The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lecture 03Decision Making under Certainty: The Tradeoff Problem
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering
Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp
August 29, 2019ME 597: Fall 2019 Lecture 03 1 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Example 1: Customer’s Decision
ME 597: Fall 2019 Lecture 03 2 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Example 2: Designer’s Decision
ME 597: Fall 2019 Lecture 03 3 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
The Multiattribute Value Problem
Driving question: Tradeoff
How much achievement on Objective 1 is the decision maker willing to giveup in order to improve achievement on Objective 2 by some fixed amount?
This is a two-part problem1 Achievability: What can we achieve in the multi-dimensional space?2 Preference structure: What are the decision maker’s preferences for the
attributes?
Today, we will only focus on deterministic scenarios.
ME 597: Fall 2019 Lecture 03 4 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Lecture Outline
1 The Multiattribute Value ProblemDefining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
2 Structuring Preferences1. Lexicographical Ordering2. Indifference Curves3. Value Functions
3 Preference Structures for Two AttributesMarginal Rate of SubstitutionAdditive Value Functions
4 Preference Structure for More than Two AttributesConditional Preferences
Chapter 3 from Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Cambridge, UK,Cambridge University Press.
ME 597: Fall 2019 Lecture 03 5 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Problem Statement
Act (alternative) space: The space, A, defined by the set of feasiblealternatives, a ∈ AConsequence space: The space defined by n evaluators X1, . . . ,Xn
A point in the consequence space is denoted by x = (x1, . . . , xn)Each point in the act space maps to a point in the consequence space,i.e., X1(a), . . . ,Xn(a)
a
Act space (A)
X1, …, Xn
x=(x1, …, xn)
Consequence space
Figure: 3.1 on page 67 (Keeney and Raiffa)
ME 597: Fall 2019 Lecture 03 6 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Problem Statement (contd.)
Decision maker’s problem
Choose a in A so that he/she is happiest with the payoff X1(a), . . . ,Xn(a)
Need an index that combines X1(a), . . . ,Xn(a) into a scalar index v ofpreferability or value, i.e.,
v(x1, . . . , xn) ≥ v(x ′1, . . . , x
′n)⇔ (x1, . . . , xn) & (x ′
1, . . . , x′n)
a
Act space (A)
X1, …, Xn
x=(x1, …, xn)
Consequence space
Figure: 3.1 on page 67 (Keeney and Raiffa)ME 597: Fall 2019 Lecture 03 7 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Choice Procedures Without Formalizing Value Trade-offs:a) Dominance
Assume:
Act a′ has consequences x′ = (x ′1, . . . , x
′n)
Act a′′ has consequences x′′ = (x ′′1 , . . . , x
′′n )
Preferences increase in each Xi (i.e., more is better)
Definition (Dominance)
x′ dominates x′′ whenever
x ′i ≥ x ′′
i , ∀ix ′
i > x ′′i , for some i
ME 597: Fall 2019 Lecture 03 8 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Dominance with Two Attributes
The idea of dominance only exploits the “ordinal” character of the numbers inthe consequence space, and not the“cardinal” character.
x’’
x1
x2
x’
Direction of
increasing
preferences
Figure: 3.2 on page 70 (Keeney and Raiffa)
Note: Dominance does not require comparisons between x ′i and x ′′
j for i 6= j
ME 597: Fall 2019 Lecture 03 9 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Choice Procedures Without Formalizing Value Trade-offs:b) The Efficient Frontier
Definition (Efficient Frontier / Pareto Optimal Set)
The efficient frontier consists of the set of non dominated consequences.
x1
x2
x1
x2
x1
x2
x1
x2
x’
x’’
x*
x(1)
x(2)
x(3)
Figure: 3.3 on page 71 (Keeney and Raiffa)ME 597: Fall 2019 Lecture 03 10 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Procedures for Exploring the Efficient Frontier
Objective
To select an act a ∈ A so that the decision maker will be satisfied with theresulting n−dimensional payoff.
Alternate procedures:1 Goal programming: Set aspiration levels xo
1 , xo2 , . . . , x
on and find points
that are closest to the aspiration levels. Update aspiration levels. Repeat.2 Standard optimization: Set aspiration levels for all attributes but one
(e.g., xo2 , x
o3 , . . . , x
on ). Seek an a ∈ A that satisfies the imposed
constraints Xi (a) ≥ xoi , for i = 2, 3, . . . , n and maximizes X1(a). Pick
another attribute and repeat.
The above procedures involve continuous interactions between what isachievable and what is desirable. The decision maker needs to constantlyevaluate what she would like to get and what she thinks is feasible.
ME 597: Fall 2019 Lecture 03 11 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Using Weighted Averages
Pose an auxiliary (optimization) problem which results in one point on theefficient frontier. Let
λ = (λ1, λ2, . . . , λn)
λi > 0, ∀in∑
i=1
λi = 1
Auxiliary Problem:
Choose a ∈ A to maximizen∑
i=1λiXi (a)
The solution to this problem must lie on the efficient frontier.
ME 597: Fall 2019 Lecture 03 12 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Defining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
Using Weighted Averages (contd.)
By moving along the efficient frontier, other points can be identified, until a“satisfactory” point is obtained.
Local marginal rates of substitution ofX1 for X2 are 1 : 4 and 3 : 7.
∆x2 = −4∆x1, and
∆x2 = −73
∆x1 respectively
These can be related to thewillingness to pay.
Note
Impact of non-convexity!
x1
x2
x’
R0.7x1+0.3x2 = constant
0.8x1+0.2x2 = constant
Figure: 3.5 on page 76 (Keeney and Raiffa)
ME 597: Fall 2019 Lecture 03 13 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Structuring Preferences
Structuring the preferences independent of whether points in theconsequence space are achievable or not.
Different approaches for structuring preferences
1 Lexicographical Ordering2 Indifference Curves3 Value Functions
ME 597: Fall 2019 Lecture 03 14 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Lexicographical Ordering
1 Widely used2 Simple and easily administered
Lexicographic ordering - Definition
Assuming that evaluators X1, . . . ,Xn are ordered according to importance,a′ � a′′ if and only if:
(a) X1(a′) > X1(a′′)or
(b) Xi (a′) = Xi (a′′), i = i . . . k , and Xk+1(a′) > Xk+1(a′′)for some k = 1, . . . , n − 1
Only if there is a tie in Xi does Xi+1 come into consideration.
Note: If x′ and x′′ are distinct points in an evaluation space, they cannot beindifferent with a lexicographic ordering.
ME 597: Fall 2019 Lecture 03 15 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Lexicographical OrderingExample
a1 a2
X1 0 0X2 10 11X3 400 12X4 56 20
ME 597: Fall 2019 Lecture 03 16 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Lexicographical Ordering with Aspiration Levels
Lexicographic Ordering with Aspiration levels
For each evaluator Xi , set an aspiration level xoi and posit the following rules:
a′ � a′′ whenever:
(a) X1 overrides all else as long as X1 aspirations are not meti.e., X1(a′) > X1(a′′) and X1(a′′) < xo
1
(b) If X1 aspirations are met, then X2 overrides all else as long as X2
aspirations are not met, i.e.,X1(a′) ≥ xo
1X1(a′′) ≥ xo
1X2(a′) > X2(a′′) and X2(a′′) < xo
2for some k = 1, . . . , n − 1
Note: In this case, two distinct points x′ and x′′ may be indifferent, providedthat x ′
j > xoj and x ′′
j > xoj , for all j .
ME 597: Fall 2019 Lecture 03 17 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Indifference Curves
Assume that any two points are comparable inthe sense that one, and only one, of thefollowing holds:
(a) x(1) v x(2), i.e., x(1) is indifferent to x(2)
(b) x(1) � x(2), i.e., x(1) is preferred to x(2)
(c) x(1) ≺ x(2), i.e., x(1) is less preferred thanx(2)
Note: All the relations v,�,≺ are assumed tobe transitive.
x ′′′ � x ′′ v x ′
x1
x2
x’’’
Direction of
increasing
preference
Indifference curves
x’
x’’
Figure: 3.6 on page 79 (Keeneyand Raiffa)
ME 597: Fall 2019 Lecture 03 18 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Value Functions
Definition (Preference Structure)
A preference structure is defined on the consequence space if any two pointsare comparable and no intransitivities exist.
Definition (Value Function)
A function v , which associates a real number v(x) to each point x in anevaluation space, is said to be a value function representing the decisionmaker’s preference structure provided that
x′ v x′′ ⇔ v(x′) = v(x′′)
and
x′ � x′′ ⇔ v(x′) > v(x′′)
ME 597: Fall 2019 Lecture 03 19 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Value Functions – Examples
Examples:
v(x) = c1x1 + c2x2, c1 > 0, c2 > 0
v(x) = xα1 xβ
2 , α > 0, β > 0
v(x) = c1x1 + c2x2 + c3(x1 − b1)α(x2 − b2)β
Using the value functions, the decision making problem can be formulated asan optimization problem:
Find a ∈ A to maximize v [X (a)]
ME 597: Fall 2019 Lecture 03 20 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
1. Lexicographical Ordering2. Indifference Curves3. Value Functions
Strategic Equivalence
The knowledge of v uniquely specifies an entire preference structure.However, the converse is not true: a preference structure does not uniquelyspecify a value function.
Definition (Strategic Equivalence)
The value functions v1 and v2 are strategically equivalent written v1 v v2, if v1
and v2 have the same indifference curves and induced preferential ordering.
Example: If xi is positive for all i , the following value functions arestrategically equivalent:
v1(x) =∑
i
kixi , ki > 0 ∀i
v2(x) =
√∑i
kixi
v3(x) = log
(∑i
kixi
)
ME 597: Fall 2019 Lecture 03 21 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Marginal Rate of Substitution
Question
If Y is increased by ∆ units, how much doesX have to decrease in order to remainindifferent?
Definition (Marginal Rate of Substitution)
If at (x1, y1), you are willing to give up λ∆units of X for ∆ units of Y , then for small ∆,the marginal rate of substitution of X for Y at(x1, y1) is λ.
Negative reciprocal of the slope of theindifference curve at (x1, y1)
Figure: 3.9 on page 83 (Keeneyand Raiffa)
ME 597: Fall 2019 Lecture 03 22 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Marginal Rate of Substitution – Example
Note: The marginal rate of substitutioncan be different for different points.
Along the vertical line, the marginalrate of substitution decreases withincreasing Y ⇒The more of Y we have, the less of Xwe are willing to give up to gain agiven additional amount of Y .
If x is money, then λ can beinterpreted as the willingness to payfor a unit increase in Y .
λc < λa < λb
λd < λa < λe
Figure: 3.10 on page 84 (Keeney andRaiffa)
ME 597: Fall 2019 Lecture 03 23 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Special Cases
1 Constant Substitution (Linear Indifference Curves)
v(x , y) = x + λy
2 Constant Substitution Rate with Transformed Variable
v(x , y) = x + vY (y)
Here, λ(y) is a function of one variable (y) only. For some reference y0,
vY (y) =
∫ y
y0
λ(y)dy
Theorem
The marginal rate of substitution between X and Y depends on y and not onx if and only if there is a value function v of the form
v(x , y) = x + vY (y)
where vY is a value function over attribute Y .
ME 597: Fall 2019 Lecture 03 24 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Additive Preference Structure
Definition (Additive preference structure)
A preference structure is additive if there exists a value function reflecting thatpreference structure that can be expressed by
v(x , y) = vX (x) + vY (y)
Is the preference structure given by the following value function additive?
v1(x , y) = (x − α1)α2 (y − β1)β2
ME 597: Fall 2019 Lecture 03 25 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Corresponding Tradeoffs Condition
Theorem
A preference structure is additive and therefore has an associated valuefunction of the form
v(x , y) = vX (x) + vY (y),
where vX and vY are value functions if and only if the corresponding tradeoffscondition is satisfied.
1 At (x1, y1) an increase of b in Y is worth apayment of a in X
2 At (x1, y2) an increase of c in Y is worth apayment of a in X
3 At (x2, y1) an increase of b in Y is worth apayment of d in X
If, at (x2, y2) an increase of c in Y is worth apayment of d in X , then we say that thecorresponding tradeoffs condition is met.
x
y
y1
x1
y2 a
c
(?)
c
a
b b
d
x2
A C
DB
Figure: 3.16 on page 90 (Keeneyand Raiffa)
ME 597: Fall 2019 Lecture 03 26 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Conjoint Scaling: The Lock-Step Procedure
1 Define origin of measurement:
v(x0, y0) = vX (x0) = vY (y0) = 0
2 Choose x1 > x0 and arbitrarily set vX (x1) = 13 Ask decision maker to provide value of y1 such that
(x1, y0) ∼ (x0, y1)
4 Ask the decision maker to give a value of X (e.g., x2) and a value of Y(e.g., y2) such that
(x2, y0) ∼ (x1, y1) ∼ (x0, y2)
Define vX (x2) = vY (y2) = 2. If the corresponding tradeoff conditionholds, then (x1, y2) ∼ (x2, y1)
5 Ask the decision maker to provide value of x3, y3 such that
(x3, y0) ∼ (x2, y1) ∼ (x1, y2) ∼ (x0, y3)
Define vX (x3) = vY (y3) = 36 Continue in the same manner as above.
Using the obtained points, define v(x , y) = vX (x) + vY (y).ME 597: Fall 2019 Lecture 03 27 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Marginal Rate of SubstitutionAdditive Value Functions
Conjoint Scaling: The Lock-Step Procedure (Illustration)
x
y
y1
x1
y2
x2
A
D
E
y0x0
a
b
a
B
b
C
cc
?
d
x
vX(x)
x1
y0
vY(y)
x2 x3
y1 y2 y3
1
2
3
0
1
2
3
Figure: 3.17-18 on pages 90-91 (Keeney and Raiffa)
ME 597: Fall 2019 Lecture 03 28 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Preferences for More than Three AttributesConditional Preferences
Consider three evaluators: {X ,Y ,Z}, e.g., Quality, Completion time, andCost.
Definition (Conditionally Preferred)
Consequence (x ′, y ′) is conditionally preferred to (x ′′, y ′′) given z′ if and onlyif (x ′, y ′, z′) is preferred to (x ′′, y ′′, z′).
Definition (Preferentially Independent)
The pair of attributes X and Y is preferentially independent of Z if theconditional preferences in the (x , y) space given z′ do not depend on z′.
If the pair {X ,Y} is preferentially independent of Z , then we can say thatif (x1, y1, z′) & (x2, y2, z′) then (x1, y1, z) & (x2, y2, z) ∀z
ME 597: Fall 2019 Lecture 03 29 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Mutual Preferential Independence
Theorem
A value function v may be expressed in an additive form
v(x , y , z) = vX (x) + vY (y) + vZ (z),
where vX , vY , and vZ are single-attribute value functions, if and only if
{X ,Y} are preferentially independent of Z ,
{X ,Z} are preferentially independent of Y , and
{Y ,Z} are preferentially independent of X .
Definition (Pairwise preferentially independent)
If each pair of attributes is preferentially independent of its complement, theattributes are pairwise preferentially independent.
ME 597: Fall 2019 Lecture 03 30 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Summary
1 The Multiattribute Value ProblemDefining the Tradeoff ProblemChoice Procedures Without Formalizing Value Trade-offs
2 Structuring Preferences1. Lexicographical Ordering2. Indifference Curves3. Value Functions
3 Preference Structures for Two AttributesMarginal Rate of SubstitutionAdditive Value Functions
4 Preference Structure for More than Two AttributesConditional Preferences
ME 597: Fall 2019 Lecture 03 31 / 32
The Multiattribute Value ProblemStructuring Preferences
Preference Structures for Two AttributesPreference Structure for More than Two Attributes
Conditional Preferences
Reference
1 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress. Chapter 3.
ME 597: Fall 2019 Lecture 03 32 / 32