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MECH 350Engineering Design I

University of VictoriaDept. of Mechanical Engineering

Lecture 8: Decision Making II

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INFORMATION UNCERTAINTY & DECISION TREESDECISION TREES

- SENSITIVITY ANALYSIS- CONDITIONAL UNCERTAINTIES- SEQUENTIAL DECISIONS

Outline:

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Detailed Design-Detailed Analysis-Simulate & Optimize-Detail Specifications-Drawings, GD&T

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Decision Making within the “General” Design Process

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Identify Need-Talk with Client-Project Goals-Information Gathering

Conceptualization-Brainstorming-Drawing/Visualization-Functional Decomp.-Morphologic Chart

Preliminary Design & Planning-Prelim. Specifications-Prelim. Analysis-Decision Making-Gantt Charts & CPM

Report/Deliver-Oral Presentation-Client Feedback-Formal Design Report

Prototyping-Prototype Fabrication-Concept Verification

Testing/Evaluation-Evaluate Performance-Are Objectives Met?-Iterate Process Steps 2 - 7 as needed

Problem Definition-Problem Statement-Information Gathering-Design Objectives(quantifiable/measurable)

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An essential part of “Design” is to make choices between alternatives. However, there are often many uncertainties for systems that are still in the ‘design stage’, such as:

- future costs- future events- future performance- _______________

Therefore, we need an “effective decision making method” that can incorporate/account-for uncertainties.

Decision Making With Uncertainty

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Once such method is the “Decision Tree”. It is ‘graphical method’ to assist in making decisions that involve uncertainty.

The method is based upon the use ‘probabilities of occurrence’ of certain events, and the “costs” associated with those events.

Decision Making With Uncertainty: Decision Trees

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Example of Decision Tree[Figure 9.10 from Hyman Textbook]

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A Decision Tree incorporates all decision alternatives and uncertainties into a convenient visual format.

They provide a complete and compact history of the decision making process, where all decisions, all uncertainties, and their assumed probabilities are shown.

They are a valuable tool for clarifying:The options being consideredThe uncertainties being accounted forThe probabilities for uncertainties occurringThe outcomes resulting from various occurrences

Decision Making: Decision Trees Advantages

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The “decision rule” used with the Decision Tree examples of these lecture notes is “costs” in terms of monetary amounts.

However, a “decision rule” can be any “measurable value or quantifiable utility” of a particular approach.

For simplicity in teaching this method, these notes will only use “costs”.

Decision Making: Decision Trees

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Basic Elements of a Decision Tree:

Branches:- Straight lines that connect different types of nodes

Decision Node:- ________________________________________

Event Node:- ________________________________________

Payoff Node:- ________________________________________

Decision Making: Decision Trees

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Making a Decision Tree:Step 1: Draw Initial NodeStep 2: Add Initial BranchesStep 3: Terminate Each Branch in a NodeStep 4: Repeat Cycle Until Completion.

4A: If the added Node is a Decision, return to Step 2 to construct branches emanating from it.4B: If the added Node is an Event, return to Step 2 to construct branches to represent each possible outcome.4C: If the added Node is a Payoff, no further branches are added.

Decision Making: Decision Tree Construction

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Example: Constructing a Decision Tree

Problem Statement: A test procedure must be done at a remote location, and you have an ‘old portable generator’ to provide power for the test. The total cost for conducting the test will be $10,000 (if the generator works). If the generator fails during the test, it will cause $25,000 of damage to the other equipment. If you overhaul the generator now (prior to test), it will cost $15,000 for the overhaul.

Question: What should be done?

Decision Making: Decision Trees

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FIRST: Must gather more information:Q: What is the probability that the generator will fail?A: The group decides there is a 30% chance of failure.

Lets draw a Decision Tree to represent this:Begin with Steps 1 & 2:

Decision Making: Decision Trees

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Use

Overhaul

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Decision Tree for Generator Problem:

Step 3:Note: Cost for Payoff Node on “Overhaul” Branch is: $15K + $10K.

Decision Making: Decision Trees

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Use

Overhaul25

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Decision Tree for Generator Problem:

Steps 4B & 4C:

Decision Making: Decision Trees

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Use

Overhaul25

Pr(W)=0.70

Pr(F)=0.30

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Decision Tree for Generator Problem:

Steps 3 & 4C:

Decision Making: Decision Trees

© N. Dechev, University of Victoria

Use

Overhaul25

Pr(W)=0.70

Pr(F)=0.3035

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Solving Decision Trees:Step 5: Trace back from the Payoff NodesStep 6: For Event Nodes, calculate the “Expected Value (EV)” at each Event Node, starting from the right and working to left:

- where: EV = Sum of all Branch Costs (BC) entering the Event Node on its right.- where: BC = Probability of that Branch (Pr()) multiplied by the EV of the Node (Event, Decision, or Payoff) on its right.

Step 7: For Decision Nodes, evaluate the BC of all Branches entering on its right. Select the best option (lowest BC) and enter that value in the Decision Node, and scratch off (with double marks) the less desirable branches.Step 8: Repeat Steps 6 and 7 until all Node values have been computed, including the starting Decision Node.

Decision Making: Decision Trees

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Solving Decision Trees: The Generator Problem:Note: EV for Node A = (0.70)(10) + (0.30)(35) = 17.5

Decision Making: Decision Trees

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Pr(W)=0.70

Pr(F)=0.3035

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17.5

Node A

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Solving Decision Trees: The Generator Problem

Decision Making: Decision Trees

© N. Dechev, University of Victoria

Use

Overhaul25

Pr(W)=0.70

Pr(F)=0.3035

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17.5

17.5

Therefore, the lowest Branch Cost (BC) stemming from the Start Decision Node is $17,500. Hence, we choose to “Use” that alternative.

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Decision Trees require “pre-requisite knowledge” of the probability of occurrence of events. However, it is often difficult to assign probability values for uncertain events.

In cases of uncertainty with probability values for events, a “Sensitivity Analysis” may be useful to perform.

Or, it may be better re-formulate the problem, and ask the question: “What is the range probability values (max, or min) for an event to occur?”

Decision Making: Decision Trees:Sensitivity Analysis

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Consider the “Generator Problem” from Lecture 9. Previously, we had a 30% probability of failure, 70% probability of success. However, what if we are not confident with these values?We can reformulate the problem as: “What is the tolerable probability of failure, before we choose to overhaul the generator?”

Decision Making: Decision Trees:Sensitivity Analysis

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Therefore, the EV at the event node is: EV = (1 - x)(10) + (x)(35) = 10 + 25xThe decision to choose one branch vs. the other branch, rests on the payoff value of 25 for the “overhaul” branch.Therefore, we equate the branches to get: 10 +25x = 25Hence, x = 0.60

Decision Making: Decision Trees:Sensitivity Analysis

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I.e. we can tolerate a 60% chance of failure, before deciding to go for an overhaul.

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Decision Making: Decision Trees:Conditional Uncertainties

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Consider the “Generator Problem” where the failure outcome, can result in the occurrence of three different events.

In some cases, there are multiple possible outcomes for an event, each with it’s own probability of occurrence.

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The solution of Decision Tree is as follows:

Decision Making: Decision Trees:Conditional Uncertainties

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In a more realistic design situation, many decisions have to be made in sequence.

Consider the “Generator Problem” with an additional possibility: The option to diagnose the generator.

Diagnosis is assumed to be 100% correct, and will cost $5,000.

Decision Making: Decision Trees:Sequential Decisions

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