D-4415-5 1 Answers to Exercises for Chapter 17 INTRODUCTION TO DELAYS from INTRODUCTION TO COMPUTER SIMULATION written by Nancy Roberts David F. Andersen Jonathan Choate Ralph M. Deal Michael S. Garet William A. Shaffer Models translated to STELLA II 3.0 and Answers provided by Kamil Msefer Mark Choudhari Prepared for MIT System Dynamics Education Project Under the Supervision of Dr. Jay W. Forrester System Dynamics Group Sloan School of Management August 5, 1994 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology Permission granted for non-commercial educational purposes2 D-4415-5 INTRODUCTION TO COMPUTER SIMULATION can be ordered from Productivity Press, Inc., Dept. 315 P.O. Box 13390, Portland, OR 97213 The Chapter has also been provided under Road Maps Five in pdf format.4 D-4415-5 Please read Chapter 17: Introduction to Delays from Introduction to Computer Simulation by Nancy Roberts et. al. However, the chapter in the book has its models built using the DYNAMO simulation language, and as you are probably using Vensim, STELLA or iTHINK, we have translated the models for your convenience. This paper contains these translated versions of the models for the first seven exercises from the book. Later, we also suggest answers to these exercises for you to measure your answers against. NOTE FROM THE TRANSLATOR: All the models in this paper have rates that flow in both directions. This is to encourage the readers to make the direction of the flow explicit in the equation, and not in the plumbing. In DYNAMO, there is no biflow-uniflow toggle. All rates are biflows. Example 1: The Martan Chemical Company We present the model of dumping of the Nobug pesticide by the Martan Chemical Company, as translated from DYNAMO to STELLA. A causal loop and flow diagram of the Martan case are shown in Figure 1. The equations for the model are as follows: NOBUG(t) = NOBUG(t - dt) + (Dump_rate - Absorb_rate) * dt INIT NOBUG = 0 OUTFLOWS: ABSORB_RATE = NOBUG/NAT NAT = 2D-4415-5 5 NOBUG DUMP RATE ABSORB RATE NOBURG ABSORP TIME Figure 1. Diagram of the Martan Case These equations show that the level of NOBUG in the Sparkill River is influenced by the DUMP RATE and the ABSORB RATE; and the ABSORB RATE is equal to the level of NOBUG divided by the Nobug Absorption Time NAT (2 days). Exercise 1: Preliminary Nobug Model a. Create a STELLA II model for the Nobug case, adding the DUMP RATE equation, and other needed specifications. Run the model, setting the initial level of NOBUG = 0 and choosing DT = 0.25 days. Set length of simulation (in the Time specs window under the Run menu) to 25 days, the range of Nobug (in the Range specs window under the Run menu) from 0 to 250 gallons and the range of the absorption rate from 0 to 100 gallons/day. What behavior does the model generate?6 D-4415-5 b. Rerun the model, setting the Nobug Absorption Time, NAT = 4. How do the results differ? Rerun the model, setting NAT = 1. How do these results differ? USING THE PULSE FUNCTION TO REPRESENT THE DUMPING RATE We use the corresponding STELLA functions instead of the DYNAMO functions in these exercises. The STELLA PULSE function permits modifying the model to represent the dumping of Nobug in once a week batches. The following rate equation indicates that 420 gallons of Nobug are dumped into the river on day 1 of the simulation, and 420 are dumped again at regular intervals of 7 days. DUMP RATE = PULSE(420,1,7) Figure 2 shows the dumping rate over the first 25 days of the simulation. Please see the chapter in the book for further reading. 1: DUMP RATE 1: 1680.00 1: 840.00 1: 0.00 1 1 1 1 0.00 6.25 12.50 18.75 25.00 Graph 3 Days 6:32 PM 7/25/94 Figure 2. Dumping rate The PULSE function can also be used to examine the response of the systemD-4415-5 7 to the dumping of just one batch of NOBUG. It is possible to do this by making the following change in the equation for DUMP RATE. DUMP RATE = PULSE(420,1,0) This equation indicates that the dumping rate rises from zero to 1680 on day one (for only a quarter of the day)and stays at zero for the rest of the simulation. Figure 3 shows a simulation of the Nobug system, with one batch of Nobug released into the river on day one. As can be seen, the level of NOBUG in the river rises sharply to 420 gallons on day one, and then drifts slowly toward zero. 1: DUMP RATE 2: NOBUG 3: ABSORB RATE 1: 2000.00 2: 500.00 3: 250.00 1: 1000.00 2: 250.00 3: 125.00 1: 0.00 2: 0.00 3: 0.00 1 2 3 1 2 3 1 2 3 1 2 3 0.00 2.50 5.00 7.50 10.00 Graph 3 Days 6:43 PM 7/25/94 Figure 3. Release of one batch of NOBUG The general form of the PULSE function is PULSE(AMOUNT, FIRST, INTERVAL) where AMOUNT indicates the amount to be inputted in the pulse; FIRST indicates the time at which the first pulse occurs; and INTERVAL indicates the time interval between pulses. Setting INTERVAL to 0, indicates to STELLA that you only want8 D-4415-5 one pulse. Exercise 2: The Halving Time for NOBUG Revise your STELLA equations to include a PULSE function for the DUMP RATE. Choose an interval of 0, in order to examine the effects of just one pulse. a. What is the halving time for the amount of NOBUG in the Sparkill River? b. Experiment with various values of NAT. How does the choice of NAT influence the halving time? c. Using your result for part (b), select a value of NAT to produce a halving time that corresponds to the data shown below in Figure 4. Exercises 3: Simulating Repeated Batches a. Select NAT equal to the value you determined in Exercise 2, part (c). Use theD-4415-5 9 PULSE function to simulate the effect of dumping 420 gallons of Nobug into the river at 7-day intervals. Compare your results with the data shown in , Figure 4 above. b. Assume Martan Chemical Inc. changed the chemical composition of Nobug, such that the halving time of Nobug was lengthened to seven days, resulting in an increase in the Nobug absorption time NAT. Change the value of NAT in your model to reflect the change in the chemical composition of Nobug. How does the system respond? NOBUG AND MATERIAL DELAYS Please read the chapter for an introduction to the concept of a first-order material delay. LEVEL INFLOW OUTFLOW ADJUSTMENT TIME Figure 5. First-order material delay When a first-order material delay is used in a model, there are two ways to write the equations. One approach is to simply to write out individual