EML4550 2007 1

EML4550 - Engineering Design Methods

Decision Theory(Decision under Uncertainty)

Hyman: Chapter 9

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Decision theory

Optimization Well-defined variables (we know what to ‘manipulate’) Well-defined objective function (we know what to

minimize/maximize) Strict mathematical framework (we know what we are doing)

Economic analysis Well-defined costs or economic benefits Decision is based on a single criterion that reduces to a dollar sign

(like optimization, single criterion)

If at all possible, design decisions should be based on models amenable to optimization or economic analysis. But more often than not, this is not possible, and the designer must reach a decision without such simple models

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Decision-making under uncertainty

Usually the decision to go with one design option or another has to be made in the middle of the design process, that is, when not all information is available and knowledge is fragmentary

Probability also plays a role, sometimes the ‘best’ decision has to be based on the odds of certain events to happen (or not)

The designer (like politicians) must be able to make decisions under uncertainty. It helps to understand the decision-making process

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Phases of Decision Making

Phase 1 - Specify all the alternatives to be included in the exercise Reduce universe of options to a manageable size, separate what has impact

from what is not all that relevant

Phase 2 - Specify all relevant events that might occur subsequent to making the decision, which could affect the outcome of the decision, but over which the decision maker has no control Will the price of oil go up? By how much? Will the government change the regulatory environment? How? When?

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Phases of Decision Making (Cont’d)

Phase 3 - Estimate the probability of occurrence of every relevant event identified in phase 2 “Crystal ball”, intuition, experience…

Phase 4 - Quantify the outcome of every possible combination of decision alternatives and relevant events What if? Expected values…

Phase 5 - Use a predetermined decision rule to select the design alternative that yields the most desirable result Maximum benefit? Minimum cost? Least risk?…

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Example: Old generator for a test in remote location

Cost to conduct the test is $10,000 IF generator works If generator fails it will cost $25,000 to repair after it failed It costs $15,000 to overhaul the generator before you take it out to

the remote location You do not know if the generator works or not until you are at the

remote location To overhaul, or not to overhaul….that is the question

What would YOU do?

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Estimate probabilities

70% probability generator will work as is (untouched), Pr(WU)=0.7 Therefore, 30% probability it will fail if not refurbished, Pr(FU)=0.3 There is no way the generator will fail if overhauled, Pr(WO)=1.0, Pr(FO)=0.0 Note Pr(WU)+Pr(FU)=1, Pr(WO)+Pr(FO)=1, complete set of relevant events

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Calculate expected values

Calculate expected values (outcomes) and apply pre-determined criterion (in this case, minimum cost to conduct the test).

EC(U) = C(WU)Pr(WU) + C(FU)Pr(FU) = (10)(0.7) + (35)(0.3) = $17.5k

EC(O) = C(WO)Pr(WO) + C(FO)Pr(FO) = (25)(1.) + (X)(0.0) = $25k

It is better not to overhaul the generator and use it as is However, variations in the probability or on the decision criterion

could lead to different outcomes

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Example: Variation in the alternatives

Suppose that now we introduce a variant in which the failure of the generator at different stages of the test would produce a different cost in damage to the equipment (see table below)

Pr(D1)+Pr(D2)+Pr(D3)=1 This more realistic case is not as amenable to tabulation

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Decision trees

When problems become more complex (as variant example), it is better to build a decision tree to visualize all the alternatives and consequences

Branches - straight lines Decision nodes - squares Events, or chance nodes - circles Payoff nodes (end results) - price tags

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Decision tree: Generator example

• List branches

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• Add events and terminations at the end of each branch

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• Continue adding branches after each event

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• Terminate new branches

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“Solve” tree backwards by calculating expected value for each decision node

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Sensitivity analysis

Use the decision tree to ‘estimate’ probabilities

Question: What probability of the generator failing would force me to overhaul instead of using as is?

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x=0.6 for an indifferent overhaul vs. use as is decisionIf Pr(F)>0.6 we need to overhaul (but we do not need to know exact probability)

25=(1-x)(10)+x(35)25=10+25xx=0.6

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Conditional uncertainties

Add to the generator example the fact that cost of damage is different depending on stage of testing when it occurs (see table below)

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68=0.1(20)+0.3(40)+0.6(90)

27.4=(0.7)(10)+(0.3)(68)

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Sequential design decisions

In the generator example, let’s add another option. Buying a $5,000 diagnostics tool that is 100% accurate in determining if the generator will fail or not (although we do not know prior to buying the tool if the generator will pass the test or not - of course)

For infallible diagnosis some of the branches on the tree can be pruned immediately (no decision dilemma). Branches are shown (see next slide) for completeness, but we know that if the generator does not pass the test we need to overhaul, and if it passes the test we do not need to overhaul

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Solve the tree

• Solve backwards• Top part of tree is identical to what we had before• Some of the branches at the bottom are shown only for completeness (infallible diagnosis)• From the tree solution we see it is worth buying the diagnostics equipment

10+5

25+5

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Value of information

We saw that paying $5k for the diagnostics tool was a good investment.

The question is, how much should we be willing to pay for a diagnostics tool?

Assign a value of “x” to the cost of an infallible diagnostic tool and solve the tree

The expected cost of the decision node is ($14.5k + x) The other lowest decision node is $25k (overhaul generator

without testing) Therefore, x=$10.5k If infallible tool costs less than that, buy it,

otherwise, do not buy diagnostics

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Imperfect information

What happens if the diagnostic tool is fallible?

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Example: Imperfect information

In the generator example, assume the diagnostic tool is 98% accurate If generator passes test, probability that it will work is 98%, and there is a

2% probability that it will not work (even if diagnosis was positive) If the diagnosis was negative, assume there is a 90% probability that the

generator will indeed fail, but a 10% probability that it will nonetheless work

Re-do and solve the decision tree

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Solve tree

• With positive diagnosis there is still a small probability of failure (0.02)• Expected cost at diagnosis node is higher with fallible tool than for the infallible case (20.3>19.5)• It is still best to purchase diagnostics tool• Sensitivity analysis is possible (on cost of tool, and/or probability of misdiagnosis)

Prob. to fail when dia. as positive

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Conclusions

Decision matrices and decision trees are very useful tools to approach problems involving decisions with multiple criteria or uncertain (probabilistic) information Reduces, but does not eliminate, subjectivity Allows for sensitivity (parametric) studies, which in turn help reduce

subjectivity

Other methods are possible (see textbook for Bayesian decision making, and risk-based decision making - utility functions)

Those that can master good decision making habits will be successful in management and business (the essence of which is sound decision making)