Mini-course 1 Decision Analysis (Dr. Mariya Sodenkamp) / Class 5 / ITB_L5_ 2015_04_27

Eigenvector (priority vector)

In the judgment matrix A, instead of assigning two numbers wi and wj and forming the ra3o wi/wj we assign a single number drawn from the Fundamental Scale. The general eigenvalue formula3on is obtained by perturba3on of the following consistent formula3on:

where A has been mul3plied on the right by the transpose of the vector of weights w=(w1,...,wn). The result of this mul3plica3on is nw.

Thus, to recover the scale from the matrix of ra3os, we must solve the problem Aw=nw. This is a system of homogenous linear equa3ons. It has a nontrivial solu3on if and only if n is eigenvalue of A.

Car Selection Example

Derive criteria priorities (importance weights)

First, add up the values in columns

Second, divide each element of comparison matrix by its Sum of Column.

Now sum of elements in each column is 1.

The matrix is normalized (distributive normalization).

Third, calculate arithmetic mean in each column (we use approximated method II, see Slide 25)

With respect to the goal, Price is the most important criterion with importance 58%, Fuel consumption is the second important factor with importance 23%, it is followed by Maintenance with 12%, and the least important criterion

is Horse power with 7%.

Sum of all elements of the priority vector is 1.

Exercise: Outsourcing Partner Selection

Based on the information above, which vendor is the most preferred (using the AHP)?

Step 3: Calculate local priorities…

Step 4: Estimate inconsistency of provided judgments

An important step in the AHP is to measure consistency of judgments that the decision maker provided during his/her pairwise comparisons.

Example: Inconsistent comparison of tower heights

Consistency Ratio = 0.19

Consistency Ratio exceeding 0.1 indicates inconsistent judgments. In such cases, the decision maker should revise his/her judgment values in the comparison matrix.