1. A. Mancusoa,b, M. Compareb, A. Saloa, E. Ziob,c
A Bayesian approach for dynamic portfolio optimization of safety barriers
A. Mancusoa,b, M. Compareb, A. Saloa, E. Ziob,c
Systems Analysis Laboratory, Department of Mathematics and Systems Analysis - Aalto University
Laboratory of Signal and Risk Analysis, Dipartimento di Energia - Politecnico di Milano
Chair on Systems Science and the Energetic Challenge - École Centrale Paris and Supelec
December 22, 2017

2. Risk-informed decision making in safety critical context
Based on Probabilistic Risk Assessment (PRA)
Fault Tree
𝑅𝑅𝑊 𝐺𝑒𝑛 = 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡) 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡|𝑁𝑜 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 𝑓𝑎𝑖𝑙𝑢𝑟𝑒)
𝑅𝑅𝑊 𝐿𝑎𝑚𝑝 = 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡) 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡|𝑁𝑜 𝑙𝑎𝑚𝑝 𝑓𝑎𝑖𝑙𝑢𝑟𝑒)
𝑅𝑅𝑊 𝑆𝑤𝑖𝑡𝑐ℎ = 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡) 𝑅(𝑁𝑜 𝑙𝑖𝑔ℎ𝑡|𝑁𝑜 𝑠𝑤𝑖𝑡𝑐ℎ 𝑓𝑎𝑖𝑙𝑢𝑟𝑒)
Concerns
Experts interpret these importance measures and choose actions
Action costs and feasibility constraints considered only afterwards
The results can be sub-optimal

3. Proposed methodology
Portfolio optimization identifies portfolios of risk management actions for the whole system, which minimize the residual risk of the system and the total cost of actions.
The methodology accounts for risk, budget and other feasibility constraints.
Methodology steps:
Step 1: Failure scenario modeling
Step 2: Definition of failure probabilities
Step 3: Specification of actions
Step 4: Optimization model

4. Step 1: Failure scenario modeling
Mapping of Fault Tree (FT) into Bayesian Network.
Advantages
Multi-state modeling
Extension of concepts of AND/OR gates
Combining expert judgement and quantitative knowledge to estimate the risk.
Reference: Khakzad N., Khan F., Amyotte P., “Dynamic safety analysis of process systems by mapping bow-tie into Bayesian network”, Process Safety and Environmental Protection 91 (1-2), pp (2013).

5. Step 1: Failure scenario modeling
The final outcome of accident scenarios can depend on the order, timing and magnitude of the component failures. If the risk analysis does not account for the dynamic behaviour of the system, it may fail to consider severe accident scenarios. We model time-dependent accident scenarios through Dynamic Bayesian Networks, which generalize Bayesian Networks by connecting nodes over successive time steps.
Reference: Zio E., “Integrated deterministic and probabilistic safety analysis: concepts, challenges,
research directions”, Nuclear Engineering and Design 280, pp (2014).

6. Step 2: Definition of failure probabilities
Information sources
Information provided by AND/OR gates in Fault Tree and Event Tree
Statistical analyses
Expert elicitation
The probabilities of events are defined as follows:
Initiating events  failure probabilities of system components
Intermediate and top events  conditional probability tables

7. Step 3: Specification of actions
Parameters of actions:
Impact on the prior and conditional probabilities at each time step
Annualized cost
Action 𝑎 on event 𝑋 𝑖 (τ) modifies the probability of occurrence of state 𝑠.
𝑃 𝑋 𝑖 (𝜏) 𝑠
𝑃 𝑋 𝑎 𝑖 (𝜏) 𝑠 𝑠 𝑠

8. Step 4: Optimization model
Action portfolio #1
Action portfolio #2
Risk acceptability
Action portfolio #3
Implicit enumeration algorithm to identify the optimal portfolios of risk management actions.
The resulting portfolios are globally optimal: they minimize the failure risk of target events (instead of selecting actions that target the riskiness of the single components).
Action portfolio #4
Action portfolio #5
Budget constraints
Action portfolio #6
Action portfolio #7
Select the Pareto optimal frontier
Action portfolio #8
Action feasibility
Action portfolio #9
Action portfolio #10
Action portfolio #11
Action portfolio #12

9. Application: Mixing tank mechanical system
Accident scenario of a vapor cloud ignition at Universal Form Clamp, Illinois, US on June
In that occasion, a flammable vapor cloud of heptane and mineral spirits overflowed from an open top mixing and heating tank. The vapor cloud ignited as it met unknown ignition sources, leading to one death, two injuries and significant business interruption.
What portfolios of risk management actions minimize the residual system risk for different total cost of risk management actions?

10. Step 1: Failure scenario modeling
To analyze the failure scenarios, the Fault Tree is mapped into a Dynamic Bayesian Network.

11. Dynamic framework
The DBN accounts for 6 time steps for the nodes following the Top Event
Vapor due to the fast dynamics of the accident scenario in case of vapor overflow.
The squared number 𝑟 = 1 on the arc connecting Sprinkler to Ignition indicates the conditional dependency of Ignition at time 𝜏 to the activation of Sprinkler at time 𝜏−1.

12. Application: Mixing tank mechanical system
Actions improve the reliability of the components, thus they mitigate the residual risk of the overall system.
Alarm
Action
Cost
Prob
Semi conductor sensor 60
0.2
Catalytic gas sensor 80
0.15
Electrochemical cells
0.1

13. Results
Minimum residual risk for the optimal portfolios of actions at different budget levels.
Larger budget  more effective actions  lower residual risk at each time step

14. Results
The core index 𝐶𝐼(𝑎) is defined as the share of non-dominated portfolios that include the safety
measure 𝑎.
The analysis of the core indexes helps identify those safety measures that should be selected
or rejected.

15. Future research
Develop computationally tractable solutions for large systems.
Develop a further extension to account for safety actions that can be activated on time according to the occurring events.
Possible collaboration with nuclear power plants with regard to inspection of fire protection barriers.

16. Thank you for your attention!
Alessandro Mancuso
System Analysis Laboratory, School of Science, Aalto University, Finland
Laboratory of Signal and Risk Analysis, Department of Energy, Politecnico di Milano, Italy

17. Results
By setting the budget to 600k€, the optimization model computes three non-dominated solutions. Portfolio 𝑧 1 dominates
the other two solutions at time steps 𝜏≥1, but at the initial time step 𝜏=0 it shows a higher risk. If these increases are not deemed to be significant, then portfolio 𝑧 1 could be
recommended as the optimal solution.
0.13%
0.45%