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ISE 562; Dr. Smith
10/24/2022
Multiattribute Decision Analysis
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Decision Theory
ISE 562; Dr.
• What is multiattribute (multicriteria) decision analysis?
• Relationship between multiattribute decision making and what we have been doing so far?
• How to define effective attributes? 10/24/2022 2
ISE 562; Dr. is multiattribute decision analysis? Also called multiattribute utility theory (MAUT).
• A methodology for providing information to decision makers for comparing and selecting from among complex alternative systems in the presence of uncertainty and risk.
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ISE 562; Dr. decision analysis?
Synthesis of techniques from: Operations Research
Statistics
Mathematics
Psychology
Definitive text: Keeney and Raiffa,
Decisions with Multiple Objectives: Preferences and Value Tradeoffs, 1976.
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ISE 562; Dr. is Multiattribute decision analysis?
• A framework that formalizes and synthesizes the relationships between uncertain technical information about alternatives and human values (preferences) that are ultimately used to evaluate the alternatives
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ISE 562; Dr. Between the System Model and the Value Model
Uncertainties:
• States of nature • Payoffs
DIFFERENT!
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Uncertainty
Alternatives
Alternative Ranking
Outcome Utilities
Outcome Descriptions
System Model
Value Model
•Effect of uncertainty on risk attitude
Sure thing vs. p
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ISE 562; Dr. example: buying a car based on cost and mpg:
System Data
Note tradeoff: cheapest car has worst mpg and most expensive has best mpg
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Alternative
ISE 562; Dr.
0c0m 20 40 16 32
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ISE 562; Dr. Smith
U(x*)=1; U(xo)=0 U(40)=0, U(20)=1 For Car A, U(20)=1.0 For Car B, U(30)=.17
For Car C, U(40)=0.0
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40c 0 16 2532m
U(x*)=1; U(xo)=0 U(16)=0, U(32)=1 For Car A, U(16)=0.0 For Car B, U(25)=.93 For Car C, U(32)=1.0
Alternative
ISE 562; Dr. Data ($ and mpg)
Value Data (attribute utilities)
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Data that describe the alternatives in terms of their physical characteristics (fact- based)
Data that describe the decision maker’s values and priorities (importance) to the decision maker (values-based)
Alternative
Alternative
ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
Answer: compute the multiattribute utility function, U(x1,…xn)=f(u(x1),..,u(xn))
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Alternative
Alternative
Now what? How do we combine the system data with the value data to evaluate the alternatives?
ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
We have the value of each attribute for each alternative which tells us how the decision maker values different levels of each attribute
But, something is missing here…
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Alternative
Alternative
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ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
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Alternative
Alternative
So, is Car A = 1+0 =1? No.
What do you think is missing?
ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
The evaluation of two attributes requires not only the utility value of the attribute, but also the relative importance of one attribute versus another.
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Alternative
Alternative
Which is more important to the decision maker, Cost or mpg?
ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
The measure of relative importance is called the attribute tradeoff scaling constant, ki.
Sometimes called an importance weight but similar to a marginal rate of substitution in economics that represents DM’s tradeoff rate for preference of attribute 1 in response to
changes in attribute 2.
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Alternative
Alternative
ISE 562; Dr. Data ($ and mpg) Value Data (attribute utilities)
As you might guess we could calculate the
utility of Car A as
U(Car A) = U(x1,x2)= .60(1)+.40(0) =0.60
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Alternative
Alternative
Suppose the attribute tradeoff scaling constants for cost and mpg are
ISE 562; Dr. Data ($ and mpg)
Value Data (attribute utilities)
U(A)= .6(1)+.4(0) U(B)= .6(.17)+.4(.93) U(C)= .6(0)+.4(1)
Alternative
Alternative
We conclude Car A > Car B > Car C
There are a variety of questions you should
be thinking about …
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U(A)=.60 U(B)=.47 U(C)=.40
= .60 = .47 = .40
ISE 562; Dr. Data ($ and mpg)
Value Data (attribute utilities)
U(A)= .6(1)+.4(0) U(B)= .6(.17)+.4(.93) U(C)= .6(0)+.4(1)
Alternative
Alternative
Why is A preferred? Not because U(A)>U(B) and U(C). • Because A has the highest utility for the most important attribute (kcost = 0.60 vs. kmpg=0.40)
whereas B and C score poorly on cost.
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U(A)=.60 U(B)=.47 U(C)=.40
= .60 = .47 = .40
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ISE 562; Dr. Smith
• What is the theoretical basis for MAU?
• What assumptions are being made to enable the multiattribute utility model?
• Where do the attribute tradeoff scaling constants come from?
• What if the attribute states are uncertain?
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ISE 562; Dr. Smith
• What is the theoretical basis for MAU?
– “Specimen Theoriae Novae De ,” 1738,
– “Theory of Games and Economic Behavior,” John von Neumann and , 1947.
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Our heros!
ISE 562; Dr. Smith
• was the son of . He was born in Groningen while his father held the chair of mathematics there. His older brother was Nicolaus(II) Bernoulli and his uncle was so he was born into a family of leading mathematicians but also into a family where there was unfortunate rivalry, jealousy and bitterness. Johann was determined that Daniel should become a merchant and he tried to place him in an apprenticeship. However Daniel was as strongly opposed to this as his own father had been and soon Johann relented but certainly not as far as to let Daniel study mathematics. Johann declared that there was no money in mathematics and so he sent Daniel back to to study medicine. This Daniel did spending time studying medicine at Heidelberg in 1718 and Strasbourg in 1719. He returned to Basel in 1720 to complete his doctorate in medicine.
• An important work which Daniel produced while in St Petersburg was one on probability and political economy. Daniel makes the assumption that the moral value of the increase in a person’s wealth is inversely proportional to the amount of that wealth. He then assigns probabilities to the various means that a person has to make money and deduces an expectation of increase in moral expectation. Daniel applied some of his deductions to insurance.
• did produce other excellent scientific work during these many years back in Basel. In total he won the Grand Prize of the Paris Academy 10 times, for topics in astronomy and nautical topics. He won in 1740 (jointly with Euler) for work on Newton’s theory of the tides; in 1743 and 1746 for essays on magnetism; in 1747 for a method to determine time at sea; in 1751 for an essay on ocean currents; in 1753 for the effects of forces on ships; and in 1757 for proposals to reduce the pitching and tossing of a ship in high seas. Another important aspect of ‘s work that proved important in the development of mathematical physics was his acceptance of many of Newton’s theories and his use of these together with the tolls coming from the more powerful calculus of Leibniz. Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newton’s basic equations. He also studied the movement of bodies in a resisting medium using Newton’s methods.
Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bernoulli_Daniel.html
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ISE 562; Dr. Smith
“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”
von Neumann was a child prodigy, born into a banking family is Budapest, Hungary. When only six years old he could divide eight-digit numbers in his head. He received his early education in Budapest, studied Chemistry, moving his base of studies to both Berlin and Zurich before receiving his diploma in 1925 in Chemical Engineering. He quickly gained a reputation in set theory, algebra, and quantum mechanics. At a time of political unrest in central Europe, he was invited to visit Princeton University in 1930, and when the Institute for Advanced Studies was founded there in 1933, he was appointed to be one of the original six Professors of Mathematics, a position which he retained for the remainder of his life.
Areas of contribution: Quantum mechanics, Hilbert spaces (mathematics), game theory, automata theory, modern computer
Source: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Von_Neumann.html
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ISE 562; Dr. Raiffa
Howard was a cofounder of the of Government, Harvard University and a cofounder of the Negotiation Program of the Harvard Law School and the of Government. He is the . (Emeritus) of Managerial Economics, Harvard University.
One of Howard’s best known books is “The Art and Science of Negotiation” (1982), which is still in print and widely used in negotiation courses in schools of business and law. He is the author, co-author or editor of innumerable books.
For his outstanding contributions to the field of decision analysis he was given the . Ramsey medal by the Operations Research Society of America. He has also been awarded the Distinguished Contributions Award from the Society of Risk analysis, and has been given honorary doctorates by , the University of Michigan, Northwestern University, and University of the Negev. From the International Association for Conflict Management, he was awarded the Lifetime Achievement Award.
Source: Duke faculty website
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ISE 562; Dr. Keeney https://ralphkeeney.com/
Source: Duke faculty website 10/24/2022
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ISE 562; Dr. Smith
• Bernoulli’s contribution—introduced the concept
• ’s contribution to Keeney-Raiffa multiattribute utility functions:
– Rationality axioms; greater outcome utility values correspond to more preferred outcomes
– The utility value to be assigned to a gamble is the expected value of the outcome utilities of the gamble (EU criterion)
• Keeney & Raiffa’s contribution: decision analysis framework and theoretical basis for multiple attributes
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ISE 562; Dr. is Multiattribute Utility Theory related to what we have learned so far?
– The payoffs at the terminating nodes of a decision tree are multiattribute utility values for the outcome of that node.
– Instead of one attribute with a single payoff or utility, the value for a vector of attributes is collapsed into a single “numeraire” or multiattribute utility for that outcome.
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ISE 562; Dr. Smith
• So instead of this:
we have this:
0 10 0.5 Event 4
Event 3 Cost
Decision 1
Decision 2
0 8 0.4 Event 4
0.5 Event 3
0.5 Event 4
0.6 Event 3
0.4 Event 4
Utility(Cost, mpg) 0.85
Decision 1 0
Decision 2 0
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ISE 562; Dr. Smith
• MAU Pro’s
– More comprehensive by considering multiple factors (attributes) and their tradeoffs
– Combines system data with decision maker preferences in a rational framework for decision making
– Organizes the decision problem into logical elements (alternatives, attributes, data, preferences, uncertainty)
– Model assumptions studied extensively
– Normative approach—how rational DM should make decision
– Theoretical foundation
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ISE 562; Dr. Smith
• MAU Con’s
– Assumes decision maker is rational when values of the decision maker may be irrational or non-rational
– Assessment of utility functions and tradeoff scaling constants can be problematic
• Labor intensive
• Subject to biases
– Normative approach—Model may not capture
(describe) how actual decision is made due to external factors (e.g., biases, politics, or hidden attributes)
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ISE 562; Dr. Smith
• Steps in a multiattribute decision problem
Specify objectives of the problem
Collect attribute data
Define alternatives
Collect utility functions and tradeoff scaling constants
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Define measurable attributes
Compute multiattribute utility of each alternative
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ISE 562; Dr. Smith
• Another view:
What does DM want to do?
What are the choices?
The numbers (data)
The DM’s preferences (values)
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How to measure the choices?
Rank ordered options
ISE 562; Dr. Smith
• Outcomes that terminate the decision tree can be described using an objectives hierarchy
– Expresses the preference structure of the DM’s from high-level objectives to measurable attributes
– Constructed in a manner consistent with the quantification and mathematical conditions required by the MAU function of the value model
Objectives Hierarchy
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Specify objectives of the problem
ISE 562; Dr. Smith
• Alternatives must be unique, separable entities whose value can be represented by measurable attributes
– Non-overlapping
– Sufficiently different to justify consideration
– Generally the choice that must be “selected” in order to achieve the goal or objective sought in the objectives hierarchy
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Define Alternatives
ISE 562; Dr. set of attributes must satisfy the following requirements in order to be a valid representative of the preference structure of the decision makers.
1. Completeness: The set of attributes should characterize all of the factors to be considered in the decision-making process.
2. Comprehensiveness: Each attribute should adequately characterize its associated criterion.
3. Importance: Each attribute should represent a significant criterion in the decision-making process, at least in the sense that the attribute has the potential for affecting the preference ordering of the alternatives under consideration.
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Define measurable attributes
ISE 562; Dr. set of attributes must satisfy the following requirements in order to be a valid representative of the preference structure of the decision makers.
4. Measurability: Each attribute should be capable of being objectively or subjectively quantified; technically, this requires that it be possible to establish an attribute utility function for each attribute.
5. Familiarity: Each attribute should be understandable to the decision makers in the sense that they should be able to identify preferences for different states of the attribute for gambles over the states of the attribute.
6. Nonredundancy: Two attributes should not measure the same criterion, thus resulting in double counting.
7. Independence: The value model should be so structured that changes within certain limits in the state of one attribute should not affect the preference ordering for states of another attribute or the preference ordering for gambles over the states of another attribute.
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Define measurable attributes
ISE 562; Dr. Smith
• Steps in a multiattribute decision problem
• Before discussing data collection, let’s examine what we are going to do with it.
Specify Define Define
objectives of alternatives measurable
• That is, how do we obtain the multiattribute
attributes
the problem
utility function, U(X1, … ,Xn)?
Collect utility functions and tradeoff scaling constants
Compute multiattribute utility of each alternative
Collect attribute data
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ISE 562; Dr. Smith
• With 2 attributes, x1, x2, the trade-offs between them are 1-1 substitutions (x1 vs. x2)
• With 3 or more attributes there may be tradeoffs among the attributes. e.g., if we have 3 attributes, the tradeoffs between any pair must consider:
• (x1,x2) vs x3, (x2, x3) vs x1, (x1,x3) vs x2 10/24/2022 37
ISE 562; Dr. 5 attributes, x1, x2, x3, x4, x5 the trade- offs between 2 attributes are:
• (x1,x2) vs {x3,x4,x5}
• (x1, x3) vs {x2,x4,x5}
• (x1,x4) vs {x2,x3,x5}
• {x1,x5} vs {x2,x3,x4}
• (x2,x3) vs {x1,x4,x5}
• (x2,x4) vs {x1,x3,x5}
(Sets of attributes)
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Tradeoff set
Complement set
ISE 562; Dr. Smith
• Three assumptions are required when there are 3 or more attributes:
1. Preferential independence
2. Utility independence
3. Mutual utility independence
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ISE 562; Dr. Smith
1. Definition: Let x, y, and z be 3 different attributes; the pair of attributes x and y is preferentially independent of z if the conditional preferences over the (x,y) space given z, do not depend on z.
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If the coffee shop I go to depends on x=cost of coffee, y=quality of coffee, and z=presence of music: and the value tradeoffs between cost of coffee and quality don’t depend on presence of music; then {x,y} is preferentially independent of z.
Similarly, if the value tradeoffs between quality of the coffee and presence of music don’t depend on the cost of the coffee, then {y,z} is preferentially independent of x.
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General Definition of Preferential Independence: Let Y1 and X be sets of attributes, YX;
Y1 is preferentially independent of its complement Y2 if the preference order of consequences involving only changes in the levels in Y1 does not depend on the levels for which attributes in Y2 are held fixed.
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Y2=complement of Y1 Y1=complement of Y2
ISE 562; Dr. 1 Y2 For example xyz
Alternative
Style (0-10)
Y1 is preferentially independent of its complement Y2 if the preference order of consequences involving only changes in the levels in Y1 does not depend on the levels for which attributes in Y2 are held fixed.
How the decision maker values mpg is not affected by differing levels of cost and style
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ISE 562; Dr. Smith
2. Definition: Attribute Y1 is utility independent of its complement, Y2, if the conditional preference order for lotteries involving only changes in the levels of attributes in Y1 does not depend on the levels at which the attributes in Y2 are held fixed.
Note that if Y1 is utility independent, then Y1 is preferentially independent (the reverse is not necessarily true).
The main difference here is that the consequences are not fixed—they are probabilistic outcomes—the independence holds when the outcomes are risky.
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ISE 562; Dr. of fixed choices for mpg = 1
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