Chapter 14 of Sterman Formulating Nonlinear Relationships by yurtgc548


									Chapter 14 of Sterman:
Formulating Nonlinear
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 additive
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
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
More Table Functions

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

n   Occurs in make-to-order
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
Delivery Delay

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

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

n   Shipments = Capacity *
    Capacity Utilization
n   Capacity Utilization is a function
    of schedule pressure
n   Capacity Utilization = Schedule
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
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
n   Defines capacity as the normal
    rate of output, not the maximum
    possible rate when heroic efforts
    are made
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
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
Plausible shapes for the
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

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