Fuzzy analytic hierarchy process and analytic network process: An integrated fuzzy logarithmic preference programming

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.     

A novel meta-heuristic based method for deriving priorities from fuzzy pairwise comparison judgments

This paper proposes a new method to derive the priority vector from fuzzy pairwise comparison matrices. Unlike several known methods, the proposed method derives crisp weights from consistent and inconsistent fuzzy comparison matrices. Therefore, the crisp weights obviate the need of additional aggregation and ranking procedures. To derive the priority vector, a Modified Fuzzy Logarithmic Least Square Model (MFLLSM) is proposed. In order to solve the MFLLSM, a framework based on genetic algorithm is proposed. In the proposed framework, a heuristic algorithm of population initialization, a heuristic algorithm for simulating fuzzy numbers and a heuristic algorithm of fitness evaluation are proposed.The solution of the prioritization problem requires finding priorities such that their ratio approximately satisfies the initial judgments. Computational results reveal the superiority of the proposed method in comparison with five well known methods of literature from the viewpoint of satisfaction of initial judgments by the obtained priority vector. It is shown by ten different examples that the deviation of the priorities ratio from initial judgments in the proposed method is less than five existing methods of literature. In addition, unlike several methods of literature, the proposed method considers fuzzy judgments represented by both triangular and trapezoidal fuzzy numbers. Furthermore, the proposed method for the first time considers judgments represented by triangular shaped fuzzy numbers and trapezoidal shaped fuzzy numbers which are discussed in the paper.     

Combining different prioritization methods in the analytic hierarchy process synthesis

A multicriteria approach for combining prioritization methods within the analytic hierarchy process (AHP) is proposed. The leading assumption is that for each particular decision problem and related hierarchy, AHP must not necessarily employ only one prioritization method (e.g. eigenvector method). If more available methods are used to identify the best estimates of local priorities for each comparison matrix in the hierarchy, then the estimate of final alternatives’ priorities should also be the best possible, which is in natural concordance with an additive compensatory structure of the AHP synthesis. The most popular methods for deriving priorities from comparison matrices are identified as candidates (alternatives) to participate in AHP synthesis: additive normalization, eigenvector, weighted least-squares, logarithmic least-squares, logarithmic goal programming and fuzzy preference programming. Which method will be used depends on the result of multicriteria evaluation of their priority vectors’ performance with regard to suggested deviation and rank reversal measures. Two hierarchies with matrices of size 3–6 are used to illustrate an approach.     

Applying fuzzy hierarchy multiple attributes to construct an expert decision making process

The main objective of this paper is to propose an approach within the AHP framework for tackling the uncertainty and imprecision of service evaluations during pre-negotiation stages, where the expert’s comparison judgments are represented as fuzzy triangular numbers. A fuzzy prioritization method, which derives crisp priorities from consistent and inconsistent fuzzy comparison matrices, is described. The fuzzy analytic hierarchy process (FAHP)-based decision-making method can provide decision makers or buyer a valuable reference for evaluating software quality. A case study demonstrates the effectiveness of the proposed scheme. Importantly, the proposed scheme can assist decision makers in assessing the feasibility of digital video recorder system to management public space, making it highly applicable for academic and commercial purposes.     

Generating consensus priority point vectors: a logarithmic goal programming approach

The analytic hierarchy process (Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, NewYork: McGraw-Hill 1980) is a popular technique for addressing multiple-criteria decision-making problems (MCDMs). Various techniques have been proposed for using the AHP in group situations. Fundamental to the AHP is the generation of priority point vectors from matrices of pairwise comparison data. In this paper, we present a logarithmic goal programming model for generating the ‘consensus’ priority point vector from the set of individual priority point vectors.Scope and purposeWithin modern organizations, multiple-criteria decision-making problems (MCDMs) often occur within a group context, and individual priorities for decision alternatives must be synthesized into a single set of priorities which represents the consensus opinion for the group. This requires a process for aggregating individual priorities into a set of group priorities. In this paper, we examine the use of the analytic hierarchy process (AHP) MCDM technique for the group situation, and present an approach for aggregating individual priorities into a set of group ‘consensus’ priorities.     

An integrated fuzzy multi criteria group decision making approach for ERP system selection

This paper aims to ease group decision-making by using an integration of fuzzy AHP (analytic hierarchy process) and fuzzy TOPSIS (technique for order preference by similarity to ideal solution) and its application to software selection of an electronic firm. Firstly, priority values of criteria in software selection problem have been determined by using fuzzy extension of AHP method. Fuzzy extension of AHP is suggested in this paper because of little computation time and much simpler than other fuzzy AHP procedures. Then, the result of the fuzzy TOPSIS model can be employed to define the most appropriate alternative with regard to this firm's goals in uncertain environment. Fuzzy numbers are presented in all phases in order to overcome any vagueness in decision making process. The final decision depends on the degree of importance of each decision maker so that wrong degree of importance causes the mistaken result. The researchers generally determine the degrees of importance of each decision maker according to special characteristics of each decision maker as subjectivity. In order to overcome this subjectivity in this paper, the judgments of decision makers are degraded to unique decision by using an attribute based aggregation technique. There is no study about software selection using integrated fuzzy AHP-fuzzy TOPSIS approach with group decision-making based on an attribute based aggregation technique. The results of the proposed approach and the other approaches are compared. Results indicate that our methodology allows decreasing the uncertainty and the information loss in group decision making and thus, ensures a robust solution to the firm.     

A measurement model for experts knowledge-based systems algorithm using fuzzy analytic network process

This study proposes an experts knowledge-based systems measurement model, the model using fuzzy analytic network process (FANP) to resolve the uncertainty and imprecision of evaluations during pre-negotiation stages, where the comparison judgments of a decision maker are represented as fuzzy triangular numbers. A novel fuzzy prioritization method, which derives crisp priorities (criteria weights and scores of alternatives) from consistent and inconsistent fuzzy comparison matrices, is also proposed. The applicability of the proposed model is demonstrated in a government purchase digital video recorder (DVR) system project study. The stability tests indicate the advantages of the proposal model in determining the value of model. Importantly, the proposed model can provide decision makers a reference material, making it highly applicable for academic and commercial purposes.     

Obtaining Crisp Priorities for Triangular and Trapezoidal Fuzzy Judgments

This paper proposes anoptimal fuzzy-based model for obtaining crisp priorities for Fuzzy-AHP comparison matrices. Crisp judgments cannot be given for real-life situations, as most of these include some level of fuzziness and complexity. In these situations, judgments are represented by the set of fuzzy numbers. Most of the fuzzy optimization models derive crisp priorities for judgments represented with Triangular Fuzzy Numbers (TFNs) only. They do not work for other types of Triangular Shaped Fuzzy Numbers (TSFNs) and Trapezoidal Fuzzy Numbers (TrFNs). To overcome this problem, a sum of squared error (SSE) based optimization model is proposed. Unlike some other methods, the proposed model derives crisp weights from all of the above-mentioned fuzzy judgments. A fuzzy number is simulated using the Monte Carlo method. A threshold-based constraint is also applied to minimize the deviation from the initial judgments. Genetic Algorithm (GA) is used to solve the optimization model. We have also conducted casestudiesto show the proposed approach’s advantages over the existingmethods. Results show that the proposed model outperforms other models to minimize SSE and deviation from initial judgments. Thus, the proposed model can be applied in various real time scenarios as it can reduce the SSE value upto 29% compared to the existing studies.     

A Fuzzy AHP Approach to Evaluating Machine Tool Alternatives

Selecting process of a machine tool has been very important issue for companies for years, because the improper selection of a machine tool might cause of many problems affecting negatively on productivity, precision, flexibility and company’s responsive manufacturing capabilities. On the other hand, selecting the best machine tool from its increasing number of existing alternatives in market are multiple-criteria decision making (MCDM) problem in the presence of many quantitative and qualitative attributes. Therefore, in this paper, an analytic hierarchy process (AHP) is used for machine tool selection problem due to the fact that it has been widely used in evaluating various kinds of MCDM problems in both academic researches and practices. However, due to the vagueness and uncertainty on judgments of the decision-maker(s), the crisp pair wise comparison in the conventional AHP seems to insufficient and imprecise to capture the right judgments of decision-maker(s). That is why; fuzzy number logic is introduced in the pair wise comparison of AHP to make up for this deficiency in the conventional AHP. Shortly, in this study, an intelligent approach is proposed, where both techniques; fuzzy logic and AHP are come together, referred to as fuzzy AHP. First, the fuzzy AHP technique is used to weight the alternatives under multiple attributes; second Benefit/Cost (B/C) ratio analysis is carried out by using both the fuzzy AHP score and procurement cost, of each alternative. The alternative with highest B/C ratio is found out and called as the ultimate machine tool among others. In addition, a case study is also presented to make this approach more understandable for a decision-maker(s).     

Make-to-order or make-to-stock decision by a novel hybrid approach

This paper presents a strategic decision-making structure to determine whether a particular product should be produced under make-to-order (MTO) or make-to-stock (MTS) strategy. A novel hybrid methodology consisting of strengths, weaknesses, opportunities and threats (SWOT) analysis and analytic hierarchy process (AHP) is proposed. Utilizing AHP in SWOT analysis diminishes one of the main deficiencies of this strategic approach that is the restriction of employing quantitative calculation to evaluate the factors that affect the decision. AHP method improves the quantitative information basis of strategic decision-making processes. The traditional AHP requires exact judgments, but due to the complexity and uncertainty involved in real world decision problems, it is sometimes unrealistic or even impossible to require exact judgments. Consequently, a commonly used decision-making method, fuzzy analytic hierarchy process (FAHP) is suggested. FAHP’s connection to SWOT yields a novel hybrid method for partitioning of MTO/MTS products. At the end, the proposed fuzzy AHP-SWOT methodology is validated through a real case study.     

Modification of the fuzzy analytic hierarchy process via different ranking methods

The use of fuzzy set theory in the analytic hierarchy process (AHP) has gained popularity in recent years as part of the multiple criteria decision-making (MCDM) process to more realistically reflect human judgment. However, due to the nature of fuzzy calculations, this situation imposes more computational load. The aim of this study is to propose methods for obtaining accurate weights from fuzzy pairwise comparison matrices with the least amount of computational load possible. In this context, two different fuzzy AHP (FAHP) methods based on fuzzy numbers ranking methods have been proposed and these proposed methods are compared with commonly accepted FAHP methods. Magnitude-based fuzzy AHP (MFAHP), which is one of the proposed methods, has outperformed all other methods according to accurate weight and computational load. Although the other proposed method, called the total difference-based fuzzy AHP (TDFAHP), gave better results than the frequently used Chang's fuzzy extent analysis method, it could not produce more accurate weight results than many other methods in general. But performance analysis shows that it is as good as the MFAHP in terms of computational load.     

Integration of AHP-TOPSIS method for prioritizing the solutions of reverse logistics adoption to overcome its barriers under fuzzy environment

Reverse logistics practices are gaining attention due to industrial ecology, enforced legislation and corporate citizenship but presence of barriers make reverse logistics (RL) implementation difficult and hence reduce the success rate. To increase RL adoption, robust and flexible strategies are required to overcome its barriers. This study focuses on identification and ranking the solutions of reverse logistics adoption in electronics industry to overcome its barriers. It aids firms to ponder on high rank solutions and develop strategies to implement them on priority. This paper proposes a methodology based on fuzzy analytical hierarchy process (AHP) and fuzzy technique for order performance by similarity to ideal solution (TOPSIS) to identify and rank the solutions of RL adoption to overcome its barriers. Fuzzy AHP is applied to get weights of the barriers as criteria by pairwise comparison and final ranking of the solutions of RL adoption is obtained through fuzzy TOPSIS. The empirical case of Indian electronics industry is shown to illustrate the use of the proposed method. This proposed method offers a more precise, efficient and effective decision support tool for stepwise implementation of the solutions due to consideration of fuzzy environment. Finally sensitivity analysis is performed to illustrate the robustness of the method.     

Regime switching fuzzy AHP model for choice-varying priorities problem and expert consistency prioritization: A cubic fuzzy-priority matrix design

The aim of this paper is to develop a regime switching design of the fuzzy analytic hierarchy process (FAHP) and to improve its functionality under the choice-varying priority (CVP) problem. In the conventional AHP decision process, priority matrices are identical and their values are invariant for a specific objective. However, in many Multi-Criteria Decision Making (MCDM) problems, the relative importance of criteria may differ according to the choices. A regime switching process is proposed for improving the CVP problem. Under the fuzzy-AHP (FAHP) framework, choice-varying priorities are presented in a cubic matrix form. Another novel contribution is suggested in the prioritization of the level of expert consistency. During the decision-making practice, experts may have different attitudes and their individual matrix consistencies might be superior or inferior in their overall practices. Individual consistency is one of the objective indicators of the quality of judgment. An expert consistency prioritization approach is proposed to deal with the classification of response stability. For the financial risk assessment part of the study, the loss probability of the intended projects is calculated by the fuzzy Monte-Carlo simulation framework.     

Railway station site selection using analytical hierarchy process and data envelopment analysis

This paper deals with the problem of finding the optimum site for a railway station for the city of Mashhad, northeast Iran, using the methods of analytical hierarchy process (AHP) and data envelopment analysis (DEA). The paper identifies a four-level hierarchy model for the railway station site-selection problem. The model uses four main criteria: (1) rail-related, (2) passenger services, (3) architecture and urbanism, and (4) economics. In addition, there are 26 subcriteria as well as five (potential) candidates or alternatives. Comparison matrices are used to obtain the local weights and priorities of the railway-station candidates. A DEA model is proposed to determine the optimum site for a railway station. It is shown that the local priorities (or weights) obtained from the AHP can be defined as the multiple outputs of a DEA model for finding the best site for a railway station.     

Aggregation of direct and indirect judgments in pairwise comparison matrices with a re-examination of the criticisms by Bana e Costa and Vansnick

In a recent paper by Bana e Costa and Vansnick [C.A. Bana e Costa, J.C. Vansnick, A critical analysis of the eigenvalue method used to derive priorities in AHP, European Journal of Operational Research 187 (3) (2008) 1422-1428], analytic hierarchy process (AHP), particularly its eigenvector method (EM) used for deriving priorities from pairwise comparison matrices, was criticized for the violation of a so-called condition of order preservation (COP). Due to this violation, the EM was considered to have a serious fundamental weakness which makes the use of AHP as a decision support tool very problematic. The consistency ratio (CR) index in the AHP was also criticized for its failure to act as an alert of this violation of COP. In this paper, we look into decision makers’ overall judgments which can be obtained through the aggregation of their direct and indirect judgments and then re-examine Bana e Costa and Vansnick’s numerical examples with a detailed analysis to show the invalidity of their criticisms.     

An intelligent approach to machine tool selection through fuzzy analytic network process

In this study, we utilize analytic network process (ANP), a more general form of AHP, for justifying stand-alone machine tools out of available alternatives in market due to the fact that AHP cannot accommodate the variety of interactions, dependencies and feedback between higher and lower level elements. However, due to the vagueness and uncertainty on judgments of a decision-maker, the crisp pair wise comparison in the conventional ANP seems to be insufficient and imprecise to capture the right judgments of the decision-maker. That is why, also in this paper, fuzzy number logic is introduced in the pair wise comparison of ANP to make up for this deficiency in the ANP. In short, here, an intelligent approach to machine tool selection (MTS) problem through fuzzy ANP is proposed to improve the imprecise ranking of company’s requirements which is based on the conventional ANP. In order to reach to final solution, a preference ratio (PR) analysis is done by using the results of the fuzzy ANP, and investment costs of alternatives. In addition, a numerical example is presented to illustrate the proposed approach.     

A GP-AHP method for solving group decision-making fuzzy AHP problems

The analytic hierarchy process (AHP) elicits a corresponding priority vector interpreting the preferred information from the decision-maker(s), based on the pairwise comparison values of a set of objects. Since pairwise comparison values are the judgments obtained from an appropriate semantic scale, in practice the decision-maker(s) usually give some or all pair-to-pair comparison values with an uncertainty degree rather than precise ratings. By employing the property of goal programming (GP) to treat a fuzzy AHP problem, this paper incorporates an absolute term linearization technique and a fuzzy rating expression into a GP-AHP model for solving group decision-making fuzzy AHP problems. In contrast to current fuzzy AHP methods, the GP-AHP method developed herein can concurrently tackle the pairwise comparison involving triangular, general concave and concave–convex mixed fuzzy estimates under a group decision-making environment.

Scope and purpose

Many real world decision problems involve multiple criteria in qualitative domains. As expected, such problems will be increasingly modeled as multiple criteria decision-making problems, which involve scoring on subjective/qualitative domains. This results in a class of significant problems for which an evaluation framework, which handles occurrences of seeming intransitivity and inconsistency will be required. Another interesting issue of group decision-making analysis is how to deal with disagreements between two or more different rankings within an alternative set. These phenomena are likely to appear in qualitative/subjective domains where the decision-making environment is ambiguous and vague. Therefore, this study proposes a GP-AHP model that is sufficiently robust to permit conflict and imprecision. Numerical examples demonstrate the effectiveness and applicability of the proposed models in deriving the most promising priority vector from a fuzzy AHP problem within a group decision-making environment.     

Synthesis of individual best local priority vectors in AHP-group decision making

An assessment of the individual judgments and AHP-produced priority vectors for involved decision-makers indicates that the individual consistencies of decision makers may vary significantly, thus making the final group decision less reliable. In this paper, an approach is proposed as to how to combine decision makers’ local priority vectors in AHP synthesis and reduce so-called group inconsistency. Instead of aggregating individual judgments (AIJ), or aggregating individually derived final priorities (AIP), we propose to perform an AHP synthesis of the best local priority vectors taken from the most consistent decision makers. The approach and related algorithm we label as MGPS after the key terms ‘multicriteria group prioritization synthesis.’ The concept is analogous to the one proposed by Srdjevic [1] for individual AHP applications where the best local priority vectors are selected based on the consistency performance of several of the most popular prioritization methods. Here, decision makers are combined instead of prioritization methods, and group context is fully implemented. After completing an evaluation of the decision makers inconsistencies in each node of the hierarchy, the selected best local priority vectors are synthesized in a standard manner, and the final solution is declared to be an AHP-group decision. Two numerical examples indicate that the developed approach and algorithm generate the final priorities of alternatives with the lowest overall inconsistency (in the multicriteria sense).     

Multicriteria decision making in energy planning using a modified fuzzy TOPSIS methodology

Energy planning is a complex issue which takes technical, economic, environmental and social attributes into account. Selection of the best energy technology requires the consideration of conflicting quantitative and qualitative evaluation criteria. When decision-makers’ judgments are under uncertainty, it is relatively difficult for them to provide exact numerical values. The fuzzy set theory is a strong tool which can deal with the uncertainty in case of subjective, incomplete, and vague information. It is easier for an energy planning expert to make an evaluation by using linguistic terms. In this paper, a modified fuzzy TOPSIS methodology is proposed for the selection of the best energy technology alternative. TOPSIS is a multicriteria decision making (MCDM) technique which determines the best alternative by calculating the distances from the positive and negative ideal solutions according to the evaluation scores of the experts. In the proposed methodology, the weights of the selection criteria are determined by fuzzy pairwise comparison matrices. The methodology is applied to an energy planning decision-making problem.     

On the order of the preference intensities in fuzzy AHP

We show that a recently discovered fundamental problem with the Analytic Hierarchy Process (AHP) concerning the meaning of the resultant preference intensities is also evident for the fuzzy AHP. We prove that if there is a judgmental inconsistency in the fuzzy pair-wise comparisons, it is impossible to ensure the preservation of the order regarding to preference intensities in the resultant priority vector. Further, it is shown with an example from the published literature that the order of the preference intensities may not be preserved even there is no inconsistency in the judgment set, albeit it is possible to comply with this order via using fuzzy preference programming (FPP) methodology. Finally, it is proved that if the interval judgments regarding to the decompositions of original judgments to - level sets are consistent, FPP guarantees the preservation of the order of the preference intensities at those levels.