A Comparative Study on AHP and FAHP for Consistent and Inconsistent Data

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1 A Comparative Study on AHP and FAHP for Consistent and Inconsistent Data Md. Ashek-Al-Aziz Department of Computer Science & Engineering Ahsanullah Institute of Information & Communication Technology (AIICT) Abstract Analytical Hierarchical Process (AHP) and Fuzzy Analytical Hierarchical Process (FAHP) are two methods both used for Multi Criteria Decision Making (MCDM). Consistency Ratio (CR) is checked for all data sets fed into both these methods and weight values are compared. As well as differences between weights determined by both methods and decision values have been compared and tried to understand the nature of both methods. Index Terms AHP, FAHP, MCDM. The motivation comes from this point that though we can implement FAHP methods quite confidently but how much it holds similarities in behavior as seen in normal AHP method as both are decision tools. The aim of our research is to observe the nature of output values of both AHP and FAHP methods for the consistent and inconsistent data, the differences and fluctuation of differences of the decision results. I. INTRODUCTION AHP and FAHP methods deal with the stochastic data that holds some sort of uncertainty. Davoudi and Shykhvand (2012) states that due to the uncertainty in the judgments of participants, the crisp pair wise comparison in the conventional AHP is insufficient and imprecise to capture the right judgments. Therefore, a fuzzy logic is introduced in the pair-wise comparison of AHP in one of steps of the method [1]. Özdağoğl & özdağoğlu (2007) state that Fuzzy Analytical Hierarchy Process (FAHP) is a synthetic extension of classical AHP method when the fuzziness of the decision makers is considered [2, 3]. Kabir & Hasin (2011) demonstrated the advantages of capturing the vagueness of human thinking and to aid in solving the research problem through structured manner and simple process in a case of inventory classification. They have successfully identified the flexibility of incorporating more criteria or removing for specific implementation. Different classification analyses can be done for different inventory records, application specific variable sets can be deployed and crisp comparison values can be substituted for fuzzy comparison values for optimization if fuzzy comparisons are not available in case [3]. Vayvay, Ozcan & Cruz-Cunha (2012) suggested from their experiments that both AHP and FAHP led to the same results, but neither of these considered the interactions within decision elements during the selection process [4]. But the major advantage of FAHP methods stated by Mishra & Thakar (2012) that it can be used for both qualitative and quantitative criteria and also enables decision-makers to deal with inconsistent judgements systematically [5]. A. AHP a a a a a a A = a a a N = a a a a II. MATHEMATICAL OUTLINES a a a a a a a a N N N N = N N N N N a a a a a a N + N N n N + N N w w W = n = w N + N N n R = max (w ) (1) (2) (3) (4) This work is an extension of part of Master s Thesis work. This manuscript is prepared and submitted in March Md. Ashek Al-Aiz is with Ahsanullah Institute of Information & Communication Technology (AIICT), Dhaka, Bangladesh ( ashek3000@ gmail.com). B. FAHP fx, x = (, ) [(, ),(, )] f / (X ) = min(x ) (5) (6) url: or 1

2 R = max [f / (X )] (7) The pair wise comparison matrix is substituted by some linguistics and equivalent numerical values for preferences like 1. Very low choice, 2. Low Choice, 3. Moderate Choice, 5. High Choice, 7. Very High Choice and 9. Extremely High Choice. In our analysis, this particular linguistics is followed for the data sets used for calculations. III. CHARTS OF COMPARISONS The values of pair wise comparison matrices are substituted using the linguistics mentioned in Section II and Weight values are calculated using the equations. The comparison of Weight values are shown in Fig 1 to Fig XX. Fig 1 to Fig 12 shows comparisons of Consistent Data Set results and Fig 13 to Fig 24 are of Inconsistent Data Sets. Fig. 4. Weight values of AHP and FAHP with attributes (nmax = 5.27, CI = 0.07, RI = 1.19, CR = 0.06) Fig. 5. Weight values of AHP and FAHP with attributes (nmax = 5.25, CI = 0.06, RI = 1.19, CR = 0.05) Fig. 1. Weight values of AHP and FAHP with attributes (nmax = 5.27, CI = 0.07, RI = 1.19, CR = 0.06) Fig. 6. Weight values of AHP and FAHP with attributes (nmax = 5.37, CI = 0.09, RI = 1.19, CR = 0.08) Fig. 2. Weight values of AHP and FAHP with attributes (nmax = 5.31, CI = 0.08, RI = 1.19, CR = 0.06) Fig. 7. Weight values of AHP and FAHP with attributes (nmax = 5.45, CI = 0.11, RI = 1.19, CR = 0.09) Fig. 3. Weight values of AHP and FAHP with attributes (nmax =5.06, CI = 0.02, RI = 1.19, CR = 0.01) url: or 2

3 Fig. 8. Weight values of AHP and FAHP with attributes (nmax = 5.31, CI = 0.08, RI = 1.19, CR = 0.07) Fig. 12. Weight values of AHP and FAHP with attributes (nmax = 5.42, CI = 0.10, RI = 1.19, CR = 0.09) Fig. 9. Weight values of AHP and FAHP with attributes (nmax = 5.06, CI = 0.02, RI = 1.19, CR = 0.01) Fig. 13. Weight values of AHP and FAHP with attributes (nmax = 6.25, CI = 0.31, RI = 1.19, CR = 0.26) Fig. 10. Weight values of AHP and FAHP with attributes (nmax = 5.31, CI = 0.08, RI = 1.19, CR = 0.07) Fig. 14. Weight values of AHP and FAHP with attributes (nmax = 6.19, CI = 0.30, RI = 1.19, CR = 0.25) Fig. 11. Weight values of AHP and FAHP with attributes (nmax = 5.29, CI = 0.07, RI = 1.19, CR = 0.06) Fig. 15. Weight values of AHP and FAHP with attributes (nmax = 6.11, CI = 0.28, RI = 1.19, CR = 0.23) url: or 3

4 Fig. 16. Weight values of AHP and FAHP with attributes (nmax = 6.64, CI = 0.41, RI = 1.19, CR = 0.34) Fig. 20. Weight values of AHP and FAHP with attributes (nmax = 6.18, CI = 0.29, RI = 1.19, CR = 0.25) Fig. 17. Weight values of AHP and FAHP with attributes (nmax = 7.59, CI = 0.65, RI = 1.19, CR = 0.54) Fig. 21. Weight values of AHP and FAHP with attributes (nmax = 7.83, CI = 0.71, RI = 1.19, CR = 0.59) Fig. 18. Weight values of AHP and FAHP with attributes (nmax = 7.53, CI = 0.63, RI = 1.19, CR = 0.53) Fig. 22. Weight values of AHP and FAHP with attributes (nmax = 8.48, CI = 0.87, RI = 1.19, CR = 0.73) Fig. 19. Weight values of AHP and FAHP with attributes (nmax = 6.93, CI = 0.48, RI = 1.19, CR = 0.41) Fig. 23. Weight values of AHP and FAHP with attributes (nmax = 7.47, CI = 0.62, RI = 1.19, CR = 0.52) url: or 4

5 Fig. 24. Weight values of AHP and FAHP with attributes (7.55, CI = 0.64, RI = 1.19, CR = 0.54) Fig. 28: Flactuation tendecy for consistent and inconsistent data Fig. 29: Result comparision of AHP and FAHP for consistet datasets Fig. 25: Set differeces for consistent data and fluctuations Fig. 30: Result comparison of AHP and FAHP for inconsistet datasets Fig. 26: Set differeces for inconsistent data and fluctuations Fig. 27: Average differeces of decision values for consistent and inconsistent data IV. CONCLUSIONS Average difference between AHP and FAHP weight values are greater in consistent data than inconsistent. As well as Standard Deviation of the result deviations are also more than 0.5 for consistent data whereas it is less than 0.5 for inconsistent data. It means that result differences are greater found for consistent data values but less differences are seen in inconsistent data sets i.e. If data sets are inconsistent then we can safely use both AHP and FAHP methods so that similar result generation possibility increases but for consistent data it is better suggestion to go with AHP instead of using FAHP method or both methods. Another point is that fluctuation tendency increases for decreases of nmax value of data set. And nmax value is more fluctuated for inconsistent data though fluctuation tendency of the url: or 5

6 differences of results of two methods are negligible. V. RECOMMENDATION Differences between the decision values of both the methods have been drawn in Fig. 29 and Fig. 30. The differences are seen more steady and constant for inconsistent datasets. Here the FAHP decisions are come out due to ceiling value 1.0 among the weight result from normalizations. But AHP generates values between 0 and 1.0. Due to this nature, we can think of a method modification of mathematical outline or normalization process so that both methods produce same decision values for results. Master of Engineering. He is now serving as Department Head of Computer Science & Engineering of Ahsanullah Institute of Information & Communication Technology (AIICT). He worked in few different organizations before current position. He attended in an International Conference organized by Rajshahi University of Engineering & Technology and presented his paper on Fuzzy based risk identifier for electronic procurement process. His research interests focus the area of Fuzzy Logic and Multi Criteria Decision Making. Md. Ashek Al-Aziz is Member of Institution of Engineers, Bangladesh (IEB), Life Member of Bangladesh Electronic Society (BES) and Life Fellow of Bangladesh Computer Society (BCS). ACKNOWLEDGMENT I like to thank Dr. Abdullahil Azeem, who supervised my thesis work during the tenure in Masters Program at Bangladesh University of Engineering & Technology (BUET), he advised me to work on method comparison. I also like to thank my friends who encouraged me to do this method comparison and prepare this article. REFERENCES [1]. Mahsa Razavi Davoudi, Kourosh Sheykhvand, An Approach towards Enterprise Architecture Analysis using AHP and Fuzzy AHP, International Journal of Machine Learning and Computing, Vol. 2, No. 1, February 2012 [2]. Aşkın ÖZDAĞOĞL, Güzin ÖZDAĞOĞLU, Comparison Of The AHP and fuzzy AHP For The MultiCriteria Decision Making Processes With Linguistic Evaluations, Istanbul Commerce University Journal of Science Year: 6 Issue: 11Bah for 2007/1 p [3]. Golam Kabir, Dr. M. Ahsan Akhtar Hasin, Comparative Analysis of AHP and Fuzzy AFP Models For Multicriteria Inventory Classification, International Journal of Fuzzy Logic Systems (IJFLS) Vol.1, No.1, October 2011 [4]. Ozalp Vayvay, Yigit Ozcan and Maria Manuela Cruz-Cunha, ERP consultant selection problem using AHP, fuzzy AHP and ANP: A case study in Turkey, E3 Journal of Business Management and Economics Vol. 3(3). pp , March, 2012 [5]. Yogesh Mishra, Dr. Girish Thakar, Comparison of Classical Fuzzy Analytic Hierarchy Process and Extended Fuzzy Analytic Hierarchy Process for Vendor Selection in Automobile Supply Chain, present at the National Conference on Emerging Challenges for Sustainable Business 2012 [6]. George J. Klir, Bo Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall of India Privat Limited (2003) ch. 15, pp [7]. Hamdy A. Taha, Operations Research: An Introduction, 8 th Edition, Prentice Hall of India Private Limited (2008) ch. 13, pp [8]. S. Rajasekaran, G. A. V. Pai, Neural Networks, Fuzzy Logic and Genetic Algorithm: Synthesis and Application, Prentice Hall of India Private Limited (2007) ch. 6, pp Md. Ashek Al-Aziz was born in 1977 in Dhaka. He studied in Computer Science & Engineering and received his Bachelor degree from Ahsanullah University of Science & Technology (AUST). He received Master degree in Computer Science from National University, Gazipur. He also studied in Advanced Engineering Management at Bangladesh University of Engineering & Technology (BUET) and achieved degree of url: or 6