By W. Busse, R. Zelazny

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If it further satisfies all the J þ K constraints in Eq. (63), then it is called a feasible solution. e. f = ( f1, f2,. ,fM)T, we have a vector-valued function that represents the objective function and whose image is in a multi-dimensional space, called the objective space, O. Introduction to Ranking Methods 45 In this way, for each x in the decision space D, there is a point in the objective space O, f(x) = (f1(x), f2(x),. ,fM(x))T, therefore mapping takes place between an n-dimensional space (solutions) and an M-dimensional space (objectives).

Further, for computing the ranks it is not necessary to square the deviations to obtain the required ranking, because the same order is achieved by ranking the absolute deviations. If we denote the squares of the ranks of these absolute deviations by ui ðxÞ ¼ jxi À x j, i = 1,. , n, and uj ðyÞ ¼ yj À y , j = 1,. , m, then the test statistic is as follows: Zcalc ¼ where T¼ 1 u¼ nþm X T À nu s Xn n i¼1 i¼1 ðui ðxÞÞ ð40Þ 2 ð41Þ Xm À Á2 ðui ðxÞÞ þ u j ð yÞ 2 j¼1 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ X ﬃ u Xm À n Á2 u 2 2 ðui ðxÞÞ þ uj ðyÞ À ðm þ nÞu u mn t i¼1 j¼1 s¼ ð m þ nÞ ð m þ n À 1 Þ ð42Þ ð43Þ If there are no ties, the test statistic T in Eq.

E. f = ( f1, f2,. ,fM)T, we have a vector-valued function that represents the objective function and whose image is in a multi-dimensional space, called the objective space, O. Introduction to Ranking Methods 45 In this way, for each x in the decision space D, there is a point in the objective space O, f(x) = (f1(x), f2(x),. ,fM(x))T, therefore mapping takes place between an n-dimensional space (solutions) and an M-dimensional space (objectives). That is the reason why sometimes multi-objective optimization is known as vector optimization.