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Last Friday we hosted the first of a four-part web seminar series titled Modeling for Environmental Sustainability. This first session was about how you can use simple bathtub dynamics to understand climate change. In the session, Chris Soderquist demonstrates how you can use embeddable simulations to help raise awareness and understanding about climate change. Using …
This is the third installment of a four-part series. The other three parts can be accessed by clicking on the links below.
Methods for Using Arrays Effectively
Modeling a Watershed with Arrays
Modeling Customers Switching Between Brands – The General Case
In the second post of this series, I showed how to selectively pull information from an array in order to route water through a watershed. In this post, I will use the exact same technique to move customers between different product brands.
Switching Customers between Different Products
Business models often need to model gaining customers from, and losing customers to, competing products in a relatively mature market (what Kim Warren, in his excellent book Strategy Management Dynamics, calls “Type 2 Rivalry”). These are often driven with statistical models developed through market research. For this application, we need a matrix describing the probability of switching from product A to product B each time unit. A sample appears in the table below.
| From\To | A | B | C | D | E |
| A | 0.000 | 0.010 | 0.030 | 0.050 | 0.001 |
| B | 0.030 | 0.000 | 0.050 | 0.070 | 0.020 |
| C | 0.010 | 0.001 | 0.000 | 0.020 | 0.015 |
| D | 0.001 | 0.000 | 0.020 | 0.000 | 0.005 |
| E | 0.001 | 0.005 | 0.020 | 0.045 | 0.000 |
switching probability (units: dimensionless)
To read this table, locate the product the customer is presently using in the left column (say, B). Read across that row (the second row, in this case) until you find the product the customer is switching to (say, C). The number in that cell (in this case, 0.05 or 5%) is the probability the customer will switch from the first product to the second (from B to C) in this time unit. If the model is running in months, as ours is, this table indicates that 5% of customers using product B switch to product C every month.
Of course, the values in the table do not need to be constant. Often each cell will contain a regression equation based on various product characteristics – including market share, marketing effort, product features, and product quality – that evolve over the course of the simulation.
Note the diagonal is zero. This means customers do not switch from one product to the same product.
Note also that the sum in any row cannot exceed 1.0, which represents 100% of the customers using that product. It is quite normal for it to be below 1.0 because we do not include people who are not switching. Some modelers find it easier to always have each row add up to 1.0. If you desire to do this, fill the diagonal with the difference between 1.0 and the sum of the other columns. For example, to do this for product A, replace the top left cell with 1.0 – (0.01 + 0.03 + 0.05 + 0.001) = 0.909 [for you Beatles fans].
This is the second installment of a four-part series. The other three parts can be accessed by clicking on the links below.
Methods for Using Arrays Effectively
Modeling Customers Switching Between Brands
Modeling Customers Switching Between Brands – The General Case
This is the second installment of a multipart series. The first part can be found by clicking here. Part 3 is available here.
In the first post of this series, I showed how to conditionally pull information from an array. In this post, I will extend this concept to show how to route information through an arrayed model. This is especially useful in spatial modeling applications.
Routing Water Through a Watershed
A common ecology application is the modeling of a watershed. Part of such a model will necessarily involve a network of stream or river segments – called reaches – which feed each other. It is desirable to implement this in a way that makes it easy to modify the reach network. Using an explicit stock-flow network makes this very difficult. However, it is relatively straightforward to use arrays of stocks and flows to build an easily configurable network.
Imagine a small watershed broken down into reaches as shown below:
![clip_image001[6]](https://blog.iseesystems.com/wp-content/uploads/2009/09/clip_image0016_thumb.gif "clip_image001[6]")
For our purposes, a new reach will need to be created at every junction point. Therefore, in this example and from a topological point-of-view, it is not strictly necessary to treat reach 4 separately from reach 2 nor reach 5 separately from reach 3, but reaches 2 and 3 must be separate from reach 1. There are, of course, other reasons to separate reach 4 from 2 and reach 5 from 3, for example, slope, channel width, length, etc.
Every reach flows into exactly one other reach at its head, but many reaches can flow into the head of the same reach. This requires a many-to-one representation of the reach network. This is accomplished quite easily with a routing map which, for each reach, contains the number of the reach that this reach flows into. We also need someway to signify the outlet. Since reach numbers start at one, we can use zero to signify the outlet. Using these rules, the above network is completely represented in the following routing map:
| Reach | Flows into |
| 1 | 0 |
| 2 | 1 |
| 3 | 1 |
| 4 | 2 |
| 5 | 3 |
The nice thing about this representation is that it fits nicely into a one-dimensional array where the array index is the reach number and the reach it flows into is the value stored in that array element.
The model itself uses one stock to represent each reach. That stock has one inflow for water entering the reach and one outflow for water leaving the reach: (Download the zipped STELLA model here)
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