Chapter 14 of Sterman: Formulating Nonlinear Relationships

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