A New Technique in Ranking of Alternatives; Keen Analysis Surge of Advantage States

This paper presents a new dimensionless technique for decision-making. The new technique is the keen Analysis Surge of Advantage States of Alternatives with respect to its criterion for appreciates the incontestable value. This technique depends on the real advantage of all alternatives with the new definitions of concordance and discordance indices especially that have not the same unit. The advantages can be summarized as follows; don’t cause a redundant evaluation and also provide a clear ranking of alternatives with priority vector, capability in introducing the actual comparisons between alternative, clarify the effective relations between alternatives, gives a priority vector with actual outranking relations and robust ranking of alternatives. Numerical example introduced with actual estimation. Keyword Concordance, Discordance, Ranking of alternatives, Outranking


A. Performance Matrix
Most of the MCDM methods require that assign the weights of importance for each attribute. Usually, these weights normalized to add up to one. Therefore, MCDM problems has expressed in matrix format. a decision matrix (performance matrix) is an * matrix in which elements indicate the performance of alternative with respect to criteria [10].
The performance of alternatives is evaluated in terms of decision criterion (for 1, 2, 3, … , and 1, 2, 3, … , ). It is also assumed that the decision maker has determined the weights of relative performance of the decision criteria (denoted as , for 1, 2, 3, … , ).This information is best summarized in the performance matrix as indicated in Figure 1. After construction of performance matrix of alternatives with respect to its criterion the used technique has been applied.

B. Keen Analysis Surge of Advantage States (KASAS)
The major difference of any MCDM technique is the pholosoghy in performance of alternatives with respect to its criterion. Therefore, this technique depends on the outcoming steps.

1) Detrmination of concordance set
The concordance set of two alternatives and ,where ∈ and 1, is defined as the set of all criteria for which is preferred to as indicated in equation 1: such that 1, 2, 3, … , From equation 1 it is shown that when the comparison occures between and , then the alternative is preffered than at criterion only if . Therefore, if comparison occures between and , then the alternative is not preffered than at criterion . Equation 1 neglicts the equality between any two alternatives with respect to each criterion to delete aredundnt of weights on each comparison and permits the correct prefrence relations between any two alternatives due to controversial without advantages.

2) Discordance Sets
The complementary subset of concordance is called the discordance set and it is described as in equation 2: From equations 1 and 2, it is shown that two complementary sets must be estimated without any intersections that resulted from the equality and the decision will be depends on the computation between two sets.

3) Construction of concordancematrix
The new technique suggested that the degree of difference between alternatives under each criterion must be known. This degree is the contribution in concordance between any two alternatives. So, the degree of difference between any two alternatives and under criterion is denoted by . . / and defined by equation 3 as follows: The positive difference between any two alternatives belongs to concordance set and this degree of difference is the contribution in concordance of these alternatives under its criterion. Therefore, the max difference under each criterion . between any two alternatives should be estimated. The contribution of relation between any two alternatives and under any criterion in concordance index is defined in equation 4.
The outranking relation of concordance of / / . . / . and so, This contribution is the partial outranking between and in concordance index. Thus, the concordance value that indicates the relative importance of alternative over with respect to all criterion is the sum of multiplication of contributions with its criterion weights as defined in equation 5 as follows: Therefore, the concordance matrix is constructed from concordance values 0 1 and is defined as follows: It should be noted that here the entries of matrix are equal zero due to the equality.

4) Construction of discordance matrix
The discordance matrix expresses the degree that a certain alternative is worse than a competing alternative relative to each criterion. From the discordance set in equation 2, it is shown that the complementary subset of concordance is the set that has the negative difference between alternatives as indicated in equation 3. The same philosophy of the degree of difference in concordance is also applied in discordance with the negative difference. Thus, the contribution of relation between the alternatives and in discordance index under each criterion is defined in equation 6 as follows: / .
; 1, 2, 3, … , / 0 The discordance value that indicates the poorness of alternative with respect to as shown in equation 7 is the sum of multiplication of contribution under criterion with criterion weight for all criterion and in indicated as follows: So, discordance matrix is constructed from discordance values 1 0 and is constructed as follows: It should be noted that here the entries of matrix are equal zero due to the equality.

5) Preference of alternatives
From the concordance matrix, the outranking of any alternative to others was identified. Where, row one indicates the outranking of alternative number one to all other alternatives with the effects of criterion weights. Thus, sum of rows is the score of dominance of row alternative to all others. So, the Preference of dominance of any alternative is constructed as follows: . ; The priority values of dominance are the priority vector of dominance which has one column and rows as indicated as in equation (9). The maximum value in priority vector of dominance candidates the alternative to be an optimal alternative depends on the required order.
In the discordance matrix, each row indicates the degree of outranked of by other alternatives. Therefore, the sum of each row is the score of outranked of each alternative by others. Thus, the discordance of any alternative is constructed as follows: . ; The Total discordance ( . ) is the priority vector of discordance which has one column and rows. The minimum value of discordance candidates the alternative to be optimal where all members are negative. .

6) Ranking of alternatives
The Ranking of alternatives generally depends on the required decision. If the required decision is the optimal alternative that ouranke all other alternatives, the alternative that has high score in priority vector of dominance is the optimal alternative (profit). If the the required decision is the optimal alternative that has less outranked from other alternatives (cost), the alternative that has the minimum value (negative value) in the priority vector of discordance is the optimal alternative.
Globally, the optimal ranking is estimated by summation of the corresponding values in the priority vector of dominance and in the priority vector of discordance and is represented in ranking vector ( . ) as follows: .
Therefore, the alternative that has high score is the optimal alternative than others and vice versa. Ranking of alternative is constructed depends on the priority vector values. Ranking of alternatives indicates the outranking of any alternative to athers, so, the outranking relation will be discussed in the next division.

7) Outranking relations
The outranking relations is constructed by means of a threshold value in the concordance index. For example, will only have a chance to dominate if its corresponding concordance index exceeds at least a certain threshold value . That is, the following is true: . The threshold value can be determined as the average concordance index. That is, the following relation is true: The threshold value can be determined as the average discordance index, where is defined as follows: Once the two indices are defined, an outranking relation was defined by: Equation 15 assures that outranks if the concordanceof exceedsconcordanceof and the discordance of is lower than discordance of .

IV. NUMERICAL EXAMPLE
It is assumed that there are ten options , , , … , to be compared using six criteria [11]. These criterions areassumed on numerical scales, and much that high values are deemed preferable to low ones. Details of performance matrix are contained in Table I with its criterion weights. To easy understanding the estimations, we will begin with the degree of deference of alternative with respect to all other alternatives as indicated in Table II.Max degrees of deference with respect to six criterions are indicated in Table III.The concordance contribution of alternative with respect to all other alternatives is illustrated in Table IV with the application of equations 4 and 6. Application of equations 8 and 10 are constructed in Table IV to extract the weighted concordance and discordance of alternative with respect to others. The weighted concordance and discordance of alternatives from to with respect to other alternatives are defined From Table VI up to Table XIV respectively.  TABLE IV  The contribution of concordance and discordance of  and  under each criterion   TABLE V  The weighted concordance and discordance values with           After construction of the concordance and discordance matrices from the previous tables, the requirements are the priority of alternatives, outranking relations between alternatives. From Tables XV and XVI, the priority vectors of concordance and discordance are estimated ( . and . ) and then the overall score of alternatives can be constructed from . (equation 15) as indicated in Table XVII. In Table XVII, the priority of concordance and discordance was summed to be the . vector that is the overall score of priority of all alternatives.
Thus, it is now necessary to start to identify patterns of dominance among the options, using the conditions in equation 19. Here, this yields the following initial dominance pattern for each alternative and the number of dominated alternatives with it and identification as in Table XVIII.
The new technique has one advantage than other methods where it constructs a complete relation between all alternatives and there is not a relation that contradict with them. Each relation has number of dominations varying than others, which is the road of ranking and outranking. In begins with the lowest domination, then is the lowest alternative, the next is the relation that has one dominated. This is found in relation one, which dominates , so is the previous of . thus alternative becomes number ten and has number nine in ranking. The next relation is the relation that has two dominated which is relation number nine which dominates and . So, alternative becomes number eight in ranking. This procedure gives that same ranking from . vector and extract that the alternative, which has a high score, outranks all alternatives lower it.
When we subtract the number of alternatives from the dominated alternatives from each alternative, this extracts the ranking of alternative. As example in relation 10, and are dominated from so, alternative has ranking equal 10 2 8, where 10 is the number of alternatives. Also, in relation 8 the alternative has a ranking number one (10 9 1). Finally, ranking of alternatives as indicated in Table XVII arranged as follows HGICEBDJAF. This analysis is true for alternatives more than three. VI. CONCLUSION From the weakness in the ELECTRE techniques and its shortages that have been indicated in most researches, this paper introduced a new dimensionless technique with conflicting criteria. This technique is a Keen Analysis Surge of Advantage States [KASAS] which able to deals with the decision problems under the presence of number of decision criteria. This technique has a power to deal with the conflicting criteria and incommensurable units. The new technique used a new adjustment of nature of concordance and discordance indicies. It is indicated the contribution of advatage of one alternative in related to ather alternatives contributions in the direction of dominance by the degree of difference. This technique is called dimensionless analysis because its structure eliminates any units of measure, which uses relative ratios insteadof actual ones. Thus, it can be used in single or multi-dimensional decision making problems. This technique ha a capability to identify the outranking relations between any two alternatives without shortage, delete the old understanding of imposibility of getting an outranking relation. Priority score is the advantage of this method offered through computation of composite priorities of the alternatives by linearly adding the weighted cocordance and dicordance indecies values. The priority vector of alternatives in this technique is the effective tool to make a correct and complete ranking of alternatives.