System Dynamics Primer

The System Dynamics Approach

The System Dynamics approach was developed in the mid 1950s as a method for analyzing complex dynamical systems, with the aim to better understand how and when systems and policies produce an unexpected and undesirable outcome. The modelling approach carves out the part of the reality that needs to be understood. The usefulness of the System Dynamics approach is the conceptualizaton of a model that can be used for decision making in various areas including business and policy simulation, project management, product development and risk management.

The modelling approach carves out the part of the reality that needs to be understood. Qualitatively, the approach involves identification feedback loops to allow capture of interdependencies between system parts. Systems are modeled as interdependent collections of stocks which represent measurable quantities that describe the state of systems. Flows represent the rate of changes that govern the change in the state variables aka the stocks. Feedback loops differ in term of impact on the dynamics of the system. Positive feedback loops lead to reinforcing behaviour and negative feedback loops lead to balancing (controlling) behaviour of systems. Further details of the elements involved are outlined as follows. 

• Graphical Icons
  The Stock icon is used to represent the quantity of a characteristic system state variable whose value we want to track and model. The stock can only change through inflows and outflows. To exemplify, consider the change of money in a bank account, or the increase of population in a city. The Flow icon is used to represent inflows to the stock and outflows from the stock. The flows are represented by valves to indicate rates of changes. The Connector arrow is used to indicate dependencies between Stock and Flow variables and Auxiliary variables, with dependencies quantified by formulas and tables.

 

• The Stock and Flow diagram
  The stock-flow diagram illustrates the dependencies that describe system behaviour. The diagram shown below displays dependencies for a simple production system consisting of inventory of machines with investments into machinery leading to increase in value and depreciation leading to decrease in value over the courseof time. The reinforcing and balancing feedback loops indicate the dynamical behaviour of this system. Precisely, the positive feedback loop presents increases in the connected stock wherea the negative feedback loop presents decreases in the connected stock. Auxiliary variables are used to establish dependencies for flows of information and material.

 

• The Quantitative Modelling of Flows
  The stock variables are typically defined as a result of the qualitative modeling and the grapical design based on stock-flow diagramms. For a system to model its behaviour the flow in and out of the stocks require quantification. Quantification is achieved through definition of rate changes linked together via differential equations. Integration over rate changes is based on the defining differential equations and takes the form

\[Stock Level _{t} =Stock Level _{t-\Delta t} +\Delta t \cdot (Inflow Rate_{t-\Delta t} - Outflow Rate_{t-\Delta t} )\]

• The Model Building Steps
  The common thread across these examples is that each model addresses a clear problem, and therefore each model has a definite purpose. With a clearly defined goal, a valuable strength of system dynamics is that it is supported by an iterative five-stage methodology. This can be used to structure projects and ensure a process for engaging with clients and problem owners in order to address problems.

Purpose definition: A clear purpose needs to be defined before the building of a System Dynamics model can proceed. As an example consider the supply-chain used by a producer and the purpose is to evaluate distribution strategies and production techniques for enhancing the performance of the supply-chain.

Hypothesis finding and Graphical design: Next is to identify a set-up of stocks, flows and feedback loops that best represent the dependencies within the system to enable further exploration of dynamics. The aim of this step in the model building process is to qualitatively represent the system dynamics based on diagramms and feedback loops.

Model quantification: With a stock and flow diagram in place and flow rates determined, the model quantification follows and a simulation can be performed.

Model Testing: Validation of model behaviour is performed to test any uncertainty of model and parameters in terms of extreme conditions and sensitivity. S

Policy evaluation: A part of the final stage new decision rules and policies are derived providing the model has passed testing and is deemed sufficiently robust. Simulation of what-if scenarios is used to quantify the potential impact of policy changes and actions before actual system changes are implemented.

 

The Quantification of System Dynamics Models

Dynamic Systems that change with time are usually complex due to the many interacting parts involved. The key outcome of the model building stage is the articulation of the model design, with state variables and assumptions spelled out and relationships between variables established via flows of material and information. Of particular practical importance is the right modelling of delays. Delays appear in many disguises, whether it is the planned increase in a firm's production capacity to meet risen demand, or  whether it is the time required to measure and report information and to action on the consequences.  

Few demonstrations of model building are illustrated in the following. 

 

• Using the Logistic Growth Model

Ordinary Differential Equations (ODE) are useful in describing growth phenomena in multiple areas. The logistic equation describes a growth pattern that is commonly observed in a context where competing forces and saturation effects lead to limitations in growth.
To illustrate, consider the growth of company revenues from selling a product until market saturation takes place. For a variable of interest y, its growth is limited by a carrying capacity. Specifically, growth is reduced with higher levels of y due to competing forces that pull the levels towards the carrying capacity. As a result, growth is initially exponential at a growth rate r. However, with increasing levels of y the growth turns negative above a carrying capacity K. The logistic ODE describing this phenomenon is written as. 

\[\normalsize \frac{d\,y}{d\,t} = r \cdot y \cdot (1-\frac{y}{K})\]

Considering two solutions to the logistic equation, each for an initial condition specified by \(y(0)=2\) and \(y=12\). Parameters are specified as \(r=1\) and \(K=10\). Displayed below are two solutions to the logistic equation. For each initial condition the matching solution is shown.

 

• Demonstrating a Limited Growth Model

With this example the effects of a positive and a negative feedback loop are demonstrated for a production process. The context is a production process with machines used to produce some economic output. A fraction from revenues is invested to purchase additional machines to further increase production of economic output.

• The positive feedback loop that describes the growth pattern:
  Any increase of machines leads to an increase in production and growth in revenues.
  Any increase in production and growth in revenues leads to an increase of investment in machines.
  Any increase of investment in machines leads to an increase in production and revenues.

• The negative feedback loop that describes the balancing pattern:
  Any increase in machines leads to an increase in defaulted machines
  Any increase in defaulted machines leads to a decrease in machines

• The number of discarded machines, per unit of time, is proportional to the current level of machines \(N(t)\). The depreciation percentage is given by \(D\)
  as 10% in the simulation.

\[N_d(t) = N(t) \cdot D \]

• The number of new purchased machines is determined by the Revenues(t) times a reinvestment percentage. Revenues are assumed to be proportional to the producion output, which in turn is assumed to be determined by a constant magnitude in labour times the square root of machines used in production. The reinvestment percentage \(I\) has a value of 20%. The inflow of new purchased machines is hence written as.

\[N_p(t) = Revenues(t) \cdot InvPercentage = 100 \cdot \sqrt {N(t)} \cdot I\]

• Any accumulation in the stock of machines above the initial stock of machines \(N(t0)\) is written below. The initial stock of machines \(N(t0)\) is set at value 100 for simulation purposes.

\[N(t) = N(t_0) + \int_{t=0}^t \, (\,N_p(u) - N_d(u)\,) \, du\]

 

• Illustrating the Impact of Delays on Systems

Delays appear naturally in systems and have a significant impact on a system’s behaviour. To illustrate with regard to a business context, consider the time that it takes for a company to reach its target production capacity after demand for products has risen. In this and similar examples any action to respond to a triggering event will not be performed instantaneously but rather after a delay has passed. A typical pattern of a slow-to-respond system is illustrated below. The system’s output overshoots and undershoots several times the target value. Many cycles of corrective actions are required to force the system to achieve the desired equilibrium. The system’s outcome trajectory is displayed below and shows a typical example of a goal-seeking dynamics.

 

The impact of delays can be illustrated with a simple Delay Differential Equation (DDE) written below. For many systems it is desirable that perturbations to the equilibrium lead to system values that remain nearby. The DDE written below has the equilibrium solution \(y(t) = 0\) and adopts a negative feedback that brings the system back to equilibrium. The negative feedback triggers values changes suitably. Consider if \(y(t) \gt 0\) then any feedback \(c(t) \lt 0\) and thus value changes are negative. In other words, a system value \(y(t)\) that is observed at time \(t\) leads to a control via \(-y(t-\tau)\) after some delay time \(\tau\) has passed.

\[\normalsize \frac{d\,y}{d\,t} = - y(t-\tau) \;\;\;,\;\; t \gt 0\]  

To find a solution in the interval \(0≤t≤3τ\)  the method of steps is applied. The initial condition is written as a history of values \(y(t)=k\) for \(t≤0\). The solutions is a polynomial of degree n in each sub-interval \(n=1, 2, ...,n\).