The Role of Constraints in Multi-Criteria Problems

In the class of problems under consideration, system resources are limited. If resources can be selected in an open area, then there are no “red lines”. In an unconstrained problem, the principle of rational organization [1] is applied and a consensus solution can be obtained. Limitations on resources give rise to “red lines” and the solution to the problem can only be a compromise. An example of the tasks under consideration would be diplomatic negotiations. Here the parties can make concessions on some aspects, but only insofar as such a concession does not bring the solution to the problem closer to the corresponding “red line”. Negotiations are considered successful when the maximum possible distance of concessions from the “red lines” is achieved. The objective function Y[y(x)] connects the vector of quality criteria with the optimization arguments. With some reservations, the optimization problem is formulated as finding such a combination of arguments from the domain of their definition at which the objective function acquires an extreme value. If, without loss of generality, we assume that “better” means “less”, then

Restrictions can be imposed both on the optimization arguments x∈ X and on the criteria of the effectiveness of the solution y∈M . Even small changes in constraints can significantly affect the solution results [2]. Moreover, the very concept of a decision-making situation is assessed by the measure of the dangerous approach of individual criteria to their maximum permissible values (“red lines”). Indeed, it is logical to consider the difference between the current value of the criterion and its maximum permissible value as a measure of the tension of the situation:

Where is the vector of maximum permissible minimized criteria. Let’s consider the logic of decisionmakers in polar decision-making situations. If some criterion y_p(x).p∈[1,...s] is dangerously approaching its limit y_pmax, that is ρ_p(x)→0, then we call such a situation tense. In a tense situation, the decision-maker pays primary attention only to this, the most “unfavorable” criterion, trying to remove it from the dangerous “red line”. In this case, with criteria of one dimension, the optimization problem is solved using the minimax (Chebyshev) model

In less tense situations, one should turn to satisfying other criteria, taking into account the contradictory unity of all interests and goals of the system. At the same time, the decision maker varies his assessment of winning according to some criteria and losing according to others, depending on the situation.

When ρ_k(x)→1,k∈[1,...s] the partial criteria are small, there is no threat of violating the restrictions. In a calm situation, the decision maker believes that a unit of deterioration in any of the criteria is fully compensated by an equivalent unit of improvement in any of the others. Here the optimization problem is solved using the integral optimality model

Thus, as a rule, the decision maker varies his choice from the integral optimality model in calm situations to the minimax model in tense situations. In intermediate cases, the decision maker selects optimality principles that provide different degrees of satisfaction of individual criteria, in accordance with the given situation. The type of objective function Y[y(x)] depends on the chosen optimality principle and reflects the scheme of compromises adequate to the given situation.

Based on the above analysis of decision maker logic, it is advisable to formalize the multi-criteria decision procedure and obtain an algorithm that reflects this logic, but does not explicitly include elements of the human factor. From the standpoint of a systems approach, it is advisable to replace the task of choosing a compromise scheme with an equivalent task of synthesizing some unified scalar convolution of criteria, which in various situations would automatically express adequate principles of optimality. Separate models of trade-off schemes are combined into a single holistic model, the structure of which is adapted to the situation of making a multi-criteria decision.

The synthesized function Y*[y(x)] must be smooth and differentiable; in tense situations it should express the minimax principle Y[y(x)]₁; in calm conditions-the principle of integral optimality Y[y(x)]₂; in intermediate cases it should lead to Pareto-optimal solutions that provide various measures of partial satisfaction of the criteria.

Our principle of optimality “away from limits” expresses the desire in any situation to ensure the greatest possible distance from the “red lines”. Therefore, it is necessary to explicitly include a characteristic of the tension of the situation ρ_k(x)=y_kmax−y_k(x),k ∈[1,...s], in the expression for the desired scalar convolution. The simplest of them in the case of minimized criteria is scalar convolution

This objective function embodies a trade-off scheme that is adequate to the entire range of possible situations. Let’s call the proposed scheme a Nonlinear Trade-Off Scheme (NTS). It corresponds to the vector optimization model

Let’s show how it works in different situations. If any of the criteria, for example y_i(x), begins to approach its “red line” imax y , i.e. the situation becomes tense, then the corresponding term in the sum being minimized will increase so much that the problem of minimizing the entire sum will be reduced to minimizing only this worst term, i.e., ultimately, the criterion y_i(x) . This is equivalent to the action of the minimax model Y[y(x)]₁.

If the criteria are far from their limits, i.e. the situation is calm, then the model Y*[y(x)] acts equivalent to the integral optimality model Y[y(x)]₂. In intermediate situations, various degrees of partial alignment of criteria are obtained. If “better” means “more”, then for the criteria to be maximized the scalar convolution according to the nonlinear compromise scheme has the form

Where y_kmin are the minimum acceptable values of the criteria to be maximized.

Note that the construction of scalar convolution Y*[y(x)] allows solving the optimization problem even in the case when the criteria have different dimensions.

A systematic approach to the problem of vector optimization made it possible to combine models of individual compromise schemes into a single holistic structure that adapts to the situation of making a multi-criteria decision.

The nonlinear compromise scheme has the property of continuous adaptation to the situation of making a multi-criteria decision. From this point of view, traditional trade-off schemes can be viewed as the result of the “linearization” of a nonlinear scheme at various “operating points” - situations. This, by the way, explains the name of the proposed nonlinear trade-off scheme, since in other respects it is no more “nonlinear” than other schemes considered in decision theory. We emphasize that the adaptation of a nonlinear scheme to the situation is carried out continuously, while the traditional choice of a compromise scheme is made discretely, which adds to the subjective errors the errors associated with the quantization of the compromise schemes.

The advantage of the concept of a nonlinear compromise scheme is the ability to make a multi-criteria decision formally without the direct participation of the decision maker, which is a distinctive feature of artificial intelligence. The apparatus of a nonlinear compromise scheme, developed as a formalized tool for studying systems with conflicting criteria, allows an artificial intelligence system to practically solve multi-criteria problems of a wide class.

Artificial Intelligence (AI) systems are created in order to replace a person as a decision maker in a given situation [3]. AI systems such as robots, decision support systems, neural networks, etc. work in conditions that a person considers unfavorable for himself. Thus, a robot carrying out mine clearance operates in an environment that is dangerous for the sapper. Decision support systems are usually used under time pressure or in hostile environments. Neural network classifiers process volumes of information that exceed the capabilities of a human operator, etc. The implementation of the outlined stages of vector optimization allows, without the direct participation of the decision maker, to determine the architecture of the neural network classifier, in which conflicting criteria for the effectiveness of its functioning are systematically linked, and the resulting architecture itself is a compromise-optimal one.

The analytical solution to the optimization problem is represented as a solution to the system of equations

If the analytical solution turns out to be difficult, then numerical methods or a computer program for multi-criteria optimization are used [1].

The proposed approach makes it possible to work out formally different scenarios of managerial decisions to solve problems with different input data. The apparatus of a nonlinear compromise scheme, developed as a formalized tool for studying systems with conflicting criteria, allows one to practically solve multi-criteria problems of a wide class.

1. Albert Voronin (2017) Multi-criteria decision making for the management of complex systems. IGI Global, USA, p. 328.
2. Lebedeva TT, Semenova NV, Sergienko TN (2020) Multi-criteria optimization problem: Resistance to disturbances in input data of a vector criterion. Cybernetics and System Analysis (6): 107-114.
3. Savchenko AS (2012) Methods and systems of artificial intelligence: Laboratory workshop. K.NAU, p. 40.