Chapter 14 of Sterman: Chapter 14 of Sterman: Formulating Nonlinear Formulating Nonlinear

RelationshipsRelationships

Why have we been focusing Why have we been focusing on linear relationships?on linear relationships? Not because the relationships Not because the relationships

were in-fact linear!were in-fact linear! Because the mathematics were Because the mathematics were

much simplermuch simpler

Nonlinear relationships are Nonlinear relationships are fundamental in the dynamics of fundamental in the dynamics of systems of all types--Examples:systems of all types--Examples: You can’t push on a ropeYou can’t push on a rope When quality goes well below When quality goes well below

market, sales go to zero even if market, sales go to zero even if the price fallsthe price falls

Improvements in health care Improvements in health care and nutrition boost life and nutrition boost life expectancy up to a pointexpectancy up to a point

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

A nonlinear Relationship Y = A nonlinear Relationship Y = f(X1, X2,…, Xn)f(X1, X2,…, Xn) Y = X1Y = X122*X2/X3*…*Xn*X2/X3*…*Xn

A way to represent the nonlinear A way to represent the nonlinear effect: THE TABLE FUNCTIONeffect: THE TABLE FUNCTION

One problem: what ordinate value to One problem: what ordinate value to return when the abscissa is outside return when the abscissa is outside the range of defined ordinate valuesthe range of defined ordinate values—either to the left or right—either to the left or right

Solution: simply return the last Solution: simply return the last remaining ordinate valueremaining ordinate value

Ano. Solution: perform linear Ano. Solution: perform linear interpolation to extrapolate an interpolation to extrapolate an ordinate valueordinate value

Table FunctionsTable Functions

Normalize the input using Normalize the input using dimensionless ratiosdimensionless ratios

Normalize the output using Normalize the output using dimensionless ratiosdimensionless ratios

Identify reference points where Identify reference points where the values of the function are the values of the function are determined by definitiondetermined by definition The point (1,1) is one such pointThe point (1,1) is one such point Causes Y = Y* when x = x*Causes Y = Y* when x = x*

More Table FunctionsMore Table Functions

Identify reference policiesIdentify reference policies Consider extreme conditionsConsider extreme conditions Specify the domainSpecify the domain Identify plausible shapes within Identify plausible shapes within

the feasible regionthe feasible region Specify values for your best Specify values for your best

estimateestimate

More Table FunctionsMore Table Functions

Run the modelRun the model Test the sensitivity of your Test the sensitivity of your

results results See Table 14-1See Table 14-1

Capacitated DelayCapacitated Delay

Occurs in make-to-order Occurs in make-to-order systemssystems

Very commonVery common Arises any time the outflow from Arises any time the outflow from

a stock depends on the quantity a stock depends on the quantity in the stock and the normal in the stock and the normal residence time but is also residence time but is also constrained by maximum constrained by maximum capacitycapacity

Delivery DelayDelivery Delay

Is the average length of time Is the average length of time that an order is in the backlogthat an order is in the backlog

= backlog/shipments= backlog/shipments

Structure for a capacitated Structure for a capacitated delaydelay

BacklogOrders Shipments

DesiredProduction

Delivery Delay

Schedule Pressure

Target DeliveryDelay

CapacityUtilization Capacity

B

CapacityUtilization

Equations in the modelEquations in the model

Delivery delay = Delivery delay = backlog/shipmentsbacklog/shipments

Backlog = INTEGRAL(orders – Backlog = INTEGRAL(orders – shipments, Backlog Initial)shipments, Backlog Initial)

Desired Production = Desired Production = backlog/Target Delivery Delaybacklog/Target Delivery Delay

Shipments = F(Desired Shipments = F(Desired Production)Production)

Equations in the modelEquations in the model

Shipments = Capacity * Shipments = Capacity * Capacity UtilizationCapacity Utilization

Capacity Utilization is a function Capacity Utilization is a function of schedule pressureof schedule pressure

Capacity Utilization = Schedule Capacity Utilization = Schedule pressurepressure

Schedule pressure = Desired Schedule pressure = Desired Production / capacityProduction / capacity

Reference pointsReference points

Capacity is defined as the Capacity is defined as the normal rate of output achievable normal rate of output achievable given the firm’s resources.given the firm’s resources.

The capacity Utilization function The capacity Utilization function must pass through the reference must pass through the reference point (1,1)point (1,1)

For simplicity, I have set…For simplicity, I have set… Capacity Utilization = Schedule Capacity Utilization = Schedule

pressurepressure

Capacity Utilization Table Capacity Utilization Table function looks like…function looks like…

Schedule PressureSchedule Pressure

Schedule pressure = Desired Schedule pressure = Desired Production / capacityProduction / capacity

Is a dimensionless ratioIs a dimensionless ratio It is normalizedIt is normalized When Schedule Pressure = 1, When Schedule Pressure = 1,

shipments = Desired Production shipments = Desired Production = Capacity= Capacity

And, the actual delivery delay And, the actual delivery delay equals the targetequals the target

Normalization of Schedule Normalization of Schedule PressurePressure Defines capacity as the normal Defines capacity as the normal

rate of output, not the maximum rate of output, not the maximum possible rate when heroic efforts possible rate when heroic efforts are madeare made

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

Reference PoliciesReference Policies

Capacity Utilization = 1Capacity Utilization = 1 Capacity Utilization = Schedule Capacity Utilization = Schedule

PressurePressure Capacity Utilization = Slope max Capacity Utilization = Slope max

* Schedule Pressure* Schedule Pressure This corresponds to the policy of This corresponds to the policy of

producing and delivering as fast producing and delivering as fast as possible, that is with minimum as possible, that is with minimum delivery delaydelivery delay

Extreme conditionsExtreme conditions

The Capacity Utilization function The Capacity Utilization function must pass through the point (0,0) must pass through the point (0,0) and the point (1,1)and the point (1,1)

(0,0) because shipment must be (0,0) because shipment must be zero when schedule pressure is zero zero when schedule pressure is zero or else the backlog could become or else the backlog could become negative—an impossibilitynegative—an impossibility

At the other extreme, capacity At the other extreme, capacity utilization must be 1 when schedule utilization must be 1 when schedule pressure is maxed out at 1pressure is maxed out at 1

Specifying the domain for the Specifying the domain for the independent variableindependent variable Should encompass the entire Should encompass the entire

domain of possible abscissa domain of possible abscissa valuesvalues

Plausible shapes for the Plausible shapes for the functionfunction Use actual data if you have anyUse actual data if you have any Otherwise, bound the Otherwise, bound the

relationship by consider what is relationship by consider what is happening at the extreme pointshappening at the extreme points

Specifying the values of the Specifying the values of the functionfunction