- Summary:
- Complex supply chains are inherently chaotic, which means demand planners need to understand Chaos Theory and the math behind it to unravel what's going on

Many years ago, when supply chain management was simply called logistics, the mode of management was paper reports, faxes and telephones. Computers, particularly PCs, followed. They were so-called ‘shadow' systems, not endorsed by IT, which provided messaging and a level of modeling via Excel. Eventually, ERP systems provided comprehensive data management and consistency, but their supply chain software was developed using static rules. Once the Internet and online shopping and B2B volume grew, the capability of these systems declined.

Supply chains provide the integration of inbound, outbound and reverse flows of products, services, and related information. Managing supply chains today requires understanding the diverse roles of the supply chain's members, their interactions, and the transaction models they use to interact with one another. Trying to optimize these flows for timeliness, yield, cost and a host of other objectives is complex. Dealing with complexity is the realm of Chaos Theory. As a result of the aggressive progress in software technology, communication and data management, uncertainty in such systems is considerable. Let's have a look at how Chaos Theory can unravel the complexity.

- Aperiodic — the same state is 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.
- "Islands of Stability" generates patterns, invalidates the reductionist view and undermines computer accuracy.
- Structure in phase space — chaotic systems display discernible patterns when viewed.

Multi-point multi-role, multi-location supply chain systems are nonlinear (In mathematics, a **nonlinear **system's output is not directly proportional to its input), dynamic and chaotic. The word chaos may conjure up all sorts of unpleasant situations, but in complexity theory, it has a very specific definition and rather pleasant outcome. Chaos is not random at all, it just appears to be, and applying the techniques of Chaos Theory, it is possible to find the things that actually control the system.

## Chaos and supply chain management

Complex systems are complex structures in time or space, which disguise simple deterministic rules. According to the theory, once these rules are found, one can make reasonable predictions and even control the complexity. Chaos self-organizes with the appearance of the ‘strange attractor' — the state of a mathematically chaotic system toward which the system trends — and offers avenues for creativity and innovation. In the self-organized state of chaos, participants are not locked into rigid roles and gradually develop their capacity for differentiation and relationships, growing progressively towards their maximum potential contribution to the organization's efficiency.

Three canonical models for Supply Chain Management are:

- The Management Model
- The Deterministic Queuing Model
- The Preemptive Queueing Model with Delays

These have been researched for decades as components of a supply chain system.

The Management Mode allocates resources to its Production and Marketing Departments, in accordance with shifts in inventory and/or backlog. It has four level variables — resources in production, resources in sales, inventory of finished products and number of customers. The rate of production is determined from resources in production through a nonlinear function, which expresses a decreasing productivity of additional resources as production approaches maximum capacity. The whole system is controlled by two interacting negative feedbacks.

These subsystems all display elements of chaos.

System complexity can be classified as (1) structural complexity related to the uncertainty of a static system, and (2) operational complexity associated with the uncertainty of a dynamic system. Systems with a higher degree of uncertainty will have a higher level of complexity.

A classical supply chain is often complicated and includes the following elements:

- Mass amounts of information, goods, and capital flowing among suppliers, manufacturers, and distributors.
- Supply chain members may also be members of other supply chain networks.
- Constantly changing network structure.
- Supply chain members have their own goals.

Various supply chain members can simultaneously interact with one another in multiple channels via multiple information flows and logistics, making the entire network a ‘complicated' system, meaning, it exhibits features of a complex adaptive system.

To understand how to deal with complex non-linear systems, a little background in Chaos Theory is helpful. Chaos Theory explains underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization within the apparent randomness of complex, chaotic systems. Chaos Theory is used to explain complex systems such as weather, astronomy, politics, and economics. Although many complex systems appear to behave randomly, Chaos Theory shows that, in reality, there is an underlying order, but it is difficult to see, to control and predict.

## In search of the Strange Attractor

There is an essential contradiction in Chaos Theory — attempting to predict the behavior of things that are unpredictable. A classic explain is the Three-Object Paradox discovered by Henri Poincaré, a French mathematician. In 1887. Poincaré demonstrated that Newton's Laws of Gravity predict two planets' mutual attraction, but adding a third made the equations unsolvable. Chaos Theory was able to unravel the motion many years later.

So how is this useful in managing complex supply chains? The mathematics of Chaos Theory can untangle the hidden structure of a chaotic system. The method employed is finding the set of behaviors known as its Strange Attractor, which was described above. A more formal definition is:

An

attractorcan be a point, a finite set of points, a curve, a manifold, or even a complicated set with a fractal structure known as astrange attractor... Describing theattractorsofchaoticdynamical systems has been one of the achievements ofchaos theory.

Jonathan Borwein and Michael Rose give an example in an article in The Explainer: What Is Chaos Theory:

Modern fighter jets achieve great maneuverability by virtue of being aerodynamically unstable — the slightest nudge is enough to drastically alter their flightpath. Consequently, they are equipped with on-board computers which constantly and delicately adjust the flight surfaces to cancel out the unwanted butterfly effects, leaving the pilot free to exploit his own. If you can tease out the pattern's underlying chaotic systems, you can effect a measure of control over randomness and turn instability into an asset by exploiting its ‘strange attractor.

A novel approach to supply chain management in organizations is the application of ‘chaotic systems theory' developed originally in physics and mathematics. Supply chain management is on the edge of chaos where innovation occurs, and chaotic social systems can no longer be effectively managed.

Despite its name, an attractor is not some sort of force or objective function; it is a representative model of the system's behavior. It merely depicts where the system is heading based on its motion rules. In supply chain management organizations that cultivate or share values of autonomy, responsibility, independence, innovation, creativity, and proactivity, the risk of short-term chaos is mitigated by external complexities that organizations are currently confronting.

Evaluating a complex supply chain involves some high-level mathematical concepts but some ordinary calculus and linear algebra. One popular approach is to apply Information Theory. According to Information Theory, the entropy of a random variable is the average level of "information," "surprise," or "uncertainty" inherent in the variable's possible outcomes. You can use entropy functions to measure uncertainty, probability, and fortuity. The math is straightforward, but the process is rather complicated. Any of your Operations Research or various applied math professionals can work it out. Here is the basic formula:

Supply chains can exhibit complex systems characteristics when external events of great magnitude occur, such as the financial collapse of 2008 or the COVID-19 pandemic of 2020. But they can also become rapidly unbalanced from just minor perturbations — the entry or exit a significant player, the introduction of a new production technique, etc. ‘Gut' or ‘legacy methods' are not adequate for managing these supply chains. Rigid rules-based systems are likewise not agile or fast enough for today's economies.

## My take

Supply chains have evolved from keeping track of things and trying to manage the flow to extremely complex systems that are subject to rapid adjustment across networks of participants. Existing approaches to supply chain management aren't able to apply the use of complexity math to find the attractors and the coherence in what they perceive as an unmanageable system.