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This powerpoint is from Chapter 6 of Sterman's Business Dynamics book used in the 3rd Class of Business Dynamics and System Modeling Class at Southern New Hampshire University
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Business Dynamics and System ModelingChapter 6: Mapping Stocks and Flows
Pard TeekasapSouthern New Hampshire University
Outline
1. Stock, Flows, and Accumulation2. Identifying Stocks and Flows3. Mapping Stocks and Flows
Diagramming Notation for Stocks and Flows
• Stock• Flow• Valve (Flow Regulator)• Source or Sink (Stocks outside model boundary)
StockInflow Outflow
StockInflow Outflow
StockInflow Outflow
Stock and Flow
Equivalent representations of stock and flow diagram
Hydraulic Metaphor:
Stock and Flow Diagram:
Stock (t ) = [Inflow (s ) - Outflow (s )]ds + Stock (t 0)t0
t
d(Stock) /d t = Net C hange in Stock = Inflow (t ) – O ut flow (t )
Integral Equation:
Differential Equation:
StockInflow Outflow
Contribution of Stocks to Dynamics
• Stocks characterize the state of the system and provide the basis for actions
• Stock provide systems with inertia and memory
• Stocks are the source of delays• Stocks decouple rates of flow and create
disequilibrium dynamics
Stocks and Flows in different disciplines
Field Stocks Flows
Mathematics, physics and engineering
Integrals, states, state variables, stocks
Derivatives, rates of change, flows
Chemistry Reactants and reaction products
Reaction rates
Manufacturing Buffers, inventories Throughput
Economics
Levels Rates
Accounting Stocks, balance sheet items
Flows, cash flow or income statement items
Biology, physiology Compartments Diffusion rates, flows
Medicine, epidemiology Prevalence, reservoirs
Incidence, infection, morbidity and mortality rates
Which one is the stock and which one is the flow?
• Units of measure in stock and flow networks• The snapshot test
Unit of measure of stocks and flows
• Stocks are usually a quantity e.g. widgets of inventory, people employed, or Dollar in an account
• The associated flows must be measured in the same units per time period e.g. the rate at which widgets are added per week to inventory, the hiring rate in people per month
The snapshot test
• Freeze the system scene with a snapshot• Stocks would be things you can count or
measure in the picture, including psychological states and other intangible variables
Identifying stocks, inflows, outflows
• Completing/Assigning/Homework• Brownies in stomach/Eating/Digesting• Expenses/Income/Money in back account• Shrinking/Pinocchio’s nose/Lengthening• Building/Nuclear Weapons/Disarming• Cavities/Developing/Filling• Sand castles/Demolishing/Constructing
Conservation of Material
• The contents of the stock and flow networks are conserved
• Items entering into a stock will remain there until they flow out
• When an item flows from one stock to another, the first stock loses precisely as much as the second gains
State-determined systems
• Stock can change only via its inflows and outflows
• Stocks determine the flows
State of theSystem (Stock)
Net Rate ofChange
Auxiliary Variables
• System requires only stocks and flows. However, for ease of communication and clarity, it is helpful to define intermediate or auxiliary variables
• The auxiliaries can always be eliminated and mathematically the same, but it is harder to explain, understand, and modify
• The use of auxiliary variables is critical to effective modeling. Ideally, each equation in the models should represent one main idea
Example of using auxiliary variables
PopulationNet Birth Rate
?
Food
+ ?
PopulationNet Birth Rate
+
Food
R
Food perCapita
FractionalBirth Rate
-
+ +
+
B
Stocks change only through their ratesCustomersWating for
ServiceCurstomer
Arrival Rate
CustomerDeparture Rate
Workweek
Service Staff
Productivity
+
-
+
--
B
B
CustomersWating for
ServiceCurstomer
Arrival Rate
CustomerDeparture Rate
Workweek
Service Staff
Productivity
+
+
+
++
B
B
Incorrect
Correct
Should I use mathematical diagram or stock and flow diagram
• Depends on the context of the modeling project you are doing and the background of your client team
• If your clients have very little technical training, the bathtub metaphor is often used to good effect
Which detail I should present and which detail I should lump it together
• Usually it is wise to identify the main stocks in a system and then the flows that alter those stocks
• It is better to start with a high-level, aggregate representation and add detail if needed to address the purpose
• Beginning with detailed process maps often leads to paralysis due to their complexity, data requirements, and rapid obsolescence rates
Example of aggregation
Parts inProcess
Assembliesin Process
Product inTesting
FinishedInventory
Assembly StartRate
Test StartRate
ProductionCompletion Rate
Production StartRate
Shipment Rate
Work in Process
Inventory
Modifying the Model
Modify the diagram to represent the case where units that fail testing are scrapped
Modify your diagram to represent the case where items failing testing are returned to assembly for rework
Guideline for aggregation
• For activities taking place serially, consider the average residence time of item in each stock. The stock with short residence times relative to the time scale for the dynamics of interest can be omitted or aggregated into adjacent stocks
• Parallel activities can be aggregated together if the individual flows are governed by similar decision rules and if the time different items spend in the individual stocks is similar
Setting the model boundary
• Are the assumptions about the “clouds” reasonable?
Are these two clouds reasonableSource:
UnlimitedSupply ofMaterial
Shipments
Production
ProductionStarts
FinishedInventory
ModelBoundary
Sink:Unlimited
AbsorptionCapacity
WIP
Extending the model scope
SupplierShipments
SupplierProduction
SupplierProduction
Starts
SupplierInventory
SupplierSector
Shipments
Production
ProductionStarts
FinishedInventory
WIP
PartsInventory
SupplierWIP
New CarSales
DealerStocks
ManufacturerSector
DealerSector
Adding the customersShipments
New CarSales
DealerSector
ScrapRate
Cars onthe Road Household
Sector
DealerStocks