Modelling the Milk Glass Game

My Process

by Nick Louca, July, 2009

In this project, I model the Milk Glass Game in the iThink programming language.  You can find the Milk Glass Game in the EcoNowMics pages. See the link, above.

I go through 3 challenges and lots of confusion while I develop the Milk Glass Game in iTHink.  I also have a number of AHA’s and key lessons about model building and system dynamics principles during the process.  I share my insights here for the reader as my process evolves.

First Challenge – Positive or Negative Feedback Loop?

I face the first challenge before I even start building the model in iThink while I go through the EcoNowMics pages. 

I look at the model and see a target so I think that it is a negative feedback system.  I know from reading Road Maps materials that filling a level to a target implies a goal and a gap, two of the basic components that make up a negative feedback loop.  I take another look at the system structure.  There is a milk flow rate into the level and the milk flow rate is increasing the level.  This seems positive so I start to feel some confusion.

Throughout my studies I do not recall a negative feedback system structure with a rate inflow into the level.  I am under the impression that negative feedback systems only contain rate outflows from the level.

I carry on.  I experiment with filling the glass with milk on Ed’s Milk Glass Game on the EcoNowMics pages.  As I fill the glass to the target line I notice that I act to minimize the discrepancy between the level and the target line.  This is negative feedback - however I am still not clear on how I can have negative feedback with the level increasing.  I recall that positive feedback has exponential growth in which the rate increases the level.

I find that I do not really know the difference between positive feedback and negative feedback.  I decide to do some research to resolve this question.

The Milk Glass Game System Structure

I go to the Road Map papers and find a negative feedback model with a rate inflow into the level.  I read through it carefully in order to better understand how the system structure behaves.  The model I find (the Widgets Model, below) has a rate inflow into the level and exhibits asymptotic growth as its approaches its target.  I have my first insight.  A system with a rate inflow into the level, a goal and a gap can also be a negative feedback system.

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The Widgets Model

Clip: Road Maps, Beginner Modeling Exercises, Section 3,

Mental Simulation of Simple Negative Feedback, Pg 15

I notice in Chapter 3, ‘Negative Feedback Structure’ of Study Notes in System Dynamics by Michael R. Goodman, another system structure with a rate inflow into the level that looks similar to the Milk Model structure.  The chapter outlines the basic elements of negative feedback systems.  These are: the target (goal), the discrepancy (gap), the action (rate) and the system state (level).  While I feel that I am getting more understanding on negative feedback loops, I also notice that I am still not clear on the difference between positive and negative feedback loops.

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Negative Feedback Structure

Study Notes in System Dynamics, Michael R. Goodman, Chapter 3, Pg 38

Ed senses my confusion about the difference between positive and negative feedback loops.  We have a discussion about the difference between the two.  I describe negative feedback systems to Ed as systems that have a goal and a gap in their system structure.  I also mention that the behavior negative feedback systems exhibit is either asymptotic growth or exponential decay.  I describe positive feedback system structure as a structure that consists of a level, a rate and a constant and that positive feedback exhibits exponential growth.

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Positive Feedback Structure

Study Notes in System Dynamics, Michael R. Goodman, Chapter 3, Pg 40

Ed tells me that both positive and negative feedback loops have a level, a reference level (target, goal) and a delta (gap) that results from comparing the level and the reference level.  This is new to me.

System Structure – Showing Policy

I do not recall thinking about positive and negative feedback systems in this way before.  Ed tells me that the main difference between positive and negative feedback systems is that negative feedback systems act to decrease the delta while positive feedback systems act to increase the delta.

Key Lesson 1

Negative feedback loops act to decrease the delta and positive feedback loops act to increase the delta.

Second Challenge – Units of Measure

I am now clear on the difference between positive and negative feedback loops so I decide to build the Milk Glass Game (Version 1, below) in iThink as per the format of the Widgets Model in Road Maps.  I make sure to include the basic elements of negative feedback systems in my model.  I run the system and get the asymptotic behaviour I see in the Excel spreadsheet.

I choose a dt value of 0.1 seconds. Note: Dt (delta time) is the time interval between calculations. This is the same value as in the Excel simulation of the Milk Glass Game.  Thus, I re-compute all the system variables every 0.1 seconds. A small dt generates a more accurate simulation and it also requires more computations. A dt value of about 1/5 of the shortest "time constant" in the system is generally small enough. In real-life, everything influences everything else continually, in the ever-evolving moment of now; in real-life dt becomes the zero-width moment of now. Choosing a very very small dt for a model can make a model very very accurate and never exactly accurate.  I mention "time constant" later in this article: 1  2.

Notes:

1. Dt is the delta time between simulation computations; the other "delta" in this model is the gap between the target (goal) and the actual milk level.

2. "Cause and Effect" models generally attempt to link events that have very long time intervals between them.  Such linkages do not and cannot actually exist. We cannot model them with system dynamics.  In System Dynamics, we do not attempt to make "Cause and Effect" claims such as "Since the empty glass occurs first, the empty glass obviously causes the full glass."  You can find more on "Cause and Effect" in the EcoNowMics pages.

 The Milk Glass Game System Structure Version 1.

I present the model to Ed and we review it together.  Ed questions me on what the Milk Inflow Fraction represents.  Is it a fraction of time?  Is it a fraction of milk pouring into the rate which then pours into the level?  Is this how filling a glass of milk works in reality?  I conclude that the Milk Inflow Fraction {cc/sec} does not represent anything real even though I get the desired asymptotic behaviour.  Also I realize that the units do not match.

Ed and I have a discussion about the importance of unit consistency in model building.  Ed tells me that if the units of measurement are inconsistent then the whole model is inconsistent.  Ed also tells me that unit inconsistency is a vital clue that something in the model is not right.  Ed advises me to take another look at the Milk Glass Game on the EcoNowMics page and rearrange the model according to what I see.

Key Lesson 2

Always make sure that units of measure in the model are consistent

and make sense.

I review the Milk Glass Game and the system structure again.  I also go through the Milk Model in the Excel spreadsheet to get a better understanding of the units of measure.  The Excel spreadsheet has a Time Constant instead of Milk Inflow Fraction that I use in my first version of the Milk Glass Game.  The Time Constant represents the time it takes to fill the milk glass to the target level in a system where there is no feedback.  In this system the Time Constant and the delta (gap) both determine the flow rate.  The unit of measure for the Time Constant in the Milk Glass Game is seconds.

I revise the model (version 2, below), replace the Milk Inflow Fraction with a Time Constant and make the necessary adjustments to the model equations.  I run the model again and get the same behavior as I get in the previous model which also reconciles with the Excel simulation of the Milk Glass Game.  In addition, all the units of measure make sense and are consistent with each other.

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The Milk Glass Game System Structure Version 2

I notice that something is still missing from my structure when I compare it to Ed’s structure on the EcoNowMics page.  I do not have a “Policy” in my model.  I am not clear about the meaning of the word Policy in a System Dynamics context.

Third Challenge – Define the Policy

I present the model to Ed and we discuss the second version of my iThink Model.  Ed asks me if I really use a Time Constant when I fill a glass of milk.  The answer is No.  Ed asks me to define the location and operation of the Policy in this model.  I draw a blank.  I realize I am not clear on the word Policy.

Ed asks me to go back to the Milk Glass Game and simulate filling the glass.  As I start to fill the glass Ed asks me what I notice about what I am doing as I fill the glass.  I notice that as I see the milk flowing into the glass I use this information to regulate the position of the button.

As the milk glass fills up and approaches the white target line I gradually release the button to reduce the flow of milk into the glass.  I have an AHA.  My hand and eyes are part of the system and part of the feedback loop for filling the glass!

My Policy for managing the valve completes and closes the feedback loop.  Ed explains the meaning of Policy and how to use it in a System Dynamics context.  The policy acts as a transducer that receives information about the level and controls the rate.

Key Lesson 4

A policy is a plan of action that receives information from a level and controls a rate. Complex policies may receive information from many levels and control many rates.

During the process of my discussion with Ed I have another AHA for model building.

Key Lesson 5

Be sure to locate and define model policies.  Make sure that polices correspond to reality.

The sixth key lesson I receive comes from Thinking in Systems by Donella H. Meadows.

Key Lesson 6

The least obvious part of the system, its function or purpose is often the most crucial determinant of system behaviour.

In the case of the Milk Glass Game the least obvious parts of the system, my nervous system, eyes and muscles are all part of my policy.

Success

I make my third attempt to build the model (version 3, below) as per my previous discussion with Ed on the meaning of “Policy”.  I am now clear on the term Policy and the role it plays in the feedback loop.  Ed also demonstrates how a valve works by filling a glass of water from a water faucet.  As the valve handle rotates, the flow rate into the glass increases or decreases accordingly.  The valve handle position (degrees) associates with the flow (cc/sec) so the valve transfer function (TF) is in cc/sec/degree.

I use millimeters (mm) of milk to represent the level and mm to represent the button position in version 3 of the Milk Glass Game model.  I use millimetres instead cubic centimetres as this is what I see on my 17" monitor as I fill the milk glass.  I respond to the height of the milk in the milk glass while I fill the glass (mm) by controlling the button position (mm).  The TF for the Policy has units of mm/mm = 1.  The TF unit for the button itself is in mm/sec ÷ mm or 1/sec.

Note: The total control path from the level to the rate has a TF with units of 1 * 1/sec = 1/sec where the number of seconds represents the “Time Constant.” 

I figure out how to implement the Policy function in iThink and include the details about what actually takes place within the Policy.  I run the model and notice that the behavior I see is the same as that in the Excel model. 

The Milk Glass Game System Structure Version 3 

Showing Policy as a Block

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The Milk Glass Game System Structure Version 3

Showing Policy Detail

Note: Some system builders who are familiar with this type of structure may condense the policy and valve function into one shorthand variable, the Time Constant.

Model System Structure in “Time Constant” Formulation

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