A Method for Incomplete Comparisons

6. A Method for Incomplete Comparisons

When not all judgments are available, either because of the limitation or because the judge is unwilling or unsure to make a comparisons of two elements, Harker [4] provides the following suggestions:

Have the decision maker provide judgments, such that at least one judgment is answered in each column, yields a matrix with some unknown ratio elements.

Enter zero for any missing judgment in that matrix, and add the number of missing judgments in each row to the diagonal element in the row, producing a new matrix A.

Calculate the weight w:

lim

= cw

Use the resulting w i /w j as a suggested value for the missing judgments to make it consistent with the judgments already provided.

If needed, the decision maker can be guided to make additional judgments, that have the greatest impact on the weight w. One computes the larger absolute value of the gradient of w with respect to the (i,j) calculated using the following formula:

x:right principal eigenvector = w, in AHP notation Ax = λ max x

t y:left principal eigenvector y A=λ

max y

A  ∂ λ max

D λ max =  | i, j  ,

 ∂ ij

[ ( y i x j ) - ( y j x i )/ a ij i > j ]

' where y is normalized so that y x = 1.

 ∂ ij

Then

is the matrix of gradients for the weights x and is given by:

I ~  λ  max D λ max x -  z

 -1 ~ ~   ~ A

where

I = n x n identity matrix

e = n dimensional row vector of ones z = (z k )=n dimensional column vector defined by:  x j if k = i

k =  - x i / a ij if k = j

  0 otherwise

~ denotes the matrix or vector with its last row deleted. The choice of the next (i,j) value is made according to:

where Q is the set of unanswered comparisons and

denotes

or the Chebyshev norm.

Harker's Example

If: If:

x = (0.5396145, 0.2969613, 0.1634241) y = (0.1634241, 0.2969613, 0.5396145)

After normalizing y so that y T x = 1, we have: y = (0.6177249, 101224806, 2.0396826) The derivatives of the principal eigenvector are:

 0 0.0320137 - 0.0213425 

D λ max =  0 0 0.0320137 

A -1 - λ

max I   - 2.0092027

and For each question, z and x/ ij are:

z =  - 0.1349036  =  - 0.0612519 

(2,3) Therefore, (1,2) should be asked next.

The decision maker may decide to stop the questioning, or continue according to: k a.)If the maximum absolute difference in the attribute weights from one question to the next (w The decision maker may decide to stop the questioning, or continue according to: k a.)If the maximum absolute difference in the attribute weights from one question to the next (w

1 i n and w ) is less than or equal to a given constant α% and ≤ ≤ then the

k+1

procedure would stop at the (k+1) comparison if

b.)The next question derived from the gradient procedure would only be asked if it appears that the ordinal ranking could be reversed. Due to the computational complexity of this task, a simplification can be made by using a sample of random spanning trees to calculate: a th ij = the current value of the (i,j) question which has just been chosen.

ij =the largest path intensity in the set of all elementary paths connecting i and j a ij = the smallest path intensity a

and define P(w) to be a function which returns the ordinal ranking inherent in the cardinal ranking

w; that is P: R n Z where Z is the n dimensional space of natural numbers. For example, if

w = (.15, .3, .2, .35) T then P(w) = (4, 2, 3, 1) . Three rankings can be defined with this function:

1 = P(w) P

2 = P(w + w/ P ij (a ij +u ij ))

3 = P(w + w/ P ij (a ij -L ij ))

1 P = current ordinal ranking

, P th P 3 =approximation to the ordinal ranking if the (i,j) comparison achieved its max and

min deviation, respectively.

If P 1 =P 2 =P 3 , the next comparison is unlikely to alter the ordinal ranking.

Nonlinear responses

In the standard theory it is assumed that a ij is an approximation to the ratio w i /w j . However, one

j could have situations in which a α is an approximation to some function of this ratio f(w /w ) with α > 0.

ij

w = λ max w

The eigenvector problem can be written as

w = ( where w 1 w , 2 ..., , w n ). Defining v to be equal to w α leads to Av = λ max v, a standard eigenvalue problem in the AHP.