Fuzzy AHP approach for the Selection of Groundwater Recharge Alternative: Sensitivity Analysis

Transcription

1 Fuzzy AHP approach for the Selection of Groundwater Recharge Alternative: Sensitivity Analysis Masengo Ilunga, Department of Civil Engineering and Chemical Engineering, University of South Africa, Pretoria, South Africa. ABSTRACT Fuzzy Analytical Hierarchical Process (FAHP) has been introduced in the recent decade to accommodate vagueness in the AHP. In a previous study by the above-mentioned author, a fuzzy set theory through AHP was used to select the most suitable groundwater artificial recharge (AR) alternative or technique for Kharkams Village, in Namaqualand of South Africa. ASR and ASTR were the two techniques used considering a normal judgment (i.e. decision maker or water resource manager s preference of 0.5). ASR and ASTR are Aquifer Storage and Recovery and Aquifer Storage Transfer and Recovery respectively. Five criteria were used in the selection, among those the legal and regulatory issues were considered as the fifth criterion associated with vagueness or uncertainty. The current study explores the impact of the different values for the water manager s preference-ranging between a pessimistic situation to an optimistic situation; on the consistency of the judgment matrix. The results showed that the weights of the five criteria remained almost unchanged as compared with the previous study. Throughout FAHP process the judgment was shown to be consistent. From a pessimistic situation to an optimistic situation, the overall preferences (recharge suitability) for ASR and ASTR remained closer to 55 % and 45 % respectively for FAHP likewise in the previous study. This supported that the decision maker or water resource manager s preference does not influence the legal and regulatory framework for selecting artificial groundwater recharge techniques. Keywords: Fuzzy analytic hierarchical process, artificial recharge, sensitivity analysis 1. INTRODUCTION For a semi-arid country like South Africa, artificial groundwater recharge alternative can be beneficial for sustainable development of groundwater. The two techniques or alternatives ASR and ASTR were evaluated in the selection of groundwater recharge for Kharkams in Namaqualand of South Africa [1], [2]. ASR and ASTR are Aquifer Storage and Recovery and Aquifer Storage Transfer and Recovery respectively. Analytic Hierarchy Process (AHP) and FAHP were used respectively by [1] and [2]. In the FAHP, the water resource manager s preferences were presented in terms interval judgment in a similar way done by [3]. A fuzzy set defuzzification technique to address vague data (i.e. fish activity data) was then proposed [3]. The application of AHP and FAHP specifically to groundwater recharge methods remains very sparse, except for example articles few studies [4] and [2]. In this study, alternatives and criteria for Kharkams Village are evaluated through a sensitivity analysis by considering cases where the decision maker or water resource manager s preferences range from pessimistic to optimistic using fuzzy triangular numbers (TFN). The fuzzy set defuzzification is carried out in a similar way done by [3]. 2. FAHP FORMULATION AHP method was developed by Prof. Thomas L. Saaty in the 1980s and is known as one of the Multi Criteria Decision Making methods (MCDM). AHP uses pairwise comparisons and derives ratio scales either from actual measurements or from subjective considerations (e.g. preference). Due to human judgment, AHP may allow some small inconsistency [5]. The technique has been applied to several fields including engineering and recently to the selection of artificial groundwater recharge techniques [1]. AHP technique can be traced generally in the following steps as replicated by [1]: 1. Formulate the problem as a hierarchy: A goal (objective), alternatives and criteria are contained in the hierarchy. 2. The pairwise comparison of the criteria based on a nine-point scale is done for the elements of the hierarchy. Table 1b shows the scale. Intensities are allocated based on the human judgmentsexperts experiences or individual experience who have knowledge of relevant topics. 3. Establish the judgment matrix for the hierarchy: The information derived from the previous step is summarized in a comparison matrix. 4. Consistency test: The validity of hierarchy structure is tested by computing the consistency ratio from the judgment matrix. The consistency ratio CR is the constant index CI divided by random index RI. CR is less than 10 % for a consistent judgment. Mathematically, CR is expressed as follows:

2 CI CR = RI Where: CI = (1) n n 1 λ MAX (2) λ MAX is the maximum Eigen value and n is the dimension of the judgment matrix Normally, RI are fixed values that exist in the literature and are associated with the size of judgment matrix, for example, see Table 1a. 5. The selection of alternative(s) is based on the computation of normalized principal priority vector (Eigen vector) obtained from a comparison as built matrix. In general, FAHP uses the same steps as listed above, however variables are expressed in terms of fuzzy data. Hence Table 2 is a similar table used by [3]. Two parameters λ and α were introduced and are the decision maker s preference and the risk tolerance respectively [3]. In this study, λ values vary from 0.1 to 0.9 to evaluate the impact on the goal of the hierarchy structure. When λ tend to 0.1, the decision maker or water resource manager is considered to be pessimistic otherwise, he is considered to be optimistic about his preference Table 1a. Random index (R I )as function size of matrix (n) n RI Table 1b. Scale for pairewise comparison in AHP analysis [1] Intensity of Importance Definition 1 Equal importance 3 Moderate importance of one element over another 5 Strong importance of one element over another 7 Very strong importance 9 Extreme importance 2,4, 6, 8 are intensities to express intermediates values Table 2. Triangulation fuzzy numbers and linguistic variables Intensity of Importance Definition TFN Linguistic variables 1 Equal importance (1,1,1) Least importance 3 Moderate importance of one element over another 5 Strong importance of one element over another (2,3,4) Moderate importance (4,5,6) Essential importance 7 Very strong importance (6,7,8) Demonstrate importance 9 Extreme importance (9,9,9) Extreme importance 2,4, 6, 8 are intensities to express intermediates values (1,2,3), (3,4,5), (5,6,7) and (7,8,9) intermediates values

3 In general, FAHP uses the same steps as listed above, however variables are expressed in terms of fuzzy data. Hence Table 2 is a similar table used by [3]. Two parameters λ and α were introduced and are the decision maker s preference and the risk tolerance respectively [3]. 3. IMPLEMENTATION OF FAHP FOR KHARKAMS Kharkams is a small village in the semi-arid Namaqualand region of South Africa and depends solely on groundwater. Namaqualand is in the northwestern part of the Northern Cape. The lowest yielding of the village s three production boreholes is artificially recharged with ASR technique whenever surface runoff is available. For more details, the reader can be referred to [6]. The implementation of FAHP for Kharmans was done through the following hierarchy structure [2]: *Goal: Defining the most suitable artificial groundwater recharge *Criteria: need for an artificial recharge scheme, water source, aquifer permeability and water quality. In addition to these four top criteria, legal and regulatory issues are considered as the fifth criterion. The legal and regulatory issues received a strong importance as compared to other criteria since in South Africa all artificial recharge schemes need to be licensed and obtaining the necessary permits is thus crucial to the success of new projects. Fuzziness is only associated with the last criterion since it is ambiguous to quantify with precision this last criterion. Fuzzy numbers and linguistic variables are presented to address the inherent vagueness, imprecision or uncertainty associated with this last criterion. *Alternatives: Aquifer Storage and Recovery (ASR) and Aquifer Storage Transfer and Recovery (ASTR). In addition to the above, the following values for the decision marker s preference λ were considered (Table 2): 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9 Each value of the decision maker s preference leads finally to the determination of the judgement matrix. Through numerical computation, the consistency ratio can be approximated using the principal Eigen value for the judgement matrix by summing the products between each element of Eigen vector and the sums of columns of the reciprocal matrix. Using MathLab, the maximum Eigen value can also be computed for each value of the water resource manager s preference. 4. PAIREWISE COMPARISONS Table 3 shows the criteria pairwise comparison with corresponding preferences. The following criteria for the selection of artificial groundwater recharge techniques were considered: need for an artificial recharge scheme (A), water source (B), aquifer permeability (C), water quality (D) and legal and regulatory issues (E). Based on the author s subjective consideration or preference, the legal and regulatory issues can be considered to have stronger importance than the rest of criteria [2]. For a subjective consideration to hold, the consistency ratio should be is less than 10%. Table 4 shows the weights of alternatives with respect to criteria. Table 3. Pairwise comparison of criteria X Y Importance Intensity A B Y 3 A C X 2 A D X 3 B C X 5 B D X 3 C D Y 2 E A X 4 E B X 4 E C X 4 E D X 4 X, Y: represent criteria in first column and second column of table 3 respectively, any particular raw. The pairwise comparison between artificial recharge techniques, i.e. ASR and ASTR with respect to criteria was carried out in a similar way done by [2]. 5. RESULTS AND DISCUSSION Table 5 shows the weights of the five criteria for selecting the artificial groundwater recharge techniques for λ = 0. 1.Calculations of criteria weight and overall preference of alternatives (which are not shown here) were conducted in a similar way as done previously (see Appendices 1, 2 and 3 of article published in [2]). The results revealed that the goal weights on the five criteria for different values of λ were very close (see Table 6). In all cases the legal and regulatory issues have a strong preference (varying between 41 % and 46 %). This could be justified by the fact that the legal and regulatory issues received highest priority since in South Africa all artificial recharge schemes need to be licensed and obtaining the necessary permits is thus crucial to the success of new projects. This criterion in particular will draw the attention of the water manager as a decision maker. Then he will prioritize other criteria. Table 7 shows the calculations of values of the consistency ratio for different values of λ. The maximum Eigen values were estimated using MathLab. For a 5 x 5 matrix, (n = 5 and CI = 1.12). In all cases, CR values were less than 1 (or 10 %). These results support that the judgment was logical and consistent during FAHP process.

4 Table 4. weights of alternatives A B C D E ASR ASTR Table 5. Criterion weights, λ = 0. 1 A B C D E Average A B C D E sum 1 Table 6. Criteria weights matrix for different values of λ A B C D E Table 7. Computed consistency ratio (CR) values for different values λ CR Table 8. Overall preference of artificial recharge techniques, i.e. ASR and ASTR ASR ASTR Table 8 shows the overall preferences for ASR and ASTR, for different values of λ. The overall preferences were closer to 55 % and 45 % for ASR and ASRT respectively during FAHP. These results confirmed the suitability of ASR for the AR groundwater for the Kharmans Village. These results translate that the choice of the groundwater artificial recharge technique is less influenced by the fact that the decision maker (water manager) is more pessimistic or optimistic with regards to the legal and regulatory issues. In other words, these results are justified by the higher priority during FAHP. In any circumstances, the water resource manager as a decision maker has to comply with the South African legal framework before any artificial recharge schemes need to be licensed and obtaining the necessary permits. This is crucial to the success of new projects. The overall preferences obtained could be explained by the fact that the legal and regulatory issues were ranked higher than the rest of criteria.

5 6. CONCLUSIONS A sensitivity analysis on FAHP was carried out in the selection of artificial groundwater recharge alternatives/techniques i.e. ASR and ASTR to Kharkams Village of Namaland region in South Africa. It was shown that the decision maker or water resource manager s preference does not influence the legal and regulatory framework for artificial groundwater projects. In general, ASR outperformed ASTR. This confirmed conclusions from previous studies. The application of FAHP in this study is a relatively simple case (i.e. Kharmans Village). For problems of multi-hierarchy and multi-variable system, the power of AHP is very much acknowledged in decision-making process. Further work may include a sensitivity analysis of FAHP with more fuzzy variables for groundwater artificial recharge problems. REFERENCES [1] M. Ilunga Application of Analytic Hierarchy Process (AHP) for the Selection of Groundwater Recharge Alternative. Proceedings of the scientific conference of the NASAC-KNAW collaboration initiative on Water management issues in Africa, march 2012, Boname Hall, Msiri, Reduit, Mauritius. Accepted. [2] M. Ilunga Fuzzy AHP approach for the Selection of Groundwater Recharge Alternative, Proceedings of the Fifth International Groundwater Conference (IGWC- 2012) on Assessment and management of groundwater resources in Hard Rock systems with special reference to Basltic Terrain at Aurangabad, Maharashtra, India, December 18-21, 2012, submitted. [4] L. Dong, M. Fanxiang, F. Qiang, Application of analytic hierarchical process in optimisation selection of groundwater artificial recharge methods in Sanjiang Plain, Proceedings of the International Conference on Management and Service Science, MASS 2009 Wuhan, 20 th September-22 September Policy and Management, Chinese Academy of Science, IEEE Wuhan Section. [3] M. A. Ashari, Using Fuzzy Analytic Hierarchy Process for Southern Johor River Ranking. Int. J. Advance, Soft Comput. Appl., Vol. 1, No. 1, July 2009 [5] M. Ilunga and E.K. Onyari Application of Hierarchical Process (AHP) for ANN model selection in streamflow prediction. The 15 th World Multiconference on Systemics, Cybernetics and Informatics (WMSCI 2011), Orlando, USA, July 19 th-22 nd, 2011, Proceedings Volume III, pp [6] Department of Water Affairs (DWA), Strategy and Guideline Development for National Groundwater Planning Requirements. Water Banking: A practical guide to using Artificial Groundwater recharge, dated November 2010, pp. 24.