Chapter 3 Methodology of Analytic Hierarchy Process
3.2 The Process of AHP
AHP was developed by Saaty (1977, 1980, 1990) as a decision supportive tool taking into account both tangible and intangible aspects. AHP is a mathematical method for formulating and analyzing complex decisions through making structure of the complex decision problems in the form of a simple hierarchy. Besides, AHP method can be applied to estimate a large number of qualitative and quantitative factors in a methodically pattern under conflicting multiple criteria.
A hierarchical decision structure was constructed to apply AHP method, begin with breaking the decision problem down to its decision components and levels. Then, the critical of the decision components are investigated through pair wise comparison in the hierarchy.
The apex of levels is the main problem or ultimate goal, following by the intermediate levels (dimensions) which relate to the major components while the lowest levels (factors) relate to sub-components. If each component of each level hinges on all the components criteria of upper level, the hierarchy is complete. The components of each level are compared pair wise with respect to a definite component in the level instantly above.
Yoo and Choi (2006) use AHP for identifying relative importance of factors to improve passenger security checks at airport. As well as Berrittella et al. (2009) made a
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ranking operating cost of low cost and full service airlines by AHP. They proposed the below five major steps in AHP process.
Step 1: Define the decision problem and goal
In this stage, the general objective of the decision must be clearly defined which can be broken down into three components:
1. Define a goal: The goal of the problem is main objective that drives the decision problem.
The goal should be a single and specific to the problem, that can be examined properly by the decision makers.
2. Define Criteria: The criteria (dimensions) of a decision problem which used to evaluate the alternatives regarding to the goal. We can go further to create sub-criteria (factors), when more differentiation is required.
3. Define an alternative: The alternatives are the different options that are weighed in the decision. Each alternative will be judged based on these criteria, to see how well they meet the goal of the problem.
With these three components, we can construct a hierarchy for the problem, where each level represents a different cut at the problem.
Step 2: Structure the hierarchy
This is the most imaginative and necessary section of decision-making. As structuring the decision problem into a hierarchy is basically to the process of AHP.
In this stage, the hierarchical model is constructed in the form of a network that modified tree structure is like a hierarchy which hierarchy shows a relationship between elements of one level with other level immediately below, and every element of manner is corresponded to every other, at least in an indirect manner.
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The hierarchical model is structured from the top through the intermediate toward the lowest level which usually holds a group of alternatives. Generally, the model composes of four levels from the top to down, including the goal, criteria, sub criteria and alternatives, shown in Figure 3.1.
(Level 1: Goal)
Figure 3.1 Decision Hierarchy (Saaty, 1980)
Saaty noted that a favorably way to structure the hierarchy is to work down from the goal as long as one can. After that work up from the alternatives until the levels of the two processes are connected in such a way as to make comparisons possible.
Step 3: Construct the pair wise comparison matrix
Pair wise comparison matrices are constructed so as to examine the important or preferences of decision elements which presented into the elements in the hierarchy.
The matrix are constructed for each of the lower levels with one matrix for each element in the level immediately above by using a pair wise comparison measurement scale to weights the important or preference on a nine point scale number that allows the conversion of qualitative judgments into cardinal values. Table 3.1 show the measurement scale for pair wise comparison.
Goal
Criteria 1 Criteria 2
Sub Criteria Sub Criteria Sub Criteria Sub Criteria
Alternative B
Alternative A Alternative C
(Level 2: Criteria) (Level 3: Sub Criteria)
(Level 4: Alternative)
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Table 3.1 A Pair Wise Comparison Measurement Scale Intensity
of important
Definition Explanation
1 Equal importance Two activities contribute equally to the objective
3 Weak importance of one over another
Experience and judgment slightly favor one activity over another
5 Essential or strong importance
Experience and judgment strongly favor one activity over another diagonal, and the number of pair wise comparison equals to 𝑛 (𝑛−1)
2 . For instance, if there is 4 elements, the number of pair wise comparison is 4×32 = 6. The pair wise comparison table on 4 elements is shown in Table 3.2
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Table 3.2 The Pair Wise Comparison Table on 4 Elements
9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9
E1 E2
E1 E3
E1 E4
E2 E3
E2 E4
E3 E4
The element on the left is more important The element on the right is more important
than the right than the left
The elements are equivalent for left and right
Table 3.3 The Result of Pair Wise Comparison Matrix
From Table 3.3
1. In term of above level, E1 is weak important than E2 and the strength is 3.
2. In term of above level, E2 is strong important than E3 and the strength is 5.
Element E1 E2 E3 E4
E1 1 3 7 9
E2 1/3 1 5 7
E3 1/7 1/5 1 3
E4 1/9 1/7 1/3 1
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The above step will be created repeatedly until the judgment is constructed for each criteria. Moreover, each element is equally important when compare to itself, thus the entries in the main diagonal need to be 1.
Then, the comparison matrix are created for the size of n× n. Assume there are n elements with weights w1, w2…, wn. Then, we have the following matrix.
w1/w1 w1/w2 … w1/wn
w2/w1 w2/w2 … w2/wn (1)
⋮ ⋮ ⋱ ⋮
wn/w1 wn/w2 … wn/wn
The above matrix above is an example of a consistent matrix.
Step 4 Compute the eigenvalue
We denote aij the number indicating the strength of ith element (criteria) when comparing with jth element. The matrix of these numbers aij is denoted A, or
A= [aij]
As aij = 1/ aij that is, the matrix A is a reciprocal matrix (a reciprocal matrix is one in which for each entry aij, aij = 1/aji). If our judgment is perfect in all comparisons, then aij = aij× ajk for all i, j, k and we call the matrix A consistent matrix, and matrix A contains no errors (the weights are already known).
An obvious of a consistent matrix is one in which the comparisons are based on extract measurements; that is, the weights w1,…,wn are already known. Then
A = [aij] =
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aij =𝑤𝑖
𝑤𝑗 i ,j=1,…n (2)
Thus
aijajk = 𝑤𝑖
𝑤𝑗 𝑤𝑗 𝑤𝑘= 𝑤𝑖
𝑤𝑘= aik
∑𝑛 𝑎𝑖𝑗𝑤𝑗
𝑗=1 = nwi i, j=1,…n (3)
Then
Aw = nw (4)
In matrix theory, an eigenvector of a square matrix A is a vector w while an eigenvalue is n (associated with an eigenvector w). Both eigenvectors and eigenvalues are very important in AHP. The matrix equation is as follows.
𝑤1 𝑤1
𝑤1
𝑤2 … 𝑤𝑤1
𝑛 w1 w1
𝑤2 𝑤1
𝑤2
𝑤2 … 𝑤𝑤2
𝑛 w2 = n w2 (5)
⋮ ⋮ ⋱ ⋮ ⋮ ⋮
𝑤𝑛 𝑤1
𝑤𝑛
𝑤2 … 𝑤𝑤𝑛
𝑛 wn wn
By making pair wise comparisons between alternatives, we can easily construct a ratio matrix into (6).
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1 a12 … a1n
1/a12 1 … a2n
⋮ ⋮ ⋱ ⋮ (6)
1/a1n 1/a2n … 1
From (4), Aw = nw, if the vector of weights is unknown, it can be evaluated from the pair wise comparison of matrix A, generated from the principal eigenvalue which refers to λmax (for a standard scale ratio matrix λmax = n, that is the largest eigenvalue of that matrix).
We could modify (4) into (7),
Aw = λmaxw (7)
Therefore, we will have reciprocal values under the diagonal (only inverting the ratio) which all ones on the diagonal when compare the alternative itself. This matrix is called reciprocal matrix. They are reciprocal stems from the fact that the ratio does not change depending on which element you compare to another. The construct will be shown in (8).
A =
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1 a12 … a1n w1 w1
1/a12 1 … a2n w2 = λmax w2 (8)
⋮ ⋮ ⋯ ⋱ ⋮ ⋮ ⋮
1/a1n 1/a2n … 1 wn wn
According to Saaty (1980), it was proved that the largest eigenvalue (λmax of a reciprocal matrix A) is always greater than or equal to n.
If the pair wise comparisons consistency, λmax = n.
If the pair wise comparisons inconsistency, λmax > n
We need to check the consistency of pair wise comparison matrix.
Step 5 Analyze the consistency and consequence weight
To maintain rational consistency when deriving priorities from pair wise comparisons, therefore, we have to measure the consistency of the judgment matrix which can be determined from the consistency ratio (C.R.), the consistency ratio C.R. ≤ 10% is acceptable.
If the value is higher, the judgments may not be accepted and should be elicited again, the consistency ratio (C.R.) is defined as (9)
C.R. = C.I.
R.I.
(9)
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C.I. is a consistency index and R.I. is a Random Index
C.I. measures the inconsistencies of pair wise comparisons calculated from (10):
C.I. = 𝜆𝑚𝑎𝑥−𝑛
(𝑛−1)
(10)
n is the matrix size, and
R.I. refers to average random C.I. in a large number of randomly generated matrices from the Table 3.4 below (Saaty and Kearns, 1991).
Table 3.4 Average Random Consistency Index (R.I. Value) Size of
matrix
1 2 3 4 5 6 7 8 9 10
RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49
After the comparison matrix passing the consistency check, we will have to do another process to test the overall consistency of the hierarchy so as to find the global consistency ratio of AHP before acquiring the priority, by defining a Consistency Ratio of the Hierarchy (C.R.H.).
C.R.H. was computed by the Consistency Ratio of Hierarchy (C.I.H.) and Random Index of Hierarchy (R.I.H.) from (11), (12), and (13)
C.I.H. = ∑ℎ𝑗=1∑𝑛𝑖=1𝑗 𝑊𝑖𝑗𝑈𝑖,𝑗+1 (11)
R.I.H. =∑ℎ𝑗=1∑𝑛𝑖=1𝑗 𝑊𝑖𝑗𝑅𝑖,𝑗+1 (12)
nj is to the number of elements contained in j level,
Wij is the comprehensive weight value of i element in j level,
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U i, j+1 is the C.I. of j+1 level toward the i element in j level
R i, j+1 is the R.I. of j+1 level toward the i element in j level.
Or in the form of easily equation below,
C.I.H. = 2nd level C.I. + vector of 2nd level Vector of 3rd level
priority weights consistency indices
R.I.H. = 2nd level R.I. + vector of 2nd level Vector of 3rd level
priority weights random indices
Then
C.R.H. = C.I.H.
R.I.H. (13)
When C.R.H. ≤0.1, it means the overall hierarchy of the developed comparison evaluation is consistency.
And when the local priorities of elements (local weights) in different levels are available, to obtain the final priorities (global weights) of the alternatives βi, the priorities should be as (14)
T(βi) = ∑𝑘𝑤𝑘𝑇𝑘(ai) (14)
×
×
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wk is the local priority of element k and Tk(ai) is the priority of alternative βi with respect to element k of the upper level. An alternative with the largest value of T(βi) is chosen as the best alternative.