Methodology Study On The ANP Based On The Perspective Of Nonlinear Complex Systems

The analytic network process (ANP) by famous Prof. Saaty is an essential decision method for articulating our understanding of complex decision issues. The ANP provides a general framework to deal with decisions without making assumptions about the independence of higher-level elements from lower lever elements and about the independence of the elements within a level as in a hierarchy, so it is a better decision tool, compared with the analytic hierarchy process (AHP). Thus far, the ANP has been widely applied in many fields. However, the ANP has the drawbacks such as the concept of weights being undefined and the relation of system structure being fuzzy. And further observed from the mode of thinking, conventional ANP embodies a thought of reductionism in essence. Since the thought of reductionism ignores the holistic characteristic and complex influence relationship among the system elements by assuming independence of primitives, it is difficult to reflect the basic characteristics such as nonlinearity, nonbalance, mutation, bifurcation and chaos, etc. Therefore, ANP has serious theory flaws with the viewpoint of methodology. It follows that a new ANP decision-making approach, which may overcome the above drawbacks and wrongs of traditional ANP, should be suggested from the perspective of nonlinear complex systems, instead of using traditional ANP. The proposed decision approach not only has evident innovation value in theory but great practical meaning to solve all kinds of complex decision problem in real life.This paper first outlines relevant theoretical bases and some inspirations are obtained from that. Then, the structure of ANP is divided again into the following six forms, i.e., analytic hierarchy process (AHP), circular system of inner independence (CSII), hierarchy system with inner circular dependence (HSCID), hierarchy system with inner multi-element dependence (HSIMD), hierarchy system with outer feedback (HSOF) and circular system with inner dependence (CSID). Then, the foregoing four typical structures is regarded as the study objects, respectively, corresponding four alternative ranking approaches for the nonlinear ANP are proposed, based on nonlinear complex-system theory and the meta-syntheses methodology of combining holism and reductionism are proposed. These approaches are derived by way of constructing the new analytic structures for the complex evaluating problem and the value systems for evaluating alternatives. Next, a new synthesis method of alternatives priorities derived from separate nonlinear ANP frameworks for benefits, opportunities, costs and risks (referred as to BOCR) with nonlinear ANP is suggested in this paper.For AHP, several specific ranking approaches to AHP alternatives based on the invariable weight system might appear the issue of incorrect rankings. To overcome the above drawbacks, with the variable weights idea of the MCDM methods based on variable weights adopted, according to the analysis idea of combining reductionism and holism, a nonlinear decision-making approach to AHP alternatives is given by the way of constructing a new analysis structure of AHP evaluating problems and its corresponding value system for evaluating alternatives. In contrast to conventional AHP, the suggested approach has following advantages. Compared with conventional AHP, not only does the proposed approach adopt the new analysis structure and the new extracting way for experts'information (i.e., the way of holistic judgments among hierarchies), which can better reflect the characteristics such as emergence and nonlinearity, but also in essence it well solves the AHP rank reversal problem that trouble experts and scholars for years. Applied to a numerical example, the suggested approach is validated to be applicable. For the evaluation problem of CSII, in our opinion, even if the ANP is feasible, an important submatrix of the supermatrix, i.e., the influence matrix of alternatives to goals (IMAG), is also constructed in ANP by the same way as the other submatrices, that is, by way of point estimates. This constructing pattern does not reasonably reflect the characteristic of high imprecision existing in holistic logic judgments, required in the IMAG constructing by the holism thought. To overcome the above disadvantages, this paper suggests that experts should adopt interval estimates to reflect their relative preferences, instead of adopting point estimates. With the above suggestion concerned, a modified approach to construct the supermatrix based on the methodology of data envelopment analysis is presented. Applied to a specific example, the modified approach is validated to be more scientific and reasonable than the conventional ANP. It should be noticed that, the novel modified approach has great innovation value in theory, but the approach is constructed based on assumptions of the concept of relative important weights being clear and of the eigenvalue method used to derive priorities in ANP being reliable, however, the two assumptions had been validated to be faulty in essence. With these views considered, in this paper, respective to the CSII system, a new ANP approach based on nonlinear complex-system theory and the metasynthesis methodology of combining holism and reductionism is proposed, which may well reflect the nonlinear relations among these elements for complex systems. The proposed approach is given by constructing a new analytic structure and adopting the way of holistic judgments among hierarchies, instead of linear analysis structure and the judgment way of paired comparisons in conventional ANP. Not only does the suggested approach realize the real reconcile of subjective preferences information and objective input-output systems, but also it overcomes the inner disadvantages of traditional ANP. An illustrating case shows the nonlinear decision approach for CSII is reasonable and can be applied to solve real complex problems.The system of HSICD is another typical system for ANP. However, conventional ANP to the alternative evaluating problem of HSICD has two drawbacks, i.e., the cluster matrix being difficult to be constructed and the concept of weights being undefined. To overcome these drawbacks, a new alternative ranking approach with stable analytic structure based on the nonlinear complex-system theory and the meta-syntheses methodology of combining holism and reductionism is proposed. The new approach is derived by way of constructing the new analytic structure for the evaluating problem of HSICD, the new expression means of experts'judgment information (i.e., probability influence matrix), and the value system for evaluating alternatives. The results of comparison analysis by a numerical case application show that the judgment information can be more easily obtained and the derived alternative ranking is more reliable and scientific for the new approach, compared with the conventional ANP.In contrast to the HSICD system, the HSIMD system considers more complex relationships among the system elements. To solve the issue that complex dependence relations are difficult to reflect in traditional ANP, in this paper a new ranking approach are given based on the perspective of nonlinear complex systems. The approach is derived by way of constructing the new analytic structure for the evaluating problem of HSIMD, the new expression means of experts'judgment information, and the value system for evaluating alternatives. In the proposed approach, complex multielement dependence relations among the same hierarchy and conjoint hierarchies can be well reflected and the characteristics of mutation and emergence are also better revealed. Furthermore, the concept of priority is clearly defined and rank preserval for evaluating alternatives may be realized in our approaches. The results of comparison analysis by a numerical case application indicate the suggested approach is more reliable and reasonable, compared with the conventional ANP.The usual synthesis way of alternative priorities for benefits (B), opportunities (O), costs(C) and risks(R) merits, obtained from separate ANP models, can be ambiguous. The reason is that the commensurability of priorities on the four factors is ignored in initial expressions suggested by Saaty. Furthermore, we realize either of synthesis formulas remains fairly arbitrary because the real combining mechanism among BOCR merits is unknown. To correct the situation, a new synthesis method for BOCR merits is proposed in this paper, with the idea of DEA adopted. The main advantages of the proposed approach are as follows. First, the commensurability of alternative priorities for BOCR merits can be overlooked, for the reason that dimension of composite weights for BOCR has no impact on final efficiency index. Second, relative efficiency for each alternative is indirect measured by DEA model, instead of direct measuring by arbitrary synthesis expressions. Thirdly, the value preferences of the decision-maker can be reflect better by the interval numbers of swing weighting rather than the crisp numbers used in traditional synthesis expressions. Finally a numerical example is given to show the merits of the new approach, compared with traditional synthesis formulas.Finally, by two real cases (one is the issue of investment risk evaluation for a steel company, the other is the evaluation issue of knowledge management ability for different enterprises) these proposed methods are validated to be reasonable and can be applied to solve real complex decision issues.