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Decision Analysis > Lecture 2: Making Decisions with Multiple Objectives under Certainty > Flashcards

Lecture 2: Making Decisions with Multiple Objectives under Certainty Flashcards

1
Q

Rational preference (2 conditions)

A
  1. complete: DM has a preference for any pair of alternatives
  2. transitive: for any three alternatives a, b and c holds: from a > b and b > c follows a > c
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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
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3
Q

Steps to solve decision problems under uncertainty

A
  1. Determine fundamental objectives, how to measure the achievement (attributes) and the set of alternatives that might achieve these goals
  2. Apply the Multi-Attribute Value Theory (MAVT)
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4
Q

Multi-Attribute Value Theory (MAVT) - Steps

A
  1. Assign value scores to each attribute level for all alternatives
  2. Determine the weight of each attribute
  3. Rank all alternatives according to weighted-average total score
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5
Q

Additive value function

A

V(a) = w1 * v1(a1) + w2 * v2(a2) + …

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6
Q

Requirements of the additive value function

A
  • mutual preferential independence
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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
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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)
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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/)
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10
Q

When can we use an additive multi-attribute value function?

A
  1. Mutual preferential independence -> ordinal value function
  2. Additive difference independence -> cardinal value function
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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 (-))
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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
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13
Q

General procedure for deriving value functions

A
  1. Choose X_min and X_max
  2. Determine some points on the value function curve
  3. Use these data points to generate the complete curve
  4. Normalize the function on the interval [0,1]
  5. Check for consistency
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14
Q

Methods for determining attribute value functions

A
  1. Direct rating method
  2. Bisection method (mid-value splitting technique)
  3. Difference standard sequence technique (DSST)
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15
Q

Direct Rating Method (Steps)

A
  1. Determine the most-preferred outcome and the least-preferred outcome
  2. Order the outcomes of all alternatives from the most preferred to the least preferred
  3. Assign 100 and 0 points to the best and worst outcomes
  4. Assign points to the intermediate outcomes, such that the point differences truly reflect the strength of preference
  5. Normalize: Divide points by 100
  6. Use linear interpolation to complete the value functions
  7. Check consistency (use different method)
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16
Q

Bisection Method (Steps)

A
  1. Determine the most-preferred outcome and the least-preferred outcome
  2. Normalize the value function by assuming lowest outcome = 0 and highest outcome = 1
  3. 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
  4. Assign the evaluation 0.5 to this outcome
  5. Determine the outcomes 0.25 and 0.75 in the same way
  6. Use linear interpolation to complete the function
  7. Check consistency (different method, different question: mid of 0.25 and 0.75)
17
Q

DSST Method (Steps)

A
  1. Determine the most-preferred outcome and the least-preferred outcome
  2. Define a unit delta that is approximately 1/5 of the length of the interval -> Define X1 = worst + 1/5
  3. Ask the DM which change produces a greater value improvement: worst -> X1 or X1 -> X2
  4. Repeat the question while changing X2 until the DM is indifferent
  5. Proceed with the same procedure until the best outcome is reached
  6. Determine the normalised values
  7. Use linear interpolation to complete the value function
  8. 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
  1. Direct-rating method provides least support to the decision maker
  2. 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)
  3. 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)

Decision Analysis (22 decks)

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