Transcript Decision Theory-1

EML4550 - Engineering Design Methods
Decision Theory
Hyman: Chapter 9
EML4550 2007
1
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 theory
 Many times a designer will be faced with decisions that cannot be
easily (or appropriately) reduced to an optimization or an
economic analysis model
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Multiple criteria
Non-quantifiable variables
Uncertainty (probabilistic variables)
‘apples and oranges’ comparison
 Many different approaches to decision-making. We will cover:
 Multiple criteria - Decision matrices (already partially covered in concept
selection)
 Decision under uncertainty (probabilistic analysis)
 Risk-based decision-making (utility functions)
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Multiple criteria
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Like in the example below (car bumper)
3 design options, 4 decision criteria
Loosely defined metric of ‘goodness’ on each criterion
How can a decision be made?
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Decision Matrix (Unweighted Pugh Matrix)
Comparison Table
Criteria
Option A
Option B
Option C
Cost
+
-
0
Damage
control
0
+
-
Recyclability
-
+
0
Drivability
0
-
+
Sum of +
1
2
1
Sum of -
1
2
1
Sum of 0
2
0
2
Total Score
0
0
0
Assign: excellent +, adequate 0, and poor EML4550 -- 2007
Decision matrix
 Identify criteria
 Develop criteria BEFORE the options are clear (prune later if necessary)
 Include only those attributes for which a differentiation exists
 Refine criteria as options are clarified or we gain further knowledge (e.g.,
eliminate, combine, etc.)
 Develop criteria metrics
 Tie criteria to quantifiable variables (e.g., ‘cost’ ---> dollars, ‘durability’ --> fatigue limits, etc.)
 Some criteria will be hard to tie to a metric (define a self-consistent metric,
e.g., ‘excellent’, ‘very good’, …, ‘poor’, etc.)
 All metrics are different (‘apples and oranges’), we need a consistent and
common evaluation scale
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Decision matrix (Cont’d)
 Evaluation scales
 Assign similar values on common scale (e/g/, ‘excellent’, ‘adequate’, and
‘poor’)
 Tie these scales to a metric (e.g., cost<$1000 ---> ‘excellent’)
 Numerical scales
 Assign numbers (e.g., from 1 to 10) instead of categories
 Even though it opens the possibility of quantitative comparison, it does not
remove the subjective nature of assigning values
 A strong correlation between the evaluation scale and the
corresponding metric is needed
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Example
 Power transmission between two
shafts
 Hierarchical tree structure to
determine criteria (and associated
metrics)
 Total 15 criteria to consider
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Focus on these two
criteria later
Example: Connecting evaluation scale with metrics
 Specify end points
 Note end points have been defined for
each metric (e.g., highest achievable
spec.)
 Numerical values (and corresponding
‘word’ values) have been assigned to
each range
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Comparison (decision) matrix
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Assign values (from scale) to each option and for each criteria
Compile scores (‘raw’ scores)
Add up the score for each option and normalize it
Torque: Max 50,000 (scale of 10), min 1,500 (scale of 0); increment: 4,850
Load: Max 5,000 (10), min 500 (0); increment: 450
(4200-500)/450=8.22
sum15+14+5=34
=15/34
(35000-1500)/4850=6.91
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Relative importance of criteria
 The previous example shows how to select one option (e.g., option
C is clearly inferior and must be discarded, but options A and B
are very similar)
 Question: should all criteria be treated as equally important?
 Before we proceed with a method to include all 15 criteria (in the
example) we must determine the relative importance of each
criterion, and assign ‘weights’ to them
 How do we determine the relative importance of criteria?
 How do we assign a ‘weight’ to each one?
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Relative importance of criteria: Pairwise Comparison
 Returning to the car bumper example
 Matrix with all four criteria in rows and columns
 For each row (criterion) include a “1” on the column where that criterion is
more important than the one on that column (ex. Damage control is more
important than cost  assign 1 to damage control row)
 Add the “1s” on each row to assess relative importance of criterion
 Normalize weights over the total for all rows
=1/6
1+2+0+3=6
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Relative importance of criteria: Pairwise comparisons
 There are 16 elements in the matrix, 4 (on the diagonal) are not necessary. Of
the 12 remaining elements, not all are independent, in this 4x4 example, 6
independent comparisons are needed
 For N criteria, N(N-1)/2 comparisons are needed for self-consistency (e.g., 15
criteria like in the simple shaft transmission example require 15(15-1)/2=105
comparisons)
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Use objective tree
3(2)/2=3
 Instead of making all 105
comparisons (tedious and prone to
errors), use the objective tree
5(4)/2=10
 Only establish relative importance
of criteria within a subgroup (in this
example that reduces the total
number of pairwise comparisons
from 105 to 23)
 (10)level 2 + (3+1+1+1+3)level 3 +
(3+1)level 4 = 23
3
1
1
1
1
3
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Using objective trees
to assign weights
Example: at level 2, a total 5 criteria with the
following weight assignment: maintenance
(k=0.1), geometry (k=0.1), health/safety (k=0.3),
operating conditions (k=0.25), power/load rating
(k=0.25). Sum(ki)=0.1+0.1+0.3+0.25+0.25=1.0
Individual weight=group weight*k
Example 1: @ level 2, power/load rating  k=0.25,
group weight from level 1 wlevel 1=1.0,
w=(1.0)(0.25)=0.25
Example 2: @level 4, speed flexibility  k=0.3,
group weight from level 3 w op. speed=0.162
w=(0.162)(0.3)=0.049
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0.4+0.5+0.1=1.0
Using objective trees to assign weights
Sum=0.925+0.825+0.35=2.1
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0.925/2.1=0.44
Analytical Hierarchy Process (AHP)
 The previous processes (relative importance or objective trees) of
assigning weights is still subjective and open to inconsistencies
 They do not take into account the options to be evaluated (‘a
priori’ selection of weights and end points for scale)
 It could be that the scale is too coarse for the options at hand, or
that one or more options are ‘off the scale’
 For very large $ value projects, it may pay to do some preliminary
engineering and to apply AHP
See Hyman, Section 9.3, for further details on this method
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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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