Sunday, 13 March 2011

Topics in systems dynamics

The elements of system dynamics diagrams are feedback, accumulation of flows into stocks and time delays.

As an illustration of the use of system dynamics, imagine an organisation that plans to introduce an innovative new durable consumer product. The organisation needs to understand the possible market dynamics in order to design marketing and production plans.
 Causal loop diagrams
Main article: Causal loop diagram

A causal loop diagram is a visual representation of the feedback loops in a system. The causal loop diagram of the new product introduction may look as follows:
Causal loop diagram of New product adoption model

There are two feedback loops in this diagram. The positive reinforcement (labeled R) loop on the right indicates that the more people have already adopted the new product, the stronger the word-of-mouth impact. There will be more references to the product, more demonstrations, and more reviews. This positive feedback should generate sales that continue to grow.

The second feedback loop on the left is negative reinforcement (or "balancing" and hence labeled B). Clearly growth can not continue forever, because as more and more people adopt, there remain fewer and fewer potential adopters.

Both feedback loops act simultaneously, but at different times they may have different strengths. Thus one would expect growing sales in the initial years, and then declining sales in the later years.
Causal loop diagram of New product adoption model with nodes values after calculus

In this dynamic causal loop diagram :

    * step1 : (+) green arrows show that Adoption rate is function of Potential Adopters and Adopters
    * step2 : (-) red arrow shows that Potential adopters decreases by Adoption rate
    * step3 : (+) blue arrow shows that Adopters increases by Adoption rate

 Stock and flow diagrams
Main article: Stock and flow

The next step is to create what is termed a stock and flow diagram. A stock is the term for any entity that accumulates or depletes over time. A flow is the rate of change in a stock.
A flow is the rate of accumulation of the stock

In our example, there are two stocks: Potential adopters and Adopters. There is one flow: New adopters. For every new adopter, the stock of potential adopters declines by one, and the stock of adopters increases by one.
Stock and flow diagram of New product adoption model

The real power of system dynamics is utilised through simulation. Although it is possible to perform the modeling in a spreadsheet, there are a variety of software packages that have been optimised for this.

The steps involved in a simulation are:

    * Define the problem boundary
    * Identify the most important stocks and flows that change these stock levels
    * Identify sources of information that impact the flows
    * Identify the main feedback loops
    * Draw a causal loop diagram that links the stocks, flows and sources of information
    * Write the equations that determine the flows
    * Estimate the parameters and initial conditions. These can be estimated using statistical methods, expert opinion, market research data or other relevant sources of information.[4]
    * Simulate the model and analyse results

In this example, the equations that change the two stocks via the flow are:

 \ \mbox{Potential adopters} = \int_{0} ^{t} \mbox{-New adopters }\,dt

 \ \mbox{Adopters} = \int_{0} ^{t} \mbox{New adopters }\,dt

List of all the equations, in their order of execution in each year, from year 1 to 15:

1) \ \mbox{Probability that contact has not yet adopted}=\frac{\mbox{Potential adopters}}{\mbox{Potential adopters } + \mbox{ Adopters}}

2) \ \mbox{Imitators}=q \cdot \mbox{Adopters} \cdot \mbox{Probability that contact has not yet adopted}

3) \ \mbox{Innovators}=p \cdot \mbox{Potential adopters}

4) \ \mbox{New adopters}=\mbox{Innovators}+\mbox{Imitators}

4.1) \ \mbox{Potential adopters}\ -= \mbox{New adopters }\

4.2) \ \mbox{Adopters}\ += \mbox{New adopters }\

\ p=0.03

\ q=0.4

 Dynamic simulation results

The dynamic simulation results show that the behaviour of the system would be to have growth in Adopters that follows a classical s-curve shape.
The increase in Adopters is very slow initially, then exponential growth for a period, followed ultimately by saturation.
Dynamic stock and flow diagram of New product adoption model

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