Dear colleagues,
I would like to use an Entropy method for calculation the weights of my criteria.
My problem:
1. matrix R, in which criteria are written in columns (j=1,2,3,..k) and variants in rows (i=1,2,3,..n)
2. one or more cases: rij = 0
One step of calculations is to calculate ln(rij), but in case that rij = 0, it cannot be calculated. How to solve it?
... I am thinking about replacing the zero with something else, e.g. 0.0000000001 (0=>0.0000000001). Can I do it? (I doubt that)
Thanks in advance for any help.
Roman
I have the following suggestions:
1) if the matrix R( m by n), which I assume is non-zero, is also non-negative ( all rij greater than or equal to 0), you can try the following translation scheme:
G(m by n) = R(m by n) + J(m by n), where J (m by n) is the matrix of all 1's
thus gij = 1+ rij for all i and j, all the zero values translated to 1, strictly positive rij values translated by 1 as well.
2) let rank (R)= t, do a thin SVD ( singular value decomposition) of the matrix R(m by n):
R(m by n) = U(m by t) *S(t by t) "V^H (t by n), here V^H denotes the hermitian conjugate of the matrix V(n by t)
by properties of SVD: U and V matrices are Orthonormal matrices, i. e.
U^H*U = V^H*V = Identity matrix (t by t)
S(t by t) = diagonal matrix, with strictly positive diagonal elements s1, s2,.., st
Trace (S) = s1+s2+.. +st = D, define f1 = s1/D, f2= s2/D .... upto ft=st/D, each of these ft terms are thus confined in the interval (0, 1)
define the associated entropy function H(R) = H(f1, f2,.., ft) = - [ f1*loge(f1) + f2*loge(f2) +.... + ft*loge(ft) ]
In my opinion, Entropy is a method that can be preferred especially in problem situations where there is uncertainty and lack of information about the criteria. We have used the entropy weighting method before and we have no objection to the adequacy of this method. However, financial data, for example, may contain negative data, and the Entropy methodology does not allow negative data. We can overcome this by normalizing twice, but this may pose some kind of risk of data loss. My suggestion for this is to use, for example, min-max or vector methods as an alternative to the "sum normalization" method used in Entropy. In my opinion "sum" normalization is not a good and comprehensive choice for Entropy anyway (for all possible data types) in all cases. The preferred type of normalization may also affect the efficiency of the Entropy result, as in the MCDM method. On the other hand, the technique you suggested as an alternative to the "zero" value in Entropy can be applied. As long as you minimize the degree to which it affects the results, I think it's fine.
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