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MULTIOBJECTIVE VALUE ANALYSIS
Reference: Kirkwood, C. W., Strategic Decision Making: Multiobjective Decision Analysis with
Spreadsheets, Belmont, California: Duxbury Press, 1997
OVERVIEW
 Terminology
 Introduction
 Illustrative Example
 Multiobjective Value Function
 Single Dimensional Value Functions
 Weights
 Alternative Evaluation
 Meaning of Value Numbers
 Multiobjective Utility Analysis
TERMINOLOGY
 Evaluation Consideration: A factor to compare alternatives
(annual income)
 Objectives: Preferred direction of attainment of an evaluation
consideration (higher annual income)
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 Goal: Threshold of achievement (> $100K)
 Evaluation Measure: Scale to measure degree we attain an
objective (annual salary in $) Also: MOE, MOM, or metric
 Level or score: specific numerical rating of the evaluation
measure ($55K)
 Value Structure: Evaluation considerations, objectives, and
evaluation measures
 Value Hierarchy (Value Tree): Pictorial representation of the
structure of the evaluation considerations and evaluation
measures
 Layer/Tier: Levels in the value hierarchy do not have to be
symmetric
 Scoring function: A function that assigns value to an
evaluation measure (single dimensional value function)
 Value Model: The entire value tree, weights, scoring functions
-- everything necessary to mathematically evaluate a set of
alternatives.
INTRODUCTION
 Multiobjective Value Analysis
 Multiple, sometimes conflicting objectives
 No uncertainty about the outcomes
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 We need to obtain
 Evaluation considerations
 Evaluation measures
 Alternatives
 Alternative scores for each evaluation measure
 We want to rank alternatives and select the most preferred
 Develop a value function
v (x )   w v (x )
n
i 1
i
i
QUESTIONS: Define:
x
x 
i
v( x) 
v (x ) 
i
i
w 
i
i
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EXAMPLE: SELECTING A NETWORK STRATEGY
(EXPANDED), Kirkwood, pp. 54-55.
 Company deciding on a networking strategy for its personal
computers.
 Three evaluation considerations and three evaluation measures
selected
 Productivity enhancement, constructed scale
 Security, constructed scale
 Cost increase, net present value
QUESTIONS:
1. What is a constructed scale?
2. Can you come up with direct, natural or proxy scales for the two
evaluation measures with constructed scales?
-1
0
1
2
Productivity Enhancement
User group productivity is diminished …
No change in user group productivity …
User group productivity is enhanced …..
Significant & perceived increase ….
-2
-1
0
1
Security
Potentially serious decrease in system security ….
Noticeable but acceptable decrease ….
No detectable change in ….
System security is enhanced …..
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The scores are given in the following matrix. Alternatives have
been added to those listed in the Kirkwood.
Evaluation Measures
Productivity
Cost
Security, Xs
Enhancement, Xp Increase, Xc
Status Quo
0
0
0
High Qual/High Cost
2
125
0.5
Avg Qual/Avg Cost
1
110
0
Med Qual/Med Cost
1
95
0
Lower Qual/Lower Cost
1
100
-1
Low Qual/Low Cost
0.5
65
-1
QUESTIONS:
1. What role does the status quo play in the three scales?
2. Which alternatives are dominated?
MULTIOBJECTIVE VALUE FUNCTION
 Tradeoffs between the evaluation measures must be made to
determine the best alternative.
 We need a single index of overall “value”  value function!
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Approach # 1: Using Simple Averaging
 Take the average of each evaluation measure
Status Quo
High Qual/High Cost**
Avg Qual/Avg Cost
Med Qual/Med Cost
Lower Qual/Lower Cost
Low Qual/Low Cost
Xp
0
2
1
1
1
0.5
Xc
0
125
110
95
100
65
Xs
0
0.5
0
0
-1
-1
Score
0
43
37
32
33
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QUESTION: What is wrong with this approach?
Approach # 2: Simple Average with Negative of Decreasing
Evaluation Measures
Status Quo
High Qual/High Cost
Avg Qual/Avg Cost
Med Qual/Med Cost
Lower Qual/Lower Cost
Low Qual/Low Cost
Xp
0
2
1
1
1
0.5
Xc
0
-125
-110
-95
-100
-65
QUESTION: What is wrong with this approach?
Xs
0
0.5
0
0
-1
-1
Score
0
-41
-36
-31
-33
-22
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The Problem of Units for Evaluation Measures
 The relative ratings of the alternatives should not change if we
change the units used for the evaluation measure
 Suppose we measure the cost increase in $m instead of $k
Status Quo
High Qual/High Cost**
Avg Qual/Avg Cost
Med Qual/Med Cost
Lower Qual/Lower Cost
Low Qual/Low Cost
Xp
0
2
1
1
1
0.5
Xc
0
-0.125
-0.11
-0.095
-0.1
-0.065
Xs
0
0.5
0
0
-1
-1
Score
0.00
0.79
0.30
0.30
-0.03
-0.19
Approach # 3 - Average the Normalized Scores
 One approach to overcoming the problem of units is to
normalize the scores. This is a proportional score.
 If higher scores are preferred we use
Rating = Score - Lowest Level
Highest Level- Lowest Level
 If lower scores are preferred we use
Rating = Highest Level - Score
Highest Level - Lowest Level
QUESTION: Calculate the score of High Cost/High Quality.
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 Here’s one of the three functions.
Linear or Proportional Scoring
1.20
1.00
Score
0.80
0.60
0.40
0.20
0.00
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Productivity Enhancem ent
Here’s the resulting scores.
Status Quo
High Qual/High Cost
Avg Qual/Avg Cost
Med Qual/Med Cost
Lower Qual/Lower Cost
Low Qual/Low Cost
Xp
0.00
1.00
0.50
0.50
0.50
0.25
Xc
1.00
0.00
0.12
0.24
0.20
0.48
Xs
0.67
1.00
0.67
0.67
0.00
0.00
Score
0.56
0.67
0.43
0.47
0.23
0.24
 We still have two problems
1. The scores on the evaluation measure depend on the range
of variation.
2. We have assumed the variations are of equal importance.
QUESTION: How can we solve these problems?
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Using Weights for Evaluation Measures
 Weights solve both of these problems.
 Assume wp = 0.5, wc = 0.3, and ws = 0.2, we get
Xp
0.00
1.00
0.50
0.50
0.50
0.25
0.5
Status Quo
High Qual/High Cost
Avg Qual/Avg Cost
Med Qual/Med Cost
Lower Qual/Lower Cost
Low Qual/Low Cost
Weights
Xc
1.00
0.00
0.12
0.24
0.20
0.48
0.3
Xs
0.67
1.00
0.67
0.67
0.00
0.00
0.2
Score
0.43
0.70
0.42
0.46
0.31
0.27
Single Dimensional Value Functions Measure Returns to Scale
 Constant returns to scale (Linear)
 Decreasing returns to scale (Concave)
 Increasing returns to scale (Convex)
 Other, e.g., S-curve (a combination of the above)
The Value Function
v (x )   w v (x )
n
i 1
i
i
i
w 1
n
i 1
i
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QUESTION: What does the overall value function measure?
DETERMINING THE SCORING FUNCTIONS (SINGLE
DIMENSIONAL VALUE FUNCTIONS)
* Single dimensional value functions measure returns to scale
 Usually scale all n functions
 0 to 1
 0 to 10
 0 to 100
QUESTION: Why do we want to scale the functions this way?
 Kirkwood uses two types of functions because they are easy to
use with a Spreadsheet and can handle qualitative evaluation
measures
 piecewise linear
 exponential
 In my consulting and research, I have used more general curves
 Linear
 Concave
 Convex
 S-curve
 Using Logical Decisions you put in any data and the program
will fit the curve
 Using DPL you can put in any functional form you can define
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PIECEWISE LINEAR SINGLE DIMENSIONAL VALUE
FUNCTIONS
 Kirkwood recommends piecewise linear when the evaluation
measure has a small number of possible scoring levels (Discrete)
-1
0
1
2
Productivity Enhancement
User group productivity is diminished …
No change in user group productivity …
User group productivity is enhanced …..
Significant & perceived increase ….
 Kirkwood’s assessment technique
 uses relative value increments for each possible score
 Value increment = degree decision-maker prefers higher to
lower score level
QUESTION: What does the single dimensional value function
look like if the 1st increment has twice the value of the 2nd and 3rd?
1
Value
0.8
0.6
0.4
0.2
0
-1
0
1
Productivity Enhancem ent
2
12
-2
-1
0
1
Security
Potentially serious decrease in system security ….
Noticeable but acceptable decrease ….
No detectable change in ….
System security is enhanced …..
QUESTION: What does the single dimensional value function
look like if the 1st increment has three times the value of the 3rd and
the 2nd increment has twice the value of the 3rd ?
1
Value
0.8
0.6
0.4
0.2
0
-2
-1
0
1
Security
 When I have continuous evaluation measures, I usually ask the
decision-maker to directly assess the evaluation measure score
that provides a specified value level, e.g., 10, 8, 3, 0.
 With a little practice, they can usually do this
 This doesn’t require any calculations
 Everyone can directly see the value
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EXPONENTIAL SINGLE DIMENSIONAL VALUE
FUNCTIONS
 Kirkwood recommends the exponential

For monotonically increasing preferences over Xi ,
 1  exp[  ( x  x ) /  ]
1  exp[  ( x  x ) /  ] ,   
v (x )  
x x

, otherwise

x x
L
i
i
i
H
i
i
i
L
i
i
L
i
i
i
H
L
i
i
 exp(y) represents the exponential function, ey
 Single dimensional value function scaled from [0,1]
  is the exponential constant
 Exponential value function for five values of 
 Excel file: exponential value.xls
Increasing Exponential Preferences
1.200
1.000
-1
0.800
Value
-5
Infinity
0.600
5
1
0.400
0.200
0.000
0
1
2
3
4
5
6
Evaluation Measure
7
8
9
10
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 See Kirkwood for monotonically decreasing form
 To use the exponential we must find 
 Realistic values of  will be greater than 1/10 the range of
the evaluation measure
 see plot above
 Usually find midvalue such that v(x) = 0.5, then we can
solve for 
 no closed form solution
 use Goal Seek in Excel
 Excel file: exponential value.xls
 Tools
 Goal Seek
 In exponential value.xls
 Block c47:e49
 Use <control> <~> to view the equations
Rho
100
x
3
v(x)
=(1-EXP(-(D47/C47)))/(1-EXP(-10/C47))
v(x)=0.5
What is rho?
QUESTION: Use Goal Seek to find
ANSWER:
Rho
5.540058
x
3
v(x)
0.500437672
v(x)=0.5
What is rho?

if v(3)=0.5.
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DETERMINING THE WEIGHTS
 Our value function is
v (x )   w v (x )
n
i 1
i
i
i
w 1
n
i 1
i
 Each single dimensional value function is scaled on [0,1]
 Assess the swing weights - the numerical importance - by
swinging each attribute using
w  v ( x ; x )  v ( x ,... x , x , x ,... x )

i
i
0
i
0
0

0
0
1
i 1
i
i 1
n
Swing Weights: Value Increment Procedure
1. Place each evaluation measure in order - smallest to largest
value increment
2. Scale each weight as a multiple of the smallest value increment
(weight)
3. Sum the weights to one and solve for the weights
NETWORKING EXAMPLE
1. Value increments
 Smallest: Productivity Enhancement
 Middle: Cost
 Largest: Security
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2. wc = 1.5 wp and ws = 1.25wc
3. Solve for the weights
 Many times we assess weights using a group of people
 It is easier to have each person spread 100 points among the
evaluation considerations.
 This usually requires discussions and revotes
Group Weights Procedure: Spreading 100 Points
1. Have each individual spread 100 points over the evaluation
measures.
2. Calculate the average weights
3. Discuss any significant differences
4. Revote
5. Go to step 2 until consensus is reached
EXAMPLE 1
Our class has three evaluation measures:
 Homework
 Midterm
 Final
Use the group weights procedure to develop weights for a course
grade value function.
CLASS EXAMPLE 2 – Trick Question
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You have decided to buy a laptop for your office. The evaluation
measures are cost, speed, and storage.
Use the group weights procedure to develop weights for laptop
evaluation value function.
DETERMINING THE OVERALL VALUE OF THE
ALTERNATIVES
 Finally, we use single dimensional value functions and the
weights to obtain the overall score of each alternative
Status Quo
High Qual/High Cost
Med Qual/Med Cost
Low Qual/Low Cost
Xp
0
2
1
0.5
Xc
0
125
95
65
Xs
0
0.5
0
-1
Status Quo**
High Qual/High Cost
Med Qual/Med Cost
Low Qual/Low Cost
Weights
Xp
0.5
1
0.75
0.63
0.22
Xc
1
0.23
0.46
0.66
0.35
Xs
0.83
0.92
0.83
0.5
0.43
MEANING OF THE VALUE NUMBERS
QUESTIONS
1. What do the value numbers mean?
2. What is the meaning of the Status Quo score of 0.82?
Score
0.82
0.70
0.68
0.58
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3. Do the overall scores have meaning apart from the ranges of the
evaluation measures?
 First, we define the hypothetical worst and best possible
alternatives
x0 = (-1, 150,-2)
x* = (2, 0,1)
 The value increment, v(x*) - v(x0 ) = 1
 Status Quo value of 0.82 means Status Quo alternative obtains
82% of the hypothetical value increment
MULTIOBJECTIVE UTILITY ANALYSIS
 Above analysis was multiobjective value analysis
 Value functions measure returns to scale
QUESTIONS
1. What do utility functions measure?
2. How do we obtain utility functions?
3. Are utility functions different from value functions?
 Multiobjective utility analysis (assuming the conditions for an
additive multiobjective utility function are met)
 Assess utility functions using lotteries
 Assess weights
 Calculate expected utility of alternatives
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SUMMARY
 Introduction to multiobjective value analysis
 Terminology
 Multiobjective Value Function
 Single Dimensional Value Functions
 Weights
 Alternative Evaluation
 Meaning of Value Numbers
 Brief comments on multiobjective utility
MAT 647, MULTIOBJECTIVE DECISION ANALYSIS
 More on the applications and theory
 Multiobjective Value
 Multiobjective Utility
 Solution using spreadsheet macros