فرآیند تحلیل سلسله مراتبی فازی و فرایند تحلیل شبکه ای : برنامه ریزی ترجیحی لگاریتمی فازی یکپارچه

This paper proposes a two-stage fuzzy logarithmic preference programming with multi-criteria decision-making, in order to derive the priorities of comparison matrices in the analytic hierarchy pprocess (AHP) and the analytic network process (ANP). The Fuzzy Preference Programming (FPP) proposed by Mikhailov and Singh [L. Mikhailov, M.G. Singh, Fuzzy assessment of priorities with application to competitive bidding, Journal of Decision Systems 8 (1999) 11–28] is suitable for deriving weights in interval or fuzzy comparison matrices, especially those displaying inconsistencies. However, the weakness of the FPP is that it obtains priorities of comparison matrices by additive constraints, and generates different priorities by processing upper and lower triangular judgments. In addition, the FPP solves the comparison matrix individually. By using multiplicative constraints, the method proposed in this paper can generate the same priorities from upper and lower triangular judgments with crisp, interval or fuzzy values. Our proposed method can solve all of the matrices simultaneously by multiple objective programming. Finally, five examples are demonstrated to show the proposed method in more detail.

The analytic hierarchy process (AHP) has been widely used in multi-criteria decision-making to measure tangible and intangible criteria [2]. AHP aims to decompose the decision-making process into a hierarchical structure, assuming that the relationships of criteria in different levels are independent. Since its development, many applications have been published, such as portfolio selection [3], facility location selection [4], reverse logistics [5], etc. Conventional AHP only uses crisp pair-wise judgments to derive weights without considering the uncertainty of human intuition. Thus, Satty and Vargas [6] proposed an interval AHP to deal with interval judgments. Unfortunately, when using this approach for interval judgments, the measurement of inconsistencies, while generating weights, becomes difficult. Subsequently, Islam et al. [3] proposed Lexicographic Goal Programming (LGP) to handle the inconsistency problems in the interval AHP by using deviation variables. Mikhailov and Singh [1] proposed the fuzzy preference programming (FPP) method, which derived crisp priorities from interval or fuzzy comparison matrices by introducing tolerance parameters. However, due to the additive constraints for generating weights in the FPP, different priorities and rankings from the upper and lower triangular judgments could be obtained [7] and [8]. Wang [9] proposed that the LGP has the same problem with different priorities and rankings from upper and lower triangular judgments. In fact, the upper and lower triangular judgments of a comparison matrix provides the same information on the preference of weights, but generates different weights. Therefore, the question arises as to which weight vector and ranking is more accurate [7], [8] and [9]. A two-stage logarithmic goal-programming method ensures that the same interval weights are generated from both upper and lower triangular judgments of an interval comparison matrix [9]. Chandran et al. [10] applied a two-stage linear programming (LP) model to determine weights, using logarithms for crisp or interval inconsistent matrices. Their approaches adopted the geometric mean for interval judgments. Unfortunately, it had too many constraints with regard to the number of weights. The performing time or complexity of the aforementioned methods is proportional to the number of comparison matrices. If more interval weights need to be obtained, a longer processing time is required, which leads to complexity. As a result, multiple objective programming has been proposed to obtain all priorities simultaneously [11]. In a real-life environment, many problems exist with interdependence relations. These are difficult to present by means of a hierarchical structure. To cope with more practical real-life problems, the analytic network process (ANP) was proposed by Satty [12]. The ANP is capable of addressing interdependent relationships among criteria and sub-criteria [13]. In fact, the ANP is a generalization of the AHP, and extends the AHP to the problems with dependence and feedback among criteria by using a network structure and supermatrix approach. The crisp ANP approach has been applied to such processes as economic forecasting [14] and successful system factor selection [15]. Moreover, both AHP and ANP are effective in regard to prediction [16]. To handle uncertain judgments with inconsistency in the ANP, Mikhailov and Singh [17], and Mikhailov [18] applied the FPP in fuzzy and interval ANP. The advantage of the FPP is that it obtains crisp priorities from interval and fuzzy judgments straightforwardly within the ANP for decision-making under uncertain conditions, while all the other known interval and fuzzy prioritization methods, which derive interval or fuzzy priorities, cannot be used directly in the ANP. The initial weight-generating process of the ANP is the same as that of the AHP. Each comparison matrix must be constructed as an individual FPP model. When the network structure is more complex, more complexity is required to perform the FPP models [11]. Because of additive constraints in the FPP [7] and [8], the issue of different weights obtained from the lower and upper triangular in the ANP or fuzzy ANP remains. Thus, a two-stage fuzzy logarithmic preference programming with multiple objective programming is proposed to simultaneously derive priorities in fuzzy AHP or ANP. Our proposed method can handle complex problem structure, regardless of the number of comparison matrices, because all of the priorities are generated simultaneously. Moreover, the proposed method can generate the same priorities from upper and lower triangular judgments by multiplicative constraints [19] instead of additive constraints, as demonstrated in the five examples. The remainder of this paper is organized as follows. Section 2 describes the FFP method in detail; Section 3 presents the proposed method; Section 4 discusses comparative results of five examples; Section 5 offers conclusions.

An integrated fuzzy logarithmic preference programming has been proposed to provide efficient and reliable priorities in both AHP and ANP problems. The proposed method improves the weakness of the FPP, which handles comparison matrices individually and generates different weights by using the lower and upper triangular judgment of the matrices. In the five examples, the results demonstrate that the FPP method obtains inferior priorities and rankings for the lower and upper triangular judgments in each case because of additive constraints. Since the upper triangular judgments provide the same information as the lower triangular judgments, the priorities should be the same. The proposed method obtains identical priorities and rankings by using multiplicative constraint. In addition, by employing the multiple objective programming, the proposed method is able to solve all comparison matrices at one time, regardless of the number of comparison matrices. For future research, the proposed method can be extended to other AHP or ANP approaches with additive constraints, and be applied to more empirical applications.