Hierarchical distancebased fuzzy approach to evaluate urban water supply systems in a semiarid region
 Tahereh Sadeghi Yekta^{1},
 Mohammad Khazaei^{1},
 Ramin Nabizadeh^{2},
 Amir Hossein Mahvi^{2, 3},
 Simin Nasseri^{2, 4}Email author and
 Ahmad Reza Yari^{1}
DOI: 10.1186/s402010150206y
© Yekta et al. 2015
Received: 2 October 2014
Accepted: 30 May 2015
Published: 14 July 2015
Abstract
Hierarchical distancebased fuzzy multicriteria group decision making was served as a tool to evaluate the drinking water supply systems of Qom, a semiarid city located in central part of Iran. A list of aspects consisting of 6 criteria and 35 subcriteria were evaluated based on a linguistic term set by five decisionmakers. Four water supply alternatives including “Public desalinated distribution system”, “PET Bottled Drinking Water”, “Private desalinated water suppliers” and “Household desalinated water units” were assessed based on criteria and subcriteria.
Data were aggregated and normalized to apply Performance Ratings of Alternatives. Also, the Performance Ratings of Alternatives were aggregated again to achieve the Aggregate Performance Ratings. The weighted distances from ideal solution and antiideal solution were calculated after secondary normalization. The proximity of each alternative to the ideal solution was determined as the final step. The alternatives were ranked based on the magnitude of ideal solutions.
Results showed that “Public desalinated distribution system” was the most appropriate alternative to supply the drinking needs of Qom population. Also, “PET Bottled Drinking Water” was the second acceptable option. A novel classification of alternatives to satisfy the drinking water requirements was proposed which is applicable for the other cities located in semiarid regions of Iran.
The health issues were considered as independent criterion, distinct from the environmental issues. The constraints of hightech alternatives were also considered regarding to the level of dependency on overseas.
Keywords
Fuzzy logic Drinking water MCDM Distribution systemIntroduction
Evaluating the alternatives to satisfy the drinking water demands of societies is a complicated issue that usually should be relied on human judgments. Furthermore, Different criteria should be considered to evaluate the alternatives available for supplying the drinking water needs, especially in populations faced with fresh water scarcity which are relied on brackish water sources [1].
Various methods based on human decisionmaking have been used to evaluate the alternatives assigned for water supply systems such as Life cycle assessment [2, 3], MCDM approach [4], Fiveparametric matrix [5], Multicriteria decision aid (MCDA) approach [6], and consumer cooperatives [7].
The major concern related to the water supply systems in developing countries is the large scale projects such as transbasin water transfer [8], and constructing the sophisticated water supply systems which may not be completed on time because of the financial deficiencies or changing in political considerations [9]. So, applying the available water supply systems as the viable alternatives can be helpful to deliver an obvious viewpoint for administrators as well as for the public sector [10]. Also, few studies, worked on evaluating the available alternatives, have drown the hierarchy of aspects directly from the other studies and did not consider the background factors in their intrinsic society which may influence the arrangement of criteria and subcriteria [4, 7, 11].
This paper outlines a methodology that evaluates the available alternatives to supply drinking water demands of Qom population, a city located in plains fed with brackish aquifers. The evaluation processes are according to a complete package of criteria and subcriteria.
A simpleminded and wellknown method of decisionmaking is adopted based on fuzzy logic to evaluate the alternatives. The presented method is known as hierarchical distancebased fuzzy multicriteria group decision making (DBF –MCDM) approach. Applying DBF–MCDM enables the decisionmaking committee to improve the identification of discrepancies and similarities of their judgments [12]. Also, the DBF–MCDM process justifies both ideal and antiideal solutions simultaneously that help the decisionmakers to have more obvious judgments [13]. A new arrangement of criteria and subcriteria to evaluate the drinking water supply alternatives is also adopted using the MCDM method under fuzzy environment.
Methodology
Various aspects should be considered when a team or organization decides to make a decision among several available alternatives. The decision making process maybe comes more complicated if the number of alternatives and criteria be increased [14]. This section dedicates a short description about the principles of multicriteria group decision making (MCDM) that is based on fuzzy set theory to resolve the decision making problems on the subject of drinking water supply alternatives.
Fuzzy sets theory
Definition 1
A fuzzy set can be defined as Ã = (X, μ _{ Ã }(x)), Where X is the space on which the fuzzy set is defined, and μ _{ Ã }(x) → [0, 1], x ∈ X, the membership function of the set [15].
Definition 2
Using the triangular fuzzy number is due to its simplicity compare with trapezoid or sigmoid fuzzy numbers and intuitively easy for decisionmakers to utilize. Furthermore, modeling according to triangular fuzzy numbers is a competent approach for organizing the decisionmaking problems [17, 16].
Definition 3
A linguistic variable is defined as a kind of variable whose values are expressed in linguistic terms. Because of the imprecise and vague nature of human judgments, it is preferred to express the expert judgments via linguistic terms. The linguistic terms are the study variables with the capability of describing the qualitative data. A linguistic variable comprises an ordinary word or phrase in natural language and so they are representatives of imprecise data whose values are not numbers. In situations that the study has been affected by ill defined or complex variables, a linguistic term can be a useful tool to prepare an approximate characterization [18].
Definition 4
Comparing and ranking the final ratings \( {\overline{r}}_1,{\overline{r}}_2\dots, {\overline{r}}_m \) are performed to judge the relevant values of the different alternatives [14].
Definition 5
If ñ be considered as a triangular fuzzy number and \( n\begin{array}{c}\hfill \propto \hfill \\ {}\hfill \ell \hfill \end{array}>\kern0.5em 0,n\begin{array}{c}\hfill \propto \hfill \\ {}\hfill u\hfill \end{array}\le \kern0.5em 1 \) for ∝ ∈ [0, 1] then ñ is called a normalized positive triangular fuzzy number [19].
Definition 6
The ideal solution A* = (r _{1}*, r _{2}*, …, r _{ n }*) and also the antiideal solution A ^{−} = (r _{1} ^{−}, r _{2} ^{−} …, r _{ n } ^{−}) are defined where r _{ j }* = (1, 1, 1) and r _{ j } ^{−} = (0, 0, 01) for j = 1, 2 …, n [20].
Definition 7
The size of the trapezoidal area is obtained by the distance formula. The larger values of a _{1} − b _{1} ora _{3} − b _{3} are the lower trapezoid base. The values of a _{2} − a _{2} determine the upper trapezoid base, and the trapezoid height is equal to one. The Closer triangular numbers \( \tilde{A}\kern0.5em and\kern0.5em \tilde{B} \) the smaller trapezoidal area.
Hierarchical distancebased fuzzy Multicriteria group decision making (DBF –MCDM) approach
The fuzzy multicriteria group decision making approach has the ability of addressing the decision problems including a multilevel hierarchical structure which has been equipped with attributes of qualitative performance [22]. The distancebased fuzzy MCDM approach has been introduced by Karsak (2002) for selecting the technology alternative [23]. The DBFMCDM is constructed according to the closeness to the ideal alternative concept. Also, DBFMCDM has the potential of including both crisp and fuzzy data.
Usually, the performance attributes can be organized in multilevel hierarchy when they are in large numbers. The multilevel hierarchy enables the analysis to be done more efficiently.
 Step 1.
Establish a decision makers team of z experts (l = 1,2…, z). Introduce the alternatives, necessary criteria, and attributed subcriteria.
 Step 2.
Assemble the decision matrices that comprise the importance weights of criteria and attributed subcriteria. The decision matrices also, should be included the fuzzy assessments in relation with subcriteria for each decisionmaker.
 Step 3.Introduce the mathematical signs used for representation the criteria, subcriteria, decision makers and alternatives and their relationships as depicted in Table 1.Table 1
Mathematical signs used for representing the equations
Definition
Description
i = (1, 2 …, m)
Set of alternatives
j = (1, 2 …, n)
Set of criteria
k = (1, 2 …, p)
Set of subcriteria
l = (1, 2 …, z)
Set of decision makers
\( {\tilde{X}}_{ijkl}=\left({X}_{ijkl}^1,{X}_{ijkl}^2,{X}_{ijkl}^3\right) \)
Alternative i attributed to subcriterion k of criterion j.
\( {\tilde{W}}_{jkl}=\left({W}_{jkl}^1,{W}_{jkl}^2,{W}_{jkl}^3\right) \)
Importance weight of subcriterion k of criterion j.
\( {\tilde{W}}_{jl}=\left({W}_{jl}^1,{W}_{jl}^2,{W}_{jl}^3\right) \)
Importance weight of criterion j for the lth decisionmaker
 Step 4.Calculate the aggregated fuzzy assessments of alternatives \( \left({\tilde{X}}_{ijkl}\right) \), the aggregated importance weight of subcriteria \( \left({\tilde{W}}_{jkl}\right) \) and the aggregated importance weight of criteria \( \left({\tilde{W}}_{jl}\right) \) based on follows:$$ {\tilde{W}}_j={\displaystyle \sum_{l1}^z{v}_l}{\tilde{W}}_{jl} $$(4)$$ {\tilde{W}}_{jk}={\displaystyle \sum_{l1}^z{v}_l}{\tilde{W}}_{jkl} $$(5)$$ {\tilde{X}}_{ijk}={\displaystyle \sum_{l1}^z{v}_l}{\tilde{X}}_{ijkl} $$(6)
Where v _{ l } ∈ [0, 1] represents weight assigned to the lth decisionmaker.
Also, ∑_{ l = 1} ^{ z } v _{ l } = 1.
So, by using above equations, aggregated ratings of alternatives with respect to each subcriterion \( \left({\tilde{X}}_{ijk}\right) \), aggregated importance weights of subcriteria \( {\tilde{W}}_{jk} \) and aggregated importance weights of criteria \( \left({\tilde{W}}_j\right) \) can be computed as (X _{ ijk,} ^{1} X _{ ijk,} ^{2} X _{ ijk,} ^{3}), (W _{ jkl,} ^{1} W _{ jkl,} ^{2} W _{ jk,} ^{3}) and (W _{ j,} ^{1} W _{ j,} ^{2} W _{ j,} ^{3}) respectively.
 Step 5.To obtain the unitfree and comparable subcriteria values, the aggregated decision matrix resulted from step 4 should be normalized. Among various methods used for data normalization [24, 17] a linear scale transformation is selected. Based on this approach, first the subcriteria are categorized in two groups known as benefitrelated (BR) and cost related (CR) ones as identified in Fig. 3. Then, the linear scale transformation is used for data normalization as follows:$$ \begin{array}{l}{\tilde{r}}_{ijk}=\left({r}_{ijk}^1,{r}_{ijk}^2,{r}_{ijk}^3\right)\\ {}=\left\{\begin{array}{c}\hfill \left(\frac{x_{ijk}^1{x}_{jk}^{}}{x_{jk}^{*}{x}_{jk}^{}},\frac{x_{ijk}^2{x}_{jk}^{}}{x_{jk}^{*}{x}_{jk}^{}},\frac{x_{ijk}^3{x}_{jk}^{}}{x_{jk}^{*}{x}_{jk}^{}}\right),\kern0.5em k\in \kern0.5em \mathrm{B}{\mathrm{R}}_j;\kern0.5em i=1,2\dots, m;\kern0.5em j=1,2\dots, n\hfill \\ {}\hfill \left(\frac{x_{jk}^{*}{x}_{ijk}^3}{x_{jk}^{*}{x}_{jk}^{}},\frac{x_{jk}^{*}}{x_{jk}^{*}},\frac{x_{ijk}^2}{x_{jk}^{}},\frac{x_{jk}^{*}{x}_{ijk}^1}{x_{jk}^{*}{x}_{jk}^{}}\right),k\in \kern0.5em C{\mathrm{R}}_j;\kern0.5em i=1,2\dots, m;\kern0.5em J=1,2\dots, n\hfill \end{array}\right.\end{array} $$(7)
Where, \( {\tilde{r}}_{ijk} \) is the normalized value of \( {\tilde{x}}_{ijk} \), x _{ jk } ^{*} denotes max_{ i } x _{ ijk } ^{3} and x _{ jk } ^{−} is min_{ i } x _{ ijk } ^{1}BR_{j} is the set of benefitrelated subcriteria of criterion j for which the higher the efficiency value the more performance of it and CR_{j} is the sets of costrelated subcriteria of criterion j for which the higher the efficiency value the less preference of it. Also, m identifies the number of alternatives and n denotes the number of criteria.
 Step 6.The performance ratings of alternatives at the subcriteria stage to criteria stage should be aggregated to compute the aggregate performance ratings (APRs) as follows:$$ {\tilde{y}}_{ij}=\left({y}_{ij}^1,{y}_{ij}^2,{y}_{ij}^{31}\right)=\frac{{\displaystyle {\sum}_{k=1}^p\kern0.5em {\tilde{w}}_{jk}\otimes}\kern0.5em {\tilde{r}}_{ijk}}{{\displaystyle {\sum}_{k=1}^p{\tilde{w}}_{jk}}},i=1,2\dots, m;j=1,2\dots, n $$(8)
Where, ỹ _{ ij } is served as the APR of alternative i in relation with criterion j. It should be added that ⊗ is the multiplication operator in fuzzy logic.
 Step 7.The APRs are normalized at criteria stage with linear normalization method again. Based on this approach and as can be recognized from the following equation, the best results acquire the value equal 1 and the worst ones obtain the value equal 0.$$ {\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}=\left({\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^1,{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^2,{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^3\right)=\left(\frac{y_{ij}^1{y}_j^{}}{y_j^{*}{y}_j^{}},\frac{y_{ij}^2{y}_j^{}}{y_j^{*}{y}_j^{}},\frac{y_{ij}^3{y}_j^{}}{y_j^{*}{y}_j^{}}\right),\kern1em i=1,2\kern0.5em \dots, \kern0.5em \mathrm{m};\kern0.5em j=1,2\kern0.5em \dots, \kern0.5em \mathrm{n} $$(9)
Where, \( {\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij} \) is the normalized APR of alternative i with respect to criterion j. y _{ j } ^{*} = max_{ i } y _{ ij } ^{3} and y _{ j } ^{−} = min_{ i } y _{ ij } ^{1}.
 Step 8.The weighted distances (WDs) from ideal solution and antiideal solution may be represented as D _{ i } ^{*} and D _{ i } ^{−} respectively. The value of WD for each alternative can be computed as follows:$$ o{D}_i^{*}={\displaystyle \sum_{j=1}^n\frac{1}{2}}\left\{ \max \left({\tilde{w}}_j^1\left{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^11\right,{\tilde{w}}_j^3\Big{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^31\right)+{\tilde{w}}_j^2\left{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^21\right\right\},\kern0.5em \mathrm{i}=1,2\kern0.5em \dots, \kern0.5em \mathrm{m} $$(10)$$ {D}_i^{}={\displaystyle \sum_{j=1}^n\frac{1}{2}}\left\{ \max \left({\tilde{w}}_j^i\left{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^10\right,{\tilde{w}}_j^3\left{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^30\right\right)+{\tilde{w}}_j^2\left{\overset{\tilde{\mathit{\hbox{'}}}}{y}}_{ij}^20\right\right\},\kern2em \mathrm{i}=1,2\dots, \mathrm{m} $$(11)
 Step 9.The proximity of the alternatives to the ideal solution is represented with Ω _{ i } ^{*} and can be calculated as follows:$$ {\Omega}_i^{*}=\frac{D_i^{}}{D_i^{*}+{D}_i^{}},\kern3em \mathrm{i}=1,2\dots, \mathrm{m}. $$(12)
By using the Ω _{ i } ^{*} concept, the distances from ideal and antiideal solutions are computed.
 Step10.
If the results of Ω _{ i } ^{*} are sorted from largest to the smallest values, the best alternative is one which has obtained the highest Ω _{ i } ^{*} value and therefore is located in the top of the descending ranking of alternatives.
Study area
Qom province has low annual precipitation and also salty marls are prevalence geological structures [27] in its plains. Consequently, like the other cities located in central part of Iran, Qom population has engaged with both water quality and quantity crisis [28, 29]. Local water sources of Qom which are flowed in public salty distribution system (PSDS) contain relatively high levels of salt and are considered only for nondrinking purposes. Dissolved solids concentration (TDS) of surface water and groundwater sources of the province is around 1800 and 4500 mg/L, respectively. To improve the quality of these brackish water sources for drinking demands, some programs have been carried out since past decades, such as Public desalinated distribution system (PDDS), Private desalinated water suppliers (PDWS), and Household desalinated water units (HDWU) [29].
Evaluating drinking water supply alternatives using DBF –MCDM approach

A_{1}: Public desalinated distribution system (PDDS)

A_{2}: PET Bottled Drinking Water (PBDW)

A_{3}: Private desalinated water suppliers (PDWS)

A_{4}: Household desalinated water units (HDWU)
Six and 35 evaluation criteria and subcriteria were defined, respectively which illustrated in Fig. 3. Also, subcriteria were classified to CostRelated and BeneficialRelated groups. The benefitrelated subcriteria are those for which the higher the performance value the more its preference, and the costrelated subcriteria are considered as subcriteria for which the higher the performance value the less its preference (Fig 3).
Linguistic term set for criteria and subcriteria
Linguistic term  Fuzzy value  

Very low(VL)  0  0  0.25 
Low(L)  0  0.25  0.5 
Moderate(M)  0.25  0.5  0.75 
High(H)  0.5  0.75  1 
Very High(VH)  0.75  1  1 
Importance of criteria
Criteria  DM_{1}  DM_{2}  DM_{3}  DM_{4}  DM_{5} 

Economic  M  H  M  H  H 
Environmental  VH  H  VH  H  H 
Public Health  VH  VH  VH  VH  VH 
Occupational Health  VH  VH  H  H  H 
Technical  H  H  H  H  VH 
Social  VH  H  VH  H  H 
Importance of subcriteria
Decision maker  DM_{1}  DM_{2}  DM_{3}  DM_{4}  DM_{5} 

Subcriteria  
CC*  H  M  M  H  H 
OC  VH  H  H  H  H 
WPPC  H  VH  H  H  H 
SREI  VH  H  VH  H  H 
WREI  VH  H  H  H  H 
AREI  M  L  VL  M  L 
Noise  M  M  L  L  VL 
TOP  VH  H  VH  H  H 
CP  VH  H  H  VH  H 
MP  VH  VH  VH  VH  VH 
RQMV  H  VH  H  H  M 
RSC  H  VH  VH  H  H 
Qual.SPW  H  H  VH  H  H 
Quan.SPW  H  H  H  H  H 
RWBD  VH  VH  VH  VH  VH 
OHOF  VH  VH  VH  H  H 
OHOI  H  VH  VH  H  H 
REL  VH  VH  VH  H  H 
WSR  H  VH  VH  H  H 
EDPC  H  M  H  M  H 
LA  H  M  M  M  H 
NSO  M  M  L  VL  VL 
NWTU  M  L  VL  L  VL 
SDOC  M  M  L  VL  VL 
EDOC  H  M  H  M  L 
TC  M  M  L  VL  VL 
DCTW  H  VH  H  H  H 
NIV  H  VH  H  M  H 
AUA  H  H  M  M  M 
NPS  H  H  H  H  H 
AEP  VH  H  VH  H  H 
PAO  H  VH  VH  H  H 
SOH  M  M  L  L  L 
ATWS  VH  H  VH  H  H 
EED  VH  VH  VH  H  H 
Ratings of the alternatives with respect to the subcriteria (The full form of abbreviations was represented in Fig. 3)
Decision Maker  DM_{1}  DM_{2}  DM_{3}  DM_{4}  DM_{5}  

Alternative  A_{1}(PDDS)  A_{2}(PBDW)  A_{3}(PDWS)  A_{4}(HDWU)  A_{1}(PDDS)  A_{2}(PBDW)  A_{3}(PDWS  A_{4}(HDWU)  A_{1}(PDDS)  A_{2}(PBDW  A_{3}(PDWS)  A_{4}(HDWU)  A_{1}(PDDS)  A_{2}(PBDW)  A_{3}(PDWS)  A_{4}(HDWU)  A_{1}(PDDS)  A_{2}(PBDW)  A_{3}(PDWS)  A_{4}(HDWU) 
Subcriteria  
CC*  VL  VL  VL  VH  L  VL  L  VH  L  L  L  H  VL  VL  VL  VH  VL  VL  VL  VH 
OC  VL  L  VL  VH  VL  M  VL  H  L  M  L  VH  L  H  VL  VH  VL  M  VL  H 
WPPC  VL  H  M  L  VL  VH  L  VL  VL  VH  M  VL  L  VH  H  L  VL  VH  M  VL 
SREI  L  VH  L  H  VL  VH  VL  VH  VL  VH  VL  M  L  H  L  H  VL  VH  VL  H 
WREI  L  L  L  H  L  VL  L  VH  VL  VL  VL  H  L  VL  L  H  L  VL  L  VH 
AREI  VL  VL  VL  VL  VL  VL  VL  L  VL  VL  VL  VL  VL  VL  VL  VL  VL  VL  VL  VL 
Noise  VL  VL  VL  M  VL  VL  VL  M  VL  L  VL  H  VL  VL  VL  M  L  VL  L  M 
TOP  L  VL  H  H  L  VL  VH  H  VL  VL  H  H  L  VL  H  H  L  L  H  M 
CP  L  VL  VH  H  L  VL  H  M  L  VL  H  M  M  VL  VH  L  L  L  H  M 
MP  VH  VL  VH  H  VH  VL  VH  M  H  VL  VH  H  VH  VL  H  H  VH  L  H  H 
RQMV  VH  VH  H  L  VH  H  M  VL  H  VH  H  VL  H  H  M  VL  VH  H  M  VL 
RSC  VH  L  VH  VH  VH  L  VH  H  VH  M  VH  H  H  L  VH  H  VH  VL  VH  H 
Qual.SPW  VH  H  L  L  VH  VH  L  VL  VH  VH  VL  L  H  VH  L  L  VH  VH  L  L 
Quan.SPW  L  M  M  VH  M  M  M  VH  M  M  L  H  L  H  L  H  L  M  M  M 
RWBD  H  VL  H  M  M  VL  VH  M  H  L  H  L  H  VL  H  M  M  L  VH  M 
OHOF  VH  L  VH  L  VH  L  VH  M  VH  M  VH  L  H  L  VH  L  H  L  H  M 
OHOI  H  VL  H  M  H  VL  H  H  H  VL  H  H  H  VL  VH  H  M  L  H  M 
REL  H  M  L  H  VH  H  VL  H  H  H  L  M  VH  VH  VL  M  M  VH  VL  H 
WSR  H  VH  H  M  VH  H  VH  L  VH  VH  VH  M  VH  H  VH  L  H  M  H  L 
EDPC  M  VL  M  VH  H  VL  H  VH  H  VL  H  VH  H  VL  H  VH  M  VL  M  H 
LA  H  VL  VL  H  H  VL  VL  H  H  VL  VL  H  H  VL  VL  H  M  VL  VL  M 
NSO  L  VL  L  VH  L  L  L  VH  VL  VL  VL  H  VL  VL  VL  VH  VL  VL  VL  H 
NWTU  H  VL  L  VH  M  VL  VL  H  M  VL  VL  VH  H  VL  VL  H  H  VL  L  H 
SDOC  M  VL  L  H  M  L  L  VH  L  L  VL  H  L  VL  L  VH  L  VL  L  H 
EDOC  L  VL  L  VH  L  VL  VL  H  M  L  VL  VH  VL  VL  VL  VH  L  VL  VL  H 
TC  L  VL  VL  VH  M  L  VL  H  M  L  VL  VH  L  VL  L  VH  L  L  L  VH 
DCTW  VL  VL  VL  VH  VL  VL  VL  VH  VL  VL  VL  VH  VL  VL  VL  VH  VL  VL  VL  VH 
NIV  VH  L  VH  VL  VH  L  VH  VL  H  M  H  L  VH  L  VH  VL  VH  L  VH  VL 
AUA  H  H  H  L  H  H  VH  VL  VH  VH  VH  VL  VH  VH  VH  L  VH  VH  VH  L 
NPS  L  L  L  VH  L  VL  L  VH  L  VL  L  VH  VL  VL  VL  H  L  VL  L  VH 
AEP  M  VH  L  H  H  VH  L  M  M  VH  M  M  L  H  L  M  M  H  M  L 
PAO  H  H  VH  M  M  H  VH  L  L  H  M  L  M  H  H  M  H  H  H  L 
SOH  M  VL  M  H  L  L  L  H  L  VL  L  VH  M  VL  M  H  L  L  L  H 
ATWS  M  L  M  VH  L  L  L  VH  M  M  M  VH  M  L  M  VH  M  L  M  H 
EED  M  VH  VH  VL  L  VH  H  VL  M  VH  M  VL  L  H  H  L  M  H  H  L 
Results and discussion
Aggregated Importance weights of criteria
Criteria/Subcriteria  Aggregated weights 

Economic  (0.40, 0.50, 0.90) 
Environmental  (0.60, 0.70, 1) 
Public Health  (0.75, 0.80, 1) 
Occupational Health  (0.60, 0.70, 1) 
Technical  (0.55, 0.60, 1) 
Social  (0.60, 0.70, 1) 
Aggregated Importance weights of subcriteria
Subcriteria  Aggregated weights 

CC  (0.40, 0.50, 0.9) 
OC  (0.55, 0.65, 1) 
WPPC  (0.55, 0.65, 1) 
SREI  (0.60, 0.70, 1) 
WREI  (0.60, 0.70, 1) 
AREI  (0.10, 0.25, 0.55) 
Noise  (0.10, 0.30, 0.55) 
TOP  (0.60, 0.70, 1) 
CP  (0.60, 0.70, 1) 
MP  (0.75, 0.80, 1) 
RQMV  (0.50, 0.65, 0.95) 
RSC  (0.60, 0.70, 1) 
Qual.SPW  (0.55, 0.65, 1) 
Quan.SPW  (0.50, 0.60, 1) 
RWBD  (0.75, 0.80, 1) 
OHOF  (0.65, 0.75, 1) 
OHOI  (0.60, 0.70, 1) 
REL  (0.65, 0.75, 1) 
WSR  (0.60, 0.70, 1) 
EDPC  (0.60, 0.70, 1) 
LA  (0.60, 0.70, 1) 
NSO  (0.60, 0.70, 1) 
NWTU  (0.60, 0.70, 1) 
SDOC  (0.60, 0.70, 1) 
EDOC  (0.30, 0.50, 0.80) 
TC  (0.10, 0.25, 0.50) 
DCTW  (0.55, 0.65, 1) 
NIV  (0.50, 0.60, 0.95) 
AUA  (0.35, 0.50, 0.85) 
NPS  (0.50, 0.60, 1) 
AEP  (0.60, 0.70, 1) 
PAO  (0.60, 0.70, 1) 
SOH  (0.10, 0.30, 0.6) 
ATWS  (0.60, 0.70, 1) 
EED  (0.65, 0.75, 1) 
Aggregated ratings of alternatives with respect to subcriteria
Subcriteria  A_{1}  A_{2}  A_{3}  A_{4} 

CC  (0.00, 0.10, 0.35)  (0.00, 0.05, 0.30)  (0.00, 0.10, 0.35)  (0.70, 0.95, 1) 
OC  (0.00, 0.10, 0.35)  (0.25, 0.50, 0.75)  (0.00, 0.50, 0.30)  (0.65, 0.90, 1) 
WPPC  (0.00, 0.05, 0.30)  (0.70, 0.95, 1)  (0.25, 0.50, 0.75)  (0.00, 0.10, 0.35) 
SREI  (0.00, 0.01, 0.35)  (0.70, 0.95, 1)  (0.00, 0.10, 0.35)  (0.50, 0.75, 0.95) 
WREI  (0.00, 0.02, 0.45)  (0.00, 0.05, 0.30)  (0.00, 0.20, 0.45)  (0.60, 0.85, 1) 
AREI  (0.00, 0.00, 0.25)  (0.00, 0.00, 0.25)  (0.00, 0.00, 0.25)  (0.00, 0.05, 0.30) 
Noise  (0.00, 0.05, 0.30)  (0.00, 0.25, 0.30)  (0.00, 0.05, 0.30)  (0.30, 0.55, 0.80) 
TOP  (0.00, 0.20, 0.45)  (0.00, 0.05, 0.30)  (0.55, 0.80, 1)  (0.45, 0.70, 0.95) 
CP  (0.05, 0.30, 0.55)  (0.00, 0.05, 0.30)  (0.60, 0.85, 1)  (0.25, 0.50, 0.75) 
MP  (0.70, 0.95, 1)  (0.00, 0.05, 0.30)  (0.65, 0.90, 1)  (0.45, 0.70, 0.95) 
RQMV  (0.65, 0.90, 1)  (0.60, 0.85, 1)  (0.35, 0.60, 0.85)  (0.00, 0.05, 0.30) 
RSC  (0.70, 0.95, 1)  (0.05, 0.25, 0.5)  (0.75, 1, 1)  (0.55, 0.80, 1) 
Qual.SPW  (0.70, 0.95, 1)  (0.70, 0.95, 1)  (0.00, 0.20, 0.45)  (0.00, 0.20, 0.45) 
Quan.SPW  (0.01, 0.35, 0.60)  (0.30, 0.55, 0.80)  (0.15, 0.40, 0.65)  (0.55, 0.80, 0.95) 
RWBD  (0.40, 0.65, 0.90)  (0.00, 0.10, 0.35)  (0.60, 0.85, 1)  (0.20, 0.45, 0.70) 
OHOF  (0.65, 0.90, 1)  (0.05, 0.30, 0.55)  (0.70, 0.95, 1)  (0.10, 0.35, 0.60) 
OHOI  (0.45, 0.70, 0.95)  (0.00, 0.05, 0.30)  (0.55, 0.80, 1)  (0.40, 0.65, 0.90) 
REL  (0.55, 0.80, 0.95)  (0.55, 0.80, 0.95)  (0.00, 0.10, 0.35)  (0.40, 0.65, 0.90) 
WSR  (0.65, 0.90, 1)  (0.45, 0.65, 0.75)  (0.65, 0.90, 1)  (0.10, 0.35, 0.60) 
EDPC  (0.40, 0.65, 0.90)  (0.00, 0.00, 0.25)  (0.40, 0.65, 0.90)  (0.70, 0.95, 1) 
LA  (0.45, 0.70, 0.95)  (0.00, 0.00, 0.25)  (0.00, 0.00, 0.25)  (0.45, 0.70, 0.95) 
NSO  (0.00, 0.10, 0.35)  (0.00, 0.05, 0.30)  (0.00, 0.10, 0.35)  (0.65, 0.90, 1) 
NWTU  (0.40, 0.65, 0.90)  (0.00, 0.00, 0.25)  (0.00, 0.10, 0.35)  (0.60, 0.85, 1) 
SDOC  (0.05, 0.20, 0.35)  (0.00, 0.10, 0.35)  (0.00, 0.20, 0.45)  (0.60, 0.85, 1) 
EDOC  (0.05, 0.25, 0.50)  (0.00, 0.05, 0.30)  (0.00, 0.05, 0.30)  (0.65, 0.90, 1) 
TC  (0.10, 0.35, 0.60)  (0.00, 0.15, 0.40)  (0.00, 0.10, 0.35)  (0.70, 0.95, 1) 
DCTW  (0.00, 0.00, 0.25)  (0.00, 0.00, 0.25)  (0.00, 0.00, 0.25)  (0.75, 1, 1) 
NIV  (0.70, 0.95, 1)  (0.05, 0.30, 0.55)  (0.70, 0.95, 1)  (0.00, 0.05, 0.30) 
AUA  (0.65, 0.90, 1)  (0.65, 0.90, 1)  (0.70, 0.95, 1)  (0.00, 0.15, 0.40) 
NPS  (0.00, 0.20, 0.45)  (0.00, 0.05, 0.30)  (0.00, 0.20, 0.45)  (0.70, 0.95, 1) 
AEP  (0.25, 0.50, 0.75)  (0.65, 0.90, 1)  (0.10, 0.35, 0.60)  (0.25, 0.50, 0.75) 
PAO  (0.30, 0.55, 0.8)  (0.50, 0.75, 1)  (0.55, 0.80, 0.95)  (0.10, 0.35, 0.60) 
SOH  (0.10, 0.35, 0.60)  (0.50, 0.10, 0.35)  (0.10, 0.35, 0.60)  (0.55, 0.80, 1) 
ATWS  (0.20, 0.45, 0.70)  (0.05, 0.30, 0.55)  (0.20, 0.45, 0.70)  (0.70, 0.95, 1) 
EED  (0.15, 0.40, 0.65)  (0.65, 0.90, 1)  (0.50, 0.75, 0.95)  (0.00, 0.10, 0.35) 
Normalized the aggregated performance ratings
Criteria/Subcriteria  Aggregated weights 

Economic  (0.40, 0.50, 0.90) 
Environmental  (0.60, 0.70, 1) 
Public Health  (0.75, 0.80, 1) 
Occupational Health  (0.60, 0.70, 1) 
Technical  (0.55, 0.60, 1) 
Social  (0.60, 0.70, 1) 
Ranking of the drinking water alternatives
Alternative  D _{ i } ^{∗}  D _{ i } ^{−}  Ω _{ i } ^{∗}  Rank 

A_{1}: Public Desalinated Distribution System (PDDS)  2.131  3.346  0.611  1 
A_{2}: PET Bottled Drinking Water (PBDW)  2.212  3.405  0.606  2 
A_{4}: Household Desalinated Water Units (HDWU)  2.279  3.482  0.604  3 
A_{3}: Private Desalinated Water Suppliers (PDWS)  2.384  3.01  0.558  4 
As can be inferred from Table 10 the Public Desalinated Distribution System (A_{1}) is the best alternative as drinking water source for Qom population.
Abrishamchi and coworkers (2004) denoted a small potable water network (less than 30 km) with public valves (water standpipes) at several points across the city of Zahidan. They considered the “Extension of the small drinking water distribution network with public standpipes” as an alternative to supply the drinking water needs of population.
Public Desalinated Distribution System (PDDS) has several benefits such as simple operation of treatment facilities and ease of health inspection process. Now, more than 180 km of potable water network has been constructed in the city of Qom which have connected to 260 public valve (water standpipes) and supply more than 4500 cubic meter of desalinated water per day [29]. The only noticeable problem dealing with the PDDS is the low extension of distribution system which tends to handle the water containers from public valves to houses by people.
Jafaripour estimated that over 36000 houses in Qom use the Household desalinated water units (HDWU) which cover more than 15 % of all population. Based on the findings of Jafaripour, more than 1000 m^{3} of brine water and up to 550 discarded filter are produced by using of Household desalinated water units (HDWU) [30].
Yari reported that 24 Private desalinated water suppliers (PDWS) are operated in the city of Qom. Their results showed that the chemical characteristics of potable water produced by PDWS could not meet the national standard criteria. Also, transferring the water containers by vendees is the other constraint of PDWS. Purchased water containers may stored in homes for a long time in uncontrolled health condition [31].
More than 18 various brands of PET Bottled Drinking Water (PBDW) are sold in the retails of Qom city [32]. Noticeable merits of PBDW are Chemical and biological acceptable quality which serve as an alternative beside the other water supply system. High price and lack of coverage for all population, in the other hand, are the essential drawbacks of PBDW.
A significant factor that should be considered in the judgment process of purchasing hightech equipment is the level of dependency to the foreign suppliers. A more appropriate strategy is to encourage the use of the alternative technologies available within the country. Hence, except for the household desalinated water units (HDWU), the other alternatives could not obtain higher levels of linguistic terms by decisionmakers for SDOC and EDOC subcriteria.
Considering the occupational and public health criteria independent of the environmental and technical criteria significantly improved the precision of the results.
Conclusions
An efficient analysis was performed by applying the evaluation criteria and their associated subcriteria on a hierarchical structure. Thirty five subcriteria associated with six criteria were structured in a multilevel hierarchy and the decision processes allowed the decisionmakers to employ linguistic concepts, and thus, decreased the cognition problems during the evaluation process.
In this study, hierarchical distancebased fuzzy multicriteria group decision making (DBF –MCDM) approach was presented to avoid the problems that may occurred when the classical decisionmaking approaches are employed for evaluating the water supply alternatives.
New arrangement of criteria and subcriteria was proposed in this study. Traditionally, four criteria including financial, environmental, technical, and social aspects have been proposed in similar works. Using a new hierarchy containing the public health and occupational health aspects as the independent criteria enabled the decisionmaking process to assign more effective evaluations.
System and equipment dependency to other countries (SDOC and EDOC) were added to the technical aspects as subcriteria for obtaining a state of compatibility with the socioeconomic condition which restrict the level of dependency on the foreign companies.
The DBF–MCDM method proposed in this research is a simple approach that can be used for similar environmental management issues only with some modifications.
Declarations
Acknowledgments
We would like to thank the professors and experts of Qom University of Medical Sciences (QUMS) and Qom Water and Wastewater Organization (QWWO) who support the study as decisionmakers.
Authors’ Affiliations
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