Lecture 2: Making Decisions with Multiple Objectives under Certainty Flashcards
1
Q
Rational preference (2 conditions)
A
- complete: DM has a preference for any pair of alternatives
- transitive: for any three alternatives a, b and c holds: from a > b and b > c follows a > c
2
Q
Conflicting objectives
A
- There is no alternative that dominates the other alternatives in all objectives
- A procedure has to be used to combine conflicting attributes into a single index
3
Q
Steps to solve decision problems under uncertainty
A
- Determine fundamental objectives, how to measure the achievement (attributes) and the set of alternatives that might achieve these goals
- Apply the Multi-Attribute Value Theory (MAVT)
4
Q
Multi-Attribute Value Theory (MAVT) - Steps
A
- Assign value scores to each attribute level for all alternatives
- Determine the weight of each attribute
- Rank all alternatives according to weighted-average total score
5
Q
Additive value function
A
V(a) = w1 * v1(a1) + w2 * v2(a2) + …
6
Q
Requirements of the additive value function
A
- mutual preferential independence
7
Q
Simple Preferential Independence
A
- Preferences over attribute levels of a particular attribute should not depend on the level of other attributes
- Required for mutual preferential independence
- Satisfied if
(White, $X, Y km/h) > (Black, $X, Y km/h) for any X and Y
OR
(Zcolor, $20,000, Y km/h) > (Zcolor, $30,000, Y km/h) for any Z and Y
8
Q
Mutual Preferential Independence
A
- Preferences over attribute levels must be preferential independent for each possible subset of attributes
- Attributes X1, …, Xn are mutually preferential independent if each possible subset of attributes is preferential independent of the complementary set
- Not satisfied if
(White, $20,000, 220 km/h) > (Black, $30,000, 220 km/h)
BUT
(Black, $30,000, 250 km/h) > (White, $20,000, 250 km/h)
9
Q
Additive Difference Independence
A
- Preferences over transitions between attribute levels of a particular attribute should not depend on the level of other attributes
- Satisfied if
(Black, $30,000, 220 km/h) -> (Black, $30,000, 250 km/)
~
(White, $20,000, 220 km/h) -> (White, $20,000, 250 km/)
10
Q
When can we use an additive multi-attribute value function?
A
- Mutual preferential independence -> ordinal value function
- Additive difference independence -> cardinal value function
11
Q
What happens if preferences are not independent?
A
- Cannot use additive value functions
- Try to redefine attributes to eliminate dependencies
- Use non-additive value functions (has a term that captures the interaction between attributes: complement (+) or substitute (-))
12
Q
What are attribute value functions doing?
A
- Convert attribute levels into levels of utility/desirability
- Shape of the function depends on the DMs preferences (no right or wrong, subjective)
- No need for perfect function; should capture the preferences well enough to analyse the situation
13
Q
General procedure for deriving value functions
A
- Choose X_min and X_max
- Determine some points on the value function curve
- Use these data points to generate the complete curve
- Normalize the function on the interval [0,1]
- Check for consistency
14
Q
Methods for determining attribute value functions
A
- Direct rating method
- Bisection method (mid-value splitting technique)
- Difference standard sequence technique (DSST)
15
Q
Direct Rating Method (Steps)
A
- Determine the most-preferred outcome and the least-preferred outcome
- Order the outcomes of all alternatives from the most preferred to the least preferred
- Assign 100 and 0 points to the best and worst outcomes
- Assign points to the intermediate outcomes, such that the point differences truly reflect the strength of preference
- Normalize: Divide points by 100
- Use linear interpolation to complete the value functions
- Check consistency (use different method)
16
Q
Bisection Method (Steps)
A
- Determine the most-preferred outcome and the least-preferred outcome
- Normalize the value function by assuming lowest outcome = 0 and highest outcome = 1
- Determine the midpoint of the total range (= best + worst / 2) and ask which change produces a greater value improvement: worst-mid or mid-best -> repeat while changing mid until DM is indifferent
- Assign the evaluation 0.5 to this outcome
- Determine the outcomes 0.25 and 0.75 in the same way
- Use linear interpolation to complete the function
- Check consistency (different method, different question: mid of 0.25 and 0.75)
17
Q
DSST Method (Steps)
A
- Determine the most-preferred outcome and the least-preferred outcome
- Define a unit delta that is approximately 1/5 of the length of the interval -> Define X1 = worst + 1/5
- Ask the DM which change produces a greater value improvement: worst -> X1 or X1 -> X2
- Repeat the question while changing X2 until the DM is indifferent
- Proceed with the same procedure until the best outcome is reached
- Determine the normalised values
- Use linear interpolation to complete the value function
- Check consistency (ask for the attribute level that is in the middle of the interval (worst, best); repeat the method with a different starting unit delta)
18
Q
DSST Method: Attribute Range
A
- preferable to use a (global) range that is wieder than the minimum and maximum values f the alternatives (local range)
If the last question in the interview results in an x value that is greater than x_best expand the range or ask for the value of the transition from x_best to this value
19
Q
Differences between the methods
A
- Direct-rating method provides least support to the decision maker
- DSST and bisection method are much simpler for the DM because he just has to state preferences with respect to some explicit transition (DSST is the simplest one)
- Bisection method forces the DM to adjust both transitions simultaneously
20
Q
What if attribute values are discrete?
A
- Cannot use bisection and DSST method
- Use direct method
21
Q
What if the value functions are non-monotonic?
A
- Split the objective into monotonic lower level objectives
- Or split the interval into subintervals on which the value function is monotonically increasing or decreasing (use method for both intervals)
