Chapter 3 The Methodology of Analytic Hierarchy Process
3.2 The Process of AHP
AHP starts with choosing a few main evaluation criteria groups. Then, it extends the main criteria into sub criteria. The evaluation begins with ranking criteria based on their importance level through pair-wise comparison of the alternatives. AHP meets the need to accommodate preference differences between criteria.
Based on Tahriri et al. (2008), there are six major steps in implementing AHP in supplier selection process:
Step 1: Define criteria for supplier selection
It is very important to decide evaluation criteria in the first stage. It is the starting point to structure interviews, questionnaires, and audits. It is also important in sorting and grouping information under each criterion for later evaluation.
Step 2: Define dimension and factor for supplier selection
Dimension and factor shall be defined under each main criterion. They are selected for more detailed weighing of main criteria to generate a more comprehensive supplier evaluation. Every system is a set of elements which are mutually related. These elements form a particular hierarchy, which is crucial for the existence and survival of many systems, both natural and human-made. A system is a multi-layer arrangement. The levels are differentiated by internal structure and functions. The functions of elements on a lower level
(Level 3 Sub-Criteria)
are subordinated to the functions of elements on a higher level. The proper functioning of the higher levels depends on the proper functioning of the lower levels.
Step 3: Structure the hierarchical model
This is the phase to construct the analytic hierarchical tree. The decision hierarchy is level. An element is a defined object such as a decision variant or an evaluation criterion. The reference point for the comparisons is an element which is higher in the hierarchy. Next, we need the following notation to construct the preference matrix A.
A = [𝑎𝑖𝑗], where i, j =1, 2, ..., n , (1)
The preference matrix with the above properties is the result of a pairwise comparison of all the elements at a given hierarchical level with respect to a defined feature (attribute, criterion). Formula (1) means that we are dealing with a matrix of dimensions n×n, where n is the number of elements compared. Formula (2) is an expression of the principle of identity, which says that two identical elements compared with each other are not differentiated by preference. A lack of difference in preferences is expressed by the number 1. Therefore, all the element values along the diagonal of the matrix are equal to 1.
In comparisons between elements at a given hierarchical level, 𝑎𝑖𝑗
indicates how much
more (or less) important the i-th element is than the j-th element. In the AHP, we also assume that preferences are reciprocal, which is expressed by property Formula (3). If we state that the i-th element is, for instance, x times more important than the j-th element (e.g. 𝑎𝑖𝑗= x),
then we automatically assume that the j-the element is 1/x as important as the i-th element (𝑎𝑗𝑖= 1/x).
A suitable evaluation scale is introduced which will allow the analyst to estimate the preferences between the sets of combined elements. In AHP, this method tries to construct the relative importance matrix of the various criteria using the nine-point scale developed by Saaty. The scale and their definitions are shown in Table 4.
Table 4 The AHP pairwise comparison measurement scales between two elements
Intensity of
importance Definition Explanation
9 Extreme importance The evidence favoring one element
over another is of the highest possible order of affirmation
7 Very strong importance One element is favored very strongly over another
5 Strong importance Experience and judgm ent strongly
favor one element over another
3 Moderate importance Experience and judgment slightly favor one element over another
1 Equal importance Two elements contribute equally to the objective
Intensity of
importance Definition Explanation
2,4,6 and 8 For compromise between the above values
Sometimes one needs to interpolate a com prom ise judgm ent num erically because there is no good w ord to with activity j, then j has the reciprocal value when compared with i
A comparison mandated by choosing the sm aller elem ent as the unit to estimate the lager one as a multiple of that unit
Rational Ratios arising from the scale
If consistency were to be forced by obtaining n numerical values to span the matrix
1:1–1:9 For tied activities
W hen elements are close and nearly indistinguishable; moderate is 1:3 and extreme is 1:9
Elvira Haezendonck, 2007; developed by Saaty
In pairwise comparison of n elements, it is sufficient for the comparison values to be entered above the diagonal in matrix A. The remaining values equal 1 (the diagonal) or are the reciprocals of the values above the diagonal. Therefore, the total number of comparisons necessary equals 𝑛(𝑛−1)
2 . Furthermore, when experts perform pairwise comparisons, they do not need to enter their results in a table like matrix A. Better results can be obtained by preparing suitable scales. For example, for 4 elements which are to be compared pairwise, it suffices to draw up the 6 appropriate combinations in the form shown in Figure 5.
9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9
Figure 5 An example of pairwise comparisons for the 4 elements being examined
Entries into the judgment matrices are expressed in terms of the importance intensities illustrated in Table 4. For instance, consider a judgment matrix comparing between elements
“A,” “B,” and “C” in terms of above element “N.” An element in the matrix is equally
important when compared with itself, and thus the main diagonal of all judgment matrices must be 1. Employing Table 5, consider the following scenario:1. In terms of above element “N,” A is very strong importance than B and the intensity of importance is 7. In practice, such a comparison would indicate that, in terms of satisfying criterion “N,” alternative A strongly outperforms alternative B.
2. In terms of above element “N,” C is moderate importance than A and its scale is 3. In practice, this comparison expresses that, in terms of criterion “N,” alternative C is slightly superior to alternative A.
At this point, the judgment matrix of criterion N appears as follows:
Table 5 Preliminary construction of judgment matrix (Criterion “N”)
N
A B CA 1 7 1/3
B 1/7
C 3
The elements on the left and right are equivalent
The element on the left is more important than the element on the right
The element on the right is more important than the element on the left
Following the aforementioned convention, notice that the relative importance from Table 5 are found in row one, while their reciprocal values are found in column one. “It is not mandatory to enter a reciprocal value, but it is generally rational to do so.” (Saaty, 1980) Furthermore, observe that, intuitively, element A is equally important when compared to itself. Now, consider the following additional constraint in the completion of the above element N judgment matrix:
3. In terms of above element “N,” C is absolutely more important than B and its scale is 9.
Such a ranking indicates that, in practice, element C is absolutely superior to element B in satisfying criterion “N.”
Table 6 Completed judgment matrix (Criterion “N”)
N
A B CA 1 7 1/3
B 1/7 1 1/9
C 3 9 1
The above step is repeated until judgment matrices are constructed for each selection criterion. As presented in Tables 5 and 6, the competing elements, in this example A, B, and
C, must be compared in terms of each element. The final task in this step is the construction
of a judgment matrix that prioritizes each selection criterion by comparing one against all other selection criteria.After constructing the pair-wise comparison matrix and making the normalization computation to form the matrix elements onto a common scale, you can obtain the priority ranking of the criteria through calculating row averages.
Ax =
𝜆
𝑚𝑎𝑥x ,𝜆
𝑚𝑎𝑥 = 1𝑛
∑
(𝐴𝑥)𝑖𝑥𝑖 𝑛𝑖=1
Where A is denoted as the pair-wise comparison matrix and X as row averages, Consistency Index (C.I.) can be calculated by:
C.I. = 𝜆𝑚𝑎𝑥 –𝑛
𝑛−1
Where n is the total number of elements in the matrix. The final step in the consistency evaluation is to examine the ratio of the calculated consistency index and the random index (R.I.) derived from the size of matrix.
The corresponding value of R.I. is found in the Saaty’s table below
Table 7 Average random consistency: the reference values of R.I. for different matrix sizes
Size of
Matrix 1 2 3 4 5 6 7 8 9 10
Random
Consistency 0 0 0.58 0.9 1.12 1.24 1.32 1.41 1.45 1.49 Alsuwehri, 2011; developed by Saaty The consistency index (C.I.) refers to the average of the remaining solutions of the characteristic equation for the inconsistent matrix A. This index increases in proportion to the inconsistency of the estimates. Table 7 shows the R.I. values for matrices with dimensions from 1 to 10. Matrices of a higher degree do not concern us because, following our assumption, the maximum number of compared elements should not exceed nine.
The C.I. values for random pairwise comparisons should vary considerably from the expert respondents. The expression of this difference is the Consistency Ratio (C.R.).
Using the responding R.I. found in Table 7, we can receive the consistency ratio C.R. = 𝐶.𝐼.
𝑅.𝐼.
Generally, C.R. ≤ 0.1 a consistency ratio of 0.10 or less is acceptable (Saaty, 1980). In the event that the consistency ratio is greater than 0.10, the operator must re-evaluate the weight assignments within the matrix violating the consistency limits.
Step 4: Prioritize the order of criteria or sub criteria
Having completed the calculative comparisons, this step will rank the criteria according their preference values to give a better grasp of evaluation emphasis.
Step 5: Measure supplier performance
This phase is to implement the evaluation model to assess every supplier’s performance under each criterion. A total score of each supplier will be generated through adding up the weighted scores – multiplied preference values with scores – with respect to criteria.
Step 6: Identify supplier priority and selection
Last step is to rank the overall supplier performance based on the mathematical results, whose priority determines the optimal alternative among those being considered. Column entries in the decision matrix are simply comprised of the principal priority obtained for each selection criteria judgment matrix. The decision matrix is of dimensions M×N, “M”
representing the number of alternatives being considered, and “N” indicating the total number of influential criteria for which judgment matrices was constructed. Considering three possible alternatives (A, B, and C), three selection criteria (i, j, and k), and adopting the following priority subscript convention:
The decision matrix would appear as follows:
(
𝐴𝑖 𝐴𝑗 𝐴𝑘 𝐵𝑖 𝐵𝑗 𝐵𝑘 𝐶𝑖 𝐶𝑗 𝐶𝑘
)
To obtain the overall ranking of the alternatives, the decision matrix is multiplied by the transpose (column version) of the row priority vector from the selection criteria judgment matrix. Considering the following subscript convention for the row priority vector of the selection criteria:
{𝐴𝑉𝐸𝑖 𝐴𝑉𝐸𝑗 𝐴𝑉𝐸𝑘} = Row priority vector for selection criteria matrix The matrix multiplication operation is then formulated as follows:
[
Executing this operation accomplishes weighting of each of the individual criteria priority vectors by the priority of the corresponding selection criteria (Saaty, 1980). The overall rank of each alternative is shown as follows:
Rank of alternative A =𝐴
𝑖𝐴𝑉𝐸𝑖 + 𝐴𝑗𝐴𝑉𝐸𝑖+ 𝐴𝑘𝐴𝑉𝐸𝑘Rank of alternative B = 𝐵
𝑖𝐴𝑉𝐸𝑖 + 𝐵𝑗𝐴𝑉𝐸𝑗 + 𝐵𝑘𝐴𝑉𝐸𝑘Rank of alternative C = 𝐶
𝑖𝐴𝑉𝐸𝑖 + 𝐶𝑗𝐴𝑉𝐸𝑗+ 𝐶𝑘𝐴𝑉𝐸𝑘The overall weighted score for each supplier. Through ranking, the supplier with the best score will be chosen as the suitable supplier as it should have a compelling performance level comparing to all alternatives and satisfy all the goals and objectives of the company.
Advantages of AHP by Tahriri & et. (2008) :
1. The AHP produces a weak linear ordering of the alternatives.
2. The AHP allows for the difference in value between any two alternatives to be quantified.
3. The methodology by which the AHP assigns weights to criteria is simpler than that used by most of the other MCDM methods that rely upon an aggregate value function.
4. Unlike multiple attribute value theory (MAVT), the AHP does not assume the complete transitivity of the decision maker’s preferences. A certain degree of inconsistency is allowed, which in most decision scenarios is realistic.
5. The methodology of the AHP is similar to that used in common sense decision making.
Consequently, this methodology is quite easy for most decision makers to understand.