System Dynamics
Homework 7
Homework 7 - Condition, Deterioration, and Shocks
Problem 1: model selection and description
We discussed in class a methodology to represent the deterioration of the condition of an inventory of assets. The inventory may have hundreds to thousands of buildings.
The structure of the model (below), follows a 2-parameter Weibull failure rate (also known as survival function).
The parameters of the Weibull (shown below), were given an interpretation in terms of maintenance and repair schedule (M&R) and retrofit operations.
where:
• λ = lifespan = mean life span + effect of M&R quality
• k = failure rate, a function of the major interventions such as retrofit
• t = time
The CDF is
And the survival function (blue curve is)
S = 1 - F(t)
Digression about characteristics of the Weibull
When choosing parameter values in the simulation, notice that even though lambda is constant, the inflection point (i.e. location of the change of dominance of feedback loop = max value of the f(x)) also changes with k. The figures below help you see how this happens in a Weibull function for different lambdas and k’s
Figure 1: 2-parameter Weibull model with lambda = 150 and changing k. (Left) survival functions with inflection points (blue dots). (Right) density functions with inflection/max points (blue dots).
An interesting fact is that the time at which there’s a change of slope is correlated with k.
Figure 2: change of location of feedback dominance as a function ofk for lambda = 150
For other cases, with different lambda, note how the inflection point changes and pi → λ for increasing k.
With these observations in mind, now consider the following questions:
Questions:
• Identify and describe both feedback loops present in the Weibull model. As shown in the figures earlier, they’re clearly divided by an inflection point (shown by blue dots) that changes as a function of k. The left (right) branch, i.e. feedback loops 1 and 2, are located to the left (right) of the blue dot. What physical interpretation this has?
• Select one other model (e.g. 3-parameter Weibull, exponential, Lognormal, Log-logistic, Gamma, Inverse Gaussian), and update the Vensim model provided on Canvas and reinterpret the model parameters in terms of M&R and retrofit. Modify the model as you see fit to incorporate more/less complexity with respect to the Weibull model.
Present the plots to behavior and comment about how each model represents the laws of deterioration.
• The model currently has not feedback loops. Is this realistic?
• Extra 20% points: develop a physically better (yet convincing!) interpretation of the
lambda and k in terms of M&R and retrofit than that in model provided. Please include references that you use.
Question 2: interventions
A more complete deterioration model of an inventory of buildings incorporates the effect of random damages induced by shocks (hurricanes, earthquakes). This model incorporates the following:
• Hurricane occurrences are described with a Poisson stochastic process. Damage
severities are described with a Uniform. stochastic process. You don't need to modify any of this too variables.
• The law of deterioration is governed by the intrinsic characteristics of the building
typology: mean life span, baseline M&R, and baseline failure rate. You don’t need to modify these variables.
• On the other hand, the deterioration is counteracted by four mechanisms: enhanced
M&R, spasmodic retrofit schedule, reactive retrofit, and reconstruction. The enhanced M&R are pre-disaster, the spasmodic retrofit is pre- disasters. Reactive retrofit is post- disaster, building sturdier constructions after a disaster, and reconstruction is also post- disaster, but with the particularity that has a Delay 1. You should adjust these.
Figure 3: Complete model deterioration
The bold assumptions, as in question 1, are that:
k ∝ f(retrofit schedules)
And
λ ∝ f(maintenance and repair schedule)
Roughly,
Lambda := Lifespan = Mean Lifespan * (1 + "Enhanced M&R" + "Baseline M&R")
k := Failure Rate = 5 - (Reactive Retrofit + Spasmodic Retrofit)
• Do you consider that it would be better to express the variables the other way around? i.e. retrofit increases lifespan, and M&R the failure rates? If so, implement, compare the before and after and justify your final choice.
• Develop an M&R policy which is based on the current Condition level, or damage severity, and modify the model.
• Assign a unit cost ($) to each operation (i.e. M&R, retrofit, and reconstruction), and conduct a simple Cost/Benefit analysis of the most adequate combination of the 3 operations (leave spasmodic retrofit out) . Benefit is defined as the damage avoided (as a function of condition, for example, or degradation). Feel free to suggest options and describe your assumptions. Cost: is simply the aggregation of the interventions times the unit cost. Adopt a consistent set values for each.
• You need to conduct a few “scenarios” of interventions, and write down the cost, benefits, and cost/benefit in a table like the one below. For example, Reconstruction intensity is a percentage of current damage, so, if damage is less because an increased M&R, that will save costs of Reconstruction. Indicate which one is the most convenient strategy.
• The problem is described generally, so if you think that there are missing inputs, feel free to make assumptions and list them clearly.
|
Scenario
|
M&R
|
Retrofit
|
Reconstruction
|
Cost
|
Benefit
|
C/B
|
|
1
|
|
|
|
|
|
|
|
2
|
|
|
|
|
|
|
|
3
|
|
|
|
|
|
|
|
4
|
|
|
|
|
|
|
• Please submit a report and the Vensim files you create.