. As systems evolve with time recognisable patterns are generated. This property enables analysts to make some predictions about the system. One can predict the qualitative nature of the patterns generated and the quantitative limits within which the pattern will move [13]. Chaos invalidates the reductionist view. One consequence of chaotic systems is that in general the reductionist view becomes invalid. The reductionist view argues that a complex system or problem can be reduced into simple forms for the purpose of analysis [24]. It is then believed that the analysis of the individual parts gives an accurate insight into the working of the whole system. This methodology of reductionism is also often applied to improvement within supply chain systems. The optimisation of the individual units, for example manufacturing, purchasing, and distribution, is believed to result in the optimisation of the global system. Goldratt and Cox [25] demonstrated that in manufacturing environments this is often not the case. One of the rules for manufacturing developed by Goldratt and Cox state: “The sum of the local optimums is not equal to the global optimum” Chaos theory states that a small change to an individual unit within a system may result in dramatic effects on the global system. These effects may not in all cases be beneficial to the operation of the global system. Chaos undermines computer accuracy. Even simple equations can behave chaotically and these can have a dramatic effect on perceived computer accuracy. To demonstrate this phenomena a simple example will be described. The example demonstrates chaos by iterating a simple equation using a standard spreadsheet package. The simple equation to be iterated is as follows: