3.4.1 How to produce ANP
AHP was proposed by Saaty as a method for MCDM. According to Alonson and Lamata (2006)., AHP is a very popular approach to establish measures in both the physical and social domains that involves qualitative data The AHP has a special concern with departure from consistency, the measurement of this departure, and with dependence within and between the groups of elements of its structure (Saaty, 2006). However, the AHP does not deal with the interdependences among elements. For dealing with the interdependences among elements, the ANP was proposed as a new MCDM method by Saaty (Saaty, 1996). The ANP is actually a modification of the AHP.
Unlike AHP, the ANP offers relations that are non-linear, non-hierarchical, but bi-directional. The ANP utilizes more elements of the complex situation to churn out values for interdependence and feedback relations among factors.
This key benefit of the ANP method allows a more optimal solution towards decision-making such that it provides a ratio-based scale summarizing the distribution of influence among each criterion and among each category. This kind of model can capture effectively the complex effects of interplay in human
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society, especially when risk and uncertainty are involved.
3.4.2 Operations Process of ANP
The AHP is a well-known technique that decomposes a problem into several levels in such a way that they form a hierarchy. Each element in the hierarchy is supposed to be independent, and a relative ratio scale of measurement is derived from pairwise comparisons of the elements in a level of the hierarchy with respect to an element of the preceding level. However, in many cases, there is interdependence among criteria and alternatives. The ANP can be used as an effective tool in those cases where the interactions among the elements of a system form a network structure (Saaty, 1996).
While AHP employs a unidirectional hierarchical relationship among decision levels, it enables interrelationships among the decision levels and attributes to be taken into consideration in a more general form. ANP uses ratio scale measurements based on pairwise comparisons; however, it does not impose a strict hierarchical structure as in AHP, and models a decision problem using a systems-with-feedback approach. Figure 3-3 a and b shows the structural difference between the hierarchy and network. Nodes of the network represent components of the system, and arcs denote interactions between them.
The directions of the arcs represent dependence, whereas loops signify inner dependence of the elements in a cluster. As we can observe, a hierarchy is a simple and special case of a network.
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Figure 3- 3: Product planning in quality function deployment using a combined analytic network process and goal programming approach
(a) A hierarchy. (b) A nonlinear network.
In ANP, the relative importance values are determined similar to AHP using pairwise comparisons with a scale of 1-9, where a score of 1 indicates equal importance between the two elements and 9 represents the extreme importance of one element compared to the other one. The relations a ji=1/aij ; where aij denotes the importance of the ith element compared to the jth element, and aii =1 are preserved in the pairwise comparison matrix to improve the consistency of the judgments.
From a general point of view, the ANP consists of two stages: the first one is the construction of the network, and the second one is the calculation of the priorities of the elements. In order to construct the structure of the problem, all of the interactions among the elements should be considered. When the elements of a component Y depend on another component X, we represent this relation with an arrow from component X to Y. All of these relations are evaluated by pairwise comparisons and a supermatrix, which is a matrix of
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influence among the elements, is obtained by these priority vectors. The supermatrix is raised to limiting powers to calculate the overall priorities, and thus the cumulative influence of each element on every other element with which it interacts is obtained (Saaty & Vargas, 1998). The supermatrix of a hierarchy with three levels is as follows
G C A W =
Goal(G) Criteria(C) Alternatives(A)
(
0 0 0
w21 0 0
0 W32 𝐼
)
Where w21 is a vector that represents the impact of the goal on the criteria, W32 is a matrix that represents the impact of the criteria on each of the alternatives, and I is the identity matrix. The supermatrix of a system of N clusters is denoted as follows
Where Ck is the kth cluster (k = 1,2,…,N), which has nk elements denoted as ek1 ,ek2, …, eknk. A ratio scale priority vector obtained by pairwise comparisons indicates the impact of a cluster’s elements on the elements of
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another cluster. These vectors are located in appropriate positions. Since W is a column stochastic matrix, its limiting priorities depend on the reducibility, primitivity, and cyclicity of that matrix. For instance, if the matrix is irreducible and primitive, the limiting value is obtained by raising W to powers (Saaty, 1996; Saaty & Vargas, 1998).
When a network consists of only two clusters apart from the goal, namely criteria and alternatives, the matrix manipulation approach proposed by Saaty and Takizawa can be employed to deal with dependence of the elements of a system (Saaty & Takizawa, 1986).
3.4.3 Operational Steps of ANP
Four major steps of ANP (Carlucci & Schiuma, 2010; Lili & Ke, 2012):
Step I-Model construction and problem structuring:
In the first step, problems should be started clearly. The goals, criteria and sub-criteria related with the problems should also be selected via brainstorming or other methods. And then, the network with the interactions between and within clusters and elements should be constructed based on all these above.
Step II-Pairwise comparisons matrices and priority vectors:
According to the network proposed in the first step, pair-wise comparisons should be made by a number of experts. And then experts’ preferences should be integrated to form comparison matrices. Consistency test should also be applied to verify the consistency of each comparison matrix.
Step III-Supermatrix formation:
In step 3, a supermatrix should be formed, which lists down all the sub-matrices consisting of all the clusters and necessary elements in order on
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the left and upper sides of the matrix. The supermatrix, whose concept resembles the Markov chain process, is a partitioned matrix, it is formed by grouping the local priority vectors obtained in step 2 and then location them in appropriate positions based on the flow of effect from one component to another, or from a component to itself as a loop in order to determine the global priorities.
Step IV-Prioritizing and Selecting Alternatives:
The main results of the ANP application were the overall priorities of the indicators obtained by synthesizing the priorities of the indicators from the entire network. The priorities for all the indicators can be read from any column of the limit supermatrix.
Selected indicators represent the most basic and important dimensions that managers have estimated to be valuable as the basis for tracking future progress and assess the current baseline performance of the process. Obviously, the appropriateness, over time, of this set of performance indicators will depend upon how the manufacturing process evolves as well as internal and external stakeholders’ information needs change.