A decision problem is considered to exist, when a planner or a decision maker (DM) perceives a discrepancy between the current and desired states of a system, and when (i) the DM has alternative courses of action available; (ii) the choice of action can have a significant effect on this perceived difference; and (iii) DM is motivated to make a decision, but he is uncertain a-priori as to which option should be selected. Multicriteria decision aid (MCA) is a branch of decision theory which deals with decision problems characterized by a number of evaluation criteria. Two fundamental parts of MCA are (i) Multiple Attribute Decision Making (MADM), and (ii) Multiple Objective Decision Making (MODM). The former approach requires that the choice (selection) be made among a limited number of options which are described by their attributes. The second approach allows the number of potential options to be nearly indefinite. The options are not defined explicitly. Instead, MODM provides a mathematical framework for designing a set of decision alternatives.
Preference is a decision maker’s notion about the available options.
Options[1] represent the different choices of action available to the decision-maker. Feasible options must fulfil the satisfaction level (constraints) given by the decision maker for a set of criteria. A non-dominated option refers to the one that is at least equal in all criterion scores and at least is better in one criterion than other (dominated) options.
A criterion is a standard of judgment to test the desirability of an option. In MCA the concept of a criterion includes both attributes and objectives referring to two main directions: multi-attribute and multi-objective decision-making. An attribute is a qualitative or quantitative property that is measurable and relevant to a decision. An objective is a statement about the desired state. Functionally related to the attributes, the objectives indicate the directions of improvement (e.g. an objective may be formulated as: minimizing the water pollution).
Decision matrix is a (M x N) matrix in which the element xij indicates the performance of the option ai evaluated in terms of the decision criterion cj. While the “raw” performances expressed in different non-comparable units and scales are represented in the so-called analysis matrix, the relative performance (uij) is constituted by the preference mapping using a value/utility function and expressed in the same scale as the evaluation matrix.
Value function (u) is a mathematical representation of human judgments. It translates the performances of the options into value scores, which represent the degree to which a decision objective is matched. I.e.:
u(a) > u(b) ↔ a ≻ b
u(b) < u(b) ↔ a ≺ b
u(a) = u(b) ↔ a ~ b
where a, b …options
u() …value function
≻… is preferred
~ …is indifferent
[1] In this document the term “option” is used where many theorists would use the term “alternative”.