D-4593-2 1 Beginner Modeling Exercises Section 5 Mental Simulation of Combining Feedback in First-Order Systems Stock OutflowInflow Inflow Fraction Outflow Fraction Prepared for the MIT System Dynamics in Education Project Under the Supervision of Prof. Jay W. Forrester by Laughton Stanley Helen Zhu May 29, 1996 Vensim Examples added October 2001 Copyright © 2001 by the Massachusetts Institute of Technology Permission granted to distribute for non-commercial educational purposes.D-4593-2 3 Table of Contents 1. INTRODUCTION 5 2. EXAMPLE 1: IDENTICAL CONSTANT FRACTIONS 3. EXAMPLE 2: DIFFERENT CONSTANT FRACTIONS 4. EXAMPLE 3: NONLINEAR FEEDBACK 5. EXAMPLE 4: NONLINEAR FEEDBACK WITH GRAPH FUNCTION 12 6. REVIEW 16 7. EXERCISES 7.1 EXERCISE 1: RUNNING A MARATHON 17 7.2 EXERCISE 2: UNREAD BOOKS 17 7.3 EXERCISE 3: RAGNAR’S SAILBOAT 19 7.4 EXERCISE 4: GALT DEVELOPMENT COMPANY 19 8. SOLUTIONS TO EXERCISES 21 8.1 SOLUTION TO EXERCISE 1 21 8.2 SOLUTION TO EXERCISE 2 21 8.3 SOLUTION TO EXERCISE 3 23 8.4 SOLUTION TO EXERCISE 4 23 9. APPENDIX: EXPONENTIATION 25 10. VENSIM EXAMLES5 7 9 17 26D-4593-2 5 1. Introduction Feedback loops are the basic structural elements of systems. Feedback in systems causes nearly all dynamic behavior. To use system dynamics successfully as a learning tool, one must understand the effects of feedback loops on dynamic systems. One way of using system dynamics to understand feedback is with simulation software on a computer. Computer simulation is a very useful tool for explaining systems; however one should also be able to use the other simulation tool of system dynamics: mental simulation. A strong set of mental simulation skills will enhance the ability to validate, debug, and understand dynamic systems and models. It is assumed that the reader is familiar with the simulation of positive feedback, negative feedback, and adding constant flows.1 To further develop these capabilities, it is necessary to study the behaviors associated with combined feedback loops in first-order systems. Here we examine four possible behaviors of this structure: equilibrium, exponential growth, asymptotic growth, and S-shaped growth. A set of exercises will reinforce understanding of these models. Solutions are included. 2. Example 1: Identical Constant Fractions Eddie is creating a model of his nursery’s tree production so that he can determine the policies that might best increase his inventory and sustain sales. Eddie looks at the behavior of the past five years and realizes that he has been using the combined feedback model shown in Figure 1. Eddie’s yearly sales have been around 8% of his inventory, so his past policy was to replenish his inventory by planting the same 8% yearly.2 1 The “Beginner Modeling Exercises, Mental Simulation” series is available in “Positive Feedback” (D-4487) by Joseph Whelan, “Negative Feedback” (D-4536) by Helen Zhu, and “Adding Constant Flows” (D-4546) by Alan Coronado. The series is part of Road Maps. 2 All models are simplifications. The assumptions in this model about the Sales_Fraction are only an approximate method of modeling Sales. In the reality it is unlikely that Sales would remain a constant percentage of inventory, but for the purposes of this paper it is instructional to model Sales in this way.D-4593-2 6 Trees Planting SalesPlanting Fraction Sales Fraction Figure 1: Eddie's Past Business Strategy Using the mental simulation skills that he had learned from Road Maps, Eddie sets out to determine the results of his current strategy. He examines the model and recognizes the presence of both positive and negative feedback loops in the system. He ponders over how the two interact to produce the final system behavior. In attempting to figure this out, Eddie takes the following steps of mental simulation: 1. Determine the nature of loop behavior in the system. Eddie looks at his model and realizes that as the number of Trees increases, so does the Planting rate. Over time, this generates exponential behavior due to positive feedback. The outflow loop, however, shows that as the number of Trees increases, Sales also increase, causing the Trees and Sales to drop again. Under just this second pattern of behavior, the number of Trees gradually approaches zero, its goal. This goal-oriented behavior is caused by negative feedback. To determine the net behavior of the system, both flows must be considered. The two following equations give the values of Planting and Sales. The line below the equation gives the units for the equation. Planting = Trees * Planting_Fraction = Trees * 0.08 trees/year = trees * (1/year) Sales = Trees * Sales_Fraction = Trees * 0.08 trees/year = trees * (1/year) In this case, the two flows are equally strong because the compounding fractions are equal. That means the inflow is always equal to the outflow; thus the net flow for the system is always zero, and the stock value is always constant.D-4593-2 7 2. Determine the initial stock value. Eddie notes that he starts out with 500 trees. 3. Determine the final stock value. What does the stock do after the initial point? The inflow at any time is the stock times the Planting_Fraction, and the outflow at any time is the stock times the Sales_Fraction. Both fractions happen to be 0.08. Eddie realizes that in this case the system is always in equilibrium because the inflow and outflow start out equal and never change. Thus the value of the stock is always 500 trees. Now that Eddie has a pretty good idea of what his Tree inventory looks like, he plots it in Figure 2. 1: Trees 2: Planting 3: Sales 1000.00 500.00 0.00 1 1 1 1 2 3 2 3 2 3 2 3 0.00 5.00 10.00 15.00 20.00 Years Figure 2: Eddie's Projected Inventory 3. Example 2: Different Constant Fractions Although business is stable, Eddie would like to design a strategy to gradually increase his inventory and expand business. He realizes that if the Planting_Fraction and the Sales_Fraction are equal, there will be no growth in his Sales. Therefore he decides to increase the Planting_Fraction in the hopes that it will give him more trees to sell. Eddie sticks with the same model structure, but changes the Planting_Fraction to 0.30 so that heD-4593-2 8 will plant 30% of his inventory each year. He then mentally simulates the system behavior using the same steps: 1. Determine the nature of loop behavior in the system. The behaviors of the