# Decision-Making Tools to Manage the Microbiology of Drinking Water Distribution Systems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Microbial Quality of Drinking Water

#### 1.2. Multi Criteria Decision-Making

## 2. Aim and Structure

## 3. Materials and Methods

#### 3.1. The Problem of Microbial Evaluation for DWDSs

#### 3.2. Integrated MCDM Approach

#### 3.2.1. Modified DEMATEL to Establish Relationships of Influence among Elements

- Collecting the non-negative input matrix, $X$, whose cells give the relation of influence ${x}_{ij}$ of one element, $i$, over another one, $j$, according to the following linguistic evaluation scale: 0 (no influence), 1 (very low influence), 2 (low influence), 3 (high influence), 4 (very high influence). The main diagonal of the matrix will be zeroes, since one element has no influence on itself.
- According to the traditional DEMATEL procedure [29,30], the previous stage is carried out by involving a decision-making group and by asking each expert to fill in their own input matrix. All these matrices are then aggregated into one, the so called direct relation matrix, $A$ (output of the third stage of the procedure), with the aim to treat the set of input data in a way as balanced and reliably as possible. In this paper, a single input matrix is used, in which, instead of subjective expert evaluations, the relations of influence for each pair of elements are derived from the related values of measured Spearman correlations. Hence, the direct relation matrix $A$ coincides with a single input matrix $X$.
- Calculating the normalised direct relation matrix $N$ as:$$N=sX,$$$s$ being a positive number slightly smaller than$$\mathrm{min}\left[\frac{1}{\underset{1\le i\le n}{\mathrm{max}}{{\displaystyle \sum}}_{j=1}^{n}{x}_{ij}},\text{}\frac{1}{\underset{1\le j\le n}{\mathrm{max}}{{\displaystyle \sum}}_{i=1}^{n}{x}_{ij}}\right].$$Matrix $N$ shows the initial influence that elements exert on and receive from the others. The next step consists of obtaining a continuous decrease of indirect effects among factors in terms of consecutive powers of $N$.
- Obtaining the total relation matrix, $T$, which collects the total interrelation, including both direct and indirect effects among elements, which can be calculated as the sum of the powers of the normalised direct relation matrix $N$, given by:$$T=N{\left(I-N\right)}^{-1},$$
- Defining the two vectors $r=\left({r}_{i}\right)$ and $c=\left({c}_{j}\right)$, respectively representing the $n\times 1$ and $1\times n$ vectors of sums of the rows and sums of the columns in the total relation matrix $T$. From these two vectors it is possible to calculate the prominence as the sum ${r}_{i}+{c}_{i}$, reflecting the general effect of element $i$ on all the other elements, and the relation as the subtraction ${r}_{i}-{c}_{i}$, helping in dividing the elements into classes of cause (if positive) and effect (if negative).
- Drawing up the final ranking of elements, ordered according to their decreasing values of prominence.

#### 3.2.2. FTOPSIS to Rank Bacteria according to the Type of Pipe Material

- Defining the fuzzy decision matrix $\tilde{X}$ collecting input data:$$\tilde{X}=\left[\begin{array}{ccc}{\tilde{x}}_{11}& \cdots & {\tilde{x}}_{1n}\\ \vdots & \ddots & \vdots \\ {\tilde{x}}_{m1}& \cdots & {\tilde{x}}_{mn}\end{array}\right]$$$${\tilde{x}}_{ij}=\left({l}_{ij},\text{}{m}_{ij},\text{}{u}_{ij}\right).$$
- Obtaining matrix $\tilde{Z}$ by weighting and normalising the fuzzy decision matrix of input with relation to each criterion. Elements of matrix $\tilde{Z}$ are calculated as:$${\tilde{z}}_{ij}=\left(\frac{{l}_{ij}}{{u}_{j}^{*}},\text{}\frac{{m}_{ij}}{{u}_{j}^{*}},\text{}\frac{{u}_{ij}}{{u}_{j}^{*}}\right)\xb7{w}_{j},\text{}j\in {I}^{\prime},$$$${\tilde{z}}_{ij}=\left(\frac{{l}_{j}^{-}}{{u}_{ij}},\text{}\frac{{l}_{j}^{-}}{{m}_{ij}},\text{}\frac{{l}_{j}^{-}}{{l}_{ij}}\right)\xb7{w}_{j},\text{}j\in {I}^{\u2033},$$$${u}_{j}^{*}=\underset{i}{\mathrm{max}}{u}_{ij}\mathrm{if}\text{}j\in {I}^{\prime},$$$${l}_{j}^{-}=\underset{i}{\mathrm{min}}{l}_{ij}\mathrm{if}\text{}j\in {I}^{\u2033}.$$
- Computing distances between each alternative and two fuzzy ideal solutions, namely the fuzzy positive ideal solution ${S}^{*}$ and the fuzzy negative ideal solution ${S}^{-}$:$${S}^{*}=\left({\tilde{z}}_{1}^{*},\text{}{\tilde{z}}_{2}^{*},\dots ,\text{}{\tilde{z}}_{n}^{*}\right),\text{}$$$${S}^{-}=\left({\tilde{z}}_{1}^{-},\text{}{\tilde{z}}_{2}^{-},\dots ,\text{}{\tilde{z}}_{n}^{-}\right),$$$$d\left({\tilde{n}}_{1},\text{}{\tilde{n}}_{2}\right)=\sqrt{\frac{1}{3}\left[{\left({l}_{1}-{l}_{2}\right)}^{2}+{\left({m}_{1}-{m}_{2}\right)}^{2}+{\left({u}_{1}-{u}_{2}\right)}^{2}\right]}.$$Then, aggregating with respect to the set of considered criteria, the distances of each alternative $i$ from ${S}^{*}$ and ${S}^{-}$ are:$${d}_{i}^{*}={{\displaystyle \sum}}_{j=1}^{n}d\left({\tilde{z}}_{ij},{\tilde{z}}_{j}^{*}\right)i=1\dots n,$$$${d}_{i}^{-}={{\displaystyle \sum}}_{j=1}^{n}d\left({\tilde{z}}_{ij},{\tilde{z}}_{j}^{-}\right)i=1\dots n.$$
- Calculating the closeness coefficient $C{C}_{i}$ to get the final ranking. The mentioned closeness coefficient $C{C}_{i}$ is calculated as:$$C{C}_{i}=\frac{{d}_{i}{}^{-}}{{d}_{i}{}^{-}+{d}_{i}{}^{*}}.$$To get the final ranking it is necessary to sort the values of the closeness coefficient related to each alternative in a decreasing way. The elements with higher $C{C}_{i}$ values will be selected.

#### 3.3. Case Study

## 4. Results and Discussion

_{1}) and iron (C

_{2}). Bacteria evaluations under the two considered criteria are triangular fuzzy numbers representing the relative abundance of bacteria detected in each type of pipe in a given interval of time. Each fuzzy number gives three values representing the lower, the medium and the higher level of bacteria relative abundance observed from three measurements performed in a given month of observation.

_{1}and C

_{2}are equally weighted. Both criteria have to be minimised since we assume as positive ideal condition the total absence of bacteria both in plastic and cast iron pipes and, as negative ideal condition, the maximum bacteria abundance.

## 5. Conclusions

- Mutual interdependencies existing among water quality parameters (e.g., iron, chlorine, phosphate etc.,) and bacterial class can be determined by the decision-making trial and evaluation laboratory, also removing the need for reliance on expert judgement.
- Bacterial classes can be ranked according to their relative abundance depending on pipe materials using the fuzzy technique for order preference by similarity to ideal solution.
- The method reveals that the critical bacterial classes, those that have the most inter-dependencies and therefore potential management impact, may not be the most abundant.
- Initial application of the approach generated new knowledge of the physicochemical and biological parameters that are most likely to influence the presence and relative abundance of bacterial classes, for the limited data set available. Such knowledge will allow water companies to inform management strategies to promote favourable bacterial communities and hence help to safeguard drinking water quality.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. DEMATEL and FTOPSIS Results

Factor | r = c | Prominence | Order | Ranking | |
---|---|---|---|---|---|

P_{1} | 1.437216965 | 2.87443393 | 37 | P_{12} | Iron |

P_{2} | 2.973143464 | 5.946286928 | 6 | P_{3} | Phosphate |

P_{3} | 3.17935789 | 6.358715781 | 4 | P_{9} | Betaproteobacteria |

P_{4} | 2.345268204 | 4.690536408 | 17 | P_{10} | Turbidity |

P_{5} | 1.726788409 | 3.453576818 | 32 | B_{2} | Manganese |

P_{6} | 2.251308008 | 4.502616016 | 23 | B_{8} | Diversity |

P_{7} | 2.030897355 | 4.06179471 | 27 | B_{4} | Spirochaetia |

P_{8} | 2.640276655 | 5.280553311 | 12 | B_{16} | Gammaproteobacteri |

P_{9} | 3.448076894 | 6.896153789 | 1 | P_{2} | Flavobacteriia |

P_{10} | 3.137029854 | 6.274059707 | 5 | B_{15} | Gemmatimonadetes |

P_{11} | 2.240798202 | 4.481596404 | 24 | B_{3} | Deltaproteobacteria |

P_{12} | 3.424555671 | 6.849111342 | 2 | B_{5} | Aluminium |

P_{13} | 1.509733521 | 3.019467042 | 36 | P_{8} | Acidobacteria |

B_{1} | 1.58008322 | 3.160166439 | 34 | B_{21} | Bacilli |

B_{2} | 3.393248734 | 6.786497469 | 3 | B_{9} | Anaerolineae |

B_{3} | 2.895992476 | 5.791984953 | 8 | B_{25} | Holophagae |

B_{4} | 2.653556974 | 5.307113947 | 11 | B_{19} | Total organic carbon |

B_{5} | 2.511551785 | 5.02310357 | 14 | B_{24} | Bacteroidete |

B_{6} | 1.395898146 | 2.791796292 | 38 | P_{4} | Sphingobacteriia |

B_{7} | 1.927236046 | 3.854472092 | 29 | B_{11} | Firmicutes |

B_{8} | 2.815970477 | 5.631940955 | 9 | B_{23} | Bacteroidia |

B_{9} | 2.303522588 | 4.607045176 | 21 | B_{12} | Chloroflexi |

B_{10} | 1.953345677 | 3.906691354 | 28 | B_{7} | pH |

B_{11} | 2.528074142 | 5.056148284 | 13 | B_{10} | Nitrate |

B_{12} | 2.341968998 | 4.683937996 | 18 | P_{6} | Planctomycetia |

B_{13} | 1.548426935 | 3.09685387 | 35 | P_{11} | Cytophagia |

B_{14} | 1.835317607 | 3.670635214 | 30 | B_{17} | Chlorine |

B_{15} | 2.201492206 | 4.402984412 | 25 | B_{20} | Cyanobacteria |

B_{16} | 2.93537933 | 5.870758661 | 7 | P_{7} | Mollicutes |

B_{17} | 2.320906655 | 4.64181331 | 19 | B_{14} | Clostridia |

B_{18} | 2.369284818 | 4.738569635 | 15 | B_{22} | Spirochaetes |

B_{19} | 2.112981146 | 4.225962291 | 26 | B_{13} | Temperature |

B_{20} | 2.351855933 | 4.703711866 | 16 | P_{5} | Verrucomicrobia |

B_{21} | 1.752601241 | 3.505202483 | 31 | B_{26} | Alphaproteobacteria |

B_{22} | 2.266608635 | 4.533217271 | 22 | P_{13} | Planctomycetes |

B_{23} | 2.314947689 | 4.629895379 | 20 | P_{1} | Sulphate |

B_{24} | 2.711514009 | 5.423028017 | 10 | B_{1} | Richness |

B_{25} | 1.652805885 | 3.305611769 | 33 | B_{6} | Actinobacteria |

ID | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ||||
---|---|---|---|---|---|---|

B_{1} | 20.49107931 | 22.98176432 | 35.19847571 | 17.49200859 | 24.68045308 | 25.27149145 |

B_{2} | 6.999646706 | 7.151476659 | 9.267970455 | 5.299802556 | 11.94256668 | 15.4431105 |

B_{3} | 4.560177834 | 5.074191839 | 15.04694037 | 6.692299699 | 10.77111383 | 13.02206152 |

B_{4} | 0.393775802 | 3.161985515 | 3.85331385 | 0.020783539 | 1.1318975 | 1.666710071 |

B_{5} | 4.930240006 | 9.649355238 | 21.37821531 | 6.791843538 | 8.375766393 | 9.982707122 |

B_{6} | 6.66573253 | 8.351857733 | 11.1596891 | 10.72682492 | 10.88049168 | 22.21760366 |

B_{7} | 0 | 0.232199692 | 7.138774214 | 0 | 0.639312477 | 11.86228808 |

B_{8} | 1.258611553 | 1.431230859 | 2.229279136 | 0.539747419 | 0.572931587 | 6.131144134 |

B_{9} | 0.176647235 | 1.144984687 | 1.390917752 | 0 | 1.582560394 | 12.63639198 |

B_{10} | 1.174976183 | 2.40606922 | 8.386327504 | 3.153727962 | 4.574752397 | 5.840174582 |

B_{11} | 0 | 0 | 0.749444268 | 0 | 0 | 1.203156332 |

B_{12} | 0.039745628 | 0.527151477 | 3.342874872 | 0.463553738 | 0.654681492 | 2.814022952 |

B_{13} | 0.080068859 | 0.387424579 | 0.463698993 | 0 | 1.013047215 | 3.018393334 |

B_{14} | 0.018015493 | 0.225225225 | 0.381073357 | 0 | 0.317716607 | 1.62972279 |

B_{15} | 0.368370911 | 2.295974538 | 5.272919979 | 0 | 0 | 0.005240266 |

B_{16} | 0.317561131 | 1.175010509 | 4.107048225 | 0 | 3.143311024 | 5.502279516 |

B_{17} | 0.260400127 | 0.980392157 | 1.937666393 | 0 | 0 | 0 |

B_{18} | 0.02540489 | 1.801801802 | 2.259943551 | 0 | 1.828172609 | 1.844573704 |

B_{19} | 0 | 0.174149769 | 0.251722311 | 0 | 0 | 0 |

B_{20} | 0 | 0.04857799 | 0.058049923 | 0 | 0 | 0 |

B_{21} | 0 | 0.379791556 | 1.373180936 | 0 | 0.01039177 | 1.294345753 |

B_{22} | 0 | 0 | 0 | 0 | 0 | 0 |

B_{23} | 0 | 0.002001721 | 0.587352058 | 0 | 0 | 0 |

B_{24} | 0 | 0.05404648 | 1.519166225 | 0 | 0 | 0 |

B_{25} | 0 | 0.438377004 | 0.507860802 | 0 | 0 | 0.282974375 |

ID | ${\mathit{C}}_{1}\text{\hspace{1em}}\left({\mathit{w}}_{1}=0.5\right)$ | ${\mathit{C}}_{2}\text{\hspace{1em}}\left({\mathit{w}}_{2}=0.5\right)$ | ||||
---|---|---|---|---|---|---|

B_{1} | 0.000014205 | 0.000021756 | 0.000024401 | 0.000019785 | 0.000020259 | 0.000028584 |

B_{2} | 0.000053949 | 0.000069915 | 0.000071432 | 0.000032377 | 0.000041867 | 0.000094343 |

B_{3} | 0.000033229 | 0.000098537 | 0.000109645 | 0.000038396 | 0.000046420 | 0.000074713 |

B_{4} | 0.000129758 | 0.000158128 | 0.001269758 | 0.000299992 | 0.000441736 | 0.0240575 |

B_{5} | 0.000023388 | 0.000051817 | 0.000101415 | 0.000050087 | 0.000059696 | 0.000073618 |

B_{6} | 0.000044804 | 0.000059867 | 0.000075010 | 0.000022504 | 0.000045954 | 0.000046612 |

B_{7} | 0.000070040 | 0.002153319 | 1 | 0.000042150 | 0.00078209 | 1 |

B_{8} | 0.000224288 | 0.00034935 | 0.000397263 | 0.000081551 | 0.000872705 | 0.000926359 |

B_{9} | 0.000359475 | 0.000436687 | 0.0028305 | 0.000039568 | 0.000315944 | 1 |

B_{10} | 0.000059621 | 0.000207808 | 0.000425541 | 0.000085614 | 0.000109296 | 0.000158543 |

B_{11} | 0.000667161 | 1 | 1 | 0.000415574 | 1 | 1 |

B_{12} | 0.000149572 | 0.000948494 | 0.01258 | 0.000177682 | 0.00076373 | 0.001078624 |

B_{13} | 0.001078286 | 0.001290574 | 0.006244625 | 0.000165651 | 0.00049356 | 1 |

B_{14} | 0.001312083 | 0.00222 | 0.027753889 | 0.000306801 | 0.00157373 | 1 |

B_{15} | 0.000094824 | 0.000217772 | 0.001357328 | 0.095415 | 1 | 1 |

B_{16} | 0.000121742 | 0.000425528 | 0.0015745 | 0.000090871 | 0.000159068 | 1 |

B_{17} | 0.000258042 | 0.00051 | 0.001920122 | 1 | 1 | 1 |

B_{18} | 0.000221244 | 0.0002775 | 0.01968125 | 0.000271065 | 0.000273497 | 1 |

B_{19} | 0.001986316 | 0.002871092 | 1 | 1 | 1 | 1 |

B_{20} | 0.008613276 | 0.010292727 | 1 | 1 | 1 | 1 |

B_{21} | 0.000364118 | 0.001316512 | 1 | 0.000386296 | 0.048115 | 1 |

B_{22} | 1 | 1 | 1 | 1 | 1 | 1 |

B_{23} | 0.000851278 | 0.249785 | 1 | 1 | 1 | 1 |

B_{24} | 0.000329128 | 0.009251296 | 1 | 1 | 1 | 1 |

B_{25} | 0.000984522 | 0.001140571 | 1 | 0.001766944 | 1 | 1 |

February 2012 | June 2012 | October 2012 | February 2013 | ||||
---|---|---|---|---|---|---|---|

$\mathit{C}{\mathit{C}}_{\mathit{i}}$ | Ranking | $\mathit{C}{\mathit{C}}_{\mathit{i}}$ | Ranking | $\mathit{C}{\mathit{C}}_{\mathit{i}}$ | Ranking | $\mathit{C}{\mathit{C}}_{\mathit{i}}$ | Ranking |

0.000021904 | B_{1} | 0.000032782 | B_{1} | 0.000020284 | B_{1} | 0.000018493 | B_{1} |

0.000050556 | B_{6} | 0.000092323 | B_{6} | 0.000054029 | B_{6} | 0.000134639 | B_{6} |

0.000064015 | B_{2} | 0.000173322 | B_{3} | 0.000314969 | B_{3} | 0.000151797 | B_{2} |

0.000064507 | B_{5} | 0.000473575 | B_{2} | 0.000460293 | B_{14} | 0.000242965 | B_{14} |

0.000071328 | B_{3} | 0.000609658 | B_{5} | 0.000489808 | B_{10} | 0.000273064 | B_{3} |

0.000198617 | B_{10} | 0.000679344 | B_{14} | 0.000551176 | B_{5} | 0.000915538 | B_{5} |

0.000533991 | B_{8} | 0.001587454 | B_{10} | 0.000618129 | B_{2} | 0.001244508 | B_{15} |

0.003518699 | B_{17} | 0.001798972 | B_{15} | 0.000916142 | B_{15} | 0.00212396 | B_{4} |

0.004021346 | B_{11} | 0.001857045 | B_{19} | 0.012192723 | B_{17} | 0.003477588 | B_{17} |

0.007296215 | B_{4} | 0.002647 | B_{11} | 0.012280875 | B_{9} | 0.005532909 | B_{9} |

0.241561775 | B_{15} | 0.004477637 | B_{9} | 0.014121489 | B_{24} | 0.028684001 | B_{10} |

0.241845487 | B_{9} | 0.006075334 | B_{23} | 0.014330545 | B_{4} | 0.241510047 | B_{19} |

0.242680267 | B_{12} | 0.011889233 | B_{25} | 0.061908005 | B_{13} | 0.243271905 | B_{11} |

0.24738986 | B_{13} | 0.241329775 | B_{16} | 0.24315878 | B_{11} | 0.243461954 | B_{12} |

0.349935656 | B_{14} | 0.241336195 | B_{4} | 0.2644993 | B_{8} | 0.243912411 | B_{8} |

0.414398757 | B_{7} | 0.241677682 | B_{8} | 0.419032998 | B_{16} | 0.342153911 | B_{20} |

0.41738494 | B_{19} | 0.242711521 | B_{17} | 0.426324756 | B_{25} | 0.342212028 | B_{25} |

0.50033904 | B_{23} | 0.414485803 | B_{20} | 0.432738399 | B_{19} | 0.342907479 | B_{22} |

0.500513044 | B_{16} | 0.416274632 | B_{13} | 0.501509865 | B_{23} | 0.346318664 | B_{13} |

0.501015736 | B_{25} | 0.422019615 | B_{24} | 0.505008398 | B_{12} | 0.346642736 | B_{16} |

0.659998225 | B_{22} | 0.500396692 | B_{12} | 0.506825384 | B_{18} | 0.500423791 | B_{18} |

0.661057314 | B_{18} | 0.502661286 | B_{18} | 0.511728433 | B_{7} | 0.501795598 | B_{24} |

0.688589598 | B_{21} | 0.503899708 | B_{7} | 0.594284831 | B_{20} | 0.667389964 | B_{21} |

0.763948195 | B_{24} | 0.659464916 | B_{22} | 0.660197068 | B_{22} | 0.759022559 | B_{23} |

1 | B_{20} | 0.659802486 | B_{21} | 1 | B_{21} | 0.759056766 | B_{7} |

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**Figure 3.**FTOPSIS results with bars heights normalised to the max $CC$ value (varying between 0 and 1).

Water Quality Parameter | Bacterial Class | ||||
---|---|---|---|---|---|

P_{1} | Richness | B_{1} | Alphaproteobacteria | B_{14} | Clostridia |

P_{2} | Diversity | B_{2} | Betaproteobacteria | B_{15} | Planctomycetia |

P_{3} | Turbidity | B_{3} | Gammaproteobacteri | B_{16} | Spirochaetia |

P_{4} | Total organic carbon | B_{4} | Deltaproteobacteria | B_{17} | Sphingobacteriia |

P_{5} | Temperature | B_{5} | Bacilli | B_{18} | Anaerolineae |

P_{6} | pH | B_{6} | Actinobacteria | B_{19} | Cytophagia |

P_{7} | Chlorine | B_{7} | Mollicutes | B_{20} | Holophagae |

P_{8} | Aluminium | B_{8} | Flavobacteriia | B_{21} | Spirochaetes |

P_{9} | Iron | B_{9} | Bacteroidia | B_{22} | Chloroflexi |

P_{10} | Manganese | B_{10} | Cyanobacteria | B_{23} | Firmicutes |

P_{11} | Nitrate | B_{11} | Acidobacteria | B_{24} | Gemmatimonadetes |

P_{12} | Phosphate | B_{12} | Bacteroidete | B_{25} | Verrucomicrobia |

P_{13} | Sulphate | B_{13} | Planctomycetes |

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**MDPI and ACS Style**

Carpitella, S.; Del Olmo, G.; Izquierdo, J.; Husband, S.; Boxall, J.; Douterelo, I. Decision-Making Tools to Manage the Microbiology of Drinking Water Distribution Systems. *Water* **2020**, *12*, 1247.
https://doi.org/10.3390/w12051247

**AMA Style**

Carpitella S, Del Olmo G, Izquierdo J, Husband S, Boxall J, Douterelo I. Decision-Making Tools to Manage the Microbiology of Drinking Water Distribution Systems. *Water*. 2020; 12(5):1247.
https://doi.org/10.3390/w12051247

**Chicago/Turabian Style**

Carpitella, Silvia, Gonzalo Del Olmo, Joaquín Izquierdo, Stewart Husband, Joby Boxall, and Isabel Douterelo. 2020. "Decision-Making Tools to Manage the Microbiology of Drinking Water Distribution Systems" *Water* 12, no. 5: 1247.
https://doi.org/10.3390/w12051247