1. Chapter 14 of Sterman: Formulating Nonlinear Relationships

2. Why have we been focusing on linear relationships?
Not because the relationships were in-fact linear!
Because the mathematics were much simpler

3. Nonlinear relationships are fundamental in the dynamics of systems of all types--Examples:
You can’t push on a rope
When quality goes well below market, sales go to zero even if the price falls
Improvements in health care and nutrition boost life expectancy up to a point

4. A Linear Relationship Y = f(X1, X2,…, Xn)
Y = aX1 + bX2 + ….. + gXn
The effects are additive
The effects are divisible

5. A nonlinear Relationship Y = f(X1, X2,…, Xn)
Y = X12*X2/X3*…*Xn

6. A way to represent the nonlinear effect: THE TABLE FUNCTION
One problem: what ordinate value to return when the abscissa is outside the range of defined ordinate values—either to the left or right
Solution: simply return the last remaining ordinate value
Ano. Solution: perform linear interpolation to extrapolate an ordinate value

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

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

9. More Table Functions Run the model
Test the sensitivity of your results
See Table 14-1

10. Capacitated Delay Occurs in make-to-order systems Very common
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

11. Delivery Delay
Is the average length of time that an order is in the backlog
= backlog/shipments

12. Structure for a capacitated delay

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

14. Equations in the model Shipments = Capacity * Capacity Utilization
Capacity Utilization is a function of schedule pressure
Capacity Utilization = Schedule pressure
Schedule pressure = Desired Production / capacity

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

16. Capacity Utilization Table function looks like…

17. Schedule Pressure Schedule pressure = Desired Production / capacity
Is a dimensionless ratio
It is normalized
When Schedule Pressure = 1, shipments = Desired Production = Capacity
And, the actual delivery delay equals the target

18. Normalization of Schedule Pressure
Defines capacity as the normal rate of output, not the maximum possible rate when heroic efforts are made

19. If ‘normal’ met maximum possible output, utilization is less than one under normal conditions, then Schedule Pressure = Desired Production/(Normal Capacity Utilization * Capacity)

20. Reference Policies Capacity Utilization = 1
Capacity Utilization = Schedule Pressure
Capacity Utilization = Slope max * Schedule Pressure
This corresponds to the policy of producing and delivering as fast as possible, that is with minimum delivery delay

21. Extreme conditions
The Capacity Utilization function must pass through the point (0,0) and the point (1,1)
(0,0) because shipment must be zero when schedule pressure is zero or else the backlog could become negative—an impossibility
At the other extreme, capacity utilization must be 1 when schedule pressure is maxed out at 1

22. Specifying the domain for the independent variable
Should encompass the entire domain of possible abscissa values

23. Plausible shapes for the function
Use actual data if you have any
Otherwise, bound the relationship by consider what is happening at the extreme points

24. Specifying the values of the function