A logarithm taken to the base of the mathematical constant e.
The natural logarithm of a number x is typically written ln(x). Like all logarithms,
ln(x) is the power to which a number must be raised to get the number x. A natural logarithm ln(x)
is the power to which e would have to be raised to equal x.
A normal probability distribution
The natural logarithm function is the inverse of the
Also called natural scale, a commonly used measure or
scale for an attribute, expressed in
original or natural units. For example, project cost is commonly measured
in dollars (or other currency), and time to complete may be measured in
net present value (NPV)
The traditional method for quantifying the financial
attractiveness of a project based on
equating project value to the present value
(PV) of the project's present and future cash flows. NPV, also called
discounted cash flow (DCF), represents the amount in today's dollars
by which all income projected from the project exceeds all costs.
Basically, NPV attempts to answer the question, "What is
the equivalent, lump-sum worth of this project?" According to NPV logic,
given two projects, the one with the larger NPV should be preferred. Also,
any project with a positive NPV should be viewed as a good investment.
NPV computes the present value for a project by
discounting estimated future incremental cash inflows and outflows. The
discount rate is chosen to
represent a required rate of return or target yield for the capital
invested, which is often chosen to the company's weighted average cost of
To accurately calculate a project's NPV, it is necessary
to estimate the life-cycle cash flows that would result from doing the
project— not just the project costs, but also all of the financial
benefits that would result from the project. For example, if a project
improves productivity, the future cost savings that would result should be
included in the estimated cash stream. Cost estimates must reflect the
total cost of ownership (TCO) perspective.
Thus, costs include project investment costs, future operating costs, and
any "exit" costs associated with ultimately phasing out whatever it is that
the project produces. Since taxes and the capital structure of the firm can
have a significant impact, cash flows should ideally be calculated
"after-tax," accounting for depreciation, working capital, and other
considerations. More precisely, what should be estimated is the impact of
the project on the company's so-called free cash flow, the value actually
available to shareholders and debt holders.
NPV cannot be used directly as a reasonable metric for ranking projects because it ignores
the size of the projects being compared. Larger projects tend to have
larger NPV's. Thus, projects with large NPV's tend to consume greater
portions of the available budget. However, NPV can be used to translate the
financial benefits expected from a project into an equivalent dollar value
which, if divided by the project cost, can be a useful metric for ranking
projects (assuming the projects are independent).
By requiring that a single, nominal cash flow be
identified for the project, NPV ignores uncertainty. This limitation may be
addressed by using NPV in conjunction with simulation. If alternative project cash
flows are simulated, they may be used to generate a range or probability distribution of
possible project NPV's. This distribution may be interpreted as describing
the uncertainty over the actual value that the project will generate. See
expected net present value (ENPV) for
The major limitation of NPV for project prioritization is
that it underestimates the true value of projects (e.g., the impact of a
project on shareholder value)
because it leaves out "intangible" project benefits that are difficult to express as
incremental cash flows. One such omitted component of value is "option
value," the value associated with options embedded in or created by the
project which may allow management to better respond to future risks and
take advantage of future opportunities. See real options analysis for an explanation
of option value. One method for addressing this bias is to add to the
estimated project cash flows additional dollar amounts that represent the
equivalent dollar value of the non-financial project benefits. See multi-attribute utility analysis for a method for
Another problem with NPV is that it is not always clear
what discount rate should be chosen. According to finance theory, the
correct rate is a risk
adjusted discount rate equal to the return available from investing in
securities equivalent to the risk of the project being evaluated. Research
on real options shows that the discount rate ought to be adjusted over time
depending on how uncertainties are resolved and on the project-management
strategy. Using a constant discount rate for a project implicitly assumes
that uncertainty increases over time in a specific way (geometrically). If
the discount rate is adjusted upwards to account for the risk of the
project, there will be a bias toward short-term, quick payoff projects
because project benefits that occur in the more distant future will be
A model composed of multiple, relatively independent
sub-models linked or interact in specified ways. Thus, network models are
often used to represent systems composed of multiple, interacting
components or subsystems.
A network model is distinct from a hierarchical
model. The latter has a top-down, tree structure such that each
subsystem is linked to at most one "parent" subsystem. With a network, each
sub-model may have links to multiple parents.
The term network model is also used to describe a
database structured as a collection of records with relationships among the
data represented by links.
nominal group technique
A group decision-making method, developed originally by
Andre Delbecq and Andrew Vandeven, that helps prevent the domination of
discussion by a single person, encourages more passive group members to
participate, and results in a set of prioritized solutions or
The basic process begins with each participant suggesting
possible solutions or choices, typically recorded on a flip chart. After
duplicate ideas are eliminated, each person ranks the solutions and then
anonymously votes points according to his or her ranking (e.g., 5 points to
the first choice, 4 points to the second choice, etc.). The total points
each solution receives determines the group ranking, and the top ranked
alternative is assumed to be the group's choice.
A multi-criteria decision making method (MCA) that is not
that poor performance of an alternative against one criterion may not be compensated for
by good performance relative to other criteria. A lexicographic method
is an example of a noncompensatory method.
Noncompensatory methods are based on the premise that certain preference attributes
are not compensatory by nature, that is, certain attributes cannot be traded off with, or compensated by, other
attributes even if they are objectively considered "good" attributes. For example, when purchasing food
you may reject any products that contain a substance to which you are allergic.
The advantage of non-compensatory
methods is that performance with respect to criteria having a disqualifying threshold can
be evaluated sequentially. Whenever you find an alternative that fails the test
you can disqualify it, there is no need to expend the effort to consider its performance
relative to any other criteria. People often employ non-compensatory rules over compensatory
methods for making purchase
decisions. For example, when purchasing a car you might be concerned with acceleration, gas mileage, and cost, but
you won't bother even considering cars whose price is above a level set by your budget. The reason being that
evaluating performance against all relevant criteria and then mentally trading off the
alternative's perceived weakness on one or more attributes with its perceived strength on other attributes
collecting and comparing all of the necessary data is simply too difficult or mentally labor intensive.
While most project portfolio management tools with capability to prioritize projects
employ compensatory methods, a few incorporate non-compensatory methods. Such methods gradually eliminate projects from
consideration based on identifying alternatives with less attractive attributes.
An often used probability distribution with
a symmetrical bell shape. It has been found to approximate the frequency
distributions for many economic, natural, social and other real world
The normal distribution has two parameters—the
mean, denoted μ, and the standard deviation, denoted σ.
The mean of the distribution determines the location of the center of the
graph, and the standard deviation determines the spread and height of the
graph. When the standard deviation is large, the curve is short and wide;
when the standard deviation is small, the curve is tall and narrow.
A normal probability distribution
With the normal distribution, roughly 68 percent of all
values lie within one standard deviation, 95 percent within two standard
deviations, and 99 percent lie within three standard deviations of the
As used here and in multi-criteria analysis, the process of multiplying a function,
series, or other mathematical item by a factor that makes it achieve some associated "normal" property, such as summing to
one unit. In the case of weihts, for example, it is just a matter of dividing each weight by the sum of the weights.
Another common usage is for the process of converting a performance measure for an
relative to a criterion
into a unitless measure (a score) on a zero-to-one scale. In this case, the purpose of
normalization is to allow performance measures expressed using different
units to be converted into a common, unitless measure, thereby allowing
direct comparisons across criteria.
Different normalization methods are used, although most
often the transformation is a linear one. If the performance measure is to
be maximized, the two most common transformations are (1) the fraction of
the maximum value of the performance measure and (2) the fraction of the
difference between the minimum and maximum values of the performance
measure. Mathematically, let the level of performance achieved by an
alternative with respect to the i'th criterion is denoted
xi. The units for xi will be whatever
unit is most natural for measuring performance relative to the criterion.
To normalize this measure, identify the best (maximum) and worst (minimum)
levels of performance attained by any alternative, denoted
ximax and ximin
respectively. The transformation of the performance measure to a score
indicating the fraction of the maximum value is:
The transformation of the performance measure to a score
indicating the fraction of the difference between the maximum and minimum
To illustrate, suppose that you are deciding which of
several automobile models to purchase. One criterion might be engine
power—you want a car with sufficient power to safety pass slower
vehicles. The natural unit of measurement for a car's power is brake
horsepower (bph). Of the several cars that you are considering, 200 bhp and
360 bhp are the least and most horsepowers. Suppose one particular car has
a horsepower of 300 bhp. Its normalized horse power, based on the second
formula above, is 0.625 (=100/160); it is 62.5% of the difference between
the most and least horse powers of the cars you are considering.
Normalization is useful because it allows performance
against all criteria to be expressed on the same, unitless scale.
According to an ideal standard; that is, what would be
considered the correct way of doing something based on norms of behavior.
The terms normative and descriptive are often used to contrast two types of
models. A normative model describes idealized behavior whereas a
description model describes typical or actual behavior. Vendors of project portfolio management software sometimes
say their tool prioritizes projects based on a normative model for project