Multi-criteria decision making

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The field of multi-criteria decision making is a full-grown branch of operations research, concerned with the mathematical and computational tools to support the subjective evaluation of a finite number of decision alternatives under a finite number of performance criteria.

Alternatives, criteria, decision makers.

A decision is a choice out of a number of alternatives, and the choice is made in such a way that the preferred alternative is the "best" among the possible candidates. Usually, there are several criteria to judge the alternatives and there is no alternative which outranks all the others under each of the performance criteria. Hence, the decision maker does not only judge the performance of the alternatives under each criterion. He/she also weighs the relative importance of the criteria in order to arrive at a global judgement. Moreover, in a group of decision makers each member faces the question of how to judge the quality of the other members and their relative power positions before an acceptable compromise solution emerges.

Screening phase.

Multi-criteria decision making starts with the screening phase, which proceeds via several inventarizations. What is the objective of the decision process? Who is the decision maker or what is the composition of the decision-making body? What are the performance criteria to be used in order to judge the alternatives? Which alternatives are in principle acceptable or not totally unfeasible? These questions are not always answered in a particular order. On the contrary, throughout the decision process new alternatives may appear, new criteria may emerge, old ones may be dropped, and the decision-making group may change. When a family selects a car, for instance, these features of the decision process also emerge. First, the members have to identify the problem. Does the old car need replacement? Next, how to judge the cars? On the basis of generally accepted criteria that other people normally use as well? Can one also use past experiences to introduce new criteria? Are there any particular cars on the market which lead to new criteria? Does one only want to compare the cars themselves or does one also consider the supporting dealer networks, both on the home market and abroad? And who are the decision makers? The parents only?

Criterion Unit $A _ { 1 }$ $A _ { 2 }$ $A _ { 3 }$
Consumer price Dfl
Fuel consumption km/l
Maintenance, insurance Dfl/year
Maximum speed km/h
Acceleration, 0–100 km/h sec
Noise and vibrations verbal
Reliability %
Cargo volume $\operatorname {dm} ^ { 3 }$
Comfort verbal
Ambiance verbal

Performance tableau.

The result of the screening phase is the performance tableau, which exhibits the performance of the alternatives. Under the so-called quantitative, or measurable, criteria, the performance is recorded in the original physical or monetary units. Under the qualitative criteria it can only be expressed in verbal terms. Table 1 shows such a possible tableau for the car selection example. The tacit assumption is that the alternatives are acceptable and that the decision makers are prepared to trade-off possible deficiencies of the alternatives under some criteria against possible benefits elsewhere in the performance tableau. The alternatives which do not appear have been dropped from consideration because their performance under at least one of the criteria was beyond certain limits. They were too expensive, too small, or too slow, for instance. One should not underestimate the importance of the tableau. In many situations, once the data are on the table, the preferred alternative clearly emerges and the decision problem can easily be solved.

Choice of the criteria.

The number of criteria in Table 1 is already quite large, and they are not independent. The consumer price, the fuel consumption, and the expenditures for maintenance and insurance are closely related. One can take the estimated annual expenditures (based on the estimated number of kilometers per year) or just the consumer price in order to measure how well the objective of cost minimization has been satisfied. Similarly, a high maximum speed, a rapid acceleration, and the absence of noise and vibrations contribute to the pleasure of driving a car. That pleasure may be the real criterion. The performance tableau could accordingly be reduced, but the performance indicators in it provide valuable information. They help the decision makers to remain down to earth, and they prevent that the decision makers are swept away by a nice car-body design, for instance. Finally, the decision makers have to convert the data of the performance tableau into subjective values expressing how well the alternatives satisfy the objectives such as cost minimization and pleasure maximization.


Several simple MCDM methods present an arithmetic scale for the assessment of the performance, such as the seven-point scale $1 , \dots , 7$, which is well-known in the behavioural sciences. Let $g_{ij}$ denote the grade or score assigned to alternative $A _ { j }$ under the criterion $C_i$, and take $c_{i}$ to represent the normalized weight of $C_i$. According to the arithmetic-mean aggregation rule, the final grade or score $f_j$ of alternative $A _ { j }$ is given by

\begin{equation*} f _ { j } = \sum _ { i } c _ { i } g _ {i j }. \end{equation*}

Preference modeling.

MCDM methods vary considerably in their attempts to model the preferences of the decision makers. In the ordinal methods, the decision makers merely rank-order their preferences. In the cardinal methods, they express the intensity of their judgement (indifference, weak, strong, or very strong preference). In the last-named category, the leading methods are:

1) the simple multi-attribute rating technique (SMART), where the performance of the alternatives under the respective criteria is expressed in grades on a numerical scale;

2) the analytic hierarchy process (AHP), where the alternatives are considered in pairs and where the relative performance is expressed as a ratio of subjective values or as a difference of grades;

3) the multi-attribute utility (MAU) method, introducing a utility function whereby to each alternative a value between zero and one is assigned, the degree in which the underlying criterion has been satisfied. In each of these methods, the analyst has to work with the proper criterion weights and the proper aggregation rules. Hence, the design of these methods is far from trivial.

Decision processes.

Many decisions take a long period of preparations, not only in a state bureaucracy or in an industrial organization, but sometimes also in a small organizational unit like a family. As soon as a problem has been identified which is sufficiently mature for action, a decision maker is appointed or a decision-making body is established. The choice of the decision maker or the composition of the decision-making body usually emerges as the result of a series of negotiations where power is employed in a mixture of subtle pressure and brute force. The composition of the decision-making body reflects the strength or the influence of various parties in the organization. In general, the members are also selected on the basis of their ability to judge at least some of the possible alternatives under at least some of the criteria. In many organizations there is a considerable amount of distributed decision making. In the mission of the decision-making committee, the relevant criteria may have been prescribed and the relative importance of the criteria may have been formulated in vague verbal terms. So, the evaluation of the alternatives under the pre-specified criteria is delegated to the experts in the committee, but the weighting of the criteria themselves is felt to be the prerogative or the responsibility of the authorities who established the committee. During the deliberations it may happen that new alternatives and/or new criteria emerge and that the composition of the decision-making body changes because new expertise is required. Nevertheless, there may be a clear endpoint of the decision process, in a particular session of the decision-making group, where each member expresses his/her judgement. At this moment, multi-criteria decision making plays a significant role.

Taxonomy of decisions and decision makers.

Numerous multi-criteria decisions are daily made, both in public and in private life: strategic decisions (in a company the choice of products and markets, for instance, and in private life the choice of a partner and a career), tactical decisions (the choice of a location for production and sales, the choice of a university or a job), and operational decisions (daily or weekly scheduling of activities). Numerous decision makers are also involved in them: charismatic leaders, cool administrators, and manipulating games-men, for instance, who all (inconsistently?) adopt widely different tactics in their style of decision making. Sometimes they defer the decisions to higher authorities, sometimes they delay a decision until there is only one alternative left or until the problem evaporates, and sometimes they deliberate the pros and cons of the alternative options before they arrive at a conclusion. Some decision makers are experienced (the physicians who are specialized in the treatments they prefer), others are totally unexperienced (the patients who have to choose a particular treatment for themselves). And although decision makers are usually not illiterate, some of them seem to be innumerate in the sense that they are insensitive to what numerical values mean within a particular context. Many decisions are made in groups by members who may have widely varying power positions. On certain occasions a brute power game seems to be acceptable, on other occasions the members of the group moderate their aspirations, out of self-interest or motivated by considerations of fairness and equity.

Objectives of multi-criteria decision making.

Methods for multi-criteria decision making have been designed in order to designate a preferred alternative, to classify the alternatives in a small number of categories, to rank the alternatives in a subjective order of preference, and/or to allocate scarce resources to the alternatives on the basis of the final grades or scores. Multi-criteria decision making is usually supposed to have some or all of the following objectives:

1) improvement of the satisfaction with the decision process, because it urges the decision makers to frame and to structure the decision problem and because it enhances the communication in a group of decision makers;

2) improvement of the quality of the decision itself, because it enables the decision makers to break down a decision problem into manageable portions and to keep an eye on the performance of all alternatives under all criteria simultaneously;

3) increased productivity of the decision makers, because it enables them to take more decisions per unit of time, both in public administration and in industrial management.


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[a2] S. French, "Decision theory, an introduction to the mathematics of rationality" , Horwood (1988)
[a3] R. Keeney, H. Raiffa, "Decisions with multiple objectives: Preferences and value trade-offs" , Wiley (1976)
[a4] F.A. Lootsma, "Fuzzy logic for planning and decision making" , Kluwer Acad. Publ. (1997)
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[a6] T.L. Saaty, "The analytic hierarchy process, planning, priority setting, and resource allocation" , McGraw-Hill (1980)
[a7] D. von Winterfeldt, W. Edwards, "Decision analysis and behavioral research" , Cambridge Univ. Press (1986)
[a8] F.A. Lootsma, "Multi-criteria decision analysis via ratio and difference judgement" , Kluwer Acad. Publ. (1999)
How to Cite This Entry:
Multi-criteria decision making. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by F.A. Lootsma (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article