Category: SystemDynamics

In a breakout in the student colloquium at ISDC 2022, we discussed the difficulty of getting a paper accepted into the conference, where the content was substantially a discrete event or agent simulation. Readers may know that I’m not automatically a fan of discrete models. Discrete time stinks. However, I think “discreteness” itself is not the enemy – it’s just that the way people approach some discrete models is bad, and continuous is often a good way to start.

On the flip side, there are certainly cases in which it’s sensible to start with a more granular, detailed model. In fact there are cases in which nonlinearity makes correct aggregation impossible in principle. This may not require going all the way to a discrete, agent model, but I think there’s a compelling case for the existence of systems in which the only good model is not a classic continuous time, aggregate, continuous value model. In between, there are also cases in which it may be practical to aggregate, but you don’t know how to do it a priori. In such cases, it’s useful to compare aggregate models with underlying detailed models to see what the aggregation rules should be, and to know where they break down.

I guess this is a long way of saying that I favor a “big tent” interpretation of System Dynamics. We should be considering models broadly, with the goal of understanding complex systems irrespective of methodological limits. We should go where operational thinking takes us, even if it’s not continuous.

This doesn’t mean that everything is System Dynamics. I think there are lots of things that should generally be excluded. In particular, anything that lacks dynamics – at a minimum pure stock accumulation, but usually also feedback – doesn’t make the cut. While I think that good SD is almost always at the intersection of behavior and physics, we sometimes have nonbehavioral models at the conference, i.e. models that lack humans, and that’s OK because there are some interesting opportunities for cross-fertilization. But I would exclude models that address human phenomena, but with the kind of delusional behavioral model that you get when you assume perfect information, as in much of economics.

I think a more difficult question is, where should we draw the line between System Dynamics and model-free Systems Thinking? I think we do want some model-free work, because it’s the gateway drug, and often influential. But it’s also high risk, in the sense that it may involve drawing conclusions about behavior from complex maps, where we’ve known from the beginning that no one can intuitively solve a 10th order system. I think preserving the core of the SD genome, that conclusions should emerge from replicable, transparent, realistic simulations, is absolutely essential.

Related:

Discrete Time Stinks

Dynamics of the last Twinkie

Bernoulli and Poisson are in a bar …

Modeling Discrete & Stochastic Events in Vensim

IM PRETTY SURE THIS IS THE FURST EVAH SYSTEM DYNAMICZ SIMULASHUN MODEL WRITTEN IN LOLCODE.

HAI 1.2
    VISIBLE "HAI, JWF!"
    
    OBTW
     ==========================================================================
     SYSTEM DYNAMICZ INVENTORY MODEL IN LOLCODE
     TOM FIDDAMAN, METASD.COM, 2021
     INSPIRED BY THE CLASSIC BEER GAME
     AND MODEL 3.10 OF MICHAEL GOODMAN'S 
     'STUDY NOTES IN SYSTEM DYNAMICS'
     ==========================================================================
    TLDR
    
    BTW FUNKTION 4 INTEGRATIN STOCKZ WITH NET FLOW INOUT
    HOW IZ I INTEGRATIN YR STOCK AN YR INOUT AN YR TIMESTEP
        FOUND YR SUM OF STOCK AN PRODUKT OF INOUT AN TIMESTEP
    IF U SAY SO
    
    BTW FUNKTION 4 CHARACTER PLOTZ
    HOW IZ I PLOTTIN YR X AN YR SYMBOL
        I HAS A STRING ITZ ""
        I HAS A COUNT ITZ 0
        IM IN YR XLOOP
            BOTH SAEM COUNT AN BIGGR OF COUNT AN X, O RLY?
                YA RLY, GTFO
                NO WAI, STRING R SMOOSH " " STRING MKAY
            OIC
            COUNT R SUM OF COUNT AN 1
        IM OUTTA YR XLOOP
        VISIBLE SMOOSH STRING SYMBOL MKAY
    IF U SAY SO
    
    BTW INISHUL TIME - DEKLARE SUM VARIABLZ AND INIT STOCKZ

    I HAS A INV ITZ 0.0         BTW INVENTORY (WIDGETS)
    I HAS A MAKIN               BTW PRODUCTION RATE (WIDGETS/WEEK)
    I HAS A SELLIN              BTW SALES RATE (WIDGETS/WEEK)
    I HAS A TIME ITZ 0.0        BTW LOL I WISH (WEEK)
    I HAS A TIMESTEP ITZ 1.0    BTW SIMULATION TIME STEP (WEEK)
    I HAS A ZEND ITZ 50.0       BTW FINAL TIME OF THE SIM (WEEK)
    I HAS A TARGET ITZ 20.0     BTW DESIRED INVENTORY (WIDGETS)
    I HAS A ADJTIME ITZ 4.0     BTW INVENTORY ADJUSTMENT TIME (WEEK)
    I HAS A ORDERIN             BTW ORDER RATE (WIDGETS/WEEK)
    I HAS A INIORDERS ITZ 10.0  BTW INITIAL ORDER RATE (WIDGETS/WEEK)
    I HAS A STEPTIME ITZ 30.0   BTW TIME OF STEP IN ORDERS (WEEK)
    I HAS A STEPSIZE ITZ 5.0    BTW SIZE OF STEP IN ORDERS (WIDGETS/WEEK)
    I HAS A INVADJ              BTW INVENTORY ADJUSTMENT NEEDED (WIDGETS)
    I HAS A WIP ITZ 0.0         BTW WORK IN PROGRESS INVENTORY (WIDGETS)
    I HAS A SHIPPIN             BTW DELIVERIES FROM WIP (WIDGETS/WEEK)
    I HAS A PRODTIME ITZ 4.0    BTW TIME TO PRODUCE (WEEK)
    
    VISIBLE "SHOWIN RESULTZ FOR PRODUKSHUN"
    
    IM IN YR SIMLOOP        BTW MAIN SIMULASHUN LOOP
        
        BTW CALCULATE RATES AND AUXILIARIES
        
        BTW STEP IN CUSTOMER ORDERS
        BOTH SAEM TIME AN BIGGR OF TIME AN STEPTIME, O RLY?
            YA RLY, ORDERIN R SUM OF INIORDERS AN STEPSIZE
            NO WAI, ORDERIN R INIORDERS
        OIC
        
        SELLIN R SMALLR OF ORDERIN AN QUOSHUNT OF INV AN TIMESTEP
        INVADJ R DIFF OF TARGET AN INV
        MAKIN R SUM OF SELLIN AN QUOSHUNT OF INVADJ AN ADJTIME
        MAKIN R BIGGR OF MAKIN AN 0.0
        SHIPPIN R QUOSHUNT OF WIP AN PRODTIME
        
        BTW PLOT
        VISIBLE SMOOSH TIME " " MAKIN MKAY
        BTW PRODUKT WITH SCALE FACTOR FOR SIZING
        I IZ PLOTTIN YR PRODUKT OF MAKIN AN 4.0 AN YR "+" MKAY
                
        BTW INTEGRATE STOCKS
        
        TIME R I IZ INTEGRATIN YR TIME AN YR 1.0 AN YR TIMESTEP MKAY
        INV R I IZ INTEGRATIN YR INV AN YR DIFF OF SHIPPIN AN SELLIN AN YR TIMESTEP MKAY
        WIP R I IZ INTEGRATIN YR WIP AN YR DIFF OF MAKIN AN SHIPPIN AN YR TIMESTEP MKAY
        
        BTW CHECK STOPPING CONDISHUN
        BOTH SAEM TIME AN BIGGR OF TIME AN SUM OF ZEND AN TIMESTEP, O RLY?
            YA RLY, GTFO
        OIC
        
    IM OUTTA YR SIMLOOP
    
    
KTHXBYE

YOU CAN RUN IT IN THE TUTORIALSPOINT ONLINE INTERPRETER, OR GET JUSTIN MEZA’S DESKTOP LCI.

SD INVENTORY LOLCODE.TXT

$lci main.lo
HAI, JWF!
SHOWIN RESULTZ FOR PRODUKSHUN
0.00 5.00
                    +
1.00 5.00
                    +
2.00 5.93
                        +
3.00 6.64
                           +
4.00 7.34
                              +
5.00 8.00
                                 +
6.00 8.62
                                   +
7.00 9.22
                                     +
8.00 9.78
                                        +
9.00 10.31
                                          +
10.00 10.82
                                            +
11.00 11.30
                                              +
12.00 11.75
                                                +
13.00 12.18
                                                 +
14.00 12.46
                                                  +
15.00 12.30
                                                  +
16.00 12.03
                                                 +
17.00 11.68
                                               +
18.00 11.29
                                              +
19.00 10.89
                                            +
20.00 10.51
                                           +
21.00 10.17
                                         +
22.00 9.89
                                        +
23.00 9.66
                                       +
24.00 9.49
                                      +
25.00 9.39
                                      +
26.00 9.35
                                      +
27.00 9.35
                                      +
28.00 9.40
                                      +
29.00 9.47
                                      +
30.00 14.56
                                                           +
31.00 15.91
                                                                +
32.00 14.12
                                                         +
33.00 14.07
                                                         +
34.00 14.45
                                                          +
35.00 14.73
                                                           +
36.00 15.01
                                                             +
37.00 15.27
                                                              +
38.00 15.51
                                                               +
39.00 15.75
                                                                +
40.00 15.97

I THINK THIS SHOULD BE A PART OF EVERY SYSTEM THINKERZ LITTERBOX TOOLBOX.

There’s a lot of art to modeling, and more generally to managing complex systems. But there’s also some craft to it: simple, mechanical steps that must be followed, almost without exception.  Woodworkers know that when you’re using a chisel or plane, you cut with the grain, not across it. Knowing that isn’t sufficient to make a nice-looking chair, but at least your funny-looking chair won’t have ugly tearout.

So what are the rules for classic System Dynamics? Here are a few:

1. Unbalanced or missing units. It’s possible to build a correct model without units, but most people (including me) are unlikely to manage it. Even if the model is right in some sense, without units it’s still unintelligible to others.
2. No FONFOO. Every physical stock needs First-Order Negative Feedback On the Outflows. This means the equations ensure that the outflow goes to 0 as the stock goes to 0 – not after a while, but now and forever. This ensures conservation of stuff: no inventory -> no sales. Nonphysical stocks often require this treatment as well, unless negative values are permitted by definition.
3. Embedded parameters. A colleague just found an equation in a spreadsheet model reading something like =A2*EXP(-C4/C1) + 4. The “4” was just an arbitrary fudge factor on the answer. This should never happen; anything more complex than the 1 in 1/x should always be exposed as a distinct, named variable with appropriate units.
   • Corollary: the embedded parameter often represents an implicit goal. For example, in inventory adjustment = (1000-inventory)/inventory adj time, the goal of 1000 units should be made explicit.
4. Discrete time. Generally, your model should be independent of the TIME STEP and simulation method. Decision rules should integrate information smoothly, not at arbitrary point lags.
5. Discrete logic. Sometimes I see equations that involve big cascades of logical statements: IF THEN ELSE( inventory < 100 :AND: price > 2, do x, IF THEN ELSE( inventory > 200 :AND: expected sales > inventory/desired coverage, do y, IF THEN ELSE( …  Constructions like this are hard to read and hard to debug, and they often fail important reality checks. They might be appropriate in tactical cases where reality has discernible, discrete rules. But they’re seldom helpful in strategic models involving the aggregate behavior of many agents or objects.
6. Overuse of delays. Every feedback loop must include a stock. This is a consequence of “time is what keeps everything from happening at once.” If there’s no integration in a loop, then feedback would run infinitely fast. Sometimes, confronted with an apparently simultaneous loop, modelers just insert a SMOOTHI or similar function that contains a stock. This may not be good enough; the stock in the loop can’t be arbitrary; it has to have real meaning.
   • It’s also possible to commit the opposite sin: underuse of delays. Perceptions lag reality, and people often underestimate the extent to which this is true. Decision rules in your model should reflect this, but I think it’s more a matter of art than craft.
7. Taking the cream out of the coffee. Suppose you have a stock of people, with a coflow of money used to keep track of the average wealth of people in the stock. It’s then tempting to handle a thought experiment like, “ok, what if all the rich people leave the country?” by siphoning off a greater-than-average share of the money alongside each departing person. This violates the assumption that a stock is the complete representation of system state. What if, for example, the rich people already left, so that the remainder are uniformly poor? If the distinction is important, you simply must disaggregate the people into classes.

Like all rules, these are made to be broken, but exceptions are rare, and require that you really know what you’re doing. They are important because they ensure compliance with Reality Checks that should remain inviolate for strong reasons. If your population model isn’t conserving people, you have a problem.

Incidentally, at least half of these are mentioned in Appendix O of Industrial Dynamics, “Beginners’ Difficulties.” However, these are not just tricks for beginners: everyone can benefit from keeping them in mind, just as professional pilots rely on checklists.

I’m eager to hear your thoughts in the comments. What rules did I miss?

See also:

How to critique a model (and build a model that withstands critique)

Towards Principles for Subscripting in Models

Dynamics of the last Twinkie

Misadventures with Little’s Law

* Update: edited slightly for parallelism of the headers.

I’ve been approached several times recently with questions about stocks behaving badly. All involved a construction something like the following:

This is a simple inventory control system, in which I’ve short-circuited the production start feedback by making Starting exogenous and equal to the desired sales rate. Therefore, there are really only two interesting equations:

Shipping=desired sales rate
Units: widgets/Month

Completing=DELAY3( Starting, production time )
Units: widgets/Month

Notice that there’s a violation of standard practice here, in that there’s a flow-to-flow connection from Starting to Completing. This is due to the DELAY3 function, which is shorthand for an explicit 3rd-order delay:

The 3rd-order delay is often a realistic compromise between a 1st-order system, in which the first completions arrive too quickly after Starting, and a pipeline delay or conveyor, which has too little dispersion to represent an aggregation of many items. (See the Delay Sandbox and Erlang models for examples.)

So, how can we break this model?

I always like to start with some tests in Synthesim. A good one is to stress the system with a step in the desired sales rate, here from 100 to 120. You can immediately spot a problem:

Inventory goes negative, because Shipping proceeds, even when inventory is exhausted. That can’t happen in reality, but it happens here because Shipping is not a function of Inventory. There’s a simple fix:

Shipping=MIN(desired sales rate, Inventory/min shipping time)
Units: widgets/Month

Above, min shipping time is a time constant representing the minimum time needed to deplete inventory. It’s common to set min shipping time = TIME STEP in situations where you want to prevent negative inventory, and the precise dynamics of inventory exhaustion are not central to the model. (If it matters, see Dynamics of the Last Twinkie.)

This is FONFOO. The “first order negative feedback” refers to the balancing loop created by the Inventory/min shipping time term in the fixed equation:

The tricky thing about this situation is that if Starting had been endogenous, the negative inventory problem would have been much harder to spot. Here’s the same model with a simple decision rule for Starting that maintains Inventory and WIP and desired levels:

Now, a modest step in sales doesn’t cause negative inventory, as long as the production process can replenish it in time. It takes a huge step (from 100 to 400 widgets/month) to reveal the problem:

This means that experiments on a model as a whole may not reveal problems that lurk in the details of the model, unless they’re quite extreme. I recommend extreme tests, but prevention is more important. Simply make it a habit to implement FONFOO everywhere, and you won’t have problems. (Note that we could automate this in Vensim, but we don’t, because doing so can easily mask other formulation problems, fall short of the control that’s really needed, or impede situations in which nonphysical stocks are intentionally negative.)

Now let’s take a look at the 3rd-order production delay surrounding WIP. As presented above, it works fine – it’s mathematically equivalent to the explicit 3rd-order aging chain. However, there are consistency issues to be aware of. Consider the following augmentation of the structure, representing stock losses (the flow of Breaking) from WIP:

Completing=DELAY3( Starting, production time )
Units: widgets/Month
Breaking=DELAY3(Starting*loss fraction,production time)
Units: widgets/Month

Completing is still a delayed function of Starting. But Completing is not directly aware of WIP and therefore unaware of the consequences of Breaking. This is a violation of FONFOO because the DELAY3 function contains internal states that are independent of the WIP stock. Consider what happens if the loss fraction is nonzero. In equilibrium, the output of DELAY3 is equal to the inflow. So, the outflow from WIP would be Breaking+Completing, which equals Starting+Starting*loss fraction, which is of course greater than starting for any nonzero loss.

A step in the loss function from 0 to 0.2 causes WIP to go negative:

Again, the remedy is simple. In most cases, you can keep the DELAY function if you ensure that the inflows and outflows are conserved. For example, adding a term:

Completing=DELAY3( Starting*(1-loss fraction), production time )
 Units: widgets/Month
 Breaking=DELAY3(Starting*loss fraction,production time)
 Units: widgets/Month

In some situations, it may be desirable to switch to an explicit aging chain in order to handle an idiosyncratic distribution of losses across the WIP process, or other complexities. Often arrays are useful for such purposes.

You may encounter the DELAY1 function in similar circumstances. DELAY1 is just like DELAY3, except that it’s first order. So, the system:

inflow = 10 ~ widgets/month
stock = INTEG(inflow-outflow, inflow*tau) ~ widgets
outflow = stock/tau ~ widgets/month
tau = 6 ~ months

is identical to the system:

inflow = 10 ~ widgets/month
stock = INTEG(inflow-outflow, inflow*tau) ~ widgets
outflow = DELAY1(inflow,tau) ~ widgets/month
tau = 6 ~ months

In this case, there’s really no reason to use the DELAY1 – it just obfuscates the first-order stock dynamics. However, there’s still a potential pitfall, which also applies to DELAY3. The initialization is important. The DELAY functions generally initialize their internal stocks in equilibrium, as if the inflow had been at its initial level historically. Therefore the stock above needs to be initialized the same way, to inflow*tau. If you want to use some other value, like zero, you need to use DELAY3i (or its equivalent) to set the stock and delay function to a consistent set of assumptions.

In reviewing other models, you may also find hybrid approaches, like:

inflow = 10 ~ widgets/month
stock = INTEG(inflow-outflow, inflow*tau) ~ widgets
outflow = DELAY1(stock/tau,tau) ~ widgets/month
tau = 6 ~ months

This is another FONFOO violation. The outflow is indeed a function of the stock, which ensures that the outflow eventually goes to zero when the stock is exhausted. But this does not create a 1st-order negative feedback loop; the DELAY1 contains an additional stock. So, this is SONFOO (second order negative feedback on the outflow), which might be useful for creating an oscillator, but won’t solve your supply chain problems.

If you make FONFOO a habit, you’ll have one less thing to worry about when you start exploring the interesting, complex behaviors of your models.