System Dynamics Models

Small System Dynamics Models

forBig Issues

Triple Jump towards Real-World Dynamic Complexity

Erik Pruyt

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Chapter 7

MCQs Part II

‘It’s so much easier to suggest solutions when you don’t know too much about the problem.’Malcolm Forbes

Which of the following statements are right and which are wrong?

1. The value of a stock variable can only be changed, during a simulation, by its flow variables.

2. An inflow cannot be negative.

3. The behavior of a stock is described by a differential equation.

4. If two causally linked variables show exactly same behavior as a function of time, then oneof these variables is a stock.

5. A stock variable always needs to be embedded in at least one loop.

6. Plugging a SD model with correction factors that do not have any relation to the real worldsystem is not considered good SD practice.

7. If a hypothetical sheet of paper that could be folded infinitely is about 0.1mm thick and themaximum distance between the earth and the moon is 405500 km, then folding the paper42 times more than bridges the maximum distance between earth and moon.

8. The units of inflow variables and outflow variables of the same stock variable always havethe same units.

9. Non-linear functions, feedback loops, or their combination can generate non-linear dynamics.

10. Underlying SD models, there is exactly one differential equation for each feedback loop.

11. Diffusion and transition models always have the same feedback loop structure consisting ofone�+ or�R and one- orB .

12. Quantitative System Dynamics simulation results need to be interpreted qualitatively.

13. If A→+ B, both variables A and B were increasing until time t, and variable A starts todecrease at time t, then variable B may either start to decrease or keep on increasing but ata reduced rate of increase.

14. If a potentially important variable is not reliably quantifiable, it should be omitted from aSD model.

15. In SD studies, conclusions can only be drawn validly if they are based on simulations.

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c⃝ 2013 by Erik Pruyt RUN-UP: MCQs Part II

Multiple Choice Question 1

The equation for the CLD on the right –with p(t) = size of the rat population, g the annualnumber of births per rat, and s the annual number of deaths per rat– is:

a. p(t) = gp(t)− sp(t)

b. p(t) = g(t)/s(t)

c. dp(t)/dt = gp(t)

d. dp(t)/dt = gp(t)− sp(t)


Consider the SD model displayed onthe right, with c > 0, d > 0, A0 > 0,en B0 > 0. The formula of the flowvariable is equal to c.d.A.B/(A + B).Which behavior corresponds to (stock)variable B?

(a) (b)

(c) (d)


Many European countries consider raising the retirement age to keep the social security systemfinancially healthy. To study this issue in its most basic form (that is: all adults work, only theretired die, first-order delays are good approximations, etc.), three groups need to be discerned:the kidsK, adults younger than the retirement age A, and retirees R (see the incomplete stock-flowdiagram below).

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RUN-UP: MCQs Part II c⃝ 2013 by Erik Pruyt

Which of the following systems of differential equations is best suited for studying the sustain-ability of the security systems? [death rate of the Retired = m; birth rate per adult = b; averageresidence time in group K = t1 and in group A = t2].

a.

dK(t)dt = bK(t)− t1K(t)

dA(t)dt = t1K(t)− t2A(t)

dR(t)dt = t2A(t)−R(t)/m

b.

dK(t)dt = K(t)

t1− A(t)

t2dA(t)dt = K(t)

b − K(t)t1

dR(t)dt = A(t)

t2−mR(t)

c.

dK(t)dt = bA(t)− K(t)

t1dA(t)dt = K(t)

t1− A(t)

t2dR(t)dt = A(t)

t2−mR(t)

d.

dK(t)dt = bK(t)− K(t)

t1dA(t)dt = K(t)

t1− A(t)

t2dR(t)dt = A(t)

t2−mR(t)


The graph on the right contains data and projections of theCBS (Dutch Central Bureau of Statistics) concerning the births(blue) and deaths (red) in the Netherlands between 1960 and2060. Assuming there is no net migration in this period, itmeans that:

a. the Dutch population remained more or less constant be-tween 1980 and 2000 because births and deaths developedmore or less in parallel;

b. the Dutch population is projected to decrease between2000 and 2032 because the number of deaths increasesfaster in that period than the number of births;

c. the Dutch population is projected to keep on growing till2032 – nevertheless with a decreasing growth rate, becausethe number of deaths remains larger than the number ofbirths;

d. none of the statements above is correct.

Blue: births x1000; Red: deathsx1000; Source: CBS in NRC Han-delsblad 17/12/2010 p4.


The unit of time in a model concerning the large-scale introduction of wind turbines is expressedin months. The production capacity of a company that produces wind turbines is modeled as astock variable with units expressed in turbine/month. The enormous growth of the demand fornew wind turbines, leads to an increase of the production capacity of the company. Which unitneeds to be used for the increase of the production capacity of the company?

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a. turbine b. turbine/month c. turbine/month2 d. (turbine/month)2


The green curve in the left hand side graph is the only inflow ‘in’ into a stock variable, and thered line is the only outflow ‘uit ’ out of the same stock variable. Which of the behaviors in theright hand side graph corresponds, given the ‘in’ and ‘uit ’ flows, with the behavior of the stockvariable?

a. curve 1 (green) b. curve 2 (red) c. curve 3 (dark blue) d. curve 4 (light blue)


The graph below shows the number of parcels entering and leaving a warehouse between 10:00amen 10:35am in time intervals of 5 minutes. It is unknown how many parcels there were in thewarehouse before 10:00am. It is unknown how many parcels left the warehouse after 10:35am. Inwhich time interval between 10:00am and 10:35am was the smallest number of parcels inside thewarehouse?

a. The number of parcels inside was smallest at 10:00am or 10:05am.

b. The number of parcels inside was smallest at 10:15am or 10:20am.

c. The number of parcels inside was smallest at 10:30am or 10:35am.

d. That cannot be determined with the information provided here.

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The left hand side graph in the above figure shows the behavior of two out of three flows of thestock-flow diagram displayed on the right. The third flow is proportional to the Stock. Which ofthe following figures displays possible behaviors for the Stock and outflow1 that could have beengenerated with the above structure?

(a) (b)

(c) (d)


What was long predicted, happened in the second quarter of 2010: China’s economy becamebigger than Japan’s. The Japanese economy amounted in the second quarter to $ 1290 billion,against $ 1340 billion for China. Economists point out that Japanese welfare is still much higherthan Chinese welfare. The average Japanese income ($ 37800 per capita) is about 10 times higher

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than the average Chinese income. Measured in Gross Domestic Product per year, the top fivecurrently looks –according to the figures of the IMF– as follows: the United States of America arethe largest economic power (e10995 billion), far ahead of China (e4319 billion), Japan (e4053billion), Germany (e2486 billion) and France (e1921 billion). Practically all analysts predict thatChina will take over the lead from the USA – the only question seems to be when. When willChina take over the lead from the USA if China keeps on growing at a rate of 11,9% per year asin the first quarter of 2011, and the USA at 2% per year?

a. 2019; b. 2021; c. 2023; d. 2027.


Given is the fully displayed sub-model of a company with a largenumber of employees on the left.The model is about gain andloss of experience, which is mea-sured in weeks. The SD modelaggregates the experience of allemployees in the company. Allrelevant variables are displayed;no constants or parameters areused that are not shown.

Employees gain experience over the course of time. If an employee leaves the company, then it isassumed that the employee leaves with his/her experience which equals the average experience.The formula for loss of experience is equal to the firing and attrition of employees times the aver-age experience of employees. The formula for average experience is equal to the total experience ofemployees divided by number of employees in the company. The number of employees is expressedin ‘person’, the unit of time is expressed in ‘week’. What units should be used for the averageexperience of employees and the total experience of employees?

a. week; person * week; b. Dmnl; person; c. week; person; d. Dmnl; week/person.


The green line in the graph on the left is the only inflow ‘in’ into a stock variable. The blueline with the positive slope is the only outflow ‘uit ’ out of this same stock variable. Which ofthe patterns of behavior on the right could, given the ‘in’ and ‘uit ’ flows, correspond to the stockvariable?

a. pattern 1 (green) b. pattern 2 (blue) c. pattern 3 (orange) d. pattern 4 (red)

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The SD model below could be used to calculate demographic scenarios. Suppose one uses themodel to project the expected demographic developments in Germany starting from followingassumptions about the flow variables (see also the graph below): The net immigration decreasesfrom 300000 in 2000 to 0 in 2008 and remains constant at that level until the year 2025 after whichan annual amount of immigrants equal to 500000 are admitted. The birth rate decreases linearlyfrom 0.0083 in 2000 to 0.0075 in 2050. The death rate increases linearly from 0.01 in 2000 to 0.02in 2050.

Given these assumptions and the model displayed above, what would be the expected evolutionof the stock variable ‘Inhabitants’?

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Consider the model below. Assume that it is in equilibrium. Inventory is measured in ‘item’,the time unit of the model is days. The variable X is defined as X = inventory/shipment rate.What is the unit of X? What would be a good name for X?

a. dimensionless, average residence time

b. day, average residence time

c. day, residence coefficient

d. 1/day, day


Pattern 1 in the graph on the left shows the only inflow ‘in’ into a stock variable and pattern 2its only outflow ‘out’. Which pattern of behavior on the right belongs to the stock variable?

a. pattern 1 (blue) b. pattern 2 (red) c. pattern 3 (pink) d. pattern 4 (orange)


Different parts of the Dutch housing market have been under attack from different sides for the pastfew years. The social housing market too. According to the European Commission, the rules andorganization of the Dutch corporation system did not correspond to the purpose of social housing.That is, the social rental market was not restricted to the poor. Although nobody complained, theEuropean Commission argued that the Dutch rental market did not comply with the rules of freecompetition, creating a situation of unfair competition with the private rental market, resultingin a distorted housing market. After long negotiations, the Dutch government and the EuropeanCommission agreed that after 1/1/2011, 90% of all social housing vacancies would be allocated tofamilies with incomes up to e28475 per year – which is below modal. However, no agreement wasreached about the majority of above modal income families in the social housing sector, resultingin the situation that above modal income families were not allowed to move (upward or sideways)inside the social housing sector any more. For them, moving meant, from then on, leaving thesocial housing sector, i.e. buying or renting on the (excessively expensive) private market. Before1/1/2011, the housing market was a pull system: after an attractive house had become vacant,another social housing family would soon move to the vacant house, freeing up their own house,which would soon be filled up by yet another social housing family, until finally, a ‘starters house’

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would become vacant and would be occupied by young entrants into the social housing system.Due to these regulations, higher income families stay put, which brought the dynamics on thesocial renting market to a halt.

Consider the SFD of a Dutch social housing simulation model displayed above. Which of thefollowing CLDs corresponds to this SFD?

(a)

(b)

(c)

(d)

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Consider the detailed CLD on the outbreak of a disease displayed above. Which of the followingSFDs corresponds best to this CLD?

(a) (b)

(c) (d)

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Consider the above simulation model and behavior of the flow variable. What is the correspondingbehavior of the stock variable?

(a) (b)

(c) (d)


What is wrong with the SFD on the right?

a. There are too many inflows into the stock.

b. There is a variable in the diagram that isnot influenced by other variables, hence itshould have been modeled as a constant.

c. There is a loop in the diagram without astock variable, which is not allowed.

d. The diagram is correct.

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Suppose a stock variable displays the behaviorover time displayed above. If the stock variableonly has one flow variable, more precisely aninflow, then what is the corresponding behaviorof the inflow?

(a) none of the behaviors below

(b) (c) (d)


Suppose you made the model on the left. By simulating the model, you obtained the behavior onthe right. What can you say about the link between model and behavior?

a. You found the critical variable values that make model behave chaotically.

b. It is impossible to obtain this behavior with that model since it is a model with only oneindependent stock variable.

c. You have chosen a bad time step: the output shown is due to a numeric solution error.

d. The model has complex eigenvectors.

Link to the answers to the 15 right/wrong questions & 20 multiple choice questions in this chapter.

Links to web based quizzes: | | | |

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Flexible E-Book for Blended Learning with Online Materials

Although this e-book is first and foremost an electronic case book, it is much more than justa set of case descriptions: it is the backbone of an online blended-learning approach. It consistsof 6 concise theory chapters, short theory videos, 6 chapters with about 90 modeling exercisesand cases, many demo and feedback videos, feedback sheets for each case, 5 overall chapters withfeedback, 5 chapters with multiple choice questions (with graphs or figures), hundreds of onlinemultiple choice questions, links to on-site lectures, past exams, models, online simulators, 126slots for new exercises and cases, and additional materials for lecturers (slides, exams, new cases).The fully hyperlinked e-version allows students (or anybody else for that matter) to learn –in arelatively short time– how to build SD models of dynamically complex issues, simulate and analyzethem, and use them to design adaptive policies and test their robustness.

ISBN paperback version:

ISBN e-book version:

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System Dynamics Models