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