Evaluating groundwater ponds for urban drinking water supply under uncertainty Open Access

Access to clean water and sanitation is one of the 17 Sustainable Development Goals to be achieved by 2030 as defined by the UN (UN 2015). The level of achievement of this goal is very different. In Serbia, for example, more than 74% of the population was using safely managed drinking water in 2020 (Our World in Data 2022). Unfortunately, this percentage can be decreased in the future for many reasons: climate change, slow adjustment of supply systems to increase in the urban population, irresponsible water use, availability of financial resources for system maintenance and operation, brain drain, and so on. When it comes to groundwater and its use for water supply, uncertainty becomes even more emphasized due to the lack of data (sparse temporally and spatially), imperfect understanding of groundwater dynamics, modeling simplifications, projections of future water demand, ecological, political, and social issues, among others. Mannix et al. (2022) identified four forms of interconnected uncertainty in groundwater planning: hydrogeologic uncertainty, modeling uncertainty, water demand uncertainty, and urban planning uncertainty, which were discussed with stakeholders to improve particularly short-term decision-making. Other authors have also tackled uncertainty problems within the participatory framework and with various decision support tools (Crowe et al. 2016; Srdjevic et al. 2017; Bakhtiari et al. 2020; Brown et al. 2020; Freeman et al. 2020; Xiang et al. 2021).

Among many tools and methods, multi-criteria analysis, especially fuzzy versions, is often used as an approach that successfully merges qualitative and quantitative criteria, uncertainty, different sectors, and different stakeholders, as well as the spatial and temporal issues in ill-structured problems. Different fuzzy-based multi-criteria analysis models have been developed, including many that more or less follow the philosophy of the analytic hierarchy process (AHP), established by Thomas Saaty in his seminal book (Saaty 1980). Original AHP uses pairwise comparisons to determine the weights of criteria and alternatives in a structured manner. Because subjective judgments made by the decision-maker during the comparison process could be imprecise for many reasons, fuzzy sets have been combined with AHP in much reported research. Commonly this combination is referred to as fuzzy AHP or FAHP (e.g. Triantaphyllou & Lin 1996; Raju & Pillai 1999).

There are hundreds of reported research and case study applications of fuzzified AHP in water resources planning and management. Scientific papers offer a broad spectrum of information but no state-of-the-art or other summary review exists which presents in one place the main findings, achievements, drawbacks, or problems in applications of fuzzified AHP. The following brief literature review is a part of the complete figure and serves as an introduction to the proposed fuzzy decision-making (FDM) approach based on the combined use of fuzzy and crisp AHP for solving the same decision-making problem – ranking by the overall quality of three major groundwater ponds in a given city.

Teklu et al. (2019) discussed the issue of proactive integrated water resources management which requires the prediction of emerging future water resources situations and formulation of multi-sector and participatory long-term strategic development plans to cope with such situations. The authors presented a method of coupling the simulation method WEAP (Water Evaluation and Planning System) with fuzzy AHP to facilitate the decision-making process and to suggest the optimal water shortage mitigation in the Awash River basin in Ethiopia. WEAP is used to predict future water availability using feasible basin development scenarios and then multi-criteria decision analysis is performed with fuzzified AHP using the WEAP's simulation output and opinion data obtained from experts' through questionnaires.

Minh et al. (2019) studied the spatio-temporal variation of groundwater quality for 10 years in An Giang Province in the Vietnamese part of the Mekong River Delta which is known as one of the agricultural intensification areas. The weighted groundwater quality index was developed based on the fuzzy AHP for assigning weighted parameters. This quality index was interesting to analyze as it has temporal variation because groundwater quality wells have improved from 2009 to 2018 in the wet season as compared to the dry season. The reason behind the improvement of groundwater quality during the wet season was the decrease in river discharge, which causes less deposition of suspended solids near the flood plains. Moreover, the filling of unused wells is the reason for the reduced movement of pollutants from unused wells to groundwater aquifers. The authors concluded that understanding groundwater quality can help policymakers to manage limited water resources on a long-term basis and that sophisticated fuzzy AHP may effectively support related management decision-making.

An AHP-based fuzzy evaluation approach proposed by Zhou & Huang (2007) is aimed to support the management of sustainable water resources. Because of the complexity and fuzziness of the water resource system, the authors combine fuzzy theory with AHP and put forward an AHP-based fuzzy evaluation approach to enable the conversion of quantitative analysis into the qualitative analysis. Several evaluation procedures are proposed. The most important is that in which the weights of several indices and their membership degree had to be calculated based on AHP and fuzzy sets theory.

In Hosseini-Moghari et al. (2017) the following questions are discussed for the study area known as the Gorganrood basin in Iran. First, due to difficulties in employing a unique drought policy for the whole basin with different land uses, what is the most pragmatic approach to handle this issue? Second, what is the best framework for considering short-term and long-term development strategies? Third, should alleviating uncertainty in drought mitigation problems be addressed as a multiple criteria problem or as a single criterion? The fuzzy AHP is proposed to, respectively, categorize land use, handle risk management, and provide crisis management. The study was associated with qualitative criteria, subjectivity, uncertainty, and synthesizing the group judgments. For civil, agricultural, and environmental sectors seven criteria are adopted to evaluate 21 alternative policies. Fuzzifying Saaty's ratio scale is made with a distance of 1 for triangular fuzzy numbers 1–9, and defuzzification is performed by the center of gravity method. Based on Bonissone (1982), the fuzzy utility of each alternative policy is determined and authors claim that fuzzified AHP performed well as a practical tool for decision-making. Sensitivity analysis of solutions is not presented regarding different fuzzification(s) of a 9-point scale, or implementation of optimism-pessimism tuning of the decision-makers' attitudes and behavior.

Alias et al. (2009) present the application of fuzzy set theory and fuzzy AHP in multi-criteria decision-making to find the most reasonable and efficient use of the Southern Johor River System in Malaysia. The assessment of this river system included qualitative and quantitative aspects, modeled via four criteria and 20 subcriteria. Fuzzy numbers and linguistic variables are used to address inherently uncertain or imprecise data and the technique is tested with real river data. A comparison between the proposed technique and some previous results obtained using conventional techniques is also presented. It is claimed that the proposed approach can deal with vague data using fuzzy triangular numbers.

Zaresefat et al. (2022) present a case study of the Iranshahr Basin in Iran aimed to identify appropriate site-specific recharge areas in the basin. They used fuzzy AHP in a single platform along with GIS fuzzy logic spatial modeling. Authors identified numerous means that can be employed to renovate groundwater resources, claiming at the same time that the accurate findings of appropriate sites for artificial recharge by traditional methods may be impossible (Sprenger et al. 2017) or difficult (Mahdavi et al. 2013). Several techniques have been also proposed for artificial groundwater recharge zoning such as remote sensing, decision-making method TOPSIS, and finally standard and fuzzy AHP. In conclusion, the authors state that ‘the most appropriate sites for artificial recharge could be determined using all available and effective parameters and combining GIS and fuzzy AHP methods as powerful decision-makers' (Zaresefat et al. 2022, p. 10). On the other hand, proper scientific investigations can assess the need and feasibility of an area for artificial recharge. Thus, this or other similar methods could be considered as the prerequisites for planning and implementation. Recent articles prove this statement (Dilekoglu & Aslan 2022; Li et al. 2022; Vasan et al. 2022).

The fuzzy decision-making approach (FDM), based on parallel use of the fuzzified and crisp analytic hierarchy process, is proposed here to assess the importance of groundwater ponds serving for urban freshwater supply. To keep the complex and unreliable process of comparing fuzzy utilities within reasonable limits, several improvements are proposed to the earlier developed FDM approach that is originally applied to the ranking of several long-term scenarios of water management in the Paraguacu River Basin in Brazil (Srdjevic et al. 2002). Improved FDM completely follows a methodology of the original AHP method in part of pairwise comparisons and uses fuzzy extent analysis and the additive weighting method in calculating utilities of criteria versus goal and alternatives across criteria. The center of gravity method and the total integral value method are used to defuzzify final utilities and rank alternatives and display a decision-maker's preference and risk tolerance. Application of crisp and fuzzy AHP and comparison of their crisp and fuzzy outcomes serve to complete a reliable environment for decision-makers while performing judgments and creating judgment matrices within the problem hierarchy. In the same way, comparisons point to some advantages and disadvantages of the two AHP versions.

Improvements in the methodology, based on the parallel use of fuzzified and crisp AHP, are aimed at creating a reliable framework for the decision-making of city planners and managers. It is expected that cross-checking of results of the two models (crisp and fuzzy AHP) will provide a base for better planning of the financing, maintenance, and extension of the urban supply system that takes into account uncertainty.

A case study for the city of Novi Sad in Serbia is presented. A set of seven most relevant criteria has been defined by an expert in hydrogeology and urban water distribution systems to evaluate the overall quality of three groundwater ponds as major city suppliers of freshwater. Different levels of his optimism (and pessimism) enabled sensitivity analysis of solutions and verification of the robustness of the decision-making process.

Basic concepts of fuzzy AHP are described in many research papers worldwide. The brief description presented here is based on published research (Srdjevic & Medeiros 2008).

Operations on fuzzy sets are based on using triangular norms T and S, which model the intersection operator in set theory (T), and the union operator (S). The min and max norms defined by Zadeh are the most frequently used, likewise the composition operators sup (supremum) and inf (infimum) for connecting fuzzy sets. Operations on fuzzy sets are enabled by combinations of norms and composition operators.

The most probable value of the fuzzy number A is the modal value ⁠, while the lower and upper bounds and define the degree of fuzziness of modal value a₂. The greater is, the fuzzier the degree is. When the value is not a fuzzy number (i.e. it is a crisp number). The fuzzy number is symmetrical, ⁠.

All (i = 1,…, n; j = 1,…, m) are triangular fuzzy numbers representing the performance of the object concerning each goal ⁠.

Notice that Equation (5) corresponds to the core equation of the additive normalization method (ANM) in standard AHP (Saaty 1980). As shown in many pieces of research (see for instance Srdjevic 2005), the ANM is, although very simple, a valuable matrix-related prioritization method when applied to the pairwise comparison matrix for calculating the weight of decision elements criteria and alternatives. The values of fuzzy extents (Equation 5) functionally and mathematically correspond to the crisp weights in ANM.

Defuzzification is the process of producing a quantifiable crisp result, given fuzzy sets and their corresponding membership degrees. It is the conversion of a fuzzy quantity to a precise quantity and is the opposite process of fuzzification. There are several heuristic defuzzification methods, but the most common one used is the center of gravity (CoG). This method determines the center of the area of the fuzzy set and returns the corresponding crisp value. Two alternative methods in defuzzification are also worth mentioning, the center of sums (CoS) method and the mean of maximum method (MoM) (Soliman & Al-Kandari 2010). Description of other defuzzification methods can also be found in Liu et al. (2020) such as the dominance measure method or the α-cut with interval synthesis method.

The fuzzy decision-making (FDM) approach used here is based on the philosophy of standard (crisp) AHP and the application of fuzzy extent analysis. FDM involves assessments of decision elements in the standard three-level hierarchy decision-making problem settled in a fuzzy framework by following several key premises and methodological steps:

• 1.

  The decision-maker uses original (crisp) AHP and Saaty's 9-point scale for performing pairwise comparisons of decision elements (criteria and alternatives) in the three-level hierarchy composed of a goal at the top, criteria set in level two, and a set of alternatives in the bottom level. The cosine maximization method CMM (Kou & Lin 2014) is used for calculating local weights of criteria versus goal and alternatives versus criteria. Standard synthesis of the priority vectors within the problem hierarchy produces final weights of alternatives versus goal. The result is crisp in logic and may serve for comparison with the fuzzy result (after defuzzification).

• 2.

  Crisp values of evaluations performed by the decision-maker in the previous step are fuzzified by using symmetrical and asymmetrical positive triangular fuzzy numbers with distances 1 or 2 and special treatment of boundary values 1 and 9 in the scale.

• 3.

  Fuzzy extent analysis is applied in all instances as an analogous method to standard AHP prioritization and synthesis.

• 4.

  The center of gravity method is used at the end for defuzzification of the fuzzy weights of alternatives.

• 5.

  The total integral value method is also used for defuzzification to shape decision-makers' attitudes toward pessimism and optimism.

Core issues of the FDM are (1) fuzzification of decision-makers' judgments given via the 9-point Saaty's scale; and (2) application of fuzzy extent analysis across the hierarchy and defuzzification of the final result. Regarding the fuzzification of Saaty's 9-point scale given in the first two columns in Table 1, some authors (such as Chang 1996; Deng 1999) used only odd integers 1, 3, 5, 7, and 9 to express the decision-maker's subjective measure of the dominance of one element over the other. In fuzzification by triangular fuzzy equivalents, the distance of 2 is commonly used; on boundaries, (1,1,1) or (1,1,3) is used for 1, and (7,9,9) or (7,9,11) is used for 9; note that in a later case upper value 11 is out of Saaty's scale and therefore unjustified (no semantic equivalent).

In the case study presented here two fuzzifications of the original Saaty's (crisp) 9-point scale are performed:

• (a)

  For fuzzy distance ⁠, all integer values in Saaty's scale from 2 to 8 are symmetrical positive triangular fuzzy numbers. At the boundaries, fuzzy numbers are (1,1,2) and (8,9,9), Table 1.

• (b)

  For fuzzy distance = 2, integer values 3 to 7 in Saaty's scale are symmetrical positive triangular fuzzy numbers. At the boundaries, asymmetrical fuzzy numbers are (1,1,3), (1,2,4), (6,8,9), and (7,9,9), Table 2.

Regarding the fuzzy extent analysis and synthesis of the result, after the multi-criteria problem has been created by using triangular fuzzy numbers and related membership functions, as defined in either Tables 1 or 2 (the third column), the ranking procedure starts with the determining of the importance of criteria concerning the goal. A fuzzy reciprocal judgment matrix for criteria importance is transformed into the triangular fuzzy weights of criteria via fuzzy extent analysis (Equation (5)). The same is repeated for alternatives concerning all criteria. The final synthesis of all fuzzy weights is performed by the additive method as in crisp AHP.

To summarize, the difference concerning the crisp decision-making (CDM) approach is that Saaty's scale is fuzzified, all operations in the fuzzy decision-making (FDM) approach are fuzzy with triangular fuzzy numbers, and the final ranking of alternatives is performed after the defuzzification. Because the defuzzification can be performed differently (Triantaphyllou & Lin 1996; Deng 1999), likewise, the final results may differ. Our case study shows that these differences might not be that significant which leads to the conclusion that the fuzzy version of AHP significantly replicates the results of the crisp version of AHP. At least, our case study may serve as proof of this claim.

The city of Novi Sad is the second largest city in Serbia, with more than 400,000 inhabitants and with a yearly increase in population. During the summer season, a public water and sewerage utility company, Vodovod i kanalizacija, is forced to apply restrictions due to the water shortage. Because it is expected that the restrictions will be more frequent in future, there is a need to carefully analyze options to mitigate the shortage. One way to do this is to define the importance of water supply ponds for the city of Novi Sad on a long-term basis and prepare a maintenance and operation action plan accordingly.

Major ponds are in a full 24-hour operation. Their exploitation is supported on a temporary and intervening basis by two other ponds known as Kamenjar and Detelinara. These two ponds were not considered in this study.

The decision problem is settled as to evaluate three major city groundwater ponds and derive their global weights concerning certain qualitative, quantitative, and mixed criteria. The top-ranked pond should be labeled as ‘the best pond’, with the understanding that the global weights of the three ponds should serve as indicators of their importance for drinking water supply to the city of Novi Sad on a long-term basis.

Namely, decision elements are:

Alternatives (groundwater ponds):

• A1 - Petrovaradinska ada

• A2 - Strand

• A3 - Ratno ostrvo

Evaluating criteria:

• C1 - Capacity

• C2 - Water quality

• C3 - Cost of water

• C4 - Natural protection

• C5 - Recharging capabilities

• C6 - Technical accessibility

• C7 - Environmental impacts

To compare characteristics and rank three major ponds by importance for strategic planning of development and preservation, an expert in groundwater hydrology was asked to define the most appropriate set of criteria and evaluate ponds by AHP. The following set of seven criteria is adopted: capacity, water quality, cost of water, natural protection, recharging capabilities, technical accessibility, and environmental impacts. The capacity of a pond is defined as the total well's capacity installed. Water quality is understood as a necessity for water treatment. The unit cost of water is defined as the cost per m³ of installed pump capacity. As far as the ‘natural protection’ criterion is considered, it was assumed, for example, that low-permeable layers such as clays or sandy clays should cover water-bearing layers with the underlying logic in evaluations by AHP: the more massive protecting layers are, the better natural protection of the pond is. Recharging capability aggregates both natural and artificial recharging possibilities that exclude any hazardous pollution. Technical accessibility of the pond is a global measure of technical characteristics of wells, pumps, local infrastructure, and so on. Finally, environmental impacts serve to include interrelations between ponds, water treatment plants, social interests, and other environmental factors; certain psychological issues are considered to be included in evaluations under this criterion, too.

Fuzzy numbers from the scale, corresponding to the expert's judgments, are inserted in the upper triangle of the matrix while reciprocal values are automatically inserted in the lower triangle (shaded part of the matrix).

Following the same procedure, groundwater ponds are compared in pairs concerning each criterion and after 7 × ((3 × 2)/2) = 21 judgments made by the expert, seven fuzzified matrices are generated as shown in Table 3. Note that rows and columns in these matrices correspond to groundwater ponds in order: A1 – Petrovaradinska ada, A2 – Strand, and A3 – Ratno ostrvo.

Table 3

^aAll entries in the tables are fuzzy triangular numbers with a given distance (1 or 2).

View Large

For the typical values λ that express the decision-maker's attitude toward risk, the final ranking of groundwater ponds is obtained by applying the integral defuzzification method (Equation (7)). The normalized values presented in Table 4 show that Pond 1 – Petrovaradinska ada is the best. It is followed by Pond 3 – Ratno ostrvo and Pond 2 – Strand, regardless of the decision-maker's level of optimism.

Table 4

Final ranking of groundwater ponds for a different index λ of decision-maker's optimism

Groundwater Pond .	Fuzzy AHP, distance = 1 .			Fuzzy AHP, distance = 2 .
	λ = 1.0 (Optimistic) .	λ = 0.5 (Moderate) .	λ = 0.0 (Pessimistic) .	λ = 1.0 (Optimistic) .	λ = 0.5 (Moderate) .	λ = 0.0 (Pessimistic) .
A1 Petrovaradinska ada	0.4121	0.4138	0.4199	0.4081	0.4092	0.4197
A2 Strand	0.2643	0.2623	0.2551	0.2770	0.2747	0.2534
A3 Ratno ostrvo	0.3236	0.3239	0.3250	0.3150	0.3161	0.3269

Groundwater Pond .	Fuzzy AHP, distance = 1 .			Fuzzy AHP, distance = 2 .
	λ = 1.0 (Optimistic) .	λ = 0.5 (Moderate) .	λ = 0.0 (Pessimistic) .	λ = 1.0 (Optimistic) .	λ = 0.5 (Moderate) .	λ = 0.0 (Pessimistic) .
A1 Petrovaradinska ada	0.4121	0.4138	0.4199	0.4081	0.4092	0.4197
A2 Strand	0.2643	0.2623	0.2551	0.2770	0.2747	0.2534
A3 Ratno ostrvo	0.3236	0.3239	0.3250	0.3150	0.3161	0.3269

View Large

By using the center of gravity method to defuzzify the F values given above, the final weights of alternatives obtained after normalization were: 0.4122 (Pond A1), 0.2647 (Pond A2), and 0.3231 (Pond A3). The final ranking is equal to the previous one.

For the sake of completeness, the original version of AHP has been used with the same judgment matrices given in Tables 3 and 5, but with crisp values used instead of fuzzy values. The final ranking of ponds was the same and the derived weights (Pond A1: 0.428; Pond A2: 0.247; Pond A3: 0.326) had only small differences.

Table 5

View Large

To conclude, both standard AHP and fuzzy AHP show that the most important groundwater pond for supplying the city of Novi Sad with fresh drinking water is A1 – Petrovaradinska ada. Out of 100 points, it takes 43. The second best is groundwater pond A3 – Ratno ostrvo with 34 points. The last one in the row is A2 – Strand with 23 points.

Fuzzy approaches to the decision-making process can be found in many sectors of human activity. In water resources, there is a significant number of published research papers, case study applications, and other documents informing interested audiences on the capability of fuzzy theory and fuzzy sets to support the solution of real-life problems related to the selection of artifacts, allocation of resources, and other related challenges. The main reason for using a fuzzy approach instead of crisp mostly lies in the assumed awareness of the decision-maker that they might not be sure about judgments made, for instance, because of insufficient information.

This case study application of the well-established multi-criteria method AHP in ranking by the overall quality of three existing sources of fresh water to the city of Novi Sad City, Serbia showed that in application of either crisp or fuzzified AHP an expert ‘generated’ unmodified weights of ponds and identified one that should be considered as most important for further improvements for the benefit of Novi Sad's citizens. The expert's satisfaction with the process and the result were clearly expressed and the proposed approach can be recommended for other decision-making problems under uncertainty.

The similarity in the results obtained by either AHP (crisp or fuzzy) can be understood as a comfortable side of the decision-making process as far as the decision-maker is concerned. From the mathematical point of view, it should be pointed out that the fuzzy extent we (and most other authors) use is mathematically analogous to the basic prioritization method in crisp AHP, the additive normalization method (ANM). Consequent use of fuzzy extent and ANM within the complete hierarchy of the problem, such as the case described in this study, assumes that results can be expected to be similar. However, this claim may not stand if other prioritization methods are used in crisp AHP instead of ANM. Worth mentioning is that different matrix and optimization techniques generally extract more or less different local priorities in the hierarchy. Consequently, crisp AHP synthesis can produce different results from the fuzzy extent analysis employed in fuzzy AHP which ‘emulates’, or, better to say, ‘replicates’ the additive normalization process. The main reason is the cumulative effect of additions of weighted priorities in crisp AHP and fuzzy weighted additions during the fuzzy AHP synthesis.

Part of the future research agenda is foreseen in exploring other fuzzy versions of AHP and their applicability within a group decision-making context with more experts getting involved. It is also, worth while to evaluate an extension of the set of criteria to account for economic, environmental, and people satisfaction issues, to mention but a few.

This work was supported by the Ministry of Education, Science, and Technological Development of Serbia (Grant No. 451-03-9/2021-14/ 200117).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.