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Analytic Hierarchy Process part-2

The analytical hierarchy process (AHP) method, developed by Saaty (1980),
is based on three principles: decomposition, comparative judgment, and synthesis of priorities. The decomposition principle requires that the decision problem be decomposed into a hierarchy that captures the essential elements of the problem.

The principle of comparative judgment requires assessment of pairwise comparisons of the elements within a given level of the hierarchical structure, with respect to their parent in the next-higher level. The synthesis principle takes each of the derived ratio-scale local priorities in the various levels of the hierarchy and constructs a composite (global) set of priorities for the elements at the lowest level of the hierarchy (i.e., alternatives).Given these principles, the AHP procedure involves three major steps:

01.Develop the AHP hierarchy-

The first step in the AHP procedure is to decompose the decision problem into a hierarchy that consists of the most important elements of the decision problem. In developing a hierarchy, the top level is the ultimate goal of the decision at hand (e.g., select the best site for a nuclear power station). The hierarchy then descends from the general to the more specific until a level of attributes is reached. This is the level against which the decision alternatives of the lowest level of the hierarchy are evaluated. Each level must be linked to the next-higher level. Typically, the hierarchical structure consists of four levels: goal, objectives, attributes, and alternatives. The alternatives are represented in GIS databases. Each layer contains the attribute values assigned to the alternatives, and each alternative (e.g., cell or polygon) is related to the higher-level elements (i.e., attributes). The attribute concept links the AHP method to GIS-based procedures. Although the hierarchical structure typically consists of goal, objectives, attributes, and alternatives, a variety of elements relevant to a particular decision and a different combination of these elements can be used to represent the decision problem. For example, the following combinations of decision elements can be incorporated in the hierarchical structure:

a. Goal, objectives, subobjectives, attributes, alternatives
b. Goal, scenarios, objectives, attributes, alternatives
c. Goal, interests groups, objectives, attributes, alternatives
d. Goal, interest groups, objectives (subobjectives), attributes, alternatives

2. Compare the decision elements on a pairwise base-

Pairwise comparisons are the basic measurement mode employed in the AHP procedure. The procedure greatly reduces the conceptual complexity of decision making since only two components are considered at any given time. It involves three steps:

(a) development of a comparison matrix at each level of the hierarchy, beginning at the top and working down;
(b) computation of the weights for each element
of the hierarchy; and
(c) estimation of the consistency ratio.

The only difference is that the criteria are replaced by the elements of a particular level of the hierarchy. Thus, if we assume that the hierarchy
consists of goals, objectives, attributes, and alternatives, the procedure would
be performed for the objective level, the attribute level, and the alternative
level. Each time, pairwise comparisons would be generated to estimate the relative importance of each element at a particular level with respect to the higher-level components. One of the fundamental assumptions of the AHP is that decision makers are inconsistent in their values and judgments concerning decision criteria and alternatives. The AHP employs a measurement of this inconsistency which can help the decision maker learn more about the decision in question and about his or her own biases and inconsistencies. Note that the pairwise comparison procedure can be employed only for a relatively small number of elements at each level of the decision hierarchy. Therefore, it can only be applied to problems involving a relatively small number of alternatives. When a large number of alternatives is considered, the AHP procedure is terminated at the attribute level, and the attribute weights are assigned to the attribute map layers and processed in the GIS environment. This approach is referred to as spatial AHP (Banai-Kashani 1989; Eastman et al. 1993; Siddiqui et al. 1996).
3. Construct an overall priority rating-

 The final step is to aggregate the relative weights of the levels obtained in the second step to produce composite weights. This is done by means of a sequence of multiplications of the matrices of relative weights at each level of the hierarchy. The sequence is one in which the relative weights matrix for the second level is multiplied by the relative weights matrix for the third level, and then this resulting matrix is multiplied by the relative weights matrix for the next-lower level. This process is continued until all levels from the second level to the bottom level have been multiplied together, forming a vector of composite weights (since there is only one goal, the highest level does not have any matrices associated with it). The vector of composite weights has a dimension of 1 by m (where m is the number of decision alternatives at the bottom level of the hierarchy). The composite weights represent ratings of alternatives with respect to the overall goal. The weights, also referred to as decision alternatives scores, are the basis from which decisions can be made. They serve as ratings of the effectiveness of each alternative in achieving the goal. The overall score R, of the ith alternative is the total sum of its ratings at each of the levels and is thus computed in the following way:

                                                 R = Σ k Wkrik                 

where Wk is the vector of priorities associated with the kth element of the
criterion hierarchical structure, Σ Wk = 1; and rik is the vector of priorities
derived from comparing alternatives on each criterion. The most preferred
alternative is selected by identifying the maximum value of Ri (i = 1,2,...,m).

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