Lecture
In what follows, for simplicity, in this section, unless stated otherwise, we shall assume that in the DMPs under consideration the state of the external environment is completely determined and is not taken into account when choosing the optimal (or best) managerial decisions from the standpoint of the DM's goals. As a rule, we shall denote the set of feasible decisions by the capital Latin letter X. It is from this set X that, in the process of solving a DMP, the selection of the subset of optimal (or best) managerial decisions from the standpoint of the DM's goals is carried out, which we shall denote C(X).
Note!
The subset C(X) of optimal (or best) managerial decisions from the standpoint of the DM's goals in a specific DMP can be empty, finite or infinite, depending on the conditions of the problem .
It is precisely the finding of the subset C(X) ⊆ X that constitutes the result of solving the DMP.
Each managerial decision is aimed at achieving one or several goals defined by the DM. Here, the choice of the needed MD from the set of feasible decisions X and the attainability of a specific goal are accomplished by means of one or several MD selection criteria.
Definition
It is said that a criterion for choosing an alternative is given on the set of feasible decisions X if on the set X there are defined an attribute (or an individual property of the elements of X) and a rule, that allows one to check unambiguously, for any element of the set X whether that element satisfies this attribute or not, i.e., it allows the given set X to be split (according to this attribute and by means of the given rule) into two disjoint and mutually complementary subsets. Here, to each criterion for choosing alternatives on the set X there corresponds a certain criterion (objective, or evaluation) function, which is defined on the set X, with values, as a rule, in the set of real numbers ℜ. It is precisely on the basis of this function that the rule for selecting elements is implemented and the set X is split into two disjoint subsets.
Note!
The definition given above of criterion and attribute on the set of feasible decisions X is a more formal and precise definition of these notions than the ones given in Section 6.2.
If in a DMP only one criterion for choosing alternatives is given and its criterion (evaluation) function f: X — ℜ is known, then the problem of choosing the optimal (or best) managerial decision from the DM's standpoint reduces simply to finding the extremum (maximum or minimum) of the criterion function f on the set X, which is considered a fairly simple optimization problem. The situation is more complicated when, in a DMP, the choice of alternatives has to be made not with respect to one but to several, say m criteria. In this case, the ability to find the extremum separately for each criterion (evaluation) function f_k: X — ℜ, where k = 1,2,m, does not, in general, allow one to hope that a point x' ∈ X will be found such that all the functions fk attain their extremum at it, i.e., for example, so that for all k= 1,2,...,m the following equalities hold: f_k(x') = max f_k(x). Consequently, in the multi-cri-
x∈X
terial DMP, when choosing the optimal (or best) managerial decision from the standpoint of the DM's goals, one must take into account several given criteria for choosing alternatives from the set X, and here one must also take into account the DM's preferences regarding the relations (importance) among the criteria. In this case, when solving a multicriteria DMP, it is mathematically convenient to deal with the so-called vector criterion.
Definition
Let a DMP be given with m criteria, to which there correspond m criterion (evaluation) functions f_k: X — ℜ, where k= 1,2,m. Then the vector criterion of a given DMP is what we shall call the mapping from the set of feasible decisions X into the m-dimensional real vector space ℜ^m, defined as follows:
In this case it is said that a multicriteria DMP or multicriteria optimization problem is given, with the set of feasible decisions X and vector criterion f. Here the vector space ℜ^m is called the criterion space or the evaluation space of this DMP, the image of the mapping f in ℜm, i.e., Im(f) ⊆ ℜ^m — the set of feasible estimates, and the value of the vector criterion f at an arbitrary element x ∈ X, i.e., f(x) = (f_1(x), f2(x), ..., f_m(x)) ∈ ℜm — the vector estimate of the feasible decision x ∈ X .
Example: buying a new car. In this problem, the DM is the car buyer. Any feasible decision (alternative) denotes an action — "buy a specific new car of model x." The result of implementing such an MD is the purchase of a specific new car of brand x,having certain characteristics and properties that can be expressed in quantitative or qualitative indicators. Thus, for example, the criteria for choosing a car from the standpoint of the DM's goals might be: price (in rubles, no more than 1 million rubles); maximum engine power (in "horsepower," no more than 120 hp); fuel consumption (in liters, no more than 7 L per 100 km on the highway); comfort (presence of power steering and climate control or air conditioning); ground clearance (at least 190 mm); ease of use (expert rating on a 10-point scale); and safety (expert rating on a 10-point scale).
Note that in this problem there is no ideal decision option at which all the criterion functions would attain their respective extrema. Thus, for example, the desire to lower the purchase price of the car immediately leads to a deterioration of such criteria as ease of use and safety. In this case, to successfully solve this DMP one must obtain from the DM information about his preferences when buying a car. It should also be noted that the outcome of solving this DMP depends significantly both on the specific DM and on his system of preferences.
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