Decision making is the most central and pervasive human activity,
intrinsic in our biology and done both consciously and
unconsciously. We need it to survive.
Everybody makes decisions all the time. Young and old, educated or
uneducated, with ease or with great difficulty. Making a decision
is not just a question of selecting a best alternative. Often one
needs to prioritize all the alternatives for resource allocation,
or to combine the strengths of preferences of individuals to form
a collective preference.

Applying mathematics to decision making calls for ways to quantify
or prioritize personal or group judgments that are mostly
intangible and subjective. And, decision making requires doing
what is traditionally thought to be impossible, comparing apples
and oranges. But we can compare apples and oranges by decomposing
our preferences into the many properties that apples and oranges
have, determining their importance to us, comparing and obtaining
the relative preference of apples and oranges with respect to each
property, and synthesizing the results to get the overall
preference.

Breaking a problem down into its constituent parts or components,
in the framework of a hierarchy or a feedback network, and
establishing importance or priority to rank the alternatives is a
comprehensive and general way to look at the problem
mathematically. This kind of concern has been loosely called
multicriteria decision making.

In Operations Research and Management Science today, decision
making is essentially thought of in this focused area of research
concerned with goals and criteria and how to measure and rank them.
The journals have specialized editors for processing papers in this
new area.

The majority of models in the literature of operations research
have been concerned with single criterion decision making. That
criterion, known in optimization as an objective function, is
necessarily a measurable quantity. The decision is made, for
example, by maximizing dollars to maximize economic success,
minimizing the amount of material used to maximize factory
efficiency, or minimizing distance to minimize travelling costs.

The tangibles of today were intangibles not too far back and how
they are measured involves the use of an arbitrary unit that is
replicated so many times in each reading to obtain a scale. In the
end, one must make some correspondence between a real world outcome
and the number from the scale.

Clearly, for many situations people will differ in what they
subjectively imagine the number means despite the much talked about
objectivity thought to inhere in the scale reading itself. In this
sense, multicriteria decision making looks beyond the manipulation
of numbers from scales into the validity of how judgments arise and
the legitimacy and accuracy of representing these judgments with
numbers. This is particularly useful in making predictions of
happenings and in assessing the likelihoods and intensity of
occurrences, and finally also in making optimal decisions that can
be easily reviewed and modified to survive the turbulence of the
future environment.

In our complex world, there are usually many solutions proposed for
each problem. Each of them would entail certain outcomes that are
more or less desirable, more or less certain, in the short or long
term, and would require different amounts and kinds of resources.
We need to set priorities on these solutions according to their
effectiveness by considering their benefits, costs, risks, and
opportunities, and the resources they need.

Our present complex environment calls for a new logic - a new way
to cope with the myriad factors that affect the achievement of the
goals and the consistency of the judgments we use to draw valid
conclusions. This approach should be justifiable and appeal to our
wisdom and good sense. It should not be so complex that only the
educated can use it, but should serve as a unifying tool for
thought in general.

There are two parts to the multicriteria problem: how to measure
what is known as intangibles, and how to combine their measurements
to produce an overall preference or ranking; and then, how to use
it to make a decision with the best mathematics we have. Learning
how to measure intangibles gave the clue to how to combine in one
framework tangibles having different scales with each other and
with intangibles.

The solution was to treat them all as intangibles and assess their
priority or utility thus producing an integrated theory of
measurement across all dimensions of preference. The measurement
of the intensity of preference introduces cardinal numbers, making
it possible to interpret objective numbers according to one's value
system and to obtain results such as constructing a group decision
function from those of the individuals involved, contrary to what
was learned from yes-no voting.

The known approaches to multicriteria decisions are few. They
include: priority theory of the Analytic Hierarchy Process, Utility
and Value Theory of economics based on the use of lottery
comparisons, Bayesian Theory based on probabilities, Outranking
Method based on ordinal comparison of concordance and discordance,
and Goal Programming that is basically a modified version of
Linear Programming. There are other methods that are variations of
these.

The youngest, and mathematically most general, is the Analytic
Hierarchy Process (AHP). It is a theory of measurement which has
been validated through numerous applications to complex decisions
around the world, and meets the challenge to solve problems in a
scientifically valid way. More significantly, it is free of
paradoxes, from which we reap two salutary results. The first is
that one does not have to make unrealistic assumptions to make the
theory work, and the second is that the theory and its applications
can be expanded to cover much wider complexity without complicating
its mathematics. It has been shown that priority functions can
be used to construct utility functions without the use of
lotteries.

It has also been shown that priorities can serve the role of
likelihood in probability theory and that Bayes Theorem is
derivable from the dependence approach of priority theory.
Finally, it has been shown that the well known Arrow Impossibility
Theorem is made possible with the use of priority functions to go
from individual to group preferences. Applications of the AHP
have been facilitated through the use of several software programs
of which Expert Choice and its extension to decisions with network
dependence, ECNet, are among the better known and user friendly
ones.

Through the use of hierarchic and network structures, the AHP
attempts to incorporate the objectives, criteria, actors, time
frames, and alternatives that have bearing on the decision. It
accommodates all the factors that people may believe should be
included in describing the decision problem. Their judgments are
then applied to relate and compare these factors in a systematic
manner that leads to priorities in the form of principal
eigenvectors and eigenfunctions (and hence also ratio scales) and
to the synthesis of these priorities to derive an overall priority
through the use of multilinear forms.

In the end it is people's personal and collective values that need
to be served. The challenge to mathematicians today is to learn
about these new ideas and create and push the models and the
mathematics as far as needed.