# Chapter 14 of Sterman Formulating Nonlinear Relationships by yurtgc548

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```									Chapter 14 of Sterman:
Formulating Nonlinear
Relationships
Why have we been focusing
on linear relationships?
n   Not because the relationships
were in-fact linear!
n   Because the mathematics were
much simpler
Nonlinear relationships are
fundamental in the dynamics of
systems of all types--Examples:
n You can’t push on a rope

n When quality goes well below
market, sales go to zero even if
the price falls
n Improvements in health care
and nutrition boost life
expectancy up to a point
A Linear Relationship Y =
f(X1, X2,…, Xn)
n   Y = aX1 + bX2 + ….. + gXn
n   The effects are divisible
A nonlinear Relationship Y =
f(X1, X2,…, Xn)
n   Y = X12*X2/X3*…*Xn
A way to represent the nonlinear
effect: THE TABLE FUNCTION
n   One problem: what ordinate value to
return when the abscissa is outside
the range of defined ordinate values
—either to the left or right
n   Solution: simply return the last
remaining ordinate value
n   Ano. Solution: perform linear
interpolation to extrapolate an
ordinate value
Table Functions

n   Normalize the input using
dimensionless ratios
n   Normalize the output using
dimensionless ratios
n   Identify reference points where
the values of the function are
determined by definition
l The point (1,1) is one such point
l Causes Y = Y* when x = x*
More Table Functions

n   Identify reference policies
n   Consider extreme conditions
n   Specify the domain
n   Identify plausible shapes within
the feasible region
n   Specify values for your best
estimate
More Table Functions

n   Run the model
n   Test the sensitivity of your
results
n   See Table 14-1
Capacitated Delay

n   Occurs in make-to-order
systems
n   Very common
n   Arises any time the outflow from
a stock depends on the quantity
in the stock and the normal
residence time but is also
constrained by maximum
capacity
Delivery Delay

n   Is the average length of time
that an order is in the backlog
n   = backlog/shipments
Structure for a capacitated
delay
Equations in the model

n   Delivery delay =
backlog/shipments
n   Backlog = INTEGRAL(orders –
n   Desired Production =
backlog/Target Delivery Delay
n   Shipments = F(Desired
Production)
Equations in the model

n   Shipments = Capacity *
Capacity Utilization
n   Capacity Utilization is a function
of schedule pressure
n   Capacity Utilization = Schedule
pressure
n   Schedule pressure = Desired
Production / capacity
Reference points

n   Capacity is defined as the
normal rate of output achievable
given the firm’s resources.
n   The capacity Utilization function
must pass through the reference
point (1,1)
n   For simplicity, I have set…
n   Capacity Utilization = Schedule
pressure
Capacity Utilization Table
function looks like…
Schedule Pressure

n   Schedule pressure = Desired
Production / capacity
n   Is a dimensionless ratio
n   It is normalized
n   When Schedule Pressure = 1,
shipments = Desired Production
= Capacity
n   And, the actual delivery delay
equals the target
Normalization of Schedule
Pressure
n   Defines capacity as the normal
rate of output, not the maximum
possible rate when heroic efforts
n   If ‘normal’ met maximum
possible output, utilization is
less than one under normal
conditions, then Schedule
Pressure = Desired
Production/(Normal Capacity
Utilization * Capacity)
Reference Policies

n   Capacity Utilization = 1
n   Capacity Utilization = Schedule
Pressure
n   Capacity Utilization = Slope max
* Schedule Pressure
l   This corresponds to the policy of
producing and delivering as fast
as possible, that is with minimum
delivery delay
Extreme conditions

n   The Capacity Utilization function
must pass through the point (0,0)
and the point (1,1)
n   (0,0) because shipment must be zero
when schedule pressure is zero or
else the backlog could become
negative—an impossibility
n   At the other extreme, capacity
utilization must be 1 when schedule
pressure is maxed out at 1
Specifying the domain for the
independent variable
n   Should encompass the entire
domain of possible abscissa
values
Plausible shapes for the
function
n   Use actual data if you have any
n   Otherwise, bound the
relationship by consider what is
happening at the extreme points
Specifying the values of the
function

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