levels and rates
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Dennis T. Beng Hui, De La Salle University-Manila
Levels and Rates
The Bath tub Example
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow Diagram (Flow Diagrams)
Stock and flow diagrams are ways of representing the structure of a system with more information than a simple causal loop diagram.Stocks (levels) are fundamental to generating behavior in a system.
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow Diagram (Flow Diagrams)
Flows (rates) causes stocks to change.Stock and flow diagram is a common step toward building a simulation model because they help define the types of variables that are important in causing behavior.
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow DiagramStocks or levels
Flows or Rates
Auxiliary
Table Function
Constant
Exogenous Variable
Variable not defined in diagram
Information Link
Material Link
Source or Sink of material
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Dennis T. Beng Hui, De La Salle University-Manila
Population Stock Flow Diagram
PopulationBirth Death
% Birth WomenGiving Birth
% of Populationdying
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Dennis T. Beng Hui, De La Salle University-Manila
Aids Stock Flow Model
AIDSHIVIncubation RateInfection Rate Death Rate
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow of Ordering system
Amount ofInvtyDelivery Orders
Demand RateAmount toreplenish
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow of Ordering system (Alternative)
Amount ofInventory
Net of Orders andDelivery
Order
Deliver
Demand
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Dennis T. Beng Hui, De La Salle University-Manila
Coffee Temperature Stock Flow Diagram
Coffee TemperatureChange in Temp
Heat Loss
Room Temp
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Dennis T. Beng Hui, De La Salle University-Manila
Stock Flow of Household Expenditure
AvailableMoney Allowance
Utilities
Income
Amount ofOvertime
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Dennis T. Beng Hui, De La Salle University-Manila
Problem and Addiction Stock Flow diagram
AddicitionLevelChange in level
Amount ofProblemNew Problems Solved Problems
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Dennis T. Beng Hui, De La Salle University-Manila
Classes of EquationsLevel equations Rate equations Auxiliary equations Supplementary equations Initial-value equations
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Dennis T. Beng Hui, De La Salle University-Manila
Level equationsLevel equations have varying contents of reservoirs of the system. They would exist even if the system is in rest and no flows existed.Examples are stocks, inventories and others.New values of levels are calculated at each of the closely spaced solution intervals.
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Dennis T. Beng Hui, De La Salle University-Manila
Level equationsLevels are assumed to change at a constant rate between solution times, but no values are calculated between those times.Levels determine rates Example:
L INVTY.K=INVTY.J+DT(MAKES.JK- SALES.JK)
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Dennis T. Beng Hui, De La Salle University-Manila
Rate equationsRate equations are decision functions.Defines the rates of flow between the levels of the system.A rate equation is evaluated from presently existing values of levels in the system, very often, including the level from which the rate comes and the one into which it goes.
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Dennis T. Beng Hui, De La Salle University-Manila
Rate equationsThe rate in turn cause the changes in the levels.Rates determine levels.Example:
R BIRTH.KL = POPN.K*0.20
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Dennis T. Beng Hui, De La Salle University-Manila
Auxiliary equationsAuxiliary Equations are components of a rate equation. These are equations that assist but are incidental. Helps in keeping the model in close correspondence to the actual system.
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Dennis T. Beng Hui, De La Salle University-Manila
Auxiliary equationsThese equations can be substituted forward into one another and hence into rate equations. Unlike rate equations, auxiliary equations must be evaluated in proper order.
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Dennis T. Beng Hui, De La Salle University-Manila
Auxiliary equationsExample:
A DRUGS.K = POPN.K * 0.1R USERS.KL = DRUGS.K * 0.2L AIDS.K = AIDS.J + DT(BIRTH.JK + USERS.JK)
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Dennis T. Beng Hui, De La Salle University-Manila
Supplementary equationsSupplementary Equations are used to define variables which are not actually part of the model structure but arise in printing and plotting values of interest about the model. These equations are denoted y “S”.
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Dennis T. Beng Hui, De La Salle University-Manila
Initial-value equationsInitial-Value Equations are used to define initial values of all levels and some rates that must be given before the first cycle of model equation computation can begin. These also be values of some constants from other constants.Example:
N INVTY = 100
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Dennis T. Beng Hui, De La Salle University-Manila
Computational Interval (Solution Interval)
DT represents “Delta Time”It is the model time elapsing between computations in the simulation model.
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Dennis T. Beng Hui, De La Salle University-Manila
Computational Interval (Solution Interval)
The solution interval must be short enough so that its value does not seriously affect the computed results. It should also be long enough as permissible to avoid unnecessary digital-computer timeDT should be between one-half to one-tenth of the smallest time constant in the model.Common values are 0.50, 0.25, and 0.125)
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Dennis T. Beng Hui, De La Salle University-Manila
Coffee Cooling Model using Dynamo
*Coffee Cooling Temperature
NOTE COFTEMP.K = Coffee Temperature in CelsiusL COFTEMP.K=COFTEMP.J+DT*(COOL.JK) N COFTEMP=100
NOTE COOL.KL = Cooling Rate of CoffeeR COOL.KL=K(ROOM-COFTEMP.K)C ROOM=25NOTE K is a constantC K=.01
SPEC DT=.25/SAVPER=.25/LENGTH=5NOTE Time is in minutesSAVE COFTEMP,COOL,ROOM