### Mathematics Awareness Week 1996

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## Mathematics and Multicriteria Decision Making

### Mathematics and Multicriteria Decision Making

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.

Dr. Thomas L. Saaty is a University Professor at the University of Pittsburgh and the creator of the Analytic Hierarchy Process. He can be reached at SAATY@vms.cis.pitt.edu.

For additional information on the Analytic Hierarchy Process, see the Expert Choice, Inc. web site.

 Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.