Determine optimal interventions strategies

Estimation of financial needs and condition of infrastructure assets

Hamed Mehranfar
Yushu An

2024-04-15

How to evaluate possible maintenance intervention strategies

flowchart TB
    step1[<font size=2>Step 1: Estimate the state of each asset type at t=0]
    step2[<font size=2>Step 2: Determine the maintenance intervention strategy]
    step3[<font size=2>Step 3: Determine the steady state probabilities]
    step4[<font size=2>Step 4: Determine the costs associated with the steady state]
    step5[<font size=2>Step 5: Generate the view of condition and costs over time]
    step6[<font size=2>Step 6: Repeat steps 2-5 for all strategies]
    step7[<font size=2>Step 7: Compare results with goals and constraints]

    step1 --> step2
    step2 --> step3
    step3 --> step4
    step4 --> step5
    step5 --> step6
    step6 --> step7

Step 1 (1/2)

Estimate the state of each asset type at \(t=0\)


Example: Reinforced concrete bridges, in terms of surface area per state.

CS1 CS2 CS3 CS4 CS5 Total
6’878 4’149 3’186 2’570 0 16’782

Step 1 (2/2)

Estimate the state of each asset type at \(t=0\)


Once you know the objects you are considering, you can also determine their deterioration transition matrix

\(CS^{t}\) \(CS^{t+1}\)
1 2 3 4 5
1 0.956 0.044 0 0 0
2 0 0.968 0.032 0 0
3 0 0 0.956 0.044 0
4 0 0 0 0.944 0.056
5 0 0 0 0 1

Step 2

Determine the maintenance intervention strategy to be considered (1/3)


Consider the maintenance intervention strategies as follows:


IS CS1 CS2 CS3 CS4 CS5
1 Do nothing Do nothing Do nothing Rehabilitation Replacement
2 Do nothing Do nothing Do nothing Do nothing Replacement

Step 2

Determine the maintenance intervention strategy to be considered (2/3)

In connection to the effectiveness vectors and unit costs of interventions

CS Interventions CS Maintenance unit
costs (\(mu/m^2\)) 		Travel time unit
costs (\(mu/m^2\)) 		Accident unit
costs (\(mu/m^2\))
1 2 3 4 5
1 Do nothing 0.96 0.04 0 0 0 0 0 0
2 Do nothing 0 0.97 0.03 0 0 10 0 0
3 Do nothing 0 0 0.96 0.04 0 50 0 0
4 Do nothing 0 0 0 0.94 0.06 100 1000 100
Rehabilitation 0.75 0.25 0 0 0 1000 2000 200
Replacement 1 0 0 0 0 3500 5000 500
5 Replacement 1 0 0 0 0 3500 5000 500

Step 2

Determine the maintenance intervention strategy to be considered (3/3)

Translation into the deterioration-intervention matrices

CS 1 2 3 4 5
1 0.96 0.4 0 0 0
2 0 0.97 0.03 0 0
3 0 0 0.96 0.04 0
4 0.75 0.25 0 0 0
5 1 0 0 0 0

Step 3

Determine the steady state probabilities (1/4)

Finite-state Markov chain

\[P(S^{t+1}=j \mid S^t=i, S^{t-1}=i^{t-1},\ldots, S^1=i^1, S^0=i^0) \] \[= P(S^{t+1}=j \mid S^t=i) \] \[= P_{ij}\]

Step 3

Determine the steady state probabilities (2/4)

A one-step transition matrix, \(P\), with the same dimention as the deterioration matrix is needed, i.e., \(P_{K\times K}\)

\(S^t\) \(S^{t+1}\)
1 2 3 \(\cdots\) k
1 \(p_{11}\) \(p_{12}\) \(p_{13}\) \(\cdots\) \(p_{1k}\)
2 \(p_{21}\) \(p_{22}\) \(p_{23}\) \(\cdots\) \(p_{2k}\)
3 \(p_{31}\) \(p_{32}\) \(p_{33}\) \(\cdots\) \(p_{3k}\)
\(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\) \(\vdots\)
k \(p_{k1}\) \(p_{k2}\) \(p_{k3}\) \(\cdots\) \(p_{kk}\)

\(\quad p_{ij} \geq 0 \qquad \qquad\) \(\sum_{j=1}^{K} p_{ij}=1\)

Step 3

Determine the steady state probabilities (3/4)

Steady State Probabilites: \[ \lim_{n \rightarrow \infty} p_{ij}^n = \pi_j \quad ; \quad \pi_j \geq 0 \quad\textbf{&} \quad \sum_{k=1}^{K} \pi_j =1 \]

Step 3

Determine the steady state probabilities (3/4)

Steady State Probabilites: \[ \lim_{n \rightarrow \infty} p_{ij}^n = \pi_j \quad ; \quad \pi_j \geq 0 \quad\textbf{&} \quad \sum_{k=1}^{K} \pi_j =1 \]

Steady states are derermined using the follownig formula: \[ \pi_j = \sum_{k=1}^{K} \pi_i \cdot p_{ij} \qquad \forall\quad j=1,2,3,\cdots,k \quad\backepsilon \sum_{k=1}^{K} \pi_i=1 \]

Step 3

Determine the steady state probabilities (4/4)

CS Contributing bits to \(\pi_j\)
1 2 3 4 5
1 0.226 0.010 0 0 0
2 0 0.420 0.014 0 0
3 0 0 0.302 0.014 0
4 0.010 0.003 0 0 0
5 0 0 0 0 0

Step 3

Determine the steady state probabilities (4/4)

CS Contributing bits to \(\pi_j\) \(\pi_j\)
1 2 3 4 5
1 0.226 0.010 0 0 0 0.237
2 0 0.420 0.014 0 0 0.434
3 0 0 0.302 0.014 0 0.316
4 0.010 0.003 0 0 0 0.014
5 0 0 0 0 0 0
Sum 0.237 0.434 0.316 0.014 0 1

Step 4

Determine the costs associated with the steady state (1/2)

If \(N\) equals a large number of transitions, and \(X_i\) is the number of times an asset enters state \(i\)

\[ N \rightarrow\infty\quad,\frac{X_i}{N}\rightarrow \pi_i \] \[ \eta_\infty=\sum_{i=1}^{K}\pi_i\cdot C(i) \qquad\qquad C(i)=C(i)^{bI}+C(i)^{dI} \]

Step 4

Determine the costs associated with the steady state (2/2)

CS Contributing bits to \(\pi_j\) \(\pi_j\) Maintenance Unit Costs \((mu/m^2)\) Travel Time Unit Costs \((mu/m^2)\) Accident Unit Costs \((mu/m^2)\) Total Costs \((mu/m^2)\)
1 2 3 4 5
1 0.226 0.010 0 0 0 0.237 0 0 0 0
2 0 0.420 0.014 0 0 0.434 4.34 0 0 4.34
3 0 0 0.302 0.014 0 0.316 15.78 0 0 15.78
4 0.010 0.003 0 0 0 0.014 13.88 27.77 2.78 44.43
5 0 0 0 0 0 0 0 0 0 0
Sum 0.237 0.434 0.316 0.014 0 1 1 27.77 2.78 64.55

Step 5

Generate the view of how the condition and the costs evolve (1/2)


Asset
state		 p
1 0.237
2 0.434
3 0.316
4 0.014
5 0.000
Sum 1.000

Step 5

Generate the view of how the condition and the costs evolve (2/2)

Steady State

Asset
state		 Maintenance
Unit Costs
\((mu/m^2)\)		 Travel Time
Costs \((mu/m^2)\)
1 0.00 0.00
2 4.34 0.00
3 15.75 0.00
4 13.88 27.77
5 0.00 0.00
Sum 34.00 27.77
Asset
state		 Accident
Costs \((mu/m^2)\)		 Total
Costs \((mu/m^2)\)
1 0.00 0.00
2 0.00 4.34
3 0.00 15.78
4 2.78 44.43
5 0.00 0.00
Sum 2.78 64.55

Step 6

Repeat steps 2-5 for all maintenance intervention strategies

At least two interventions strategies for each asset subcategory.

  • Asset
    • Category 1
      • Intervention strategy 1
      • Intervention strategy 2
    • Category 2
      • Intervention strategy 1
      • Intervention strategy 2
      • Intervention strategy 3
    • Category 3
      • Intervention strategy 1
      • Intervention strategy 2

Step 7

Compare the results with respect to your goals and constraints (1/2)

Costs \((mu/m^2)\) IS1 IS2
Maintenance 18’782 39’030
Travel time 18’510 131’263
Accident 1’851 13’126
Total 40’142 183’419

Step 7

Compare the results with respect to your goals and constraints (2/2)

Steady state IS1 (prob.) IS1 \((m^2)\) IS2 (prob.) IS2 \((m^2)\)
1 0.237 3’972 0.238 3’991
2 0.434 7’281 0.327 5’488
3 0.316 5’296 0.238 3’991
4 0.014 233 0.187 3’136
5 0.000 0 0.010 176

Next steps

  • This session:
    • Finalize possible interventions and estimated intervention cost
    • If possible, determine optimal intervention strategy
  • Next session(s):
    • Finalize determining optimal intervention strategy for every asset category
    • Develop intervention programs

Thank you for your attention!

Questions?




Yushu An
Doctoral student
yushan@ethz.ch

ETH Zürich
Institute of Construction and Infrastructure Management (IBI)
Chair of Infrastructure Management
HIL G 32.1, Stefano-Franscini-Platz 5
8093 Zürich

www.ibi.ethz.ch