AN INTRO TO DECISION ANALYSIS 5/16/2016 1 What do Coin Tosses, Decision Making under Uncertainty , The Vessel Traffic Risk Assessment 2010 and Average Return Time Uncertainty have in common? SAMSI Workshop Presentation May 16 – May 20, 2016 Presented by: J. Rene van Dorp Jason R.W. Merrick (VCU) and J. Rene van Dorp (GW)
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AN INTRO TO DECISION ANALYSIS
5/16/2016 1
What do Coin Tosses, Decision Making under Uncertainty, The Vessel Traffic Risk Assessment 2010 and
Average Return Time Uncertainty have in common?
SAMSI Workshop Presentation May 16 – May 20, 2016 Presented by: J. Rene van Dorp
Jason R.W. Merrick (VCU) and J. Rene van Dorp (GW)
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams? 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
6. Return Time Uncertainty 5/16/2016 2
OUTLINE
AN INTRO TO DECISION ANALYSIS
5/16/2016 3
1. Imagine we have a coin and we flip it repeatedly
2. When heads turns up you “win” when tails turns up you “lose”
Suppose we flip the coin four times, how many times do you expect to win?
Suppose we flip the coin ten times, how many times do you expect to win?
2 times
5 times
WHAT ASSUMPTION(S) DID YOU MAKE?
AN INTRO TO DECISION ANALYSIS
5/16/2016 4
Conclusion: you made reasonable assumptions – 1. The coin has two different sides 2. When flipping it, each side turns up 50% of the time “on average”.
Would it have made sense to assume the coin had only one face
i.e. both sides show heads (or tails)? No
Assuming both sides show heads or tails is equivalent to making
a worst case or best case assumption.
AN INTRO TO DECISION ANALYSIS
5/16/2016 5
Suppose you actually flip the “fair” coin ten times How many times will “heads” turn up?
Answer could vary from 0 to 10 times, for example, First ten times : 3 times heads turns up Second ten times : 7 times heads turns up Third ten times : 6 times heads turns up Fourth ten times : 4 times heads turns up etc.
We say “on average” 5 out of ten times heads turns up
AN INTRO TO DECISION ANALYSIS
5/16/2016 6
0% 1%
4%
12%
21%
25%
21%
12%
4%
1% 0%
0%
5%
10%
15%
20%
25%
30%
0 1 2 3 4 5 6 7 8 9 10
Approximately 90% of ten throw series will have 3, 4, 5, 6 or 7 times heads turn up
Conclusion: While we expect 5 times heads to turn up, the actual number is uncertain!
AN INTRO TO DECISION ANALYSIS
5/16/2016 7
0%
5%
10%
15%
20%
25%
-2 0 2 4 6 8 10 12
Prob
abili
ty
Probabilities for Decision Tree '10 Tosses Coint 1'Optimal Path of Entire Decision Tree
0%
20%
40%
60%
80%
100%
-2 0 2 4 6 8 10 12
Cum
ulat
ive
Prob
abili
ty
Cumulative Probabilities for Decision Tree '10 Tosses Coint 1'Optimal Path of Entire Decision Tree
Probability Node
Risk Profile (RP) – Probability Mass Function (PMF)
Cumulative Risk Profile (CRP) – Cumulative Distribution Function (CDF)
Decision Analysis Software: Precision Tree
AN INTRO TO DECISION ANALYSIS
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams? 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
6. Return Time Uncertainty 5/16/2016 8
OUTLINE
AN INTRO TO DECISION ANALYSIS
5/16/2016 9
1. Imagine we have two coins: Coin 1 shows heads 50% of the time Coin 2 shows heads 75% of the time
2. When heads turns up, you win a pot of money. When tails turns up, you do not get anything.
You have to choose between Coin 1 and Coin 2 Which one would you choose? Coin 2
WHAT ASSUMPTION DID YOU MAKE? You assumed that the pot of money you win is
the same regardless of the coin you chose!
Coin 1 Coin 2
AN INTRO TO DECISION ANALYSIS
5/16/2016 10
1. Imagine we have two coins: Coin 1 shows heads 50% of the time Coin 2 shows heads 75% of the time
2. Each time heads turns up, you win the same pot of money. When tails turns up you do not get anything, regardless of the coin you throw.
You have to choose between two alternatives Alternative 1: Throwing ten times with Coin 1 Alternative 2: Throwing five times with Coin 2
Alternative 1 you expect to win 5 times and Alternative 2 you expect to win 3.75 times
Which alternative would you choose? CHOOSE
ALTERNATIVE 1
Coin 1 Coin 2
AN INTRO TO DECISION ANALYSIS
5/16/2016 11
Our objective is to maximize pay-off. So faced with uncertainty of pay-off outcomes we choose the alternative with largest average pay-off..
Reference Nodes
Decision Node
Probability Nodes
A DECISION TREE: The Basic Risky Decision
AN INTRO TO DECISION ANALYSIS
5/16/2016 12
0%
20%
40%
60%
80%
100%
-2 0 2 4 6 8 10 12
Cum
ulat
ive
Prob
abili
ty
Cumulative Probabilities for Decision Tree 'Coin Choice'Choice Comparison for Node 'Decision'
Flip Coin 1 10 Times
Flip Coin 2 5 Times
1. Deterministic Dominance 2. Stochastic Dominance 3. Make Decision Based on
Averages
Pr 𝑋 ≤ 𝑥 𝐶𝐶𝐶𝐶 1 ≤ Pr 𝑋 ≤ 𝑥 𝐶𝐶𝐶𝐶 2 ⇕
Pr 𝑋 > 𝑥 𝐶𝐶𝐶𝐶 1 ≥ Pr 𝑋 > 𝑥 𝐶𝐶𝐶𝐶 2
Observe from CRP’s on the Right
Chances of an “Unlucky” Outcome Increase going from 1, 2 to 3
Cumulative Risk Profiles of both Alternatives
AN INTRO TO DECISION ANALYSIS
5/16/2016 13
1. Imagine we have two coins: Coin 1 shows heads 50% of the time Coin 2 shows heads 75% of the time 2. Each time heads turns up with Coin 1 you win $2. Each time heads turns up with Coin 2 you win $4. When tails turns up you do not get anything.
You have to choose between two ALTERNATIVES Alternative 1: Throwing ten times with Coin 1 Alternative 2: Throwing five times with Coin 2
Alternative 1 you average 5 * $2 = $10 Alternative 2 you average 3.75 * $4 = $15
Which alternative would you choose? CHOOSE
ALTERNATIVE 2
Coin 1 Coin 2
AN INTRO TO DECISION ANALYSIS
5/16/2016 14
0% 1%4%
12%
21%25%
21%
12%
4%1% 0%0% 1%
9%
26%
40%
24%
0 2 4 6 8 10 12 14 16 18 20
Prob
abili
ty
Pay - Off Outcome
Alternative 1 Alternative 2Average Pay-Off Alt. 1: $10
Average Pay-Off Alt. 2: $15
Our objective is to maximize pay-off. So faced with uncertainty of pay-off outcomes we choose the alternative with largest average pay-off.
AN INTRO TO DECISION ANALYSIS
5/16/2016 15
1. Deterministic Dominance 2. Stochastic Dominance 3. Make Decision Based on
Averages
Pr 𝑋 ≤ 𝑥 𝐶𝐶𝐶𝐶 2 ≤ Pr 𝑋 ≤ 𝑥 𝐶𝐶𝐶𝐶 1 ⇕
Pr 𝑋 > 𝑥 𝐶𝐶𝐶𝐶 2 ≥ Pr 𝑋 > 𝑥 𝐶𝐶𝐶𝐶 1
Observe from CRP’s on the Right
Chances of an “Unlucky” Outcome Increase going from 1, 2 to 3
CRP’ S of both Alternatives
0%
20%
40%
60%
80%
100%
-5 0 5 10 15 20 25
Cum
ulat
ive
Prob
abili
ty
Cumulative Probabilities for Decision Tree 'Coin Choice'Choice Comparison for Node 'Decision'
Flip Coin 1 10 Times
Flip Coin 2 5 Times
Please Note Optimal Choice And Stochastic Dominance “Schwitched”
AN INTRO TO DECISION ANALYSIS
5/16/2016 16
Conclusion? When choosing between two alternatives entailing a series of coin toss trials, the following comes into play: 1. The number of trials N in each alternative 2. The probability of success P per trial 3. The pay-off amount W per trial
AVERAGE PAY-OFF = N × P × W Is it required to know the absolute value
of N, P and W to choose between these two alternatives?
AN INTRO TO DECISION ANALYSIS
5/16/2016 17
1. Imagine we have two coins: Coin 2 shows heads 1.5 times more than Coin 1 2. When heads turns up with Coin 2 you win 2 times the amount when heads turns up with Coin 1.
You have to choose between Two Alternatives Alternative 1: Throwing 2*N times with Coin 1 Alternative 2: Throwing N times with Coin 2
Average Pay – Off Alternative 2 : N × 1.5× P × 2 × W Average Pay – Off Alternative 1 : 2 × N × P × W
P = % Heads turns up with Coin 1, W = $ amount you win with Coin 1.
Average Pay-Off Alt. 2/Average Pay-Off Alt. 1 = 1.5
AN INTRO TO DECISION ANALYSIS
5/16/2016 18
Conclusion? When choosing between two alternatives entailing a series of trials, we can make a
choice if we know the multiplier between the average pay-offs, even when the absolute pay-off values over the two alternatives are unknown/uncertain
AN INTRO TO DECISION ANALYSIS
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
3.00
Pay-
Off
Fact
or
Probability Factor
-20-0 0-20
1.00 1.30 1.60 1.90 2.20 2.502.80
-20
0
20
1.00
1.15
1.30
1.45
1.60
1.75
1.90
Diffe
renc
e in
Pay
-Off
-20-0 0-20
5/16/2016 19
Coin 2 Alternative
Coin 1 Alternative
2D – Strategy Region Diagram
2D – Strategy Region Diagram
AN INTRO TO DECISION ANALYSIS
5/16/2016 20
Conclusion? When choosing between two alternatives
entailing a series of trials, we can make a choice if we know the sign of the difference between the average pay-offs, even when only ranges are available for the pay-off probability factors
using a strategy region diagram.
AN INTRO TO DECISION ANALYSIS
0.00
0.20
0.40
0.60
0.80
1.00
Utili
ty
Pay-Off
5/16/2016 21
What if your Value for Money depends on the amount you win per Coin Toss?
AN INTRO TO DECISION ANALYSIS
0.00
0.20
0.40
0.60
0.80
1.00
Utili
ty
Pay-Off
Scenario 1: Winning $2 with “Heads” Coin 1
1 at Max
0 at Min
1 at Max
0 at Min
Scenario 2: Winning $20,000 with “Heads” Coin 1
Concave: Risk Averse Linear: Risk Neutral
5/16/2016 22
What if your Value for Money Changes depends on your wealth?
AN INTRO TO DECISION ANALYSIS
• Linear Utility Function implies the Decision Maker (DM) is Risk Neutral. A DM is Risk Neutral if he/she is indifferent between a bet with an expected pay-off and a sure amount equal to the expected pay-off.
• Concave Utility Function implies a Decision Maker (DM) is Risk Averse. A DM is Risk Averse if he/she is willing to accept less money for a bet with a certain expected pay-off than the expected pay-off.
• Convex Utility Function implies a Decision Maker (DM) is Risk Seeking. A DM is Risk Seeking if he/she is willing to pay more money for a bet with a certain expected pay-off than the expected pay-off.
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
3.00
Pay-
Off
Fact
or
Probability Factor
-0.5-0 0-0.5
1.00 1.30 1.60 1.90 2.20 2.502.80
-0.5
0
0.5
1.00
1.15
1.30
1.45
1.60
1.75
1.90
Diffe
renc
e in
Util
ity
-0.5-0 0-0.5
5/16/2016 23
Coin 2 Alternative
Coin 1 Alternative
2D – Strategy Region Diagram
2D – Strategy Region Diagram
AN INTRO TO DECISION ANALYSIS
Now Max. Exp. Utility
5/16/2016 24
AN INTRO TO DECISION ANALYSIS
Now Max. Exp. Utility
0.00
0.20
0.40
0.60
0.80
1.00
Utili
ty
Pay-Off
For how much money are you willing to sell this decision? 0.87
$142,018 Called Certainty Equivalent (CE)
$142,018
Provides for an Operational Interpretation
of the Utility Concept.
< $150,000
5/16/2016 25
AN INTRO TO DECISION ANALYSIS
Now Max. Exp. Utility
0.00
0.20
0.40
0.60
0.80
1.00
Utili
ty
Pay-Off
How much money are you willing to give up to not play?
0.87
$150,000 - $142,018 =
$142,018 < $150,000
$7,982
Called Risk Premium
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams? 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
6. Return Time Uncertainty 5/16/2016 26
OUTLINE
AN INTRO TO DECISION ANALYSIS
5/16/2016 27
Decision Trees or Influence Diagrams?
AN INTRO TO DECISION ANALYSIS
Lot of Detail, but become Unwieldy
Coin 1 Coin 2
Coin Series Choice
Pay Throw Coin 1
2*N Times
Pay Throw Coin 2
N times
Lack of Detail, Higher level View And Makes Dependence Explicit
Max Pay-Off
5/16/2016 28
Some Basic Influence Diagram Examples
AN INTRO TO DECISION ANALYSIS
Business Result
Investment Choice
Basic Risky Decision
Arc? Yes or No?
Return on Investment
Source: Clemen and Reilly (2014), Making Hard Decisions, Cengage Learning
5/16/2016 29
Some Basic Influence Diagram Examples
AN INTRO TO DECISION ANALYSIS
Evacuate?
Imperfect Information
Consequence
Hurricane Path
Weather Forecast
Time Sequence
Arc
Reverse Influence
Arc?
Source: Clemen and Reilly (2014), Making Hard Decisions, Cengage Learning
5/16/2016 30
Influence Diagram Example – EPA Decision
AN INTRO TO DECISION ANALYSIS
Allow Chemical?
Max. Net
Value
Two Imperfect Information Diagrams in one Influence Diagram
Exposure to Usage
Max. Economic
Value
Min. Cancer Cost
Carcino-genic
Potential
Usage Survey
Lab Tests
Multiple Conflicting Objectives
Source: Clemen and Reilly (2014), Making Hard Decisions, Cengage Learning
5/16/2016 31
AN INTRO TO DECISION ANALYSIS
Release 1? Release 2? Final Release?
Current Reliability
Outcome Test 1
Outcome Test 2
Reliability after Test 1
Reliability after Test 2
Profit if released after 1
Cost of Test 1 & Redesign
Profit if released after 2
Cost of Test 2 & Redisgn
Profit if released after final
FINAL PROFITS
Influence Diagram Example – Reliability Growth Decision
Multiple Sequential Decisions
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams? 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
6. Return Time Uncertainty 5/16/2016 32
OUTLINE
AN INTRO TO DECISION ANALYSIS
5/16/2016 33
Elements of Decision Analysis (DA)
AN INTRO TO DECISION ANALYSIS
• Multiple Decisions: The immediate one and possibly more. Decisions are sequential in time. The DP is called dynamic.
• Multiple Uncertainties: Each uncertainty node requires a probability model. Multiple uncertainty nodes may be statistically dependent.
• Multiple or Single Objectives: In case of multiple conflicting objective the trade-off between objectives needs to be modelled.
• Multiple values: Evaluation of achievements of each individual objective requires description of a utility function for each one
(linear, concave, convex?)
DA’s are Complex!
5/16/2016 34
Skill Set/Techniques for Decision Analysis (DA)
AN INTRO TO DECISION ANALYSIS
• Decision Tree/Influence Diagrams: To structure and visualize DP’s, identify its elements and prescribe the method towards evaluation.
• Expert Judgement (EJ) Elicitation: To describe/specify probability models of “on-off” uncertainty nodes and to combine expert judgements.
• Statistical Inference: In DA the inference is typically Bayesian in nature. Is used when uncertainties reveal themselves over time to refine/update probability models or combine available data with Expert Judgement.
• Utility Theory: To describe “The Decision Maker’s” risk attitude/ appetite for the evaluation of a single objective and to formalize trade-off between multiple objectives.
Thus, a DA’s is Normative in Nature !
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams? 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
• BP Cherry Point Refinery • Ferndale Refinery • March Point Refinery
VTRA 2010 Study Area
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
5/16/2016 39
What was The Objective in Coin Toss Example? Maximize Average Pay-Off
What is the Objective in a Maritime Risk Assesment? Minimize Average Potential Oil Loss
Truth be told, for some the objective is to Maximize Average Pay-Off, for some it is to Minimize Average Potential Oil Loss
and for others it is to Achieve Both.
For sake of argument, lets take in Maritime Risk Assessment a focus towards Minimizing Average Potential Oil Loss, while
recognizing the Maximize Average Pay-Off Objective is also at play.
ciii xlsR },,{ ><=Risk Analysis Objective: Evaluate Oil Spill System Risk described by a “complete” set of traffic situations
Situations Incidents Accidents Oil Spill
Maritime Simulation
Traffic Situations
Expert Judgment + Data
Incident Data
Likelihoods
Oil Outflow Model
Consequences
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
An Oil Spill is a series of cascading events referred to as a Causal Chain
Coin Toss Analogy: Trials % of Heads (P) Winnings ($) Pay-off Risk was defined by N identical Trials
5/16/2016 40
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
5/16/2016 41
VTRA 2010 Analysis Approach In light of uncertainties inherent to any risk analysis, we choose not to focus on; • absolute evaluations of risk levels, but to focus on • relative risk changes from a base case scenario by adding or removing traffic to or from that base case.
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
5/16/2016 42
VTRA 2010 Analysis Approach A Base Case (BC) Analysis Framework is constructed while; • making reasonable assumptions (not worst or best case), and • What-if (WI), Bench-Mark (BM) and Risk Mitigation Measure (RMM) cases are analyzed within that framework.
• Base Case (BC) system wide risk levels are set at 100%, and • System wide % changes up or down are evaluated for What-if (WI), Bench-Mark (BM) and Risk Mitigation Measure (RMM), moreover • Location-Specific Multipliers are evaluated for 15 Waterway Zones.
VTRA 2010 Analysis Approach Collision System Exposure in Base Case:
• Approximately 10,000 grid cells of 0.5 x 0.5 mile in VTRA study area with Vessel to Vessel traffic situations. • Approximately 1.8 Million Vessel to Vessel Traffic Situations per year generated by VTRA 2010 Model. • Vessel to Vessel Traffic Situations per cell per year range from 1 – 7,000 (or on average about 0 – 20 per day per cell) .
Recall Coin Toss – Traffic Situation Analogy: “1.8 Million Coin Tosses with very small probability of Tails”
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
5/16/2016 50
VTRA 2010 Analysis Approach Grounding System Risk in Base Case:
• Approximately 4,000 grid cells of 0.5 x 0.5 mile in VTRA study area with Vessel to Shore traffic situations. • Approximately 10 Million Vessel to Shore Traffic Situations per year generated by VTRA 2010 Model. • Vessel to Shore Traffic Situations per cell per year range from 1 – 55,000 (or on average about 0 – 150 per day) .
Recall Coin Toss – Traffic Situation Analogy: “10 Million Coin Tosses with very small probability of Tails”
1. Coin Tosses 2. Decision Making under Uncertainty 3. Decision Trees or Influence Diagrams 4. Elements of Decision Analysis 5. VTRA 2010 Case Study
• Base Case Traffic Description • What-If and Benchmark Cases
6. Return Time Uncertainty 5/16/2016 51
OUTLINE
AN INTRO TO DECISION ANALYSIS
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
P: Base Case 3D Risk Profile MAP TO DISPLAY - Vessel Time Exposure
23-24 22-23
21-22 20-21
19-20 18-19
17-18 16-17
15-16 14-15
13-14 12-13
11-12 10-11
9-10 8-9
7-8 6-7
5-6 4-5
3-4 2-3
1-2 0-1
Neah Bay
Victoria Seattle
Bellingham
Tacoma
VESSEL TIME EXPOSURE (VTE) = Annual amount of time a location is exposed to a vessel moving through it
5/16/2016 52
P: Base Case 3D Risk Profile ALL TRAFFIC - Vessel Time Exposure: 100%Total VTE
23-24 22-23
21-22 20-21
19-20 18-19
17-18 16-17
15-16 14-15
13-14 12-13
11-12 10-11
9-10 8-9
7-8 6-7
5-6 4-5
3-4 2-3
1-2 0-1
ALL VTRA TRAFFIC – VTOSS 2010 TRAFFIC + SMALL VESSEL EVENTS
VESSEL TRAFFIC RISK ASSESSMENT (VTRA) 2010
Neah Bay
Victoria Seattle
Bellingham
Tacoma
VESSEL TIME EXPOSURE (VTE) = Annual amount of time a location is exposed to a vessel moving through it
5/16/2016 53
P: Base Case 3D Risk Profile NON FV - Vessel Time Exposure: 75%Total VTE
Evaluating average return uncertainty Recall VTRA 2010 Maritime Simulation Model generated • 1.8 Million Vessel to Vessel Traffic Situations per Year • 10 Million Vessel to Shore Traffic Situations per Year
Accident Probability per Traffic Situation
(1000 - 7500] (7500 - 15000] (15000 or More)
1 e -10 N1 N2 N3
1 e -9 N4 N5 N6
1 e -8 N7 N8 N9
POTENTIAL OIL LOSS VOLUME (m3) CATEGORY
Used VTRA 2010 Model to create table of following format