SOLUTION OF A MULTI-CRITERIA DECISION- MAKING PROBLEM ON BASE OF PROMETHEE METHOD

Multi criteria decision making problem was considered. Review of existing multi criteria decision making methods was presented. Methods of solving this problem can be divided into two large groups: methods using the aggregation of all alternatives according to all criteria and the solution of the obtained one-criterion problem, the second group is associated with the procedure of pairwise comparisons. Promethee method have been considered with details. This method is based on the pairwise comparison of alternatives and specific aggregation procedures. The preference function are considered for minimization and maximization cases. As practice problem the job selection is considered. Three important criteria are used: salary, time, risk. The results of all computations are presented.

Introduction. The problem of multi-criteria decision-making (MCDM) is one of the actual one in the theory of decision-making /1-2/. From a mathematical point of view, it belongs to the class of vector optimization problems. The criteria can be divided into two groups: the criteria for which the maximum value is optimal and the criteria for which the minimum value is optimal. MCDM problems can be solved to within a plurality of non-dominated set of alternatives or set of compromises. Obtaining a single solution can be realized only on the basis of some compromise scheme that reflects the preferences of the decision maker (DM). Methods for solving this problem can be divided into two large groups: methods using the aggregation of all alternatives according to all criteria and the solution of the obtained one-criterion problem, the second group is associated with the procedure of pairwise comparisons and stepwise aggregation. In the first group include the methods: weighted average sum, weighted product and their various modifications /3-4/, in a second group are − Analytical Hierarchy Process (AHP), Elimination and Choice Translating Reality (ELECTRE), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Preference Ranking Organization Method (PROMETHEE) /5-20/. The work /3/ provides information on the popularity of various methods of multi-criteria decision-making.
The description of method.
Consider the algorithm of the PROMETHEE method Method PROMETHEE developed by J.P. Brans and B. Mareschal in 1982 and has been further improved. This method uses a special heuristic scheme for determining pairwise preferences between alternatives.
As known, MCDM problem is specified by a matrix of evaluating alternatives by criteria.
Here is С − criterion for evaluating alternatives − alternative − assessment of the alternative by criterion First of all, for each criterion, the difference between the estimates of all pairs of alternatives and is calculated = − The distance is a measure of the dominance (preference) of one alternative over another. Than more distance, than more dominance, if distance close to zero, then there is no dominance. Distances are calculated according to all criteria. For each criterion, we have a distance matrix (Table 1).
Distance matrix Table 1 A For more convenient normalized measure preferences first introduced a special preference function ( ). This function should be monotonic and is determined for each criterion individually. For the maximum criteria, the function must be monotonically non-decreasing, for the minimum criteria, it must be monotonically non-increasing.
Preference function must have the following properties: 0 ≤ ( , ) ≤ 1 ( , ) = 0, ≤ 0, no preference or indifference ( , ) ≈ 0, > 0, weak preference ( , ) ≈ 1, ≫ 0, strong preference ( , ) = 1, ≫> 0, absolute preference For criteria where the maximum of the function is optimal, it will have the form For the criteria, where it is optimal at minimum, the function will have the form As a rule, this function is set of parametric and depends on two parameters и . The parameter defines the level of indifference, and the parameter sets the preference threshold.
If the distance between the two alternatives is less than , then this difference is considered insignificant and the preference for the alternative is 0.
If the distance between two alternatives is greater than , then this difference is considered significant and there is a strong preference for the alternative , ( , ) i.e. is equal to 1. In the interval ≤ < there is a weak preference.
Various preference functions have been developed. There is no formal criterion for selection values, they are selected from the context of the corresponding criteria. For example, parameters can be selected using the formulas: q = 0.05 (maxU -minU), p = 0.2 (maxU -minU) for maximization criteria, and q = 0.05 (minU -maxU), p = 0.2 (minU -maxU) for the minimization criteria.

. The preference function for minimization criteria
In the general case, each criterion has its own preference function and its own parameters p and q.
For each criterion and for each distance matrix, preference matrices (d). are calculated. As a result, we obtain preference matrices for any criteria. Based on the preference matrices, the matrix of aggregated indices ( , ) is calculated for all criteria For this, the weighting coefficients of the criteria are set ( , ) = ∑ ( , ) =1 where ∑ =1 =1 Next, global estimates are calculated (preference coefficients of each alternative) coefficients of positive Ф + ( ) and negative Ф − ( ) preferences. The positive preference coefficients are calculated as the sum of the values of the preference index matrix by rows, and the negative preference coefficients are calculated as the sum of the columns.
Next, the total preference function is calculated The alternative with the maximum value is recognized as the best.
3. Research results. Consider the problem of selection of a job (job selection). Uses 3 criteria on the basis of which the choice salary (salary), time to get to work (time), work-related risk (risk). There are 5 alternatives, of which the selection of the optimal variant should be made based the PROMETHEE method. Obviously, for the salary criterion, the maximum is optimal, for the remaining two criteria, the minimum is optimal. All calculations were performed in MS Excel. The problem is solved in 4 stages: 1. At the first stage, paired distances are calculated for each criterion which form matrices salary, time, risk For the salary criterion, function (1) is used, and for the time and risk criteria, function (2). Applying these functions to each element of the corresponding matrices, we obtain preference matrices.

Fig. 4. Preference matrices by criteria
3. At this stage, aggregation is performed by criteria into a single matrix of preference indices. For this, weights of the criteria must be specified.
In our case 1 = 0.4 2 = 0.3 3 = 0.3. As a result, we have a matrix of aggregated preference indices. 4. Computing global scores Ф + , Ф − , Ф According to the corresponding formulas, the coefficients of positive Ф + and negative Ф − and total Ф preferences are calculated and the optimal alternative is determined, i.e. alternative with the maximum value in this case it will be alternative A2 Conclusions. The article deals with the problem of multi-criteria decision making based on the PROMETHEE method. The classification of methods of multi-criteria decision making is given. The PROMETHEE method is considered in detail. An example of solving the problem of selection a job according to three criteria is given.