# Sensitive Analysis

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## Introduction

Sensitivity analysis is used to determine how “sensitive” a model is to changes in the value of the parameters of the model and to changes in the structure of the model. In this paper, we focus on parameter sensitivity. Parameter sensitivity is usually performed as a series of tests in which the modeler sets different parameter values to see how a change in the parameter causes a change in the dynamic behavior of the stocks.

By showing how the model behavior responds to changes in parameter values, sensitivity analysis is a useful tool in model building as well as in model evaluation. Sensitivity analysis helps to build confidence in the model by studying the uncertainties that are often associated with parameters in models. Many parameters in system dynamics models represent quantities that are very difficult, or even impossible to measure to a great deal of accuracy in the real world. Also, some parameter values change in the real world.

Therefore, when building a system dynamics model, the modeler is usually at least somewhat uncertain about the parameter values he chooses and must use estimates. Sensitivity analysis allows him to determine what level of accuracy is necessary for a parameter to make the model sufficiently useful and valid. If the tests reveal that the model is insensitive, then it may be possible to use an estimate rather than a value with greater precision. Sensitivity analysis can also indicate which parameter values are reasonable to use in the model.

If the model behaves as expected from real world observations, it gives some indication that the parameter values reflect, at least in part, the “real world. ” Sensitivity tests help the modeler to understand dynamics of a system. Experimenting with a wide range of values can offer insights into behavior of a system in extreme situations. Discovering that the system behavior greatly changes for a change in a parameter value can identify a leverage point in the model— a parameter whose specific value can significantly influence the behavior mode of the system. 2 2

In this paper, the term “behavior mode” refers to the general kind of behavior, such as exponential growth, asymptotic growth, S-shaped growth, or oscillation. 48 D-4526-2 Exploratory Exercises In this section we look at two models and explore how sensitive they are to changes in parameters and initial values of stocks. The first exploration shows that parameter changes produce some change in the appearance of behavior of the system, but they do not change the behavior mode. The second exploration demonstrates that changes in different parameters create different types of changes in the behavior of the system.

In the first exploration, we conduct sensitivity analysis on all the constant parameters in the model. However, in a large model, such an extensive treatment of sensitivity analysis is often impossible. The modeler must pick the parameters he expects to have most influence on the behavior, or the ones that he is most uncertain about, and only use those in the sensitivity analysis. We will see in the second exploration that an examination of the structure of the model can indicate, without running the sensitivity tests, what kind of effect changes in some parameters would have.

An explanation of how to choose the parameters for testing will be offered in later papers in the sensitivity analysis series. Exploration 1: Lemonade Stand In the first exploration, let’ look at a lemonade stand located on a college campus. s As usual, we are particularly interested in the behavior of the stock, the number of cups of lemonade that are ready to be sold to customers. The stand is open eight hours every day. Howard, the owner, is the only person working in the stand. We encourage you to try to build the model and then compare it with the model suggested in Figure 1.

To obtain the equilibrium value of 50 cups, one only needs to multiply the new value of “Buying lemonade,” 25 cups per hour, by the “Lemonade coverage,” 2 hours. Exercise Using the change in “Buying,” we can now experiment with the model to see how the behavior resulting from disturbance changes under different parameter settings. When building a model, we are often uncertain about the exact value of a parameter. However, it is important to be able to pick reasonable values for the parameters. This means that the values chosen should stay in a plausible range that could occur in the real world system.

For example, in the Lemonade stand model it would not be reasonable to expect the “Time to correct amount of lemonade” to be just a few minutes. Howard needs a longer time to perceive the change in the stock, and then to correct the amount of lemonade so that it is ready to be sold. On the other hand, a value of 5 or 10 hours to correct the amount of lemonade would prevent him from responding to a change in his selling rate. The stock of lemonade might deplete quickly, resulting in a loss of customers, or overproduction.

Although the three curves do not look exactly the same, these parameter changes do not affect the general mode of behavior of the system. All three curves show a small decrease in the stock right after the step increase and then a slow approach to the new equilibrium value of 50 cups. The similarity of the results shows that Howard does not have to be certain how long it would take him to perceive a change in his “Selling” rate to be able to estimate the overall behavior of his stock of lemonade.

The curves indicate that the faster Howard adjusts his expectations, the faster the stock of lemonade will approach equilibrium. The initial decrease in the stock is greater for larger values of this parameter. If it takes Howard longer to perceive a decline in the stock, the stock declines for a longer time and to a lower value. 5 The parameter values used in this paper are of course not the only reasonable values, and they were only picked as examples. We encourage you to use other values if you feel that they would represent the situation more realistically. D-4526-2 53 Time to correct amount of lemonade” is another parameter about whose value Howard is uncertain. Three values that seem plausible are one hour, one hour and a half, and two hours.

As before, this means that Howard only needs to know an approximate range of values for this parameter to be confident in the behavior simulated by the model. The effect of changing the value of “Time to correct amount of lemonade” is similar to the “Time to average lemonade buying. ” A lower value of “Time to correct amount of lemonade” makes the stock reach the equilibrium value faster because the “Correction in amount of lemonade” is greater, so Howard prepares the lemonade faster. Also, an increase in the value of this parameter makes the initial decrease in the stock larger.

Contrary to the changes in the first two parameters that we examined, changing “Lemonade coverage” does not influence the time it takes the stock to approach equilibrium. Instead, it changes the equilibrium value of the amount of “Lemonade ready in stand. ” The equilibrium values can be obtained by multiplying “Buying lemonade” by the respective values of “Lemonade coverage. ” Therefore, if Howard changes his “Lemonade coverage,” only the equilibrium value of the stock of “Lemonade ready” will be affected.

Even when Howard starts out with a larger or smaller amount of lemonade, the behavior of the stock of Lemonade ready does not change greatly. When the initial amount of Lemonade ready is lower than what the equilibrium value should be, that is less than 40 cups, the stock will be increasing during the first hour, as in curve 1. When the initial amount is larger than 40 cups, the stock will decrease for the first hour, as in curve 3. However, after the step increase, all the curves immediately decline slightly, and then start slowly increasing to approach the same equilibrium value of 50 cups nearly at the same time.

Debrief As expected, changing the value of parameters in the model does make some difference in the behavior observed. Also, the sensitivity tests indicate that some parameter changes result in “greater,” or more significant, changes than others. For example, compare Figures 5 and 6. In Figure 5 the changes in “Time to correct amount of lemonade” produce little difference in the behavior, while in Figure 6 the curves show the same behavior, but at different values of the stock. This measure of more significant changes is studied through sensitivity analysis.

In all cases, however, it is the structure of the system that primarily determines the behavior mode. In general, but with exceptions, parameter values, when altered individually, only have a small influence on behavior. Exploration 2: Epidemics In the second exploration we look at an epidemics model. The model was already used in a previous chapter in Road Maps, so it is possible that you have already built it. 8 8 For more information about the model, see: Terri Duhon and Marc Glick, 1994. Generic Structures: SShaped Growth I (D-4432), System Dynamics in Education Project, System Dynamics Group, Sloan 6 D-4526-2 Duration of Illness Recovering Rate Probability of Catching Illness Healthy People Sick People Catching Illness Population Interactions Probability of Contact with Sick People Figure 8: The Epidemics Model The model shown in Figure 8 has two stocks, “Healthy People” and “Sick People. ” The total population is a constant 100 people in all the simulations, so the initial value of “Healthy People” is: Initial number of Healthy People = 100-Sick People In the base run, the initial value of “Sick People” is 1. The flow of “Catching Illness” converts the “Healthy People” to “Sick People. It is affected by three variables: “Probability of Contact with Sick People,” “Probability of Catching Illness,” and “Population Interactions. ” The “Probability of Contact with Sick People” is the ratio of “Sick People” to the total population of 100 people. Therefore, the more “Sick People” there are in the population, the greater the probability of coming into contact with a sick person. When a healthy person comes into contact with a sick person, there is a certain probability that the healthy person will catch the illness. This is the “Probability of Catching Illness. The value of “Population Interactions” determines the total number of interactions or contacts that any one person has with other people in the population during a month. The “Recovering Rate” converts “Sick People” back to “Healthy People,” with a time constant called “Duration of Illness. ” See section 7. 2 of the Appendix for the model documentation and equations. Figure 9 shows the base run of the model, using the parameter values indicated in the Appendix. School of Management, Massachusetts Institute of Technology, June 22, 25 p. ” The number of “Healthy People,” on the other hand, decreases, at first exponentially and then asymptotically to the equilibrium value of 40 “Healthy People. ” Exercise The initial value of “Sick People,” as well as three parameters— “Duration of Illness,” “Population Interactions,” and “Probability of Catching Illness”— can be used to evaluate the sensitivity of this model.

In this exercise, we use some parameter values or initial values that correspond to extreme cases. Such tests are very helpful in better understanding the dynamics of the system. For example, the extreme values of the initial number of “Sick People” would be 0 or 100 people. An extreme value of “Duration of Illness” would be a very large value, say one million months, illustrating the case of a terminal disease with no recovery. When using such extreme values, however, it is important to make sure that they stay in the range of values that are possible in the real-world system (e. g. do not use a negative initial value for a stock that cannot be negative). Sensitivity tests using extreme values are performed mostly to learn about the dynamics of the system. The initial value of “Sick People” represents the number of people who are sick when the simulation begins. The stock can, for example, start with no sick people, one sick person, 20, 50, or even 100 sick people. The behavior of both stocks, Healthy People and Sick People, is shown in

Thus, it is obvious that there must be at least one sick person to obtain the characteristic Sshaped growth behavior. Curve 2 corresponds to the case of one sick person initially. The presence of only one sick person makes a great difference in the behavior of the system. The one person starts spreading the disease to more people. Just like in the base run, the two stocks follow the S-shaped growth pattern, and stabilize at the base run equilibrium values. Curves 3 and 4 behave similarly to curve 2, but approach their respective equilibrium values faster because their initial values are closer to equilibrium.

D-4526-2 In curve 5, the behavior of the stocks is reversed— the number of “Healthy People” increases to its equilibrium value of 40, while the number of “Sick People” decreases to 60. As long as there is initially at least one sick person present in the population the equilibrium value does not depend on the initial number of “Sick People. ” 59 The parameter “Duration of Illness” determines how long it takes a sick person to become healthy again.

The only sick person becomes healthy before spreading the disease to anyone else, and all the “Healthy People” stay healthy. Curve 2 corresponds to the base run. Notice from comparing curves 2, 3, 4, and 5, that increasing the “Duration of Illness” results in higher equilibrium values of “Sick People” and lower equilibrium values of “Healthy People. ” The reason for this behavior is that when the disease is longer, each sick person can spread it to more people. In particular, curve 5 shows that when the disease is extremely long, or terminal (no recovery), all 100 people become sick.

The longer the “Duration of Illness,” the sooner the stocks approach equilibrium. This result should be expected— the longer the disease, the more people get sick and have a longer time to spread the disease further and faster, so equilibrium occurs sooner. The parameter “Population Interactions” determines how many contacts any person has with other people during a unit of time. The value of a parameter such as “Population Interactions” is quite uncertain— it can be different for each person and can even vary over time.

Therefore, the value chosen for the model can only be an estimate of the average of the actual value. A sensitivity test that uses different values of “Population Interactions” indicates how the behavior of the system is affected when its value changes. Depending on the nature of the population, the number of “Population Interactions” can vary significantly. A person can, for example, meet only one other person each month, 5 people, 10 people, 25 people, or even all the rest of the population— 99 people a month.

By comparing curves 2, 3, 4, and 5, notice that increasing the number of interactions increases the equilibrium value of “Sick People” and decreases the equilibrium value of “Healthy People. ” Therefore, if you want to study the spread of an epidemic in a specific population, and if you are interested in the equilibrium number of people who will be affected, you will have to have an idea of the value of “Population Interactions” in that population. Also, an increase in “Population Interactions” makes the stocks reach their equilibrium values faster— it increases the value of “Catching Illness,” and thus “Healthy People” become sick sooner.

Before looking at the effects of changing “Probability of Catching Illness,” turn back to Figure 8 showing the Epidemics model. Notice that both “Population Interactions“ and “Probability of Catching Illness” enter the rate of “Catching Illness” in a similar way. The rate equation is: Catching Illness = Healthy People * Probability of Contact with Sick People * Population Interactions * Probability of Catching Illness Both “Population Interactions” and “Probability of Catching Illness” are linear factors of the rate equation— they both simply multiply “Healthy People” and “Probability of Contact with Sick People. One would therefore expect that their effect on the system behavior should be similar, and that they should produce the same kind of changes in the behavior when their value changes. We encourage you to run sensitivity tests with different values of “Probability of Catching Illness. ”

Sensitivity analysis again showed that changing the value of parameters makes some difference in the behavior of the model, while the general behavior mode is relatively insensitive to parameter changes. Some parameter changes affect the behavior to a larger extent than others. Changes in some parameters affect the equilibrium values, such as “Lemonade coverage” from the Lemonade stand model. Other parameters affect the time necessary to approach equilibrium— for example “Time to correct amount of lemonade” from the Lemonade stand model, or the initial value of “Sick People” from the Epidemics model.

Others again have influence over both— “Population Interactions” for example. When two parameters have similar roles in the model, they usually affect the behavior in a similar way, such as “Population Interactions” and “Probability of Catching Illness. ” Using extreme values in sensitivity tests is an excellent tool for gaining a deeper understanding of the behavior generated by the structure of the model as well as for testing the assumptions about the model. Independent Exploration: Coffeehouse We now return to Howard, the owner of the lemonade stand on a college campus.

Howard realized that it could be more profitable for him to sell coffee because students tend to drink more coffee than lemonade, and they drink it at any time of the day and night. Therefore, he closed his lemonade stand and opened a 24-hour Coffeehouse. Howard bases the Coffeehouse model on the model he used in his lemonade stand to model the number of cups of “Coffee ready. ” We will run the simulation over a period of two days, or 48 hours. Figure 13 shows the full model. The model documentation and equations are in section 7.

Each worker is able to prepare a certain number of cups every hour, determined by the parameter called “Productivity. ” In the base run, “Productivity” is 20 cups per worker per hour. When Howard divides the “Desired making of coffee” by “Productivity,” he obtains the “Desired Workers,” or the number of workers necessary to prepare the desired amount of coffee. Howard then compares the “Desired Workers” to the actual number of “Workers. ” The difference between these two values, divided by a time constant called “Time to Correct Workers,” gives his hourly “Correction for Workers. The “Time to Correct Workers,” 3 hours in the base run, is the time constant that he needs to compensate for the difference between the desired and actual number of workers. It is the time in which he wants to call them on the phone and have them come to work. 64 D-4526-2 Howard then adds the Correction to the number of workers who leave the Coffeehouse every hour. He has to make sure that a corresponding number of workers also come to work at the Coffeehouse each hour— this is the inflow called “Coming to Work” that increases the number of “Workers. The outflow, “Going Home,” decreases the number of present “Workers. ” “Going Home” is the number of “Workers” divided by 4 hours, the “Average Length of Working. ” Finally, Howard multiplies the number of “Workers” by their “Productivity” to obtain the flow of “Making Coffee. ” Without any outside disturbance, the system starts out and remains in equilibrium at 40 cups (Coffee coverage * Buying coffee) during the 48 hours of simulation.  The additional structure added to the Lemonade stand model significantly changes the behavior of the system— the stock of coffee now exhibits “damped oscillations. ” After the increase in buying, “Coffee ready” decreases because “Selling” steps up together with demand. However, “Making Coffee” has not changed yet. It first takes Howard a certain time to perceive the change in buying as opposed to random noise and to find out how much coffee the workers should be making.

He then determines how many more workers should be working in the Coffeehouse, and calls them to make them come to work. However, because “Desired making of coffee” is high, Howard keeps calling more workers than the equilibrium number. As they come to work, the stock of coffee starts increasing, reaches its new equilibrium value of 50 cups, 11 overshoots it, and continues to increase until the number of workers decreases again to its equilibrium value. The oscillations continue, but they become smaller and smaller, until both stocks eventually approach their equilibrium values. 2 Choosing Parameters for Sensitivity Analysis Howard is interested in the behavior of “Coffee ready. ” What parameters and initial values should he use in a sensitivity analysis of the Coffeehouse model? Performing Sensitivity Tests Now take the parameters and initial values that you determined in section 4. 1 one by one. For each of them, choose three values that seem reasonable to you and that you 11 As in the first exploration, the equilibrium value can be obtained by multiplying “Coffee Coverage” (2 hours) by “Buying Coffee” (25 cups per hour). 2 For a more detailed explanation of the causes of oscillations, see: Kevin Agatstein, 1996. Oscillating Systems II: Sustained Oscillation (D-4602), System Dynamics in Education Project, System Dynamics Group, Sloan School of Management, Massachusetts Institute of Technology. 66 D-4526-2 believe would offer insights about the behavior of the system. BEFORE running the sensitivity tests, think about the parameter as part of the structure of the model. Compare it to the parameters from the Lemonade stand and Epidemics models, and try to guess the effect of changing it on the behavior of “Coffee ready. Why would Howard want to know the effect? Then simulate the model and compare the graphs to your predictions. If a behavior seems surprising at first, try to explain it through the structure of the model. Conclusion Specific parameter values can change the appearance of the graphs representing the behavior of the system. But significant changes in behavior do not occur for all parameters. System dynamics models are in general insensitive to many parameter changes. It is the structure of the system, and not the parameter values, that has most influence on the behavior of the system.

Sensitivity analysis is an important tool in the model building process. By showing that the system does not react greatly to a change in a parameter value, it reduces the modeler’ uncertainty in the behavior. In addition, it gives an opportunity for a better s understanding of the dynamic behavior of the system. We encourage you to experiment with the three models from this paper (as well as any other models that you have built) on your own. For example, try to change several parameters at the same time, observe the behavior produced, and compare it to the conclusions in this paper.

Can you suggest any parameter values that would produce the “optimal,” or most desirable behavior? The use of sensitivity analysis in such policy analysis will be explored in a later paper in this series. D-4526-2 Suggested Solutions to Section 4 67 Solution to Section 4. 1 The initial value of “Coffee ready,” as well as six parameters— “Time to average coffee buying,” “Time to correct amount of coffee,” “Coffee coverage,” “Time to correct workers,” “Productivity,” and “Average length of working”— can be used to explore the sensitivity of this model. 3 Solution to Section 4. 2 The initial value of “Coffee ready” represents the number of cups ready in the Coffeehouse before the beginning of the simulation. It is usually set to “Coffee coverage * Buying coffee,” but Howard may want to find out what would happen if he prepared more or less coffee at the beginning of the simulation. Three possible initial values are 30, 40, and 50 cups.

An increase in the “Time to average coffee buying” increases the initial decline of the stock, but makes the overshoot smaller. When the “Time to average coffee buying” is long, it takes the stock longer to reach equilibrium. As in the first exploration, the “Time to correct amount of coffee” indicates how long it takes to change the amount of coffee as a result of a sudden change in buying. The value of this parameter is also quite uncertain, so Howard wants to know how precisely he needs to know it to support the conclusions of his model.

Increasing the “Time to correct amount of coffee” increases the initial decline of the stock because the workers are not able to respond quickly enough and start preparing more coffee. A longer “Time to correct amount of coffee” also increases the time it takes the stock to reach equilibrium. The parameter “Coffee Coverage” determines how many hours worth of coffee Howard would like the workers to keep ready in the Coffeehouse. It has the same role as the “Lemonade coverage” in the first exploration. As with lemonade, Howard wants to make sure that he always has a sufficient amount of “Coffee ready,” while it is still fresh.

The “Time to correct workers” indicates how long it takes Howard to make more workers come to work. Because the workers in the Coffeehouse are students, they are usually not available immediately, and it can take Howard up to a few hours to make the necessary number of workers come to work. But he is not completely certain about the exact time required to make them come over, so he wants to try out different values to see how the behavior of the system changes. He knows that it usually takes him somewhere between 2 and 4 hours to call them up and make sure they come.

If the Coffeehouse workers were able to come to work immediately, the structure and thus the behavior of the Coffeehouse model would be the same as that of the Lemonade stand model. It is therefore the additional structure in the “Workers” sector that creates a time lag between the decision to make coffee, the time when the workers are available, and the time when the coffee is made; thus, it causes the Coffeehouse model to exhibit oscillation. However, a situation in which the workers would be available immediately is unlikely to happen, and the higher values of “Time to correct workers” are probably more reasonable.

Howard thus does not need to know exactly how long it takes him to find workers who are available to come to work. The higher the value of “Time to correct workers,” the greater the initial decrease of the stock, and the higher the magnitude of oscillations. Also, increasing the value of Time to correct workers increases the time it takes the stock to approach the same equilibrium value of 50 cups of coffee. “Productivity” measures the amount of coffee that one worker can prepare in one hour. Because the workers also have other responsibilities in the coffeehouse, they cannot spend the entire hour preparing coffee.

The oscillations occur at the same time, but their magnitude decreases as “Productivity” increases because with a higher “Productivity,” the workers can react better to sudden changes in buying. The curves approach their equilibrium values at the same time— curve 1 approaches the value of 2. 5 workers, curve 2 approaches 1. 25 workers, and for curve 3 the equilibrium value is 5/6 of a worker. Therefore, it is the number of “Workers” that compensates for the changes in “Productivity” so that the amount of “Coffee ready” remains unchanged. The “Average length of working” indicates the average number of hours a worker spends in work. Howard can easily find out this value, but he would like to know whether there would be any differences if his workers stayed in the Coffeehouse for a longer time.

Figure 23: The effect of changes in “Average length of working” A change in the value of “Average length of working” does not influence the amount of “Coffee ready. ” Let us again look at the behavior of “Workers” for an explanation. Of course, in the real world, there could not be 2/3 of a worker working in the Coffeehouse. But a model is just a representation of the real world. In this case, when our primary purpose is to test the model for sensitivity, we do not have to be concerned about this problem. The manager would have to round up the number of workers to the next whole number. 7 Again, the manager should just round up the equilibrium values to the next whole number.  The effect of changes in “Average length of employment” on the number of “Workers” Unlike “Productivity,” changes in the “Average length of working” do not influence the number of “Workers” presently working in the Coffeehouse. The reason is that “Average length of working” affects both flows into the stock of “Workers. ” Increasing the value of Average length of employment will simultaneously increase the value of “Coming to Work” and “Going Home,” so the net flow into “Workers” will not change. In addition, when the number of “Workers” does not change, the amount of “Coffee ready” will not change either (assuming that all the other variables in the model remain unchanged). Debrief

The exercise confirmed the conclusions established in the two explorations: changing parameter values makes a difference to the behavior of the model, but the overall behavior mode stays the same. A change in behavior corresponds to a change in structure. Also, some parameters cause a more significant or different change in the behavior than others. Changes in three different parameters (“Coffee coverage,” “Time to correct workers,” and “Productivity”) can produce different changes in the behavior of the stock of “Coffee ready,” or no change at all, as in the case of “Productivity. However, for large values of Average length of employment, starting at approximately 14 hours, the behavior starts to change— the oscillations become larger as Average length of employment increases. You should still keep in mind that the values in the sensitivity tests should always stay in a range that is possible in the real world. Fourteen hours does not seem to be a reasonable value for the number of hours that a worker spends in work.

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