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3 䄀 3.2 EQUATIONS FOR FLOWS 23 equations for a process model you need to give 1) the initial values of each stock, and 2) the equations for each ow. We will now apply this approach to the advertising stock and ow diagram in Figure 2.1b. To do this, we will use some elementary calculus notation. However, fear not! You do not have to be able to carry out calculus operations to use this approach. Computer methods are available to do the required operations, as is discussed below, and the calculus discussion is presented for those who wish to gain a better understanding of the theory behind the computer methods. The number of Potential Customers at any time t is equal to the number of Potential Customers at the starting time minus the number that have \ owed" out due to sales. If sales is measured in customers per unit time, and there were initially 1,000,000 Potential Customers, then Potential Customers(t) = 1; 000; 000 Z t 0 sales(; ) d; ; (3:1) where we assume that the initial time is t = 0, and ; is the dummy variable of integration. Similarly, if we assume that there were initially zero Actual Customers, then Actual Customers = Z t 0 sales(; ) d; : (3:2) The process illustrated by these two equations generalizes to any stock: The stock at time t is equal to the initial value of the stock at time t = 0 plus the integral of the ows into the stock minus the ows out of the stock. Notice that once we have drawn a stock and ow diagram like the one shown in Figure 2.1b, then a clever computer program could enter the equation for the value of any stock at any time without you having to give any additional information except the initial value for the stock. In fact, system dynamics simulation packages automatically enter these equations. 3.2 Equations for Flows However, you must enter the equation for the ows yourself. There are many possible ow equations which are consistent with the stock and ow diagram in Figure 2.1b. For example, the sales might be equal to 25,000 customers per month until the number of Potential Customers drops to zero. In symbols, sales(t) = 25; 000; Potential Customers(t) > 0 0; otherwise (3:3) A more realistic model might say that if we sell a product by advertising to Potential Customers, then it seems likely that some speci ed percentage of Potential Customers will buy the product during each time unit. If 2.5 percent of the Potential Customers make a purchase during each month, then the equation for sales is sales(t) = 0:025 ȴPotential Customers(t): (3:4) (Notice that with this equation the initial value for sales will be equal to 25,000 customers per month.)

4 24 CHAPTER 3 SIMULATION OF BUSINESS PROCESSES 3.3 Solving the Equations If you are familiar with solving di±erential equations, then you can solve equations 3.1 and 3.2 in combination with either equation 3.3 or equation 3.4 to obtain a graph of Potential Customers over time. However, it quickly becomes infeasible to solve such equations by hand as the number of stocks and ows increases, or if the equations for the stocks are more complex than those shown in equation 3.3 and 3.4. Thus, computer solution methods are almost always used. We will illustrate how this is done using the Vensim simulation package. With this package, as with most PC-based system dynamics simulation systems, you typically start by entering a stock and ow diagram for the model. In fact, the stock and ow diagram shown in Figure 2.1b was created using Vensim. You then enter the initial values for the various stocks into the model, and also the equations for the ows. Once this is done, you then tell the system to solve the set of equations. This solution process is referred to as simulation, and the result is a time-history for each of the variables in the model. The time history for any particular variable can be displayed in either graphical or tabular form. Figure 3.1 shows the Vensim equations for the model using equations 3.1 and 3.2 for the two stocks, and either equation 3.3 (in Figure 3.1a) or equation 3.4 (in Figure 3.1b) for the ow sales. These equations are numbered and listed in alphabetical order. Note that equation 1 in either Figure 3.1a or b corresponds to equation 3.2 above, and equation 4 is either Figure 3.1a or b corresponds to equation 3.1 above. These are the equations for the two stock variables in the model. The notation for these is straightforward. The function name INTEG stands for \integration," and it has two arguments. The rst argument includes the ows into the stock, where ows out are entered with a minus sign. The second argument gives the initial value of the stock. Equation 5 in Figure 3.1a corresponds to equation 3.3 above, and equation 5 in Figure 3.1b corresponds to equation 3.4 above. These equations are for the ow variable in the model, and each is a straightforward translation of the corresponding mathematical equation. Equation 3 in either Figure 3.1a or b sets the lower limit for the integrals. Thus, the equation INITIAL TIME = 0 corresponds to the lower limits of t = 0 in equations 3.1 and 3.2 above. Equation 2 in either Figure 3.1a or b sets the last time for which the simulation is to be run. Thus, with FINAL TIME = 100, the values of the various variables will be calculated from the INITIAL TIME (which is zero) until a time of 100 (that is, t = 100). Equations 6 and 7 in either Figure 3.1a or b set characteristics of the simulation process.

5 3.5 SOME ADDITIONAL COMMENTS ON NOTATION 25 (1) Actual Customers = INTEG( sales, 0) (2) FINAL TIME = 100 (3) INITIAL TIME = 0 (4) Potential Customers = INTEG( - sales, 1e+006) (5) sales = IF THEN ELSE ( Potential Customers > 0, 25000, 0) (6) SAVEPER = TIME STEP (7) TIME STEP = 1 a. Equations with constant sales (1) Actual Customers = INTEG( sales, 0) (2) FINAL TIME = 100 (3) INITIAL TIME = 0 (4) Potential Customers = INTEG( - sales, 1e+006) (5) sales = * Potential Customers (6) SAVEPER = TIME STEP (7) TIME STEP = 1 Figure Solving the Model b. Equations with proportional sales Vensim equations for advertising model The time histories for the sales and Potential Customers variables are shown in Figure 3.2. The graphs in Figure 3.2a were produced using the equations in Figure 3.1a, and the graphs in Figure 3.2b were produced using the equations in Figure 3.1b. We see from Figure 3.2a that sales stay at 25,000 customers per month until all the Potential Customers run out at time t = 40. Then sales drop to zero. Potential Customers decreases linearly from the initial one million level to zero at time t = 40. Although not shown in this gure, it is easy to see that Actual Customers must increase linearly from zero initially to one million at time t = 40. In Figure 3.2b, sales decreases in what appears to be an exponential manner from an initial value of 25,000, and similarly Potential Customers also decreases in an exponential manner. (In fact, it can be shown that these two curves are exactly exponentials.) 3.5 Some Additional Comments on Notation In the Figure 2.1b stock and ow diagram, the two stock variables Potential Customers and Actual Customers are written with initial capital letters on each word. This is recommended practice, and it will be followed below. Similarly, the ow \sales" is written in all lower case, and this is recommended practice.

6 26 CHAPTER 3 SIMULATION OF BUSINESS PROCESSES CURRENT Potential Customers 2 M 1.5 M 1 M 500,000 0 sales 40,000 30,000 20,000 10, Time (Month) a. Constant sales CURRENT Potential Customers 2 M 1.5 M 1 M 500,000 0 sales 40,000 30,000 20,000 10, Time (Month) b. Proportional sales Figure 3.2 Time histories of sales and Potential Customers While stock and ow variables are all that are needed in concept to create any stock and ow diagram, it is often useful to introduce additional variables to clarify the process model. For example, in a stock and ow diagram for the Figure 3.1b model, it might make sense to introduce a separate variable name for the sales fraction (which was given as in equation 3.4. This could clarify the structure of the model, and also expedites sensitivity analysis in most of the system dynamics simulation packages. Such additional variables are called auxiliary variables, and examples will be shown below. It is recommended that an auxiliary variable be entered in ALL CAPITAL LETTERS if it is a constant. Otherwise, it should be entered in all lower case letters just like a ow variable, except in one special case. This is the case where the variable is not a constant but is a prespeci ed function of time (for example, a sine function). In this case, the variable name should be entered with the FIRst three letters capitalized, and the remaining letters in lower case. This notation allows you to quickly determine important characteristics of variable in a stock and ow diagram from the diagram without having to look at the equations that go with the diagram.

7 3.6 REFERENCE Reference J. D. W. Morecroft and J. D. Sterman, editors, Modeling for Learning Organizations, Productivity Press, Portland, OR, 1994.

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