You label arrows with a '+' if the goal of the arrow is increasing whenever the source is increasing and vice versa, and with a '-' if the goal is decreasing whenever the source is increasing and vice versa. Then you count the number of '-' in a loop in order to tell me if the loop is reinforcing or balancing. Balancing means "goal seeking". Now look at a loop consiting in a stock and an inflow, that on its part depends of the stock.
I have two questions. First: Let flow=1/(stock*stock). This arrow is certainly labeled by a '-'. The stock is then the third root of 3*time. This increases for ever and is not goal seeking. So this loop, that has one '-' arrow is rather a reinforcing loop than a balancing one.
Second: Normally you label a pipe between an inflow and its stock with a '+'. But this is wrong in general! Take the "school example" of a reinforcing loop, with the capital as stock and the interest as inflow. The inflow depends of the capital and the interes rate. About 30 or 40 years ago, you got about 5% interest rates. Today you only get about 1%. So the interest - that is the inflow - decreases in the last 30 years. But still your capital is increasing. Not so fast as 30 years ago, but anyway.
So, the labeling with '+' and '-' seems to me as rather arbitraily, even though System Dynamics builds up on this concept.
balancing or reinforcing?
balancing of reinforcing
Hi
About the first question.
Flow = 1/ (stock * stock) is a negative arrow arrow from the stock to the flow.
If the stock increases the flow decreases.
But if the flow increases, the corresponding stock increases, so the arrow from
the flow to the stock is a positive arrow, so the loop is a negative loop and has one arrow negative plus the flow that is positive. That the loop is a balancing one is not forbidden in S.D.
Second question:
A flow is always positive, because let us say that there is an inflow of I1, and a corresponding stock S1. If you replace the I1 by an I2, so that I2 > I1, then the corresponding Stock S2 will be greater than the stock S1. So I2 > I1 => S2 > S1. This is what is meant by a positive influence.
Of course if the inflow is decreasing in time, the stock can still be increasing but it is not the same thing, as comparing two options in the same interval of time.
Regards.
J.J. Laublé
About the first question.
Flow = 1/ (stock * stock) is a negative arrow arrow from the stock to the flow.
If the stock increases the flow decreases.
But if the flow increases, the corresponding stock increases, so the arrow from
the flow to the stock is a positive arrow, so the loop is a negative loop and has one arrow negative plus the flow that is positive. That the loop is a balancing one is not forbidden in S.D.
Second question:
A flow is always positive, because let us say that there is an inflow of I1, and a corresponding stock S1. If you replace the I1 by an I2, so that I2 > I1, then the corresponding Stock S2 will be greater than the stock S1. So I2 > I1 => S2 > S1. This is what is meant by a positive influence.
Of course if the inflow is decreasing in time, the stock can still be increasing but it is not the same thing, as comparing two options in the same interval of time.
Regards.
J.J. Laublé
Dear Mr. Laublé
Thanks for your help. To the first topic: That's what I said, the loop is balancing because it contains of 1 negativ and one positiv arrow (if we assume that the inflow is indeed positiv). But the stock grows over all limits. How can a balancing loop grow over all limits?
May be that the answer to the second question is also the answer to the first question. As the flow decreases (but remains greater that zero for ever) the stock increases. So actually the flow is negative influencing too, such that my considered loop is in fact an reinforcing one. That would mean that an inflow that is always greater than zero must not always be positive influencing the stock. Indeed in "Business Dynamics" Sterman doesn't label flows but every other arrows.
As I understood the notation "A --> B positive influenced" B is increasing then and only then if A is increasing. In other words: If and only if the first derivative of A is greater that zero then the first derivative of B is greater than zero.
Regards,
Peter
Thanks for your help. To the first topic: That's what I said, the loop is balancing because it contains of 1 negativ and one positiv arrow (if we assume that the inflow is indeed positiv). But the stock grows over all limits. How can a balancing loop grow over all limits?
May be that the answer to the second question is also the answer to the first question. As the flow decreases (but remains greater that zero for ever) the stock increases. So actually the flow is negative influencing too, such that my considered loop is in fact an reinforcing one. That would mean that an inflow that is always greater than zero must not always be positive influencing the stock. Indeed in "Business Dynamics" Sterman doesn't label flows but every other arrows.
As I understood the notation "A --> B positive influenced" B is increasing then and only then if A is increasing. In other words: If and only if the first derivative of A is greater that zero then the first derivative of B is greater than zero.
Regards,
Peter
balancing loops
Hi Peter.
I have joined a tiny model that represents the problem.
You will first see that I am obliged to join a special variable to correct the unit inconsistency in your formulation. It may be then assumed that your example is not a real life example, otherwise unit consistency would be correct.
It seems true that mathematically a negative loop can have no goal, without converging towards a stable state. I should look into Sterman’s book about the exact formulation. It seems to me that in general negative loops have a stabilizing effect. But in that case Bob or an administrator will certainly have a better answer than mine.
I do not think that the second question is related to the first one.
The exact definition of positive influence is:
if Yn = f(Xn,a,b,c,d,etc…) meaning that the variable depends at TIME n,
I insist on the fact that it is in the same n interval of time, on the value of X at the SAME TIME n, and on other variables a,b,c,d etc…
then
xn < Xn => yn = f(xn,a,b,c,d,etc…) will be less then Yn, all the other variables staying the same, at the same time n.
So the increase or decrease concerns the same time period, and not successive time periods.
The derivative you talk about is a derivative where the variable is time.
In my example Time does not vary, and the variable that varies is Xn and not the time.
You mix up variation of a variable upon time, and variation of a depending variable in the same period, time being fixed, from another variable that can increase or decrease, always in the same period of time.
There is indeed the possibility to misinterpret the concept of positive influence.
Regards.
J.J. Laublé
I have joined a tiny model that represents the problem.
You will first see that I am obliged to join a special variable to correct the unit inconsistency in your formulation. It may be then assumed that your example is not a real life example, otherwise unit consistency would be correct.
It seems true that mathematically a negative loop can have no goal, without converging towards a stable state. I should look into Sterman’s book about the exact formulation. It seems to me that in general negative loops have a stabilizing effect. But in that case Bob or an administrator will certainly have a better answer than mine.
I do not think that the second question is related to the first one.
The exact definition of positive influence is:
if Yn = f(Xn,a,b,c,d,etc…) meaning that the variable depends at TIME n,
I insist on the fact that it is in the same n interval of time, on the value of X at the SAME TIME n, and on other variables a,b,c,d etc…
then
xn < Xn => yn = f(xn,a,b,c,d,etc…) will be less then Yn, all the other variables staying the same, at the same time n.
So the increase or decrease concerns the same time period, and not successive time periods.
The derivative you talk about is a derivative where the variable is time.
In my example Time does not vary, and the variable that varies is Xn and not the time.
You mix up variation of a variable upon time, and variation of a depending variable in the same period, time being fixed, from another variable that can increase or decrease, always in the same period of time.
There is indeed the possibility to misinterpret the concept of positive influence.
Regards.
J.J. Laublé
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Dear Mr. Laublé
Oh yes, I see what you mean. I indeed mixed up the variables. I also found Sterman's definition of arrow polarity. Unfortunately, he gives the definition before he intoduces stocks and flows. Nevertheless he write a few sentences about the pitfalls of flow polarity. But his definition of link polarity is not very helpful in case of flow pipes. Sterman defines the polarity of A --> B as positiv if an only if dB/dA>0 (he notated it as partial differential, but we assume here that B depends only by A and t as well). For a flow pipe he define B = Intgral (AdA) + an initial value of B. But this is no condition like dB/dA>0. And I expect that a condition is universal, so dB/dA>0 shuld be sufficient. If I apply it on our situation - the stock is s and the flow f is 1/(s*s) - then the stock becomes s=1/(sqrt f) and hence ds/df = -0.5/sqrt(s*s*s) and this is everytime smaller than zero if the stock is greater than zero such that the flow had indeed negative polarity.
In fact it's confusing!
Regards, Peter
Oh yes, I see what you mean. I indeed mixed up the variables. I also found Sterman's definition of arrow polarity. Unfortunately, he gives the definition before he intoduces stocks and flows. Nevertheless he write a few sentences about the pitfalls of flow polarity. But his definition of link polarity is not very helpful in case of flow pipes. Sterman defines the polarity of A --> B as positiv if an only if dB/dA>0 (he notated it as partial differential, but we assume here that B depends only by A and t as well). For a flow pipe he define B = Intgral (AdA) + an initial value of B. But this is no condition like dB/dA>0. And I expect that a condition is universal, so dB/dA>0 shuld be sufficient. If I apply it on our situation - the stock is s and the flow f is 1/(s*s) - then the stock becomes s=1/(sqrt f) and hence ds/df = -0.5/sqrt(s*s*s) and this is everytime smaller than zero if the stock is greater than zero such that the flow had indeed negative polarity.
In fact it's confusing!
Regards, Peter