Description of the method

The analytic hierarchy process (AHP) was developed by Saaty (1980) in the early 1970s to structure and analyse complex decisions. One important application of the method is the support of decision-making involving multiple objectives.

The AHP splits the decision process into partial problems in order to structure and simplify it. A hierarchy containing multiple target levels, such that the main target is broken down into sub-targets. At the lowest level(s) of the hierarchy, the alternatives (here, the investment projects) are included.

Using the AHP, both qualitative and quantitative criteria can be considered. In each case, the relative importance (weightings) of the different criteria, and the relative profitability of alternatives, is determined with respect to each element of the higher level by using pair comparisons. Then, a total value is calculated for subtargets to determine their relative importance for the whole hierarchy, and, ultimately, to assess the overall profitability of the alternative investment projects.

The AHP is carried out using the following steps:

1. Formation of the hierarchy.

2. Determination of the priorities.

3. Calculation of local priority vectors (weighting factors).

4. Examination of the consistency of the priority assessments.

5. Determination of (global) priorities for the sub-targets and alternatives with respect to the whole hierarchy.

Under certain circumstances some of these steps must be repeated, particularly where priority estimations are inconsistent. Evaluation of the subjective priority assessments for consistency is another characteristic feature of the method.

The initial formation of the hierarchy requires segmentation and hierarchical structuring of the decision problem. In this step, an unambiguous demarcation must be drawn between different alternatives and sub-targets. Relevant relationships should exist between the elements of successive levels only. This implies that no (or only minor) relationships exist between the elements of a single level. In addition, the elements of a single level should be comparable and belong to the same category of importance. Finally, assessments should be independent of other assessments at the same and other levels. Usually, it is also assumed that all relevant alternatives and target measures will be considered. The measurability of target criteria has not to be considered in this step of the AHP.

The second step is the determination of priorities for all elements of the hierarchy. This involves estimating and quantifying the relative importance of every element in relation to each element of the hierarchy immediately above. This is done using pair comparisons with other elements at the same level. Thus, each element's relative importance for fulfilling target criteria is ranked at each level, as a contribution to the fulfilment of the overall target. For alternative investment projects, this relative importance represents a degree of profitability.

With regard to the pair comparisons, it is assumed that the decision-maker is able to determine values vic for all pairs i and c from the set A (target criteria or alternatives) on a relational scale. This will indicate, for an element at the next level up, the relative importance of i and c, and must be estimated for all elements of the higher level and for all levels. Reciprocity should apply for the estimated values. That is, the comparative value of i relative to c must equal the reciprocal of the comparison between c and i. Then, for an element at the next level up it applies:

Moreover, a comparative value vic should never be infinite. An infinite relative importance would mean the target criteria or alternatives regarded were not comparable, and a renewed target and problem analysis would be required.

For the pair comparisons, the nine-point scale suggested by Saaty (1980) and illustrated in Figure 6-4 may be used.

Scale value |
Definition |
Interpretation |

1 |
Equal importance |
Both compared elements have the same importance for the next higher element. |

3 |
Slightly greater importance |
Experience and estimation suggest a slightly greater importance of one element in comparison with the other element. |

5 |
Considerably greater importance |
Experience and estimation suggest a considerably greater importance of one element in comparison with the other element. |

7 |
Very much greater importance |
The very much greater importance of one element in comparison with the other element has been shown clearly in the past. |

9 |
Absolutely dominating |
The maximum difference of importance between two elements. |

2,4,6,8 |
Intermediate values |

Fig. 6-4: Saaty's nine-point scale for pair comparisons

This scale has the advantage of converting verbal comparisons into numerical values, so that measurability on a relational scale is possible. A more detailed scale is not regarded as meaningful. Using this scale, comparisons can yield only values between one and nine, or their reciprocals (which apply where an element is of lesser importance than the other element).

The results of pair comparisons related to an element of the next level up may be shown in the form of a C x C matrix [denoted 'V'] with C elements being compared. The values along the main diagonal of this pair comparison matrix are always 1.

To obtain a pair comparison matrix for C elements being compared, 0.5 ■ C ■ (C - 1) pair comparisons must be made, since the values across the main diagonal are 1 and reciprocity is assumed. Therefore, the determination of a comparative value vic is not required if the reciprocal value vci is known. The required number of pair comparisons increases steeply with an increasing number of elements at a single level; this should be considered when determining a hierarchy.

A perfect (i.e. consistent) execution of all pair comparisons has been made if, for every matrix element vic, and all elements j different to i and c, the following equation is valid:

If such a consistent execution of the pair comparisons can be assumed, some values can be derived from prior assessments, and the required number of pair comparisons may be reduced to C - 1.

In the third step, local priority vectors (weighting factors) are calculated for every pair comparison matrix. From the totality of the pair comparisons, the relative importance of the elements (alternatives, target criteria) is determined and summarised in the form of a priority vector. Accordingly, every component of this vector indicates the relative importance of its associated element to the relevant element at the next level up.

The calculation of the priority vectors [denoted 'W'] may be carried out by means of the eigenvector method, as explained below. Based on the pair comparison matrix V, and (temporarily) assuming that the estimations are perfect and the relative importance wc of all the separate elements of c is known, the matrix elements vic can be calculated as follows:

Moreover, on account of the reciprocity condition: 1 1

vci wc wi

Additionally:

V vic —- = Y—- —— = C is valid for all i e A

And also:

Because this relationship applies to all lines i (i = 1,...,C) of the pair comparison matrix, the following system of C equations can be formulated:

v11 v21

v12 v22

vC1 vC2

v1C v2C

v1C v2C

A |
/ \ w1 |
/■ \ w1 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

w2 |
= C • |
w2 |
(6.12a) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

V wC ) |
V wC ) |
This system of equations represents a specific so-called eigenvalue problem. Such a mathematical problem is generally defined as follows: for a C x C matrix (B), real numbers L and corresponding vectors X must be found which fulfil the following system of equations: The numbers (L) are called eigenvalues of B, and the assigned vectors (X) are called eigenvectors. The sum of the eigenvalues in an eigenvalue problem equals the sum formed by the elements of the main diagonal. As for the pair comparison matrices considered here, these elements are each equal to 1 and so the sum of the eigenvalues is the same as the dimension (C) of the matrix. If all assessments are consistent, there is only one positive eigenvalue with the value C. However, in a multi-criteria decision problem priority estimates are often inconsistent and the weighting vectors are not known. Therefore, in the following discussion the corresponding assumptions must be abandoned. If priority estimates are inconsistent, several eigenvalues and eigenvectors will result. Thus, the maximum eigenvalue Lmax of the pair comparison matrix and the associated eigenvector must be determined. The latter should be standardised so that the sum of its components is 1, then it can be regarded as the weighting vector W. The calculation of such a weighting vector is meaningful, even with an inconsistent pair comparison matrix, as small inconsistencies will show only a slight effect on the weighting vector. To determine the maximum eigenvalue and the weighting vector, the following eigenvalue problem must be solved: Here, U represents a C x C unit matrix. For the eigenvalues L in this problem, the determinant of the matrix (V - L • U) is zero, i.e.: The maximum value L fulfilling this condition is the maximum eigenvalue Lmax. By inserting this value in the equation system given above, the eigen- or weighting vector may be calculated. For this vector it applies: And: The calculation of the maximum eigenvalue and weighting vector involves substantial computational effort. Therefore, approximations are suggested, e.g. the weighting vector can be approximated from the pair comparison matrix V by using the following arithmetical rule to generate matrix products gradually: V • U; V2 • U;V3 • U;...; Vo • U (6.18) Where: V = C x C (pair comparison matrix) U = C x 1 (unity vector) With a sufficiently high value o, the vector Vo • U is a good approximation for the eigenvector. An examination of the consistency of priority assessments takes place in the fourth step of the AHP for all pair comparison matrices. This step is necessary because the consistency of all estimates cannot be taken for granted. If all the assessments are totally consistent, the maximum eigenvalue is C. Where there are inconsistencies, however, a higher eigenvalue Lmax arises. This value Lmax might not be known exactly if, in the third step, the eigenvectors were calculated using an approximation. Then, Lmax can only be approximated (e.g. using the well known Newton method to determine zero points). The difference between Lmax and C increases with increasing inconsistency, so it provides an indication of the consistency of the estimates. An index of consistency (IoC) can be formulated using an additional calculation: In assessing consistency, the matrix dimension should also be taken into account, since it influences the extent of typical inconsistencies. To do this, a value of consistency (VOC) is calculated. The VOC indicates the relationship between the index of consistency (IOC) and an average value of indices of consistency (RI) derived from reciprocal matrices of the same size, which are produced randomly based on Saaty's nine-point scale: Figure 6-5 shows the average values, calculated by Saaty, in dependance on the matrix dimension.
Fig. 6-5: Average values of indices of consistency Saaty suggests 0.1 as a critical limit for the value of consistency. Accordingly, pair comparison matrices with a consistency value VOC < 0.1 are regarded as being sufficiently consistent, while matrices with VOC > 0.1 require an examination and revision of the pair comparisons. Up to this point in the analysis, each estimated priority has been related to only one element at the next level up the hierarchy. In the fifth step of the AHP, the determination of target and alternative priorities for the whole hierarchy, the weighting vectors are aggregated with respect to all elements in the next level up and all other higher levels. This facilitates the assessment of both the global priority (or relative importance) of each target criterion and the ultimate profitability of alternatives. As a result of the pair comparisons for the second level of the hierarchy, a weighting vector is generated. This indicates the importance of target criteria at this level relative to the overall target, thereby showing both the local and global priority of the targets. The weighting vector is a starting point for the calculation of global priorities for the elements of each subsequent level. It is multiplied by a weighting matrix, which incorporates the weighting vectors of the level subsequent to it. The product is also a weighting vector, whose components represent the global priorities of the elements of the subsequent level. The successive continuation of this step leads to the calculation of the global priority for the alternatives at the lowest level of the hierarchy. This procedure for determining global priorities for alternatives may also be interpreted as the additive calculation of a utility measure Ua for each alternative Ai with the formula: The index c refers to the elements of the next level up, which here represents target criteria. The symbol wc indicates the global priority of these target criteria, and uic is the relative importance (profitability) of the alternative i concerning the criterion c. Therefore, the global priority (as for the utility value analysis described in the previous subchapter) is calculated as a sum of weighted partial priorities. The global priorities determined in this step represent weightings of the target criteria. Concerning the alternatives under consideration, they estimate the contribution made to the fulfilment of the overall target. In assessing the relative profitability of (investment) alternatives when the overall target is to be maximised, the following key concept applies: Key Concept: Relative profitability: an investment project is preferred if its global priority is higher than that of every other project under consideration. The isolated assessment of absolute profitability by AHP is not possible, as the procedure is based on pair comparisons and, therefore, assessment of one alternative depends on the other alternatives selected. However, the alternative of not investing may be included in the procedure. In that case, an estimate of absolute profitability can be made by comparing the global priority of not investing with that of the remaining alternatives. Example 6-2 ## The following example draws on Example 6-1.The first step of the AHP is the formation of the hierarchy. In this example, the target system is drawn from the previous section. Figure 6-6 depicts this target system and contains, in addition to the previous example, the location alternatives Ai, A2 and A3 as elements of the lowest hierarchy level. "C lsl level: Target 2nd level: Sub- targets 3rd level: Target criteria 4 th level: Alternatives
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