
Term

Explanation

U




user interface

See graphic user interface (GUI).


utile

Also called util, the unit of measurement for a utility function. For
example, if the utility function has been defined such that 100 utiles are assigned to the desirability of the best available
alternative and zero utiles have been assigned to the least desirable alternative, 1 utile represents one percent of the total
satisfaction, pleasure, or happiness gained by deciding to choose the best alternative rather than the worst available
alternative.


utility

A quantitative measure of a decision maker's subjective preference for the alternatives to a decision or the outcomes of those
decisions. The concept is central to utility theory; also called the theory of rational
choice. Utility is a number representing the level of satisfaction, pleasure or happiness as it relates to the decisions that
people make.


utility function

Also called preference function, cardinal utility function, Von Neumann utility function, and
probabilistic utility function, a mathematical function that assigns numbers to attributes (also called performance measures)
selected to describe the utility (usefulness) of the outcomes to a decision. The number assigned by the utility function
indicates the decision maker's preferences for the outcome bundle as described by the
specified attribute levels (the higher the number the more preferred the bundle of attribute outcomes). The number assigned is
expressed in arbitrary units called "utiles," however, in many instances (when the function expresses utility on a cardinal scale), a utility function can be scaled so that the utility number is the
equivalent monetary value of the outcome; that is, what obtaining those attribute outcomes
is worth to the decision maker.
Utility functions are the central concern of utility theory. According to utility theory,
attributes (also called performance measures) are selected to measure
performance relative to each of the decision maker's objectives. For example, the
decision maker might be an engineer tasked with selecting the best approach for constructing a large building. One objective
might be safety, another objective might be cost, and so forth. The attribute selected to measure performance relative to the
safety might be the number of injuries that occur during construction. The attribute selected for the cost objective might be
the total cost of constructing the building. A utility function is then constructed to denote how much the decision maker
prefers outcomes as described by the selected attributes. For example, for specified levels for cost and for the other
attribute outcomes, the utility function would assign the highest utility to an outcome with zero injuries. Utility functions
are typically denoted U(x), where x may represent a single attribute, in which case the utility function is
referred to as a single attribute utility function or x may represent multiple attributes
x_{1},x_{2},...x_{N}, in which case the utility function is called a multiattribute utility
function. Regardless, the utility function has the important property that the most preferred alternative will be the one
that produces the attribute outcomes with the highest value for U. Also, if there is uncertainty and probability distributions are assigned to indicate the decision maker's beliefs
about how likely the various possible attribute outcomes are, the most preferred alternative will be the one that maximizes
the expected value of U.
The utility function concept was first developed by the mathematicians John von Neumann and Oscar Morgenstern who were
concerned about how one should choose among alternative gambles, also called lotteries.
Von Neumann and Morgenstern showed that a utility function exists provided that the individual (the decision maker) accepts
certain assumptions or axioms meant to define "rationality." Most people find the axioms, which may be expressed in
various ways, easy to accept:

Orderability: Two items A and B are always comparable; that is, you must be able to tell if you prefer A to B, B to
A, or that you are indifferent between the them.

Transitivity: If you prefer item A to item B, and you prefer item B to item C, then you must prefer item A to item C.

Continuity: If you prefer item A to item B and item B to item C, then there must be some probability p for which you
are indifferent between item B and a lottery that provides item A with probability p and item C with probability 1p.

Substitutability: If you are indifferent between two items A and B, then for any lottery that contains A as a
possible outcome, A may be replaced by B without affecting your preferences.

Monotonicity: If items A and B are the only possible outcomes for alternative lotteries, and you prefer A to B, then
you must prefer lotteries with the higher probability of winning A.


Decomposability: (The "no fun in gambling axiom") Suppose you are faced with the compound lottery illustrated by the
event tree shown to the right: The lottery will provide either item A or a subsequent
lottery that will provide either item B or item C. Suppose when you consider the second stage lottery independently, you
conclude that you are indifferent between it and some item D (figure to the left). Then, you must be indifferent between
the original twostage lottery and the one stage lottery where item D replaces the lottery between B and C.
Decision analysts have developed assessment methods for encoding (i.e., deriving) a person's utility function.
Among other things that affect preferences, a utility function may account for the decision maker's willingness to accept
risk. Since utility functions, by definition, are determined empirically, there is no
obvious reason to expect that a particular mathematical relationship would emerge. However, it has been shown that an
exponential equation nearly always provides a good approximation:
Exponential utility function:
This equation is often alternatively written with constants added so that U goes from zero to one when V goes
from the minimum to maximum values assigned to the decision outcomes.
It can be shown that if a condition known as the delta property holds, then the
utility function must have this exponential form (or a linear form). The delta property applies if the following is true:
whenever there is uncertainty over the outcome of some uncertain choice, if the value of every possible outcome were increased
by the same amount (same delta), then the value of the uncertainty (its certain equivalent)
would be increased by the same amount (by delta).
The exponential utility function scales the possible outcomes to a decision in a way that accounts for willingness to accept
risk, and the coefficient R in the exponent determines the amount of scaling. R is termed the risk tolerance, and the lower the risk tolerance the less desirable the utility function
will show outcomes that involve uncertainty to be. A method for assessing risk tolerance (and therefore, for deriving the
utility function from a value function) is provided in the section of the paper chapter on risk tolerance, where there is also an example illustrating how to use a utility function
to value uncertain project outcomes.


utility independence

An independence condition similar to preferential independence, except
that the assessments are made with uncertainty present. Attribute Y is said to
be utility independent of attribute Z if preferences over lotteries involving
different levels of Y do not depend on a fixed level of Z.
Note that utility independence (in contrast to additive independence) is not
symmetric: it is possible that attribute Y is utilityindependent of attribute Z and not vice versa.
Utility independence is a slightly weaker independence condition than additive independence. If attributes are mutual utility
independent, the multiattribute utility function will either have an additive form:
or it will have a multiplicative form:
where the U_{i}(x_{i}) are single attribute utility functions.
Since utility independence refers to lotteries and a deterministic outcome is a special case of a lottery, utility
independence implies preferential independence. However, the converse is not necessarily true.
The twovariable case of the multiplicative equation is known as a multilinear form, and it is often useful in
situations where a compromise choice is sought that would be attractive to multiple stakeholders:
U(X_{1},X_{2}) = w_{1}U_{1}(X_{1}) + w_{2}U_{2}(X_{2}) + (1
 w_{1}  w_{2})U_{1}(X_{1})U_{2}(X_{2})
The w's are weighting factors and U_{1} and U_{2} are utility functions representing the
preferences of stakeholder 1 and stakeholder 2 (or the preferences of two distinct stakeholder groups). The decision maker's
utility function is additive with respect to these two terms and represents a desire to seek an alternative that each will
desire. The third, multiplicative term, may be thought of as representing the decision maker's desire for an equitable
distribution of value between the two stakeholders.


utility theory

A theory of how individuals should make decisions, related to the concept of "rationality" used in economics. Also called
subjective expected utility theory, or, with reference to the primary developers of the theory, von Neumann
Morgenstern utility theory.
Utility theory is an "axiomatic" theory in that it is derived from a set of axioms (hypotheses) defining how rational people
behave. It has been shown that utility theory's axioms can be expressed in a number of different ways, but in all cases the
axioms seem, at least to most people, to be quite reasonable. For example, one axiom (transitivity) states that if a person
prefers outcome A to outcome B and outcome B to outcome C, that person should prefer outcome A to outcome C. Another axiom
(substitution) states that if a person is participating in a lottery where the prize is A, and if that person is completely
indifferent between receiving prize A and some alternative prize C, then that person should not care if the lottery is
modified by substituting prize C for the equally desirable prize A. Other typical statements of utility theory's axioms
include diminishing marginal utility and diminishing rate of substitution, which imply that as a person acquires more and more
of a given good, their marginal value for another unit of that good, becomes less relative to other goods. An axiom termed
nonsatiation states that people do not have so much of everything they desire that they are at the point of not wanting any
more.
Utility theory shows that if a decision maker accepts any one of the various ways of expressing the axioms of rationality,
then it can be proven that there is a mathematical function, called a utility function,
typically denoted U, with the capability of aggregating all of the different considerations that must be taken into account
when deciding among alternatives. It also shows that under certain specified conditions, the function will have various simple
mathematical forms, such as additive, multiplicative, or exponential forms. Most significantly, the theory proves that
provided the axioms apply, the best alternative (the one that is most preferred) will be the one that maximizes the value of U
(or, if there are uncertainties, the expected value of U).
The rational model is, in effect, a conceptualization wherein decision making is regarded as simply a matter of choosing from
among sets of alternatives. Alternatives are viewed as leading to outcomes that can be evaluated by applying the decision
maker's preferences. The theory recognizes that the outcomes that follow the selection of alternatives are not generally
precisely known, due to uncertainties. However, in the face of uncertainties, rational decision makers judgmentally evaluate
possible decision outcomes in terms of their preferences and assess probabilities to indicate their beliefs about
uncertainties. This, as shown by the theory, maximize their expected preference satisfaction.
Numerous doubts have been raised about utility theory's descriptive validity. In other words, it is clear that people do not
always (maybe event often) behave according to the rational model. Much has been writing criticizing utility theory based on
showing that the rational model does not reflect how people typically make decisions. The proponents of utility theory,
however, see the fact that real world decision makers fail to follow the prescriptions of utility theory as the main argument
for its application. In other words, the fact that utility theory leads to recommendations for actions that differ from what
people typically do means that following utility theory is likely to produce significant improvements in the degree to which
outcomes desired by decision makers are obtained.
The collection of techniques and methods for applying utility theory to realworld decisions is known as decision analysis. Those techniques that apply in the special case of decisions
involving multiple decision objectives is called multiattribute utility analysis (MUA), or
multiobjective decision analysis (MODA).

V




valuation

The process of determining the value of something, such as a project or asset. In a business context, valuations typically seek to determine monetary worth,
and that is the meaning ascribed to the term throughout this website. However, in much of the literature on project portfolio management, valuation means assigning a number representing some concept of
attractiveness. Regardless, various theories and techniques are available for conducting valuations, and the methods typically
involve both objective and subjective components.


value

As used on this website, value means monetary worth. The value of something to someone is the maximum amount that individual
would be willing to pay to acquire it. Thus, the value of an asset to an organization is the maximum amount that the
organization's decision makers would be willing to pay for that asset. Likewise, the value of a project is the maximum amount the organization's decision makers would be willing to pay for the
opportunity to conduct that project; that is, what they would be willing to pay to obtain the consequences of doing the
project. If the consequences of doing the project are uncertain, the value of the project is the maximum the organization
would spend for the gamble over project consequences. The net value of a project is the difference between the value of
the project and its cost.
Although my definition of value is intuitive, it is usually not practical to ask decision makers to estimate the maximum
amount they would pay for things, including projects. Fortunately, there are wellestablished methods for quantifying project
value. In particular, decision theory and its subfield multiattribute utility analysis (MUA) provide methods for building models for estimating project
value based on consideration of business objectives, the impacts of projects on those objectives. and the willingness of the
organization's decision makers to make tradeoffs.
Be aware that word value appears often in project portfolio management (PPM) literature and
that many authors either don't define the term or define it in a way that is unrelated to the concept of worth. Many PPM
tools, for example, allow users to define criteria, score projects against those criteria, and then rank projects by the
weighted, summed scores. Since weighted summed scores do not measure value, such tools are unable to find valuemaximizing
project portfolios.
Compared to other definitions, defining the value of a project as its monetary worth to the organization has two significant
advantages. First, since project value may be expressed in monetary units, project value can be directly compared to project
cost. Second, value defined as worth exactly maps to organizational preferences—given two projects competing for the
same resources, we know that the organization will prefer Project A to Project B if and only if it views the worth of Project
A's consequences to be greater than the worth of Project B's consequences. This critical mapping does not hold for most other
definitions of project value.
Despite the arguments for expressing project value in monetary units, there are situations where doing so could create
problems. If expressing value in dollar units isn't possible, then, at minimum, project value should be expressed as a
cardinal utility; that is, a number that not only correctly measures the relative preferences of a decision maker for
candidate projects, but also correctly measures the increment in preference obtained from of a change from a decision to not
conduct a project to a decision to conduct the project.


value at risk (VaR)

A metric that describes the potential for loss, typically the potential for loss from a
portfolio of financial investments. The term is sometimes similarly applied in the context of project portfolio management to describe the risk associated with a portfolio of projects. Typically, VaR is defined as the amount or percentage of available portfolio value such
that there is a 95% (or 99%) probability of the portfolio losing less than that amount over a specified time horizon.
VaR is popular because it addresses the concept of "maximum potential loss." A major weakness is that it is not additive; that
is, the VaR for a set of portfolios is not the sum of the VaR's of the individual portfolios.


value function

Also called worth function or measurable value function, a function similar to utility function but only applicable to situations of certainty rather than uncertainty.
A value function is a mathematical representation of preferences; it maps the attributespecific measurement scale onto numerical scale of preference. Value functions are used in multiattribute utility analysis (MUA). Note that a value function is different from a value model. For example, a value function expresses preference in utiles, whereas a value model often expresses value in financial
units (e.g., equivalent dollars). A value function (or utility function) is a component of a value model which may include a
mathematical transformation to convert utiles to financial (e.g., dollar) units.
The independent variables for value functions are called attributes. A value function
that has only a single attribute is called a single attribute value function If the value function has multiple
attributes, it is called a multiattribute value function.
When used to measure preference for projects, a multiattribute value function, denoted
V(x_{1},x_{2},...x_{N}), assigns a number V to a project based on attributes of the project, denoted x_{1},x_{2},...x_{N},
typically chosen so as to describe the outcomes that would result if the project were to be conducted. The number V
indicates the decision maker's relative preference for the indicated outcomes. V is typically scaled between zero and
one so that higher values indicate more preferred outcomes. V is interval scaled, so that differences in the value of
V indicate differences in the levels of preference (a definition of interval scale is provided under scale).
Some authors use the term ordinal value function to denote a different type of value function, one that assigns values
on an ordinal scale rather than on a cardinal scale. An ordinal value function is
capable of correctly ranking attribute combinations in terms of preference, but the differences in the numbers so assigned may
not correctly indicate the differences in preference. An increasing monotonic
transformation can be applied to any ordinal value function and the result will continue to be an ordinal value function
that ranks attribute combinations in exactly the same way. The term measurable value function is sometimes used to
distinguish a value function that measures value on a cardinal scale. Unless otherwise specified, the discussion of value
function in the papers on this website refer to measurable value functions.
Because the goal of project selection is to choose projects that create maximum value, creating value/utility functions is the
key step for designing formal methods for selecting optimal project portfolios. See
additive utility function for a brief description of a popular method for constructing a value function.


value judgment

Also referred to as a valuebased judgment, personal, subjective judgment of how desirable or undesirable, good or bad,
better or worse, useful or not useful something is.


value of information (VoI)

A term used in decision analysis to describe the maximum amount a decision maker should logically be willing to pay for information prior to making a
decision. It is the amount of money that, when subtracted from the decision maker who has the information, makes the utility of that situation equal to the utility of the decision situation without the information.
The value of information is computed in the third phase, the information phase, of the decision analysis cycle. In the probabilistic phase, the decision analyst assigns probabilities to uncertain outcomes and utilities to quantify
the decision maker's preference for those outcomes. Determining the VoI then requires determining how probabilities would
change depending on the information and, how, therefore, the utilitymaximizing choice would change depending on the
information. It can then be determined how much the decision maker should be willing to spend such that the utilities of the
decision situations with and without the information would be equal. By comparing the VoI with the cost of actual
informationgathering alternatives, the decision analyst can determine whether the decision maker ought to be advised to
consider purchasing the information prior to having to commit to choosing a decision alternative.
Although the VoI may be computed whether or not having the information would eliminate uncertainty, it is easier to compute
the value of perfect information (VoPI) (information that eliminates uncertainty). The VoPI then serves as an upper
bound on the value of any potentiallyavailable information gathering opportunities. When
information is less than perfect, determining the value of the information is more complicated because it requires using
Bayes theorem to compute how having the information would change the probabilities of the
possible outcomes to the decision. Decision trees are often used to compute VoI and
VoPI.


value of life

Also called the value of a statistical life, a value judgment derived or recommended
for use in
cost benefit analyses.
An unavoidable question for health and
safety risk management is, "What is the maximum amount that ought to be spent to obtain small reductions in
risk?" Suppose, hypothetically, that a city council serving a city with a population of 100,000 is considering a proposal to spend
$5 million per year to treat the municipal water supply to remove a naturally occuring carcinogen. Suppose a risk assessment
concludes that water treatment would, on average, prevent 1 cancer death per year. To help shed light on the decision, a
poll might be conducted to ask the maximum amount each person was willing to pay for a
reduction in the individual risk of dying of 1 in 100,000, or 0.001%, over the next year. The reduction,
one less death among the 100,000 person population, is
sometimes described as "one statistical life saved.”
Suppose that the average response to the willingnesstopay question
happended to be $20. Then, ignoring likely biases in people's responses, the maximum dollar amount
that the city's population would be willing to
spend to save one statistical life in a year is
$2 million = $20 per person × 100,000 people. The value just computed is sometimes referred to as the "value of a
statistical life.” The simplified analysis concludes that the maximum amount the city's population would be willing to spend to save one
statistical fatality, $2 million,
is less than the cost to save one statistical fatality, $5 million. Based on costbenefit reasoning, the treatment should not be purchased
unless the cost can be
reduced to $2 million per year or less.


value model

A model for computing the value of an alternative,
outcome, project, project portfolio, or
uncertainty, expressed in monetary units or in relative units. A value model represents value judgments for choosing among alternatives (e.g., what are the decision objectives?) and technical judgments (e.g., what metrics
will be used to measure the degree to which objectives are achieved?). A value model may be constructed using techniques from
decision analysis and multiobjective decision
analysis (MODA). Like most decision analysts, if the value model is meant to
compute the value of a project to an organization, I recommend that the necessary value judgments be obtained from the
organization's senior decision makers and that technical judgments be obtained from
technical experts that have the confidence of those decision makers.
Terminology and mathematics for constructing value models for projects is not standardized. My terminology and recommendations
are as follows. In general, a project value model consists of two parts: a performance model and a preference model. I
recommend using a utility function for representing preferences. The utility
function may be a singleattribute utility function, in which case
it represents that there is only a single objective (e.g., maximize financial return) or it may represent multiple decision
objectives. A multiattribute utility function may be used to
represent multiple decision objectives and to express willingness to tradeoff
achievement of the different objectives. A utility function may also be used to represent risk aversion (I recommend using an exponential utility function, in which case risk preference is captured by a
single parameter called risk tolerance). If a decision produces outcomes that occur
over time, time preference is typically represented by using discounting to
collapse those outcomes into an equivalent present value, in which case the discount rate is a necessary value judgment.
The performance model establishes the performance measure used to measure the
achievement of each objective. It may also include models that compute performance, most often, a model for computing
performance over time (e.g., incremental cash flows resulting from the decision to conduct a project). If no performance model
is included, estimated impacts of projects on objectives (as quantified by performance measures) are provided directly as
inputs to the preference model.



value tree

Another name for an objectives hierarchy. Sometimes the term is used to
describe an objectives hierarchy with the performance measures arrayed below
the corresponding lowestlevel objectives in the diagram.


vaporware

An upcoming software product that has been announced but is not yet available. Software developers sometimes provide
information about future products or upgrades months or even years in advance. They may do so as a marketing ploy—if
current customers believe the supplier will release a breakthrough product soon, those customers may be willing to stick with
the supplier's aging software products longer. Also, announcing a phantom product may cause potential customers to perceive
the products currently offered by competitors to be less attractive. The vaporware may or may not exist, and may not ever be
available with the indicated capabilities and features. Regardless, spreading information about possible future products helps
the software provider.


variance

A discrepancy or deviation, as in schedule variance. Also, a measure of variability that indicates how much spread there is in
a probability distribution or set of numbers. The variance is the square of the standard deviation.


vision statement

A formal statement developed by an organization to define what it wants to achieve over time. For example, this is the vision
statement of the Alzheimer Association: "Our Vision is a world without Alzheimer's disease." The vision statement is typically
written succinctly and in an inspirational manner to make it easy for employees to remember and repeat to others. Like an
organizational mission statement, the vision statement, if it has been
developed, can be found on the organization's website.


virtual machine

A software program that mimics the behavior of computer hardware. A virtual machine is capable of performing tasks such as
running applications and programs like a separate computer. The end user has the same experience on a virtual machine as they
would have on dedicated hardware.


virtual market

Also called predictive market, a mechanism that allows people to make real or simulated decisions related to the
purchase or sale of items or events. The results, such as the market prices that are produced, are used to infer collective
preferences or beliefs or to make predictions. Though a relatively new concept, numerous companies are reportedly using
virtual markets to generate information to guide decision making. Virtual markets are can be attractive because participating
in a virtual market is simple and the collective information produced by participants can be quite useful. One application
area for a virtual market is prioritizing projects (explained here).

